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Chapter 8
Half and Full Spectrum Plots
points and the limitations of digital sampling. The bottom plot shows the spectrum when a Hanning window is applied to the sample record. Note that the Y2X spectral line is narrower and higher, and the residual noise floor has virtually disappeared. A digitally calculated spectrum consists of discrete frequency bins, or lines, of finite width. The width of these lines, the resolution of the spectrum, is an important consideration. The maximum resolution of a spectrum is determined by the ratio of the spectrum span (the range of displayed frequencies) to the number ofspectrum lines that are displayed:
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----"~---
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Thus, each frequency line will, ideally, represent only the spectral energy in a 0.5 Hz (30 cpm) wide band from 0.25 Hz below to 0.25 Hz above the center frequency of the line. Accuracy in the displayed amplitude and frequency of a spectrum line will depend on where the actual vibration frequency is with respect to the center frequency and which window function is used. The limited resolution of spectrum plots means that there is always an uncertainty associated with any frequency we wish to measure. In the example above, a frequency actually located at, for example, 99.75 Hz, is displayed at 100 Hz. A spectrum plot with poor resolution will have a corresponding large uncertainty in the measured frequency. Even good resolution spectra may not be able to discriminate between vibration frequencies of exactly Y2X and 0.49X, an important distinction for malfunction diagnosis. Higher resolution (zoomed) spectra can help, but orbits with Keyphasor dots can sometimes be superior to spectrum plots for making this kind of discrimination (see Chapter 5). Noise can be a problem in spectrum plots. The Fourier transform of a spike is a series of spectrum lines extending to very high frequency. Thus, anything that produces a sharp corner in the signal will produce a series of spectrum lines. Sharp corners can result from shaft rebound at a rub contact point or from
137
138
Data Plots
an inadequate sampling frequency (causing a corner where a smooth transition really exists), among other things. Spikes or steps in the signal can originate from electrical noise problems or from scratches on the shaft. Spectrum plots are calculated from uncompensated waveforms, which may contain significant slow roll or glitch content. In general, the appearance of spectrum lines as a series of harmonics should be viewed with caution. Use timebase, orbit, or cascade plots (below) to validate the data. The Full Spectrum
The half spectrum is a spectrum of a single timebase waveform. The full spectrum is the spectrum ofan orbit. It is derived from the waveforms from two, orthogonal, shaft relative transducers, combined with knowledge of the direction of rotation. The information from the two transducers provides timing (phase) information that allows the full spectrum algorithm to determine the direction of precession at each frequency. Because the timing information is critical, the two waveforms must be sampled at the same time. The full spectrum is calculated by performing an FFT on each transducer waveform. The results are then subjected to another transform that converts the data into two new spectra that represent frequencies of precession, one spectrum for X to Y precession and one for Y to X precession. The last step uses the direction of rotation information to determine which of the spectra represents forward and which represents reverse precession frequencies. When this process is completed, the two spectra are combined into a single plot, the full spectrum plot (Figure 8-3). Figure 8-4 shows the relationships among timebase waveforms, half spectra, the orbit, and the full spectrum. The Y and X timebase waveforms and their half spectra are at the top. The two waveforms combine to produce the orbit at bottom left. The data used to generate the half spectra are further processed to produce the full spectrum at bottom right. Note that you cannot generate the full spectrum by combining the two half spectra. In the full spectrum plot, the spectrum of forward precession frequencies is on the positive horizontal axis and the spectrum of reverse precession frequencies is on the negative horizontal axis. Thus, for each frequency, there are two possible spectrum lines, one forward, and one reverse. The relative length of the spectrum lines for each frequency indicates the shape and direction of precession of the orbit filtered to that frequency. Figure 8-5 shows four, circular, IX orbits, with different directions of precession, indicated by the blank/dot sequence, and different directions of rotor rotation, indicated by the arrow. To the right of each orbit is its full spectrum. Since each orbit is circular, there is only one line, which is the peak-to-peak amplitude
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Figure 8-7 shows the progression of IX orbit shapes from forward circular through line to reverse circular and the associated full spectra. The relative size of the forward and reverse line heights correlate with the shape and precession direction of the orbit: 1. A single component, whether forward or reverse, means a circular
orbit. 2. The largest component determines the direction of precession. 3. The smaller the difference between the components, the more elliptical the orbit. 4. Equal components mean a line orbit. Complicated orbits will have forward and reverse components at many frequencies. Each pair of components represents a set of vectors that rotate in forward and reverse directions at a specific frequency. The most complex orbit can always be described by set of such vectors and full spectrum lines. The lines in the full spectrum represent the precessional structure of the orbit. Each pair of forward and reverse precession frequency components describes an orbital component, a suborbit (circular. elliptical, or line) with a particular precession frequency and direction. The entire orbit can be expressed as the sum of its orbital components in the same way that a timebase waveform can be expressed as the sum of its sine wave components. Figure 8-8 shows a complex orbit from a steam turbine with a Y2X rub. The orbit contains Y2X. IX, and some higher order vibration frequencies. The full spectrum helps clarify the complexity. Note that the IX spectral line pair shows that the IX component is largest. forward. and mildly elliptical. The ¥2X line pair shows that this component is nearly a line orbit. Also, there is 2X vibration present that is also a line orbit. Some small %X, the third harmonic of the YZX fundamental, is also visible. At first glance, the full spectrum might seem abstract. What is significant about pairs of vectors with forward and reverse precession? It lets us easily identify key orbit characteristics that might otherwise be obscured. Precession direction and ellipticity provide insight into the state of health of a machine. More importantly. some rotor system malfunctions can have characteristic signatures on a full spectrum plot that are not available on half spectrum plots. These char-
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Chapter 8
Half and Full Spectrum Plots
Summary The conventional, or half spectrum plot displays amplitude of vibration on th e vertical axis versus frequency of vibration on the horizontal axis. It is const ructed using the sampled timebase waveform from a single transducer. Spectrum plots can be used to identity the frequencies of running sp eed , harmonics of running speed. sub- and supersynchronous vibration frequencies, gear mesh frequencies, gear defect frequencies, rolling element bearing defect frequencies. vane and blade pass frequencies, sidebands. glitch. and line frequency noise. The full spectrum uses the waveforms from an orthogonal pair of vibration transducers (usually shaft relative). The full spectrum displays frequency and dire ction of precession on the horizontal axis. Forward precession frequencies are displayed to the right of the origin and reverse precession frequencie s are displayed to the left of the origin. The full spectru m is the spectrum of an orbit, and the forward and reverse frequency component pairs represent orbit components (filtered orbits). The ratio of the amplitudes of full spectrum component pairs gives information about the ellipticity and direction of precession of the components, important characteristics for malfunction diagnosis. However, there is no information about the orientation of the orbit. Spectrum cascade plots are sets of spectra that are collected during the startup or shutdown of a machine. Cascade plots can be constructed from either half spectra, or full spectra. Cascade plots have important information associated with vert ical, horizontal, and diagonal relationships. Waterfall plots are collections of spectra obtained. usually, during steady state operating conditions and plotted versus time. They also can use either half spectra or full spectra. The spectru m plot is a powerful tool when carefully applied. Because of its wide availability, there is a temptation to use the spectrum plot to the exclusion of other plot formats. But the spectrum, however powerful, is not a substitute for the information that can only be obtained in other plots: the filtered amplitude and phase in polar and Bode plots, the sh aft position information in average shaft centerline plots, the shape and frequency information in the orbit, and the waveform information in the timebase plot. All of this information is needed for comprehensive machinery management.
153
155
Chapter 9
Trend and XY Plots
IN
CHAPTER 7 WE DISCUSSED THE Amplitude-PHase-Time (APHT) plots, a form of polar and Bode plots, which are used for trending vector data. Besides vector data, there are many other parameters that we would like to trend: vibration levels, position data, process data, or any other parameter that can be useful for machine condition monitoring. The trend plot shows changes in and the rate of change of parameters that may signal a de veloping or impend ing problem in a machine. This information can be used to set limits or thresholds for action. We may also wish to examine how any of these variables change with respect to others. Such correlation is the heart of a good diagnostics methodology. No single variable or plot type can reveal everything about a machine. Data is usually correlated with many other pieces of information to arrive at a diagnosis. This kind of correlation is often done with multiple trend plots, where several variables are plotted against the same time scale. Correlation can also be done with XY plots, where two parameters are plotted against each other. This chapter will deal with the construction and uses of the trend plot and provide several examples. We will briefly discuss problems that can arise when the data sample rate is too low: Finally, we will discuss XYplots, a special type of trend plot used for correlation of two parameters.
