M. Rades - Dynamics of Machinery 1

Dynamics of Machinery I Mircea Radeş Universitatea Politehnica Bucureşti

2007

Preface This textbook is based on the first part of the Dynamics of Machinery lecture course given since 1993 to students of the English Stream in the Department of Engineering Sciences (D.E.S.), now F.I.L.S., at the University Politehnica of Bucharest. It grew in time from a postgraduate course taught in Romanian between 1985 and 1990 at the Strength of Materials Chair. Dynamics of Machinery, as a stand alone subject, was first introduced in the curricula of mechanical engineering at D.E.S. in 1993. To sustain it, we published Dynamics of Machinery in 1995, followed by Dinamica sistemelor rotor-lagăre in 1996 and Rotating Machinery in 2003. As seen from the Table of Contents, this book is application oriented and limited to what can be taught in an one-semester (28 hours) lecture course. It also contains many exercises to support the tutorial, where the students are guided to write simple finite element computer programs in Matlab, and to assist them in solving problems as homework. The course aims to: (a) increase the knowledge of machinery vibrations; (b) further the understanding of dynamic phenomena in machines; (c) provide the necessary physical basis for the development of engineering solutions to machinery problems; and (d) make the students familiar with machine condition monitoring techniques and fault diagnosis. As a course taught for non-native speakers, it has been considered useful to reproduce, as language patterns, some sentences from English texts. Finite element modeling of rotor-bearing systems and hydrodynamic bearings are treated in the second part. Analysis of rolling element bearings, machine condition monitoring and fault diagnosis, balancing of rotors as well as elements of the dynamic analysis of reciprocating machines are presented in the third part. No reference is made to the vibration of discs, impellers and blades.

August 2007

Mircea Radeş

Prefaţă Lucrarea se bazează pe prima parte a cursului de Dinamica maşinilor predat din 1993 studenţilor Filierei Engleze a Facultăţii de Inginerie în Limbi Străine (F.I.L.S.) la Universitatea Politehnica Bucureşti. Conţinutul cursului s-a lărgit în timp, pornind de la un curs postuniversitar organizat între 1985 şi 1990 în cadrul Catedrei de Rezistenţa materialelor. Dinamica maşinilor a fost introdusă în planul de învăţământ al F.I.L.S. în 1993. Pentru a susţine cursul, am publicat Dynamics of Machinery la U. P. B. în 1995, urmată de Dinamica sistemelor rotor-lagăre în 1996 şi Rotating Machinery în 2005, ultima conţinând materialul ilustrativ utilizat în cadrul cursului. După cum reiese din Tabla de materii, cursul este orientat spre aplicaţii inginereşti, fiind limitat la ceea ce se poate preda în 28 ore. Materialul prezentat conţine multe exerciţii rezolvate care susţin seminarul, în cadrul căruia studenţii sunt îndrumaţi să scrie programe simple cu elemente finite în Matlab, fiind utile şi la rezolvarea temelor de casă. Cursul are un loc bine definit în planul de învăţământ, urmărind: a) descrierea fenomenelor dinamice specifice maşinilor; b) modelarea sistemelor rotor-lagăre şi analiza acestora cu metoda elementelor finite; c) înarmarea studenţilor cu baza fizică necesară în rezolvarea problemelor de vibraţii ale maşinilor; şi d) familiarizarea cu metodele de supraveghere a stării maşinilor şi diagnosticare a defectelor. Fiind un curs predat unor studenţi a căror limbă maternă nu este limba engleză, au fost reproduse unele expresii şi fraze din lucrări scrise de vorbitori nativi ai acestei limbi. În partea a doua se prezintă modelarea cu elemente finite a sistemelor rotor-lagăre şi lagărele hidrodinamice. În partea a treia se tratează lagărele cu rulmenţi, echilibrarea rotorilor, măsurarea vibraţiilor pentru supravegherea funcţionării maşinilor şi diagnosticarea defectelor, precum şi elemente de dinamica maşinilor cu mecanism bielă-manivelă. Nu se tratează vibraţiile paletelor, discurilor paletate şi ale roţilor centrifugale.

August 2007

Mircea Radeş

Contents Preface

i

Contents

iii

1. Rotor-bearing systems

1

1.1 1.2 1.3 1.4 1.5 1.6

Evolution of rotating machinery Rotor-bearing dynamics Rotor precession Modeling the rotor Evolution of rotor design philosophy Historical perspective

2. Simple rotors in rigid bearings

1 22 24 26 29 32

39

2.1 Simple rotor models

39

2.2 Symmetric undamped rotors

40

2.2.1 Equations of motion

41

2.2.2 Steady state response

43

2.3 Damped symmetric rotors

46

2.3.1 Effect of viscous external damping

47

2.3.2 Effect of viscous internal damping

54

2.3.3 Combined external and internal damping

62

2.3.4 Gravity loading

65

2.3.5 Effect of shaft bow

66

2.3.6 Rotor precession in rigid bearings

67

2.4 Undamped asymmetric rotors

68

2.4.1 Reference frames

69

2.4.2 Inertia torques on a spinning disc

69

2.4.3 Equations of motion for elastically supported discs

72

2.4.4 Natural frequencies of precession

75

2.4.5 Response to harmonic excitation

81

2.4.6 Campbell diagrams

87

2.4.7 Effect of gyroscopic torque on critical speeds

97

2.4.8 Remarks on the precession of asymmetric rotors

98

MECHANICAL VIBRATIONS

iv

3. Simple rotors in flexible bearings 3.1 Symmetric rotors in flexible bearings

101 101

3.1.1 Effect of bearing flexibility

102

3.1.2 Effect of external damping

109

3.1.3 Effect of external and internal damping

117

3.1 4 Effect of bearing damping

119

3.1.5 Combined effect of bearing damping and shaft mass

131

3.2 Symmetric rotors in fluid film bearings 3.2.1 Unbalance response 3.2.2 Stability of precession motion

3.3 Asymmetric rotors in flexible bearings

136 136 142

145

3.3.1 Equations of motion

145

3.3.2 Natural frequencies of precession

148

3.3.3 Unbalance response

152

3.3.4 Effect of bearing damping

156

3.3.5 Mixed modes of precession

158

3.4 Simulation examples

168

4. Rotor dynamic analysis

207

4.1 Undamped critical speeds 4.1.1 Effect of support flexibility

207

4.1.2 Critical speed map

209

4.1.3 Influence of stator inertia

4.2 Damped critical speeds

4.3 4.4 4.5 4.6

Index

207

217

219

4.2.1 Linear bearing models

219

4.2.2 Equations of damped motion

220

4.2.3 Eigenvalue problem of damped rotor systems

220

4.2.4 Campbell diagrams

222

4.2.5 Orbits and precession mode shapes

223

Peak response critical speeds Stability analysis Simulation examples Planar modes of precession

224 227 231 273

283

1. ROTOR-BEARING SYSTEMS

The first part of the Dynamics of Machinery is devoted to rotor-bearing systems, including the effects of seals and bearing supports. The flexibilities of discs and blades are neglected, so that the Rotor Bearing Dynamics does not include the vibration analysis of impellers and bladed-disc assemblies.

1.1. Evolution of rotating machinery Interest in the vibration of rotating machinery has been due primarily to the fact that more than 80 percent of the problems involve vibration. In the continuing effort to develop more power per kilogram of metal in a machine, designs have approached the physical limits of materials and vibration problems have increased. These, together with the extremely high cost associated with forced outages, for machines with continuous operating regime, have determined the development of research activity and design procedures in two fields of primary practical interest: the Dynamics of Rotor-Bearing Systems and the Vibrations of Bladed Disc Assemblies.

1.1.1 Steam turbines Of significance for the technical advancement in this field is the development of steam turbines in Europe [1]. From the first single stage impulse turbine built in 1883 by the Swedish engineer Gustaf de Laval (with a speed of 30000 rpm reduced to 3000 rpm by gearing), and the first multistage reaction turbine built in 1884 by Charles Parsons (having a speed of 18000 rpm and an output of 10 HP), to the turbines of today nuclear power stations, the evolution has been spectacular. Early in 1901 the Brown Boveri Company built a steam turbine of 250 kW at 3000 rpm, coupled directly to an a.c. generator. From 1907 onwards, a double impulse Curtis wheel (invented in 1896) was mounted before the reaction

2

DYNAMICS OF MACHINERY

stage, which was replaced by single-row versions on two to three impulse wheels. In 1914, a turbine of 25 MW at 1000 rpm was the largest single-cylinder steam turbine in the world. The first systematic studies of Rotor Dynamics started in 1916, carried out by professor Aurel Stodola at the Swiss Federal Institute of Tehnology in Zürich. After 1920, the high price of coal imposed the increase of steam turbine efficiency. Among other means, this was achieved by the reduction in the diameter and the increase in the number of stages, hence by the increase of the shaft length, a major incentive for developing the Dynamics of Rotor-Bearing Systems. The maximum unit output of a turbine is largely dependent on the available last-stage blade length. The permissible blade length to diameter ratio has an influence on the machine efficiency. Shafts should be as slender as possible, to ensure small rotor diameter and large blade length. Otherwise, increased shaft weight gives rise to an increase in the average specific bearing loading. Increasing the cross-section of a machine is limited by the mechanical stresses and the size of pieces that can be transported. This is compensated by the increase of the active length, eventually with a tandem arrangement, having a long shaft line, in which the mechanical power is produced in several turbine cylinders. The first super-pressure three-cylinder (high, intermediate and low pressure) turbine was built by BBC in 1929, and had an output of 36 MW at 3000 rpm. The steam flowed through high pressure and intermediary pressure rotors in opposite directions, to balance the thrust. Rotors, which previously were composed of keyed and shrunk-on wheels on a continuous shaft, started to be welded from solid discs, allowing larger rotor diameters and increased ratings. The increased efficiency of steam turbines lowered the amount of coal required for producing 1 kWh of electrical energy from 0.75 kg during the war to 0.45 kg in 1927. The output of the largest turbines in Europe had reached 50 to 60 MW by the mid twenties, when, for large units, turbines of 1500 rpm were coupled to four-pole generators. A 165 MW two-shaft turboset was built in 1926-1928, with the highpressure shaft rotating at 1800 rpm, and the low-pressure shaft at 1200 rpm. In 1948, the largest steam turboset of single-shaft design (Fig. 1.1) had four cylinders, a length of 27 m (without the station service generator), an output of 110 MW and speed of 3000 rpm [2]. In 1950, turbosets of 125 MW were built in Europe and of 230 MW in the U.S.A., then, in 1956 - with ratings of 175 MW, and in 1964 - with ratings of 550 MW and two shafts. In 1972, the first 1300 MW cross-compound turboset was built at 3600 rpm, provided with two shaft lines for two 722 MVA generators. Figure 1.2 shows a longitudinal section of the high-pressure turbine of a 1300 MW unit at 1800 rpm. Current designs have generators of 1635 MVA at 1500 rpm, and of 1447 MVA at 3000 rpm. At present time, turbosets of 1700-2000 MW at 1500 or 1800 rpm, and of 1500-1700 MW at 3000 or 3600 rpm are currently built.


3


4

Generally, the shaft line has a length of 8 to 20 m in turbosets of 1 to 50 MW, between 25 and 30 m in those of 100 to 150 MW, and exceeds 75 m in turbosets beyond 1000 MW.

Fig. 1.2 (from [3]) The increase of the rotor length has been accompanied by the increase of the number of stages (or discs on a shaft), and the number of bearings and couplings between shafts in a line. Adding the increase of seal complexity and the problems raised by the non-uniform thermal expansion at start-up, all doubled by strength of materials problems raised by the increase in size, one can easily understand the complexity of the dynamic calculations of the rotors of such machines. Figure 1.3 shows a typical axial section in an industrial back-pressure turbine of an early design [4]. The steam is expanded in the turbine from the livesteam pressure to the exhaust pressure in two principal parts. In the first part, the steam is accelerated in the nozzle segments 1, thus gaining kinetic energy, which is utilized in the blades of the impulse wheel 2. The disc of the impulse stage is integral with the shaft. Usually, the nozzles are machined into several segments fixed into the cylinder by a cover ring. The blades of the impulse wheel are milled from chromium steel bars. The roots are fixed into the slot in the impulse wheel with spacers gripping the upset feet of the blades. In some designs, the flat outer ends are welded together in groups, thus forming an interrupted shroud. The second or reaction part consists of stationary and moving rows of blades 3 fixed with suitably shaped spacers into slots in the casing and rotor. The glands 4 prevent the steam flowing out of the casing along the shaft. Labyrinth seals allow a very small amount of steam to escape into specially


5

provided channels. Due to the turbulence of the steam, the pressure drop is sufficiently high to allow the gland to be made relatively short. The labyrinth strips are caulked into grooves in the rotor shaft whereas the corresponding grooves are machined into a separate bushing of the casing. The risk of damaging the rotor by distortion caused by friction in the seals is avoided, as the heat transfer from the tips of the thin labyrinth strips to the shaft is very small.

Fig. 1.3 (from [4]) The balancing piston 5 is positioned between the impulse wheel and the gland at the steam inlet end. The chamber between is interconnected with the exhaust. Generally, the balancing ring is integral with the shaft. In older designs it was shrunk-on but this design can give rise to instability due to rotating dry friction. This arrangement counteracts the axial forces imposed on the rotor by the steam flow. The bearing 6 at the steam inlet end is a combined thrust and journal bearing, to reduce the rotor length. The thrust part of it acts in both axial directions on the thrust collars 7 to absorb any excess forces of the balancing piston. Usually tilting bronze pads are fitted on flexible steel rings according to the Mitchell principle. The journal bearing of the combined bearing and that at the opposite end 8 are lined with white metal cast into separate shells. Tilting pad bearings are used in some designs. The rotor 9 is machined from high-quality steel forging. After the blades are fitted, the rotor is balanced and subjected to a 20 percent overspeed test for a few minutes. A high-alloy chromium steel is used for high pressures and temperatures. Figure 1.4 shows presently used steam turbine rotor designs [5].


6

Turbines running at high speeds require reduction gearing to drive alternators with 2 or 4 poles, running at 3000 or 1500 rpm (for 50 Hz). As a rule, the pinion and gear wheel shafts are connected to the driving and driven machines by means of couplings. They must be able to compensate for small errors in alignment and thermal expansion in the machine without affecting the reduction gearing. The coupling hubs are integral with the forged shafts.

Fig. 1.4 (from [5]) The first steam turbine built in Romania in 1953 at Reşiţa, was a 3 MW at 3000 rpm turbine. In 1967, the first two-cylinder 50 MW turbine was built. Twenty years later, the 330 MW four-cylinder condensing turbine was manufactured at I.M.G. Bucureşti, under a Rateau-Schneider license. Rotors have a monoblock construction, having the discs in common with the shaft. At present, General Turbo S.A. manufactures 700 MW turbines.

1.1.2 Gas turbines The development of gas turbines is more recent. From the first gas turbine for airplanes, designed by Whittle in 1937, and the first stationary turbine built by Brown Boveri in 1939, turbines of 80 MW at 3000 rpm and 72 MW at 3600 rpm are found in power plants, while 16 MW turbines are working with blastfurnace gases. The progress is mainly due to blade cooling and limitation of the effects of corrosion and erosion. State-of-the-art gas turbines built by ABB have 265 MW at 3000 rpm and 183 MW at 3600 rpm. The simplest type of open circuit stationary gas turbine installation comprises a compressor, a combustion chamber, and a gas turbine. In the


7

arrangement from Fig. 1.5, the compressor and turbine rotors form a single shaft line, while the generator 7 is coupled via a clutch 6. The starter 9 is used to launch the generator when operating as a compensator. The starter 5 is used to launch the turbine while the generator turns. Part of the compressed air is used for the fuel combustion. The remainder (approx. 70%) is used for cooling the shell of the combustion chamber and some components of the turbine, and is mixed with the hot gases.

Fig. 1.5 (from [6]) The volume of the expanded gas in the turbine is much larger than the volume of the compressed air in the compressor, due to the heating in the combustion chamber. The difference between the work produced by the turbine and the work absorbed by compressor and friction losses is the work supplied to the electrical generator. It is a function of the compressor and turbine thermodynamic efficiencies and the turbine inlet temperature.

Fig. 1.6 (from Power, Jan 1980, p.27) A design with concentric shafts, resembling the aircraft gas turbines, is shown in Fig. 1.6.

8


Figure 1.7 shows the Rolls-Royce RB.211 turbofan rotors. The threestage low pressure (LP) turbine drives the single-stage LP fan which has no inlet guide vanes. The single-stage intermediate pressure (IP) turbine drives the sevenstage IP compressor. The single-stage air-cooled high pressure (HP) turbine drives the six-stage HP compressor.

Fig. 1.7 (adapted from [7]) The eight main bearings are located in four rigid panels (not shown). The three thrust ball bearings are grouped in a stiff intermediate casing. Oil squeezefilm damping is provided between each roller bearing and housing to reduce engine vibration. The short HP system needs only two bearings located away from the combustion zone for longer life. The single-stage LP fan has 33 blades with mid-span clappers and fir-tree roots. The seven-stage IP axial compressor has drum construction. It consists of seven discs electron beam welded into two drums of five and two stages bolted together between stages 5 and 6. The blade retention is by dovetail roots and lockplates. The six-stage HP compressor consists of two electron beam welded drums bolted through the stage 3 disc with blades retained by dovetail roots and lockplates. The three-shaft concept has two basic advantages: simplicity and rigidity. Each compressor runs at its optimum speed, thus permitting a higher pressure ratio per stage. This results in fewer stages and fewer parts, to attain the pressure ratio, than in the case of alternative designs. The short, large diameter shafts give good vibration characteristics and a very smooth engine. The short carcase and the positioning of the engine mounting points give a very rigid structure. This allows the rotors to run with smaller tip clearances and thus improved efficiency. Gas turbines manufactured in Romania are: 1) the Viper 632-41, RollsRoyce license, 8-stage axial compressor and 2-stage turbine at 13,800 rpm; 2) the Alouette III B, Turbomeca license, 422 kW, 33,480 rpm; and 3) the Turmo IV CA, Turbomeca license, 1115 kW.


9

1.1.3 Axial compressors Although patents for axial compressors were taken out as long ago as 1884, it is only in the early 1950's that they become the most versatile form for gasturbine work. In the aircraft field, where high performance is at a premium, the axial compressor is now used exclusively. It is only for some industrial applications that other compressor types offer serious competition.

Fig. 1.8 (from [8]) The axial-flow compressor resembles the axial-flow steam or gas turbine in general appearance. Usually multistage, one observes rows of blades on a single shaft with blade length varying monotonically as the shaft is traversed. The difference is, of course, that the blades are shorter at the outlet end of the compressor, whereas the turbine receives gas or vapour on short blades and exhausts it from long blades. In Fig. 1.8 the numbers have the following designations: 1 and 13 bearings, 2 - seals, 3 - prewhirler, 4 - intake duct, 5 - rotor blades, 6 - stator blades, 7 - straightener stator blades, 8 - discharge duct, 9 - diffuser, 10 - coupling, 11 - gas turbine shaft, 12 - drum-type rotor, 14 - stator casing. In practically all existing axial compressor designs, the rotor is supported by one bearing at the gas inlet end and by a second bearing at the gas delivery end. In aircraft practice, ball and roller bearings are universally used, on account of their


10

compactness, small lubricating oil requirements, and insensitivity to momentarily cessations of oil flow as may occur during acrobatic flying.

1.1.4 Centrifugal compressors Although centrifugal compressors are slightly less efficient than axialflow compressors, they are easier to manufacture and are thus preferred in applications where simplicity, ruggedness, and cheapness are primary requirements. Additionally, a single stage of a centrifugal compressor can produce a pressure ratio of 5 times that of a single stage of an axial-flow compressor. Thus, centrifugal compressors find application in power station plants, petrochemical industry, gas injection and liquefaction, ground-vehicle turbochargers, locomotives, ships, auxiliary power units, etc.

Fig. 1.9 (from [9]) A typical high-pressure compressor design is shown schematically in Fig. 1.9. Apart from shaft, impellers, bearings and coupling, modeled as for other machines, items of major concern in rotor dynamic analyses are the gas labyrinths, the oil ring seals and the aerodynamic cross coupling at impellers. Furthermore, squeeze film dampers are used to stabilize compressors with problems. Multistage centrifugal compressors have relatively slender shafts. Usually, impellers are mounted on almost half of the rotor length, the other part being necessary for the centre seal, the balance drum, the oil seals, the radial bearings and the thrust bearing. The shaft diameter is kept small to increase the impeller eye. In comparison with the drum rotor of axial compressors, the shaft of centrifugal compressors is more flexible, having relatively low natural frequencies which favour instabilities.


11

Vibrations of a centrifugal compressor are controlled by: bearings, shaft geometry, gas seals and oil bushings, fluid forces on impellers, and other factors. Squeeze film dampers are used in centrifugal compressors to eliminate instabilities or to alter the speed at which they occur. In the case of centrifugal compressors, undamped critical speed maps are of little interest. For typical compressor precession modes which are heavily damped, second mode in particular, the damped natural frequency can be as much as 2 to 9 times lower than the expected peak response speed. Shop testing, carried out after compressor is constructed but before it is commissioned, can reveal problems prior to start-up. Bode plots, obtained during run-up measurements, are used to check that the critical speeds are not within the operating speed range. Separation margins of the critical speeds from the intended operating speed range are defined in API Standard 617; resonances must be 20 percent above the maximum continuous speed and/or 15 percent below the operating speeds [10]. Compliance with present specifications requires calculation of deflections at each seal along the rotor, as a percentage of the total clearance. Modern multistage compressors are typically designed to operate through and above several critical speeds so as to maximize the work done by a given size machine. For example, a 425 mm diameter impeller for an industrial centrifugal compressor can be designed for a work load well in excess of 2000 HP by running at speeds approaching 9000 rpm. Up to eight stages are used to obtain the required pressure rise. Process compressors and units used for natural gas injection can have discharge pressures of the order of 650 bar and can drive gases with high density. The result of this combination of supercritical speed, high pressure and high work load has been an increasing tendency for such machines to exhibit problems of nonsynchronous rotor whirling. This is why stability analysis is of prime interest. While many rotating machines operate below the first critical speed (point A in Fig. 1.10), turbomachinery operate above the first critical speed (point B). Until mid seventies any further shift of the resonance - and hence any increase in the maximum number of stages per casing - was precluded by the bearing stability limit. This was then raised by means of stronger bearing designs until operation above the second critical speed became possible (point C). High pressure compressors operating on fixed lobe bearings could generate a violent shaft whip condition just above twice the first natural frequency. By going to tilting-pad bearings, that threshold speed can be raised to well over two times the first natural frequency. Attempts to raise speed further came up against another stability limit: rotor instability due to gap excitation. Using vortex brakes before labyrinths this boundary has been pushed back and the way is open in principle to still higher speed ratios (point D).

12


Exhaust-gas turbocharging is used to increase the mean effective pressure (m.e.p.) of diesel engines. It has applications in stationary plants for electricity generation, in ships' auxiliary and propulsion machinery and in railway traction.

Fig. 1.10 (from [11]) One of the oldest applications was in marine engines. In 1923, BBC and the Vulkan shipyard manufactured turbochargers for the 10-cylinder four-stroke engines from the vessels 'Preussen' and 'Hansestadt Danzig'. The engines, which were designed for an uncharged performance of 1700 HP each at 235 rpm provided, when charged, a cruising power of 2400 HP at 275 rpm and a temporary overload of 4025 HP at 320 rpm (for a m.e.p. = 8.4). Turbocharging of two-stroke marine engines began after 1950. For the relatively short turbocharger rotors, which are almost always equipped with single-stage compressor and turbine wheels, two bearings are sufficient. One of these is a combined radial-axial bearing, the other a pure radial bearing. Two bearing layouts have proved successful on the market: 1) bearings at the shaft ends (external bearings), used predominantly in large machines, and 2) bearings between the compressor and turbine wheel (internal bearings) used mainly for small turbochargers. In both arrangements the axial bearing is located near the compressor wheel, to keep the axial clearance in that region small.


13

In the variant with external bearings (Fig. 1.11, a), the large distance between the bearings reduces the radial bearing forces and requires smaller clearances at the compressor wheel and turbine wheel. The frictional losses in the bearings are smaller, particularly at part load. The shaft ends can be kept small in diameter and are simple to equip with a lubricating oil pump and centrifuge, thus rendering rolling-contact bearings and self-lubrication possible.

Fig. 1.11 Internal bearings (Fig. 1.11, b) offer advantages in fitting a turbocharger with axial air and gas inlets to the engine. Small turbochargers do not, however, have an axial-flow turbine, but a radial-flow turbine with axial gas outlet. For specific applications internal bearings have advantages, which relate mainly to the wider variety of ways of fitting the turbocharger to the engine. In automotive applications, a floating bush bearing is used due to size and cost considerations. This type of bearing has a thin bush rotating freely between the journal and the fixed bush, forming two hydrodynamic oil films [12]. This turbocharger shows peculiar behaviour yet to be explained theoretically: 1) it has stable operation at very high shaft speeds, though at lower speeds it can exhibit instability in either a conical mode or an in-phase bending mode; and 2) some designs have a third flexible critical speed, very difficult to balance out; with a high amplification factor, leading to rubbing and bearing distress.


14

1.1.5 Fans and blowers Fans can be either radial-flow or axial-flow machines. The ratio discharge pressure vs. suction pressure is defined as the pressure ratio. Fans are designed for pressure ratios lower than or equal to 1.1. Centrifugal fans absorb powers between 0.05 kW and 1 MW, have flow rates up to 3·105 m3/h and discharge pressures up to 1000 mm H2O (~104 N/m2). Blowers are single-stage uncooled compressors with pressure ratios between 1.1 and 4, and discharge pressures up to 3.5·105 N/m2. Compressors have pressure ratios larger than 4, so they usually require interstage cooling.

Fig. 1.12 (from [13])

Fig. 1.13 (from [13])


15

The design from Fig. 1.12 is a medium-pressure blower, with labyrinth seals, and overhung design. The arrangement from Fig. 1.13 is with double suction and single exhaust. The symmetrical rotor has a disc at the middle. Centrifugal fans used for forced- or induced-draft and primary-air service generally have large diameter rotors, operating from 500 to 900 rpm in pillowblock bearings, supported on structural steel or concrete foundations. As a rule, the major problem with fans is unbalance caused by 1) uneven buildup or loss of deposited material; and 2) misalignment. Both are characterized by changes in vibration at or near the rotational frequency.

1.1.6 Centrifugal pumps Centrifugal pumps are used in services involving boiler feed, water injection, reactor charge, etc. Instability problems encountered in the space shuttle hydrogen fuel turbopumps and safety requirements of nuclear main coolant pumps have prompted research interest in annular seals. It is now recognized that turbulent flow annular seals in multi-stage pumps and in straddle-mounted single-stage pumps have a dramatic effect on the dynamics of the machine. Stiffness and damping properties provided by seals represent the dominant forces exerted on pump shafts, excluding the fluid forces of flow through the impellers, particularly at part-flow operating conditions. For these systems, the hydrodynamics of oil-lubricated journal bearings is dominated by seal properties. Typical multi-stage centrifugal pumps have more inter-stage fluid annuli than they have journal bearings. The fluid annuli are distributed between the journal bearings where precession amplitudes are highest and can therefore be 'exercised' more as dampers than can be the bearings. In typical applications, shaft resonant critical speeds are rarely observed at centrifugal pumps because of the high damping capability afforded by seals. Problems encountered with boiler feed pumps have been produced by excessive wear in seals, yielding a decrease in the dynamic forces exerted by the seals. Centrifugal pumps have comparatively slender shafts and relatively flexible cantilevered bearing housings (Fig. 1.14). Fine clearance annular seals are used in pumps primarily to prevent leakage between regions of different pressure within the pump. The rotordynamic behaviour of pumps is critically dependent on forces developed by annular seals, between the impeller shroud and the stator, between the impeller back disc and the stator, and between the impeller and diffuser.


16

Fig. 1.14 (from [14])

1.1.7 Hydraulic turbines Hydraulic turbines have traditionally been used to convert hydraulic energy into electricity. The first effective radial inward flow reaction turbine was developed around 1850 by Francis, in Lowell, Massachusetts. Around 1880 Pelton invented the split bucket with a central edge for impulse turbines. The modern Pelton turbine with a double elliptic bucket, a notch for the jet and a needle control for the nozzle was first used around 1900. The axial flow turbine, with adjustable runner blades, was developed by Kaplan in Austria, between 1910-1924. The horizontal bulb turbines have a relatively straighter flow path through the intake and draft tube, with lower friction losses. In the Straflo (straight flow) design, the turbine and generator form an integral unit without a driving shaft. With hydraulic turbines, despite the low rotating speeds (200-1800 rpm), problems occur owing to the vertical position of most machines, due to transients and cavitation. Rotors are very robust and stiff, problems being raised by bearings and the supporting structure.


17

Fig. 1.15 The hydro power plant at Grand Coulée (U.S.A.) has a 960.000 hp Francis turbine driving a synchronous generator of 718 MVA at 85.7 rpm. The rotor has a diameter in excess of 9 m and a weight exceeding 400 tons, the main shaft having 3.3 m diameter and more than 12 m length. The world’s largest hydroelectric plant Itaipu, on the Rio Paraná, which forms the border between Brazil and Paraguay, near the city of Foz do Iguaçu, consists of 18 generating sets of 824/737 MVA, driven by Francis turbines, with a total rating of 12,600 MW. Turbines have rotors of 300 tons and 8 m diameter, the main shaft has 150 tons and 2.5 m diameter, while the synchronous generator has 2000 tons and 16 m diameter, running at respectively 90.9 rpm for 50 Hz generators, and 92.3 rpm for 60 Hz generators (Fig. 1.15). The hydro power plant at Ilha Solteira, Brazil, has sets of 160 MW at 85.8 rpm. The rotor shaft has 6.33 m length, 1.4 m outer diameter and 0.4 m inner diameter. The generator has 495 tons and the Francis turbine has 145 tons. The first critical speed is about 222 rpm. The hydroelectric power plant at Corbeni-Argeş has four Francis turbines with nominal speed 428.6 rpm, gross head 250 m, nominal water flow 20 m3/s and individual rated power 50 MW. An axial cross-section of a vertical axis Kaplan turbine is presented in Fig. 1.16 where 1 – runner with adjustable blades, 2 – draft tube, 3 – guide vanes, 4 – lower guide bearing, 5 – stay vanes and ring support, 6 – concrete spiral casing, 7 – control ring with servo-motor for the stay vanes, 8 – thrust bearing, 9 – upper guide


18

bearing, 10 – servo-motor for adjustment of runner blades, 11 – runner blades control rod inside the turbine shaft, and 12 – generator.

Fig. 1.16 (from [15]) The Porţile de Fier I hydroelectric power plant has eight Kaplan turbines of 194 MW, head 27 m, nominal water flow 840 m3/s, speed 71.43 rpm, 6 blades and rotor diameter 9.5 m. The Porţile de Fier II hydroelectric power plant has eight doubleregulated bulb units type KOT 28-7.45, with the bulb upstream and the turbine overhung downstream. The unit has three guide bearings and a thrust bearing, 16 stator blades and 4 rotor blades, and the following parameters: head 7.45 m, nominal water flow 432 m3/s, rated power 27 MW, rotor diameter 7.5 m.

1.1.8 Turbo-generators The turbo-alternator was developed by C. E. L. Brown and first marketed by Brown Boveri in 1901. With a cylindrical rotor having embedded windings, it has proven to be the only possible design for high speeds, as when driven direct by a steam turbine. Such alternators are available for ratings between 500 kVA and


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20,000 kVA and higher, but are not normally used below 2500 kW, because salient-pole machines with end-shield bearings are more economical. Beyond 2500 kW, an alternator running at 3000 (or 3600) rpm permits a more economical gear to be used than a 1500 (or 1800) rpm alternator for the same turbine [16]. The marked increase in the unit ratings of turbo-generators has not, for the most part, been accompanied by a corresponding increase in the size of machines because of the increase in the specific electric ratings. For example, between 1940-1975, the maximum power of electric generators increased from 100 to 1600 MVA, whereas in 1940 a 3000 rpm turbo-generator weighed 2 kg per kW of output, and its 1975 counterpart weighed only 0.5 kg/kW. Alternator rotors have been also designed to be progressively longer and more flexible. The forging of a 120 MW rotor had approximately 30 tons and 8 m distance between bearing centres, while a 500 MW rotor had 70 tons and 12 m. Modern rotors have two or three critical speeds below their operating speed of 3000 rpm.

Fig. 1.17 (from [16]) The rotor of small units is a solid cylindrical forging of high-quality steel with slots milled in it to accommodate the field winding. For larger units, several hollow cylinders are fitted over a central draw-bolt threaded at both ends, to which the two shaft extensions are fastened by shrinking. The specially formed winding is a single layer of copper strip insulated with glass-fibre which is pressed and baked into the slots. To secure the end sections, end-bells forged from solid-drawn nonmagnetic steel with ventilation holes or slots are used.


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Rotors of electrical machines are different from rotors with bladed discs or impellers, being more massive, but occasionally rising problems due to asymmetrical stiffness properties. Figure 1.17 is a cutaway perspective drawing of a 400 MVA, 3000 rpm generator with water-cooled stator winding and forced hydrogen direct cooling in the rotor. Due to the high flux density and current loadings, generators of over 500 MW employing these cooling methods must have their stator cores mounted in a flexible suspension. This is necessary in order to isolate the foundations from the enormous magnetic vibration forces arising between rotor and stator. Two-pole generator rotors have axial slots machined to match more closely the principal stiffnesses. They are intended to reduce the parametric vibrations induced by the variation of the cross-section second moment of area about the horizontal axis, during rotation. The second order (or 'twice per revolution') forced vibration which arises from the dual flexural rigidity is virtually inescapable in a two-pole machine; where the motion is excited by the weight of the rotor. This is a source of considerable difficulty, largely because it can be cured only at the design stage and cannot be 'balanced'. Certain 'trimming' modifications can be made but these present problems of their own. In fact it would be very difficult to design accurately an alternator rotor so as to have axial symmetry in a dynamical sense. The rotor is, in effect, a large rotating electromagnet, having a north pole and a south pole on opposite sides of the rotor and having slots cut in it, in which copper conductors are embedded to provide the magnetic field.

Fig. 1.18 (from [17]) The cross-section of a 120 MW alternator rotor after slotting is shown in Fig. 1.18, a. It is clear from the figure that the flexural rigidity of the shaft is


21

unlikely to be the same for bending about the horizontal and the vertical neutral axes, even after copper conductors and steel wedges have been placed in the slots. In attempts to equalize these rigidities, one of two schemes is usually adopted. In the first, the pole faces are slotted as shown in Fig. 1.18, b. In order to maintain the magnetic flux density, the slots in the pole faces are filled with steel bars that are wedged in. The second technique is to build a rotor in the manner of Fig. 1.18, a and then to cut lateral slots across the poles at intervals along the length of the rotor. Figure 1.19 shows the different cross-sections in a turbo-generator rotor: A-A rectangular slots for field winding and smaller slots in the pole area, and B-B cross-cuts to ensure uniform flexibility with respect to the vertical and horizontal cross-section principal axes.

Fig. 1.19 (from [18]) Alternator rotors are supported in plain bearings. These hydrodynamic bearings present unequal dynamical stiffnesses in the vertical and horizontal directions. Asymmetry of the bearings introduces a split of critical speeds but cannot by itself cause second order vibration. For small machines, e.g. electrical motors, having relatively low rotational speeds and rolling-ball bearings, the balancing and the dynamic calculation of the rotor does not generally raise problems. On the contrary, large machines, having long and flexible rotors, sliding bearings, seals, pedestals and relatively flexible casings, with high speeds, have determined the continuous development and improvement of dynamic calculations and vibration measurement.


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1.2 Rotor-bearing dynamics Rotor-Bearing Dynamics has got its own status, apart from Mechanical Vibrations and Structural Dynamics, becoming an interdisciplinary research field, as soon as the importance of the effects of bearings and seals on the rotor dynamic response has been recognized.

The scope of Rotor-Bearing Dynamics is the study of the interaction between rotor, stator and the working fluid, for the design, construction and operation of smooth-running machines in which allowable vibration and dynamic stress levels are not overpassed, within the whole operating range. Smooth machine operation is characterized by small, stable rotor precession orbits, and by the absence of any instability throughout the machine operating range. In order to understand the dynamic response of a rotating machine it is necessary to have, early in the design stage, information on the following aspects of its behavior: 1. Lateral critical speeds of the rotor-bearing-pedestal-foundation system; effects of the stiffness and damping of bearings, seals, supporting structure and foundation on the location of critical speeds within the machine operating range. 2. Unbalance response: orbits of the rotor precession as a response to different unbalance distributions, throughout the whole operating range of the machine, and vibration amplitudes due to rotor unbalance. 3. Rotor speed at onset of instability: the threshold speed for unstable whirling due to the rotor/bearing and/or working fluid interaction, as well as the consequences of its crossing. 4. Time transient response analysis, to a blade loss, mainly for gas turbine engines operating at supercritical speeds, or when passing through a critical speed. 5. System torsional critical speeds, especially at geared rotors, eventually the transient response of the shaft line to electric disturbances applied to the generator. Practical measures regarding the balancing and the monitoring of the dynamic state of rotors are added to these: 6. Balancing of rotors: calculation and attachment (removal) of correction masses such that the centrifugal forces on the rotor due to these additional masses and the inherent unbalance forces are in equilibrium. 7. Machinery monitoring: measurement of the parameters characterizing the dynamic state of machines and trending their time evolution, in order to detect any damage, to anticipate serious faults, determining the outage.


23

The capability of predicting the performances of a rotor-bearing system is dependent firstly on the information about bearing properties, fluid-rotor interaction and the unbalance distribution along the rotor. In this respect, in recent years, important progress has been achieved in determining the dynamic coefficients of bearings and seals, and in the identification of the spatial distribution of unbalance for flexible rotors. The direct result is the development of computer programs helping in modeling most of the dynamic phenomena occurring during the operation of rotating machinery. Generally, the following dynamic characteristics of rotating machinery are of interest: a. Rotor lateral critical speeds in the operating range. b. Unbalance response amplitudes at critical speeds. c. Threshold speed of instabilities produced by bearings, seals or other fluid-structure interactions. d. Bearing transmitted forces. e. The overshoot ratio, of maximum transient response relative to the steady-state response. f. System torsional critical speeds. g. Gear dynamic loads. h. Vibration amplitudes in casing and supporting structure. The following can be added to this list: i. Natural frequencies of bladed discs, impellers, wheels. j. Frequencies and mode shapes of blades and blade buckets. k. Blade flutter frequencies. l. Rotating stall and surge thresholds. m. Noise radiated by rotating machinery. In the following, only the first three issues are treated. Problems not treated in this book are: a. Shafts with dissimilar principal moments of inertia; b. Cracked rotors; c. Reverse precession due to dry-friction contact between rotor and stator; d. Partial rubbing conditions; e. Transient critical-speed transition.


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1.3 Rotor precession The most important sources of machinery vibration are the residual rotor unbalance and rotor instability. Most rotors have at least two bearings. With horizontal rotors, the rotor weight is distributed between all the bearings. The rotation axis is coincident with the static elastic line under the own weight. If the weight effect is neglected, the rotation axis coincides with the line connecting the bearing centres. Any rotational asymmetry due to manufacturing, or produced during operation, makes the line connecting the centroids of rotor cross-sections not to coincide with the rotation axis. Hence, as the rotor is brought up in speed, the centrifugal forces due to dissymmetry cause it to deflect. For example, a 50 tons rotor, with its mass centre off-set by 25 μm from the axis of rotation, experiences a force of approximately 13 tons force, when rotating at 3000 rpm. The rotating centrifugal forces are transferred to the bearings and their supports, and produce unwanted vibrations. While the bearings and the casing vibrate, the rotor has a precession motion. For isotropic bearings, at constant speed, the deflected shape of the rotor remains unchanged during the motion, any cross-section traces out a circular whirling orbit. The motion appears as a vibration only when the whirl amplitude is measured in any fixed direction. Despite the analogy often used in describing vibration and precession, their practical implications are different. The remedy for resonance – internal damping – is totally inefficient in the case of critical speeds, since the shape of the deflected rotor does not change (or changes very slightly) during the precession motion at constant speed. Moreover, at a critical speed, if the deflections are not limited, a rotor bends rather than damages by fatigue, phenomenon produced by the lateral vibrations. Instead, journal bearings, small clearance liquid seals, or viscous sleeves are the major source of damping in most cases. Without this damping or a similar source, it would be very difficult to pass through a critical speed. That is why bearings and seals play a major role in the dynamics of the rotor systems. If identical orbits are traced out with successive rotor rotations, the motion is said to be stable precession. If the orbit increases in size with successive rotations, the motion is an unstable whirl. It may subsequently grow until the orbit becomes bounded either by system internal forces, or by some external constraint, e.g. bearing rub, guard ring, shut-down, etc. Some typical orbits are shown in Fig. 1.20. The circular orbit (Fig. 1.20, a) represents the synchronous whirling of a rotor in isotropic radial supports. The absence of loops within the orbit denotes synchronous whirl.


25

An elliptical orbit (Fig. 1.20, b) may arise from orthotropic supports, i.e. from dissimilar bearing or pedestal stiffnesses in the horizontal and vertical directions. Inclination of ellipse axes occurs due to cross-coupled stiffnesses and damping properties.

a

b

c

d

e

f

Fig. 1.20 (from [19]) If the precession is non-synchronous, i.e. the rotor whirls at a frequency other than the rotational frequency, the orbit will contain a loop as in Fig. 1.20, c, characteristic for the half-frequency whirl due to the instability of motion in hydrodynamic bearings ("oil whirl"). An internal loop indicates that the precession is in the direction of rotation. Other non-synchronous excitations may occur at several times rotational frequency, giving rise to multi-lobe whirl orbits depicted in Fig. 1.20, d, as in the case of multi-pole electrical generators. Instabilities such as the half-frequency whirl are frequently bounded. The whirl is initiated by crossing the onset of instability speed, and then it develops in a growing transient, whose radius increases until a new equilibrium orbit is reached (Fig. 1.20, e). Another type of transient condition is shown in Fig. 1.20, f. The rotor is initially operating in a small stable unbalance whirl condition. The rotor system then receives a transverse shock, and the journal displaces abruptly in a radial


26

direction within the bearing clearance, but without contacting the bearing surface. Following the impact, the rotor motion is a damped decaying spiral transient, as it returns to its original small unbalance whirl condition. Many other types of whirl orbits have been observed, such as those associated with system non-linearities and nonsymmetric clearance effects.

1.4 Modeling the rotor For the mechanical design of rotor, bearings and supporting structure, one has to take into account that they work as a whole, responding together to the dynamic loading, and interacting. The rotor is part of a dynamic system, its behavior being determined by the location and stiffness of bearings, seals, pedestals and foundation, as well as their damping properties. The casing and foundation masses also play an important role. The rotor is the main part in any piece of rotating machinery. Its function is to generate or transmit power. It consists of a shaft on which such components as turbine wheels, impeller wheels, gears, or the rotor of an electric machine may be mounted. The rotor is never completely rigid and in many applications it is actually quite flexible. However, in practice, rigid rotors are considered to be those running below 1/3 of the first bending critical speed. Elastic rotors operate near or beyond the first bending critical speed, so that the centrifugal forces due to the residual unbalance cause it to deflect. In most machines, rotors have shafts with axisymmetric cross-section. If, in some parts, the cross-section is not symmetrical, then the bending stiffness with respect to a fixed axis is variable during the rotation giving rise to non-synchronous motions and instabilities (e.g. two-pole generators and cracked rotors). The rotor shaft can be modeled as a Timoshenko-type beam, accounting for the shear and rotational inertia, including also the effect of gyroscopic couples. The discs – usually rigid – are included by lumped parameters: the mass, and the polar and diametral mass moments of inertia. More advanced calculations consider the disc flexibility. Rotors of individual machines are joined by couplings (locked spline, double-hinged, sliding spline, flex plate). Bearings are selected as a function of static load and speed, taking into account the dynamic loading, available space, energy losses, simplity of design solution, as well as durability and reliability requirements. In early studies, bearings were considered as rigid supports (Fig. 1.21, a). Later, their radial stiffness and damping has been taken into account (Fig. 1.21, b). In rolling bearings and air bearings, the damping is usually neglected. The stiffness and damping characteristics of journal bearings are functions of running speed and loading. At rolling bearings, the stiffness is considered independent of speed and


27

loading. Generally, only the bearing translational radial stiffness is taken into account, the angular stiffness being relatively small (one tenth). With journal bearings, under steady-state hydrodynamic conditions, the total pressure force equals the static load on the bearing. If the centre of the rotating journal is in motion, as for instance during synchronous precession, additional pressures are set up in the lubricant film, which act as dynamic forces on the journal in addition to the static forces. The dynamic force depends on both the relative displacement and the velocity of the journal centre motion but, in contrast to conventional elastic forces, the dynamic force does not have the same direction as the imposed motion, being phase shifted in space and time.

Fig. 1.21 Resolving the dynamic force into two components along fixed coordinate axes in the bearing, say Oy and Oz, and likewise resolving the journal centre motion into y and z displacements, the dynamic force components may be expressed by: ⎧ f y ⎫ ⎡k yy k yz ⎤ ⎧ y ⎫ ⎡c yy c yz ⎤ ⎧ y& ⎫ ⎨ ⎬ = ⎢k ⎥⎨ ⎬ + ⎢ ⎥ ⎨ ⎬. ⎩ f z ⎭ ⎣ zy k zz ⎦ ⎩ z ⎭ ⎣ c zy c zz ⎦ ⎩ z& ⎭

(1.1)

The above equations are exact only for very small amplitudes, but in practice they prove to be valid even for amplitudes as large as a third of the bearing clearance.


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The four stiffness coefficients k yy , k yz , k zy , k zz and the four damping coefficients c yy , c yz , c zy , c zz

are calculated from lubrication theory by linearizing the non-

linear bearing forces. They are properties of the particular bearing, being functions of the bearing configuration and the lubricant properties. More important, they belong to a given steady-state journal center position, changing with the speed of rotor. The inequality of cross-coupling stiffnesses k yz ≠ k zy is the source of a certain type of self-excited precession known as oil whirl, fractional frequency whirl, or half-frequency whirl. Because of the speed dependence of the eight bearing coefficients, the effective damping is negative at low speeds and may become positive at higher speeds. Active magnetic bearings are applied in industrial centrifugal compressors, turbo expanders and centrifugal pumps. The principle is an electromagnetic shaft suspension, without physical contact between rotor and stator. Sensors located near the electromagnets observe the rotor position. This rotor position signal feeds into an electronic controller which feeds, in a closed loop, the power amplifiers of the electromagnets. Short annular seals with gas or fluid are usually considered isotropic. The diagonal terms of their stiffness and damping matrices are equal, while the offdiagonal terms are equal, but with reversed sign. The two force components by which the seals act upon the rotor can be written as

⎧ f y ⎫ ⎡ K k ⎤ ⎧ y ⎫ ⎡ C c ⎤ ⎧ y& ⎫ ⎨ ⎬=⎢ ⎥⎨ ⎬ + ⎢ ⎥⎨ ⎬ + M ⎩ f z ⎭ ⎣− k K ⎦ ⎩ z ⎭ ⎣− c C ⎦ ⎩ z& ⎭

⎧ &y&⎫ ⎨ &&⎬. ⎩z⎭

(1.2)

The inertial term is negligible at gas seals, where direct stiffnesses can be very small, even negative. For long annular clearance seals, like those used to break down large pressure differences in multi-stage pumps, angular dynamic coefficients are introduced, because the rotor is acted upon by couples, and forces give rise to tilting shaft motions, and moments produce linear displacements. Radial seals in centrifugal pumps are either balance disks or the radial gap of mechanical seals. Impellers generate motion-dependent forces and moments in the flow fields between impeller tip and casing (volute or diffuser) and in the leakage flow fields developed between impeller shrouds and casing. Squeeze-film dampers are used in gas turbines as a means of reducing vibrations and transmitted forces due to unbalance. A squeeze film is an annulus of oil supplied between the outer race of a rolling-element bearing (or the bush of a sleeve bearing) and its housing. It can be considered as a parallel element of a vibration isolator, or as a series element in a bearing housing.


29

Flexible pedestals are considered in the dynamic response of machines especially for blowers and fans, centrifugal pumps and turbosets with flexible casings and cantilevered bearings. The calculation model (Fig. 1.21, c) includes the stiffness and damping of bearing supports and, in some cases, also their equivalent mass. The foundation, the sole plate and the soil are seldom included in the calculation model (Fig. 1.21, d), their influence on the rotor response being generally smaller. However, in some cases, especially for large fans on concrete pedestals, the elasticity of the subgrade is taken into account. Generally, for bearings and pedestals, even with frequency independent characteristics, the horizontal stiffness is lower than the vertical stiffness. This anisotropy doubles the number of critical speeds. In some cases, due to the high damping level, the separation of the two criticals in a pair due to orthotropy does not show up in the unbalance response of rotors. It is also possible for some whirl modes, especially the backward ones, to be overcritically damped, thus appearing neither in the natural frequency diagrams nor in those of the unbalance response

1.5 Evolution of rotor design philosophy Calculation methods and the interpretation of the results of the dynamic analysis of rotors had a spectacular evolution. Until the late 1950's, calculations were made numerically or using the graphic method developed by Mohr. It was common practice to assume rigid supports and to treat one span at a time in the model. Analysis was limited to the determination of undamped critical speeds and the objective was to avoid having a running speed at a critical speed, in other words, each span was 'tuned' to avoid certain frequencies. The purpose of an undamped analysis was to provide a close, initial estimate of the critical speeds. Many specifications explicitly require that operating speeds differ from critical speeds by safe margins. In the API Standard 610, the critical speed is required to be at least 20% greater or 15% less than any operating speed [20]. Compliance with such specifications requires that critical speeds be calculated as part of design and selection procedures of rotating machines. In some cases well established procedures are used. In other cases, e.g. machines that handle liquids, like centrifugal pumps, specific calculation procedures are used, the ‘dry’ running critical speeds being different from the ‘wet’ running criticals. It was recognized that large discrepancies existed between calculations and tests, and efforts were made to improve the analyses. The rigid supports were replaced by elastic springs with stiffness equal to that of the oil film in the bearings. Later, the effect of pedestals was added. It was recognized that entire


30

rotor-bearing systems, rather than single spans, should be analyzed, taking advantage of the advent of high-speed digital computers. After whole systems were studied, it became clear that a change in philosophy was required, involving a switch from tuning to response. In some cases, the large discrepancies between calculations and tests were diminished introducing the effect of damping and determining the damped critical speeds. Also, calculating the unbalance response, it came out that the speeds at which the radius of synchronous whirling orbits is a maximum – referred to as peak response critical speeds – are different from both the undamped and damped critical speeds, approaching the latter. It was recognized that not all potentially critical speeds are indeed critical, the large damping in bearing smoothing the unbalance response curves so that the passage through a critical speed may take place without an increase in the response amplitude. At present, numerical simulations are used in the predictive design stage, and rotor designs are accepted or rejected on the basis of the unbalance response at the journals as a function of running speed. In contrast to lateral vibration, torsional natural frequencies are tuned to avoid coincidence with running speed and known exciting frequencies.

Fig. 1.22 Figure 1.22 shows the response of a rotor journal versus the ratio of natural frequency to running speed. Values calculated for rigid supports are denoted by R1 and R2, while values for bearings treated as elastic springs are indicated by E1, E2 and E3. The continuous line shows the unbalance response calculated considering both the stiffness and the damping in bearings. D1, D2, D3


31

are the actual critical speeds, measured by the peaks in the unbalance response characteristic. It can be seen that critical speeds based on rigid support calculations can be seriously in error, that criticals calculated assuming elastic supports can be more accurate and that neither calculation can be used to determine a response level because damping has been neglected. Figure 1.23 shows an analysis of a complete system consisting of high pressure turbine (HPT), intermediate pressure turbine (IPT), low pressure turbine (LPT), generator (G) and exciter (E).

Fig. 1.23 Assuming elastic supports at the bearings, 22 critical speeds were calculated between zero and the running speed. With such a large spectrum of natural frequencies, the desired separation between the operating speed and the critical speeds has limited application, so that there is a need to change the basic philosophy of design and commissioning of a rotating machinery. In the field of compressors and turbines for industry or power plants, the actual trend to increase the size and the operating speeds has lead to a new generation of machines for which, inevitably, one or two critical speeds are within the range of operating speeds. As machinery become larger, the elasticity of the lubricant film in bearings and the flexibility of supports play a more important role with respect to


32

the rotor stiffness, determining a decrease of the critical speeds and their interference with the range of operating speeds. As the bearing dynamic characteristics are not exactly known, critical speeds cannot be determined accurately, so that the traditional criterion aiming at operation at or near to a critical cannot offer the necessary safe limit In practice it has been found out that one can operate perfectly safe and reliable at well damped critical speeds if the vibration levels do not exceed the allowable levels and if the rotor has not a pronounced sensitivity to mass unbalance. It means that the unbalance response of the rotor can give the most useful information about the soundness of a design solution. Carrying out this calculation for different unbalance distributions, judiciously chosen so as to enhance the deflection at different unbalance critical speeds, it can be established how critical each of the critical is and what measures have to be taken so that the vibration amplitudes remain within normal limits, even in the presence of unbalances that occur during the normal operation (erosion, deposits, component failures, thermal strains, etc.).

1.6 Historical perspective The first analysis of critical speeds of a uniform elastic shaft has been made in 1869 by Rankine [21], who devised the term ‘critical speed’. The phenomenon was incorrectly thought to be an unstable condition, the rotor being unable to run beyond that speed. In this case art preceded science, for in 1895 some commercial centrifuges and steam turbines were already running supercritically. Gustaf de Laval first demonstrated experimentally that a (single stage steam) turbine could operate above the rotor’s lowest bending resonance speed and supercritical operation could be smoother than subcritical. In many European papers, the rotor model consisting of a central disk, mounted on a massless flexible shaft supported at its ends, is referred to as the Laval rotor. Although the first correct solution for an undamped model has been given by Föppl [22], who was the first to demonstrate analytically that that a rotor could operate supercritically, the confusion persisted until the publication in 1919 of Jeffcott’s paper [23] using a model with damping. This simple model is called the Jeffcott rotor in recent papers. In 1894 Dunkerley [24] published results of his studies on critical speeds of shafts with many discs, and gave his well-known method with its experimental verification. In 1916 Stodola [25] published an analysis of the bearing influence on the flexible shaft whirling. He also introduced the gyroscopic couples on disks. Hysteretic whirl was first investigated by Newkirk [26] in 1924 during studies about a series of failures of blast-furnace compressors. It was observed that at speeds above the first critical speed, these units would enter into a violent


33

whirling in which the rotor centerline precessed at a rate equal to the first critical speed. If the unit rotational speed was increased above its initial whirl speed, the whirl amplitude would increase, leading to eventual rotor failure. During the course of the investigation, Kimball in 1924 suggested that internal shaft friction could be responsible for the shaft whirling. In 1925, Kimball and Lovell performed extensive tests on the internal friction of various materials. At the end Newkirk concluded that the internal friction created by shrink fits of the impellers and spacers is a more active cause of the whirl instability than the material hysteresis in the rotating shaft. In 1925, Newkirk and Taylor [27] observed oil film whirl and resonant whipping. The true upper limit for safe operating speeds has been thus revealed, namely the threshold speed of rotor-bearing instability. This typically occurs at speeds between two and three times the lowest resonant frequency, wherefrom the name of half-frequency and sub-synchronous whirl. The phenomenon was explained only in 1952 by Poritsky [28], who showed that the destabilizing influence comes from the hydrodynamic journal bearing which loses its ability to damp the lowest rotor-bearing bending resonant mode. In 1933 Smith [29] published a review of the basic rotor dynamics problems, discussing qualitatively the effect of gyroscopic coupling, and simultaneous asymmetries of the bearing and shaft flexibilities. Between 19321935, Robertson [30] presented a series of papers on the subjects of bearing whirl, rotor transient whirl, and hysteretic whirl. In 1946, Prohl [31] published a transfer matrix procedure for determining the critical speeds of a multi-disc single shaft rotor, allowing for the inclusion of gyroscopic effects, but restricted to isotropic elastic supports. Between 1955-1965, Hagg and Sankey [32], Sternlicht [33], Lund [34] and others have developed the theory of hydrodynamic bearings, Yamamoto [35] studied the rolling bearings and Sternlicht [36], Pan and Cheng investigated the rotor instability in gas bearings. In 1948 Green [37] studied the gyroscopic effect of a rigid disc on the whirling of a flexible overhang rotor, being credited with the initial generalization of Jeffcott’s model to account for rigid-body dynamics. In 1957 Downham [38] has experimentally confirmed the existence of backward whirling. Between 1963-1967, Lund [39] and Glienicke [40] presented values of the linearized stiffness and damping coefficients for a series of hydrodynamic bearings, first presented by Sternlicht in 1959 [33]. Lund [41, 42] expanded the transfer matrix method of Myklestad and Prohl for calculating damped unbalance response and damped natural frequencies of a flexible rotor with asymmetric supports. Ruhl [43] and Nordmann [44] have first used the finite element method for the dynamic analysis of rotor-bearing systems in their doctoral theses, but the first papers using this method were published by Ruhl and Booker [45] in 1972, and Gasch [46] in 1973. Reduction of the finite element model has been used


34

starting in 1980 by Rouch and Kao [47], and Jäcker [48], the latter introducing also the effect of foundation on the rotor response. The study of the effect of annular fluid seals was initiated by Lomakin [49] in 1958, and then developed by Black [50] and Childs [51]. The effect of gas seals has been studied by Benckert and Wachter [52], and Iwatsubo [53]. The study of instabilities due to unequal gaps between rotor and stator as a result of the rotor eccentricity was initiated by Thomas [54] and Alford [55]. In Romania, the first book with elements of machinery dynamics was published in 1958 by Gh. Buzdugan and L. Hamburger [56]. The lubrication theory has been developed by N. Tipei [57] and V. N. Constantinescu [58-60]. Books on sliding bearings were published by Tipei et al [61] and Constantinescu et al [62]. The first PhD thesis on Rotordynamics was presented in 1971 by M. Rădoi [63], using a computer program developed at INCREST [64], based on Lund’s transfer matrix method [65].

References 1. Hohn, A. and Spechtenhauser, A., Present state and possible applications of turbosets for industrial and medium-sized power plants, Brown Boveri Review, Vol.63, No.6, pp 321-332, June 1976. 2. Hard, F., 75 years of Brown Boveri steam turbines, Brown Boveri Review, Vol.63, No.2, pp 85-93, 1976. 3. Somm, E., Developing Brown Boveri Steam Turbines to Achieve Still Higher Unit Outputs, Brown Boveri Review, Vol.63, No.2, pp 94-105, 1976. 4. * * * Back-Pressure Turbosets for Industrial Use, Brown Boveri Publication 3090 E, 1967. 5. Bertilsson, J. E., and Berg, U., Steam Turbine Rotor Reliability, EPRI Workshop on Rotor Forgings for Turbines and Generators, Palo Alto, California, Sept 1317, 1980. 6. * * * Turbine à gaz de 6000 kW de l'Electricité de France (E.D.F.) à St-Dizier, Revue Brown Boveri, Vol.47, No.1/2, pp 37-42, 1960. 7. * * * RB.211 Technology & Description, Rolls-Royce Publ. TS2100, Issue 18, Nov.1977. 8. Kostyuk, A. G., and Frolov, V. V., Steam and Gas Turbines (in Russian), Energoatomizdat, Moskow, 1985. 9. Wachel, J. C., Rotordynamic Instability Field Problems, NASA CP 2250, pp 119, 1982.


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10. API Standard 617, Centrifugal Compressors for Petroleum, Chemical and Gas Service Industries, American Petroleum Institute, Washington, 1995. 11. Meiners, K., Compressors in Energy Technology, Sulzer Technical Review, Vol.62, No.4, pp 143-148, 1980. 12. Shaw, M. C., and Macks E. F., Analysis and Lubrication of Bearings, McGraw Hill, New York, 1949. 13. Eck, B., Ventilatoren, Springer, Berlin, 1957. 14. Pfleiderer, C., and Petermann, H., Strömungsmaschinen, 6.Aufl., Springer, Berlin, 1990. 15. Siekmann, H., Wasserturbinen, Dubbel. Taschenbuch für den Maschinenbau, 17. Aufl., Springer, Berlin, pp R30-R36, 1990. 16. Krick, N., and Noser, R., The Growth of Turbo-Generators, Brown Boveri Review, Vol.63, No.2, pp 148-155, 1976. 17. Bishop, R. E. D., and Parkinson, A. G., Second Order Vibration of Flexible Shafts, Phil. Trans. Royal Society, Series A, Vol.259, A.1095, pp 1-31, 1965. 18. * * * Caractéristiques de construction des alternateurs de grande puissance, Revue ABB, No.1, 11 pag. 1989. 19. Rieger, N. F., and Crofoot, J. F., Vibrations of Rotating Machinery. Part I: Rotor-Bearing Dynamics, The Vibration Institute, Illinois, Nov 1977. 20. API Standard 610, Centrifugal Pumps for General Refinery Services, American Petroleum Institute, Washington, 1979. 21. Rankine, W. J. M., On the centrifugal force of rotating shafts, The Engineer, Vol.27, p.249, Apr.1869. 22. Föppl, A., Das Problem der Laval'schen Turbinewelle, Civilingenieur, Vol.41, pp.332-342, 1895. 23. Jeffcott, N., Lateral vibration of loaded shafts in the neighbourhood of a whirling speed – The effect of want of balance, Philosophical Magazine, Series 6, Vol.37, pp.304-314, 1919. 24. Dunkerley, S., On the whirling and vibration of shafts, Trans. Roy. Soc. (London), Vol.185, Series A, pp.279-360, 1894. 25. Stodola, A., Neuere Beobachtungen uber die Kritischen Umlaufzahlen von Wellen, Schweizer.Bauzeitung, Vol.68, pp.210-214, 1916. 26. Newkirk, B. L., Shaft whipping, General Electric Review, Vol.27, pp.169-178, 1924.

36


27. Newkirk, B. L. and Taylor H. D., Oil film whirl – An investigation of disturbances on oil films in journal bearings, General Electric Review, Vol.28, 1925. 28. Poritsky, H., Contribution to the theory of oil whip, Trans. ASME, Vol.75, pp.1153-1161, 1953. 29. Smith, D. M., The motion of a rotor carried by a flexible shaft in flexible bearings, Proc. Roy. Soc. London, Series A, Vol.142, pp.92-118, 1933. 30. Robertson, D., The vibration of revolving shafts, Phil. Mag. Series 7, Vol.13, pp.862, 1932; The whirling of shafts, The Engineer, Vol.158, pp.216-217, 228231, 1934; Transient whirling of a rotor, Phil. Mag., Series 7, Vol.20, pp.793, 1935. 31. Prohl, M. A., A general method for calculating critical speeds of flexible rotors, Trans ASME, Vol.67, J. Appl. Mech., Vol.12, No.3, pp.A142-A148, Sept.1945. 32. Hagg, A. C. and Sankey, G. O., Some dynamic properties of oil-film journal bearings with reference to the unbalance vibration of rotors, Trans. ASME, J. Appl. Mech., Vol.23, pp.302-306, 1956. 33. Sternlicht, B., Elastic and damping properties of cylindrical journal bearings, Trans. ASME, J. Basic Eng., Series D, Vol.81, pp.101-108, 1959. 34. Lund, J. W., The stability of an elastic rotor in journal bearings with flexible damped supports, Trans. ASME, J. Basic Eng., Vol.87, 1965. 35. Yamamoto, T., On the critical speed of a shaft supported in ball bearing, Trans. Soc. Mech. Engrs. (Japan), Vol.20, No.99, pp.750-760, 1954. 36. Sternlicht, B., Gas-lubricated cylindrical journal bearings of the finite length, Trans. ASME, J. Appl. Mech., Paper 61-APM-17, 1961. 37. Green, R. B., Gyroscopic effects on the critical speeds of flexible rotors, Trans. ASME, J. Appl. Mech., Vol.70, pp.369-376, 1948. 38. Downham, E., Theory of shaft whirling. A fundamental approach to shaft whirling, The Engineer, pp.519-522, 552-555, 660-665, 1957. 39. Lund, J. W., Rotor Bearing Dynamics Design Technology. Part III, Design Handbook for Fluid-Film Type Bearings, M.T.I. Report AFSCR 65-TR-45, 1965. 40. Glienicke, J., Feder- und Dämpfungskonstanten von Gleitlagern für Turbomaschinen und deren Einfluss auf das Schwingungsverhalten eines einfachen Rotors, Dissertation, T. H. Karlsruhe, 1966. 41. Lund, J. W., Stability and damped critical speeds of a flexible rotor in fluidfilm bearings, Trans. ASME, J. Engng. Ind., Series B, Vol.96, No.2, pp.509517, May 1974.


37

42. Lund, J. W. and Sternlicht, B., Rotor-bearing dynamics with emphasis on attenuation, Trans. ASME, J. Basic Engng., Vol.84, No.4, pp.491, 1962. 43. Ruhl, R. L., Dynamics of distributed parameter turborotor systems: Transfer matrix and finite element techniques, Ph. D. Thesis, Cornell Univ., Ithaca, N. Y., Jan.1970. 44. Nordmann, R., Ein Näherungsverfahren zur Berechnung der Eigenwerte und Eigenformen von Turborotoren mit Gleitlagern, Spalterregung, ausserer und innerer Dämpfung, Dissertation, T. H. Darmstadt, 1974. 45. Ruhl, R. L. and Booker J. F., A finite element model for distributed parameter turborotor systems, Trans. ASME, Series B, J. Eng. Industry, Vol.94, No.1, pp.126-132, Febr.1972. 46.

Gasch, R., Unwucht-erzwungene Schwingungen und Stabilität Turbinenläufern, Konstruktion, Vol.25, Heft 5, pp.161-168, 1973.

von

47. Rouch, K. and Kao, J., Dynamic reduction in rotor dynamics by the finite element method, J. Mechanical Design, Vol.102, pp.360-368, 1980. 48. Jäcker, M., Vibration analysis of large rotor-bearing-foundation systems using a model condensation for the reduction of unknowns, Proc. Second Int. Conf. "Vibration in Rotating Machinery", Cambridge, U.K., Paper C280, pp.195-202, 1980. 49. Lomakin, A., Calculation of critical number of revolutions and the conditions necessary for dynamic stability of rotors in high-pressure hydraulic machines when taking into account forces originating in sealings, Power and Mechanical Engineering, April 1958 (in Russian). 50. Black, H., Effects of hydraulic forces on annular pressure seals on the vibrations of centrifugal pump rotors, Journal of Mechanical Engineering Science, Vol.11, No.2, pp.206-213, 1969. 51. Childs, D. and Kim C.-H., Analysis and testing of rotordynamic coefficients of turbulent annular seals, J. of Tribology, Vol.107, pp.296-306, 1985. 52. Benckert, H. and Wachter, J., Studies on vibrations stimulated by lateral forces in sealing gaps, AGARD Proc. No.237 Conf. Seal Technology in Gas-Turbine Engines, London, pp.9.1-9.11, 1978. 53. Iwatsubo, T., Evaluation of instability forces of labyrinth seals in turbines or compressors, Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP No.2133, pp.139-167, 1980. 54. Thomas, H., Instabile Eigenschwingungen von Turbinenläufern angefacht durch die Spaltstromungen Stopfbuschen und Beschauflungen, Bull. de l'AIM, vol.71, pp.1039-1063, 1958.

38


55. Alford, J., Protecting turbomachinery from self-excited rotor whirl, Trans. ASME, J. Engng. Power, pp.333-344, 1965. 56. Buzdugan, Gh. and Hamburger, L., Teoria vibraţiilor, Editura tehnică, Bucureşti, 1958. 57. Tipei, N., Hidro-aerodinamica lubrificaţiei, Editura Academiei, Bucureşti, 1957. 58. Constantinescu, V. N., Lubrificaţia cu gaze, Editura Academiei, Bucureşti, 1963. 59. Constantinescu, V. N., Aplicaţii industriale ale lagărelor cu aer, Editura Academiei, Bucureşti, 1968. 60. Constantinescu, V. N., Teoria lubrificaţiei în regim turbulent, Editura Academiei, Bucureşti, 1965. 61. Tipei, N., Constantinescu, V. N., Nica, Al., and Biţă, O., Lagăre cu alunecare, Editura Academiei, Bucureşti, 1961. 62. Constantinescu, V. N., Nica, Al., Pascovici, M. D., Ceptureanu, Gh., and Nedelcu, Şt., Lagăre cu alunecare, Editura tehnică, Bucureşti, 1980. 63. Rădoi, M., Contribuţii la studiul dinamicei şi stabilităţii rotorilor, cu considerarea influenţei reazemelor, Teză de doctorat, Inst. Politehnic Timişoara, 1971. 64. Biţă, O., Program pentru calculul răspunsului dinamic al unui rotor, INCREST, Bucureşti, 1973. 65. Lund, J. W., Rotor-Bearings Dynamic Design Technology Part III: Design Handbook for Fluid-Film Bearings, Mechanical Technology Inc. Report AFAPL-Tr-65-45, 1965.

2. SIMPLE ROTORS IN RIGID BEARINGS Simple single-disc rotors with massless shafts supported in rigid bearings are considered in this chapter. The effect of damping and gyroscopic couples on the rotor precession is examined in detail.

2.1 Simple rotor models The simplest flexible rotor consists of a rigid disc, fixed on a flexible shaft of axi-symmetric cross section, supported at the ends in identical bearings. The symmetric rotor, with a massless shaft supported in rigid bearings (Fig. 2.1) is known as the Laval-Jeffcott model [1, 2]. Generally, only the first precession mode is studied for which, because of the symmetry, the disc rotary inertia can be neglected. The model serves to the introduction of the concepts of critical speed and synchronous precession.

Fig. 2.1

Fig. 2.2

The Stodola-Green model [3-5] consists of a flexible shaft with an overhung disc, not necessarily thin (Fig. 2.2). The model is used to examine the influence of the disc rotary inertia and gyroscopic torques on the rotor precession, the concepts of forward and backward precession, as well as the effect of the unbalance due to the skew mounting of the disc on the shaft.


40

In the following, the rotor shaft is considered to be rigidly supported. This is possible when the shaft stiffness is much lower than (less than 10% of) the combined stiffness of bearings and pedestals. The model simplification allows the stepwise introduction of the influence of mass unbalance, external and internal damping, and gyroscopic coupling, neglecting the bearing flexibility and damping.

2.2 Symmetric undamped rotors Consider a rotor which consists of a flexible shaft of circular crosssection, supported at the ends in rigid bearings, and carrying a thin rigid disc in the symmetry plane, at mid distance between bearings (Fig. 2.3, a).

Fig. 2.3 Let the point G be the disc mass centre and point C - the disc geometric centre, where the geometric axis of the shaft intersects the disc plane. The disc has mass m and polar moment of inertia J G . The bearing line intersects the disc at point O. Denote C G = e the offset of the disc mass centre G with respect to the point C.

2. SIMPLE ROTORS IN RIGID BEARINGS

41

The shaft stiffness coefficient is denoted by k (the ratio of a force applied to the shaft middle and the static deflection produced at the same point). For the symmetric rotor, it is k = 48 EI / l 3 where l is the span between bearings, I is the shaft cross-section second moment of area, and E is the shaft material Young's modulus. In this section, the shaft mass, the damping forces and the static deflection (of the horizontal shaft) under the disc weight are neglected. It is assumed that the non-rotating shaft is rectilinear and the rotor motion is studied with respect to this static equilibrium position. Later on (Section 2.3.4) the effect of gravity on the horizontal rotor will be studied. An inertial coordinate system with the origin in O is considered. The Ox axis coincides with the bearing line (axis of the non-rotating shaft). The horizontal axis Oz and the vertical axis Oy are in the disc median plane (Fig. 2.3, b). The disc motion in its own plane can be described by the variation in time of either the coordinates yC and zC of the geometric centre C, or the coordinates yG and zG of the mass centre G. Under the action of an external torque M ( t ) , the disc turns and, at a given time t, the line CG makes an angle θ (positive anti-clockwise) with the axis Oy.

2.2.1 Equations of motion The disc equations of motion can be written using d'Alembert's principle. The disc is isolated and subjected to the elastic restoring force due to the shaft flexibility, the external torque, the inertia force and the inertia torque (Fig. 2.3, c) that must be in dynamic equilibrium [6]. The resulting equations of motion are

m &y&G + k yC = 0 , m &z&G + k zC = 0,

(2.1)

J G θ&& + k yC e sin θ - k zC e cos θ = M ( t ). The coordinates of the points C and G are related through yG = yC + e cos θ , zG = zC + e sin θ . Substituting (2.2) into (2.1) we obtain

(2.2)


42

m &y&C + k yC = meθ& 2 cos θ + meθ&& sin θ , m &z& + k z = meθ& 2sin θ − meθ&& cos θ , C

C

(2.3)

J G θ&& + k e ( yC sin θ − zC cos θ ) = M ( t ). Denoting J G = m iG2 , where iG is the disc gyration radius with respect to the spinning axis, the third equation (2.3) can be written

θ&& =

⎞ e z M ( t ) k ⎛ yC sinθ − C cosθ ⎟⎟ . − ⎜⎜ JG m ⎝ iG iG ⎠ iG

Because e << iG and yC , zC << iG , the second term in the right hand side can be neglected with respect to the first one. For steady-state motion, when the active and resistant torques balance each other, M (t ) = 0, hence θ&& ≅ 0. As a result, the running speed is constant, θ& = Ω = const ., and

θ = Ω t + θ0 .

(2.4)

The first two equations (2.3) become m &y&C + k yC = meΩ 2cos (Ω t + θ0 ) , m &z&C + k zC = meΩ 2sin (Ω t + θ0 ).

(2.5)

Equations (2.5) are uncoupled. For each equation, the total solution is the sum of the solution of the homogeneous equation and the particular solution corresponding to the right-hand side. The solutions of the homogeneous equations (for e = 0 , hence for a perfectly balanced disc) have the form yC (t ) = YC cos (ωn t + θ y ) , zC (t ) = Z C sin (ωn t + θ z ) ,

(2.6)

where k m is the natural frequency of the rotor free precession.

ωn =

(2.7)

The particular solutions, corresponding to the harmonic excitation due to the mass unbalance, describe the steady-state motion and have the form yC = ˆyC cos (Ω t + θ 0 ) , zC = ˆzC sin (Ω t + θ 0 ).

(2.8)


43

Because, in reality, the free motion decays due to the inherent damping and dies out after a short time interval, in the following only the forced steady-state motion will be studied.

2.2.2 Steady state response Substitution of (2.8) into (2.5) gives the magnitudes of the displacements of point C along the axes Oy and Oz, respectively: ˆyC = ˆzC =

(Ω / ω n ) 2 Ω2 m eΩ 2 e e. = = k − mΩ 2 ωn2 − Ω 2 1 − (Ω / ω n ) 2

(2.9)

Due to the disc mass unbalance, the point C moves along a circle of radius rˆC =

ˆyC2 + ˆzC2 = e

(Ω / ω n ) 2 1 − (Ω / ω n ) 2

,

(2.10)

with the angular speed Ω . The above calculations can be written in a more compact form using complex numbers.

Fig. 2.4 In the plane yOz, Oy is taken as the real axis and Oz as the imaginary axis (Fig. 2.4). The following notations are used rC = yC + i zC , rG = yG + i zG ,

(2.11)

where i = − 1. Multiplying the second equation (2.2) by i, by addition to the first one and using the notation (2.4), we find


44

yG + i zG = yC + i zC + e [ cos (Ω t + θ 0 ) + i ⋅ sin (Ω t + θ 0 )] or rG = rC + e ei ( Ω t +θ 0 ) .

(2.12)

Analogously, equations (2.5) are written under the form m ( &y&C + i &z&C ) + k ( yC + i zC ) = m eΩ 2 [ cos (Ω t + θ 0 ) + i ⋅ sin (Ω t + θ 0 )] or m &r&C + k rC = m eΩ 2 ei ( Ω t +θ 0 ) .

(2.13)

The steady-state solution of equation (2.13) is rC =

(Ω/ωn ) 2 1 − (Ω/ωn )

2

e ei ( Ωt + θ 0 ) .

(2.14)

Inserting (2.14) into (2.12) we obtain the displacement of the disc mass centre rG =

1 e ei ( Ωt + θ 0 ) . 1 − (Ω/ωn ) 2

(2.15)

The point G moves along a circle of radius rG = O G , with the angular speed Ω . Considering the line C G as a vector in the complex plane, we can write CG = e ei ( Ωt + θ 0 ) .

(2.16)

Denoting

rC = rˆC ei (Ω t +θ 0 ) ,

rG = rˆG ei (Ω t +θ 0 ) ,

(2.17)

because from equations (2.14) and (2.15) it comes out that rˆC and rˆG are real quantities, the vectors O G and O C are collinear with C G , hence the points O, C and G are collinear. At Ω = const. the relative position of these points is fixed. The shaft deflected axis is located in the plane defined by the axis Ox and the line O C G , having a constant bend. The shaft rotates around the Ox axis in this deflected shape so that, at any section, the bending stresses are time invariable. As the motion of point C around the bearings line is executed with the same angular speed Ω as the rotation of point G around the shaft axis (point C), the motion is referred to as a synchronous precession.


45

Figure 2.5 shows the speed dependence of the radius rˆC of the orbit of point C (solid line) and the radius rˆG of the orbit of point G (broken line).

Fig. 2.5 For Ω = ωn , the radii rˆC and rˆG become infinite, the shaft bend grows indefinitely, as do the bending stresses. This state has been considered critical and the corresponding speed 30 ωn 30 k = (2.18) ncr = π π m has been referred to as the rotor undamped critical speed. The critical angular speed Ω cr = ωn corresponds to the natural circular frequency of the rotor undamped lateral vibrations. In the undercritical range, for Ω1 < ω n , O1G1 > O1C1 , point G1 is outside the segment O1C1 , moving as in Fig. 2.6, a along a circle of radius larger than that of point C1 . In the overcritical range, for Ω2 > ωn , O2G2 < O2C2 , the point G2 is located between the points O2 and C2 , moving as in Fig. 2.6, b along a circle of radius smaller than that of point C 2 . At very large angular speeds, for Ω3 >> ωn , the point G3 coincides with O3 , hence the disc mass centre tends to the bearing line (Fig. 2.6, c). The shaft deflection becomes practically equal to the offset e. It is said that the rotor is selfbalanced. This is the optimal operating regime in the overcritical range, since the dynamic forces in bearings have the minimum value k e . The results of the above analysis are of theoretical interest. In practice, large (but not infinite) rotor shaft deflections could be anticipated at the rotor critical speed, when the rotor speed coincides with its flexural natural frequency.


46

The findings are limited to the synchronous motion, condition in which equal precession and rotation rates are assumed for the rotor. The orbits of the rotor points are circles only if the shaft is circumferentially symmetric.

Fig. 2.6

2.3 Damped symmetric rotors The rotor motion takes place in the presence of friction forces arising due to the rotor interaction with its stationary environment and due to the relative motion of its particles and components during bending. In the following, distinction will be made between external and internal friction forces. The external friction forces, producing the "external damping", limit the precession radius at the critical speed and stabilize the motion. The internal friction forces act at joints, between components mounted with shrink fits, or arise from the internal friction in the shaft material. They produce the "internal damping", attenuating the magnitude of the precession motion at the critical speed, and being able to produce, at higher speeds, unstable motions, as a result of the character of tangential follower forces.


47

2.3.1 Effect of viscous external damping It is considered that, due to the rotor motion relative to the stationary environment, the disc from figure 2.3 is acted upon by a viscous damping force proportional to the absolute tangential velocity of the disc centre. Let (−ce y& C ) and (−ce z&C ) be the components of this force along the axes of the inertial coordinate system yOz, where ce is the coefficient of external viscous damping. 2.3.1.1 Equations of motion

The equations of motion of the symmetrical rotor (2.5) become m &y&C + ce y& C + k yC = m eΩ 2cos (Ω t + θ 0 ) , m &z&C + ce z&C + k zC = m eΩ 2sin (Ω t + θ 0 ).

(2.19)

Denoting rC = yC + i zC and adding the first equation (2.19) to the second one, multiplied by i = − 1 , we obtain

m &r&C + ce r&C + k rC = m e Ω 2ei ( Ω t + θ 0 ) .

(2.20)

2.3.1.2 Free damped precession

Substituting solutions of the form rC (t ) = R e λ t into the equation (2.20) with zero right-hand side, we find the characteristic equation

mλ2 + ce λ + k = 0 , whose solutions are

λ 1,2 = −

ce ± 2m

2

k ⎛ ce ⎞ 2 ⎜ ⎟ − = ωn ⎛⎜ − ζ e ± i 1 − ζ e ⎞⎟ ⎠ ⎝ 2 m m ⎝ ⎠

(2.21)

where ωn = k m is the natural frequency of the undamped system, referred to as the undamped natural frequency, and

ζe = is the external damping ratio.

ce c = e 2 k m 2ωn m

(2.22)


48

The roots (2.21) can also be written

λ 1,2 = α e ± i ωd e

(2.23)

where α e = ω nζ e is a negative attenuation (decay) factor and ωd e = ωn 1 − ζ e2 is the damped natural frequency, i.e. the frequency of the damped free motion of the perfectly balanced rotor. The general solution is

rC (t ) = R 1 eα e t e

i ωd e t

+ R 2 eα e t e

− i ωd e t

,

(2.24)

where the integration constants R 1 and R2 are determined from the initial conditions for the displacement and velocity, rC ( 0 ) and r&C ( 0 ). In order to determine the orbit of point C, the free undamped motion will be considered first, when ce = ζ e = α e = 0. The solution (2.24) becomes

rC (t ) = R 1 ei ω n t + R 2 e − i ω n t .

(2.25)

In the complex plane, this represents the sum of two vectors of length R 1 and R 2 , respectively, rotating in opposite directions with angular velocity

ωn . The tip of the resultant vector moves along an ellipse (Fig. 2.7, a). The major semiaxis a = R 1 + R 2

is directed along the bisector of the angle between the

two vectors. The minor semiaxis is b = R 1 − R 2 .

a

b

Fig. 2.7 In the case of the damped motion, the solution (2.24) represents the sum of two vectors rotating in opposite directions with angular speed ωd e ≅ ωn ( for ζ e << 1 ) . For α e < 0 , the factor eα e t produces a decrease in magnitude, hence a motion along a converging spiral (Fig. 2.7, b).


49

2.3.1.3 Steady-state precession due to unbalance

Substitution of the steady-state solution rC (t ) = ~ rC ei ( Ω t +θ 0 )

(2.26)

into equation (2.20) gives (−m Ω 2 + i Ω ce + k ) ~ rC = m e Ω 2 . After transformations, the expression of the displacement of point C is obtained as: ~ rC = =

e (Ω / ωn ) 2 = 1 − (Ω / ωn ) 2 + i 2ζ e (Ω / ωn )

[

e (Ω / ωn ) 2 1 − (Ω / ωn ) 2 − i 2ζ e (Ω / ωn ) [1 − (Ω / ωn ) 2 ] 2 + [ 2ζ e Ω / ωn ] 2

]

(2.27) = rC R + i rC I .

The solution (2.26) is written under the form rC = ~ rC ei ( Ω t +θ 0 +θ C )

(2.28)

where ~ rC =

e ( Ω/ωn ) 2

[1 − ( Ω/ωn ) 2 ] 2 + [ 2ζ e (Ω/ωn ) ] 2 tan θC = −

2ζ e (Ω/ωn ) 1 − (Ω/ωn ) 2

,

,

so that the vector O C = rC is not collinear with C G = e ei ( Ω t + θ0 ) .

Fig. 2.8

(2.29)


50

Figure 2.8 shows the relative position of these vectors in the case of rotation at undercritical speeds (Ω < ωn ). Because tan θ C < 0, the angle θ C is negative and the vector O C lags the line C G . The real part ℜe (rC ) = O C ' represents the component in phase with C G and the imaginary component ℑm (rC ) = C C ' represents the component in quadrature with (perpendicular to) CG . The radius of the disc mass centre orbit is rG = rC + e ei ( Ωt + θ 0 ) hence ~ rG = ~ rC + e =

1 + i 2ζ e (Ω / ωn ) 1 − (Ω / ωn ) 2 + i 2ζ e (Ω / ωn )

e = rG R + i rG I

(2.30)

so that rG = ~ rG ei ( Ω t +θ 0 +θ G )

(2.31)

where e 1 + 4ζ e2 (Ω / ωn ) 2

~ rG =

[1 − (Ω / ω ) ] n

tan θ G = −

2 2

+ [2ζ e (Ω / ωn )] 2

,

(2.32)

2ζ e (Ω / ωn )3 1 − (Ω / ωn ) 2 + 4ζ e2 (Ω / ω n ) 2

In figure 2.8, for Ω < ω n , tan θG < 0 and the vector O G lags the line C G with an angle θ G . 2.3.1.4 Unbalance response diagrams

Figures 2.9, a and b illustrate the speed-dependence of the modulus and rC and ~ rG , for a value ζ e = const . The phase angle of the complex quantities ~ maximum displacements in the two orbits occur at speeds which are different from the critical speed of the undamped system. However, because practically ζ e << 1, it can be considered that the maximum displacement is e /( 2ζ e ) and takes place at

Ω = ωn =

k m.


51

The peak response speeds are slightly different from the undamped and the damped critical speeds, and are different for points C and G.

a

b

Fig. 2.9 Eliminating (Ω / ωn ) between the expressions of the real and imaginary rC , we obtain the locus of the tip of the vector ~ rC in the complex components of ~ plane, referred to as the polar plot, Nyquist plot or the hodograph of the respective vector. Such a curve, of equation

(r

2 CR

+ rC2I

)

2

(

)

+ e rC R rC2R + rC2I −

e2 2 rC = 0 4 ζ e2 I

(2.33)

is plotted in Fig. 2.10 with solid line, for a value ζ e = const . At Ω =0, point C coincides with the origin O. With increasing speed, point C moves along the curve clockwise. At Ω = ωn point C lies on the negative imaginary semiaxis, and for Ω = ∞ , it lies on the negative real semiaxis, at a distance e from the origin.


52

rG is drawn with thin In Fig. 2.10, the polar diagram of the displacement ~ line. The relative positions of the vectors O C and C G at three different rotational speeds are also plotted.

Fig. 2.10 At Ω < ωn , vectors O C and C G are drawn in the lower part of the figure. In this relative position, they turn at uniform angular velocity Ω around the point O, anticlockwise, the points C and G moving in circles. At Ω = ωn the vector C G is perpendicular to O C (the position on the left of Fig. 2.10). At Ω > ωn , the vector O G becomes smaller than O C (the upper position) and at Ω = ∞ the point G coincides with O, the rotor is "selfbalanced". 2.3.1.5 Synchronous precession

The relative orientations of points C and G are sketched in Fig. 2.11 for three speeds. Note the difference compared to Fig. 2.6, drawn for undamped rotors.


53

As the rotor rotates, the eccentric mass tends to pull the rotor toward the bearings on the side the point G is located, called the ‘heavy spot’. The point where the precession radius is a maximum is called the ‘high spot’. The opposite point is the ‘low spot’.

Fig. 2.11 When the damping is neglected (Fig. 2.6), the angle θ C = 0 at Ω < ωn , and θ C = 1800 at Ω > ωn . There is a sudden change from 0 to 1800 at Ω = ωn . The heavy point G coincides with the ‘high spot’ at speeds Ω < ωn , and then suddenly coincides with the ‘low spot’ above the critical speed. In the presence of damping, there is a continuous change of angle θ C with speed, as given by (2.29) and shown in Fig. 2.9, a, with a higher rate around the critical speed. The heavy spot S no more coincides with the high spot H. At very low speeds, the high spot is almost in phase with the unbalanced (heavy) mass, and the “heavy side flies out”. As speed increases, the high spot begins to lag the heavy spot. At Ω = ωn , the phase lag is 900 , and for at Ω > ωn it tends to 1800 , and “the heavy side flies in” (Fig. 2.11). As point C travels in a circle with angular speed Ω , line C G rotates at the same angular speed around C, so that, in the steady state precession, the line segments O C and C G have no relative motion with respect to each other. They rotate about O as though they were a rigid body. This implies that, in the synchronous precession, the shaft supported in rigid bearings does not bend back and forth during the motion, but simply revolves in a bowed position, with constant orbit radii. At constant rotor speed, stresses are constant in a given point. This means that internal damping in the shaft material


54

will be ineffective against the forced precession. The solution is to introduce damping in bearings and/or bearing supports. Hence, the external damping reduces the synchronous rotor response and determines a gradual change, with the speed increase, of the relative position of lines O C and C G , that are no more collinear. By increasing the speed, a rotor can be "passed through the critical speed" and can operate beyond the critical speed. There are at least three ways to reduce the amplitude of synchronous precession: a) balancing the rotor; b) changing either the operating speed or the critical speed; and c) adding damping.