Trend Plots The trend plot is a rectangular or polar plot on which the value of a measured parameter is plotted versus time. Trend plots can be used to display an y kind of data versus time: direct vibration, nX amplitude, nX phase (the API-IT plot is a trend plot that displays both), gap voltage (radial or thrust position),
156
Data Plots
rotor speed, and process variables, such as pressure, temperature, flow, or power. Trend plots are used to detect changes in these important parameters. They are used for both long and short term monitoring of machinery in all types of service and are, typically, an example of a steady state (constant speed) plot. The data for a trend plot can be collected by computer or by hand. Figure 91 shows a trend plot of hand-logged gap voltage from a fluid-film bearing at the discharge end of a refrigeration compressor. Due to improper grounding, electrostatic discharge gradually eroded 280 J.lm (11 mil) of the bearing, allowing the rotor shaft to slowly move into the babbitt. The trend plot alerted the operators to the fact that something was wrong, and they scheduled a shutdown in time to prevent serious damage. This is an example of how a very simple data set provided valuable information that saved the plant from an expensive failure. Even though a trend plot may look like a continuous history, it is not. Parameters are assumed to be slowly changing, so the data to be trended is sampled at intervals that depend on the importance of the machinery and the data. If a sudden change in behavior of the parameter occurs between samples, the data will be missed. Some data values may fluctuate periodically. For example, IX amplitude and phase may change periodically due to a thermal rub. The period of change for this kind of malfunction can be on the order of minutes to hours. Amplitude and phase modulation can occur in induction motors, due to uneven air gap, at twice the slip frequency; the period here is usually a fraction of a second. If the sampling frequency is less than twice the frequency of interest (does not satisfy the Nyquist criterion), then the frequency of the changes in the trend plot will be incorrect, an effect known as aliasing. The trend plot from an induction motor (Figure 9-2) looks like, at first glance, a timebase plot. However, it is a trend of unfiltered, peak-to-peak vibration that is changing periodically. This motor, which drives a boiler feed pump, has an uneven air gap problem. The vibration amplitude is modulated at a beat frequency equal to twice the slip frequency of the motor. The data in blue was sampled very rapidly, at about 10 samples per second. This produced a high resolution trend plot. The data in red is a portion of another trend plot from about two hours earlier, when the motor was experiencing the same problem. The sample rate was one sample every 10 seconds, a factor of 100 slower. Note that the frequency of the change of the red is much lower, the result of aliasing. This would not be obvious unless the data taken at the higher sample rate was available. The observed modulation frequency is much lower than the true modulation frequency. Parameters which change periodically like this are relatively rare, but this example demonstrates how the sample rate can produce a misleading picture of machine behavior. This effect can also happen
Chap ler 9
Trend . nd XV Plots
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160
Dau Plot.
XY Plou Where Ih.. tre t>d p lO! di'PJa~ o ne o r mo re para .....te. . .......u. tim... th e XY plot (n ol to be confu..-d ..ith Xl' ........ or XY t ra n.duee.. ) ca n be u..-d 10 di .play a ny two pa ram.. t<> rul. each O! he~ Co ....lat ion. be t""""n t he pa ramet ..... will . how a diagonal alio n.hip. ,\ complete laclr of corre lation will ei th er a horizo nt al or .....rt ical . ...atio n"'ip. Figu . e 9--5 .how. a n :x.... plot of vibra tion a mplit ude ,...... u. ga p ."Oltage from a 125 :\1\\' oUea m tu thine gene r" to•• II P{lP u nit ru nn mg a t 36CX) 'I'm. The p~ .t at I"" nl I ..t...n th .. mach i n~ u ndergoes • loa d ch a ng... Ilrl......... P.' ml$ I a nd 2 (mil. th~ r iO! d ....rly . hnW1; a co. .......l io n be twee n ehanging . hafl .....iti..n (measu. <-d by th .. gap ~nlt.~) a nd IX vih ",tion a mp litud... A. I.... s haft m. ....es (hlue) to r " int 3 .b u inll th .. nU l t h~ hlllJ r'$. th.. ' i h'at ion dC'C l'P"-"" whil.. th e gar voltage rem a in••p .....x;m.,..ly cunst.nl. It i. P'""ihl.. th.1 a gra .i ty b" ", may be ""\Irking it....f OUI o. Ih.. m..a'U ....mffi t p robe tna~· be vi..,.;ng a d ill......nt ...ction " f Ih.. , haft as th .. mach in.. ...aeh n th ...-ma l "'Iuilihriu m. Th.. ""aft u k... o nly about 33 minut... to mo'... from poinl 3 10 point 4 (~n ), I1lmt likely in ....P" n... a no th l"T load ch a nll".
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s ereeeeees I. Ei...n ma nn. Roht-rt C . Sr~ .rnd l'iw nma n n. ko l>ert C . Jr.. •t /achmn y Malfu nction Diagnosi. aad l .f.JTTOClio'l (L:p pe. Saddle Riw l: p ..... nu ...... II..U. l nc~ 19')8 ).PI'. i.~ I·i5S.
161
The Static and Dynamic Response of Rotor Systems
165
Chapter 10
The Rotor System Model
of forces. Forces can act in radial and axial directions, and torques and moments can act in angular directions. These forces can be static, or unchanging in direction and time, or th ey can be dynamic, where they can change in magnitude or direction with time. Static forces acting on the rotor system produce static deflections of rotor system elements. For example, a static radial load applied to the midspan of a rotor shaft will cause the shaft to deflect in a direction away from the applied load. Or, when a torque is applied to the shaft of an operating machine, the shaft will twist to some extent in response to the torque. Dynamic forces acting on the rotor system produce vibration (Chapter 1). Vibration can appear in the form of radial, axial , and torsional vibration. Usuall y, we measure radial vibration in machinery because radial vibration is the most common vibration problem. Axial vibration is less frequently encountered but can produce machine problems. Torsional vibration is very difficult to measure and tends to be overlooked. Both torsional and axial vibration can produce radial vibration through cross-coupling mechanisms that exist in machinery. Unbalance is the most common example of a dynamic force (the force direction rapidly rotates) that produces radial vibration. How do dynamic forces act on the rotor system to produce vibration? Somehow, the rotor system acts as an energy conversion mechanism that changes an applied force into observed vibration. The rotor system can be viewed as a very complicated "black box" that takes dynamic force as an input and produces vibration as an output (Figure 10-1). If we can understand the nature of this black box, we should be able to understand how forces produce vibration. We should also be able, by observing the vibration and knowing the ROTOR SYSTEMS ARE SUBJECTED TO MANY KINDS
166
l1w SIal ic and Dyn.amic R.. ~ol ROlo .
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wo rkin!/." o f th .. black box. to dl.'d oce soml'1hin g ahout Ih.. fofCt'<> th ai p.odu,'t" the , i bralion. We ca.n t r,' 10 gu e•• th.. ront ..nls o f th .. black box by shaki ng it us ing a I('(:hn iqu.. ca llro penurbalwtr a nd obS<'"i ng th.. bdla, i o. o f III.- .,..t..m. Th i. is th.. ""m... I""h niqu.. som.. p<'OpI.. uS<' wh..n " , in!! 10 g""'" I .... ront.. nl~ of a "Tapped gift. Th..,. shako> th.. gifl a nd ""aluat .. th.. w..ight.. hala nce. a nd sound. Wh..n appliro 10 rotor s,·st..ms . Ih hak ing a""lifos a known forc e tn th .. ro to r .,"t..m. a nd Ih.. vihra tion .....pom to th.. fo. C('is m....s urPd. W.. ca o also tr,. to ....timat e th.. w nten t. n f the bl....k bo.- by d e,...lopinj( a mal ....ma tkal modP/ of th.. rolo • •,.."t..m..~ !l'uj m..d el " i ll a ll..w us 10 relat e oo....r..-l.'d ,ib.al io" to th .. fo thai ac l " n lhe ,,·slem. TIl;" ..ill allow lUI to d..tect.. id..ntify. a nd ro•......,l p" ntia] p",hl..m. in Ih.. ro l..r 'pt.-m. A Ilooo"t a sysl..mat ic approoch ba.ro on knowledg.. o f roto r I a'.;ot. acrural.. bala ncing wo ,,1d bt- "rtua ll,. imposs ibl... A mod'" gh..,,; u' th.. fnu ndat ion fo r a .y.t..-matic. efficirnt.. and d J""ti,... balancing tec h· niqo... In thiS chapt..r we "ill .........I..p a si mple mod d of a rot o r s,"t..m. Th .. pri · ma ry .....ull of th.. mod el. o,·tromic Slijfn... ,. is the solution 10 11K- mpt••) · ins i"" It... black hnx. fl i. a f"n<.Lom..nlal a nd important rona-pt fo r u nd...rs tandi nll roto r "..ha\int. it p",vidn a p""... rful t....l for mal funct ion d iallnos is. a nd il is l he k..,. to s lJC\-,.,., ful ba lancing. In d....-..lop ing th is mod d . Ih..... is no "lIy to ",'oid th .. mal ....malic. of d if· f..... ntial ""l"at ions a nd romp l num lx>rs. w.. ",ill ma ko> ....wy l'IJon 10 "'-"'I' tt... m al h..mali... a. si mp!.. and d , a. J"OS'ibl... TIlOS<' who d o not h3\'" th.. ma th · ....,...tica.l bad
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-
Chapter 10
The Rotor System Model
Introduction to Modeling Everyone has seen scale models of aircraft. A model airplane mimics important features of the real thing. When viewed from different directions, a wellbuilt model airplane can look very much like the real thing, but it includes only certain features of the full-sized airplane. However, even though it looks the same, a model airplane does not behave in all ways as a real airplane; if it does not have a working engine, it cannot fly. In fact, all models are simplifications designed to represent particular features, and they will not function properly when pushed beyond the limits of their applicability. Our rotor system model is a mathematical representation that is designed to mimic certain important features of the real rotor system. The rotor model is an attempt to describe the function of the black box that transforms dynamic forces into vibration. Because it is only a model, it will have limited applicability. The limits of the model are stated in the assumptions used to derive the model. Assumptions almost always involve simplifications that make the solution of the model easier. Applying a model beyond the limits expressed in the assumptions will usually lead to error. Sometimes the error can be tolerated, sometimes not. Often the amount of error is unknown. Most real rotor systems are very complex machines. We cannot, given the current state of computers and mathematics, construct a model that duplicates the behavior of these machines in every detail. We can construct complicated computer models that do a good job of mimicking some aspects of complex behavior, but the results of such models tend to be very narrowly focused and are difficult to generalize. Simple equations can be more easily understood and interpreted. The benefit of the simple approach is clarity and, we hope, an intuitive understanding of basic rotor system behavior. Because of this, we will develop a model that will yield some relatively simple algebraic equations. The price we pay for this simplicity is that the model may not have the capability to accurately represent the behavior of complex systems. We must always keep in mind that a simple model will be limited in its application. In essence, we will trade detail for insight. The model we are going to develop is a variation of an early rotor model, often called thejeffcott rotor, developed by Henry jeffcott [1] in 1919. Our model extends the Jeffcott model by including the effects of fluid circulation around the rotor and by using complex number notation to simplify the mathematics. It is necessary to include fluid circulation effects if we want our model to predict the rotor response in machines with fluid-film bearings, seals, and other areas where fluid is in circumferential motion. Our model will be a slight simplification of th e model presented by Bently and Muszynska in reference [2].