2.3.2 Effect of viscous internal damping Internal damping in rotors is produced by either the material hysteretic damping or by the Coulomb damping due to rubbing at the interface of shrinkfitted parts. To emphasize the difference between external and internal damping, they are often referred to as, respectively, stationary and rotating damping. 2.3.2.1 Rotating damping

The force due to internal rotor damping is defined as Fi = −ci r& ,

(2.34)

where ci is the coefficient of internal viscous damping, and r& is the time rate of change of the shaft deflection at the point of disc attachment. For a non-rotating shaft, this force is proportional to the absolute velocity of the respective point, hence the internal viscous damping plays the same role as the external damping. For the rotating shaft, the rate of change of shaft deflection, equal to the velocity of the relative displacement of its points, is different from the absolute velocity. Equation (2.34) holds only in a rotating coordinate system fixed to the rotor, hence the name of rotating damping. Figure 2.12 [6] shows a simple model which illustrates the action of internal viscous damping. If the rotor has a synchronous precession, r = const . , and the damping force (2.34) is zero. Internal damping forces are produced only by alternating bending stresses and strains, when the rotor orbits are non-circular.


55

Fig. 2.12 (from [6]) An interesting description of the nature of internal friction can be found in an early paper by A. L. Kimball [7] and is adapted in the following. “When a horizontal shaft is supported at its two ends and sags upward in the middle due to a centrifugal force, its lower fibers evidently are in compression and its upper ones are in tension. If the shaft is revolving, every fiber must have its length alternately decreased and increased once every revolution, due to the alternating compression and tension to which the fiber is subjected. The amount of this alternating increase and decrease of length of the fibers depends upon how much the shaft bends and how far the fiber is from the centre of the shaft. At the centre of the shaft the change of length of the fibers is zero as they lie on the axis of the shaft. In all metals, a frictional resistance to this change of length exists in a greater or less degree. When a fiber is shortening, a frictional compression is set up. When it is lengthening a frictional tension arises. These frictional stresses are very different from the elastic stresses. The elastic stresses are proportional to the amount that the length of the fibers is changed, and have their maximum and minimum values at the bottom and top of the shaft. The frictional stresses arise during the change of length of the fibers; a frictional tension results when the length of the fibers is increasing and a frictional compression results when the length of the fibers is decreasing. Fig. 2.13 shows a cross-section of the shaft near its middle point. All of the fibers in the lower half are in elastic compression, C E , and those in the upper half are in elastic tension, TE . When the shaft is rotating anticlockwise, all the fibers in the right hand half are increasing in length, resulting in a frictional tension TF and the fibers in the left half of the shaft are decreasing in length, producing a frictional compression C F . The elastic stresses produce an inward restoring force indicated by Fel in the figure, whose direction is from the elastic tension toward the elastic compression side of the shaft. So also the frictional stresses produce a corresponding transverse reaction Fi . The magnitude of Fi is far smaller than that


56

of Fel , however. If the shaft is placed on ends, a whirling motion of the shaft is likely to build up provided the rotational speed exceeds the natural whirling speed.

Fig. 2.13 In the preceding discussion, the frictional reaction Fi has been shown to arise in the fibers during compression and elongation. This is not the only sort of friction which may cause the reaction Fi , however. Any frictional resistance which arises within a revolving deflected rotor, while one-half of the cross section is stretching and the other half is shortening, also must produce a frictional reaction component Fi . For example, the rotor may be a shaft with rings shrunk on it. In this case, friction may take place between the surface of the shaft and the inner surface of the rings, due to a working of the shaft in the rings as it revolves. The surface fibers of the deflected shaft go through a cycle of elastic lengthening and shortening for every complete revolution of the shaft. This produces a friction against the inside surface of the rings which may be as great as to cause the shaft to take a slight permanent set when deflected a small amount”. 2.3.2.2 Motion in the rotating coordinate system

Consider the coordinate system Oξηζ fixed to the disc, rotating at the running speed Ω (Fig. 2.14). The rotating coordinate frame is selected so that the axis Oη makes an angle θ with respect to C G (the unbalance). The axis Oζ is perpendicular to Oη and in the disc plane (see Fig. 2.3). For the symmetric rotor, the axis Oξ coincides with the axis Ox of the stationary coordinate system. At a given time t, the axis Oη makes an angle Ω t with the axis Oy . The coordinates of any arbitrary point P, in the two coordinate systems, are related by the following equations


⎧ y ⎫ ⎡ cos Ω t ⎨ ⎬ = ⎢ ⎩ z ⎭ ⎣ sin Ω t

− sin Ω t ⎤ ⎧ η ⎫ ⎨ ⎬ . cos Ω t ⎥⎦ ⎩ ζ ⎭

57

(2.35)

Introducing the complex variables [6] r = y + i z,

(2.36)

ρ =η + iζ ,

(2.37)

r = ρ eiΩt ,

(2.38)

ρ = r e− i Ω t .

(2.39)

it is found that

Note that r = OP in the stationary Oxyz system, while ρ = OP in the rotor-fixed Oξηζ system.

Fig. 2.14 For undamped rotors, the equation of motion of the disc centre, in stationary coordinates, has the form (2.13): m &r&C + k rC = m e Ω 2ei ( Ω t +θ 0 ) .

(2.40)

In order to change to rotating coordinates, we use equation (2.38)

rC = ρC ei Ω t

(2.41)

and, by successive differentiation with respect to time, we obtain r&C = ( ρ C + i Ωρ C ) ei Ω t ,

&r&C = ( &ρ&C + i 2Ω ρ& C − Ω 2 ρC ) ei Ω t . On substituting (2.41) and (2.43) into (2.40) we obtain

(2.42) (2.43)


58

m ρ&&C + i 2mΩ ρ& C + ( k − mΩ 2 ) ρC = meΩ 2eiθ 0 .

(2.44)

In order to take into account the internal damping, a term proportional to the relative velocity, ci ρ& C , is added in equation (2.44). The equation of motion with internal damping is

m ρ&&C + ci ρ& C + i 2mΩ ρ& C + (k − mΩ 2 ) ρ C = m eΩ 2 eiθ 0

(2.45)

⎛ ci ⎞ + i 2Ω ⎟ ρ& C + (ωn2 − Ω 2 ) ρC = eΩ 2eiθ 0 , ⎝m ⎠

(2.46)

or

ρ&&C + ⎜

where the notation (2.7) has been used. Expressing ρC in complex form as ρC = ηC + i ζ C , separating the real and the imaginary parts, equation (2.46) is splitted into a set of two coupled equations c η&&C + i η&C − 2 Ω ζ&C + (ωn2 − Ω 2 ) ηC = e Ω 2cosθ0 , m (2.47) &ζ& + ci ζ& − 2 Ω η& + (ω 2 − Ω 2 ) ζ = e Ω 2sinθ . C C C n C 0 m Because the right-hand sides contain constant terms, the particular solutions of equations (2.47) will be also constants, namely

ηC =

e Ω 2 cos θ0 , ωn2 − Ω 2

ζC =

e Ω 2sin θ0 , ωn2 − Ω 2

(2.48)

hence

ρC =

eΩ 2 . ωn2 − Ω 2

(2.48, a)

It means that, in the case of rotation with constant angular speed, the disc centre C has a fixed position with respect to the rotating coordinate system. In the steady-state regime, when the shaft deflected shape remains unchanged, the internal damping has no influence on the magnitude of the rotor precession. 2.3.2.3 Motion in the stationary coordinate system

Because the internal damping force has a fixed position with respect to the rotating coordinate system, and rotates with the same angular speed, its expression in the stationary coordinate system is

Fi = − ( ci ρ&C ) ei Ω t .

(2.49)


59

Equation (2.39) yields

ρ& C = ( r&C − iΩ rC ) e − i Ω t ,

(2.50)

ρ& C ei Ω t = r&C − i Ω rC .

(2.51)

hence

Substituting (2.51) into (2.49) we obtain − Fi = ci ( r&C − i Ω rC ) = ci [ y& C + i z&C − i Ω ( yC + i zC ) ] = = ci ( y& C + Ω zC ) + i ci ( z&C − Ω yC ) ,

(2.52)

so that the components of the internal damping force along the axes of the stationary coordinate system are Fi y = − ci ( y& C + Ω zC ) ,

Fi z = − ci ( z&C − Ω yC ) ,

(2.53)

or, in matrix form,

⎧ Fi y ⎫ ⎡ ci ⎨ F ⎬ = −⎢ ⎣0 ⎩ iz ⎭

0 ⎤ ⎧ y& C ⎫ ⎡ 0 ⎨ ⎬− ci ⎥⎦ ⎩ z&C ⎭ ⎢⎣− Ω ci

Ω ci ⎤ ⎧ yC ⎫

⎨ ⎬. 0 ⎥⎦ ⎩ zC ⎭

(2.54)

Considering that the force (2.52) acts upon the disc, the above terms are added with opposite sign in the left-hand side of equations (2.5) to obtain the equations of the motion with internal damping ⎡m 0 ⎤ ⎧ &y&C ⎫ ⎡ci ⎢ 0 m ⎥ ⎨ &z& ⎬ + ⎢ 0 ⎣ ⎦⎩ C ⎭ ⎣

0 ⎤ ⎧ y& C ⎫ ⎡ k ⎨ ⎬+ ci ⎥⎦ ⎩ z&C ⎭ ⎢⎣− Ω ci

2 ⎧cos (Ω t + θ 0 ) ⎫ ⎬. ⎨ ⎬ = m eΩ ⎨ ⎥ k ⎦ ⎩ zC ⎭ ⎩ sin (Ω t + θ 0 ) ⎭ (2.55)

Ω ci ⎤ ⎧ yC ⎫

Internal damping yields skew-symmetric (cross-coupled) terms in the stiffness matrix. They produce destabilizing tangential forces. Equations (2.55) can be written in complex notation as m &r&C + ci r&C + (k − i Ω ci ) rC = me Ω 2ei ( Ωt + θ0 ) or &r&C +

ci c ⎞ ⎛ r&C + ⎜ ωn2 − i Ω i ⎟ rC = e Ω 2 ei ( Ωt + θ 0 ) . m m⎠ ⎝

(2.56)


60 2.3.2.4 Rotor stability

The study of the motion of the perfectly balanced rotor is carried out substituting e = 0 in equation (2.56) and looking for a solution of the form rC = RC e λ t for the homogeneous equation. Denoting

ζi =

ci , 2mωn

Λ=

λ , ωn

η=

Ω , ωn

(2.57)

we obtain the characteristic equation

Λ 2 + 2ζ i Λ + ( 1 − i 2ζ i η ) = 0

(2.58)

Λ 1,2 = −ζ i ± ζ i2 − 1 + i 2ζ iη ,

(2.59)

with the roots

where ζ i is the internal damping ratio. Because ζ i2 is small with respect to the other terms, it can be replaced by

ζ i2 η 2 on condition that η is not much greater than 1. This gives Λ 1 = −ζ i ( 1 − η ) + i ,

(2.60)

Λ 2 = −ζ i ( 1 + η ) − i .

The solution of the homogeneous equation has the form

rC (t ) = RC1 e

λ 1t

+ RC 2 e

λ 2t

.

For Ω > ωn , the real part of the root Λ1 is positive and its associated motion is divergent. At the passage through the 'resonance', when Ω ≥ ωn , the motion becomes unstable, the shaft deflection increasing suddenly. Equations (2.60) show that for η < 1 the motion associated with Λ1 is forward, while the motion associated with Λ 2 is backward. For η > 1 the motion associated with Λ1 is divergent. Increasing the rotor speed Ω , the real part of Λ1 decreases and the real part of Λ 2 increases. The stability of the forward component decreases but that of the backward component increases. This is a general result. Forces which tend to destabilize forward precession modes of a rotor, generally stabilize the backward precession modes.


61

2.3.2.5 Rotor whirling due to internal friction

The previous analysis has shown that in the presence of internal damping the rotor becomes unstable for angular speeds greater than Ω s = ωn - the onset speed of instability and the displacement rC grows unbounded in time. A simple physical explanation of this phenomenon can be given considering that the point C moves in a circle of radius RC with angular speed ω n , hence

rC (t ) = RC eiω n t . In this case, the internal damping force is

Fi = − ci ( r&C − i Ω rC ) = −i ci ( ωn − Ω ) rC .

(2.61)

This force is proportional to the shaft deflection rC but rotated 90 0 behind, i.e. it is a tangential force. At Ω < ωn , Fi < 0 , so it is a genuine damping force against the rotor revolving motion. When the rotor traverses the critical speed, the sign of the damping force changes. At Ω > ωn , Fi > 0 and a "negative damping" force occurs. The work of this force is positive, energy is introduced into the system, and the disc displacement grows unbounded. The force acts tangentially in the direction of motion, giving rise to a diverging spiral orbit, hence to instability. The instability due to internal rotor friction was studied during the 1920’s by B. L. Newkirk [8] in connection with a series of failures of blast furnace compressors designed to operate above the first critical speed. The following are adapted from a paper by Gunter and Trumpler [9]. “It was observed that at speeds above the first critical speed, these compressors would enter into a violent whirling in which the rotor centreline precessed at a rate equal to the first critical speed. If the unit rotational speed was increased above its initial whirl speed, the whirl amplitude would increase, leading to eventual rotor failure” [8]. It was “concluded that the internal friction created by shrink fits of the impellers and spacers was the predominant cause of the observed whirl instability. In tests on an experimental test rotor, when all shrink fits were removed, no whirl instability would develop. A special test rotor was constructed with rings on hubs shrunk on the shaft [7]. Measurements showed that the frictional effect of shrink fits is a more active cause of shaft whirling than the internal friction within the shaft itself and long clamping fits always lead to trouble with supercritical speed rotors” [10]. “For the case of a hub or a sleeve which is fastened to a shaft, which is afterwards deflected, either the surface fibres of the shaft must slip inside the sleeve as they alternately elongate and contract, or the sleeve itself must bend along with the shaft. Usually both actions occur simultaneously to an extent which

62


depends upon the tightness of the shrink fit and the relative stiffness of the two parts”. Robertson (1935) reported that even short, highly stressed shrink fits are not entirely devoid of problems [11]. He stated that even small, tight shrink fits may develop whirl instability provided the rotor is given a sufficiently large initial disturbance or displacement to initiate relative internal slippage in the fit. If long shrink fits such as compressor wheels and impeller spacers must be employed, it is important that these pieces be undercut along the central region of the inner bore so that the contact area is restricted to the ends of the shrink fit. A similar effect can be produced by any friction which opposes a change of the deflection of the shaft, such as the friction which exists at the connections of flexible couplings, and even in “rigid” couplings. This group of friction forces was referred to as “hysteretic forces” and the corresponding instability – “hysteretic whirl”. Extensive testing using an experimental test rotor uncovered the following features of this phenomenon [8]: 1) the onset speed of whirling or whirl amplitude was unaffected by refinement in rotor balance; 2) whirling always occurred above the first critical speed, never below it; 3) the whirl threshold speed could vary widely between machines of similar construction; 4) the precession (or whirl) speed was constant regardless of the unit rotational speed; 5) whirling was encountered only with built-up rotors; 6) increasing the foundation flexibility would increase the whirl threshold speed; 7) distortion or misalignment of the bearing housing would increase stability; 8) introducing damping into the foundation would increase the whirl threshold speed; 9) a small disturbance was sometimes required to initiate the whirl motion in a well balanced rotor. If the foundation flexibility is increased, the rotor stability will be improved only if damping is incorporated into the system” [12]. In order to avoid the instability induced by internal friction, the balancing pistons of steam turbine rotors are no more shrunk on but machined integral with the shaft, built-up rotors with shrunk-on discs are used only in the low pressure turbines, and spline teeth couplings between the rotors of the turbine sections are replaced by other designs without Coulomb friction.

2.3.3 Combined external and internal viscous damping Considering both external and internal damping, the equations of motion of the disc centre with respect to the fixed coordinate system become [13]

m &y&C + (ce + ci ) y& C + ci Ω zC + k yC = m e Ω 2cos (Ω t + θ 0 ) , m &z&C + (ce + ci ) z&C − ci Ω yC + k zC = m e Ω 2sin (Ω t + θ 0 ).

(2.62)


63

Denoting

ωn2 =

k , m

ζi =

ci , 2mωn

ζe =

ce , 2 mω n

ζ = ζe +ζi ,

(2.63)

equations (2.62) can be written in matrix form as

⎧ &y&C ⎫ ⎧ y& C ⎫ ⎡ ωn2 2 + + ζ ω ⎨ ⎬ ⎬ ⎢ n⎨ ⎩ &z&C ⎭ ⎩ z&C ⎭ ⎢⎣− 2ζ i ωn Ω

2ζ i ωn Ω ⎤ ⎧ yC ⎫ 2 ⎧cos (Ω t + θ 0 ) ⎫ = e Ω ⎥ ⎨ ⎬ ⎨ ⎬. ωn2 ⎥⎦ ⎩ zC ⎭ ⎩ sin (Ω t + θ 0 ) ⎭ (2.64)

Introducing the following dimensionless quantities

Λ=

λ , ωn

η=

Ω , ωn

(2.65)

the study of the motion of the perfectly balanced rotor (e=0) leads to the characteristic equation

(

)

(

)

Λ 4 + 4ζ Λ 3 + 2 2ζ 2 + 1 Λ 2 + 4ζ Λ + 1 + 4ζ i2 η = 0 .

(2.66)

If equation (2.66) is written under the form

Λ4 + B3 Λ3 + B2 Λ2 + B1 Λ + B0 = 0, then, according to the Routh-Hurwitz stability criterion [6], the roots do not have positive real parts (the system is stable) on condition that

B0 > 0, B1 > 0, B2 > 0 , B3 > 0, B1 B2 − B0 B3 ≥ 0 ,

B1 B2 B3 − B12 − B0 B32 ≥ 0. This yields the stability condition ⎛ζ ⎜⎜ ⎝ζi

or 1+

2

⎞ ⎟⎟ − η 2 ≥ 0 , ⎠

ζe Ω − ≥ 0. ζ i ωn

(2.67)


64

⎛ ζ ⎞ Hence, for angular speeds Ω > ωn ⎜⎜1 + e ⎟⎟ the rotor motion is unstable, ⎝ ζi ⎠ and the displacement of the point C with respect to O increases unbounded if there is no damping.

A perfectly balanced rotor, rotating with the angular speed Ω , cannot operate beyond the onset speed of instability ⎛

Ω s = ωn ⎜⎜1 + ⎝

ζe ⎞ ⎟. ζ i ⎟⎠

(2.68)

In the presence of external damping, Ω s > ωn , hence the external damping extends the range of stable operation conditions (Fig. 2.15). As in all self-excited motions, the rotor tends to whirl at its natural frequency ωn . The tangential force due to internal friction (2.61) is balanced by the force due to external damping ce ωn rC so that ci ( ωn − Ωs ) rC = ce ωn rC , wherefrom the onset speed of instability (2.68) is easily obtained. Below the onset speed of instability, the rotor motion is stable and synchronous. Above this speed, the rotor motion has a subsynchronous component which diverges exponentially with time. The associated precession motion is forward. Though the above derivations hold for zero eccentricity, it has been found that the occurrence of rotor instability is rather independent of the state of rotor balance.

Fig. 2.15 The case of internal damping produced by hubs or sleeves clamped to the shaft, shrink fits or spline teeth couplings, is treated in a similar way [3]. The onset speed of instability always exceeds the rotor first critical speed.


65

2.3.4 Gravity loading For horizontal rotors, the own weight changes the location of the centre of the disc orbit. If we add the rotor weight m g (g is the acceleration of gravity) to the right-hand side of equations (2.56) with external damping, we obtain the equation of motion [6] m &r&C + (ce + ci ) r&C + (k − i Ω ci ) rC = m eΩ 2 ei ( Ω t +θ 0 ) + m g .

(2.69)

The total solution is obtained summing the solution of the homogeneous equation, the particular solution due to unbalance and the particular solution rC g due to the gravity. The latter has the form rC g =

1 mg g = . k − i Ω ci ωn2 1 − i 2ζ Ω i

(2.70)

ωn In the absence of internal damping, for ζ i = 0, we obtain rC g = rC st =

g

ωn2

=

mg k

which corresponds to the shaft static deflection under the disc weight.

Fig. 2.16 In the presence of viscous internal damping, the point C moves in a circle whose centre location depends on the running speed. When the running speed Ω increases, the point O' moves along a semicircle of radius rCst / 2 (Fig. 2.16).


66

Within the range 0 ≤ Ω ≤ ωn the displacement of the point O' is however very small, due to the generally very low value of the damping ratio. At Ω = ωn , the vector radius makes an angle 2ζ i with the vertical line OO'.

2.3.5 Effect of shaft bow Similar precession motions can occur due to an initial bend in the shaft, sometimes referred to as "elastic unbalance" [14]. The solution is identical to that of equation (2.27) except the factor eΩ 2 is replaced by aωn2 , where a denotes the initial bend of the shaft at the point of disc attachment. When the shaft is stationary, the point C is displaced a distance a from the bearing line, but the mass centre G is no longer offset from C. This initial bend must not be confused with any sag due to the disc weight. If the shaft is turned slowly, then the bend rotates with the rotor, whereas a sag induced by the gravity remains approximately vertically downwards. The rotor behaves as if the previous mass unbalance force meΩ 2 is replaced by a force

m aωn2 = k a . The difference between the two forms of motion necessitates a modification of the concept of a "balanced" rotor. Thus, if the rotor suffers from mass unbalance, the resulting precession can be balanced for all rotor speeds by attaching a small mass m1 to the circumference of the disc at the appropriate location (diametrally opposite to CG), such that m1R Ω 2 = me Ω 2 , where R is the radius of the disc. The magnitude of the required balancing mass is m1 = me / R and this mass will completely cancel the vibration at all rotor speeds. If a similar mass m1 is attached to the initially bent shaft at a radius R, however, the net force is of magnitude m aωn2 − m1R Ω 2 . Thus the exciting force cannot be removed for all speeds Ω . The best that can be done is to select m1 = m a / R , so that the rotor does not whirl at its critical speed. The only way to balance such a rotor for all speeds would be to straighten the initial bend, but this is not possible in practice.


67

2.3.6 Rotor precession in rigid bearings Although there is an obvious analogy between the analytical results of Section 2.3.1, on the one hand, and the familiar expressions for the linear vibration of a simple single-degree-of-freedom system, on the other, the forced motion of the rotor is not a true vibration. A flexible shaft with a concentrated mass bows out in a simple bend. The magnitude of the deflection and its direction relative to the radial plane containing the unbalance are determined by the speed of rotation and by the external damping. The shaft does not experience any alternating stresses while precessing in rigid bearings because its points move along circular orbits. Hence there is no concern regarding the fatigue. The centrifugal forces due to the unbalance cause the rotor to deflect. The bent rotor whirls around its neutral axis at the running speed. This is a synchronous precession which is actually not a vibration of the rotor in the normal sense of the word. The deflected shape of the rotor remains unchanged during the precession in rigid bearings. Only when the whirl amplitude is measured in any fixed direction, the motion appears as a vibration. The rotor does appear to vibrate only when the projection of the forced motion on any fixed radial plane is examined. Moreover, the bearings experience an oscillatory force in any such plane. The remedy for resonance - the internal damping, does not contribute to limit the amplitude of motion, since the shape of the deflected rotor does not change during the whirl motion. As explained later, the major sources of damping for rotors are the journal bearings, small-clearance liquid seals or viscous sleeves. For the Laval-Jeffcott rotor, the bend is greatest when the frequency corresponding to the rotational speed is equal (or nearly equal) to the natural frequency of transverse vibration that the rotor would have if it did not rotate and were simply executing forced undamped flexural vibrations. This will be referred to as a natural frequency of precession. Because the precession is synchronous, i.e. the angular speed of precession is equal to the angular speed of rotation, this frequency corresponds to the critical speed. The large shaft bow at the critical speed can produce stresses in the plastic range that can be limited by radial stops. The external damping is a ‘stationary’ damping which limits the magnitude of the precession radius. The internal damping is a ‘rotating’ damping which can produce unstable whirling above the critical speed. At a certain speed, the internal friction force changes the direction and becomes a destabilizing tangential force acting in the sense of whirling, against the external damping force. Note that the above analysis neglects the effects of the disc mass moments of inertia and the disc pitching motion when mounted off-centre. These will be treated in the following.


68

2.4 Undamped asymmetric rotors In this section, simple rotors are considered with shafts supporting a rigid disc attached either in-board off-centre, as in Fig. 2.17, or overhung, as in Fig. 2.2. The disc rotates with the angular speed Ω . If the rotor speed and the disc mass moments of inertia are relatively small, than the disc can be modeled by a concentrated mass, and the problem can be reduced to the study of the lateral vibrations of a beam carrying a point mass. In practical cases, the angular precession of the disc axis (tangent to the shaft axis) adds to the orbital motion of the centre of the shaft cross-section. This gives rise to inertia torques that influence the parameters of the rotor whirling motion.

Fig. 2.17 The disc rotary inertia due to the disc transverse mass moment of inertia resists any local angular acceleration due to the change of slope of the rotor. This contributes to the overall inertia of the rotor and tends to lower the system critical speeds. The gyroscopic couple resists any change in the angular momentum of the disc. For forward precession, this acts in opposition to rotatory inertia and introduces a so-called ‘gyroscopic stiffening’ effect, proportional to the disc polar mass moment of inertia and rotation speed. The gyroscopic coupling yields pairs of forward and backward precession modes whose natural frequencies are, respectively, larger and lower than the associated zero-speed natural frequencies. Because the natural frequencies depend on the rotor speed, distinction should be made between rotor natural frequencies and critical speeds. Base excitation and harmonic forces with fixed direction in space excite both forward and backward critical speeds. The occurrence of backward precession is not desirable in practice, producing alternating bending stresses, which can shorten the fatigue life of the rotor. Rotor unbalance cannot excite backward modes.


69

2.4.1 Reference frames It is convenient to utilize several different reference frames: a) a stationary reference frame Oxyz; b) a rotating rectangular coordinate frame Gx'y'z' whose axes are collinear with the Oxyz axes, and c) a reference frame Gx1 y1 z1 which is translating and rotating with respect to Oxyz but is not fixed to the moving disc. The bearing line crosses the non-rotating disc at point O. The moving axes Gx'y'z' have the origin at the disc mass centre G . They rotate around the point O, but remain collinear with the axes of the stationary frame. The axis Gx1 coincides with the rotor spin axis, while Gy1 , Gz1 do not rotate about Gx1 . Axes Gx1 y1 z1 can be considered to be the principal axes of inertia of the disc.

Fig. 2.18 It is assumed that, due to the shaft deflection, the disc spinning axis Gx1 makes an angle ϕG with the plane yOx (hence with y'Gx') and an angle ψ G with the plane zOx (hence with z'Gx') as shown in Fig. 2.18.

2.4.2 Inertia torques on a spinning rigid disc The principal mass moments of inertia of the rotor disc with respect to the coordinate frame G x1 y1 z 1 are denoted :

J x1 = J P ,

J y 1 = J z 1 = JT ,

(2.71)

where J P is the polar mass moment of inertia and J T is the diametral (transverse) mass moment of inertia.


70

In order to determine the expressions of the torques acting on the disc (from the shaft), the angular momentum principle with respect to the point G is used. The projections of the angular momentum vector along the axes of the G x1 y1 z 1 frame, taken to coincide with the principal axes of inertia of the perfectly balanced disc, can be written

K y 1 = J T ϕ&G ,

K z 1 = J T ψ& G ,

K x 1 = J P Ω.

(2.72)

The components K x , K y , K z of the angular momentum along the axes of the stationary frame O x y z are equal to the components along the corresponding axes of the frame G x′ y′ z ′ (Fig. 2.19).

Fig. 2.19 From Figs. 2.19, a and b it comes out that

K z' = K z 1 cosϕG − K x 1 sinϕG , K y' = K y 1 cos ψG + K x 1 sin ψG . For small angles, cosψ ≅ 1, sinψ ≅ ψ , so that K z = K z' = K z 1 − K x 1ϕG , K y = K y' = K y 1 + K x 1 ψG .

(2.73)

On inserting expressions (2.72) into (2.73), we obtain K y = J T ϕ&G + J P Ω ψG , K z = J T ψ& G − J P Ω ϕG .

(2.74)


71

Fig. 2.20 Using the angular momentum principle, the components along the axes Oy and Oz, respectively, of the torque applied to the disc, as a result of the shaft deflection, can be written M G y = K& y = J T ϕ&&G + J P Ω ψ& G , M G z = K& z = J T ψ&&G − J P Ω ϕG

(2.75, a)

or, in matrix form, ⎧M G y ⎫ ⎡ J T ⎨M ⎬ = ⎢ ⎩ Gz ⎭ ⎣ 0

0 ⎤ ⎧ϕ&&G ⎫ ⎨ ⎬+Ω J T ⎥⎦ ⎩ψ&&G ⎭

⎡ 0 ⎢− J ⎣ P

J P ⎤ ⎧ϕ&G ⎫ ⎨ ⎬. 0 ⎥⎦ ⎩ψ& G ⎭

(2.75, b)

When the disc is part of a rotor, the system equations of motion can be obtained from d'Alembert's principle if the right-hand sides of (2.75, b) are introduced with opposite signs as inertia torques acting on the disc (Fig. 2.20): ⎧ M G y ⎫ ⎡ JT ⎨ M ⎬−⎢ ⎩ Gz ⎭ ⎣ 0 ↓ applied diametral torques

0 ⎤ ⎧ ϕ&&G ⎫ ⎨ ⎬−Ω J T ⎥⎦ ⎩ ψ&&G ⎭

⎡ 0 ⎢− J ⎣ P

J P ⎤ ⎧ ϕ&G ⎫ ⎧ 0 ⎫ ⎨ ⎬ = ⎨ ⎬. 0 ⎥⎦ ⎩ ψ& G ⎭ ⎩ 0 ⎭

↓

↓

angular acceleration inertia torques

gyroscopic torques

(2.75, c)

The last term in the left-hand side of equation (2.75, c) describes the gyroscopic torques acting on the disc. They couple the equations of motion. The


72

torque M y about the Oy axis is proportional to the angular velocity ψ& about the Oz axis and vice versa.

From Fig. 2.20 it can be seen that a torque M y , which produces a rotation ϕ , gives rise to a gyroscopic torque J P Ω × ϕ& directed along the Oz axis, which "tends to rotate the spin axis Ox toward the Oy axis". A torque M z , which produces a rotation ψ , gives rise to a gyroscopic torque J P Ω × ψ& directed along the negative Oy axis, hence "tends to rotate the spin axis Ox toward the Oz axis". The general rule, also given by the vector product from the expression of the gyroscopic torque, can be stated as follows: "the spin vector Ω tries to move into the torque vector".

2.4.3 Equations of motion for elastically supported discs The shaft is acted upon by torques of the same magnitude but opposite direction as those applied to the disc M C y = −M Gy , M C z = − M Gz .

(2.76)

Moreover, the shaft is acted upon by the disc inertia forces, whose projections on the fixed frame axes have the following expressions FC y = − FG y = − m &y&G , FC z = − FGz = − m &z&G ,

(2.77)

where yG , zG are the coordinates of the disc mass centre. Figure 2.21 illustrates the forces and torques acting on the shaft at the disc attachment point, as well as the corresponding deformations (positive, by the right hand rule, in the positive direction of the coordinate axes). The equations of motion can be written using the (flexibility) influence coefficient method. The following notation is used: δ11 - deflection (at a point) produced by a unit force (applied at the same point); δ 21 - rotation (of a crosssection, or the slope of the elastic line) produced by a unit force (applied at the same point); δ 22 - rotation (of a cross-section) produced by a unit couple (applied at the same cross-section); δ12 - deflection (at a point) produced by a unit couple


73

(applied at the same cross-section). According to Maxwell's reciprocity theorem, δ12 = δ 21 .

Fig. 2.21 The displacements of the disc centre can be written yC = FC y δ11 + M C z δ12 ,

ψ C = FC y δ 21 + M C z δ 22

(2.78)

and zC = FC z δ11 + ( − M C y ) δ12 , − ϕC = FC z δ 21 + ( − M C y ) δ 22 .

(2.79)

Substituting the expressions of forces (2.77) and torques (2.76), (2.75a) into equations (2.78) and (2.79), the following equations of motion are obtained

yC = − m &y&G δ11 − ( J T ψ&&G − J P Ω ϕ&G ) δ12 , ψ C = − m &y&G δ 21 − ( J T ψ&&G − J P Ω ϕ&G ) δ 22 , zC = −m &z&G δ11 + ( J T ϕ&&G + J P Ω ψ& G ) δ12 , − ϕC = −m &z&G δ 21 + ( J T ϕ&&G + J P Ω ψ& G ) δ 22 , or m δ11 &y&G + J T δ12 ψ&&G − J P Ω δ12 ϕ&G + yC = 0 , m δ11&z&G − J T δ12 ϕ&&G − J P Ω δ12 ψ& G + zC = 0, m δ 21 &y&G + J T δ 22 ψ&&G − J P Ω δ 22 ϕ&G + ψ C = 0 , m δ 21&z&G − J T δ 22 ϕ&&G − J P Ω δ 22 ψ& G − ϕC = 0.

(2.80)


74

In matrix form ⎤ ⎡m 0 ⎡δ11 δ12 ⎥ ⎢0 J ⎢δ T ⎥⎢ ⎢ 21 δ 22 ⎢ δ11 δ12 ⎥ ⎢ ⎢ ⎥⎢ δ 21 δ 22 ⎦ ⎣ ⎣ ⎡δ11 δ12 ⎢δ δ 22 + Ω ⎢ 21 ⎢ δ11 ⎢ δ 21 ⎣

⎤ ⎧ &y&G ⎫ ⎥ ⎪ ψ&& ⎪ ⎥ ⎪⎨ G ⎪⎬ + m 0 ⎥ ⎪ &z&G ⎪ ⎥ 0 J T ⎦ ⎪⎩− ϕ&&G ⎪⎭ (2.80, a) 0 0 ⎤ ⎧ y& G ⎫ ⎧ yC ⎫ ⎧0⎫ ⎤⎡ ⎥⎢ 0 J P ⎥⎥ ⎪⎪ ψ& G ⎪⎪ ⎪⎪ ψC ⎪⎪ ⎪⎪0⎪⎪ ⎥⎢ ⎬ = ⎨ ⎬. ⎨ ⎬+⎨ ⎥ ⎪ z&G ⎪ ⎪ zC ⎪ ⎪0⎪ δ12 ⎥ ⎢0 0 ⎥⎪ ⎥⎢ δ 22 ⎦ ⎣0 − J P ⎦ ⎩− ϕ&G ⎪⎭ ⎪⎩− ϕC ⎪⎭ ⎪⎩0⎪⎭

In order to reduce the dimensionality of the governing equations, the following complex variables are introduced yG + i zG = rG ,

yC + i zC = rC ,

ψG − i ϕG = α G ,

ψC − i ϕC = α C .

(2.81)

The second equation (2.80) is multiplied by i = − 1 and added to the first equation. The fourth equation (2.80) is multiplied by i and is added to the third equation. This produces the following set of two coupled equations m δ11 &r&G + J T δ12 α&&G − i J P Ω δ12 α& G + rC = 0, m δ 21 &r&G + J T δ 22 α&&G − i J P Ω δ 22 α& G + α C = 0.

(2.82)

Generally, if the disc has a running speed Ω and an offset e, then the mass G and the geometric centre C of the shaft cross-section do not coincide. The radii of the orbits of these points are related by rG = rC + e e i ( Ω t +θ o ) .

(2.83)

If the disc is attached at an angle α to the shaft, the slope of the shaft axis α C and the inclination of the disc spinning axis (perpendicular to its plane) α G are related by

α G = α C + α e i ( Ω t +θ α ) .

(2.84)

Eliminating rG and α G between equations (2.82)-(2.84), the differential equations of the motion of the disc geometric centre C are obtained as:


75

m δ11 &r&C + J T δ12 α&&C − i J P Ω δ12 α& C + rC = = m δ11 eΩ 2e i ( Ω t +θ o ) + ( J T − J P ) δ12 α Ω 2e i ( Ω t +θα ) , m δ 21 &r&C + J T δ 22 α&&C − i J P Ω δ 22 α& C + α C =

(2.85)

= m δ 21 eΩ 2e i ( Ω t +θ o ) + ( J T − J P ) δ 22 α Ω 2e i ( Ω t +θα ) , or, in matrix form,

⎡δ11 δ12 ⎤ ⎡m ⎢δ ⎥⎢ ⎣ 21 δ 22 ⎦ ⎣ 0

0 ⎤ ⎧ &r&C ⎫ ⎡δ11 δ12 ⎤ ⎡0 0 ⎤ ⎧ r&C ⎫ ⎨ ⎬+ ⎨ && ⎬ + ⎢ ⎢ ⎥ ⎥ J T ⎦ ⎩α C ⎭ ⎣δ 21 δ 22 ⎦ ⎣0 − i J P Ω ⎥⎦ ⎩α& C ⎭

⎧ rC ⎫ + ⎨ ⎬ = Ω2 ⎩α C ⎭

⎡δ11 δ12 ⎤ ⎪⎧ me eiθ o ⎨ ⎥ ⎢δ iθ ⎣ 21 δ 22 ⎦ ⎪⎩ ( J T − J P ) α e α

⎫⎪ iΩt ⎬e ⎪⎭

(2.86)

Introducing the stiffness matrix as the inverse of the flexibility matrix ⎡δ11 δ12 ⎤ ⎢δ ⎥ ⎣ 21 δ 22 ⎦

−1

k ⎤ ⎡k = ⎢ 11 12 ⎥ , ⎣k 21 k 22 ⎦

the equation (2.86) has the simpler form 0 ⎤ ⎧ r&C ⎫ ⎡ k11 k12 ⎤ ⎧ rC ⎫ ⎡m 0 ⎤ ⎧ &r&C ⎫ ⎡0 ⎥⎨ ⎬= ⎢ 0 J ⎥ ⎨α&& ⎬ + ⎢0 − i J Ω ⎥ ⎨α& ⎬ + ⎢k P ⎦ ⎩ C ⎭ ⎣ 21 k 22 ⎦ ⎩α C ⎭ T⎦⎩ C⎭ ⎣ ⎣ ⎧⎪ m e e iθ o =Ω ⎨ ⎪⎩ ( J T − J P ) α eiθα 2

⎫⎪ i Ω t ⎬e . ⎪⎭

(2.87)

In the following, the rotor precession natural frequencies, the response to mass unbalance and the response to a harmonic force fixed in space will be studied [15]. The index C will be dropped to simplify the notation.

2.4.4 Natural modes of precession To study the free precession of the asymmetric rotor, consider equations (2.85) with zero right-hand side m δ11 &r& + J T δ12 α&& − i J P Ω δ12 α& + r = 0, m δ 21 &r& + J T δ 22 α&& − i J P Ω δ 22 α& + α = 0.

(2.88)

The system is undamped, so that the solutions are of the form

r = R e iω t ,

α = Α eiω t .

(2.89)


76

Substituting the solutions (2.89) into (2.88), we obtain the following homogeneous algebraic equations (1 − ω 2 m δ11 ) R − ( ω 2 J T δ12 − ω Ω J P δ12 ) Α = 0 , (−ω 2 m δ 21 ) R + ( 1 − ω 2 J T δ 22 + ω Ω J P δ 22 ) Α = 0.

(2.90)

The condition to have non-trivial solutions is

ω Ω J P δ12 − ω 2 J T δ12 1 − ω 2 m δ11 =0 − ω 2 m δ 21 1 − ω 2 J T δ 22 + ω Ω J P δ 22

(2.91)

and represents the characteristic equation, also termed the frequency equation. It is a quartic in the variable ω , and has four roots (two positive and two negative), which correspond to the four natural frequencies ω i ( i = 1,..,4 ) of an elastically-supported disc. They are functions of the rotational angular speed Ω .

Fig. 2.22 In order to draw the graph of this function it is useful to re-write equation (2.91) under the form

Ω=

2 1 − ω 2 ( m δ11 + J T δ 22 ) + ω 4 m J T ( δ11δ 22 − δ12 )

[

2 ω J P ω 2 m ( δ11δ 22 − δ12 ) − δ 22

]

.

(2.92)

Substituting values of ω , equal to the natural frequencies, gives the corresponding values of Ω .


77

Figure 2.22 presents the dependence ω = ω (Ω ) for a rotor whose disc has the mass moments of inertia J P > J T . The four curves in the diagram correspond to the four roots ω i of equation (2.91). They are anti-symmetrical, two by two, with respect to the coordinate axes. The labeling is arbitrary. The intersection points with the ω axis locate the natural frequencies at zero running speed (Ω = 0 ) . It can be noticed that the rotor rotation determines a variation of the natural frequencies with respect to those of the (non-rotating) rotor in transverse vibrations. The ordinates of the horizontal asymptotes correspond to the natural frequencies of a rotor with zero disc angular precession. When Ω → ∞ , the axial angular momentum becomes so large that the disc cannot be tipped out of its own plane and α remains zero during precession.

Fig. 2.23 For a complete description of the phenomenon, it is sufficient to use the curve branches located in the positive semiplane of ω (Fig. 2.23). The running speed Ω is considered to vary between (–∞) and (+∞) and the natural frequencies ω to vary between 0 and (+∞). It must be noticed that equations (2.88) admit also solutions of the form

r = R e − iω t ,

α = Α e − iω t ,

which implies the substitution of ω by (− ω ) in equation (2.87), or substitution of Ω by (− Ω ) . When ω and Ω have the same sign (in this case - positive), the rotation of the deflected shaft around the bearing line has the same direction as the disc spinning motion. The motion is a forward precession, i.e. the precession motion is in the direction of rotation.


78

When ω and Ω have opposite signs (in this case, ω > 0 , Ω < 0 ), the lines O C and C G rotate in opposite directions; the motion is a backward precession. The directions of the rotor precession and rotor rotation are opposite. Points on the curves ω = ω (Ω ) , in the quadrant with ω > 0 , Ω > 0 , correspond to the forward precession; points in the quadrant ω > 0 , Ω < 0 correspond to backward precession (Fig. 2.23).

Fig. 2.24 Figure 2.24 illustrates the dependence ω = ω (Ω ) for a rotor whose disc has the mass moments of inertia J P < J T (so-called ‘stick’ case). For a disc JP =

m 2 R , 2

JT =

(

)

m 3R 2 + H 2 , 12

(2.93)

where R is the disc radius, H is the disc length and m is the disc mass. For thin discs, H << R and J P = 2 J T . For H = 3R , the mass moments of inertia are equal, J P = J T . In numerical simulations, it is useful to use dimensionless quantities [16]. The characteristic equation (2.91) can be written

⎛ J T 2 Ω ⎞ δ 12 l ⎜ l η − η⎟ ⎜ JP ω 0 ⎟⎠ δ 11 ⎝ =0 2 δ 21 l 2 1 J P ⎛⎜ J T 2 Ω ⎞⎟ δ 22 l − 1− η η − η l δ 11 ω 0 ⎟⎠ δ 11 m l 2 ⎜⎝ J P 1 −η 2

−

JP m l2

(2.91, a)


79

where

η=

ω , ω0

(2.94)

l is rotor’s length (or span), and

ω0 =

1

(2.95)

m δ11

is the natural frequency when the disc is replaced by a concentrated mass. Equation (2.91, a) becomes a4η 4 − a3

Ω 3 Ω η − a2η 2 + a 1 η + 1 = 0 , ω0 ω0

(2.96)

where 2 J J J P δ 22 l , a2 = 1 + a 1 T , a3 = P2 a1 = 2 δ JP ml ml 11

(δ

11δ 22

)

2 − δ12 l2

2 δ11

, a4 =

JT a3 . JP (2.97)

Substituting the natural frequencies ω i

( i = 1,..,4 )

in the first equation

(2.90) we obtain the amplitude ratios which define the mode shapes Ai =

1 − ω 2i m δ11

⎛ ⎞ ⎜ J T − Ω J P ⎟ δ12 ω 2i ⎜ ⎟ ωi ⎝ ⎠

Ri .