167
168
The Static and Dynamic Response of Rotor Systems
Here are some basic definitions of terms that will be used in the derivation and discussion of our model and throughout the book: The rotor system includes all parts of the machine that are involved with vibration. This includes the shaft with any attached disks, the bearings that support the shaft, the structures that support the bearings, the machine casing, the foundation system, coupled machines, and attached piping systems or unsecured cabling. The rotor system can also include all of the plant equipment that is involved in the process in which the machine is imbedded. When working with simple models it is easy to forget that real rotor systems include all of these components. The rotor is the rotating shaft assembly that is supported by bearings. The rotor may be rigidly coupled to other rotors in other machines, effectively forming a large, extended rotor. The stator is the stationary part of the machine that contains the rotor. The rotor rotates in and is supported by bearings in the stator. The purpose of bearings is to eliminate friction while preventing unwanted contact between the rotor and the stator. Forces that act on the machine will be divided into internal and external forces. Internal forces are those that appear from the machine's interaction with parts of itself. Support forces in bearings, forces resulting from shaft deflection, and forces due to interaction of the rotor with the surrounding fluid are examples of internal forces. Externalforces are forces that are applied to the rotor system and produce some sort of perturbation, or disturbance of the system, such as impact forces due to rotor-stator contact, static radial loads, or deliberately induced perturbation forces. Even though rotating unbalance is generated internally, it will be treated as an external force. The term synchronous refers to anything that is rotating at the same frequency as the rotor. Unbalance is an example of a synchronous rotating force. The term IX is used to describe a synchronous frequency (-IX is also considered to be a synchronous frequency). The term nonsynchronous refers to any frequency other than synchronous. A nonsynchronous frequency may be either supersynchronous (higher than running speed) or subsynchronous (lower than running speed). A rotor system parameter is a property of the system that affects system response. Mass, stiffness, and damping are examples of rotor system parameters. The term isotropic describes the properties of a system that are radially symmetric. For example, isotropic stiffness means that the stiffness of the system is the same in all radial directions (Figure 10-2). The term isotropic is distinct from the term symmetric, which implies a geometric (shape) symmetry.
Chapter laThe Rotor System Model
Isotropy
Anisotropy
Figure 10-2. Isot ropic and anisot rop ic systems. A system propert y is isotropic if it is the same in all radial directions. A system property is anisotropic if it has different values in different radial direct ions.
A system property is anisotropic if it has different values in different radial directions. Fluid-film bearing stiffness is isotropic when journals operate at low eccentricity ratios and anisotropic at high eccentricity ratios, where the stiffness in the radial direction is typically much higher than the stiffness in the tangential direction. On the other hand, rolling element bearing stiffness is usually isotropic. Modeling of physical systems usually follows a structured process: 1.
State the assumptions that will be used. These usually involve simplifications that allow easier solution of the problem, but limit the application of the model. These limits must be kept in mind when applying the model to the real world.
2. Define a coordinate system. The rotor system moves in space, and there must be a measurement system to describe the motion. 3. Describe the forc es that act on the system. Forces ar e modeled as physical elements which depend on displacements, velocities, or accelerations. For example, a rolling element bearing support force can be described as a spring element where the force is proportional to the displacement of the spring.
169
170
The Static and Dynamic Response of Rotor Systems
4. Develop a free body diagram. This diagram contains the rotor mass (or masses) and all of the forces acting on it (or them). 5. Derive the equation of motion. This is the differential equation that combines the forces and mass elements with a physical law that describes how the system must behave. 6. Solve the equation of motion. The result will be an expression that describes the position of the system over time. 7. Compare the predicted behavior to the observed behavior of the machine. Theoretical behavior is compared to the results of experiments. 8. Adjust the model
if the description is not adequate.
Assumptions The assumptions define the limitations of our model. They make the model easier to solve at the expense of detail in the final results. 1.
The rotor system will have one degree offreedom in the complex plane (l-CDOF). One degree of freedom implies that there is one, independent, lateral position measurement variable (r, which will be complex), no angular deflection, and one differential equation to describe the system. This will produce a model capable of only one forward mode, or resonance.
2. The rotor system parameters will be isotropic. This will allow us to use a more compact mathematical description for the model. The effects of anisotropy will be discussed in Chapter 13. 3. Gyroscopiceffects will be ignored. Gyroscopic effects can cause a speed-dependent shift in rotor system natural frequencies. This can be very important for overhung rotor systems, but we can ignore gyroscopic effects and still gain a good understanding of basic behavior.
Chapter 10 The Rotor System Model
4. The rotor system will have significant fluid interaction. All the fluid interaction will be in an annular region; that is, a fluid-film bearing, seal , impeller, or any other part of the rotor that is equivalent to a cylinder rotating within a fluid-filled cylinder. 5. Damping will be viscous and due only to fluid interaction. There will be no other source of damping in the system. This reduces the number of parameters in the equation and simplifies the mathematics. 6. The model will be linear. This is a difficult term to define briefly. A useful definition is that a system is linear if a multiplication of the input by a constant factor produces a multiplication of the output by the same factor. For example, if we multiply the unbalance by a factor of two, then the vibration of the machine will increase by a factor of two. Also, if a linear system is subjected to a dynamic input at a particular frequency, only that frequency will appear as an output. While machines can and do behave in nonlinear ways, nonlinear mathematics can be very difficult to solve algebraically. Fortunately, most machinery behavior is approximately linear, and the linear models approximate real machine behavior well enough to be quite useful. (That's good , because balancing techniques depend on linear behavior.) 7. A fluid will completely surround the rotor. Fluid-film bearings will befully lubricated (360°, or 2'Tr). While normal hydrodynamic bearings operate in a partially lubricated (180°, or n) condition, misalignment can unload the bearing, resulting in a transition to full lubrication and fluid-induced instability. Also, seals are designed to operate concentrically with the rotor; thus, they operate, by design, in a fully "lubricated" condition. For these reasons, the fully lubricated assumption is both realistic and necessary to adequately describe fluid -induced instability problems. 8. A nonsynchronous, rotating, ex ternalforce will be applied to the rotor system. We will see that rotating unbalance is a special case of this general nonsynchronous force.
171
172
The Static and Dynamic Responseof Rotor Systems
The Coordinate System and Position Vector
Figure 10-3 shows a basic physical description of the rotor system. The rotor can be described as a single, concentrated, perfectly balanced rotor mass, M, located in the center of a fully lubricated, fluid-film bearing that is fixed in place. A massless shaft is supported at the left end by an infinitely stiff bearing that provides only lateral constraint (no angular constraint). Thus, all of the rotor mass is concentrated in the disk and is supported by the bearing. The only stiffness element in the system is associated with the fluid bearing (which can also represent a seal). The rotor rotates at an angular speed, in rad/s, in a counterclockwise (X to Y) direction, as shown in the section view. The bearing clearance is greatly exaggerated for clarity. Note that, even though this description implies that the rotor mass and shaft can pivot in the small bearing, the rotor is assumed to be constrained to move only in the plane of the bearing with no angular deflection. The most accurate graphical description of the model would eliminate the shaft and small bearing altogether, leaving only the rotor mass and the fluid-film bearing. Figure 10-4 shows the coordinate system for the measurement of the lateral motion of the rotor mass. The X and Yaxes represent the real and imaginary axes of the complex plane. The terms real and imaginary come from the mathematics of complex numbers. Both of these directions are quite real, and nothing about the rotor position is imaginary. The origin of the coordinate system is the equilibrium position of the rotor when no external forces are applied to the rotor, and, in our model, it is located at the exact center of the fluid-film bearing. The rotor position vector, r, represents the position of the center of the rotor relative to the equilibrium position. It is defined in the rectangular complex plane as
n,
r=x+ jy
(10-1)
where x is the position of the rotor in the X direction, y is the position in the Y direction, and
j=~
(10-2)
The length, or magnitude, of r is A, where (10-3)
Chap'.., 10
Thl! Rotor System Model
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174
The Static and Dynamic Response of Rotor Systems
Th e angul ar po sition of r is measured as a po siti ve angle in a counterclockwise direction from the po sitive X axis. This angle, e(Greek lower case theta), is given by
e= arctan ( ~ )
(10-4)
x = Acos e y = A sine
(10-5)
Also,
Notice the similarity between this position vector notation and the vibration vector notation in Chapter 3. The two are very closely related; in fact, the solution of the model's equation of motion will yield vibration vectors. Leonhard Euler (1707-1783) showed that the po sition vector can be described using an exponential notation, which is very compact:
r = x+ jy=Ae jil
(10-6)
where e = 2.71828... is th e base of natural logarithms. The elements in front of the exponential function (in this case, A , but th ere will be other elements) define the length, or amplitude (magnitude) of the vector, r. The exponential function defines the angle of r. If r rotates around the ori gin with constant, nonsynchronous circular frequ ency, w (in rad/s), then the angle, e, becomes a fun ction of time:
e =wt +o:
(10-7)
where 0: (Greek lower cas e alpha) is the absolute phase angle at time t = 0, when the Keyphasor event occurs, and r is located at an angle 0: with the horizontal axis. The Keyphasor event acts like a strobe, momentarily illuminating the rotat ing vector at the angle 0: . If we substitute Equation 10-7 into Equation 10-6 we obtain a general expression for r that will be very useful for our purposes: r
= Aej(wt+a )
(10-8)
Equation 10-8 des cr ibes a po sition vector that rot ates; the tip of the vector and the center of the rotor precess about the origin in a circular orbit. We obtain the velocity (the rate of cha nge of position) by differentiating the po sition with
Chapter laThe Rotor System Model
respect to time, assuming constant amplitude, A, and constant angular velocity, w:
v = dr dt
= r = j wAe j(""t+n)
(10-9)
We differen tiate once more to obtain the acceleration,
a
= dv = r = -w2 A eJ(wt+o ) dt
(10-10)
A few words about j are in order. Whenever j appears outside the exponential, it basically means "change phase by 90° in the leading direction:' In Equation 10-9. j orients the velocity vector 90° ahead of the precessing position vector. Thi s makes sense if you realize that, as r precesses in an X to Y direction, the instantaneous velocity of the tip of r points (for circular motion) 90° from r in the direction of precession. Note also that in Equation 10-9, the amplitude of the velocity, wA, is proportional to the circular frequency, w. In the acceleration expression in Equation 10-10, the negative sign indicates that the direction of acceleration is opposite to the direction of r. The negative sign is the product of j .j (P = -1), so acceleration must lead displacement by 90° + 90° = 180°. The amplitude of the acceleration is proportional to w 2 • Finally, note that the mathematical angle measurement convention is that for positive w, r precesses in a counterclockwise (X to Y) direction, and the measured angle is positive. This is opposite of the Bently Nevada instrumentation con vention. where phase lag is measured as a positive number in a direction opposite to precession (see Appendix 1). This difference is very important when trying to relate the results of the model to measured vibration. Lambda (.\): A Model of Fluid Circulation
Whenever a viscous fluid is contained in the annular region between two , concentric cylinders which are rotating at different angular velocities, the fluid will be dragged into relative motion. Thi s motion can have a complicated behavior. What we need is a simple way of quantifying this behavior. .\ (Greek lower case lambda) is a model of fluid circulation that reduces this complexity to a single parameter. Though our discussion of .\ will focus on fluid-film bearings, keep
175
in mir>d thai th...., con(' e pt~ can be a ppl iN to a ny M mi l~ r p h)-..icaJ _ituat ion, such u seaJ~or pump ;mp...le..... lmagi r>e ...."(>. infini te llal plain ...pa nott'd by a n u id-fiIkd II"P CFigu n: 10-5 ). 1 h.. UJ'J""T p la t.. m." wilh a con' la nl Iin....r ,...I" c ily. ". a nd th.. lo"'....- ..]at.. ...._ z..ro ,...Ioei" : fkcau o f fricl io n. Ihe lir>ea r ",locity " f Ih.. llu id r>e~t I" the ~u rfa.... of II>.- m","ing p la t.. "'il l be v. wh il.. lh.. wloci t~· of tl>.- n uid nn t to tl>.- ... rfaet." of th.. . tati o na ry pl~te "'"ill be zero. lh.......Iociti e. in Ih.. n uid will form a linear velocity p rofik .. the ,...locit:< smoot hly rna nll'" from on.. ... rf,..... 10 th e o ther. Th.. a'''''''!l.. hn..ar ,...Ioeit:< o f th.. llo id (rN) must be som..wh..r.. berwee n ze ro a nd ". a nd . fo r this . it Udtio n. it i. O.5". :'I:ow ima¢ne wrap p in!l th.. t....o plates into ...."(>. conc..nt ric, in fin i tel~' lonll C)iind...... as , h....... at th.. bottom of th.. figu.... This i• • imilar toa roto roperat · inlt in.id.. a flu id · fiI m bearinll- The fluid i~ trap ped in IlK> a nnular region be........ n m.. cyIir>d... . . t h.. in r>er cyIind... rot a te. at so me angular ,...Ioeity. fl. a nd the o " t... cylinde r .e mai ns mot lonles•..-\ ,l h the n at pial .... t he n u id ne~t to ( he .... rf.."... of m.. cyIind.... m,," ha,... th " rfa.... ,-..Iocit:< o f th.. C)'linde ...
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Chapte, 10
n.. R oto ' S Y'I ~mMode I
l he a,,!!-u lar .....loci h· of l he fluid next to t he inner q 1inder is fl. ..nd th" angu l.., .'"loc ily of th.. fl" id n..xt to l he ~urfa ..... o f t he ou le , cylinde, i. ,,' ro. The fluid in lhe an nula' It'gion ....ill han ' a n a..... "'ll" a "/lula , •·...ocity betw n ""ro a nd !J. Fu, t....... mfi"""(~ {Ollg cyIind..... th.. a",rag<> a ngul ar ,...Ioci ty ill he a lmos t 0.5 f!. R.... I jou' na l. a nd bea rinll" a ,.. not infinil ..ly long. In ,..al hearing.,< t .... fluid is 1' ''' 1 tI " .. lo ..nd I""k"ll" a nd has to he re placed. Th i. i. u.ua lly accomp liAh.-d m' ra";all~ injectinlt mak.-u p flu id int o th.. hea ring t h, ou gh on e or mote po, ts. lf Ille po rt . a re radial Ih..n when t his flu id fi"'l e nt "O'$ the bea ' lng. it will ha'", zero a"!lulu .....loc;ly. The new fluid p ad ually und..rg."'s a ngula ' an't'I",al ion du.. lo lhe . h"" rinll act ion o f lhe moving n uid tha I is already in th.. bea ' ;n!!. Rut. al Ihe "" me tim... bee a..... ofthe p......"'e d ilf......ntiaJ .............,..n th.. injm ion point a nd Ille end of lh .. bearing. Ihe nuid . tart ' mm i ng axially. As a It. th.. n " id path l race . o " t a s pi. al IFil{ure 10-6 ) a nd may he ejo'CI.-d lwfo.... it c lle. Ih.. a nllu, lar ,...loci ty ......n in the infinil e cylinder. Fo. th is ...a'o n. t ile ..' a g<' flu id anllu , lar >....toe;ty ;n 1}1' icaJ. fully flood.-d. h~-d.ody oamir; beario!!, is Ivp iea lly ""'..... than o.so.
.i I J.
. • I
F''9'''e 10 ·6 Ru
EIec"",.
117
178
The Static and Dynamic Response of Rotor Systems
If the fluid average angular velocity is vavg ' then we define A as the ratio of the average angular velocity to the angular velocity of rotor rotation:
(10-11) Thus, the fluid circumferential average velocity is
V avg
=Afl
(10-12)
A is called the Fluid Circumferential Average Velocity Ratio. It is a dimensionless measure of the fluid circulation around the rotor and is a powerful tool for understanding rotor interaction with fluid-film bearings and seals. Typical values of A for a fully flooded, hydrodynamic bearing with only radial injection of fluid are between 0.35 and 0.49. Hydrostatic bearings, because of their higher injection pressures and decreased exit time (less circumferential flow), can have values of A less than 0.1. If the injected fluid has a tangential angular velocity component, it will affect the value of A. If the fluid enters the bearing or seal with an angular velocity component in the direction of rotor rotation (a condition called preswirl), then A can have values considerably greater than 0.5. This can happen when fluid is preswirled by a previous stage in the machine. If fluid enters the bearing or seal with an angular velocity component opposite to rotor rotation (antiswirl), then Acan have a value much less than 0.5, even approaching zero. Fluid is sometimes deliberately injected tangentially against rotation (antiswirl injection) in order to control a fluid-induced instability problem. This will be discussed in more detail in Chapter 22. Bearing geometry can also affect A. Plain cylindrical bearings tend to have the highest values of A: 0.43 < A < 0.49. Many bearing designs have been developed to break up circumferential flow and reduce A. Examples include tilting pad, lemon bore, pressure dam, multi-lobe, and elliptical bearings. Eccentricity ratio also affects A, and this will be discussed later in this chapter. For the purposes of modeling, A will be assumed to be constant.