(2.98)

For calculations, equation (2.98) is written under the form Ai =

1 − η 2i JP m l2

Ri

⎛ JT 2 Ω ⎞ δ 12 l l ⎜ ηi − ηi⎟ ⎜ JP ω 0 ⎟⎠ δ 11 ⎝

,

(2.99)

where

ηi =

ωi . ω0

(2.100)


80

Example 2.1 Consider as an example a simply supported uniform shaft carrying an offcentre thin disc (Fig. 2.25). Determine the precession mode shapes taking a = 0.4 l , J P = 2 J T = 0.16 ml 2 , Ω = 0.8 ω 0 . The flexibility matrix is

[δ ] =

1 3E I l 2

⎡ a 2b 2 l l ⎡0.0576 l 2 ab (b − a ) l ⎤ = ⎢ ⎥ ⎢ a 3 + b 3 ⎥⎦ 3E I ⎢⎣ 0.048 l ⎢⎣ab (b − a ) l

0.048 l ⎤ ⎥. 0.28 ⎥⎦

Fig. 2.25 The frequency equation (2.96) has the form

η 4 − 1.6η 3 − 4.1667η 2 + 1.8667η + 3 = 0 with roots

η 1 = −0.8206 , η 2 = 1.0137 , η 3 = −1.3217 , η 4 = 2.7286 . The corresponding precession mode shapes are defined by (2.99) A1 = 2.4664 R1 l , A 2 = 0.6955 R2 l , A 3 = −2.9013 R3 l , A 4 = −31.3939 R4 l . The four natural frequencies are

ω 1 = −0.8206 ω 0 , ω 2 = 1.0137ω 0 , ω 3 = −1.3217ω 0 , ω 4 = 2.7286ω 0 . The mode shapes are shown in Fig. 2.26, considering R i = 0.1 l . For a stationary shaft (Ω = 0 ) , the characteristic equation is

η 4 − 4.1667η 2 + 3 = 0 , with roots

− η 01 = η 02 = 0.9621 ,

− η 03 = η 04 = 1.8003 .


81

Fig. 2.26 The corresponding mode shapes are defined by A 01 = A02 = 1.2052 R01 l , A 03 = A04 = −10.3719 R03 l .

2.4.5 Response to harmonic excitation Generally, the rotor forced precession motions are produced by external periodic perturbations. Unbalance, misalignment, blade passing and gear mesh have the excitation frequency a function of the running speed. Coincidence of an excitation frequency with a rotor natural frequency gives rise to a state similar to the resonance, wrongly referred to as ‘critical speed’ or simply ‘critical’, phenomenon characterized by very large values of the radius of the precession motion. 2.4.5.1 Unbalance response

The precession motions due to the unbalance have circular frequencies equal to the angular speed of the rotor. When a natural angular frequency ω i , computed at a given rotational angular speed Ω , becomes equal to the rotor angular speed, a critical state takes place produced by the mass unbalance. The


82

corresponding angular speed is called a critical angular speed. Measured in "rotations/minute" it is referred to as the peak response critical speed. For α = 0 , i.e. for an excitation by mass unbalance only, and taking θ 0 = 0 , the disc steady-state response has the form

⎧r ⎫ ⎨ ⎬= ⎩α ⎭

⎧ rˆ ⎫ iΩ t , ⎨ ⎬e ⎩αˆ ⎭

(2.101)

which describes a synchronous precession of angular speed Ω . Equations (2.87) become ⎡k11 − m Ω 2 ⎢ k 21 ⎣⎢

⎤ ⎧ rˆ ⎫ 2 ⎧1 ⎫ = m e Ω ⎥ ⎨ ⎬ ⎨ ⎬. k 22 − ( J T − J P )Ω 2 ⎦⎥ ⎩αˆ ⎭ ⎩0⎭ k12

(2.102)

The critical angular speeds are solutions of the equation k11 − m Ω 2 k 21

k12 =0, k 22 − ( J T − J P )Ω 2

(2.103, a)

which is obtained by cancelling the denominator in the solutions of the precession produced by unbalance. They satisfy equation Ωcr = ω (Ωcr ). When Ω = Ωcr , the angular velocity of the disc precession motion is equal to the shaft rotational speed. Substituting ω = Ω into (2.91), the equation of the critical speeds can also be written 2 m ( J T − J P ) ( δ11 δ 22 − δ12 ) Ω4 −

− [ m δ11 + ( J T − J P ) δ 22 ] Ω 2 + 1 = 0.

(2.103, b)

It comes out that, in Figs. 2.23 and 2.24, the intersection points of the synchronous excitation line ω = Ω with the curves ω i (Ω ) correspond to the critical speeds of (synchronous) forward precession. In forward precession, the rotor with J P > J T has only one critical speed (Fig. 2.23) and the rotor with J P < J T has two critical speeds (Fig. 2.24). The unbalance can excite a response at critical speeds only in the modes with forward precession. In order to determine the synchronous response to mass unbalance excitation, consider equations (2.85)


83

m δ11 &r& + J T δ12 α&& − i J P Ω δ12 α& + r = m δ11 eΩ 2 e i Ω t , m δ 21 &r& + J T δ 22 α&& − i J P Ω δ 22 α& + α = m δ 21 eΩ 2e i Ω t .

(2.104)

Substituting the solutions (2.101) into (2.104), we obtain the following set of algebraic equations (1 − Ω 2 m δ11 ) rˆ − Ω 2δ12 ( J T − J P ) αˆ = Ω 2δ11 m e ,

[

]

(−Ω 2 m δ 21 ) rˆ + 1 − Ω 2δ 22 ( J T − J P ) αˆ = Ω 2δ 21 m e.

(2.105)

Equations (2.105) can be written in terms of dimensionless quantities as (1 − η 2 ) rˆ − η 2

JP m l2

⎛ JT ⎞ δ 12 l ⎜⎜ l αˆ = η 2 e , − 1⎟⎟ δ J ⎝ P ⎠ 11

⎡ δ 12 l 1 J −η rˆ + ⎢ 1 − η 2 P2 δ 11 l ml ⎢⎣

2 ⎛ JT ⎞ δ 22 l ⎜⎜ − 1⎟⎟ ⎝ J P ⎠ δ 11

2

⎤ δ l1 ⎥ αˆ = η 2e 21 . δ11 l ⎥⎦

(2.106)

Using notations (2.94), (2.95) and (2.97), the solution for rˆ can be written

[

]

1 − ( a4 − a3 )η 2 η 2 rˆ = . e ( a4 − a3 )η 4 − ( a2 − a1 )η 2 + 1

Fig. 2.27

(2.107)


84

The plot of the amplitude ratio rˆ e versus frequency ratio η = Ω ω 0 gives the unbalance response curve, with peak(s) at the forward critical speed(s). For the rotor from Example 2.1, the unbalance response curve is shown in Fig. 2.27. As expected, there is a single peak at about 1.0137ω 0 , corresponding to the first forward mode. 2.4.5.2 Response to a harmonic force fixed in space

A harmonic force having a fixed direction in space

F (t ) = F0 cos ω t

(2.108)

can be written under the form F (t ) =

(

)

F0 i ω t e + e− i ω t . . 2

(2.109)

The first component produces a response of the form ⎧r ⎫ ⎨ ⎬= ⎩α ⎭

⎧ rˆ f ⎫ iω t , ⎨ˆ ⎬e ⎩α f ⎭

(2.110)

which describes a forward precession of angular speed ω . When the circular frequency becomes equal to the rotor angular speed, , the first component can produce resonance in the modes with forward ω=Ω precession. The second component produces a response of the form ⎧r ⎫ ⎨ ⎬= ⎩α ⎭

⎧ rˆb ⎫ − iω t , ⎨ˆ ⎬e ⎩α b ⎭

(2.111)

which describes a backward precession of angular speed (− ω ). For ω = −Ω , equations (2.87) become ⎡k11 − mΩ 2 ⎢ ⎣⎢ k 21

⎤ ⎧ rˆb ⎫ F0 ⎧1⎫ ⎨ ⎬= ⎨ ⎬ . k 22 − ( J T + J P )Ω ⎦⎥ ⎩αˆ b ⎭ 2 ⎩0⎭ k12

2⎥

(2.112)

The backward (or asynchronous) critical speeds [4] are solutions of the equation


85

k11 − m Ω 2

k12

k 21

k 22 − ( J T + J P )Ω 2

=0 .

(2.113)

The backward critical speeds can be excited only by forces rotating in a direction opposite to the rotor rotation. Such a force can be either a component of a force fixed in space, or the result of a kinematic excitation of the bearings or bearing supports. In Figs. 2.23 and 2.24, the intersection points of the frequency curves with the asynchronous excitation line ω = −Ω , locate the backward critical speeds. In order to determine the frequency response to a harmonic unidirectional force, the unbalance excitation is replaced in the right hand side of equations (2.104) by the harmonic force (2.109), resulting in

(

)

F0 i ω t e + e− i ω t , 2 F m δ 21 &r& + J T δ 22 α&& − i J P Ω δ 22 α& + α = δ 21 0 e i ω t + e − i ω t . 2

m δ11 &r& + J T δ12 α&& − i J P Ω δ12 α& + r = δ11

(

)

(2.114)

Because the forward and backward solutions are decoupled, they can be considered separately [15]. For the forward excitation component and solutions (2.110), the equations of motion are ⎡ 1 − mω 2 δ11 ω Ω J P δ12 − ω 2 J T δ12 ⎤ ⎧ rˆ f ⎫ F0 ⎧ δ11 ⎫ ⎢ ⎥⎨ ˆ ⎬= ⎨ ⎬ . (2.115, a) 2 2 ⎣⎢ − mω δ 21 1 − ω J T δ 22 + ω Ω J P δ 22 ⎦⎥ ⎩ α f ⎭ 2 ⎩ δ 21 ⎭ For the backward excitation component and solutions (2.111), the equations of motion are ⎡ 1 − mω 2 δ11 − ω Ω J P δ12 − ω 2 J T δ12 ⎤ ⎧ rˆb ⎫ F0 ⎧ δ11 ⎫ ⎢ ⎥⎨ ⎬= ⎨ ⎬ . (2.115, b) 2 2 ⎢⎣ − mω δ 21 1 − ω J T δ 22 − ω Ω J P δ 22 ⎥⎦ ⎩ αˆ b ⎭ 2 ⎩ δ 21 ⎭ The disc precession radius has a forward component rˆ f =

F0 δ11 [1 − ω (JT ω − J P Ω )δ 22 2Δ f

]

(2.116, a)

F0 δ11 [1 − ω (JT ω + J P Ω )δ 22 ] , 2Δ b

(2.116, b)

and a backward component rˆ b = where (2.91)


86

Δf =

1 − mω 2δ11 ω Ω J P δ12 − ω 2 J T δ12 − mω 2δ 21 1 − ω 2 J T δ 22 + ω Ω J P δ 22

(2.117)

Δb =

1 − mω 2δ11 − ω Ω J P δ12 − ω 2 J T δ12 . − mω 2δ 21 1 − ω 2 J T δ 22 − ω Ω J P δ 22

(2.118)

and

The disc precession orbit is an ellipse with the major and minor semiaxes a=

(

1 rˆ f + rˆb 2

),

b=

(

1 rˆ f − rˆb 2

).

(2.119)

Using notations (2.94), (2.95) and (2.97), the above solutions can be written rˆ f =

1 + a3

Ω η − a4η 2 ω0

F0 δ11 , 2 a η4 − a Ω η3 − a η2 + a Ω η +1 4 3 2 1

ω0

(2.120)

ω0

and

rˆb =

1 − a3

Ω η − a4 η 2 ω0

F0 δ11 . 2 a η4 + a Ω η3 − a η2 − a Ω η +1 4 3 2 1

ω0

Fig. 2.28

ω0

(2.121)


87

For the rotor from Example 2.1, the major semiaxis of the disc unbalance response, a (F0δ11 2) , is plotted against excitation frequency in Fig. 2.28, with solid line, for Ω = 0.8 ω 0 and with broken line, for Ω = 0.4 ω 0 . For Ω = 0.8 ω 0 , the abscissae of the four peaks correspond to the four natural frequencies calculated in Example 2.1.

2.4.6 Campbell diagrams The diagram of the natural frequencies of precession ω i = ω i (Ω ) is usually plotted, in a condensed form, only in the first quadrant of the frame ω OΩ , as a Campbell diagram. If the mirror images of the curves from the second quadrant of Figs. 2.23 and 2.24, corresponding to the backward precession, are drawn in the first quadrant, the Campbell diagrams from Fig. 2.29 are obtained. The notations are F for forward precession and B for backward precession. Note that the index of the natural frequencies has been changed.

a

b

Fig. 2.29 The intersections with the line ω = Ω are the points whose abscissae define the critical angular speeds Ω cr i . In practice, the excitation frequencies can be multiples of the runningspeed frequency. The intersections of the curves representing the natural frequencies of forward precession with the lines of slope n Ω , where n = 1,2,3,4,....., help locating the possible critical angular speeds (Fig. 2.30). The


88

magnitude of the response at the respective speed depends on both the magnitude of the forcing harmonic and the system damping.

Fig. 2.30 When the critical speeds are determined using the Campbell diagram, the remark made in Section 2.4.4 has to be taken into account: the unbalance can produce large deflections only at or near a forward critical speed.

Fig. 2.31 For the rotor from Example 2.1, the Campbell diagram is shown in Fig. 2.31. The synchronous excitation line is drawn with broken line.


89

Example 2.2 Determine the mode shapes and plot the Campbell diagram for the cantilevered rotor shown in Fig. 2.32. Take J P = 2 J T = 0.16 ml 2 , Ω = 0.8 ω 0 .

Fig. 2.32 The flexibility matrix is

[δ ] =

l 6E I

⎡ 2l 2 ⎢ ⎢⎣ 3l

3l ⎤ ⎥. 6 ⎥⎦

The frequency equation (2.91, a) is

− 0.12η ( η − 1.6 ) l =0 1 − 1.5η 2 1 − 0.24η ( η − 1.6 ) l 1 −η 2

or

0.06η 4 − 0.096η 3 − 1.24η 2 + 0.384η + 1 = 0 with roots

η 1 = −0.7868 , η 2 = 1.0491 , η 3 = −3.8742 , η 4 = 5.2119 . The corresponding mode shapes are defined by A1 = 1.6905 R1 l , A 2 = 1.4497 R2 l , A 3 = −5.5047 R3 l , A 4 = −11.5821 R4 l . The four natural frequencies are

ω 1 = −0.7868 ω 0 , ω 2 = 1.0491ω 0 , ω 3 = −3.8742ω 0 , ω 4 = 5.2119ω 0 . The mode shapes are shown in Fig. 2.33, considering R i = 0.1 l . For a stationary shaft (Ω = 0 ) , the characteristic equation is

3η 4 − 62η 2 + 50 = 0 , with roots − η 01 = η 02 = 0.9169 , − η 03 = η 04 = 4.4526 .


90

Fig. 2.33

Fig. 2.34


91

The corresponding mode shapes are defined by A 01 = A02 = 1.5797 R01 l , A 03 = A04 = −7.9130 R03 l . The Campbell diagram is illustrated in Fig. 2.34.

Example 2.3 Determine the precession mode shapes and plot the Campbell diagram for the simply supported rotor with overhang disc shown in Fig. 2.35. Take J P = 2 J T = 0.16 ml 2 , c = 0.2 l , Ω = 0.8 ω 0 .

Fig. 2.35 The flexibility matrix is

[δ ] = 1 3E I

c ⎤ ⎡ 2 ⎢ c (l + c ) 2 (2l + 3c )⎥ l ⎡0.048l 2 = ⎢ ⎥ ⎢c ⎢ (2l + 3c ) l + 3c ⎥ 3E I ⎣⎢ 0.26l ⎦ ⎣2

0.26l ⎤ ⎥. 1.6 ⎦⎥

The characteristic equation (2.96) is

0.3194η 4 − 0.5111η 3 − 3.6667η 2 + 4.2667η + 1 = 0 with roots

η 1 = −0.2008 , η 2 = 1.3273 , η 3 = −3.1985 , η 4 = 3.6721 . The corresponding precession mode shapes are defined by A1 = 6.1241 R1 l , A 2 = 4.8552 R2 l , A 3 = −1.3879 R3 l , A 4 = −3.7863 R4 l . The four natural frequencies are

ω 1 = −0.2008 ω 0 , ω 2 = 1.3273ω 0 , ω 3 = −3.1985ω 0 , ω 4 = 3.6721ω 0 . The mode shapes are shown in Fig. 2.36, considering R i = 0.1 l .


92

Fig. 2.36 For a stationary shaft (Ω = 0 ) , the characteristic equation is

η 4 − 11.4783η 2 + 3.1304 = 0 , with roots

− η 01 = η 02 = 0.5287 , − η 03 = η 04 = 3.3464 .

Fig. 2.37


93

The corresponding mode shapes are defined by A 01 = A02 = 5.9478 R01 l , A 03 = A04 = −2.1016 R03 l . The Campbell diagram is given in Fig. 2.37.

Example 2.4 Determine the precession mode shapes and plot the Campbell diagram for the symmetric rotor with a disc at the middle shown in Fig. 2.38. Take

J P = 2 J T = 0.16 ml 2 , Ω = 0.8 ω 0 .

Fig. 2.38 This is a Laval-Jeffcott rotor with includes the effects of the disc mass moments of inertia. The translational and rotational motions of the disc are elastically decoupled, since the disc is located at the centre of the shaft. The translational equations of the free motion are ⎡ m 0 ⎤ ⎧ &y& ⎫ ⎡ kT ⎢ 0 m ⎥ ⎨ &z& ⎬ + ⎢ 0 ⎣ ⎦⎩ ⎭ ⎣

0 ⎤ ⎧ y ⎫ ⎧0⎫ ⎨ ⎬=⎨ ⎬, kT ⎥⎦ ⎩ z ⎭ ⎩ 0 ⎭

where kT = 48 E I l 3 is the translational stiffness. The rotational equations of the free motion are ⎡ JT ⎢ 0 ⎣

0 ⎤ ⎧ ϕ&& ⎫ ⎡ 0 ⎨ ⎬+Ω ⎢ ⎥ J T ⎦ ⎩ ψ&& ⎭ ⎣− J P

J P ⎤ ⎧ ϕ& ⎫ ⎡ k R ⎨ ⎬+ 0 ⎥⎦ ⎩ ψ& ⎭ ⎢⎣ 0

0 ⎤ ⎧ϕ ⎫ ⎧0⎫ ⎨ ⎬ = ⎨ ⎬, k R ⎥⎦ ⎩ ψ ⎭ ⎩ 0 ⎭

where k R = 12 E I l is the rotational stiffness. When the rotor is not rotating (Ω = 0 ) , there are two independent modes of lateral vibration of equal natural frequency ωT = kT m . One of the modes is a straight-line vibration in the z direction, and the other is a straight-line vibration in the y direction (Fig. 2.39).


94

Fig. 2.39 If these two modes are given the same amplitude, they can be superposed with proper phasing to form a circular precession mode which is either forward or backward with respect to the rotor spin. For this model, these frequencies are independent of the rotor spin speed, ω 1,2 = ±ω T . For the angular motion, substituting solutions

ϕ = Φ e iω t ,

ψ = Ψ e iω t

into the equations of motion, we obtain the following homogeneous algebraic equations ( k R − ω 2 J T )Φ + i ω Ω J P Ψ = 0, − i ω Ω J P Φ + ( k R − ω 2 J T )Ψ = 0. The characteristic equation is ( k R − ω 2 J T ) 2 − (ω Ω J P ) 2 = 0 . Denoting [17]

ω R2 =

kR , JT

γ=

JP , JT

we obtain the natural frequencies 1 2

ω 3,4 = ωΩ m γ Ω , or

ωΩ2 = ω R2 +

1 (γ Ω )2 , 4


ω 3,4 =

Ω JP 2 JT

±

95

⎛ Ω JP ⎜⎜ ⎝ 2 JT

2

⎞ k ⎟⎟ + R . JT ⎠

Figure 2.40 shows the dependence of the two natural frequencies on the rotor speed. The synchronous excitation line as well as the asymptotes ω = γ Ω 1 and ω = γ Ω are also drawn in the figure. 2

Fig. 2.40 The synchronous line intersects the two natural frequency lines at

Ω3 = ωR

1+ γ ,

Ω 4 = ωR

1− γ .

The mode shapes are given by the amplitude ratio

Ψ k − ω 2JT =i R = ±i . Φ ω Ω JP

(Ψ

For ω = ω 3 , (Ψ Φ )3 = −i , the precession is backward. For ω = ω 4 ,

Φ )4 = +i , the precession is forward. The deflected shaft is planar. The rotation

axis of the disc describes a cone with circular cross-section. For a comparison with the previous examples, a solution based on notations (2.94), (2.95) and (2.97) is given in the following. The flexibility matrix is


96

[δ ] =

l ⎡0.0625l 2 ⎢ 3E I ⎣⎢ 0

The characteristic equation (2.96) is

( 0.32η

2

− 0.512η − 1

0 ⎤ ⎥. 0.25⎦⎥

) (η

2

)

−1 = 0

with roots

η 1 = −1 , η 2 = 1 , η 3 = −1.1404 , η 4 = 2.7404 . For a stationary shaft (Ω = 0 ) , the characteristic equation is

(η

2

− 3.125

) (η

2

)

−1 = 0 ,

with roots

η 01,02 = ±1 ,

η 03,04 = ±1.7678 .

Fig. 2.42 The Campbell diagram is shown in Fig. 2.42. The frequency ratio, ω i ω 0 , is plotted versus the speed ratio, Ω ω 0 , where ω 0 = ω T . The overlaid lines of the translatory modes, ω 1,2 = ±ω T , correspond to the Laval-Jeffcott rotor model. The only one critical speed excited by rotating unbalance is located at the intersection of the synchronous excitation line (dotted) with the line ω 2 at Ω cr = ωT .


97

2.4.7 Effect of the gyroscopic torque on critical speeds According to equations (2.76) and (2.75, a), in the case of forward precession, the disc acts on the shaft with a torque of components M C y = − ( J T ϕ&&G + J P Ω ψ& G ) , M C z = − ( J T ψ&&G − J P Ω ϕ&G ).

(2.122)

Generally, if the precession angular velocity is ω and the orbit is circular, as well as if the disc is perpendicular to the shaft axis, then

ϕG = ϕC = −α C sinω t , ψ G = ψ C = α C cosω t .

(2.123)

The components of the torque applied to the shaft are

Ω ⎞ ⎛ M C y = −⎜ J P − J T ⎟ ω 2ϕC , ω ⎝ ⎠ Ω ⎛ ⎞ M C z = −⎜ J P − J T ⎟ ω 2ψ C , ⎝ ω ⎠

(2.124)

and tend to decrease (for positive parenthesis) the slopes, hence to stiffen the shaft. For synchronous forward precession, Ω = ω , and expressions (2.124) become

M C y = ( J T − J P ) Ω 2ϕC , M C z = ( J T − J P ) Ω 2ψ C .

(2.125, a)

The gyroscopic torque produces an apparent decrease of J T (or even the reverse effect) raising the critical speed. For backward asynchronous precession, with Ω = −ω , we obtain M C y = ( J T + J P ) Ω 2ϕC , M C z = ( J T + J P ) Ω 2ψ C ,

(2.125, b)

so that there is an apparent increase of J T , which lowers the respective critical speed.

98


This explains the shape of the curves from figures 2.23 and 2.24, which indicate an increase of the forward precession natural frequencies and a decrease of the backward precession natural frequencies with the increase of the running speed.

2.4.8 Remarks on the precession of asymmetric rotors The new phenomena introduced by the asymmetric rotor are the following: a) The transverse disc inertia (rotary inertia) doubles the number of critical speeds. There is an 'inertial' effect, i.e. the reduction of the lowest natural frequency of the rotor in comparison to the natural frequency of the rotor with the disc modeled as a concentrated mass. The additional eigenvalue is associated with the additional rotational degree of freedom, viz. the disc rotation around its diameter. In the first mode of precession, the translatory and the angular motions are in phase. In the second mode, the translatory and the angular motions are out of phase. The rotary inertia effect acts also at zero running speed. b) Due to the gyroscopic effects, the rotor natural frequencies depend on the running speed. Generally, gyroscopic torques double the number of natural frequencies. They occur in pairs corresponding to forward and backward precession. The gyroscopic effect does not act at zero running speed. The number of critical speeds can be different from the number of natural frequencies of precession. Forward critical speeds can be encountered only in the case of corotating excitation. They are synchronous critical speeds. Backward critical speeds can be encountered only in the case of counter-rotating excitation. Backward critical speeds are referred to as asynchronous critical speeds. A constant direction harmonic force can produce both forward and backward precession. The number of critical speeds depends on the disc inertia ratio J P / J T . For a thin disc, J P > J T , there is only one forward synchronous critical speed. For a thick disc, J P < J T , there are two critical speeds in forward synchronous precession. When J P = J T , the system cannot pass through the second critical. Some drum washing machines are designed to have nearly equal axial and transverse moments of inertia. There are always two asynchronous critical speeds irrespective of the disc inertia ratio. c) Discs mounted inclined on the shaft produce the so-called 'skewunbalance' which is a source of synchronous rotor excitation analogous to the mass unbalance.


99

1 (J P − J T ) sin 2α , the effect of disc skewness can be 2 considered as a 'product of inertia' unbalance.

Since J xy =

d) When the disc is attached at the middle of the shaft, it has a planar motion in the cylindrical modes of precession, there is no gyroscopic effect, the forward and backward natural frequencies coincide and are independent of the running speed. The corresponding curves in the Campbell diagram are overlapped straight lines. e) For rotors in rigid bearings, all points move in circular orbits. The rotors have circular precession modes with planar deflected shapes. f) Internal rotor damping from shrink-fit rubbing or material hysterezis can produce rotor instability. Below the onset speed of instability the rotor's motion is stable and synchronous. Above this speed, destabilizing forces which are normal to the radial displacement, and in the direction of shaft rotation, produce large whirl amplitudes which may result in damaged or destroyed equipment. Whirl amplitudes grow until they achieve a steady-state limit cycle.

References 1. Föppl, A., Das Problem der Lavalschen Turbinenwelle, Der Civilingenieur, Vol.4, pp.335-342, 1895. 2. Jeffcott, N., Lateral vibration of loaded shafts in the neighbourhood of a whirling speed - The effect of want of balance, Philosophical Magazine, Series 6, Vol.37, pp.304-314, 1919. 3. Childs, D., Turbomachinery Rotordynamics. Phenomena, Modeling and Analysis, Wiley, New York, 1993. 4. Stodola, A., Neuere Beobachtungen uber die Kritischen Umlaufzahlen von Wellen, Schweizer.Bauzeitung, Vol.68, pp.210-214, 1916. 5. Green, R., Gyroscopic effects on the critical speeds of flexible rotors, J. Appl. Mech, Vol.15, pp 369-376, 1948. 6. Gasch, R. and Pfützner, H., Rotordynamik, Springer, Berlin, 1975. 7. Kimball, A. L., Jr., Measurement of internal friction in a revolving deflected shaft, General Electric Review, Vol.28, No.8, pp.554-558, Aug 1925. 8. Newkirk, B. L., Shaft whipping, General Electric Review, Vol.27, p.169, 1924. 9. Gunter, E. J., Jr., and Trumpler, P. R., The influence of internal friction on the stability of high speed rotor with anisotropic supports, ASME Journal of Engineering for Industry, Series B, Vol.87, pp.1105-1113, Nov 1969.

100


10. Kimball, A. L., Internal friction as a cause of shaft whirling, Philosophical Magazine, Vol.49, pp.724-727, 1925. 11. Robertson, D., Transient whirling of a rotor, Philosophical Magazine, Series 7, Vol.20, p.793, 1935. 12. Gunter, E. J., Jr., The influence of internal friction on the stability of high speed rotors, ASME Journal of Engineering for Industry, Series B, Vol.89, pp.683688, Nov 1967. 13. Dimentberg, F., Flexural Vibrations of Rotating Shafts, Butterworths, London, 1961. 14. Bishop, R. E. D., and Parkinson, A. G., Vibration and balancing of flexible shafts, Appl. Mech. Reviews, Vol.21, No.5, 1968, pp.439-451. 15. Gasch, R., Nordmann, R., and Pfützner, H., Rotordynamik, Springer, Berlin, 2002. 16. Krämer, E., Dynamics of Rotors and Foundations, Springer, Berlin, 1993. 17. Ewins, D. J., Modal Testing: Theory, Practice and Applications, 2nd ed., Research Studies Press, Baldock, 2000.

3. SIMPLE ROTORS IN FLEXIBLE BEARINGS This chapter considers the effect of bearing flexibility and damping on the precession of flexible rotors. Only single-disc rotors supported in flexible bearings will be examined. Both isotropic and orthotropic constant parameter bearings, as well as hydrodynamic bearings with speed-dependent spring and damping coefficients, are considered.

3.1 Symmetric rotors in flexible bearings This section considers single-disc rotor models, with a radially and longitudinally symmetric shaft, supported in identical flexible bearings and/or bearing supports. The disc rotary inertia is neglected and only the planar translatory disc precession is analyzed.

a

b

Fig. 3.1 It is supposed that the bearing flexibilities and damping are uncoupled as in rolling bearings on flexible supports, tilting pad journal bearings and squeezefilm supports with retaining springs designed to operate in the linear range.


102

3.1.1 Effect of bearing flexibility Consider a symmetric rotor supported in two identical anisotropic flexible bearings (Fig. 3.1, a). Such bearings have two principal directions of stiffness along which the radial stiffness has extreme values. For orthotropic bearings only the principal stiffnesses are considered. The axes Oy and Oz are along the bearing principal directions of stiffness. Let k1 and k 2 be the principal stiffnesses. Let y B , z B be the components of the displacement of the journal centre along the axes of the stationary reference frame Oxyz (Fig. 3.1, b). The other notations are as for the rotors in rigid bearings (Fig. 2.3).

a

b

Fig. 3.2 3.1.1.1 Equations of motion

Using d'Alembert's principle, the dynamic equilibrium of forces and torques acting on the disc (Fig. 3.2, a) is written as [1]: m &y&G + k ( yC − y B ) = 0 , m &z&G + k ( zC − z B ) = 0 , J G θ&& + k ( yC − y B ) e sin θ − k ( zC − z B ) e cos θ = M (t ) ,

(3.1)

and the equilibrium of forces acting on the shaft (Fig. 3.2, b): 2k1 y B = k ( yC − y B ), 2k 2 z B = k ( zC − z B ).

(3.2)

3. SIMPLE ROTORS IN FLEXIBLE BEARINGS

103

The coordinates of points C and G are related by yG = yC + e cosθ ,

(3.3)

zG = zC + e sinθ .

In steady-state conditions, if M (t ) = 0 , then θ&& ≅0, the angular speed

θ& = Ω =const. and the angular position θ = Ω t + θ 0 . By a convenient selection of the time origin,

θ =Ωt.

(3.4)

Eliminating coordinates y B , z B and yG , zG between equations (3.1)(3.3), and taking into account (3.4), the equations of motion of point C can be written

m &y&C + k y yC = m eΩ 2cosΩ t , m &z&C + k z zC = m eΩ 2sinΩ t ,

(3.5)

where ky =

2k1k , 2k1 + k

kz =

2k 2 k . 2k 2 + k

(3.6)

Equations (3.5) differ from equations (2.5), established for rotors in rigid bearings, only by the equivalent stiffnesses (3.6), which are different along Oy and Oz. Because of the system symmetry, bearings are represented by springs connected in parallel, and the flexible shaft is connected in series with the bearings. The equivalent stiffnesses are computed from

1 1 1 , = + k y k 2k1

1 1 1 . = + k z k 2k 2

Substituting k by k y in the first equation (2.5), and by k z in the second equation (2.5), we obtain equations (3.5). The complete solutions of equations (3.5) are yC (t ) = YC cos (ω y t + θ y ) +

eΩ 2 cos Ω t , ω y2 − Ω 2

eΩ 2 zC (t ) = Z C sin (ω z t + θ z ) + 2 sin Ω t , ωz − Ω 2

(3.7)


104

where

ωy =

ky m

,

ωz =

kz m

(3.8)

are the natural frequencies of the lateral vibrations along Oy, and Oz, respectively. Generally ω y ≠ ω z and if k 2 < k1 then

ω z < ω y < ωn =

k m.

The natural frequencies of the rotor in flexible bearings are lower than the natural frequency of the rotor in rigid bearings. 3.1.1.2 Unbalance response

The steady-state motion is described by the particular solutions of equations (3.5) eΩ 2 cosΩ t , yC ( t ) = ˆyC cosΩ t = 2 ωy − Ω 2 (3.9) eΩ 2 zC ( t ) = ˆzC sinΩ t = 2 sinΩ t . ωz − Ω 2 Figure 3.3 shows the magnitudes of the two motion components of point C as a function of the angular speed Ω . When Ω = ω z and Ω = ω y , the amplitude grows unbounded. The Laval-Jeffcott rotor in orthotropic bearings has two critical speeds. Because yˆ C ≠ zˆC , equations (3.9) describe an ellipse. Eliminating the time between the two equations yields ⎛ yC ⎜⎜ ⎝ ˆyC

2

2

⎞ ⎛ zC ⎞ ⎟⎟ + ⎜⎜ ⎟⎟ = 1 . ⎠ ⎝ ˆzC ⎠

(3.10)

The orbit of point C is an ellipse whose axes are collinear with the bearing principal stiffness axes. Point C completes the ellipse in a time interval T = 2π Ω , equal to the disc rotation period, hence its motion is a synchronous precession. At speeds Ω < ω z and Ω > ω y the point C moves along the ellipse in the same direction as the disc running speed; the precession is forward. At speeds ω z < Ω < ω y the point C moves along the ellipse in the opposite direction; the precession is backward.


105

At Ω = Ω * and Ω = Ω * * → ∞ , the orbit is circular. At speeds Ω < Ω * the ellipse major semiaxis is collinear with Oz and at speeds Ω > Ω * it is collinear with Oy. The bearing orthotropy doubles the number of critical speeds and produces the synchronous precession with elliptical orbits.

Fig. 3.3 Although the precession is synchronous, the angular velocity of the point C along the ellipse is variable, and Ω is the angular velocity in the circular motions that generate the ellipse. As the rotor moves along the elliptical orbit, it speeds up or slows down to conserve energy and angular momentum. The precession speed is not the angular speed of the rotor along the ellipse. It is equal to the constant angular speed of the forward and backward uniform circular motions that, compounded, generate the elliptical motion. Using complex representation, the radius vector of the disc centre precession orbit can be written as

rC = yC + i zC = ˆyC cos Ω t + i ˆzC sinΩ t , or rC =

(

)

(

(3.11)

)

ˆyC iΩ t ˆz e + e − iΩ t + i C eiΩ t − e − iΩ t , 2 2i


106 rC =

ˆyC + ˆzC iΩ t ˆyC − ˆzC − iΩ t e + e = r f eiΩ t + rb e − iΩ t , 2 2

(

)

(

)

rC = r f + rb cos Ω t + i r f − rb sin Ω t ,

(3.11, a) (3.11, b)

where rf =

ω y2 + ω z2 − 2Ω 2 eΩ 2 , 2 (ω y2 − Ω 2 ) (ω z2 − Ω 2 )

ω y2 − ω z2 eΩ 2 rb = − . 2 (ω y2 − Ω 2 ) (ω z2 − Ω 2 )

(3.12)

The first term in equation (3.11, a) represents (in the complex plane) a vector of length r f which rotates in the same direction as the rotor rotation. The second term represents a vector of length r b which rotates in the opposite direction, with the same angular speed. Addition of the two circular counterrotating motions (Ω = const .) yields an ellipse (Fig. 3.4).

Fig. 3.4 At t = 0 , rˆC (0) = r f + rb = ˆyC = a = major semiaxis. At t = π /(2 Ω ),

rˆC (π / 2Ω ) = r f − rb = ˆzC = b = minor semiaxis. The direction of rotation of vectors rC (hence of the motion of point C along the ellipse) depends on the relative magnitude of the two vector components. If r f > rb , then the point C 'rotates' in the same direction as the disc; its motion is a forward precession. If r f < rb , then the point C 'rotates' in the


107

opposite direction, the motion is a backward precession. If r f = rb , the ellipse degenerates to a line and point C has a rectilinear harmonic motion. Based on Fig. 3.3, where ω z < ω y , and on equations (3.12), the following can be said. For speeds Ω < ω z and Ω > ω y , the precession is forward when r f > rb . For speeds ω z < Ω < ω y , the precession is backward when r f < rb . At Ω = ω z the major semiaxis becomes (theoretically) infinite and as the running speed traverses the rotor first critical speed, the motion changes from forward to backward precession. Analogously, at Ω = ω y , when the running speed traverses the second critical speed, the backward precession changes into a forward precession. The motion along elliptical orbits produces variable stresses in the shaft even at constant running speed. During the synchronous forward precession, the part of the cross-section in tension remains in tension, and the part in compression remains in compression, but the bending stresses vary cyclically due to the variation of the orbit radius. During the backward precession, the bending stresses vary in an alternating non-symmetric cycle, having two reversals per rotation (Fig. 3.3). The motion of journal centres is defined by the variation in time of the coordinates of point B (Fig. 3.1). Equations (3.2) yield ( 2k1 + k ) y B = k yC ,

( 2 k 2 + k ) z B = k zC .

(3.13)

Based on equations (3.9), the steady-speed solution is

y B (t ) =

k Ω2 e 2 cos Ω t , 2k1 + k ω y − Ω 2

k Ω2 z B (t ) = e 2 sin Ω t . 2k 2 + k ω z − Ω 2

(3.14)

Point B has an elliptic orbit, whose semiaxes are smaller than those of the point C. Points B and C have a synchronous motion, the largest amplitudes occurring at ω y and ω z , the rotor critical speeds. Equations (3.3), (3.9) and (3.14) show that the points O, B, C and G are collinear. This is due to the neglecting of damping. As will be shown in the following, in damped rotors the lines O B , B C and C G are not collinear.


108

When the shaft is much stiffer than the bearings, it can be considered that k→∞ and k y = 2k1 , k z = 2k 2 . Equations (3.13) give y B = yC , z B = zC . The disc centre precession orbit is identical to the precession orbit of the journal centres. 3.1.1.3 Natural modes of precession

Dropping the index C, the equations of the free precession, obtained for e = 0 in (3.5), are m &y& + k y y = 0,

(3.15)

m &z& + k z z = 0.

Using the complex representation r = y+i z,

r = y −i z,

(3.16)

equations (3.15) become m &r& + k r + Δ k r = 0,

(3.17)

where k=

k y + kz 2

,

Δk =

k y − kz 2

>0.

(3.18)

The precession behaviour can be analyzed in terms of the forward and backward componets of the motion. Substituting r = r f eiω t + rb e −iω t ,

r = rb eiω t + r f e −iω t

(3.19)

into (3.17), we obtain the homogeneous set of equations

( k − mω ) r + Δ k r Δ k r + ( k − mω ) r 2

f

2

f

b

= 0,

b

= 0.

(3.20)

The characteristic equation is

( k − mω )

2 2

ω2 =

− (Δ k ) 2 = 0 ,

k m Δk . m

(3.21) (3.21, a)

The natural frequencies are

ω 12,2 =

k − Δk kz = = ω z2 , m m

ω 23,4 =

k + Δk k y = = ω y2 . m m

(3.22)


109

From the first equation (3.20) we obtain the amplitude ratio rb k − mω 2 =− . rf Δk

(3.23)

For ω 1,2 = ±ω z , omitting indices, we obtain rb = − r f , ˆy = r f + rb = 0 , ˆz = r f − rb = b .

(3.24, a)

The motion is a (horizontal) vibration along the z-axis, with amplitude b. For ω 3,4 = ±ω y , omitting indices, we obtain rb = + r f , ˆy = r f + rb = a , ˆz = r f − rb = 0 .

(3.24, b)

The motion is a (vertical) vibration along the y-axis, with amplitude a. The modal orbits of the four natural modes of precession degenerate into straight lines. They may be thought of as being made up of two circular orbits of equal radii, where one has forward motion and the other one has backward motion.

3.1.2 Effect of external damping In this section, the effect of external damping on the response of the flexibly supported rotor is considered. In a first approximation, it is assumed that the external damping is isotropic and viscous, giving rise to forces which are proportional to the disc absolute velocity. The main effects are the finite amplitude steady state response to unbalance and the inclination of precession elliptical orbits. 3.1.2.1 Unbalance response

For the calculation of the rotor damped precession, new terms proportional to the disc centre velocity c y& C , c z&C are added in equations (3.5). The following equations of motion are obtained: m &y&C + c y& C + k y yC = m eΩ 2 cos Ω t , m &z&C + c z&C + k z zC = m eΩ 2 sin Ω t .

(3.25)

In steady motion, the solutions of equations (3.25) are yC = ˆyC cos (Ω t + θ y ) , zC = ˆzC sin (Ω t + θ z ) ,

(3.26)


110

where e ˆy C =

Ω2 ω y2

e

2

2 ⎞ ⎛ ⎞ ⎛ ⎜1 − Ω ⎟ + ⎜ 2ζ Ω ⎟ y 2 ⎜ ⎜ ω y ⎟⎠ ω y ⎟⎠ ⎝ ⎝

2

− 2ζ y tanθ y =

,

Ω ωy

Ω2 1− 2 ωy

ˆz C =

Ω2 ω z2

(3.27)

2

2 ⎞ ⎛ ⎛ ⎞ ⎜1 − Ω ⎟ + ⎜ 2ζ z Ω ⎟ ⎜ 2 ⎟ ⎜ ω z ⎟⎠ ωz ⎠ ⎝ ⎝

2

Ω ωz tanθ z = Ω2 1− 2 ωz − 2ζ z

,

(3.28)

In equations (3.27) and (3.28), the notations (3.8) and

ζy =

c c = , 2 k y m 2mω y

ζz =

c c = 2 k z m 2mω z

(3.29)

have been used. In the following, for the simplicity of presentation, the bearing vertical stiffness is taken four times larger than the horizontal stiffness [2]: k1 = k ,

k2 =

1 k. 4

This gives ky =

2 k, 3

1 kz = k . 3

The system vertical total stiffness is two times the horizontal total stiffness. The external viscous damping coefficient is taken [2]

c = 0.1 ⋅ 2mωn = 0.1 ⋅ 2 k m . The rotor undamped natural frequencies are

ωy =

2 3

k = 0.816 ωn , ω z = m

1 3

k = 0.577ωn . m

The damping ratios are

ζy =

c 0.1 = = 0.12 , 2mω y 0.816

ζz =

c 0 .1 = = 0.17. 2mω z 0.577


111

Using the dimensionless frequency

η=

Ω , ωn

(3.30)

equations (3.27) and (3.28) become

ˆyC = e

η2 2

⎛2 2⎞ 2 ⎜ − η ⎟ + ( 0 .2 η ) 3 ⎝ ⎠ tanθ y =

,

ˆzC = e

η2

(3.31)

2

⎛1 2⎞ 2 ⎜ − η ⎟ + (0.2η ) 3 ⎝ ⎠

− 0.2η − 0.2η , tanθ z = . 2 1 −η 2 −η 2 3 3

(3.32)

Figure 3.5, a illustrates the speed-dependence of the disc unbalance response components, based on equations (3.31). Unlike the curves from Fig. 3.3, plotted for the undamped rotor, finite amplitudes result at the critical speeds:

⎛ ˆyC ⎞ = 4.08 , ⎜ ⎟ ⎝ e ⎠η = 0.816

⎛ ˆzC ⎞ = 2.885. ⎜ ⎟ ⎝ e ⎠η = 0.577

a

b

Fig. 3.5 The speed-dependence of the phase angles θ y and θ z

is illustrated

in Fig. 3.5, b. The speeds where the phase difference is 90o are denoted Ω1∗ and

Ω 2∗ . As shown in the following, the phase difference Δθ = θ z − θ y

yields


112

inclined elliptical orbits for the precession of damped rotors, in contrast with the undamped rotors whose elliptical orbits have vertical and horizontal semiaxes. 3.1.2.2 Disc precession orbit

Substitution of y C and zC by y and z in equations (3.26) yields y = ˆyC cosθ y cos Ω t − ˆyC sinθ y sin Ω t = yc cos Ω t + ys sinΩ t , z = ˆzC cosθ z sin Ω t + ˆzC sinθ z cos Ω t = zc cos Ω t + zs sinΩ t.

(3.33)

Equations (3.33) define an ellipse. Elimination of time gives the orbit equation

( zc2 + zs2 ) y 2 − 2 ( yc zc + ys zs ) y z + ( yc2 + ys2 ) z 2 = ( ys zc − yc zs )2 .