( twptl!'t
10
The ROlor S)'$le m Model
Fluio.film Bearing ForCIH ~nd StiffMsSof'i VI. t-", di>.cuo.wd how a C)til>def rotallng irWde a l>O(iwl.lb>id-filIcod cyIi!t" ..... can ott the nWlt Into moIlOn. TIm io a ~ rn
o.,ord ..m.
"at.."""•.
10. 7. Tho 100<... on • ftuod·fim b"f\0"9 A ~. Os apphfollO fOt
II """rroq _
""lJI"
boo.-...., ---.._"".. . . . __.. _--,. . . <#.-
tt.<_dlhe-...g._ •• .,. _""......... _ ... " - . .. .... '" _ .d _ """'"" .. ~ _ •• lID tnt - - . " bote_ dtt.
... -..;._<_
The ......d-...rwo
119
180
The Static and Dynamic Response of Rotor Systems
duced by the pressure wed ge exactly opposes the force applied to the rotor. The position vector, r, extends from the center of the bearing (the equilibrium po sition and origin of the coordinate system) to the center of the rotor and is not precessing for this static load example. Thi s pre ssure wed ge is the primary means of rotor support in hydrodynamic journal bearings. The force due to the pressure wedge can be resolved into two components (Figure 10-7, right), a radial component that points in the opposite direction of r toward the center of the be aring, and a tangential component that points 90 from r in the direction of rotation. Both forces are assumed to act through the center of the rotor. Th e vec tor sum of these two forces is equal and opposite to the applied force vector. These two force components behave like forces due to springs. The radi al force component, F B , can be modeled as 0
(10-13)
where KB is the bearing spring stiffness constant in N/m or lb/in, FB is proportional to the displacement and the stiffness, and the minus sign indicates that it points in the opposite direc tion of r. Such a force is also known as a restoring force, because it always acts in the direction of the original po sition, attempting to restore the system to equilibrium. The tangential force component can be modeled as (10-14) where D is the damping const ant of the bearing. The j indicates that the dire ction of F T is 90 leading relative to r (in the direction of rotation, D). In the figure , FT is rotated 90° from r in an X to Y direction (the rotation direction of the rotor). If the rotor were rotating in a Y to X dire ction, th en FTwould point in the opposite direction. The term jDAD is called the tangential stiffness. See the Appendix for a discussion of the origin of this expression. The tangential stiffness is proportional to th e fluid damping. More importantly, it is proportional to the average fluid angular velocity, AD. Thus, the strength of the tangential force depends on both the rotor speed and A; it will get stronger with increasing rotor speed and increasing A. 0
Other Sources of Spring Stiffness
Besides fluid bearings, many other elements of th e rotor syst em behave like springs. The shaft acts like a beam that is supported at two points by bearings. When a force is applied to th e shaft, it will defle ct, producing a restoring force
(l\ii pl~'
10 The Rot",
S,.I~m ModoN
d' l'ffted I",..a n! th~ appli<'d fol'C1'. Shatu. ca n be ""Iati"ely Ilexible••uch a. in .......od.. ri~ative ga~ lu rnin..... o r very s tiff. Mlch as in ..I<'ct ric mo to ..... ll.-ca...... lhe .ha ll acu like a ht-am. be-am defled ,on .... ualio n. un be u..-d to ....tima te ils st iffne.., Som.- be-aring. <'an act like a p we . pri ng. ..i lh lit tle or no ta ng..ntial "' iff· ness. Rolling element bt'ari ng~ a nd low·.pe<'d bu
no- rs: Like t lie bea. in!! ~pri ng foret', f j is proportiona l to K. and tn.. m in" . s ign ind io cates l hat it a l,,-ap pu int. in t he opposite d irect ion o f r (Figu re l o-8j.
Fig u' . 10-3 '"'" 'P""9 """~_ Th<'.CII
0"''''...:1
Nt """"""'- Th<' 'P""9 ""~ i>....Y' b.d ' oward thi, ong;na1 poootiof\ _ i>"""",",-
llONI ,,, _
m. ~ 01 ..... di.place-
_ _ ...... . prr.g
~ K. Kcon
........ t
""""" UIIlroo>s conmt>.Jt>ons in tbe fOIOt ' Y''"'" ,,,",,, bea<,ng ..,.ing "ofIne». be"ing ",ppot! ,,""""\ _
fourdonon """"",-
181
182
The
Sl ~ Ii( ~nd
Dynam ic Respo nse 01Rotor Sy>IINl'lS
Th.. Oam p ing For c.. All ro tor . y.t..... . run... int ..rnal fore thai ca u", th.. d i, s lpation ol ..n"<> ro tor ..i lh Ih.. . ta to Lt h.. mm l.'TTl('ni o f attacl1 ro pa ris o n tl><> ' 0 10 '. o. fro m in t" m al frict ion in th .. roto r mal..rial il ...ll(al"" calkd h.'sl""tic d",npl1Jg ). .~11 0111><>... frict ion fo ill be 1I1T1<'d to be . mall Th" da mpin!! fo iA !!"""ra tro h..n a ' . ...0 11. fluid i•• h..a red be"'"l'E't1 two su.fa~"l"S in Iock a"so ""'" hit, a p is to n Iha t fo.e... Irapl'ro flu id th roll¢! a n n"f, con· ....-rttn!! m....' hit.lIcal .. nergy into h..at in th" .." rkin!! fluid. A" ..tl>t-. .. u m l'le i. a he"' l bein!! pro.... lkd th rou gh th at.... If th.. ..nllin.. s top"- th.. vi"" "ity 01 th....."t" . ..il l d i;..'lip.at" Ih" kin"' ic ..n /lY of Ih.. boat, a nd it ..ill rom.. tu a st up. T\t.o> foret'S acl ing h...... a .... a co mbinal ion of . h....rinll a nd p.....,.'u.... d.ag_ Simila r da mpi n!: fo""," OC<"U ' i" a flu id-film I"'a ri"!! " h" o th.. rolor ...." . ........... in Ih.. bea ri"¢. Put .impl~; t da mpi,,!! (,."'.... ' l<'C'Jr th ",ugh a combina· tion o f . h..arinll o f th.. flu id a nd p ,.... drafl. T\t.o> a m"u nt " f da mpIng 10"'"
tI,,,
Fi
"*
"'.. to t!w w locity _ .
~all"d .. m.1..a to tho- \rioOI ~' of the JOUrn;d in th~ """'ring. .'l.ddilk>n.o.l damp "lIt Can en"'" [n lm ro l." int..,-",1i'>fl .., th th~ _ ,king flu id tha t ... m lun.u il. ~ute lluol r ......." .... d r"!t ..IT..eta M"" n..t, in a
rff"".
n... da mpmlt fo.... i. p roponiu " a1 lo ,.... d amp ,"ll oonManL I J,
.PP'.......,..
...........-ution caliai-"" prod~ ~ ~ pnl1P'bltt...... ,~
oftto.._.,...-
HiHWH
UlHPH widf! l !'f!'!FJi u 'f r.. tjHt ~Hi
\.
~
f
"f i i =,'
/"
,;: ....
l,t1I'
[! i .
~.-nn q ~ ' P"',·.lH
;' 3 -:t' ;'''' "'2!:l -2,. , ~ , s e sl p~ ~
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" i r· ~. "~~ '.l s~ lr •~ •! lj·i i ! : ·l ! iI' , ! I '" : ;; p, ! "q '~ !r ,~ j1 f \~ '<'3,l ., ", .. h. e !i! -\ '!:;!i' ~: ~P' P ' i hl B . i"l p
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~ 5; ~ . T if ' ,j, .. _~.w; "T' 'I~i!.! l"' II ",:;; I:~ '~.III t-=t '~ ~_! ~ :; 'i ' ".l l;"l _' . " '" ,;! " ':r;~"3 ' ' ' 11- { ~!" ,~
·'. l;::J ---_~ 3 _, .. 1; 0I. ;r ",. .·l5.;0•~ --...e ,.-=- .-='~
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, if
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.
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x"" _ _ "..
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bm. F•. _
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_OOOI~, _
_ ~ ~
... . -
,
'86
The Static and Dynamic Response of Rotor Systems
The Equation of Motion The rotor must obey physical laws in response to the forces acting on it. We apply Newton's Second Law: The sum ofthe forces acting on a body is equal to the mass ofthe body times the acceleration ofthe body: (10-18)
Substituting the expressions for the forces , and being careful to keep track of the signs, we obtain (10-19)
Rearranging, we obtain the equation of motion in the standard form of a differential equation, (10-20)
This is a second order, linear, differential equation with constant coefficients. The appearance of this equation differs from a standard mechanical (or electronic) oscillator equation because of the presence of the tangential stiffness term,jD)..fl. This term is what makes rotor system behavior so interesting, and it this term that is ultimately responsible for fluid-induced instability. If there is no fluid circulation, ). is zero, and the tangential term disappears, making the form of this expression identical to the equation for a simple oscillator. Note that the rotor system internal elements appear on the left side of the equation, and the applied, external force appears on the right. If there were no forcing term on the right, the resulting equation (called the homogeneous equation) would be equal to zero and would describe the free (unforced) behavior of the system. For example, if we moved the rotor system to some position away from the equilibrium position and suddenly released it, the resulting rotor motion would follow the rules defined by the homogeneous equation. Free vibration will be discussed in Chapter 14. The equation ofmotion is a set ofrules that governs the behavior ofthe rotor system. The acceleration, the velocity, and the displacement of this rotor system must be related to the applied force by this equation. However, the equation of motion tells us very little about the vibration behavior of the system. We would like to know how this rotor system behaves over time, or what the amplitude and phase relationships are between the force and response. We now need to convert the set of rules into something more useful. This conversion process is called the solution of the equation of motion.
Chapter 10 The Rotor System Model
Solution of the Equation of Motion
To solve Equation 10-20, we must find a displacement function that, when differentiated (converted to velocity and acceleration) and substituted into the equation, makes the equation true. Since this is a linear system, perturbation at a frequency w must produce a vibration at the same frequency w. We will assume that the amplitude of vibration will have some nonzero value and that the absolute phase of the vibration response will be different from the phase of the perturbation force, 8. Fortunately, the solution of this type of differential equation is well known. We assume a solution of the form, r
= Aej(wt+o )
(10-21)
where A is the length of the rotating displacement response, or vibration vector (the zero-to-peak amplitude of vibration), and ex is the phase, or angle, of the response vector when the Keyphasor event occurs at t = 0. We differentiate Equation 10-21 to find the velocity and acceleration. This has already been done in Equations 10-9 and 10-10. The displacement, velocity, and acceleration terms are substituted into Equation 10-20, and, after some algebra, we obtain
(10-22)
In this form, the expression on the left side of the equation represents the rotating response vector. The terms in the numerator of the expression on the right represent the rotating force vector. Note that each exponential term has been separated into a time term and a phase term. The time terms are responsible for the rotation of the vectors; the phase terms convey information about the absolute phase of the force and response vectors. Because the time terms are identical on both sides of the equation, we can eliminate them:
(10-23)
This equation describes the amplitudes and absolute phases of the vectors at the moment that the Keyphasor event occurs (Figure 10-12). In the figure, the response vector is shown lagging the force vector, but we will see below that the
187
188
The SI"Ii( ....,;l ()yni>mi ( 11<>""""", of ROlo, Syste m.
"""P"'n'" ' -("ClIO' u n lead Ih.. forCf' •....,10' und.., wm.... ifCum~I'1U1('''''' Th .. d .."" m;"" to. of Eq ualio " 10-23 i. cal loo I.... no nsynchro>wu5I1yn(""ic St,jJnt'JJ,5. In ito simpk.t fOl m. Equallo n 10·23 .Iat.... l h.. foll"", n!\, FOIc..
\ , bm lion is Ih.. ral io o f t.... " 1'p1ioo f"....., 10 Ih.. f)yna mic St iffne••. When""e."~ mea.,,", .,b.a ho n (fo. elam ple u
,
\ f_
e ltl-l l fOI<:.. _
'01>- The two _
.. br..,.,.., 'floPCI..... _ ·
w.m
ttn ........ h """'" .ngulor ~~, .'" """'" ~n .. _ m
_
_tor (II'>< " " ' _
force). but , ..... ~ .. GO~ be ''''''_
'''9'
Chapter 10 The Rotor System Model
Nonsynchronous Dynamic Stiffness
The nonsynchronous Dynamic Stiffness, K N , is
K N =K -Mw 2 + jD(w->.fl)
(10-24)
Dynamic Stiffness is a complex quantity that consists of two parts, the Direct Dynamic Stiffness, (10-25) and the Quadrature Dynamic Stiffness,
K Q = jD(w->.fl)
(10-26)
Direct Dynamic Stiffness acts in line with the applied force; Quadrature Dynamic Stiffness, because of the i acts at 90 to the applied force. Dynamic Stiffness is a very important result of the model. It is a function of the perturbation frequency, w, and contains all of the rotor parameters in our model, including the rotor speed, fl. Dynamic Stiffness is the black box that transforms the dynamic input force to the output vibration ofthe rotor system. It is a major key to understanding machine behavior, and it will be discussed in detail in Chapter 11. 0
Amplitude and Phase of the Vibration Response
Equation 10-23 can be manipulated into expressions for the amplitude and phase. First, let (10-27) Multiply and divide the right side of Equation 10-23 by the complex conjugate of the denominator to eliminate the j terms in the denominator: . Fej6 D - jKQ ] AeJO = -- - [K ----"-
K D + jKQ K D - jKQ
(10-28)
189
190
The Static and Dynamic Response of Rotor Systems
Now, multiply through and combine the exponential terms on the left,
(10-29)
We now have an expression that mixes exponential and rectangular notation. The exponential form on the left already separates the amplitude and phase. The amplitude of the rectangular part is found by taking the square root of the sum of the squares of the direct and quadrature parts: A F
(10-30)
which reduces to F A=-,====
~Kb+K3
(10-31)
Dynamic Stiffness controls the difference in absolute phase (the relative phase) between the force (the heavy spot) and the vibration response, {; - 0:. This change is found by taking the arctangent of the ratio of the quadrature and the direct parts of Equation 10-29:
(10-32)
Solving for
0:,
the absolute phase angle of the vibration response,
0:
= s- arctan [
~~)
(10-33)
Thus, 0: differs from the location of the heavy spot, {; (the phase angle of the forcing function) by the effect of the Dynamic Stiffness. The negative sign indicates that when K Q and K D are positive, the rotor vibration vector lags the heavy spot.
Ch.o pc~
10
The Rolor Syu.... MocHI
The Attrtuc» A"9Ie: Rotor Fl"9O"W to ;I St;lti( R;Jdi;lllNd .\ ,.... toral oJ. ttwo ~ t. to .... bow thr rocOl" l'HJIO'odo to ttwo,opplicatJocl of;l!Ut>c IWiw kl.d. \\" .....u ~ w rut;II l"fl un~1>« wtth ;l o.e.tIC loMl. f -. W~ tiO' d"'t 1M ~ i. ;lJ>plied .wt>e:ally do..-nW1lrd .. tuk 1M _ i. rot;lti"~ ;lt m~ I f"""d. Jl. F IS 1101 rotati ng. 10 th is ... !I.f""CW catoe of . nonsync hronous p"Ml1rbat ion ru l'U' '" t h~ fJl"<[U<>" CY. w', i. , ..ro. Th"... th~ rom· po".."t. of th.. J)~"II.mk St iffn bo>«>m~
)I;,c,u!hlt. k>r tlus _ ic "-:I catoe. thr Du'K1 D)-....mic Stiff...... to idmli<:al to tlMo opnllfl.hlf_.nd tlMo Qaldratu ... 0, .., Stiffnroo it idmticll (nnpc lOr thr loitlnl to the- lanfto>ntw ~ ,ffnno. I»t it lltill({ I,"","" ~Jo.mo>nu into EqlJltion l6-ll givn ... th~ prp
" '"' ~ + u eta"
KMII [O
(16-35)
pIIOM'd the- nrpt_1lipt Ih~ 1M ~ I ful\ction. TIm rn
wtwft _
h..~
_r_.. . .
Fog"'" to-t ) llotor_fO • ...., ,...,... lood "'-_ac1""9~ T ~. ~~ec:
...,
,",,- , ..... F "'" " ' - ongIt. ... d r .. '-90' "'"" tho pt,.. ongIt. ~ d F '"""_.
....
enceWl ..,, _ _ ~ .. _ l .. . ....
~-
i
191
192
The Static and Dynamic Response of Rotor Systems
a rotating, statically loaded journal in a fluid-film bearing. The pressure wedge forces the shaft to an equilibrium position, which is in the direction of rotation. The amount of the phase lead will depend on the relative strength of the tangential and spring stiffnesses. Note that this situation is identical to the basic definition of the attitude angle that was defined in Chapter 6. Thus, the attitude angle, ![t, is defined by the model as
![t
D>.D ) = arctan (K
(10-36 )
Synchronous Rotor Response A good definition of synchronous is at the frequency of rotor rotation. If we lock our nonsynchronous perturbation force to the rotor, it will rotate at the rotor speed and become a synchronous perturbation , where w = n. If we substitute D for w in Equation 10-23, we obtain an expression de scribing the synchronous rotor response to unbalance, . Ae JO =
mr.
u
sr.»