(3.34)

Equation (3.34) is more often expressed in terms of the major and minor semiaxes, a and b, and the inclination angle α . In a principal coordinate frame y1Oz1 , taking the Oy1 and Oz1 axes along the ellipse axes (Fig. 3.6), the motion is described by y1 = a cos (Ω t + γ − α ) , z1 = b sin (Ω t + γ − α ) ,

(3.35)

where γ is the phase angle at t = 0 .

Fig. 3.6 The ellipse equation in principal coordinates is 2

2

⎛ y1 ⎞ ⎛ z1 ⎞ ⎜ ⎟ + ⎜ ⎟ = 1. ⎝a⎠ ⎝b⎠

(3.36)


113

The coordinate transformation y = y1 cosα − z1 sinα ,

(3.37)

z = y1 sinα + z1 cosα , leads to parametric equations of the form (3.33).

Combining equations (3.33), (3.35) and (3.37) it is possible to obtain a, b, and α in terms of yc , y s , zc , z s . The result is

a2 =

1 2 ( yc + y s2 + zc2 + z s2 ) + 2 , 1 2 2 2 2 2 2 + ( yc + y s + z c + z s ) − ( yc z s − y s z c ) 4 b=

(3.38)

1 ( yc z s − y s z c ) , a

tan 2α =

(3.39)

2 ( yc z c + y s z s ) . + y s2 − ( zc2 + z s2 )

(3.40)

yc2

Using notations (3.33), the equations of motion (3.25) become ⎡k y − Ω 2 m 0 ⎢ kz − Ω 2m 0 ⎢ ⎢ −Ωc ky 0 ⎢ ⎢⎣ −Ωc 0

⎤ ⎧ yc ⎥⎪ Ω c ⎥ ⎪ zc 0 ⎥⎨ y − Ω 2m 0 ⎥⎪ s 2 k z − Ω m ⎥⎦ ⎪⎩ z s 0 Ωc

0

⎫ ⎧1⎫ ⎪ ⎪ ⎪ ⎪ 2⎪ 0 ⎪ m e Ω = ⎨ ⎬. ⎬ ⎪ ⎪0⎪ ⎪⎭ ⎪⎩ 1 ⎪⎭

(3.41)

They are two-by-two decoupled. In the considered particular case, equations (3.41) become

⎡(2 3) − η 2 ⎢ 0 ⎢ ⎢ − 0.2 η ⎢ 0 ⎣⎢

0 (1 3) − η 2

0.2 η 0

0 − 0.2 η

(2 3) − η 2 0

⎤ ⎧ yc ⎥⎪ ⎥ ⎪ zc ⎥ ⎨ ys 0 ⎥⎪ 2 ⎪ z (1 3) − η ⎦⎥ ⎩ s 0 0.2 η

The four ellipse parameters are ⎛2 2⎞ 2 ⎜ −η ⎟ η yc ⎝ 3 ⎠ = , e Δ1

y s 0.2η ⋅ η 2 , = e Δ1

⎫ ⎧1⎫ ⎪ ⎪ ⎪ ⎪ 2 ⎪0⎪ η e = ⎨ ⎬. ⎬ ⎪0⎪ ⎪ ⎪⎩ 1 ⎪⎭ ⎪⎭


114

2

zc 0.2η ⋅ η , =− Δ2 e

⎛1 2⎞ 2 ⎜ −η ⎟ η zs ⎝ 3 ⎠ , = e Δ2

where

Δ1 = (2 3 − η 2 ) 2 + (0.2η ) 2 , Δ 2 = (1 3 − η 2 ) 2 + (0.2η ) 2 . The inclination of the major axis is given by tan 2α =

0.4 η . 1 − 2η 2

The minor semiaxis is zero, b = 0 , when z s ys = . z c yc

(3.42)

Equation (3.33) gives ys = − tan θ y , yc

zc = tan θ z , zs and condition (3.42) becomes tan θ z = −

1 tan θ y

or

θz − θ y =

π 2

(3.43)

.

When the phase difference Δθ = θ z − θ y

between the projections of

the precession motion on the axes Oy and Oz is 90 0 , the elliptic orbit degenerates into a straight line. In fact the two motions are in phase and the 90 0 angle shows the spatial lag between the two directions. Condition (3.43) defines the limits between forward and backward precession. On inserting (3.28) into (3.43) we obtain the threshold angular speeds Ω1∗ and Ω 2∗ . Figure 3.5, b shows that there are two speeds at which condition (3.43) holds and these are different from the peak response critical speeds. For undamped rotors (Fig. 3.3), the change from forward to backward precession and vice versa takes place at the system undamped natural frequencies, hence at the undamped critical speeds. For damped rotors, the precession reversal, possible only when the orbit degenerates into a straight line, occurs at speeds which are different from the


115

peak response critical speeds, where the motion components have maximum amplitude.

Fig. 3.7 (from [2]) In the considered particular case, substitution of (3.32) into (3.43) yields

η 4 − 0,96η 2 +

2 = 0, 9

with solutions

η1∗ = 0 ,624 ,

η 2∗ = 0,755.

Figure 3.7 depicts the orbits at several rotor speeds. Figure 3.8 shows the rotor unbalance response presented as diagrams of the ellipse semiaxes as a function of running speed.


116

Peak response critical speeds are located at the peaks in the major semiaxis curve a / e . The minor semiaxis curve, b / e , crosses the speed axis at the threshold speeds between forward and backward precession, Ω1∗ and Ω 2∗ .

Fig. 3.8 3.1.2.3 Decomposition into two circular motions

If the motion along the ellipse is represented as the sum of two counterrotating circular motions, as in equation (3.11, a), then rf =

a+b , 2

rb =

a −b 2

(3.44)

and, unlike the Fig. 3.4, vectors r f and rb have non-zero phase angles at t = 0. Figure 3.9 shows the diagrams of radii r f

and rb as a function of

speed for the analyzed system. Because for r f > rb the precession is forward, and for r f < rb the precession is backward, the intersections of the two curves locate (for r f = rb ) the threshold (dimensionless) speeds η1∗ and η 2∗ . Resuming, the unbalance response can be illustrated by three kinds of frequency response diagrams: a) diagrams of the motion projections y and z onto the coordinate axes (Fig. 3.5, a); b) diagrams of the semiaxes a and b of the elliptic


117

orbit (Fig. 3.8); and c) diagrams of the radii of circular motions which generate the ellipse (Fig. 3.9).

Fig. 3.9 Because the maximum relative displacement between rotor and stator is given by the major semiaxis a, the diagram from Fig. 3.8 is the most useful in practice. It is used together with Fig. 3.7 which represents the evolution of the precession orbit with the change of speed.

3.1.3 Effect of external and internal damping Considering both external and internal damping, the equations of motion of the disc centre with respect to the stationary coordinate system become m &y&C + (ce + ci ) y& C + ci Ω zC + k y yC = m e Ω 2 cos (Ω t + θ 0 ) , m &z&C + (ce + ci ) z&C − ci Ω yC + k z zC = m e Ω 2sin (Ω t + θ 0 ).

(3.45)

Denoting k=

k y + kz

, Δk =

k y − kz

2 2 ci ce ζi = , ζe = , 2mωn 2mωn

, q=

Δk k

,

ωn2 =

ζ = ζe +ζi ,

equations (3.45) can be written in matrix form as

k , m

(3.46)


118

⎧ &y&C ⎫ ⎧ y& C ⎫ ⎡ ωn2 ( 1 + q ) 2 + ζ ω ⎨ ⎬ ⎬+ ⎢ n⎨ ⎩ &z&C ⎭ ⎩ z&C ⎭ ⎣⎢− 2ζ i ωn Ω

2ζ i ωn Ω ⎤ ⎧ yC ⎫ 2 ⎧cos (Ω t + θ 0 ) ⎫ ⎥ ⎨ ⎬ = eΩ ⎨ ⎬. 2 ωn ( 1 − q )⎦⎥ ⎩ zC ⎭ ⎩ sin (Ω t + θ 0 ) ⎭ (3.47)

Denoting

Λ=

Λ , ωn

η=

Ω , ωn

(3.48)

the study of the motion of the perfectly balanced rotor (e = 0) , leads to the characteristic equation

(

)

(

)

Λ 4 + 4ζ Λ 3 + 2 2ζ 2 + 1 Λ 2 + 4ζ Λ + 1 + 4ζ i2 η 2 − q 2 = 0 .

(3.49)

A comparison of equation (3.49) with (2.66) shows a difference only in the last term, due to the stiffness asymmetry coefficient q. Application of the Routh-Hourwitz criterion [1], yields the stability condition 4ζ 2 − 4ζ i2 η 2 + q 2 ≥ 0 .

(3.50)

The onset speed of instability is (Smith, 1933)

Ω s = ωn

⎛ ζ ⎜⎜ 1 + e ζi ⎝

2

⎞ ⎛ q ⎟⎟ + ⎜⎜ ⎠ ⎝ 2ζ i

2

⎞ ⎟⎟ . ⎠

(3.51)

A comparison with equation (2.68) shows that the bearing support stiffness orthotropy can be used to increase the rotor onset speed of instability. For rotors supported in rolling bearings this is achieved with unequal support stiffnesses in two directions, while for hydrodynamic bearings - by increasing the eccentricity ratio. The physical explanation of the effect of bearing stiffness orthotropy in restraining instability due to rotating damping is that “since the natural frequencies of the rotor system are different in the two principal transverse directions, there is no tendency to set up a whirl of the type which can be dragged forward by rotating damping until the rotating damping forces have been so far increased by rising speed that they are commensurate with the difference between elastic restoring forces in the two principal directions”. Analysis of the unbalance response reveals that, with asymmetrical bearing stiffness, the amplitude of steady motion due to unbalance is restricted by both internal and external damping, but internal damping has smaller influence in this respect, especially if there is only slight dissymmetry of bearing stiffness.


119

3.1.4 Effect of bearing damping In order to reveal the effect of bearing damping on the dynamics of rotors, the rotor from Fig. 3.1 is supported in damped isotropic bearings as in Fig. 3.10. The bearings are assumed to have the same stiffness constant k1 in all radial directions. The bearing damping forces are assumed to be proportional to the journal absolute velocity. The viscous damping coefficients c are the same in all radial directions [3].

Fig. 3.10 At constant running speed Ω =const., the equations of motion are written as for undamped bearings (see § 3.1) but adding the damping forces. For the shaft: 2 c y& B + 2k1 y B = k ( yC − y B ), 2 c z& B + 2k1 z B = k ( zC − z B )

(3.52)

and, for the disc m &y&G + k ( yC − y B ) = 0 , m &z&G + k ( zC − z B ) = 0,

(3.53)

where yG = yC + e cos Ω t ,

(3.54)

zG = zC + e sin Ω t . Using complex notation rB = y B + i z B ,

rC = yC + i zC ,

rG = yG + i zG ,

(3.55)


120

equations (3.52)-(3.54) produce the equations of motion of the disc centre and journal centre 2c r&B + 2k1rB + k (rB − rC ) = 0, m &r&C + k (rC − rB ) = m e Ω 2ei Ω t .

(3.56)

The natural frequency ωn of the rotor in rigid bearings and the ratio N between the shaft stiffness and the support (bearings in parallel) stiffness are

ωn =

k , m

N=

k . 2k 1

(3.57)

The damping ratio

ζ =

2c 2c = 2 mωn 2 k m

(3.58)

is defined with respect to the critical damping of the rigidly supported rotor. The resulting equations of motion are 2 ζ ωn r&B +

1 2 ωn rB + ωn2 (rB − rC ) = 0, N

(3.59)

&r&C + ωn2 (rC − rB ) = e Ω 2 ei Ω t . 3.1.4.1 Damped natural frequency

For zero right-hand side in (3.59) and substituting solutions of the form

rB = RB e λ t ,

rC = RC e λ t ,

(3.60)

we obtain the homogeneous algebraic set of equations 1 2 ⎛ 2⎞ 2 ⎜ 2ζω n λ + ωn + ωn ⎟ RB − ωn RC = 0 , N ⎝ ⎠ − ωn2

(

2

RB + λ

+ ωn2

)R

C

= 0.

The requirement for non-trivial solutions is ⎛1 ⎞ 2ζω n λ + ω n2 ⎜ + 1⎟ − ω n2 ⎝N ⎠ 2 λ2 + ω n2 − ωn

= 0.

(3.61)


121

This yields the characteristic equation 3

2

⎛ λ ⎞ λ N +1 ⎛ λ ⎞ 1 ⎟⎟ + ⎜⎜ ⎟⎟ + ⎜⎜ + = 0. 2 ζ N ⎝ ωn ⎠ ωn 2ζ N ⎝ ωn ⎠

If ζ = 0 , then

(3.62)

λ 1 . The critical speed of the rotor supported in =i ωn N +1

undamped flexible bearings is

ωel =

ωn N +1

.

(3.63)

If ζ ≠ 0 , then equation (3.62) has positive coefficients and can be written ⎞ ⎛ λ ⎞⎛ λ2 λ ⎜⎜ + A ⎟⎟ ⎜ 2 + 2 B + B 2 + C 2 ⎟ = 0. ⎜ ⎟ ωn ⎝ ωn ⎠ ⎝ ωn ⎠

(3.64)

There is a negative real root ( λ ωn )1 = − A and two complex conjugate roots with negative real part ( λ ωn ) 2 ,3 = − B ± iC , so that the system motion is always stable. The free damped motion of point C is described by a solution of the form rC (t ) = RC1 e − Aω n t + RC 2 e − Bω n t ei Cω n t + RC3 e − Bω n t e −i Cω n t .

(3.65)

The frequency of the damped free precession (of the perfectly balanced rotor) is

ω d = Cω n

(3.66)

where C is the imaginary part of the complex roots of the characteristic equation (3.62). 3.1.4.2 Unbalance response

For the steady-state motion due to mass unbalance, the solutions are of the form

rB (t ) = ~ rB eiΩ t ,

rC and ~ rB are complex amplitudes. where ~

rC (t ) = ~ rC eiΩ t ,

(3.67)


122

Using the dimensionless frequency (3.30) and substituting (3.67) into (3.59) we obtain ⎡⎛ 1 ⎤ ~ ~ ⎞ ⎢⎜ N + 1⎟ + i 2ζη ⎥ rB − rC = 0, ⎠ ⎣⎝ ⎦ ~ − r + 1 −η 2 ~ r = eη 2 . B

(

)

(3.68)

C

The solutions are ~ rB =

eη 2

(

1 ⎛1 ⎞ − ⎜ + 1⎟ η 2 + i 2ζη 1 − η 2 N ⎝N ⎠

)

(3.69)

,

⎛1 ⎞ 2 3 ⎜ + 1⎟ η + i 2ζη N ⎝ ⎠ ~ rC = e . 1 ⎛1 Ω ⎞ − ⎜ + 1⎟ η 2 + i 2ζ 1 −η 2 ωn N ⎝N ⎠

(

(3.70)

)

The motion of the journal centre B in the complex plane (Fig. 3.11) is represented by the vector O B .

Fig. 3.11 Its magnitude is ~ rB =

eη 2 2

(3.71)

(

N +1 2 ⎞ ⎛ 1 η ⎟ + 4ζ 2η 2 1 − η 2 ⎜ − N N ⎝ ⎠

)


123

and the phase shift with respect to the unbalance vector C G is θ B = tan −1

(

)

− 2ζ η 1 − η 2 . 1 N +1 2 η − N N

(3.72)

The motion of the disc centre C is represented by the vector O C , of magnitude

~ rC =

e

( N + 1)2 4 η + 4ζ 2η 6 2 N 2

(3.73)

(

⎛ 1 N +1 2 ⎞ η ⎟ + 4ζ 2η 2 1 − η 2 ⎜ − N ⎝N ⎠

)

2

and phase angle θ C = tan −1

1 ⎛ ⎞ − 2 ζ η ⎜1 − η2⎟ N +1 ⎝ ⎠ . N +1 2 ⎞ N ⎛ 1 2 − η ⎟ + 4ζ η 2 1−η 2 ⎜ N N +1 ⎝N ⎠

(

)

(3.74)

The points B and C have circular orbits around the point O, but the points O, B, C and G are not collinear. The vector O C has a phase lag θ C with respect to the excitation vector C G and the vector O B has a phase lag θ B with respect to C G (Fig. 3.11). If the radii of precession orbits (3.71) and (3.73) are plotted against the dimensionless speed Ω ωn , for given values of N and ζ , the peak values of the displacements of points B and C occur at speeds Ω B , and Ω C respectively, different from ωel and ωd (Fig. 3.12). Differentiating with respect to η 2 the expressions of these displacements, yields two different equations. The condition of maximum journal displacement d ~ rB = 0 2 d (η ) gives the equation ⎡ 1⎛ 1 ⎞⎤ 1 2ζ 2η 6 − ⎢2ζ 2 − ⎜1 + ⎟⎥ η 2 − 2 = 0, N⎝ N ⎠⎦ N ⎣

(3.75)


124

Fig. 3.12 (from [3]) and the condition of maximum disc centre displacement d ~ rC = 0 d (η 2 ) gives the equation ⎡ − 16 ζ 4η 6 + 4 ζ 2 ⎢ 4 ζ ⎣

2

⎡⎛ 2 4 ⎞ + ⎢⎜1 + + 2 ⎟ 2ζ N N ⎠ ⎢⎣ ⎝

2

−

2 ⎛ 1 ⎞⎤ 4 ⎜1 + ⎟ η + N ⎝ N ⎠ ⎥⎦

3 2 1 ⎛ 1 ⎞ ⎤ 2 1 ⎛ 1 ⎞ − ⎜1 + ⎟ ⎥ η + 2 ⎜1 + ⎟ = 0. N ⎝ N ⎠ ⎥⎦ N ⎠ N ⎝

(3.76)


(Ω B

125

The physically acceptable solution of equation (3.75) is denoted ωn ) and that of equation (3.76) is denoted (Ω C ωn ) . Generally, the different critical speeds are in the following order

ωel < ωa < Ω B < Ω C < ωn . If the damping in the shaft is taken into account, it is possible that

Ω C > ωn .

The angular speeds Ω C and Ω B , at which the radial displacements of the damped rotor have maximum values, are referred to as peak response critical speeds. Sometimes, they considerably differ from the critical speed ωel of the undamped system, being much larger. Therefore, a computation neglecting the bearing damping can result in erroneous values of the critical speeds.

Example 3.1 Consider the rotor from Fig. 3.10, with the following characteristics: disc mass m = 500 kg, shaft stiffness constant k = 2 ⋅10 5 N/mm, bearing stiffness

k1 = 105 N/mm, bearing viscous damping coefficient c = 316.225 Ns/mm [3]. The computations yield:

N =1, ζ =1,

ωn =

k m = 632.45 rad/sec,

ω el = ω n / 2 = 447.2 rad/sec. The critical speed of the rigidly supported rotor is

nn = 6040 rpm . The undamped critical speed of the rotor supported in flexible bearings is

nel = 4270 rpm . Equation (3.62) is written as (λ ωn )3 + (λ ωn ) 2 + ( λ ωn ) + 0.5 = 0 and the roots are


126

λ 1 ωn = − 0.6478 ,

λ 2 ,3 ωn = −0.1761 ± i 0.8607.

From the imaginary part we obtain

ωd = 0.8607ωn = 544.35 rad/sec. The rotor damped critical speed is

nd = 5198 rpm . Equation (3.75) is written 2 (Ω ω n ) 6 − 1 = 0 , hence

Ω B = 0.8909 ωn = 563.4 rad/sec. The peak response critical speed computed from the journal unbalance response is nB = 5380 rpm = 1.26 nel . Equation (3.76) is written

− 8 (Ω ω n ) 6 + 3 (Ω ω n ) 2 + 2 = 0 , hence

Ω C = 0.9076 ωn = 574 rad/sec. The peak response critical speed computed from the disc centre unbalance response is

nC = 5481 rpm = 1.28 nel . 3.1.4.3 Equivalent model

The dependence of critical speeds on the bearing damping can be simply explained noticing that the rotor-bearing system can be represented by the simplified model from Fig. 3.13. The mass m is supported by a spring of stiffness constant k (representing the shaft), connected in series to an element consisting of the dashpot of constant 2c and the spring 2 k1 connected in parallel (representing the bearings). The mass is acted upon by a force F ( t ) = m e Ω 2 ei Ω t .


127

Fig. 3.13 Two limit cases are first considered. If the damping coefficient is infinite, c = ∞ , the lower spring is blocked and the journal displacement is zero, rB = 0. The frequency response curve of the disc centre (Fig. 3.14) has a(n infinite) peak at the natural frequency of the system consisting of the mass and the upper spring, ωn = k m . The case corresponds to the rigidly supported rotor.

Fig. 3.14 If the damping coefficient is zero, c = 0 , the springs of stiffness constants k and 2k1 = k / N are connected in series, and the equivalent spring rate is k ( N + 1 ).


128

The frequency response curve has a(n infinite) peak at the undamped critical speed ωel = ωn N + 1 . The case corresponds to the rotor supported in undamped flexible bearings. For intermediary values of the bearing damping coefficient, denoted c', and c", respectively, the frequency response curves have peaks at the peak ′ , and Ωcr ′′ , respectively, within the range [ωel , ωn ] . response critical speeds Ωcr They correspond to the rotor supported in damped flexible bearings.

Fig. 3.15 There is an optimum value copt of the bearing damping coefficient, for which the maximum precession amplitude has the lower value, equal to the ordinate of the crossing point of all frequency response curves, plotted for different values of the bearing damping coefficient. The model from Fig. 3.13 can be replaced by an equivalent model, having a single spring in parallel with a dashpot (Fig. 3.15). The equivalent stiffness constant kech and the equivalent viscous damping coefficient cech can be expressed in terms of the parameters k, c and N of the initial system as follows:


129

k2 ( N + 1) + 4 Ω 2 c 2 2 N kech = k , 2 2 ( N + 1) 2 2 k + 4Ω c N2 2c k 2 cech = . 2 2 ( N + 1) 2 2 k + 4Ω c N2

(3.77)

(3.78)

Figure 3.15 shows the variation of these quantities as a function of the damping coefficient c. The stiffness constant kech increases with c, hence the natural frequency kech m also increases with c, fact that explains the increase of critical speeds with the bearing damping. The stiffness increase is higher when the natural frequencies ωel and ωn are relatively more distanced, hence when the ratio N (of the shaft stiffness to the bearing stiffness) is larger. The equivalent viscous damping coefficient cech has a maximum value for the optimal c, fact that explains the lowest value of the maximum amplitude in this case.

Example 3.2 Consider a rigid rotor ( k → ∞ ) supported by identical orthotropic bearings with the following characteristics: ny =

30

ky

π

m

cy 2 kym

=

= 600 rpm, nz =

1 , 16

30

π

kz = 500 rpm, m

cz 1 . = 20 2 kzm

Plot the unbalance response diagrams and several precession orbits for an eccentricity e = 10 μm [4]. Figure 3.16, a shows the plot of major and minor semiaxes a and b, and forward and backward circle radii r f and rb as a function of speed. Figure 3.16, b shows the plot of the y and z displacement components and the minor semiaxis b versus speed, as well as the precession orbit at eight different speeds. The range with backward precession is marked by the threshold speeds

n1∗

and

n2∗ ,

where the orbit degenerates into straight lines.


130

a

b Fig. 3.16 (from [4])


131

3.1.5 Combined effect of bearing damping and shaft mass In the following, the effect of bearing damping on the dynamics of rotors is analyzed taking into account the shaft distributed mass. The analysis is simplified lumping the shaft mass at the ends of each half. This results in a three mass model, with a quarter of shaft mass at each bearing and half of the shaft mass at the disc location (Fig. 3.17). At constant running speed Ω =const., the equations of motion are written as for undamped bearings (see § 3.1) but adding the damping forces. For the shaft: 2 c y& B + 2 k1 y B + 2 m1 &y&B = k ( yC − y B ), 2 c z& B + 2 k1 z B + 2m1 &z&B = k ( zC − z B )

(3.79)

and, for the disc 2m1 &y&C + m &y&G + k ( yC − y B ) = 0 , 2m1 &z&C + m &z&G + k ( zC − z B ) = 0, where

yG = yC + e cos Ω t , zG = zC + e sin Ω t .

(3.80)

(3.81)

Fig. 3.17 Using the complex notation (3.55), equations (3.79)-(3.81) yield 2m 1 &r&B + 2c r&B + 2k1rB + k ( rB − rC ) = 0,

( m + 2m 1 ) &r&C + k ( rC − rB ) = m e Ω 2ei Ω t .

(3.82)


132

Denoting the ratio of half the shaft mass to the disc mass

μ=

ms 2 m 1 = , 2m m

(3.83)

and using (3.57) and (3.58), we obtain the equations of motion

μ &r&B + 2 ζ ωn r&B +

(1 + μ )

1 2 ωn rB + ωn2 (rB − rC ) = 0 , N

&r&C + ωn2

2 iΩt

( rC − rB ) = e Ω e

(3.84)

.

3.1.5.1 Damped natural frequency

For zero right-hand side in (3.84) and substituting solutions of the form (3.60), we obtain the homogeneous algebraic set of equations 1 2 ⎛ 2 2⎞ 2 ⎜ μ λ + 2ζω n λ + ωn + ωn ⎟ RB − ωn RC = 0 , N ⎝ ⎠ − ωn2

RB +

[

ωn2

+ ( 1 + μ )λ

2

]R

C

(3.85)

= 0.

The requirement for non-trivial solutions yields the characteristic equation 4

⎛ λ ⎞ μ ( 1 + μ ) ⎜⎜ ⎟⎟ + 2ζ ( 1 + μ ⎝ ωn ⎠

3

2

⎞ ⎞ ⎡ ⎛ ⎤⎛ )⎜⎜ λ ⎟⎟ + ⎢ N + 1 ( 1 + μ ) + μ ⎥ ⎜⎜ λ ⎟⎟ + 2ζ λ + 1 = 0. ωn N ⎦ ⎝ ωn ⎠ ⎝ ωn ⎠ ⎣ N (3.86)

We denote

λ = α + i ωd ,

(3.87)

where α is a negative attenuation factor and ωd is the damped natural frequency. Two particular cases are considered: N = 1 , μ = 1 , and N = 2.5 , μ = 1 , respectively, both corresponding to a relatively heavy shaft. Root locus diagrams are presented in Fig. 3.18, using dimensionless coordinates α ωn and ωd ωn . For the rotor with N = 1 (Fig. 3.18, a), the first damped natural frequency ωd 1 increases from ωel 1 = 0.4682 ωn (for ζ = 0 ) to ω rig = 0.707ω n (for ζ = ∞ ). The second natural frequency ωd 2 decreases from ωel 2 = 1.5102 ωn (for ζ = 0 ) to zero (for ζ ≅ 1.325 ) when the second mode becomes overdamped.


a

133

b

Fig. 3.18 (from [5])

Fig. 3.19 (from [5])


134

The rotor with N = 2.5 (Fig. 3.18, b) has a different behaviour. The second damped natural frequency, ωd 2 , decreases first from ωel 2 = 1.3372 ωn (for

ζ = 0 ) to 0.676 ωn (for ζ = 1 ) then increases to ω rig = 0.707ω n (for ζ = ∞ ). The first damped natural frequency ωd 1 diminishes from ωel 1 = 0.3344 ωn (for ζ = 0 ) to zero (for ζ ≅ 0.98 ) when the first precession mode becomes overdamped. Figure 3.19 shows (for μ = 1 ) the variation of the dimensionless damped natural frequencies ωd 1 ωn (solid lines) and ωd 2 ωn (broken lines) as a function of the bearing stiffness, for different values of bearing damping. Generally, the increase of ζ and 1 N makes the first mode overdamped for values ζ < 1 , and the second mode overdamped for ζ > 1 . For N = 2 and ζ = 1 the system has equal eigenvalues. 3.1.5.2 Unbalance response

For the steady-state motion due to mass unbalance, substituting the solutions (3.67) into (3.84), we obtain the algebraic set of equations 1 2 ⎛ 2 2⎞ rB − ωn2 ~ rC = 0, ⎜ − μ Ω + i 2ζ ωn Ω + ωn + ωn ⎟ ~ N ⎝ ⎠ − ω2 ~ r + ω 2 − ( 1 + μ )Ω 2 ~ r = eΩ 2 . n B

[

n

]

(3.88)

C

The solutions to (3.88) are

e ~ rB =

Ω2 ωn2

,

(3.89)

2 ⎞ ⎛1 ⎜ + 1 − μ Ω + i 2ζ Ω ⎟ ⎜N ωn2 ωn ⎟⎠ ⎝ ~ rC = . 2 2⎤ ⎛1 ⎞ ⎡ Ω Ω Ω ⎜ +1− μ ⎟ ⎢1 − ( 1 + μ ) + i 2ζ ⎥ −1 ⎜N ωn ⎟⎠ ⎣⎢ ωn2 ωn2 ⎦⎥ ⎝

(3.90)

2 2⎤ ⎛1 ⎞ ⎡ ⎜ + 1 − μ Ω + i 2 ζ Ω ⎟ ⎢1 − ( 1 + μ ) Ω ⎥ − 1 ⎜N ωn ⎟⎠ ⎢⎣ ωn2 ωn2 ⎥⎦ ⎝

e

Ω2 ωn2

Figures 3.20 illustrate the variation of ~ rC e and ~ rB e as a function of

Ω ωn , for μ = 1 and N = 2.5 . With increasing ζ , the first response peak is shifted to higher speeds, while the second response peak is shifted to lower speeds.


135

The peaks of the unbalance response diagrams occur at the angular speeds Ω B and Ω C . The peak response speeds Ω B are obtained from condition d ~ r d (Ω ω ) 2 = 0 . The peak response speeds Ω are obtained from condition B

d ~ rC

n

C

d (Ω ωn ) = 0 . 2

For

μ =1,

N =1

and

ζ = 0.4 , we obtain

Ω B 1 = 0.4851 ωn ,

Ω B 2 = 1.3745 ωn , Ω C 1 = 0.4837 ωn , Ω C 2 = 1.5589 ωn . For these values of system parameters, the different critical angular speeds can be ordered as follows

ωel 1 < ωd 1 < Ω C 1 < Ω B 1 < ωrig < Ω B 2 < ωd 2 < ωel 2 < Ω C 2 . ωd 2

For other values of N, μ and ζ , the order can be different. For ζ = 1.3 , < ωd 1 .

a

b

Fig. 3.20 (from [5]) For relatively high damping, the angular speeds ωd 1 and Ω B 1 tend to

ωrig . This explains why the measured critical speeds are nearer those calculated for the rigidly supported rotor than those determined for the rotor on undamped flexible bearings.


136

3.2 Symmetric rotors in fluid film bearings As mentioned earlier, the linear theory of hydrodynamic bearings allows expressing the force exerted by the lubricant film on the rotor journal, resolved into two components f B y , f B z along the coordinate axes, under the form [6]

⎧ f B y ⎫ ⎡ k y y k y z ⎤ ⎧ y B ⎫ ⎡ c y y c y z ⎤ ⎧ y& B ⎫ ⎨f ⎬=⎢k ⎥ ⎨ ⎬ , (3.91) ⎥⎨ ⎬ + ⎢ ⎩ B z ⎭ ⎣ z y k z z ⎦ ⎩ z B ⎭ ⎣ c z y c z z ⎦ ⎩ z& B ⎭ where y B , z B are the projections along the axes of the fixed coordinate frame of the journal centre displacement, and y& B , z& B are the corresponding velocities. For many types of radial bearings, the stiffness matrix is non-symmetric, k yz ≠ k zy . It is not possible to determine stiffness principal directions, with respect to which the off-diagonal elements of the stiffness matrix vanish. The bearings are anisotropic, k yy ≠ k zz , and the stiffness matrix non-symmetry produces unstable precession motions. In the following, only Laval-Jeffcott rotors are considered, neglecting the disc rotary inertia. The bearing damping matrix is generally symmetric, c yz = c zy .

3.2.1 Unbalance response Consider a Laval-Jeffcott rotor as in Fig. 3.10, but supported in hydrodynamic bearings, characterized by the eight dynamic coefficients defined by equation (3.91). The steady-state motion produced by the disc unbalance is examined. The equilibrium equations for the shaft are 2 f B y = k ( yC − y B ), 2 f B z = k ( z C − z B ),

(3.92)

and the motion equations for the disc are

m &y&G + k ( yC − y B ) = 0 , m &z&G + k ( zC − z B ) = 0, where

(3.93)


137

yG = yC + e cos Ω t,

(3.94)

zG = zC + e sin Ω t. Substituting (3.94) into (3.93) and (3.91) into (3.92) we obtain m &y&C + k ( yC − y B ) = m e Ω 2 cos Ω t,

(3.95)

m &z&C + k ( zC − z B ) = m e Ω 2 sin Ω t ,

and

k ( yC − y B ) = k y y y B + k y z z B + c y y y& B + c y z z& B , 2 k ( zC − z B ) = k z y y B + k z z z B + c z y y& B + c z z z& B . 2

(3.96)

The steady-state solutions have the form y B (t ) = A cos Ω t + B sin Ω t,

(3.97)

z B (t ) = E cosΩ t + F sin Ω t ,

(3.98)

yC (t ) = C cosΩ t + D sin Ω t ,

(3.99)

zC (t ) = G cosΩ t + H sin Ω t .

(3.100)

On inserting expressions (3.97) and (3.99) into the first equation (3.95) and identifying the coefficients of the terms in cos Ω t and sin Ω t , we obtain a non-homogeneous algebraic set of equations, in which C and D are expressed in terms of A and B. Substitution into (3.99) yields yC =

Aωn2 + eΩ 2

ωn2 − Ω 2

cos Ω t +

Bωn2

ωn2 − Ω 2

sin Ω t ,

(3.101)

where

ωn =

k m

(3.102)

is the critical speed of the rigidly supported rotor. Analogously, inserting (3.98) and (3.100) into the second equation (3.95), identifying the coefficients and solving the algebraic equations, G and H are expressed in terms of E and F, yielding


138 zC =

Eωn2 Fωn2 + eΩ 2 cos t sinΩ t , + Ω ωn2 − Ω 2 ωn2 − Ω 2

(3.103)

The solutions (3.97), (3.98), (3.101) and (3.103) are then substituted into (3.96). Identifying the coefficients of the terms in cosΩ t and sinΩ t , we obtain the algebraic set of equations (k y y − χ ) A + Ω c y y B + k y z E + Ω c y z F = χ e , − Ω c y y A + (k y y − χ ) B − Ω c y z E + k y z F = 0,

(3.104)

k z y A + Ω c z y B + (k z z − χ ) E + Ω c z z F = 0, − Ω c z y A + k z y B − Ω c z z E + (k z z − χ ) F = χ e , where

χ=

k Ω2 . 2 ωn2 − Ω 2

(3.105)

In the following, in order to simplify the solution, the eight dynamic bearing coefficients are reduced to four coefficients k y , k z , c y , c z , defined by the equations:

⎧ f B y ⎫ ⎡k y 0 ⎤ ⎧ y B ⎫ ⎡c y 0 ⎤ ⎧ y& B ⎫ ⎨f ⎬=⎢ ⎥⎨ ⎬ + ⎢ ⎥⎨ ⎬ ⎩ B z ⎭ ⎣ 0 k z ⎦ ⎩ z B ⎭ ⎣ 0 c z ⎦ ⎩ z& B ⎭

(3.106)

The four equations (3.104) become ( k y − χ ) A + Ω c y B = χ e, − Ω c y A + ( k y − χ ) B = 0, ( k z − χ ) E + Ω cz F = 0,

(3.107)

− Ω c z E + ( k z − χ ) F = χ e. It is necessary now to establish the relationships between the four equivalent bearing coefficients and the eight dynamic coefficients defined by (3.91). In (3.104) the second equation is added to the third and the fourth equation is subtracted from the first equation. This yields


139

( k z y − Ω c y y ) A + ( k y y − χ + Ω cz y ) B = = −( k z z − χ − Ω c y z ) E − ( k y z + Ω c z z ) F , ( k y y − χ + Ω cz y ) A − ( k z y − Ω c y y ) B =

(3.108)

= −( k y z + Ω c z z ) E + ( k z z − χ − Ω c y z ) F . Solving equations (3.108) in terms of A and B we obtain E=−

μ A +ν B , χy

F=

ν A−μ B χy

(3.109)

where

μ = ( kz y − Ω cy y ) ( kz z − χ − Ω cy z ) + + ( k y y − χ + Ω cz y ) ( k y z + Ω cz z ),

(3.110)

ν = ( k y y − χ + Ω cz y ) ( k z z − χ − Ω c y z ) − − ( k z y − Ω c y y ) ( k y z + Ω cz z ),

χ y = ( k z z − χ − Ω c y z )2 + ( k y z + Ω cz z )2 . Putting (3.109) into the first (second) equation (3.104), by identification to the first (second) equation (3.110), we find

k y = k yy −

1

χy

Ω c y = Ω c yy −

( μ k yz − ν Ω c yz ) , 1

χy

(3.111) (ν k yz + μ Ω c yz ).

Solving equations (3.108) in terms of E and F yields A=

ν F −μE , χz

B=−

μ F +ν E , χz

(3.112)

where

χz =

μ 2 +ν 2 = ( k yy − χ − Ω c zy ) 2 + ( k zy + Ω c yy ) 2 . χy

(3.113)

Substituting (3.112) into the third (or the fourth) equation (3.104), by identification to the third (or the fourth) equation (3.107), we obtain


140

k z = k zz −

1

χz

Ω c z = Ω c zz +

( μ k z y + ν Ω cz y ), 1

χz

(3.114) (ν k z y − μ Ω c z y ).

Equations (3.111) and (3.114), together with (3.110) and (3.113), allow the reduction of the eight bearing coefficients to only four, equations (3.107) being coupled two by two. Solving equations (3.107), the coefficients A, B, E, F are determined as: A=

eχ ( k y − χ ) 2

( k y − χ ) + (Ω c y )

E=−

2

,

B=

eχ Ω c y ( k y − χ ) 2 + (Ω c y ) 2

, (3.115)

eχ Ω c z eχ ( k z − χ ) , F= . (3.116) 2 2 ( k z − χ ) + (Ω c z ) ( k z − χ ) 2 + (Ω c z ) 2

The solutions (3.97) and (3.98) define the journal centre motion and can be written as y B (t ) = ˆy B cos (Ω t + θ y B ) ,

(3.117)

z B (t ) = ˆz B sin (Ω t + θ z B ) , where ˆy B = tan θ y B

A2 + B 2 =

eχ (k y − χ ) 2 + (Ω c y ) 2

, (3.118, a)

Ω cy B , =− =− A ky − χ

and ˆz B = E 2 + F 2 = tan θ z B

eχ (k z − χ ) 2 + (Ω c z ) 2

,

E Ω cz = =− . F kz − χ

(3.118, b)

The parametric equations (3.117) define an ellipse. Eliminating the time, the orbit equation is obtained as (3.34) ( E 2 + F 2 ) y B2 − 2 ( AE + BF ) y B z B + ( A2 + B 2 ) z B2 − ( AF − EB ) 2 = 0.


141

The distance from the origin of the stationary coordinate frame O to the journal centre B is represented in the complex plane by the vector O B = rB , which can be written as the sum of two counter-rotating vectors

rB = y B + i z B = ( A + i E ) cos Ω t + ( B + i F ) sin Ω t = E − B ⎞ iΩ t ⎛ A − F E + B ⎞ −i Ω t ⎛ A+ F = r f ei Ω t + r b e − i Ω t . +i =⎜ +i ⎟e ⎟e +⎜ 2 2 2 2 ⎝ ⎠ ⎝ ⎠ (3.119) The magnitudes of the two components are rf =

1 2

( A + F )2 + (E − B )2 =

1 2

A2 + B 2 + F

2

+ E 2 + 2 AF − 2 EB ,

(3.120)

1 2

A2 + B2 + F

2

+ E 2 − 2 AF + 2 EB .

(3.121)

respectively rb =

1 2

( A − F )2 + (E + B )2 =

They rotate in opposite directions with the same speed Ω . As shown in sections 3.1.1.2 and 3.1.2.2, the end of vector rB moves along an elliptic orbit, of major semiaxis a = r f + rb

(3.122)

b = r f − rb

(3.123)

and minor semiaxis

where a and b are functions of the running speed Ω . The inclination of the major axis on the Oy axis is defined by (3.40) tan 2α =

2(AE + B F) . ( A + B2 ) − (E 2 + F 2 ) 2

(3.124)

If b > 0 , the journal centre has a forward precession, and if b < 0 , it has a backward precession. Similar conclusions are obtained from the analysis of the motion of point C, the disc geometric centre. The forces acting on the bearing supports have the following components


142

f B y = k y y B + c y y& B = = eχ

( )2 ( k y − χ )2 + ( Ω cy )2 k y2 + Ω c y

)

(3.125, a)

sin ( Ω t + θ zB + φ z ) ,

(3.125, b)

(

cos Ω t + θ yB + φ y ,

f B z = k z z B + cz z& B = = eχ

k z2 + ( Ω c z

)2

( k z − χ ) 2 + ( Ω cz ) 2

where tan φ y =

Ω cy ky

tan φ z =

,

Ω cz kz

.

(3.126)

3.2.2 Stability of precession motion Experience has shown that the rotor synchronous precession in hydrodynamic bearings becomes unstable at a given value Ω s of the running speed, when the orbit radius has a sudden increase. Analytically, this is studied using the equations of motion (3.95) and (3.96) for the perfectly balanced rotor (e = 0) . For e = 0 , hence for yG = yC , equations (3.95) become m &y&C + k yC = k y B , m &z&C + k zC = k z B .

(3.127)

The solutions have the form yC = YC eν ω n t ,

zC = Z C eν ω n t ,

y B = YB eν ω n t ,

z B = Z B eν ω n t ,

(3.128)

where

ωn =

k m

(3.129)

is the critical angular speed of the rigidly supported rotor (3.102). Substituting solutions (3.128) into equations (3.127) yields YC =

1 YB , 1 +ν 2

ZC =

1 ZB . 1 +ν 2

(3.130)


143

Inserting (3.128) and (3.130) into (3.96) we obtain the homogeneous algebraic equations ( X + k y y + c y y ν ωn ) YB + (k y z + c y z ν ωn ) Z B = 0 , (k z y + c z y ν ωn ) YB + ( X + k z z + c z z ν ωn ) Z B = 0 ,

(3.131)

where X=

k ν2 . 2 1 +ν 2

(3.132)

At the stability threshold, ν is pure imaginary. Substituting

ν =iΛ

(3.133)

into equations (3.131), the requirement for non-trivial solutions produces X + k y y + i Λ c y y ωn k z y + i Λ c z y ωn

k y z + i Λ c y z ωn =0. X + k z z + i Λ c z z ωn

(3.134)

Canceling the real and the imaginary parts of the determinant (3.134) gives the equations X

2

+ (k y y + k z z ) X + (k y y k z z − k z y k y z ) − − Λ 2ω n2 ( c y y c z z − c y z c z y ) = 0 ,

(3.135)

ω n Λ [ X (c y y + c z z ) − − (c z y k y z + c y z k z y − c y y k z z − c z z k y y ) ] = 0 ,

which can also be written as

Λ

2

ω n2

=

X =

X

2

+ (k y y + k z z ) X + (k y y k z z − k z y k y z ) c y ycz z − c y zcz y

(c z y k y z + c y z k z y ) − (c y y k z z + c z z k y y ) c y y + cz z

,

(3.136) ,

where X=

k Λ2 . 2 Λ2 −1

It is useful to use the dimensionless coefficients

(3.137)


144

2 So ΔR 2 So ΔR Ω, (3.138) , Ci j = ci j mg mg where So is the inverse of the usual Sommerfeld number S [6], ΔR is the bearing clearance (difference between the bearing radius and the journal radius) and g is the acceleration of gravity. K i j = ki j

Equations (3.136) become Ω s2 = Λ 2ω n2 = =

X=

X 2 + (K y y + K z z ) X + (K y y K z z − K z y K y z ) C y yC z z − C y zC z y

Ω 2,

m g ( Cz y K y z + C y z K z y ) − ( C y y K z z − Cz z K y y ) . 2 So ΔR C y y + Cz z

(3.139)

(3.140)

The onset speed of instability Ω s can be computed using an iterative approach. A value Ω is first selected. The corresponding Sommerfeld number So and the eight bearing coefficients are then computed. They are given in tabular or graphic form, as functions of So for given values of the bearing clearance and length-to-diameter ratio (see Chapter 6).

Fig. 3.21 Equation (3.140) delivers X, which is inserted into (3.139), wherefrom Ω s is obtained. If Ω < Ω s , a new value Ω is considered and the computations are repeated until Ω = Ω s . The results are plotted as in Fig. 3.21 [7].


145

A more detailed analysis of the stability of precession for rotors supported in hydrodynamic bearings is presented in Chapter 7.

3.3 Asymmetric rotors in flexible bearings The previous sections 3.1 and 3.2 examined single-disc Laval-Jeffcott rotors, considering only the disc motion in the rotor plane of symmetry, hence neglecting the effect of disc rotary inertia. In the following, the two single-disc rotor models from Table 3.1 are analyzed. The bearings are orthotropic and dissimilar.

3.3.1 Equations of motion Consider the asymmetric rotor from Fig. 3.22, supported in orthotropic flexible bearings.

Fig. 3.22 In this case, the flexibility influence coefficients δ ij in equations (2.78) are different from those used in (2.79).


146

The force-deflection equations can be written: ⎧y ⎪z ⎪ ⎨ ⎪ϕ ⎪⎩ ψ

⎡ ⎫ ⎢ ⎪ ⎪ ⎢ = ⎬ ⎢ ⎪ ⎢ ⎪⎭ C ⎣

δ11 0 0

δ 41

0

0

δ 22 δ 23 δ 32 δ 33 0

0

δ14 ⎤ ⎧ Fy ⎫

0 ⎥⎥ ⎪⎪ Fz ⎪⎪ ⎬ ⎨ 0 ⎥ ⎪My ⎪ ⎥ δ 44 ⎦ ⎪⎩ M z ⎪⎭C

(3.141)

Table 3.1 Model I

δ11 =

Model II

l3 2 2 β 2 α 2 + α β + 3EI k A1 k B1

δ11 =

γ 2 ( 1 + γ )2 l3 2 γ (1 + γ ) + + 3EI k A1 k B1

β α l 2αβ (β − α ) + − 3EI k A1l k B1l

δ14 = −

γ 1+ γ l 2γ ( 2 + 3γ ) − − 6 EI k A1l k B1l

1 1 l + (α3 + β 3) + 2 3EI kA1l kB1l 2

δ 44 =

l 1 1 (1 + 3γ ) + + 2 EI k A1l k B1l 2

l3 2 2 β 2 α 2 + α β + 3EI k A2 k B 2

δ 22 =

γ 2 (1 + γ ) 2 l3 2 γ (1 + γ ) + + 3EI k A2 kB2

δ 23 =

l2 β α αβ ( β − α ) − + k A2l k B 2 l 3EI

δ 23 =

γ 1+ γ l2 γ (2 + 3γ ) + + 6 EI k A 2l k B 2l

δ 33 =

l 1 1 + (α 3 + β 3 ) + 2 3EI k A2 l k B 2l 2

δ 33 =

l 1 1 + (1 + 3γ ) + 2 EI k A2l k B 2l 2

δ14 = −

δ44 =

δ 22 =

α=

a , l

β=

b l

γ=

c l

The flexibility influence coefficients δ ij = δ ji of the two rotor models are listed in Table 3.1 [7].


147

The inverse of the flexibility matrix is the stiffness matrix given by k11 = k 22

δ44 δ δ 2 , k14 = − 14 , k 44 = 11 , Δ1 = δ11δ44 − δ14 , Δ1 Δ1 Δ1

δ δ δ 2 = 33 , k 23 = − 23 , k33 = 22 , Δ2 = δ22 δ33 − δ23 . Δ2 Δ2 Δ2

(3.142)

The equations of motion (2.80) are written in the form m &y&G + k11 yC + k14 ψ C = F1( t ), m &z&G + k 22 zC + k 23 ϕC = F2 ( t ),

J T ψ&&G − J P Ω ϕ&G + k 41 yC + k 44 ψ C = M 1( t ),

(3.143)

J T ϕ&&G + J P Ω ψ& G + k32 zC + k33 ϕC = M 2 ( t ). Using expressions (3.54) and (2.123) to eliminate the coordinates of the mass centre G, equations (3.143) are written in matrix form as ⎡m 0 ⎢0 J T ⎢ ⎢ ⎢ ⎣

0 0 ⎤ ⎧ y& C ⎫ ⎤ ⎧ &y&C ⎫ ⎡ ⎥ ⎪ ψ&& ⎪ ⎢ 0 J P ⎥⎥ ⎪⎪ ψ& C ⎪⎪ ⎥ ⎪⎨ C ⎪⎬ + Ω ⎢ ⎨ ⎬+ ⎢0 ⎥ ⎪ z&C ⎪ m 0 ⎥ ⎪ &z&C ⎪ 0 ⎥ ⎢ ⎥⎪ 0 J T ⎦ ⎪⎩− ϕ&&C ⎪⎭ ⎣0 − J P ⎦ ⎩− ϕ&C ⎪⎭ ⎡ k11 k14 ⎤ ⎧ yC ⎫ ⎧ F1 ⎫ ⎢k ⎥⎪ ψ ⎪ ⎪ M ⎪ 41 k 44 ⎢ ⎥ ⎪⎨ C ⎪⎬ = ⎪⎨ 1 ⎪⎬ + ⎢ k 22 − k 23 ⎥ ⎪ zC ⎪ ⎪ F2 ⎪ ⎢ ⎥ − k32 k33 ⎦ ⎪⎩− ϕC ⎪⎭ ⎪⎩− M 2 ⎪⎭ ⎣

(3.144)

or ⎡[ m ] ⎢ 0 ⎣

0 ⎤ ⎧ { &y& } ⎫ ⎡ 0 + [ m ]⎥⎦ ⎨⎩ { &z& } ⎬⎭ ⎢⎣− [ g ]

[ g ]⎤ ⎧ { y& } ⎫ ⎨ ⎬ 0 ⎥⎦ ⎩ { z& } ⎭

[ ]

⎡ k +⎢ y ⎣⎢ 0

{ }

⎧ fy ⎫ 0 ⎤ ⎧ { y }⎫ ⎥⎨ ⎬ = ⎨ ⎬ [ k z ]⎦⎥ ⎩ { z } ⎭ ⎩ { f z } ⎭

(3.145) which in shorthand has the form [ M ]{ &x&} + [ G ] { x& } + [ K ] { x } = { f }.

(3.146)


148

The vector in the right-hand side has the form

{ f } = { Fc } cos Ω t + { Fs } sin Ω t + { F } = ⎧ ⎪ (J ⎪ = Ω2 ⎨ T ⎪ ⎪⎩ ( J T

⎫ − J P ) α c ⎪⎪ 2 ⎬ cos Ω t + Ω m es ⎪ − J P ) α s ⎪⎭ m ec

− m es ⎧ ⎪ − (J − J ) α ⎪ T P s ⎨ m e c ⎪ ⎪⎩ ( J T − J P ) α c

⎫ ⎧ F1 ⎫ ⎪ ⎪ M ⎪ (3.147) ⎪ ⎪ 1 ⎪ ⎬ sin Ω t + ⎨ ⎬, ⎪ ⎪ F2 ⎪ ⎪⎭ ⎪⎩− M 2 ⎪⎭

where ec , es and α c ,α s are the projections of e and α along the coordinate axes.

3.3.2 Natural frequencies of precession The displacements y and z can be written as in (3.33)

(

)

y = yc cosω t + y s sinω t = ˆy cos ω t + θ y ,

z = zc cos ω t + z s sin ω t = ˆz cos ( ω t + θ z ) ,

{

y = ℜe ˆy e

{

iθ y iω t

e

}= ℜe { ( y − i y ) e },

}

z = ℜe ˆz eiθ z eiω t = ℜe

{( z

c

c

s

iω t

}

− i z s ) e iω t .

For { f } = { 0 } , equations (3.146) describe the rotor free precession. The solutions of these equations can be expressed in terms of complex phasors as [8]

{x }=

⎧{ y }⎫ ⎧ { yc } − i { ys }⎫ iω t = {Φ } e iωt . ⎬ e ⎬ = ⎨ ⎨ { } { } i z z − { } z s ⎭ ⎭ ⎩ c ⎩

(3.148)

Substituting (3.148) into (3.145) with zero right-hand side, leads to the eigenvalue problem

[ ]

⎡ k y − ω2 [ m ] ⎢ ⎢⎣ − iω Ω [ g ]

⎧{ 0 }⎫ iω Ω [ g ] ⎤ ⎧{ yc } − i { y s }⎫ = ⎥ ⎬ ⎨ ⎬ ⎨ [ k z ] − ω2 [ m ]⎥⎦ ⎩ { zc } − i { z s }⎭ ⎩{ 0 }⎭

(3.149)

which delivers the following four equations coupled two by two

( [ k ]− ω ( [ k ]− ω ( [ k ]− ω ( [ k ]− ω y

y z

z

2

[ m ] ){ yc } + ω Ω [ g ]{ z s } = { 0 },

(3.150)

2

[ m ] ){ ys } − ω Ω [ g ]{ zc } = { 0 },

(3.151)

2

[ m ] ){ zc } − ω Ω [ g ]{ ys } = { 0 },

(3.152)

2

[ m ] ){ z s } + ω Ω [ g ] { y c } = { 0 } .

(3.153)


149

Equations (3.150) and (3.151) yield

( [ k ]− ω } = −ω ( [ k ] − ω

{ yc } = −ω

y

{ ys

y

2 2

[m ])

[m ])

−1

Ω [ g ] { z s },

(3.154)

−1

Ω [ g ] { zc }.

(3.155)

Inserting (3.154) and (3.155) into (3.152) and (3.153) we obtain ⎛ [ k ] − ω 2 [ m ] − ω 2Ω [ g ] ⎜ z ⎝ ⎛ [ k ] − ω 2 [ m ] − ω 2Ω [ g ] ⎜ z ⎝

( [ k ]− ω [ m ] ) ( [ k ]− ω [ m ] ) y

y

2

2

−1

Ω [ g ]⎞⎟ { zc } = { 0 }, ⎠ −1 Ω [ g ]⎞⎟ { z s } = { 0 }. ⎠

(3.156) (3.157)

Comparing equations (3.156) and (3.157) it can be noticed that the two solutions are proportional to one another where β is a real constant.

{ zc } = β { zs },

(3.158)

Substituting equation (3.158) into equation (3.154) and comparing the result with equation (3.155) yields

{ ys } = − β { yc }.

(3.159)

Inserting (3.158) and (3.159) into (3.148) we obtain ⎧{ y } − i { ys }⎫ {Φ } = ⎨ c ⎬ ⎩ { zc } − i { z s }⎭

{ }

ay ⎫ ⎧ { yc } ⎫ iγ ⎧ = ( 1 + iβ ) ⎨ ⎬ =e ⎨ ⎬ ⎩− i { a z }⎭ ⎩− i { z s }⎭

(3.160)

where { a y } and { a z } are real vectors. Equation (3.160) shows that, by proper normalization, the elements of vectors { Φ } become real in the xOy plane and pure imaginary in the xOz plane, hence the precession modes are planar. From equations (3.148) and (3.149) we obtain y z β =− s = c yc z s yc zc + y s z s = 0.

(3.161)

According to expression (3.40), the inclination angle α = 0 and the ellipse axes are collinear with the coordinate axes. For undamped rotors, the eigenvectors are complex quantities due to the spatial character of the precession and the gyroscopic coupling, but the natural modes of precession are planar.


150

Using the transformation to real vectors

{ }

{ }

⎧ a y ⎫ ⎡[ I ] [ 0 ] ⎤ ⎧ a y ⎫ ⎨ ⎬=⎢ ⎬, ⎥ ⎨ ⎩− i { a z } ⎭ ⎣[ 0 ] − i [ I ]⎦ ⎩{ a z }⎭ where [ I

(3.162)

] is an identity matrix, and pre-multiplying by ⎡⎢

[0] ⎤ ⎥ , equation (3.149) ⎣[ 0 ] i [ I ]⎦ [I ]

becomes ⎡[ k y ] − ω 2 [ m ] ω Ω [ g ] ⎤ ⎧{ a y }⎫ ⎧{0}⎫ ⎢ ⎥⎨ ⎬ = ⎨ ⎬. [ k z ] − ω 2 [ m ]⎦⎥ ⎩{ a z }⎭ ⎩{0}⎭ ⎣⎢ ω Ω [ g ]

(3.163)

The requirement for non-trivial solutions is [ k y ] − ω 2 [ m]

ω Ω [g] =0 [ k z ] − ω 2 [ m]

ω Ω [g]

(3.164)

or k11 − ω 2 m

k14

k 41

k 44 − ω J T

0 0

0 ω Ω JP

2

0 0 ω Ω JP 0 = 0. 2 k 22 − ω m − k 23 − k32 k33 − ω 2 J T

The frequency equation has the form ω8 − ( A6 + B6 Ω 2 ) ω6 + ( A4 + B4 Ω 2 ) ω 4 − ( A2 + B2 Ω 2 ) ω 2 + A0 = 0.

(3.165)

Figure 3.23 presents a plot of the natural frequencies of precession ω as a function of the running speed Ω , for J P > J T . The symmetry with respect to the ω and Ω axes is due to the odd powers in equation (3.165). The horizontal asymptotes correspond to zero angular precession of the disc for Ω → ∞ . Inserting ω = Ω into (3.163), the synchronous precession critical speeds are obtained from ⎡[ k y ] − Ω 2 [ m ] Ω 2 [ g ] ⎤ ⎧{a y }⎫ ⎧{0}⎫ ⎢ ⎥⎨ ⎬ = ⎨ ⎬ 2 [ k z ] − Ω 2 [ m ]⎦⎥ ⎩{a z }⎭ ⎩{0}⎭ ⎣⎢ Ω [ g ]

which can be written as a generalized eigenvalue problem

(3.166)


⎡[ k y ] [ 0 ] ⎤ 2 ⎢ [ 0 ] [ k ]⎥ {Ψ r } = Ω r z ⎦ ⎣

⎡ [ m ] − [ g ]⎤ ⎢− [ g ] [ m ] ⎥ {Ψ r }. ⎣ ⎦

151 (r = 1,..,4 )

(3.167)

The eigenvalues Ω r give the synchronous critical speeds. The eigenvectors {Ψ r } define the semiaxes of the disc precession orbit at the related critical speed and the directivity of precession (forward or backward).

Fig. 3.23 Substituting ω = −Ω into (3.163), the off-diagonal elements of the matrix from equation (3.166) become negative, but the same critical speeds are obtained. Unlike the rotor supported in isotropic bearings, the unbalance will also excite the backward critical speeds. The Campbell diagram is shown in Fig. 3.24, b (first quadrant of Fig. 3.23). For comparison, Fig. 3.24, a depicts the Campbell diagram for the same rotor supported in rigid bearings. The motions with angular speeds ω1 and ω3 are backward precessions, while the motions with angular speeds ω2 and ω4 are forward precessions. The points where the line ω = Ω intersects the precession natural frequency lines define the critical speeds. The shape of the curves in the Campbell diagram from Fig. 3.24, b is typical for lightly damped rotors. As will be shown in section 3.3.4, the consideration of damping can substantially modify the shape of the diagram.


152

a

b

Fig. 3.24 Figure 3.25 presents separately the effects of bearing flexibility, disc diametral mass moment of inertia and gyroscopic coupling on the Campbell diagram of single-disc asymmetric rotors, supported in isotropic bearings (Fig. 3.25, a) and in orthotropic bearings (Fig. 3.25, b) [2]. The rigid disc has J P > J T . The horizontal asymptotes correspond to natural frequencies of pure translatory precession.

3.3.3 Unbalance response Considering equations (3.146), for a synchronous excitation

{ f } = {Fc } cos Ω t + {Fs } sin Ω t ,

(3.168)

the steady-state response has the form { x } = { X c } cos Ω t + { X s } sin Ω t .

(3.169)

Substituting (3.168) and (3.169) into (3.146) we obtain the algebraic set of equations

([ K ] − Ω

2

) } + ([ K ] − Ω

[ M ] { X c } + Ω [ G ]{ X s } = { Fc },

− Ω [ G ]{ X c

2

)

[ M ] { X s } = { Fs }.

(3.170)

The two components of the disc translational displacements are given by equations of the form (3.33). They are utilized for the calculation of the elliptic orbit parameters of the unbalance response, using equations (3.38) to (3.40) and (3.44).


Fig. 3.25

153


154

In order to calculate the finite amplitudes of motion at the peak response critical speeds, some form of damping has to be taken into account. In the left-hand side of equation (3.146), a diagonal damping matrix [ D ] = diag [ c11 c44 c22 c33 ] is added to the gyroscopic matrix. Its elements are calculated assuming given values of the damping ratios

ζ1 =

c33 c22 c44 c11 , ζ2 = , ζ3 = . , ζ4 = 2 k11m 2 k 22 m 2 k33 J T 2 k 44 J T

(3.171)

Usually, it is considered that ζ 1 = ζ 2 = ζ 3 = ζ 4 = ζ .

Example 3.2a Consider a rotor with an overhung disc (Model II) with the following J T = 4260 kg m 2 , l = 4 m, parameters: m = 8000 kg , J P = 8520 kg m 2 , ζ = 0.02 , k A1 = 333 N/μm , c = 0.8 m, E = 210 GPa , d = 0.3 m, k A2 = 667 N/μm, k B1 = 83.3 N/μm, k B 2 = 167 N/μm .

Fig. 3.26 Figure 3.26 presents the Campbell diagram with the running speed on the horizontal axis. The intersections with the synchronous line determine the damped critical speeds: n1 = 437 rpm, n2 = 761 rpm, n3 = 1282 rpm. The second critical speed is in forward (synchronous) precession. The others are in backward (asynchronous) precession.


155

Figure 3.27 shows the unbalance response diagrams. The peaks in the major semiaxis diagram (Fig. 3.27, a) locate the peak response critical speeds. Although the first and the third critical speed correspond to backward precession modes, they are excited by the rotating unbalance. The maximum value of the major semiaxis in the operating speed range is usually compared to admissible limits.

a

b

c

d Fig. 3.27

In figure 3.27, b the minor semiaxis diagram is added. The two ranges with negative values define the operation speeds with backward precession produced by the unbalance. The crossing points with the horizontal axis locate the threshold speeds where the precession orbit degenerates to a straight line. At these speeds the orbit changes from forward to backward precession and vice versa. Note that the threshold speeds are different from the critical speeds. Figure 3.27, c shows the diagrams of the y and z components of the disc centre displacement. Figure 3.27, d presents the diagrams of the radii r f and rb of the two circular counter-rotating motions that generate the ellipse. The ranges where the radius rb of the circle with backward motion is larger than the radius


156

rf

of the circle with forward precession define the speed ranges with backward

precession. It is easy to see that they correspond to ranges with negative minor semiaxis in Fig. 3.27, b.

3.3.4 Effect of bearing damping Including the effect of bearing damping, equations (3.146) become [ M ]{ &x&} + [ C ]{ x& } + [ K ]{ x } = { f }.

(3.172)

where

⎡ [ c yy ] [ c yz ] ⎤ [C ] = ⎢ ⎥+Ω ⎣ [ c zy ] [ czz ] ⎦

⎡ [0] [ g ] ⎤ ⎢− [ g ] [ 0 ] ⎥ ⎣ ⎦

(3.173)

is the sum of the damping and gyroscopic matrices. For { f } = { 0 } , trying solutions of the form { x } = { u } eλ t ,

(3.174)

the following quadratic eigenvalue problem is obtained ( λ2r [ M ] + λr [C ] + [ K ] ) { ur } = { 0 } ( r = 1,..,4 ) .

(3.175)

The eigenvalues λr are real numbers for overdamped modes and complex numbers for underdamped modes. The complex eigenvalues have the form

λ r = α r + i ω r , λr = α r − i ω r

(3.176)

and are functions of the running speed Ω . The imaginary part ωr is the damped natural frequency (of precession) and the real part α r is an attenuation ( or growing) constant. Usually, the damping is expressed in terms of the modal damping ratio

ζr = −

αr α r2 + ωr2

≅−

αr . ωr

(3.177)


157

The Campbell diagram is a plot of the dependence ωr = ωr (Ω ). Sometimes it is presented together with the stability diagram α r = α r (Ω ) or the damping ratio diagram ζ r = ζ r (Ω ) . For underdamped systems, the complex eigenvectors have the form { u r } = { ar } + i { br },

{ ur } = { ar } − i { br }.

(3.178)

The solution for the free precession can be written as

{ xr (t ) } = 2 eα r t ( { ar } cos ωr t − { br } sin ωr t ) ,

(3.179)

and describes spiralling orbits. However it is agreed to represent the orbits as incomplete (open) ellipses, considering α r = 0 and approximating the expression (3.179) by { xr (t ) } = { uc } cos ωr t + { u s } sin ωr t (3.180) where { uc } = { ar } = ℜe { u r }, { u s } = − { br } = −ℑm { ur }.

Fig. 3.28

Example 3.3 Consider the cantilevered rotor from Fig. 3.28, with l = 0.4 m, d = 0.02 m, E = 210 GPa . The end carrying a thin rigid disc with m = 16.5 kg,

J T = 0.094 kg ⋅ m 2 , J P = 0.186 kg ⋅ m 2 , is flexibly supported. The support has principal stiffnesses k1 = 5 ⋅ 105 N/m and k 2 = 2 ⋅ 105 N/m, and damping coefficients proportional to the related stiffness, c1 = β k1 and c2 = β k 2 .


158

Figures 3.29 depict the Campbell diagrams and the diagrams of the modal damping ratio for three values of the coefficient β .

a

b

c

Fig. 3.29 It can be seen that the curves corresponding to the first mode of precession change significantly with the bearing damping. For large damping levels (Fig. 3.29, c), the first precession mode becomes overdamped within a certain speed range. This mode is not 'seen' in the unbalance response diagrams, where the corresponding peak is missing.

3.3.5 Mixed modes of precession The rotor precession is usually described by modal characteristics associated with forward and backward modes. However, the directivity of precession is a local, not a global property. A rotor can have mixed modes, with both forward and backward precession coexistent at different stations, at a given speed. Hopefully, the precession of rotors in almost isotropic bearing systems can be classified as pure forward or pure backward, the motion at all stations of interest having the same direction. This enables a logical mode labeling. In most cases, mixed modes are predominantly forward or backward, with limited zones of reverse precession, so that if the number of stations in the model is small, then the mixed character of the precession is lost.


159

In anisotropic rotor-bearing systems, bearing orthotropy yields different rotor deflected shapes in two orthogonal directions, at the same rotational speed. The coupling of two (even slightly) different rotor orthogonal eigenforms yields mixed precession modes. This can be easier explained for conservative rotor-bearing systems that have planar precession modes. In this case, the deflected shapes in two orthogonal planes correspond to mode shapes plotted at two instants with a quarter of period time difference. At the station where only one modal form crosses the rotor longitudinal axis, the precession orbit degenerates into a straight line, separating portions of forward and backward motion along the rotor [9]. In a rotor with oil-lubricated bearings, the slightest asymmetry yields small differences in load and oil temperature between the two bearings. Even with physically identical bearings, the stiffness and damping coefficients are different at the both ends. For reasonable amounts of dissymmetry and coupling effects, the damped natural frequency curves in the Campbell diagram do not cross, giving rise to curve veering, denoting modal coupling and compound modes. The abrupt continuous change of mode shapes within the speed interval of natural frequency curve veering yields mixed modes. Along a natural frequency curve, a mode can be forward over a given speed interval, then mixed in the region of curve veering, changing to backward away of that region. Simultaneous plotting of the speed dependence of modal damping ratios helps understanding the nature of mixed modes. For a large class of actual rotor systems, precession modes occur in pairs, the mode at lower frequency has backward precession and the mode at higher frequency has forward precession. Inclusion of bearing damping can change the sequence. Overdamped modes can transform into underdamped modes and appear in the Campbell diagram only in limited speed regions. The bearing cross-coupling stiffnesses increase the gap between the natural frequencies of a backward-forward pair. This way, a forward mode from a lower pair approaches a backward mode from a higher pair, yielding either a crossing or a curve veering in the Campbell diagram. The compounding of two different modes gives rise to mixed precession. In some academic examples, the mixed nature of some precession modes is lost if the motion is analyzed at a relatively reduced number of stations along the rotor. As a first example, a simple rotor system is taken, consisting of a rigid disc attached to a massless rigid shaft supported by two identical bearings at the ends. Three cases are considered: a) a symmetric rotor with isotropic bearings; b) a symmetric rotor with orthotropic bearings; and c) an asymmetric rotor with orthotropic bearings. The massless rigid shaft was modeled with values of E = 2 ⋅ 1015 Pa and ρ = 1 kg m3 in the computer simulation [9].


160 Example 3.4 a

A rigid disc is mounted at the centre (l 1 = l 2 = 0.35m ) of a massless rigid shaft and the shaft is supported by identical isotropic bearings at both ends. The disc mass and mass moments of inertia are m = 30 kg, J T = 1.2 kg m 2 ,

J P = 1.8 kg m 2 .

The

bearing

stiffness

and

damping

coefficients

are

6

k yy = k zz = 7 ⋅ 10 N/m and c yy = c zz = 200 Ns/m . The Campbell diagram is shown in Fig. 3.30. Forward modes are labeled ‘F’ while backward modes are labeled ‘B’. The two ‘cylindrical’ modes at 103.48 Hz have natural frequencies independent of the rotational speed, hence overlaid straight lines. The disc has a translational motion not influenced by gyroscopic effects and decoupled from the angular motion. The third and fourth ‘conical’ modes, labeled 2B and 2F, are decoupled from the cylindrical modes. As the rotor speed increases, the natural frequency of the backward mode decreases and crosses the line of the cylindrical modes, due to gyroscopic effects. The natural frequency of the forward mode increases with rotor speed. Due to bearing isotropy the two curves start from the same point at zero rotational speed.

Fig. 3.30 The synchronous excitation line is plotted with dotted line. The critical speeds are determined as the abscissae of the crossing points with the natural frequency lines, at 6209 rpm and 6876 rpm. In the case of unbalance excitation, the only one critical speed is located at the intersection with the line of mode 1F. For rotor systems with isotropic bearings, backward modes cannot be excited by synchronous excitation.


a

161

b

c

d

Fig. 3.31 The precession mode shapes at 10000 rpm are shown in Fig. 3.31. Due to bearing isotropy, the orbits at any station are circles. They are plotted as incomplete (“open”) orbits to help recognizing the motion directivity. The mode shape at t = 0 is plotted with solid line and the mode shape at t = π 2Ω is drawn with broken line, so that the motion along the orbit takes place from the point lying on the solid line, at t = 0 , to the point lying on the broken line, a quarter of a period later.

Fig. 3.32


162

The radius of the unbalance response orbit at the disc location is plotted in Fig. 3.32 as a function of the rotational speed, for a 30 g mm unbalance of the disc. As expected, only one peak occurs in the diagram, at the natural frequency of mode 1F. Example 3.4 b

Consider the symmetric rotor of Example 3.4 a, but with orthotropic identical bearings . The bearing vertical stiffness coefficients are k yy = 5 ⋅ 106 N/m , the horizontal stiffness coefficients are k zz = 7 ⋅ 106 N/m , and the damping coefficients are c yy = c zz = 2 ⋅10 2 Ns/m [10].

Fig. 3.33 The Campbell diagram is shown in Fig. 3.33. The two ‘cylindrical’ modes 1B and 1F have different natural frequencies at 88.63 Hz and 103.48 Hz due to the bearing anisotropy. They are independent of the rotational speed due to system symmetry. The third and fourth ‘conical’ modes, are decoupled from the cylindrical modes. As the rotor speed increases, the natural frequency of the mode 2B decreases and crosses the lines of the cylindrical modes, due to gyroscopic effects. The natural frequency of the mode 2F increases with rotor speed. Due to bearing anisotropy the two curves in a pair start from different points at zero rotational speed.


163

The synchronous excitation line intersects the natural frequency lines at the points whose abscissae determine the damped critical speeds at 5318 rpm , 6209 rpm and 6341 rpm.

Fig. 3.34 The damping ratio diagram is shown in Fig. 3.34. Backward modes are more damped than the forward modes of the same pair. The curves for the conical modes cross those of cylindrical modes, denoting no coupling effects.

a

b

c

d

Fig. 3.35


164

The precession mode shapes at 10000 rpm are shown in Fig. 3.35. As before, the mode shape at t = 0 is plotted with solid line and the mode shape at t = π 2Ω is drawn with broken line. The motion along the orbit takes place from the point lying on the solid line, to the point lying on the broken line. The orbits of the two ‘cylindrical’ modes 1B and 1F at 88.63 Hz and 103.48 Hz are almost straight lines due to the strong bearing anisotropy and decoupling of the two motions. The orbits of modes 2B and 2F are elliptical.

a

b

Fig. 3.36

Fig. 3.37 The unbalance response curves calculated at the disc station are shown in Fig. 3.36, for a 30 g mm unbalance of the disc. In Fig. 3.36, a , curve a is for the major semiaxis and curve b is for the minor semiaxis. In Fig. 3.36, b , curve r f is


165

for the forward circle radius and rb is for the backward circle radius. The two peaks indicate that only two of the three possible damped critical speeds become peak response critical speeds due to the high damping of mode 2B. Between the two peaks there is a speed range with backward precession indicated by negative values of the orbit minor semiaxis, or by rb > r f . Useful information is given by the root locus diagram (Fig. 3.37). This is a plot of the damped natural frequency versus negative damping ratio, for each mode of precession. When the curves are distant of each other, as in Fig. 3.37, there is no coupling between modes and no compound or mixed modes of precession can occur. Example 3.4 c

Consider the rotor of Example 3.4 b, but with the rigid disc mounted off the shaft centre (l 1 = 0.3 m, l 2 = 0.4 m ) . The shaft is rigid and massless. The bearing stiffness coefficients are k yy = 5 ⋅ 106 N/m , k zz = 7 ⋅ 106 N/m , and the damping coefficients are c yy = c zz = 2 ⋅10 2 Ns/m [10].

Fig. 3.38 The Campbell diagram is shown in Fig. 3.38. Curve 2B no more crosses the lines 1B and 1F and, near the rotational speed of 8000 rpm , veers away from the line 1F. The rotor translational and angular motions are coupled. With increasing rotational speed, mode 2B becomes a mixed mode and tends to change into the first forward mode, while mode 1F becomes a mixed mode and tends to change into the first backward mode. The synchronous excitation line intersects the


166

natural frequency lines at the points whose abscissae determine the damped critical speeds 5236 rpm , 6051 rpm and 6532 rpm. The damping ratio diagram is shown in Fig. 3.39. With increasing rotational speed, curve 2B transforms into the former 1F, while 1F transforms into the former 1B and curve 1B follows the former line 2B. These transformations take place in the speed range with curve veering in the Campbell diagram.

Fig. 3.39 The root locus diagram is presented in Fig. 3.40. Modes are labeled as before, according to their shapes at low rotational speeds. When the root loci are close to each other, two modes with nearly the same natural frequency and different mode shapes can combine to yield a compound mode which has mixed backward and forward precession due to the coupling between modes.

Fig. 3.40


a

167

b

c

d

Fig. 3.41 The precession mode shapes at 10000 rpm are shown in Fig. 3.41. For mixed modes, the precession along the ellipse is marked by B (backward) or F (forward) and takes place from the point lying on the solid line, at t = 0 , to the point lying on the broken line, a quarter of a period later.

a

b

Fig. 3.42 Along the rotor, the portions of backward and forward motion are separated by a location where the precession orbit degenerates into a straight line. Such lines do not appear in Fig. 3.41 due to the small number of stations where orbits have been drawn.


168

The unbalance response curves calculated at the disc station are shown in Fig. 3.42, for a 30 g mm unbalance of the disc. The abscissae of the three peaks indicate the peak response critical speeds. Again, the speed range with backward precession is indicated by negative values of the orbit minor semiaxis or values of rb larger than r f . When b = 0 and rb = r f the orbit degenerates into a straight line.

3.4 Simulation examples Example 3.5 a

A rigid disc is mounted at the middle of a uniform shaft (Fig. 3.43) of length 0.44 m , diameter 90 mm , Young’s modulus 2 ⋅ 1011 N m 2 and mass density 7800 kg m 3 .

Fig. 3.43 The mass of the disc is 560 kg, while the diametral and polar mass moments of inertia are 18 kgm 2 and 32 kgm 2 , respectively. The shaft is supported at the ends by identical bearings with the following constant coefficients: ′ = k zz ′′ = 1.1 ⋅108 N m , c′yy = c′yy ′ = 2.2 ⋅ 108 N m , k zz ′ = 2.2 ⋅10 4 Ns m , and k ′yy = k ′yy

c′zz = c′zz′ = 1.11 ⋅ 10 4 Ns m [9]. The Campbell diagram is shown in Fig. 3.44. Due to the system symmetry, the disc translational and angular motions are decoupled. Modes 1B and 1F have natural frequencies independent of rotational speed. As the rotor speed increases, the natural frequency of the mode 2B decreases and crosses the lines of the cylindrical modes, due to gyroscopic effects. Due to bearing orthotropy, the two curves in a pair start from different points at zero rotational speed.


169

Fig. 3.44 The damping ratio diagram is shown in Fig. 3.45. As in Example 3.4 a, backward modes are more damped than the forward modes of the same pair. The curves for the conical modes do not cross those of cylindrical modes.

Fig. 3.45 Example 3.5 b

Consider the rotor of Example 3.5 a supported by bearings with slightly different stiffness and damping coefficients (Fig. 3.43): k ′yy = 2.15 ⋅108 N m ,

′ = 1.15 ⋅ 108 N m , c′yy = 2.15 ⋅ 10 4 Ns m , c′zz = 1.15 ⋅104 Ns m , k ′yy ′ = 2.25 ⋅ 108 N m , k zz ′′ = 1.05 ⋅ 108 N m , c′yy ′ = 2.25 ⋅ 10 4 Ns m and c′zz′ = 1.05 ⋅ 10 4 Ns m [9]. k zz


170

The Campbell diagram is shown in Fig. 3.46 for the first four modes of precession. The curves in the diagram are labeled in the usual way, 1B, 1F, 2B and 2F, as for Example 3.5 a, though there are speed intervals with mixed modes. Curve 2B crosses the line 1F at 600 rpm and veers away from line 1B at 2065 rpm.

Fig. 3.46 In Fig. 3.47 the damping ratio curves of modes 2B and 1F have a trough, respectively a peak, at 600 rpm, not crossing each other, while curves 2B and 1B do cross each other at about 2065 rpm.

Fig. 3.47


171

With increasing rotational speed, mode 2B becomes a mixed mode and changes into the first backward mode, while mode 1B changes into 2B. The root locus diagram (Fig. 3.48) indicates the possible coupling of modes 1B and 2B, whose loci are close to each other. Also modes 2B and 1F have a range of equal natural frequencies and this can also produce compound modes with mixed precession.

Fig. 3.48 A closer look at the shape of precession modes is useful, especially at their evolution within the speed intervals with modal interaction.

a

b

c

Fig. 3.49 Figure 3.49 shows the evolution of mode 1M between 1700 and 3000 rpm. Mode 1M results from the coupling of a vertical conical mode 1B with a horizontal cylindrical mode 1F. With increasing rotational speed, the latter becomes a conical horizontal mode.


172

a

b

c

Fig. 3.50 Figure 3.50 shows the evolution of mode 2M between 200 and 1000 rpm. Despite the crossing of natural frequency curves (Fig. 3.46) the mode is mixed. It is the result of the compounding of a cylindrical vertical mode and a conical horizontal mode. Mixed modes exist even when there is no curve veering in the Campbell diagram.

a

b

c

Fig. 3.51 Figure 3.51 presents the evolution of mode 3M between 250 and 1000 rpm. It is basically the second backward mode 2B, but at low speeds, the vertical and horizontal conical components cross the rotor longitudinal axis at different locations. At these points the precession orbit degenerates into straight lines that mark the change from backward to forward or vice versa.

a

b

Fig. 3.52

c


173

Figure 3.52 shows the evolution of mode 3M between 1700 and 2500 rpm. At 1700 rpm the mode is apparently still backward 2B. Its mixed nature is overlooked due to the small number of stations at which the orbit is drawn. A closer look at Fig. 3.52, a shows that the vertical and horizontal conical components cross the rotor longitudinal axis at different locations so that there is a portion with forward precession not revealed with only five stations. At 2100 rpm the horizontal mode becomes cylindrical. Because the vertical component remains conical, the precession mode is mixed. A similar rotor system with slightly different parameters is presented in the following, to illustrate the above statements. Horizontal stiffnesses are larger in this case than the vertical stiffnesses. Example 3.6

A uniform shaft of length 0.437 m , diameter 91 mm , Young’s modulus

2 ⋅ 1011 N m 2 and mass density 7750 kg m 3 carries at the middle a rigid disc of mass 566 kg , diametral and polar mass moments of inertia 18.1 kg m 2 and

36.2 kg m 2 , respectively. The shaft is supported at the ends by orthotropic bearings with the following constant stiffness and damping coefficients: (Fig. 3.43): k ′yy = 1.14 ⋅108 N m ,

′ = 2.14 ⋅108 N m , k ′yy ′′ = 2.24 ⋅108 N m , c′yy = 1.14 ⋅ 10 4 Ns m , ′ = 1.04 ⋅108 N m , k zz k zz ′ = 1.04 ⋅10 4 Ns m and c′zz′ = 2.24 ⋅104 Ns m [11]. c′zz = 2.14 ⋅ 10 4 Ns m , c′yy

Fig. 3.53


174

The Campbell diagram is shown in Fig. 3.53 for the first four modes of precession. In Fig. 3.54 the damping ratio curves of modes 2B and 1F have a trough, respectively a peak, at 400 rpm, not crossing each other, while curves 2B and 1B do cross each other at about 1800 rpm.

Fig. 3.54 With increasing rotational speed, mode 2B becomes a mixed mode and changes into the first backward mode 1B, while mode 1B changes into 2B.

a

b

c

d

e

f

Fig. 3.55


175

Figure 3.55 shows the evolution of mode 3M between 1000 and 2000 rpm. Mode 3M is obtained from the coupling of a horizontal conical mode 2B with a vertical cylindrical mode 1B. With increasing rotational speed, the latter becomes a vertical cylindrical mode 1F.

a

b

c

Fig. 3.56 Figure 3.56 shows the evolution of mode 1M between 1900 and 2400 rpm.

Fig. 3.57 Example 3.7

Consider a rotor with two bearings and a single disc overhung at one end (Fig. 3.57). The rigid disc, with mass 8000 kg , polar mass moment of inertia 8520 kgm 2 and diametral mass moment of inertia 4260 kgm 2 , is located at station 7, at the right end. The shaft with Young’s modulus 2.1 ⋅ 1011 N m 2 and mass density 7800 kg m 3 has four different sections with the following lengths and diameters: l 1 = 0.7 m , d1 = 0.1 m , l 2 = 2.9 m , d 2 = 0.3 m , l 3 = 0.4 m , d3 = 0.32 m , l 4 = 0.8 m , d 4 = 0.34 m , and is modeled by 6 beam elements. The bearings are located at stations 1 and 6 having the following constant stiffness and ′ = (1 12 ) ⋅ 109 N m , damping coefficients: at station 1, k ′yy = (1 6 ) ⋅109 N m , k zz

′ = (2 3) ⋅ 109 N m , k zz ′′ = (1 3) ⋅ 109 N m , c′yy = c′zz = 105 Ns m ; at station 6, k ′yy ′ = c′zz′ = 105 Ns m [7]. c′yy


176

The Campbell diagram is presented in Fig. 3.58 for the first six natural modes. Modes are numbered in ascending order and labeled with their index without mentioning the directivity.

Fig. 3.58 The damping ratio diagram is shown in Fig. 3.59 for the same six modes.

Fig. 3.59 The shape of the first six modes of precession at 2400 rpm is shown in Fig. 3.60. The system has 4 mixed modes, although there is neither curve veering nor curve crossing at 2400 rpm in figure 3.58. The natural frequencies of modes 2


177

and 3, as well as those of modes 4 and 5, belonging to different pairs, are approaching each other. Mode 2 is predominantly forward (2F) and its mixed character is the result of the different crossing points of the vertical and horizontal component modes with the rotor axis.

a

b

c

d

e

f

Fig. 3.60 The unbalance response curves calculated at the bearing stations are shown in Fig. 3.61, for a 80 g mm unbalance of the disc. The abscissae of the five peaks indicate the peak response critical speeds. The peak due to the first mode is barely noticeable at 413 rpm.

a

b

Fig. 3.61


178 Example 3.8

Consider a rotor with two bearings and an overhung disc (Fig. 3.62). The rigid disc, with mass 7.5 kg , polar mass moment of inertia 0.04 kgm 2 and diametral mass moment of inertia 0.02 kgm 2 , is located at station 5, at the right end. The shaft with Young’s modulus

2 ⋅ 1011 N m 2

and mass density

8000 kg m 3 has diameter d = 50 mm and total length l = 1 m , and is modeled by 4 equal length beam elements. The identical bearings are located at stations 1 and 3 having the following constant stiffness and damping coefficients k yy = 2.5 ⋅107 N m , k zz = 4 ⋅ 107 N m , and c yy = c zz = 5 ⋅ 103 Ns m [12].

Fig. 3.62

Fig. 3.63


179

The Campbell diagram for the first six modes is presented in Fig. 3.63. Mode 4 is mixed, due to the interaction of modes 2F and 3B. There is a curve veering in Fig. 3.63 and a curve crossing in Fig. 3.64, around 13000 rpm. The damping ratio diagram is shown in Fig. 3.64 for eight modes.

Fig. 3.64 The root locus diagram is shown in Fig. 3.65 for the first six modes and speeds up to 30000 rpm.

Fig. 3.65


180

The evolution of the mixed mode with the rotor speed is shown in Fig. 3.66.

a

b

c

d

e

f

Fig. 3.66 The first six mode shapes at 10000 rpm are presented in Fig. 3.67.

a

b

c

d

e

f

Fig. 3.67 The unbalance response curves at the bearing stations 1 and 3 are shown in Fig. 3.68 for a disc unbalance of 15 g mm . Peaks occur at the eigenfrequencies of forward modes, because backward modes are relatively highly damped.


a

181

b

Fig. 3.68 Figure 3.69 shows the variation of the undamped natural frequencies of modes 3 to 6, as a function of the bearing stiffness, at 12000 rpm. The vertical lines indicate the bearing vertical and horizontal stiffness coefficients

k yy = 2.5 ⋅107 N m and k zz = 4 ⋅ 107 N m . Modes 4 (2F) and 5 (3B) have different shapes but almost equal natural frequencies. They interact, giving rise to a compounded mixed mode.

Fig. 3.69 In the following examples, the rotors are carried in oil-film journal bearings. The Campbell diagrams of these systems have specific features. The first two ‘rigid body’ backward modes are overdamped and do not appear in the diagram. The curves of the first two forward modes follow closely the halffrequency excitation line. The two-node flexural forward mode interacts with the


182

cylindrical forward rigid body mode giving rise to compounded modes, sometimes referred to as ‘convex cylindrical’ and ‘concave cylindrical’. In some cases, even the forward modes become overdamped and disappear from the diagram. The stability diagrams are useful to locate the onset speed of instability. The damping ratio diagrams help locating this threshold speed and show also when some modes are overdamped. The root locus diagrams give an overview of the eigenvalue variation with the rotor speed and can be used to explain the occurrence of mixed modes of precession. Generally, the mode labeling for these systems is more difficult than for rotors carried by supports with constant coefficients, and the pattern of mode pairs with backward and forward precession is either changed or difficult to recognize. Example 3.9 a

Consider the rotor from Fig. 3.70 supported in two identical journal bearings. The rigid disc has the mass 20 kg , the polar mass moment of inertia 1 kg m 2 and the diametral mass moment of inertia 0.7 kg m 2 . The massless flexible shaft of diameter 25.4 mm and Young’s modulus 2.1 ⋅ 1011 N m 2 has lengths l12 = 85 mm and l 23 = 255 mm [13] and is modeled with only two elements.

Fig. 3.70 The bearings have diameter D = 25.4 mm , length L = 16 mm , radial clearance C = 35.2 μm , and oil dynamic viscosity μ = 0.02 Ns m 2 . The static loads on bearings are W1 = 142.4 N and W2 = 53.8 N . The speed dependence of the stiffness and damping coefficients, calculated based on Ocvirk’s short bearing assumptions [14], with a fully cavitated film, i.e. with the oil film extending only 180 0 , is shown in Fig. 3.71.


a

183

b

Fig. 3.71 The Campbell diagram is presented in Fig. 3.72 for the first six natural modes of precession. At the crossing points with the synchronous excitation line, the damped critical speeds are determined as 2186, 6047 and 9442 rpm.