[K -Mf2 2 + jD (I->' )f2]
(10-37)
Because we are now modeling synch ro nous (IX) behavior, {; represents the location of the heavy sp ot, the unbalance in the rotor; the vibration response ab solute phase, o, represents the high spot of the rotor. We can see that synchronous rotor response is actually a special case of gen eral, nonsynchronous rotor response; however, it is the most important from a practical point of view. Ordinary, unbalance-induced IX rotor vibration is a synchronous response. The rotor behavior described by Equation 10-37 will be thoroughly explored in the next chapter. Synchronous Dynamic Stiffness The denominator of Equation 10-37 is called the synchronous Dynamic Stiffness:
K s =K -MD2 + jD (I->. )D
(10-38)
Chapter 10
The Rotor System Model
The synchronous Direct and Quadrature Dynamic Stiffnesses are given by K D=K - M ft 2 K Q =D(l->.)D
(10-39)
The Direct Dynamic Stiffness is identical in form to th e nonsynchronous case in Equation 10-25, but the Quadrature Dynamic Stiffne ss has a different form than Equation 10-26. These differences will be discussed in the next chapter. Predicted Rotor Vibration
We have already examined the predicted behavior of the model to a static load , and we have found that it produces a reasonable result. We will now examine the predicted rotor vibration response for both nonsynchronous and synchronous perturbation over a wide range of perturbation frequencies. First , we will look at rotor system behavior for a system with low (subcritical) damping. A system like this is also referred to as an underdamped system. Low damping means that the Quadrature Dynamic Stiffness term, which depends on damping, is also low; this is a typical condition for most rotating machinery. The parameters used in the model are summarized in Table 10-1.
Table 10-1. Mo del pa ram et ers Low KQ
Parameter
52.7 x 10 3 N . s/ m (30 1 lb · s/i n) 900 rpm 1000 kg (5.7 lb · s2/in) 25.0 x 106 N/ m (143 x 10 3I b/ in)
D
n
M
K A m
s
0,48 ru
0.0 1 kg ' m (13.9 oz · in)
45°
High K Q 1.58 x 10 6 N · s/rn (9.02 x 10 3 1b · s/i n) 1500 rpm same same same same sam e
193
194
The Static and Dynamic Response of Rotor Systems
With two minor differences (zero-to-peak amplitude and the mathematical phase convention), the model output is equivalent to a set of startup or shutdown IX vibration vectors taken from a single transducer. In this simulation, since all angles are measured relative to horizontal right, the vibration transducer is mounted at horizontal right. The rotating heavy spot of the perturbator is located at 45° from the transducer, in the direction of rotation, when the Keyphasor event occurs. Figure 10-14 shows the Bode and polar plots of the predicted behavior of the model. For the nonsynchronous perturbation, the rotor is operated at a constant speed, n, of 900 rpm. The perturbation frequency, w, is swept from zero to 4000 cpm, and the model produces vibration vectors that are filtered to the perturbation frequency. For the synchronous case, the perturbation frequency is set equal to the rotor speed (the unbalance heavy spot is now the perturbator), the rotor speed is varied from zero to 4000 rpm, and the model produces response vectors that are filtered to rotor speed. Thus, the horizontal axes represent cpm for nonsynchronous perturbation and rpm for synchronous perturbation. The phase produced by the model is in mathematical form, where phase lead is positive and phase lag is negative. However, the phase in the plots in the figure is presented using the instrumentation convention, where phase lag is positive, increasing downward. Phase is measured relative to the positive X-axis of the system (the location of the transducer). The amplitude produced by the model is zero-to-peak. On the plots, the amplitude has been doubled to produce peak-to-peak to conform to the instrumentation measurement convention for displacement vibration. Note that, at frequencies near zero, the nonsynchronous response phase leads the heavy spot location slightly. However, at zero speed the synchronous response phase (the high spot) is equal to the heavy spot location. This is an important finding with application to balancing. A resonance amplitude peak is clearly visible and is accompanied by a significant increase in phase lag. For this low damping case, it can be shown that the nonsynchronous resonance occurs near
w=~
(10-40)
Equation 10-40 is also called the undamped naturalfrequency, or, less accurately, the natural frequency of the system. This is sometimes referred to as the mechanical resonance.
Chapt ... 10 Th,. Rotor Sf""" Model
----
.
'
.,'... J:: ••
il
.'
_..,,,,-';~ ' _ ....t w _ "" ,..., ,Oi: _ .... _ ".._ .... ..,_ 'lOll - .
".
,-
f ig u... ' O-l a. Bode MId p<>Lor plo'" of p'.,licl<'d 1Olo< vibraTion fol low Quad''''~ Oynornl[ "',.,.,...'- 80lh nonsynchron
prod"".
"""SO' ..
_
..
""'''''''od
19S
196
The Static and Dynamic Response of Rotor Systems
The synchronous resonance occurs near
n=Ji:
(10-41)
A commonly used term in the industry for a synchronous balance resonance is a critical, or critical speed, but the term balance resonance is preferred when speaking of a synchronous resonance due to unbalance. At this frequency, the phase of the response lags the heavy spot location by 90°. When damping increases to a supercritical (overdamped) value, the predicted vibration changes dramatically. Figure 10-15 shows plots for both nonsynchronous (blue) and synchronous (green) perturbation. For the nonsynchronous case, the rotor speed, was set to 1500 rpm. At low speed, the nonsynchronous response phase leads the heavy spot by about 80°. This is the attitude angle ofthe system. Using the parameters in Table 10-1 and Equation 10-36, it is calculated as 78°. Because the system is overdamped, there is no synchronous resonance. However, there is a nonsynchronous resonance near
n,
w=>.. n
(10-42)
This resonance is sometimes referred to as the fluid-induced resonance. At this perturbation frequency, the phase of nonsynchronous vibration is equal to the heavy spot location. Both the mechanical resonance and the fluid-induced resonance are different manifestations of the same thing. Recall that our model is only capable of one resonance. The frequency of this resonance depends on whether the system is underdamped or overdamped, and on whether the perturbation is nonsynchronous or synchronous. The fluid-induced resonance is only visible when 1. The rotor system is overdamped, and 2. The rotor system is subjected to a nonsynchronous perturbation. Both conditions must be true. Because operating machines are typically underdamped and subjected primarily to synchronous perturbation due to unbalance, the fluid-induced resonance will never be visible under normal operation. For a typical machine, the resonance will occur at the balance resonance speed given by Equation 10-41.
Chapll'f 10
'" ," .
I"',. I
I
o·
•, • ,. ,.
--
\ ------------:-----'''-~
,
- -'.
.',
#-
•
,
~r' ..'· .......
---
",,, """......""" """ """ , •• -..
• I ••,. I
Th RotCII' S~ .... Mod...
--....._. - .... -
•• "" "... "'" ~ "
"..
•
'O ......... ~ ...
,...
F;g.- 10- 15 Bodo . Cod POI.' pIoU
phno _
pt>."'...., . .
PO>IIiYe wnword.P_ "' _ · rolt>o!_ X-4'i.of tho ' y>l~ m . rn. 1ow .~ non,ynch'''''''''''
u ~ f@l
,
,...., 10 tho hN vy spot ,~o"ItS~ . no(Udo,
' ''9~
of ..... ' Y" e m.
197
198
Th ~
Slal;c a <>
Nonlinea ritie. T",.., basic aM"mplions uoed in I.... d..ri' -al io n of l h.. m. odd ...... thaI Ih.. 'JOSle m isli n r a nd Ihat Ihe rotor sr"I ..m p
Chapter 10 The Rotor System Model
Bearing center
Q) I~ ~
ro
, -, , -
~ u
'
:J
.
•
:
:
:
i' I
1
. Q) ~ -- _ . ". --
ro
A ........... .. /
~ U
:J
Iz ' c J 1
.s
0
.s
1
Eccent ricity ratlo. s
Figure 10-16. Qualit ative plot s of flui d-fil m bearing parameters versus journa l eccentricity ratio. Stiffness, K. and damp ing, D, are minimum w hen the journa l is at the center of t he bearing, and t hey are approximat ely constant for low eccentr icity ratios. As the jo urnal nears th e bearing surface, stiff ness and damping increase drarnaticallyA behaves in the opposite way. When the jo urnal is at t he cente r of the bearing, ), is maximum . As th e j ournal nears the wall, t he fl uid flow is increasingly restricted, until ), nears zero at the wall.
will use this insight to examine synchronous rotor behavior in more detail in th e next chapter. As we have mentioned, vibration is a ratio, and changes in Dynamic Stiffness produce changes in vibration. Dynamic Stiffness is a function of the rotor parameters of mass, stiffness, damping, lambda, and rotor speed. By relating vibration behavior to changes in rotor system parameters, the model provides a conceptual link between observed vibration behavior and root cause malfunctions. Thi s is a major advantage of this modeling approach when compared to matrix coefficient methods. The basic relationships between simple rotor parameters and malfunctions will be exploited throughout this book to solve practical machinery problems. The model provides a n excellent description of the lowest mode of a rotor system and can provide some information about higher system modes. However, accurate treatment of multiple modes requires a mo re complicated model with additional degrees of freedom.
199
200
The Static and Dynamic Response of Rotor Systems
Extending the Simple Model
The simple, isotropic model provides an excellent description of the lowest mode of a rotor system. However, it does not adequately describe the behavior of rotor systems with anisotropic stiffness or with multiple modes. In this section, we will present examples of a single mass, anisotropic rotor model with two reaL degrees offreedom (2-RDOF), and a two mass, two-mode, isotropic model with two compLex degrees of freedom (2-CDOF). Readers can skip this section with no loss of continuity. The anisotropic rotor model is similar to the l-COOF model that we have been discussing in this chapter; the rotor is modeled as a lumped mass with significant fluid interaction. There are two primary differences: 1.