Fig. 3.72 Modes 1B and 2B are overdamped and do not appear in the diagram. Modes 1F and 2F are ‘rigid body’ modes controlled by the hydrodynamic bearings and follow closely the half-frequency excitation line ω = Ω 2 . If one sliding bearing is replaced by a rigid bearing, one of these lines disappears. If both sliding bearings are replaced by rigid bearings then both lines disappear. The curves of modes 3F and 4B cross each other at about 12600 rpm but the two modes do not interact.


184

Fig. 3.73 The damping ratio diagram is shown in Fig. 3.73 for the same 6 modes.

Fig. 3.74 The stability diagram is plotted in Fig. 3.74 for only four modes. Mode 1F becomes unstable at 10331 rpm. Looking at the associated point in the Campbell diagram, it can be seen that the whirling takes place at a frequency of about half the spin speed, describing the bearing instability known as the ‘oil whirl’.


185

a

b

c

d

e

f

Fig. 3.75 The shape of the first six eigenmodes at 15000 rpm is shown in Fig. 3.75. The forward modes, with larger relative displacements in bearings, have higher damping ratio values.

Fig. 3.76 The root locus diagram for the first six modes and for speeds up to 15000 rpm is shown in Fig. 3.76. The curve of mode 1F crosses the vertical at zero damping ratio, indicating the loss of stability.


186 Example 3.9 b

Solve the problem of Example 3.9 a using the Moes’ impedance model [15] for plain cylindrical bearings. The speed dependence of the bearing stiffness and damping coefficients is shown in Fig. 3.77.

a

b

Fig. 3.77 The Campbell diagram is presented in Fig. 3.78 and the damping ratio diagram in Fig. 3.79. The damped critical speeds are 2179, 6047 and 9322 rpm.

Fig. 3.78 Modes 1B and 2B are overdamped and do not show up in the Campbell diagram. Modes 1F and 2F are ‘rigid body’ modes and their curves follow closely


187

the half-frequency excitation line ω = Ω 2 . The curves of modes 3F and 4B cross each other twice, but the modes do not interact.

Fig. 3.79 The stability diagram for only four modes is given in Fig. 3.80. Mode 1F becomes unstable at 10016 rpm, which is lower than the onset speed of instability calculated for Ocvirk bearings. Thus, use of the short bearing approximation is not recommended in stability analyses.

Fig. 3.80


188

The shapes of the first six eigenmodes at a rotor speed of 15000 rpm are shown in Fig. 3.81. With only three nodes in the model, their shape is approximate.

a

b

c

d

e

f

Fig. 3.81 The root locus diagram in presented in Fig. 3.82 for the first six modes and speeds up to 15000 rpm.

Fig. 3.82


189

Example 3.9 c

Consider the rotor of Example 3.9 a supported now by two-lobe bearings with L = 12.7 mm . The speed dependence of the bearing stiffness and damping coefficients, calculated based on data from Someya’s book [16] for L D = 0.5 and a preload factor m p = 3 4 , is shown in Fig. 3.83.

a

b

Fig. 3.83 The Campbell diagram is presented in Fig. 3.84. The damped critical speeds are 2182, 7638 and 9619 rpm. Modes 1F and 2F do not follow the halffrequency line. The curves of modes 3F and 4B do not cross each other.

Fig. 3.84


190

The damping ratio diagram in presented in Fig. 3.85 for only two modes. Mode 3F becomes unstable at 13854 rpm, which is much higher than the onset speed of instability for cylindrical bearings.

Fig. 3.85 The same information is given by the stability diagram from Fig. 3.86.

Fig. 3.86


191

The root locus diagram in presented in Fig. 3.87 for the first six modes and speeds up to 16000 rpm. Modes 1F and 2F are highly damped. The curve of mode 3F intersects the zero damping line at the point marking the damped natural frequency at the instability threshold.

Fig. 3.87 The unbalance response curves calculated at the left bearing and disc locations are presented in Fig. 3.88 for an unbalance of 20 g mm on the disc.

a

b

Fig. 3.88 Example 3.10 a

Consider the rotor of Fig. 3.43 supported in two identical plain cylindrical bearings. The rigid disc has the mass 9.07 kg , the polar mass moment of inertia 0.0468 kg m 2 and the diametral mass moment of inertia 0.0305 kg m 2 . The


192

massless

flexible

shaft

of

diameter

22 mm

and

Young’s

modulus

2.145 ⋅ 1011 N m 2 has the total length 0.508 m and is divided into four equal elements [17]. The bearings have diameter 25.4 mm , length 25.4 mm , clearance 203.2 μm , and oil viscosity 0.0241 Ns m 2 . The static loads on bearings are W1 = W2 = 44.49 N . The speed dependence of the stiffness and damping coefficients, calculated based on Moes’ impedance model, is shown in Fig. 3.89.

Fig. 3.89 The Campbell diagram for the first four modes is presented in Fig. 3.90. The damped critical speeds are 384, 1141, 1739 and 2790 rpm.

Fig. 3.90


193

The damping ratio diagram in presented in Fig. 3.91. Mode 3F∗ becomes unstable at 4061 rpm.


Fig. 3.92


194

The root locus diagram in presented in Fig. 3.93 for speeds up to 6000 rpm. The curve of mode 3F∗ intersects the zero damping line at the point marking the damped natural frequency at the instability threshold.

Fig. 3.93 The first six mode shapes at 3000 rpm are presented in Fig. 3.94.

a

b

c

d

e

f

Fig. 3.94


195

Mode 3 ( 3F∗ ) is a convex-cylindrical mode, while mode 2 (2F) is a concave-cylindrical mode.

a

b

Fig. 3.95

a

b

Fig. 3.96

a

b

Fig. 3.97


196

The unbalance response curves are calculated at locations 1 and 3 for an eccentricity of 1.084 ⋅ 10 −4 m of the disc mass. Figures 3.95 show the diagrams of the ellipse semiaxes, Fig. 3.96 presents the diagrams of the vertical and horizontal components, while Fig. 3.97 gives the diagrams of the radii of the forward and backward generating circles. Around 3000 rpm, the orbits in bearings are circular, while the disc orbit is elliptical. Example 3.10 b

Consider the rotor of Example 3.10 a, with small modifications. The massless flexible shaft has the diameter 22.2 mm and Young’s modulus 2.038 ⋅ 1011 N m 2 . The plain cylindrical bearings have diameter D = 25.4 mm , length L = 25.4 mm , radial clearance C = 1.8796 ⋅10 −4 m , and oil dynamic viscosity μ = 960 ⋅ 10 −5 Ns m 2 , as in [18]. The speed dependence of the stiffness and damping coefficients, calculated based on Moes’ impedance model, is shown in Fig. 3.98.

Fig. 3.98 The Campbell diagram for the first four modes of precession is presented in Fig. 3.99. The damped critical speeds are 805, 876, 1778 and 2817 rpm. Mode 1F becomes overdamped beyond 1000 rpm.


197

Fig. 3.99 The damping ratio diagram in presented in Fig. 3.100 for only three modes. Mode 3F∗ becomes unstable at 5180 rpm.



198

Fig. 3.101 The root locus diagram in presented in Fig. 3.102 for speeds up to 6000 rpm. The curve of mode 3F∗ intersects the zero damping line at the point marking the damped natural frequency at the instability threshold.

Fig. 3.102 The first three mode shapes at 2500 rpm are presented in Fig. 3.103.


a

199

b

c

Fig. 3.103 The unbalance response curves are calculated at locations 1 and 3, for an eccentricity of 1.084 ⋅ 10 −4 m of the disc mass. Figures 3.104 show the diagrams of the ellipse semiaxes, Figs. 3.105 present the diagrams of the vertical and horizontal components, while Figs. 3.106 give the diagrams of the radii of the forward and backward generating circles.

a

b

Fig. 3.104

a

b

Fig. 3.105


200

Around 3000 rpm, the orbits in bearings are circular, while the disc orbit is elliptical.

a

b

Fig. 3.106 Between about 2200 and 2800 rpm, the steady state response due to unbalance is a mixed mode (Fig. 3.107, a), with backward precession at the disc and forward precession at bearings. At 5200 rpm the steady state precession is forward (Fig. 3.107, b).

a

b


The rotor rig of Fig. 3.108 is carried by an Oilite (oil impregnated, sintered bronze) bush supported on a rubber O-ring at the left inboard end, and by an oil lubricated Lucite plain cylindrical journal bearing at the outboard end. The rigid disc has the mass 0.81 kg , the polar mass moment of inertia 5.7835 ⋅ 10 −4 kg m 2 , the diametral mass moment of inertia 3.3572 ⋅ 10 −4 kg m 2 and is located at l = 22.4 mm from the right end. The flexible shaft of diameter


201

9.525 mm , density 7860 kg m 3 and Young’s modulus 2.06 ⋅ 1011 N m 2 has total length 0.59 m and is divided into six elements [19].

Fig. 3.108 (from [19]) The

parameters

of

the

left

bearing

are

k yy = 4.76 ⋅ 105 N m ,

k zz = 4.54 ⋅105 N m , c yy = 26.87 Ns m and c zz = 23.1 Ns m . The journal bearing parameters are D = 24.91 mm , L = 13 mm , C = 120 μm , μ = 0.02784 Ns m 2 . The static loads on bearings are W1 = 2.38 N and W2 = 8.8 N .

a

b

Fig. 3.109 The speed dependence of the stiffness and damping coefficients, calculated based on Moes’ impedance model, is shown in Fig. 3.109, b. The Campbell diagram for the first three modes of precession is presented in Fig. 3.110. The damped critical speeds are 2648, and 2867 rpm.


202

Fig. 3.110 The damping ratio diagram in presented in Fig. 3.111. Mode 3F becomes unstable at 5070 rpm.



203

Fig. 3.112 The root locus diagram in presented in Fig. 3.113 for speeds up to 8000 rpm. The curve of mode 3F intersects the zero damping line at the point marking the damped natural frequency at the instability threshold.

Fig. 3.113 The first four mode shapes at 4000 rpm are shown in Fig. 3.114.


204

a

b

c

d

Fig. 3.114 Mode 3 is a mixed mode, predominantly backward.

a

b

Fig. 3.115 The unbalance response curves are calculated at the disc location 5 (Fig. 3.115, a) and at the oil-film bearing 7 (Fig. 3.115, b) for an eccentricity of 0.0305 m of the disc mass.


205

References 1. Gasch, R. and Pfützner, H., Rotordynamik, Springer, Berlin, 1975. 2. Wölfel, H. P., Maschinendynamik, Umdruck zur Vorlesung, T. H. Darmstadt, 1989/90. 3. Radeş, M., On the effect of bearing damping on the critical speeds of flexible rotors, Buletinul Inst. Politehnic Bucureşti, Vol.42, No.3, pp.101-112, 1980. 4. Kellenberger, W., Elastisches Wuchten, Springer, Berlin, 1987, p.59. 5. Radeş, M., Influenţa amortizării lagărelor asupra turaţiilor critice ale rotorilor elastici, St. Cerc. Mec. Apl., Vol.30, No.6, pp.903-911, 1980. 6. Constantinescu, V. N., Nica, Al., Pascovici, M. D., Ceptureanu, Gh., and Nedelcu, Şt., Lagăre cu alunecare, Editura tehnică, Bucureşti, 1980. 7. Krämer, E., Maschinendynamik, Springer, Berlin, 1984. 8. Wang, W., and Kirckhope, J., New eigensolutions and modal analysis for gyroscopic/rotor systems. Part 1: Undamped systems, J. Sound Vib., Vol.175, No.2, pp 159-170, 1994. 9. Radeş, M., Mixed precession modes of rotor-bearing systems, Schwingungen in rotierenden Maschinen III, (Irretier, H., Nordmann, R. and Springer, H., eds.), Vieweg, Braunschweig, pp. 153-164, 1995. 10. Jei, Y.-G. and Kim, Y.-J., Modal testing theory of rotor-bearing systems, ASME J. of Vibration and Acoustics, Vol.115, pp.165-176, April 1993. 11. Radeş, M., Dynamics of Machinery, Vol.2, Univ. Politehnica Bucureşti, 1995. 12. Lee, C.-W., Vibration Analysis of Rotors, Kluwer Academic Publ., Dordrecht, 1993. 13. Genta, G. and Vatta, F., A lubricated bearing element for FEM rotor dynamics, Proc. Int. Modal Analysis Conf., pp 969-975, 1991. 14. Ocvirk, F., Short bearing approximation for full journal bearings, NACA TN 20808, 1952. 15. Childs, D., Moes, H., and van Leeuwen, H., Journal bearing impedance descriptions for rotordynamic applications, J of Lubrication Technology, pp.198-219, 1977. 16. Someya, T., (ed.), Journal-Bearing Databook, Springer, Berlin, 1988. 17. Bhat, R. B, Subbiah, R., and Sankar, T. S., Dynamic behavior of a simple rotor with dissimilar hydrodynamic bearings by modal analysis, ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol.107, pp.267-269, April 1985.

206


18. Subbiah, R., Bhat, R. B., Sankar, T. S., and Rao, J. S., Backward whirl in a simple rotor supported on hydrodynamic bearings, NASA CP 2409, Instability in Rotating Machinery, 1985. 19. Van de Vorst, E. L. B., Fey, R. H. B., De Kraker, A., and Van Campen, D. H., Steady-state behaviour of flexible rotor dynamic systems with oil journal bearings, Proc. WAM of ASME, Symposium on Nonlinear and Stochastic Dynamics, (A.K.Bajaj, N.S. Namachchivaya, R.A.Ibrahim, eds.), AMDVol.192, DE-Vol.78, New York, pp.107-114, 1994.

4. ROTOR DYNAMIC ANALYSIS

In order to understand the dynamic response of a rotating machine it is necessary to have information on the following aspects of its behavior: 1) the lateral critical speeds of the rotor-bearing-pedestal-foundation system; 2) the precession orbits as a response to different unbalance distributions, over the whole operating range of the machine; 3) the rotor onset speed of instability, i.e. the threshold speed for stable whirling due to the rotor/bearing and/or working fluid interaction; and 4) the response to transient excitation such as blade loss. In the prediction (design) phase of rotor dynamic analysis, the main concern is the placement of critical speeds with respect to the machine operating speed. In order to ensure smooth and safe operation, most standards require at least 15% separation margin between the operating speed and the critical speeds. When the critical speeds are within the undesirable range, they can be shifted outside this range by modifying bearing and support stiffnesses, the bearing span, the disc mass properties or the shaft geometry.

4.1 Undamped critical speeds For lightly damped rotor systems, the associated undamped isotropic system is used for critical speed computation. When the bearings have different stiffnesses, a single mean value is considered.

4.1.1 Effect of support flexibility The simplest rotor with distributed mass consists of a shaft with uniform cross-section, simply supported at the ends on flexible bearings of equal stiffness k B 2 . Figure 4.1 shows the influence of bearing flexibility on the first four lateral critical speeds of the undamped rotor system.


208

In Fig. 4.1, the ratio of the rotor critical speed on flexible bearings to the first critical speed on rigid bearings ω 1 ( k B = ∞ ) is plotted against the dimensionless flexibility parameter

[δ ( g B

ω 12

)]

12

, where δ B is the static

deflection of bearings. The first and second critical speed ratios show a continuing decrease with increasing bearing flexibility. At the same time, the associated mode shapes gradually transform, from those corresponding to rigid bearings to the ‘cylindrical’ and ‘conical’ rigid-body mode shapes.

Fig. 4.1 The third and fourth critical speed ratios first exhibit a rapid decrease with increasing bearing flexibility and then level out and asymptotically approach values of 2.27 and 6.25. These two modes are referred to as the first and second flexible “free-free” modes of the system.

4. ROTORDYNAMIC ANALYSIS

209

In Fig. 4.2, the first four critical speed ratios are plotted against the

[ (g

dimensionless stiffness parameter δ B

ω 12

)]

−1 2

.

Fig. 4.2

4.1.2 Critical speed map A convenient means for analyzing the influence of rotor support dynamic properties on the dynamic performance of a rotor-bearing system is by a critical speed map, as shown in Fig. 4.3. In such a chart, the horizontal scale represents support stiffness, N m , and the vertical scale is rotor speed, rpm. The curves are drawn calculating the first few lateral critical speeds for different values of an average constant stiffness of the bearings. For relatively highly damped rotors, an equivalent dynamic stiffness is used, calculated as


210

kd =

k2 + ( Ω c )2 ,

(4.1)

where k and c are the average stiffness and damping coefficients, respectively, and Ω is the rotor angular speed. But few actual bearings retain constant stiffness with speed change. To determine the actual critical speeds, it is necessary to plot the bearing stiffness versus speed characteristic over the rotor critical speed lines. The intersections between the rotor curves and the bearing characteristic are the critical speeds for the actual rotor in its bearings. Figure 4.3 shows a rotor having a “soft” support in the horizontal direction, k zz , and a “harder” stiffness in the vertical direction, k yy . The lower stiffness mode shape in the horizontal direction at the first critical speed is almost cylindrical. In the vertical direction, higher support stiffness causes this rotor to be almost simply supported.

Fig. 4.3 (from [1]) In order to minimize the rotor precession radius, hence the vibration amplitude, one should not design the machine to operate at the critical speed. There are two ways that this can be avoided: 1) change bearing stiffness; 2) change rotor geometry. The first method works when the support stiffness curves cross the critical speed curves in the sloped region (Fig. 4.4, a). In this position, an undesirable resonance can be shifted upward and downward in frequency by changing the support stiffness. Such a shift can be accomplished by a small geometric change in


211

the bearings such as decreasing the clearance or increasing the preload to raise the critical speed (10 to 20%).

a

b

Fig. 4.4 (from [1]) High bearing stiffness (Fig. 4.4, b) is undesirable. The support stiffness curves are in the flat part of the critical speed curves, where simple geometric bearing changes have minor effects on resonance. The shaft is much more flexible than the bearing supports.

Fig. 4.5 (from [1]) Rotors with stiff bearings operating near criticals change the engineer options to the rotor shaft. He can modify the shaft diameter (Fig. 4.5) or change the


212

bearing span. An increase in resonance in the asymptotic region of stiffness-speed curves is dependent on rotor construction. Attempts to alter resonance conditions in this region by bearing stiffness changes are often doomed to failure. In some applications, the insensitivity of the third critical speed to support stiffness permits a range of operating speeds that does not traverse any of the critical speeds. Many machines are designed to work between the second and the third critical speed where, in the region of soft bearings, there is the largest speed range between criticals. The critical speed map provides a useful guide to the dynamic performance of rotor-bearing systems, especially for lightly damped machines such as those in rolling element bearings, and for externally pressurized gas bearings. Apart from showing just undamped critical speeds, it shows only possible critical speeds. For other bearings having high damping, e.g. oil lubricated hydrodynamic journal bearings, bearing stiffness alone does not determine rotor behaviour. Damping plays an equally important part, especially at high speeds. The first two ‘rigid-body’ precession modes have relatively large orbit radii in soft bearings so that the motion at the critical speeds can be completely damped and the rotor passes through the critical without noticeable vibrations. The critical speed map is particularly useful for rotors in tilting pad bearings. Bearings with vertical preloading on the pad are ‘more anisotropic’ than the bearings with loading between pads, as shown in Fig. 4.6.

Fig. 4.6 (from [1]) Critical speed maps are also constructed for machines on more than two bearings. Figure 4.7 shows such a diagram for a 15 MW synchronous electrical


213

motor without slip rings. In the expected range of bearing flexibility α = 1− 2 mm MN , the first critical speed is 2100 − 2400 rpm and the other critical speeds are above the maximum operating speed of 5000 rpm [2].

Fig. 4.7 (from [2]) Another critical speed map for a generator rotor on three bearings (used for balancing) is presented in Fig. 4.8, where the mode shapes are also shown for various bearing flexibilities. The hatched area marks the speed range below onethird the first critical speed for rigid supports ( α = 0 ).

Figure 4.9 shows the differences in the mode shape forms of a turbogenerator, coupled and uncoupled. The first five mode shape forms and the corresponding critical speeds of the shaft line of the generator G coupled with the turbine T are shown in Fig. 4.9, a. The first two mode shapes of the generator in three bearings (as it is balanced) are shown in Fig. 4.9, b. The first and second mode shapes of the generator alone are recognized in the first and fifth mode shapes of the shaft line, but occur at different speeds. The rigid coupling places a firm restraint on the shaft end, changing the critical speed, but the difference between the corresponding mode shapes is small.


214

Fig. 4.8 (from [3])

Fig. 4.9 (from [3])


215

Figure 4.10 shows the critical speed map for an industrial turbine rated 50 MW. The mode shapes of the shaft line show the predominance of each component – generator G, turbine T and exciter E - at the respective critical speed. Comparing the operating speed of 3600 rpm with the critical speed lines it is concluded that the practical region of bearing flexibility is between 5 and 10 μm tf .

Fig. 4.10 (from [4])

216


In practice, the following important conclusions are useful. Undamped critical speeds calculated taking into account the bearing support stiffness are lower than those predicted by rigid support analysis. The less stiff the bearings are relative to the shaft, the more likely the mode shape nodes will be displaced from the bearings. With the nodes displaced from the bearings, relative motion between the bearing and shaft will occur at the critical speeds. This relative motion produces a velocity-dependent force. The larger the displacement, the larger the velocity and, thus, the larger the damping force. Bearing damping (inherent in hydrodynamic bearings) has the effect of raising the undamped critical speeds. Damped critical speeds will be determined using the Campbell diagrams. The amount of increment depends on the degree to which the undamped critical speeds are depressed from the rigid support criticals and on the location of the nodal points with respect to the bearing centre lines. Bearing damping does not raise the undamped critical speeds if these are close to rigid support critical speeds. It is of paramount importance to distinguish among undamped critical speeds, damped critical speeds and peak response speeds. Undamped and damped critical speeds are different from the speeds of peak steady-state unbalance response. This requires unbalance response analyses. In general, the percentage deviation of the first critical lies within a range of ± 6% . For higher critical speeds, the variations may be significantly greater than 6 percent. The plotting of undamped critical speeds versus static stiffness in order to predict peak response speeds can be very misleading and should be avoided. This is especially true when trying to specify the location of second and third modes for the purpose of meeting specification on their closeness to maximum continuous operating speeds, as required by, for example, API Standards. According to the API Standard 617 [5], to avoid excessive vibration, the first lateral critical speed of rigid-shaft compressors must be at least 20 per cent higher than the maximum continuous speed. Flexible-shaft compressors must operate with the first critical speed at least 15 per cent below any operating speed. The second lateral critical speed must be 20 per cent above the maximum continuous speed. The same conditions are imposed by API Standard 613 [6] for gear units. For special-purpose steam turbines, the API Standard 612 [7] requires a 10 per cent margin for rigid-shaft rotors while the first critical speed of a flexible-shaft rotor should not exceed 60 per cent of the maximum continuous speed, nor should it be within 10 per cent of any operating speed.


217

4.1.3 Influence of stator inertia An interesting application of a critical speed map is the selection of bearings for a balancing facility [3]. Figure 4.11 shows a resonance curve measured on a bearing pedestal using a vibration exciter. The driving point displacement is plotted against the excitation frequency for constant amplitude of the sinusoidal excitation force. The narrow bandwidth and the sharp phase change at resonance indicate practically negligible damping, the quality factor being Q = 14 .

Fig. 4.11 (from [3]) Neglecting the damping, the bearing dynamic flexibility (ratio of deflection to force amplitude) can be plotted versus speed, instead of frequency, as in Fig. 4.12, a. Plotting the speed versus flexibility in semi-logarithmic coordinates (Fig. 4.12, b), the bearing dynamic characteristic encompassing both stiffness and mass effects is obtained. It has the same format as the critical speed map of a rotor mounted in idealized elastic bearings (springs), so that they can be overlaid to obtain the critical speeds of the combined system. Figure 4.13, a shows the critical speed map for the rotor of a turbo generator rated 130 MVA ( 60 Hz ) . The speed-flexibility curve of the bearing from Fig. 4.11, transformed as in Fig. 4.12, is overlaid.


218

a

b

Fig. 4.12 The cross-over points of the two curves give the critical speeds for the combined rotor-bearing system. In the range from zero to slightly above overspeed (4320 rpm) there are four critical speeds, at 1000, 2900, 4100 and 4400 rpm. Unfortunately, two critical speeds lie very close to the overspeed, which is undesirable, making the balancing difficult.

a

b

Fig. 4.13 (from [3]) The bearings are too ‘soft’. If the bearing pedestal is the softest element in the chain, then it should be stiffened.


219

Figure 4.13, b shows the critical speed map of the same rotor in which the speed-flexibility characteristic of another stiffer bearing is overlaid, having a higher natural frequency of 128 Hz (7690 rpm). In this case, the bearing dynamic flexibility is almost constant 5 μm t f . This leaves only two critical speeds, at 1000 and 2900 rpm, well below the operating speed and the overspeed. Operating a rotor at or near a forward critical speed causes large and potentially damaging deflections and should be avoided.

4.2 Damped critical speeds Bearing damping shifts the critical speeds to larger values. When the damping is significant, as for most hydrodynamic bearings, the main concern is the computation of damped critical speeds and their associated damping ratios. They are determined solving the eigenvalue problem of the linearized damped rotor system.

4.2.1 Linear bearing models The nonlinear characteristics of the sliding bearings can be linearized at the static equilibrium position, as shown in Chapter 6. The dynamic characteristics of a bearing can be represented by four stiffness and four damping coefficients. The amplitude of the forces acting on the shaft journal can be expressed as in (1.1) ⎧ f y ⎫ ⎡k yy k yz ⎤ ⎧ y ⎫ ⎡c yy c yz ⎤ ⎧ y& ⎫ ⎨ ⎬ = ⎢k ⎥⎨ ⎬ , ⎥⎨ ⎬ + ⎢ ⎩ f z ⎭ ⎣ zy k zz ⎦ ⎩ z ⎭ ⎣ c zy c zz ⎦ ⎩ z& ⎭

(4.2)

where f y , f z are the components acting in the y , z directions, respectively, y , z and y& , z& are the journal displacements and velocities, in the same directions. For journal bearings, the eight dynamic coefficients k yy ,..., k zz , c yy ,.., c zz are functions of the Sommerfeld number (see Chapter 6). For rolling element bearings, the bearing coefficients can be assumed to be constant and with the cross coupling coefficients equal to zero. In the following applications, three kinds of data will be used for hydrodynamic bearings: a) the Moes’ impedance model [8] and b) the Ocvirk short bearing model [9] for plain circular bearings, and c) tabular values of the eight dynamic dimensionless coefficients as a function of either speed or the Sommerfeld number, as given in Someya’s book [10]. Spline interpolation is used in the last case for calculations at speeds not included in the initial data.


220

4.2.2 Equations of damped motion Rotor-bearing systems are modelled as an assemblage of rigid discs, distributed mass and stiffness shaft elements, and discrete bearings and seals. Once the element mass, stiffness and damping matrices are established (see Chapter 5), they can be assembled to obtain the global system matrices. The damped critical speeds and system stability limits are determined from the homogeneous form of the equations of motion [ M ]{ &x&} + [ C ]{ x& } + [ K ]{ x } = { 0 } ,

(4.3)

where { x } is the global vector of nodal coordinates and [ C ] = [ C D ] + Ω [ G ] , where [ G ] is the skew-symmetric gyroscopic matrix, Ω is the rotational speed, and [ C D ] is the damping matrix. In (4.3), [ M ] is the symmetric mass matrix, the stiffness matrix [ K ] is usually unsymmetrical due to the cross coupling terms of hydrodynamic bearings and the clearance excitation factors. The damping matrix [ C D ] is usually symmetric, but it can be unsymmetrical when the internal damping or structural damping due to shrink fits is taken into account.

4.2.3 Eigenvalue problem of damped rotor systems It is convenient to write the system equation (4.3) in the state space form [ A ]{ q& } + [ B ]{ q} = { 0 } ,

(4.4)

where the matrices [ A ] , [ B ] and the vector { q } are defined as

[ A ] = ⎡⎢

M ⎣0

0⎤ , I ⎥⎦

[ B ] = ⎡⎢

C ⎣− I

K⎤ , 0 ⎥⎦

{ q } = ⎧⎨

x& ⎫ ⎬. ⎩x⎭

(4.5)

On trying a solution to equation (4.4) of the form

{ q } = { y } eλ t

(4.6)

we obtain the linear eigenvalue problem

( λ [ A ] + [ B ] ){ y } = { 0 } .

(4.7)


221

Since matrix [ A ] is a positive definite real symmetric matrix and [ B ] is an arbitrary real matrix, the generalized eigenvalue problem (4.7) can be reduced to the standard form

(− [ A]

−1

[ B ] ){ y } = λ { y } ,

(4.8)

where

(− [ A]

−1

[ B ])= ⎢ − [ M ] [C ] [I ] ⎢⎣ ⎡

− [M

] −1 [ K ] ⎤ ⎥ [0 ] ⎥⎦

−1

(4.9)

is an unsymmetrical real matrix. Due to the generally unsymmetrical matrices [ C ] and [ K ] , the eigenvalues λr of equation (4.8) are real numbers for overdamped modes and complex numbers for underdamped modes. Because the matrix has real elements, the complex eigenvalues must occur in complex conjugate pairs and have the form

λr = α r + iωr , λr = α r − iωr

(r = 1, 2,3,....)

(4.10)

and are functions of the rotational speed Ω . The imaginary part ω r is the damped natural frequency (of precession) and the real part α r is an attenuation (or growth) constant. Usually, the damping is expressed in terms of the modal damping ratio

ζr = −

αr α r2 + ωr2

≅−

αr . ωr

(4.11)

For underdamped systems, the complex eigenvectors have the form ⎧ λr { ur } ⎫ ⎬, ⎩ { ur } ⎭

⎧ λr { u r } ⎫ ⎬, ⎩ { ur } ⎭

(4.12)

{ ur } = { ar } − i { br }

(4.13)

{ yr } = ⎨

{ yr } = ⎨ where

{ ur } = { ar } + i { br },

so that only the lower half is considered.


222

4.2.4 Campbell diagrams Plots of the damped natural frequencies ωr as a function of the rotor speed Ω are called precession speed maps. When these plots contain the excitation lines overlaid they are referred to as interference diagrams or Campbell diagrams. It is common practice to plot also the damping ratios versus the rotor speed. These plots will be referred to as the damping ratio diagrams. Figure 4.14 shows a typical Campbell diagram for a multi-mass flexible rotor. As shown in Section 3.3.5, the rotor precession is usually described by forward (F) and backward (B) modes. The stator anisotropy gives rise to pairs of backward and forward modes. The gyroscopic effect splits a B and F mode pair, softening the B mode (lowering its natural frequency) and stiffening the F mode (increasing its natural frequency). For actual rotors, the normal sequence of B and F modes can change. The speed-dependent bearing coefficients, the high damping levels and the formation of compounded modes, with mixed B and F precession, require the calculation and use of precession mode shape forms for the proper labeling of modes in the Campbell diagram.

Fig. 4.14 In some cases, it is better to use a mode numbering based on the index of the eigenfrequencies (sorted in ascending order) and not based on the mode directivity B or F. Mixed modes (M) are difficult to label. One can either mention


223

the percent of F and B motion along the rotor stations, or indicate the two basic components of a compounded mode which influence each other to give rise to the M mode. A critical speed of order κ is defined as the rotor speed for which a multiple of that speed coincides with one of the system natural frequencies of precession. An excitation frequency line has an equation ω = κ Ω . It is a line of slope κ passing through the origin of the Campbell diagram. The intersection of this line with the damped natural frequency curve ωr defines the damped critical speed Ω r . When Ω equals Ω r , the excitation frequency κ Ω r creates a resonance (critical) condition. One approach for determining critical speeds is to use the diagram of damped natural frequencies versus speed and overlap all excitation frequency lines of interest, marking the intersection points of the two families of curves. Their abscissae determine the damped critical speeds. For κ = 1 , ω = Ω is the synchronous excitation line, usually due to mass unbalance. For κ = 2 , ω = 2Ω is the misalignment excitation line. For κ ≅ 1 2 , ω = Ω 2 is the half-frequency subharmonic excitation line due to oil whirl in plain bearings.

4.2.5 Orbits and precession mode shapes A damped mode shape can be solved from equation (4.4) with a particular root substituted

{ xr (t ) } = 2eα r t ( { ar } cos ωr t − { br } sin ωr t ) ,

(4.14)

which describes spiralling orbits. However it is agreed to represent the orbits as incomplete (open) ellipses, considering α r = 0 and approximating the expression (4.14) by { xr (t ) } = ℜe { ur } cos ωr t − ℑm { ur } sin ωr t . (4.15) It is common practice to plot spatial precession mode shapes by first drawing an ellipse at each station, then connecting the points on the ellipses at all stations along a rotor (at a given time t). Figure 4.15 shows a typical precession mode shape. In the following examples, the points corresponding to t = 0 are connected by a solid line, while the points at a quarter of a period later are connected by a broken line. This way, the directivity of the motion along each orbit is from the solid line to the broken line.


224

Fig. 4.15 There are ellipses with forward precession (denoted F), ellipses with backward precession (denoted B), and ellipses degenerated into straight lines. Mixed modes are described in Section 3.3.5.

4.3 Peak response critical speeds Undamped and damped critical speeds have been defined based on the coincidence of an excitation frequency with a rotor natural frequency. Both are ‘possible’ critical speeds, as far as nothing is said about the level of damping. Of practical interest are the rotational speeds at which the rotor response has the largest value. The abscissae of peaks in the plots of the unbalance response at a rotor station as a function of the rotational speed determine the so-called peak response critical speeds. As shown in Example 3.1, these criticals are slightly different from the damped critical speeds and depend on the location along the rotor where they are calculated. In machines with journal bearings, the relative motion between journal and bearing is measured with proximity transducers. The largest orbit radius (major semiaxis for ellipses) is an indication of the severity of rotor precession. The peaks in the diagrams of the steady state synchronous response at a rotor section, for a given unbalance magnitude and location, indicate the peak response critical speeds. Figure 4.16 shows an example of unbalance response numerical simulation for an industrial turbine rotor. First, the total unbalance is estimated based on existing standards on permissible residual unbalance values (ISO 1940) [11]. For a turbine shaft, a quality grade G2.5 is usually selected. [12].


225

The value G = 2.5 mm/s corresponds to a centre of gravity offset

e=

G

ω

=

2.5 ⋅ 1000 23,885 24 ,000 = ≈ π nN nN nN 30

[ μm ]

(4.16)

where n N is the operating speed , rpm. The total unbalance is

U = me =

24,000 ⋅ m nN

[ mm ⋅ g ] ,

(4.17)

where m is the mass of the rotor section between two bearings, kg. For a turbine operating at 3000 rpm, the permissible residual unbalance is U = 8m [mm g]. Using a finite element model of the rotor, the first modes of lateral vibration are calculated, usually all modes below the trip speed and the mode just above the trip speed. Then, the worst unbalance distribution for each mode is considered, as in Figs. 4.16, a − d , subdividing the total unbalance into suitable individual unbalance components. Apart from the first three modes of vibration, the vibration produced by the overhang half couplings is also considered.

Fig. 4.16 (from [12]) The result of the unbalance response calculation at the left bearing location is shown in Fig. 4.16, e. Each unbalance distribution results in a different amplitude versus speed curve. The amplitude A is calculated as the major semiaxis of the precession elliptical orbit.


226

The positions of the amplitude peaks along the horizontal axis indicate the peak response critical speeds. In figure 4.16, e, n 1,1 and n 1,2 are the critical speeds for the first unbalance distribution, n2,1 and n2,2 are the critical speeds for the second unbalance distribution. The amplitudes of the unbalance response have to be compared with limit values given by guidelines and standards for the operating speed n N . The guideline ISO 7919-2 [13] indicates, as a limit of 'good' vibration performance, a maximum value of the journal orbit radius

smax A =

2400 [μm] . nN

(4.18)

Taking a safety factor of 1.7, the limit value is established at A lim =

smax A 1 .7

=

1400 nN

[μm] .

(4.19)

For a turbine operating at 3000 rpm, A lim = 26 μm . The condition A ≤ A lim must be satisfied at all speeds up to n N . In Fig. 4.16, e, the peak at n2 ,2 higher than A lim is beyond the operating speed range.

Fig. 4.17 (from [12]) The effectiveness of the unbalance response calculation is illustrated in the following by means of an example concerning a turbine rotor used for mechanical drive with variable operating speeds [12].


227

The critical speed map for the first two modes of vibration is shown in Fig. 4.17, a. The average value of oil film flexibilities of both bearings for the first mode is α v =1.3 mm/MN in the vertical direction and α h =3.7 mm/MN in the horizontal direction. According to Fig. 4.17, a, there are two undamped critical speeds within the speed range from 2600 to 5800 rpm, the first at 3250 rpm with a predominant horizontal response, and the second at 5000 rpm with a predominant vertical response. The turbine must therefore be operated at both critical speeds which is inadmissible according to the conventional critical speed design considerations. The commissioning report, however, stated: "Turbine operating behavior is very good irrespective of load and speed. Turbine shaft amplitudes are 8 − 10 μm , bearing housing vibrations are 1 − 3 μm , in the horizontal and vertical directions. Critical speeds could not be determined" [12]. The unbalance response calculation confirmed the observed operating behavior. Shaft vibration amplitudes of both bearings show absolutely no resonance peaks near the critical speeds (solid lines in Fig. 4.17, b). Despite this, there are two critical speeds in the operating speed range which can be determined if the unbalance response is calculated with the oil film damping reduced to 10% (dotted lines). As shown in Fig. 4.17, b, there are two unbalance response peaks, the first at 3250 rpm and the second at approximately 5000 rpm. The operating behavior of the shaft was smooth, since in this case it was short and quite rigid and was also provided with a soft oil film. The softer the oil film is in relation to the shaft rigidity, the more the shaft moves in its bearing so that oil film damping becomes fully effective.

4.4 Stability analysis The stability of a linear rotor-bearing system depends on the damping exponent α r (4.10). A positive damping exponent indicates instability. The solution (4.14) shows that α r > 0 is a growth factor, the whirling motion is along a spiral with growing radius, while α r < 0 is a decaying factor, the whirl being along a spiral with decreasing radius. When the real part of one of the roots changes from negative to positive, α r = 0 , the rotor reaches the instability threshold. The corresponding imaginary part defines the precession frequency of the incipient rotor instability. The rotational speed at the threshold of instability, ns , is called the onset speed of instability.


228

In most applications, when the system becomes unstable, the rotor whirls in its first forward precession mode. The motion associated with an instability becomes unbounded in time. Below ns the rotor motion is stable and synchronous. Above this speed, there is a subsynchronous component to the rotor motion whose amplitude diverges exponentially with time and is a forward motion. The onset speed of instability always exceeds the rotor first critical speed. Most of the destabilizing forces in rotor systems are “cross-coupled” in two directions. A radial deflection of the shaft, away from its equilibrium position, gives rise to a tangential force which, if it is larger than the opposite damping force, drives the rotor in an orbital motion. The destabilizing force is proportional to the shaft deflection and grows larger as the radius of the whirl grows. The self-excited motion is along a spiral with increasing radius until is limited by nonlinear effects. The results of the linear theory predict the onset speed of instability but do not indicate the degree of instability, i.e. neither the violence of the motion at onset nor the growth of the unstable motion with increasing speed. A common type of instability for rotors with oil-lubricated bearings is the “oil whirl”. The oil film wedge drives the journal within the bearing at slightly less than half the running speed, hence the name of “half-frequency whirl”. A sliding bearing can be made stable by increasing its natural frequency or by increasing the bearing eccentricity ratio ε = e C , where e is the journal eccentricity and C is the radial clearance. Changing the natural frequency of the bearing is considerably more difficult than adjusting the ε ratio. The latter can be increased by decreasing the Sommerfeld number (see Chapter 6) or by decreasing the bearing length/diameter ratio. The Sommerfeld number is defined as S=μ

2

NLD⎛ R ⎞ ⎜ ⎟ , W ⎝C ⎠

(4.20)

where R = D 2 is the bearing radius, L is the bearing length, C is the bearing radial clearance, μ is the oil dynamic viscosity and N = Ω 2π is the journal rotational frequency. The following changes have a tendency to decrease the Sommerfeld number: 1) increasing the bearing average pressure p = W L D by: a) grooving the bearing circumferentially to reduce the surface loading area, using pressure dams and pockets; b) misaligning the bearing purposely to achieve greater loading; and


229

c) changing the valve-opening sequence to increase the loading due to partial admission bearing reaction. 2) increasing the bearing clearance; 3) increasing the bearing temperature which reduces the oil viscosity. For plain cylindrical bearings, the difference of cross-stiffness coefficients k yz − k zy decreases with decreasing Sommerfeld number, which improves the stability (see Chapter 6). At high Sommerfeld numbers, the coefficient k zy has negative values, increasing the difference. Bearings operating at speeds with positive k zy values are recommended. The eccentricity ratio can be increased by preloading. This observation led to the construction of multilobe bearings, where the circular arcs are displaced towards the bearing centre to obtain the preloading. For each circular sector an oil wedge is formed producing a radial pressure distribution giving rise to forces that centre the rotor, stabilizing it. Plain cylindrical bearings have been replaced by two-, three- or four-lobe bearings with improved stability. Changing from sleeve type journal bearings to tilting-pad bearings also eliminates oil whirl. This type of bearing has zero cross-coupling stiffness coefficients so that it is theoretically completely stable. With increasing speed, the oil whirl frequency increases until it reaches the natural frequency of the rotor-bearing system when it dwells on this frequency, becoming an oil whip, also called resonant whirl. The identifying frequency of this condition is either slightly less than or slightly greater than half the running-speed frequency.

Fig. 4.18 The increase of the onset speed of instability by changing the bearing type was studied [14] for the rotor from Fig. 4.18. The uniform shaft of diameter 80 mm has lengths l1 = l 3 = 0.3 m , l 2 = l 4 = 0.2 m , Young’s modulus 2 ⋅ 1011 N m 2 and mass density 8000 kg m 3 .


230

The three identical discs have the following mass and mass moments of inertia: m = 15 kg, J T = 0.05 kg m 2 , J P = 0.1 kg m 2 . The bearings have the length/diameter ratio 0.5, radial clearance C p = 300 μm and oil dynamic viscosity 5 ⋅ 10 −3 N s m 2 . When applicable, the preload factor is m p = 1 − Cb C p = 3 4 , where Cb is the assembled clearance and C p is the machined clearance. The speed dependence of the bearing stiffness and damping coefficients was taken from [10].

Fig. 4.19 (from [14]) The stability diagrams for six different bearings are overlaid in Fig. 4.19 for the given rotor configuration. The onset speeds of instability are 9,155 rpm for cylindrical bearings without axial grooves, 9,220 rpm for cylindrical bearings with two axial grooves, 11,760 rpm for 2-lobe bearings, 12,244 rpm for 4-lobe bearings, 12,389 rpm for 3-lobe bearings and 14,332 rpm for cylindrical bearings with pressure dam.


231

4.5 Simulation examples In the following, examples of rotor dynamic analysis are given for selected rotor models taken as benchmark examples. The numerical results are dependent on the spatial discretization error (number of finite elements in the model) and the speed resolution (number of speeds in a given interval). Example 4.1

A simply supported rotor with three discs (Fig. 4.20) is considered in this example [15]. The shaft of length 1.3 m and diameter 0.1 m has the Young’s modulus 2 ⋅ 1011 N m 2 and the mass density 7800 kg m 3 . It was divided into 13 beam finite elements of length 0.1 m each.

Fig. 4.20 The three rigid discs, located at stations 3, 6 and 11, have the following masses and mass moments of inertia: m 1 = 14.58 kg, J T 1 = 0.0646 kg m 2 , J P1 = 0.123 kg m 2 ,

m 2 = 45.94 kg,

J T 2 = 0.498 kg m 2 ,

J P 2 = 0.976 kg m 2 ,

m 3 = 55.13 kg, J T 3 = 0.602 kg m 2 , J P 3 = 1.171 kg m 2 . The two identical orthotropic bearings are located at stations 1 and 14, and have the following stiffness and damping coefficients: k yy = 7 ⋅ 107 N m , k zz = 5 ⋅ 107 N m , c yz = c zy = 0 .

k yz = k zy = 0 ,

c yy = 7 ⋅10 2 Ns m ,

c zz = 5 ⋅ 10 2 Ns m ,


232

Fig. 4.21 The Campbell diagram for the first 10 natural modes is shown in Fig. 4.21. The damping ratio diagram is shown in Fig. 4.22.

Fig. 4.22 The precession modes occur in pairs, the lower mode in a pair with backward precession, and the upper with forward precession. With increasing mode index, the gyroscopic effect makes the two lines in a pair to become more


233

divergent. The line 5B crosses the line 4F so that beyond 20000 rpm the mode ranking is changed. Modes are labeled according to their order at low rotational speeds. The synchronous excitation line (dotted line) intersects the natural frequency lines at the points whose abscissae determine the damped critical speeds 3620 , 3798 , 10017, 11278, 16769, 24397, 26604 rpm, etc.

a

b

c

d

e

f

g

h

i

j

k

l

Fig. 4.23 Twelve precession mode shapes at 25000 rpm are shown in Fig. 4.23. The orbits at any station are ellipses due to bearing anisotropy. The mode shape at


234

t = 0 is plotted with solid line and the mode shape at t = π 2Ω is drawn with broken line, so that the motion along the orbit takes place from the point lying on the solid line, at t = 0 , to the point lying on the broken line, a quarter of a period later. Modes 1B and 1F are almost ‘cylindrical’. Modes 2B and 2F are almost ‘conical’. Modes 3B and 3F are ‘two-node’ flexural, modes 4B and 4F are ‘threenode’ flexural, etc.

Fig. 4.24 The unbalance response curves calculated at station 6 are shown in Fig. 4.24 for the orbit major semiaxis (solid line) and minor semiaxis (broken line). A mass unbalance of 200 g mm on the disc at station 6 was considered. Example 4.2

Consider the rotor of Example 4.1, but with the following bearing stiffness and damping coefficients [16]:

′ = 5 ⋅ 107 N m , k ′yz = k ′zy = −4 ⋅ 107 N m , k ′yy = 7 ⋅ 107 N m , k zz c′yy = 7 ⋅103 Ns m , c′zz = 4 ⋅ 103 Ns m , c′yz = c′zy = 0 ,

′′ = 4 ⋅ 107 N m , k ′yz ′ = 6 ⋅ 107 N m , k zz ′ = k zy ′′ = −4.5 ⋅ 10 7 N m , k ′yy ′ = c′zy ′ =0. ′ = 6 ⋅ 103 Ns m , c′zz′ = 5 ⋅ 103 Ns m , c′yz c′yy The Campbell diagram for the first eight modes is shown in Fig. 4.25. The conservative cross-stiffness increases the interval between the eigenvalues in a pair corresponding to the same modal index. When a lower index forward mode approaches a higher index backward mode, as for modes 4 and 5, the result is a


235

curve veering in the Campbell diagram (at about 28000 rpm) with a corresponding crossing of damping ratio curves. The relative departure of the two curves in a pair gives rise to mixed modes. Because all modes are mixed, they are labeled in ascending order, with their eigenvalue index at very low rotational speeds.

Fig. 4.25

Fig. 4.26 The damping ratio diagram is shown in Fig. 4.26. Predominantly backward modes (like 1 and 3) are more damped that their (predominantly) forward pair (2 and 4).


236

a

b

c

d

e

f

g

h

i

j

k

l

Fig. 4.27 Twelve precession mode shapes at 25000 rpm are shown in Fig. 4.27. For mixed modes, the precession along the ellipse takes place from the point lying on the solid line, at t = 0 , to the point lying on the broken line, a quarter of a period later. The bearings have the principal axes of stiffness oriented at + 450 and − 450 , respectively, relative to the vertical axis, so that the elliptical precession orbits have inclined axes.


237

The root locus diagram is presented in Fig. 4.28. When the root loci are close to each other, two modes with nearly the same natural frequency (4 and 5) and different deflected shapes can combine to yield a compounded mode which has mixed backward and forward precession due to the coupling between modes.

Fig. 4.28 The unbalance response curves calculated at station 6 are shown in Fig. 4.29 for the orbit major semiaxis (solid line) and minor semiaxis (broken line). A mass unbalance of 200 g mm on the disc at station 6 was considered. There is no peak corresponding to the 5th mode, due to the relatively high damping. The ordinate in Fig. 4.29, a is logarithmic and in Fig. 4.29, b is linear. The latter shows better the regions with backward precession, where the minor semiaxis has negative values.

a

b

Fig. 4.29


238 Example 4.3

A

multi-stepped

11

2.078 ⋅ 10 N m

2

rotor

(Fig.

4.30)

has

Young’s

modulus

3

and mass density 7806 kg m . At station 5 it carries a rigid

disc of mass 1.401 kg , diametral and polar mass moments of inertia 0.00136 kgm 2 and 0.00203 kgm 2 , respectively. The shaft is supported at stations 11 and 15 by isotropic bearings with the following constant stiffness and damping coefficients: k yy = k zz = 4.378 ⋅107 N m , and c yy = c zz = 1.752 ⋅103 Ns m . The geometric data are given in Table 4.1 [17].

Fig. 4.30 Table 4.1 Element no.

Length, mm

1 2 3 4 5 6 7 8 9

12.7 38.1 25.4 12.7 12.7 5.1 7.6 12.7 7.6

Outer radius, mm 5.1 10.2 7.6 20.3 20.3 30.3 30.3 25.4 25.4

Inner radius, mm 0 0 0 0 0 0 15.2 17.8 0

Element no.

Length, mm

10 11 12 13 14 15 16 17 18

30.5 25.4 38.1 38.1 20.3 17.8 10.2 30.4 12.7

Outer radius, mm 12.7 12.7 15.2 15.2 12.7 12.7 38.1 20.3 20.3

Inner radius, mm 0 0 0 0 0 0 0 0 15.2

The Campbell diagram is shown in Fig. 4.31 for the first six modes of precession. The damped critical speeds, determined at the intersections with the synchronous excitation line, are also shown in the diagram.


239

Fig. 4.31 The damping ratio diagram is presented in Fig. 4.32.

Fig. 4.32


240

The first six precession mode shapes at 50000 rpm are shown in Fig. 4.33.

a

b

c

d

e

f

Fig. 4.33 Figure 4.34 shows the unbalance response orbit radius versus speed at station 15 (right bearing) for an unbalance of 200 g mm on the disc at station 5. In the considered speed range, there are only two peak response critical speeds at the frequencies of modes 1F and 2F.

Fig. 4.34


241

Example 4.4

A rotor is modeled as a 13 station (12 elements) assembly with stations as indicated in Fig. 4.35 [18]. It is supported by bearings at stations 3, 6 and 13. The four rigid discs which represent the fan, the low and high pressure compressors and the turbine are located at the stations 1, 4, 5, and 12. Details of the rotor configuration are listed in Table 4.2.

Fig. 4.35 Table 4.2 Inner diameter, mm 28.4

Element no.

Length, mm

42.9

Outer diameter, mm 59

152.4

Outer diameter, mm 59

Inner diameter, mm 53.8

7

2

46.0

59

28.4

8

152.4

59

53.8

3

16.0

59

28.4

9

152.4

59

53.8

4

96.8

59

28.4

10

152.4

59

45.2

5

75.2

59

39.2

11

149.8

59

28.4

6

165.1

59

53.8

12

78.0

59

46.2

Element no.

Length, mm

1

The shaft has Young’s modulus 2.069 ⋅ 1011 N m 2 and mass density

8193 kg m 3 . The disc data are given in Table 4.3. Table 4.3 Disc no. 1 2 3 4

Station 1 4 5 12

Mass, kg 11.38 7.88 7.7 21.7

Polar mass moment of inertia,

Diametral mass moment of inertia,

10 2 ⋅ kg m 2 19.53 16.70 17.61 44.48

10 2 ⋅ kg m 2 9.82 8.35 8.80 22.44


242

The isotropic bearing data are given in Table 4.4. Table 4.4 Bearing no.

Station

Stiffness coefficients,

Damping coefficients,

10 −6 ⋅ N m

10 −3 ⋅ N s m

1

3

1.751

1

2

6

96.95

1

3

13

13.368

1

The Campbell diagram is presented in Fig. 4.36 for the first 8 natural modes. At the crossing points with the synchronous excitation line, the damped critical speeds are determined as 2807, 3670, 10631, 10841 and 17278 rpm. The line 3F crosses the lines 4B and 4F.

Fig. 4.36 The damping ratio diagram is shown in Fig. 4.37 for the same 8 modes. The curve 3F has a peak and the curve 4F has a trough at the speed where the corresponding lines cross each other in the Campbell diagram. The two modes do not interact.


243

Fig. 4.37 The shape of the first six modes of precession at 25000 rpm is shown in Fig. 4.38.

a

b

c

d

e

f

Fig. 4.38 The unbalance response curves calculated at the three bearing stations 3, 6 and 13 are shown in Fig. 4.39, for a 200 g mm unbalance on disc 1.


244

a

b

c


A uniform shaft (Fig. 4.40) is supported by plain cylindrical bearings at the ends, at the stations 1 and 9 [19].

Fig. 4.40

Fig. 4.41 The shaft has a material with Young’s modulus 2.068 ⋅ 1011 N m 2 and mass density 7833.6 kg m 3 . The plain cylindrical bearings have length 25.4 mm ,


245

diameter 101.6 mm , radial clearance

51μm

and oil dynamic viscosity

6.94 ⋅ 10 −3 N s m 2 . The static loads on bearings are 395.6 N . The speed dependence of the bearing stiffness and damping coefficients is shown in Fig. 4.41.

Fig. 4.42

Fig. 4.43 The Campbell diagram is presented in Fig. 4.42 for the first 4 natural modes. Modes 1B and 2B are overdamped and do not show up. Modes 1F and 2F


246

follow closely the half-frequency excitation line (lower dotted line). The damping ratio diagram is shown in Fig. 4.43 for the same 4 modes. For mode 1F it becomes negative at 9060 rpm, the onset speed of instability. The shape of the first four modes of precession at 6000 rpm is shown in Fig. 4.44.

a

b

c

d

e

f

Fig. 4.44


247

Fig. 4.45 The stability diagram is shown in Fig. 4.45. At 9060 rpm, mode 1F becomes unstable. The precession frequency at the onset of instability is very close to one-half of the rotor speed (lower dotted line in Fig. 4.42), which is characteristic for the type of instability called ‘oil whirl’ or ‘half-frequency whirl’.

Fig. 4.46 The root locus diagram is presented in Fig. 4.46.


248

The unbalance response curves calculated at the stations 1 and 5 are shown in Fig. 4.47, for an unbalance of 512 g mm at station 2. The major semiaxis has maximum values at different speeds in the two bearings. The peak response critical speeds cannot be predicted from the Campbell diagram.

a

b

Fig. 4.47

Example 4.6

A solid rotor is mounted in plain cylindrical bearings. The shaft consists of 12 elements (Fig. 4.48) and has Young’s modulus 2.06 ⋅ 1011 N m 2 and mass density 7850 kg m 3 [20].

Fig. 4.48 The shaft dimensions are given in Table 4.5


249

Table 4.5 Element 1 and 12

Diameter, mm 660

Length, mm 260

Element 4 and 9

Diameter, mm 657

Length, mm 340

2 and 11

590

570

5 and 8

970

1100

3 and 10

550

280

6 and 7

1100

990

The bearings are located at nodes 3 and 11. The eight bearing dynamic coefficients are given in Table 4.6 [21] and Fig. 4.49. Note that c yz ≠ c zy ! Table 4.6

n

k yy

k yz

k zy

k zz

c yy

10 −9 N m

rpm

c yz

c zy

c zz

10-6 Ns m

800

5.17

2.67

0.712

1.4

38.7

7.15

13.7

11.0

1000

4.72

2.55

0.57

1.36

29.7

6.12

11.4

9.44

1300

4.28

2.44

0.422

1.34

22.5

4.97

8.98

7.84

1500

3.90

2.38

0.3

1.33

19.3

4.5

7.75

7.0

1700

3.7

2.32

0.232

1.325

16.76

4.02

6.82

6.52

2100

3.17

2.16

0.0416

1.322

13.25

3.49

5.5

5.86

2600

2.7

2.0

-0.175

1.32

10.34

3.02

4.58

5.49

3000

2.5

1.9

-0.3

1.315

9.0

2.7

4.0

5.25

3400

2.3

1.81

-0.437

1.31

8.08

2.49

3.51

5.08

3500

2.28

1.75

-0.45

1.306

7.8

2.4

3.4

5.0

3600

2.25

1.74

-0.5

1.303

7.65

2.35

3.3

4.9

4000

2.14

1.71

-0.56

1.3

7.0

2.21

2.98

4.81

4200

2.05

1.68

-0.6

1.295

6.7

2.15

2.85

4.65

4500

2.0

1.65

-0.65

1.29

6.4

2.05

2.7

4.5

5000

1.91

1.66

-0.75

1.295

5.9

1.91

2.38

4.37

5500

1.8

1.67

-0.8

1.3

5.45

1.8

2.2

4.2

6000

1.75

1.685

-0.9

1.32

5.1

1.7

2.05

4.0

6600

1.7

1.7

-0.95

1.35

4.63

1.59

1.955

3.84

The rotor is symmetrical, so that the physically identical bearings have also identical dynamic properties. They have different cross stiffnesses, k yz ≠ k zy , which produce unstable whirling.


250

Fig. 4.49 For the first six modes of precession, the Campbell diagram is shown in Fig. 4.50. Modes 1B and 2B are overdamped at low rotational speeds.

Fig. 4.50


251

The damping ratio diagram is given in Fig. 4.51. Mode 1F becomes unstable at 4911 rpm.

Fig. 4.51 The root locus diagram is shown in Fig. 4.52. Curves are labeled with both the eigenvalue index and the mode index showing the directivity.

Fig. 4.52


252

Despite the absence of discs on the shaft (no gyroscopic effects), the damped natural frequencies have a strong variation with the rotor speed, due to the speed variation of bearing coefficients.

a

b

c

d

e

f Fig. 4.53

In Fig. 4.50, the synchronous excitation line, ω = Ω , is drawn with dotted line. The abscissae of its crossing points with the curves in the Campbell diagram give the damped critical speeds at 1865, 2823, 3324 and 4171 rpm. Mode 1B is highly damped so that it will not produce a peak in the unbalance response.


253

The shape of the first six precession modes at 5576 rpm is shown in Fig. 4.53. Modes 1B and 1F are ‘cylindrical’, while modes 2B and 2F are ‘conical’. They correspond to the rotor so-called rigid body precession in flexible bearings. Modes 3B and 3F are ‘two-node flexural’.

Fig. 4.54 Mode 1F becomes unstable at 4911 rpm, where the damping ratio becomes negative. This can also be seen in the stability diagram from Fig. 4.54, where the real part of four relevant eigenvalues is plotted versus speed. Curve 1F crosses the zero ordinate line at 4911 rpm – the onset speed of instability. In the Campbell diagram, the point on 1F at 4911 rpm (81.8 Hz) has a damped natural frequency of about 39 Hz, which is a little less than half the driving frequency 40.9 Hz. This is known as the ‘half-frequency’ or ‘oil-whirl’ type of instability. It transforms into ‘oil-whip’ at the natural frequency of mode 1F and remains almost constant with increasing rotational speed. Note the atypical behaviour of this rotor, for which the lines 1F and 2F follow closely the synchronous excitation line and not the half-frequency excitation line. This is probably due to the values of the bearing coefficients. Example 4.7 a

A multi-stepped rotor used in laboratory experiments (Fig. 4.55) is supported by hydrodynamic bearings at stations 6 and 23. Two balancing discs are located at stations 12 and 17 [22].


254

Fig. 4.55 The shaft has Young’s modulus 2 ⋅ 1011 N m 2

and mass density

7850 kg m 3 . The plain cylindrical bearings have length 30 mm , diameter 100 mm , radial clearance 125 μm and oil dynamic viscosity 9 ⋅ 10 −3 N s m 2 . The static loads on bearings are 272.45 N and 251.77 N , respectively. Table 4.7 Element

Length,

Element

Length,

mm

Outer diameter, mm

no.

Outer diameter, mm

no.

mm

1

6.35

38.1

13

76.2

38.1

2

25.4

77.6

14

76.2

109.7

3

25.4

38.1

15

76.2

38.1

4

25.4

38.1

16

25.4

102.9

5

101.6

100

17

25.4

102.9

6

101.6

100

18

44.45

38.1

7

44.45

38.1

19

44.45

38.1

8

44.45

38.1

20

44.45

38.1

9

44.45

38.1

21

44.45

38.1

10

44.45

38.1

22

101.6

100

11

25.4

116.8

23

101.6

100

12

25.4

116.8

Details of the rotor configuration are listed in Table 4.7.


255

The disc data are given in Table 4.8. Table 4.8 Disc no.

Station

Mass, kg

Polar mass moment of inertia,

Diametral mass moment of inertia,

10 2 ⋅ kg m 2

10 2 ⋅ kg m 2

1

12

4.33

2.97

1.51

2

17

4.808

3.118

1.585

The speed dependence of the stiffness and damping coefficients for bearing 1 is shown in Fig. 4.56.

Fig. 4.56 The Campbell diagram is presented in Fig. 4.57 for the first 6 natural modes. Modes 1B and 2B show up only above 5700 rpm. At the crossing points with the synchronous excitation line, the damped critical speeds are determined as 549, 2400, 2949, and 4446 rpm. The first forward mode is denoted 1F∗ because, with increasing rotational speed, it changes from cylindrical to a two-node flexural mode. The third forward mode is denoted 3F∗ because it changes from a two-node flexural to an almost cylindrical mode. Note again the lines 1F∗ and 2F following closely the synchronous excitation line.


256

Fig. 4.57 The damping ratio diagram is shown in Fig. 4.58 for the same 6 modes.

Fig. 4.58 The shape of the first six modes of precession at 7000 rpm is shown in Fig. 4.59. Modes 1F∗ and 3F∗ , whose lines intersect in the Campbell diagram, are in fact almost interchanged.


257

a

b

c

d

e

f

Fig. 4.59

Fig. 4.60


258

Mode 3F∗ becomes unstable at 4876 rpm, where the damping ratio becomes negative. This can also be seen in the stability diagram from Fig. 4.60, where curve 3F∗ crosses the zero ordinate line at 4876 rpm.

a

b

c

d

Fig. 4.61

Fig. 4.62


259

The unbalance response curves calculated at the bearing stations 6 and 23 are presented in Fig. 4.61, for a 433 g mm unbalance on disc 1. Figures 4.61, a and b show the speed variation of the major and minor ellipse semiaxes. The peak value of the major semiaxis defines the peak response critical speed. It is compared with admissible limits given in standards and recommendations. Figures 4.61, c and d show the speed-variation of the radii of forward and backward circles which generate the elliptical precession. The root locus diagram is presented in Fig. 4.62 for speeds up to 7000 rpm. Curve 3F∗ crosses the zero damping vertical, marking the threshold of instability. Example 4.7 b

An alternate set of simulation results has been obtained for the rotor of Example 4.7, a using a ten times smaller oil viscosity 9 ⋅ 10 −4 N s m 2 . The speed dependence of the bearing stiffness and damping coefficients is shown in Fig. 4.63. The stiffness coefficient k yz has only positive values. This means that the selected speed range corresponds to relatively low values of the Sommerfeld number. The difference k yz − k zy being smaller than in the previous

(

)

example, the rotor is more stable with these bearings.

a

b

Fig. 4.63 The Campbell diagram for the rotor with modified oil viscosity is presented in Fig. 4.64. The damped critical speeds are located at 2241, 2961, 3551, 4733, 4947 and 5143 rpm. Modes 1B and 2B are overdamped at relatively low running speeds. The curves of modes 1F and 2F start with slopes higher than the synchronous excitation line.


260

Fig. 4.64 The damping ratio diagram is shown in Fig. 4.65 for the same 6 modes. There is no negative value in the considered speed range.

Fig. 4.65 The unbalance response curves calculated at the bearing stations 6 and 23 are shown in Fig. 4.66, for a 433 g mm unbalance on disc 1.


261

a

b

c

d

Fig. 4.66 The three peaks occur near the natural frequencies of modes 1F, 3B and 3F. The second peak is narrower due to the low damping of mode 3B. Example 4.8

A rotor test rig, designed for rotor dynamic experiments, is presented in Fig. 4.67 [23]. The finite element model of the rotor-bearing system is shown in Fig. 4.68. The shaft is modeled with 19 axisymmetric beam elements (4 DOFs/node) with consistent mass and gyroscopic matrices. The shaft has a material with Young’s modulus 2 ⋅ 1011 N m 2 and mass density 7850 kg m 3 . The two discs, located at nodes 6 and 19, and the coupling located at node 1 have masses 35.19 kg , 61.65 kg , 2.08 kg , respectively, polar mass moments of inertia 0.6422 kg m 2 , 1.9389 kg m 2 , 0.00208 kg m 2 , and diametral mass moments of inertia 0.3258 kg m 2 , 0.9789 kg m 2 , 0.00192 kg m 2 .


262

Fig. 4.67

Fig. 4.68


263

The two journal bearings, located at nodes 2 and 17, are plain cylindrical, with length 30 mm , diameter 40 mm , radial clearance 17.5 μm , and oil dynamic viscosity 0.00345 kg ms . The static loads on bearings are 179.37 N and 925.44 N , respectively. The speed dependence of the bearing stiffness and damping coefficients, calculated using Moes’ [5] impedance model, is given in Fig. 4.69.

a

b

Fig. 4.69 The Campbell diagram is presented in Fig. 4.70 for the first 6 natural modes. Modes 1F and 2F are controlled by bearings. The associated backward modes are overdamped. At the crossing points with the synchronous excitation line, the damped critical speeds are determined at 847, 1304, 2210, and 2687 rpm.

Fig. 4.70


264

The damping ratio diagram is shown in Fig. 4.71 for only four modes.

Fig. 4.71 Mode 3F becomes unstable at 3146 rpm, where the damping ratio becomes negative. This can also be seen in the stability diagram from Fig. 4.72, where curve 3F crosses the zero ordinate line at 3146 rpm.

Fig. 4.72


265

The shapes of the first six modes of precession at 10000 rpm are shown in Fig. 4.73.

a

b

c

d

e

f

Fig. 4.73 Unbalance response diagrams at bearing 2 are shown in Fig. 4.74 for eccentricities of 35 μm at 90 0 on the disc at location 6, and 44 μm at − 900 on the disc at location 19.


266

a

b

Fig. 4.74 The root locus diagram for the first six modes is presented in Fig. 4.75, a. Modes 1F and 2F are highly damped. The root locus diagram for selected four modes is presented in Fig. 4.75, b. Mode 3F becomes unstable at the point marked by a circle.

a

b

Fig. 4.75 Example 4.9 a

Consider the rotor with three discs from Fig. 4.76 carried by journal bearings at the ends. The uniform shaft of diameter 80 mm has lengths

l1 = l 3 = 0.3 m , l 2 = l 4 = 0.2 m , Young’s modulus 2 ⋅ 1011 N m 2

and mass

density 8000 kg m 3 . The three identical discs have the following mass and mass moments of inertia: m = 15 kg, J T = 0.05 kg m 2 , J P = 0.1 kg m 2 .


267

The bearings have the length/diameter ratio 0.25, radial clearance C = 50 μm and oil dynamic viscosity 7.038 ⋅ 10 −3 N s m 2 [24]. The static loads on bearings are 403.25 N and 432.68 N , respectively.

Fig. 4.76 The speed dependence of the bearing stiffness and damping coefficients, calculated using Ocvirk’s short bearing model [6], is given in Fig. 4.77.

a

b

Fig. 4.77 The Campbell diagram for the first four natural modes is shown in Fig. 4.78. Curves 1F and 2F start along the synchronous excitation line. Curves 1F and 3F merge at a speed about 2700 rpm, where there is an apparent switching. This is due to the short bearing approximation and will be explained in Example 4.9 b. The damping ratio diagram is presented in Fig. 4.79. The curves 1F and 3F cross each other at the speed where their corresponding pairs merge in the Campbell diagram. The damping ratio of mode 1F becomes negative at 7964 rpm,


268

the threshold of instability. The corresponding point in the Campbell diagram is very near the 66.6 Hz point on the half-frequency excitation line.

Fig. 4.78

Fig. 4.79


269

Fig. 4.80 The stability diagram is shown in Fig. 4.80. The onset speed of instability is marked at 7964 rpm, where the damping constant becomes positive.

Fig. 4.81


270

The root locus diagram is shown in Fig. 4.81 for speeds up to 10000 rpm. Curve 1F crosses the zero damping vertical at the onset speed of instability.

a

b

c

d

e

f

Fig. 4.82 Six precession mode shapes at 3000 rpm are shown in Fig. 4.82. Modes 1F and 3F are compounded cylindrical and two-node forward. Mode 2F is ‘conical’. Mode 3B is ‘two-node’ backward. Modes 4B and 4F are three-node flexural.

a

b

Fig. 4.83 The unbalance response curves calculated at stations 1 and 3 are shown in Fig. 4.83 for a mass unbalance of 150 g mm on the disc at station 3.


271

Example 4.9 b

Consider the rotor of Example 4.9 a and use Moes’ impedance model [8] for the calculation of bearing stiffness and damping coefficients.

Fig. 4.84 The Campbell diagram for the first four natural modes is shown in Fig. 4.84. Note that the curves 1F and 3F∗ are clearly separated now.

Fig. 4.85


272

The damping ratio diagram is shown in Fig. 4.85. Curves 1F and 3F∗ are crossing each other. Curve 1F crosses the zero damping line at 7826 rpm.

Fig. 4.86 The stability diagram is shown in Fig. 4.86. The onset speed of instability is marked at 7826 rpm, where the damping exponent of mode 1F becomes positive.

Fig. 4.87 The root locus diagram is shown in Fig. 4.87 for speeds up to 10000 rpm.


273

4.6 Planar modes of precession Undamped rotors in isotropic bearings exhibit planar modes of precession with circular orbits. A planar unbalance distribution yields planar deflected shapes. Undamped rotors in orthotropic bearings have spatial precession modes with elliptical orbits. With suitable scaling, the precession modes are planar in the principal planes of orthotropy, and so is the deflected shape due to planar unbalance. Damped gyroscopic systems have complex modal vectors describing spatial deflected shapes with “damped” elliptical orbits, rotated with respect to each other at different rotor stations. It was shown [25] that a certain class of damped gyroscopic systems has planar modal vectors describing elliptical orbits with coinciding principal directions and the same phase angle of the motions at different rotor stations. The characteristic phase angles and the mode shapes are speed-dependent. By a proper transformation, the complex monophase modal vectors can be transformed into equivalent real vectors, solutions of a real eigenvalue problem in the configuration space. There is no need to expand the problem and to solve it in the state space. The planar modes of precession form an orthogonal basis and can be used to decouple the equations of motion. This approach gives an alternative modal analysis solution for the steady-state response of the considered damped gyroscopic systems, otherwise treated by a perturbation technique.

4.6.1 Response of undamped gyroscopic systems For rotor-bearing systems having: a) axi-symmetric rotor; b) conservative cross-coupling forces; and c) orthotropic bearings with coincident principal directions of stiffness and damping, the equations of the free precession can be written in the form (3.145). The solutions have the form (3.148)

{ x } = {Φ } eiωt where (3.160) ⎧{ y } − i { ys }⎫ {Φ } = ⎨ c ⎬ ⎩ { zc } − i { z s }⎭

{ }

ay ⎫ ⎧ { yc } ⎫ iγ ⎧ = ( 1 + iβ ) ⎨ ⎬. ⎬ =e ⎨ ⎩− i { a z }⎭ ⎩− i { z s }⎭

The above equation shows that, with appropriate scaling, the elements of vectors { Φ } become real in the xOy plane and pure imaginary in the xOz plane, hence the precession modes are planar.


274

The rotor deflected line in the xOy plane has a + 900 or − 900 phase shift with respect to the deflected line in the xOz plane, which corresponds to a quarter of rotation of the rotor. This implies that, as a phasor, − i { a z } has a 900 phase lag with respect to a y . The orbits of the rotor stations are ellipses with axes

{ }

coincident with the y-z axes. The inclination angle is either 00 or 900 . By proper scaling of eigenvectors, the ellipse points at the reference time t = 0 are on the yaxis, where the phase angle γ = 0 .

4.6.2 Response of damped gyroscopic systems The equation of motion of a damped gyroscopic system can be written as

{ }

⎡[m] [0] ⎤ ⎧{&y&}⎫ ⎛ ⎡[c y ] [0] ⎤ ⎡ [0] [ g ]⎤ ⎞⎧{y& }⎫ ⎡[k y ] [0] ⎤ ⎧{ y }⎫ ⎧ f y ⎫ ⎢ [0] [ m]⎥ ⎨{&z&}⎬ + ⎜⎜ ⎢ [0] [c ]⎥ + Ω ⎢− [ g ] [0] ⎥ ⎟⎟⎨{z&}⎬ + ⎢ [0] [k ]⎥ ⎨{ z }⎬ = ⎨ { f } ⎬ z ⎦ z ⎦⎩ ⎭ ⎩ z ⎭ ⎣ ⎦⎩ ⎭ ⎝ ⎣ ⎣ ⎦ ⎠⎩ ⎭ ⎣ (4.21) where

{ f y } and { f z }

are the forcing vectors in the xOy and xOz planes,

respectively, [ c y ] and [ c z ] are positive definite damping matrices. Using phasor notation, consider the unbalance excitation

{ }

⎧ fy ⎫ 2 ⎧ {U } ⎫ iΩ t , ⎨ ⎬=Ω ⎨ ⎬e ⎩− i {U }⎭ ⎩ { fz }⎭

(4.22)

where {U } is the unbalance complex subvector. The synchronous unbalance response is ⎧{ y }⎫ ⎧ { ~y } ⎫ iΩ t ~ = X e iΩ t ⎨ ⎬=⎨ ~ ⎬e ⎩{ z }⎭ ⎩ { z } ⎭ ~ where X is a complex vector.

{ }

(4.23)

{ }

It is of interest to find an excitation of the form (4.22), with {U } a real vector, able to produce a synchronous planar response. This is a particular kind of precession, expressed by equation (3.160), in which all displacements have the same phase angle γ with respect to the unbalance plane:

{ y } ⎫ ⎧ {a y } ⎫ iγ ⎬=⎨ ⎬e . ⎩ { z } ⎭ ⎩− i { a z }⎭

{ X } = ⎧⎨

(4.24)


275

The response vector (4.24) is a planar mode of precession, defined by elliptical orbits whose axes coincide with the y-z coordinate axes, and whose generating radius at t = 0 has a phase angle γ with respect to the plane of unbalance. Substituting equations (4.22)-(4.24), and using the transformation (3.162), equation (4.21) becomes

{ }

⎛ ⎡[k ] − Ω 2 [ m ] [0] ⎤ ⎞⎟⎧ a y ⎫ iγ ⎧⎪Ω 2 {U }⎫⎪ ⎡Ω [ c y ] Ω2 [g ] ⎤ ⎜⎢ y + i ⎥ ⎬ ⎢ [ 0] ⎥ ⎟⎨{ a }⎬ e = ⎨ 2 2 ⎜⎢ Ω2 [ g ] c Ω [ ] ⎪⎩Ω {U }⎪⎭ [ ] − [ ] m k Ω ⎥ z ⎣ ⎦ ⎠⎩ z ⎭ z ⎦ ⎝⎣ (4.25) or

( [ BR (Ω ) ] + i [ BI (Ω ) ] ){ q }eiγ = { f }, where { q } and

(4.26)

{ f } are real vectors.

Separation of real and imaginary parts in equation (4.26) yields

( [ BR ] cosγ − [ BI ] sinγ ){ q } = { f }, ( [ BI ] cosγ + [ BR ] sinγ ){ q } = { 0 }.

(4.27, a) (4.27, b)

If cos γ ≠ 0 , denoting

λ = tan −1γ ,

(4.28)

the homogeneous equation (4.27, b) can be written as a generalized eigenvalue problem

[ BR ] {Φ r } = −λr [ BI ] {Φ r } .

(4.29)

Both the eigenvalues λr and the modal vectors {Φ r } are real and speed dependent. Vectors {Φ r } , referred to as planar response modal vectors, represent a specific type of precession, in which all stations execute synchronous motions along elliptical orbits, having the same phase shift γ r with respect to a reference unbalance plane. Their spatial shape varies with the speed. They are produced only by the external forcing defined by the planar excitation modal vectors { Fr } derived from equation (4.27, a):

{ Fr } =

1+ λ2r [ BI

] {Φ r }.

(4.30)

The planar response vectors satisfy the bi-orthogonality conditions

{Φ s }T [ BI ] {Φ r } = 0,

(r ≠ s )

(4.31)


276

{Φ s }T [ BR ] {Φ r } = 0,

{Φ s }T { Fr } = 0 .

They can be conveniently normalized so that

{Φ r }T [ BI ] {Φ r } = −sin γ r ,

{Φ r }T [ BR ] {Φ r } = cosγ r ,

(4.32)

{Φ r }T { Fr } = 1.

The coordinate transformation 4n

{ q } = ∑ {Φ r } ν r

(4.33)

r =1

simultaneously diagonalizes the matrices [ BR ] and [ BI ] . This way, a spectral decomposition of the system response is obtained in terms of the planar response vectors 4n

{q } = ∑

r =1

eiγ r {Φ r

}{Φ r }T { f }.

The transformation (3.162) is then used to obtain { X

(4.34)

} from { q } .

If cos γ = 0, then γ r = −90° and λr = 0. Equations (4.29) and (4.30) become

[ BR ] {Φ r } = { 0 }, [ BI ] {Φ r } = { Fr }.

(4.35, a) (4.35, b)

Equation (4.35, a) coincides with equation (3.166) so that the undamped normal mode {Ψ r } is the r-th planar response vector {Φ r } calculated at Ω = Ω r , which corresponds to λr = 0 in equation (4.29). The eigenvalues λr of the generalized problem (4.29) vary with the rotor angular speed. Each eigenvalue cancels at, and only at, the corresponding undamped critical speed, i.e. λr (Ω r ) = 0 . Plotting λr against speed, each curve crosses only once the speed axis, so undamped critical speeds can be easily located. This diagram can be used as a Real Mode Indicator Function (RMIF).

4.6.3 Planar precession modes A planar precession mode is defined by three speed-dependent elements: a) a planar response vector, whose elements are the semiaxes of the elliptical orbits, and the corresponding slopes at the rotor stations; b) a planar forcing vector, whose elements give the planar unbalance distribution which produces the planar


277

response; and c) a characteristic phase angle, the same at all stations, between the response plane and the unbalance plane. A planar response mode is shown in Fig. 4.88. The orbit axes are along the coordinate axes. The phase angle γ is measured in the positive direction of rotational speed Ω , between the point in the plane of unbalance and the point at t = 0 on the major generating circle. The construction presented in the next section helps understanding the physical meaning of the characteristic phase angle.

Fig. 4.88 The rotor finite element model has 4n degrees of freedom and is excited by planar unbalance forces applied at the n rotor stations. At any rotational speed Ω , there exist 2n independent sets of unbalance distributions, each of which excites the corresponding planar precession response, in which all points have the same phase lag with respect to the unbalance plane. The phase angle between the unbalance and the generating radius is different for each mode. As the rotor speed changes, so do the characteristic phase angles, planar response vectors and planar forcing vectors. At an undamped critical speed, one of the characteristic phase angles becomes − 900 and the corresponding planar response mode coincides with the undamped normal mode of precession.

4.6.4 Ellipse from two concentric circles The method of two concentric circles for constructing an ellipse is illustrated in Fig. 4.89.


278

Consider an ellipse with semiaxes a and b, and the angle α between the major semiaxis and the y-axis. The y1Oz1 coordinate system has the axes along the ellipse axes.

Fig. 4.89 First, two circles are drawn, with the centre at the origin of the coordinate system y1Oz1 and radii equal to the ellipse semiaxes. Second, the circles are intersected with a line O M passing through the origin and which rotates anticlockwise with an angular speed Ω , equal to the rotor speed. From the crossing point with the small circle, P1 or P2 , a line is drawn perpendicular to the Oz1 axis. From the crossing point with the large circle, M, a line is drawn perpendicular to the Oy1 axis. These two orthogonal lines cross each other at a point on the ellipse, C1 or C2 . When the crossing points M and P1 are on the same side of the origin, point C1 moves along the ellipse in a forward precession. When the crossing points M and P2 are on both sides of the origin, point C2 has a retrograde motion called backward precession. In principal coordinates y1Oz1 , the ellipse is defined by the parametric equations (3.35). If O D1 is the ellipse vector radius at the reference time t = 0 ,


279

then the phase angle γ 1 defines the position of the generating line ON1 , at t = 0 , for forward precession. The phase angle γ 2 defines the position of the generating line ON 2 , at t = 0 , for backward precession. Note that the angular frequency of the precession motion is equal to the angular speed of the generating points M and P1 or P2 on the two circles, and not to the angular speed of the ellipse vector radius, which is not constant.

Example 4.10

Consider the three-disc rotor of Example 4.1 with c yy = c zz = 500 Ns m .

Fig. 4.90 (from [25]) The Campbell diagram of the associated undamped system is presented in Fig. 4.90. The undamped critical speeds are determined at the crossing points with the synchronous excitation line.


280

For comparison, the RMIF diagram (eigenvalues λr versus speed) is shown in Fig. 4.91. Undamped critical speeds are located at the intercepts with the horizontal axis.

Fig. 4.91 (from [25])

Fig. 4.92 (from [25])


281

Precession mode shapes at the first six undamped critical speeds are illustrated in Fig. 4.92. They have been determined from the monophase modal vectors calculated at the corresponding natural frequency.

References 1. Jackson, Ch., and Leader, M. E., Turbomachines: How to avoid operating problems, Hydrocarbon Processing, Nov. 1979. 2. Meyer, A., Schweickardt, H., and Strozzi, P., The converter-fed synchronous motor as a variable-speed drive system, Brown Boveri Rev., Vol.69, No.415, pp.151-156, 1982. 3. Kellenberger, W., Weber, H., and Meyer, H., Overspeed testing and balancing of large rotors, Brown Boveri Rev., Vol.63, No.6, pp.399-411, 1976. 4. Hohn, A., The mechanical design of steam turbosets, Brown Boveri Rev., Vol.63, No.6, pp.379-391, 1976. 5. API Standard 617, Centrifugal Compressors for Petroleum, Chemical and Gas Industry Services, 1995. 6. API Standard 613, Special-Purpose Gear Units for Refinery Services, 1979. 7. API Standard 612, Special Purpose Steam Turbines for Petroleum, Chemical and Gas Industry Services, 1995. 8. Childs, D., Moes, H., and van Leeuwen, H., Journal bearing impedance descriptions for rotordynamic applications, J of Lubrication Technology, pp.198-219, 1977. 9. Ocvirk, F., Short bearing approximation for full journal bearings, NACA TN 20808, 1952. 10. Someya, T., (ed.), Journal-Bearing Databook, Springer, Berlin, 1988. 11. ISO 1940, Balance Quality of Rotating Rigid Bodies, 1973. 12. Busse, L., and Heiberger, D., Aspects of shaft dynamics for industrial turbines, Brown Boveri Rev., Vol.67, No.5, pp 292-299, 1980. 13. ISO 7919-2, Mechanical Vibration of Non-Reciprocating Machines Measurements on Rotating Shafts and Evaluation Criteria - Part 2: Large LandBased Steam Generator Sets, 1996. 14. Scarlat, G., Predicţia stabilităţii rotorilor în diferite tipuri de lagăre hidrodinamice cu ajutorul analizei modale, Buletinul Conferinţei Naţionale de Dinamica Maşinilor CDM97, Braşov, 29-31 mai 1997.

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15. Lalanne, M., and Ferraris, G., Rotordynamics Prediction in Engineering, 2nd ed, Wiley, Chichester, 1998, p.125. 16. Radeş, M., Mixed precession modes of rotor-bearing systems, Schwingungen in rotierenden Maschinen III, (Irretier, H., Nordmann, R. and Springer, H., eds.), Vieweg, Braunschweig, pp. 153-164, 1995. 17. Nelson, H. D., and Meacham, W. L., Transient analysis of rotor-bearing systems using component mode synthesis, ASME Paper No.81-GT-110, 1981. 18. Chen, W. J., Rajan, M., Rajan, S. D., and Nelson, H. D., The optimal design of squeeze film dampers for flexible rotor systems, ASME J. of Mechanism, Transmission and Automation in Design, Vol.110, pp.166-174, 1988. 19. Lund, J. W., Stability and damped critical speeds of a flexible rotor in fluidfilm bearings, ASME J. of Engineering for Industry, Series B, Vol.96, No.2, pp.509-517, 1974. 20. Radeş, M., Analiza modală a rotorilor elastici în lagăre cu alunecare, Bul. Conf. Naţ. Dinamica Maşinilor CDM94, Braşov, 24-25 Nov 1994, pp 17-24. 21. Bigret, R., Vibrations des machines tournantes et des structures, tome 2, ch.10, Technique et Documentation, Paris, p.40, 1980. 22. Friswell, M. I., Garwey, S. D., Penny, J. E. T., and Smart, M. G., Computing critical speeds for rotating machines with speed dependent bearing properties, J. Sound Vib., Vol.213, No.1, pp.139-158, 1998. 23. Kreuzinger-Janik, T., and Irretier, H., On modal testing of flexible rotors for unbalance identification, Proc. 16th Int. Modal Analysis Conf., Santa Barbara, California, pp.1533-1539, 1998. 24. Lee, C.-W., Vibration Analysis of Rotors, Kluwer Academic Publ., Dordrecht, 1993. 25. Radeş, M., Use of monophase modal vectors in rotordynamics, Schwingungen in Rotierenden Maschinen IV, (Irretier, H., Nordmann, R. and Springer, H., eds.), Vieweg, Braunschweig, pp.105-112, 1997.

Index Angular momentum 70 − precession 68 − speed 42 Axial compressor 9 Backward precession 77, 105, 116 Bearing 26 − damping 119, 131, 156 − flexibility 102 Blowers 14 Campbell diagram 87, 152, 222 Centrifugal compressors 10 − pumps 15 Complex eigenvalues 221 Coordinate system 154 − rotating 56 − stationary 56, 58 Critical speed 67 − backward 84 − damped 219 − forward 82 − map 209 − peak response 82, 114, 224 − undamped 45, 207 Damped rotors 158 − asymmetric 154 − symmetric 46 Damping 62 − coefficient 128 − external 47, 62, 109, 117 − internal 54, 62, 117 − hysteretic 50 − optimum 127 − ratio 47, 60, 110, 154 − rotating 54 − stationary 67 − viscous 47 Decay factor 48 Eigenvalue problem 220 Eight bearing coefficients 138

Elliptical orbit 105, 112 Ellipse 104 Equations of motion 41, 72, 102, 147 Equivalent stiffness 103 External damping 47, 62, 109, 117 Fans 14 Forward precession 77, 105, 116 Free precession 47 − − damped 47, 121 Gas turbines 6 Gravity loading 65 Gyroscopic torques 71, 97 Harmonic force 84 Hydraulic turbines 18 Hysteretic damping 62 Inertia torques 69 Influence coefficients 146 Internal damping force 54 − − ratio 60 Major semiaxis 86, 106 Mass unbalance 43 Minor semiaxis 86, 106 Mixed modes of precession Matrix, damping 59, 154 − flexibility 75 − stiffness 59, 75 Mode of precession 75 − shapes 90, 92 Model 29 − Laval-Jeffcott 39 − Stodola-Green 39

158

Natural frequency 67, 111, 148 − damped 48, 120, 132, 221 − undamped 47 Nyquist plot 51 Onset speed of instability 64, 118, 144 Orbits 25, 111, 223 Overdamped modes 183, 186

MECHANICAL VIBRATIONS

284 Phase angle 50, 123 Planar modes 273 Polar diadram 52 Precession 23, 67 − backward 78, 105 − forward 77, 105 − free damped 47 − radius 50 − synchronous 44, 52 Reference frames 69 Rotor 6 − asymmetric 145 − rigid 26 − symmetric 101 Rotor bearing dynamics 22 Routh-Hourwitz criterion 63, 118 Rigid body modes 208 Rotating damping 54 Shaft 26 − bow 66 − mass 131 Sommerfeld number 228 Spiral 48 Spot, heavy 53 − high 53 Stability 59, 142, 227 Stator inertia 217 Steady state precession 49 − response 43 Stiffness coefficients 40 − matrix 59, 75 Symmetric rotors 101 − damped 46 − in flexible bearings 101 − in fluid film bearings 136 − in rigid bearings 40 − undamped 40 Synchronous excitation 88 − precession 44, 52 Threshold speed 113 Turbo-generators 18 Unbalance response 104, 121, 134, 152 − diagrams 50, 225 Undamped critical speed 45, 207 − natural frequency 47 Undamped rotors 40

− asymmetric 68 − symmetric 40 Underdamped modes

221

Viscous damping 47, 62 − coefficient 47 Whirling

61

M. Rades - Dynamics of Machinery 1

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