The displacement is measured using two, independent, reaL variables, x and y (two degrees of freedom). The complex plane used for the simple model is not used (although complex notation will be used to simplify the mathematics of the solution process).
2. The rotor parameters are different (anisotropic) in the X and Y directions. In general, any parameter can be anisotropic, but in this discussion, we will only treat the spring stiffness (Kx and K) as anisotropic; all other parameters will be assumed to be isotropic. The rotor free body diagram of the anisotropic system (Figure 10-17) is similar to that for the isotropic system (Figure 10-11), with two exceptions: the force components are now shown aligned with the measurement axes and the tangential stiffness terms appear without the j that is used in the complex plane. The tangential stiffness terms cause a response at right angles to the displacement. Imagine that the rotor, which is rotating here in an X to Y direction, is deflected a distance x in the positive X direction. Because of the fluid circulating around the rotor, a pressure wedge will form that will push the rotor up, in the positive Y direction. Thus, the tangential force in the Y direction is +D>"[2x.
Similarly, if the rotor deflects a distance y in the positive Y direction, the pressure wedge will try to push the rotor to the left, in the negative X direction. Thus, the tangential force in the X direction is - D>"ily . The tangential force terms cross coupLe the X and Y responses. As long as the tangential force term is nonzero, any deflection in one direction will create a force that produces a response in the other direction.
- 0.
,.
,-~ _
10-11 ~rr...bodyd"· 9'_ "" a 2~ _""I'ic _ _,Tho bU' """-"
- ~r
•
*""
"""" _ "'-'" aIoryood II-. - - . . - ~ Tho tangIt<"O-aI .. 11-. J ..... _ _ " .... U>, . . .
... ~ ...
--.r. N._ ill
+OAflx
,.,. -~""'''''''''-'9
-~""""' _ ""'""'" }( -._Iorte.... .... ...- 0ft9I0 t. tu. ... . ""'" "'" r .... _ k>rr;e .... Ilhaw 0ft9I0 ~ - ~ 2
x
202
The Static and Dynamic Response of Rotor Systems
The perturbation force is also expressed in terms of X and Y components:
F; = mru w 2 cos (wt + 8) Fy = mru w 2 sin (wt + 8)
(10-43)
However, even though we are modeling the system with real numbers, it is mathematically simpler to use complex notation and take the real part of the result. Then, the perturbation force can be expressed as F
x
= mr:u w 2 e j (wt+8)
Fy =mru w2 e
j[wt+8-~]
(10-44)
2
where the 7r/2 is the angular difference between the two coordinate system axes. These two expressions identify the same rotating unbalance vector, which is referenced to each coordinate axis (see the figure). The 2-RDOF system requires two differential equations in x andy: Mx + Dx + K x x
+ D>'[2y = mru w 2 e j (wt+8)
My+Dy+K yy-D>'[2x=mru w2e
j[wt+8-~]
(10-45)
2
We assume two solutions of the form:
= Aej(wt+a) y = Bej(wt+.3) x
(10-46)
where A and B are the amplitudes of the rotating response vectors, and a and (3 are the phases. The solutions will provide a set of rotating response vectors, each of which is measured relative to its own axis. The instantaneous physical position of the rotor is formed from the combination of the real part of these vectors:
x(t) = Re[ Aej(wt+a)] y (t) =
Re[ Bej(wt+(3)]
Solution of the system of equations 10-45 leads to
(10-47)
Chapter 10 The Rotor System Model
Aejn
= mr w2e j8 u
2 K y -Mw + jD(w + >..n) (K x -Mw2 + jDw)(fS, -McJ + jDw)+(D>..n/ K x -Mw 2 + jD(w+>..n)
oc, -Mw2 + jDw)(Ky -McJ + jDw)+(D>..n)2 (10-48) For each vector, the amplitude is found by taking the absolute value of the expression; the phase of the response is the arctangent of the ratio of the imaginary part to the real part,
a
= 8 + arctan
Im(Ae jn)
.
(10-49)
Re(Ae P )
The 2-RDOF, anisotropic model (in scalar form) can be converted to the simple, isotropic model quite easily, a procedure that validates the anisotropic modeling of the tangential force. We make the system isotropic by setting K x = Ky = K. Equations 10-45 are modified to use the perturbation forces of Equations 10-43, and the y equation is multiplied by j:
Mx + Dx + Kx + D>..ny = mruw 2cos(wt +8) j(Mj + Djr+ Ky- D>..nx) = j(mruw 2 sin(wt +8))
(10-50)
When the equations are added, we obtain
M(x+ jj)+D(x+ jjr)+K(x+ jy)- jD>..n(x+ jy) = mruw 2 [cos(wt +8)+ jsin(wt +8)]
(10-51)
This reduces to
My + Dr + (K - jD>..n)r = mruw 2e j (wt+8)
(10-52)
which is identical to Equation 10-20, the equation of motion for the simple, isotropic model.
203
204
The Static and Dynamic Responseof Rotor Systems
In the two-mode, isotropic system, the rotor is modeled with a complex displacement vector in each of two, axially separated, complex planes (two complex degrees of freedom, or 2-CDOF). There are many ways a system like this can be modeled; what follows here is only one possibility. The rotor is separated into two, lumped masses, M I and M 2 (Figure 10-18, top). A midspan mass, M I , is connected through a shaft spring element, K 1, to a stiff bearing at left. The mass experiences some damping, D 1• The mass is also connected through a shaft spring element, K 2 , to a journal mass, M 2, at right. The journal operates in a fluid-film bearing with damping, DB' bearing stiffness, KB , and A. The resulting free body diagrams are shown at the bottom. As in the anisotropic model, a two degree of freedom system requires two differential equations, this time in two, independent, complex displacement vectors, r i- and r 2 : M1rl
+ D1rl +(K1 +K2 )rl -K2 r2 =m1rU1w2e}(wt+/\)
M 2 r2 +DB r2 +(K2 +KB - jD BA!?)r2 -K2rl
=m 2 ru 2 w 2e}(wt+IJ,)
(10-53)
Note that there are two, independent unbalance masses, each with its own mass, radius, and phase angle. We assume a solution of the form r r
= A1e}(wt+o\) = A 2e}(wt+o 2
l
2 )
(l0-54)
The solution is, again, two expressions:
(10-55)
ehdple, 10
K >
,
K
•
n..,Rotor S)'S' ..... M ~
,
D,
-D,',
m , " '~. 1~A J(. ' +~ )
,
- ", (',- ',)
F"og_ 16-18 2-axJF ' olOt ~ .....:I h~ body
'0.
m"""""",,",..
""""
lOS
206
The Static and Dynamic Responseof Rotor Systems
As with the ani sotropic example, the amplitude is found by taking the absolute value of the expressions. The phase is found using th ese expressions :
(l0-56)
Summary Lambda (A), the Fluid Circumferential Average Velocity Ratio, is a nondimensional number that represents the average angular velocity of the circulating fluid as a fraction of the angular velocity of the rotor. Using assumptions of a single, complex degree of freedom; isotropic parameters; no gyroscopic effects; significant fluid interaction; and linear behavior, a set of forces were defined that act on the rotor system. The se forces are the spring force, the tangential force due to a pressure wedge in fluid-film bearings and seals, the damping force , and an external perturbation force. The forces were combined in a free body diagram, and used with Newton's Second Law to obtain the differential equation of motion. The solution of the equation of motion provided the rotor system Dynamic Stiffness, an important result. Dynamic Stiffness is the "black box" that relates input force to output vibration. The general, nonsynchronous Dynamic Stiffn ess was found to be
K N = K -Mw2
+ jD (w-.\ fl)
The response of the rotor to a static radial load led to an expression for the attitude angle of the rotor in terms of rotor parameters. The atti tude angle was found to be equal to the arctangent of the tangential stiffness divided by the spring stiffness. By se tting the non synchronous perturbation frequency, w, equal to the rotor sp eed , fl, an expression for synchronous rotor vibration resp on se was obtained. Synchronous rotor response, which is the mo st commonly observed mode of
Chapter laThe Rotor System Model
operation of machinery, wa s found to be a special ca se of the general, nonsynchronous model. The model behavi or over frequency or speed was explored. Th e model clearly shows a resonance, coupled with a 180 ph ase change in th e lagging direction. The frequency of the resonance depends on the spring stiffness and mass of the system, th e type of perturbation used (nonsynchronous or synchronous), and, for nonsynchronous response, on the Quadrature Dynamic Stiffness of the system. For underdamped rotor systems, the resonance occurs near the undamped natural frequency, 0
w=Jf; References 1. jeffcott, H. H., "The Lateral Vibration of Loaded Shafts in the Neighbourhood of a Whirling Speed.-The Effect of Want of Balance;' Philosophi cal Magazin e 6, 37 (1919): pp . 304-314. 2. Muszynska, A, "One Lateral Mode Isotropic Rotor Respon se to Nonsynchronous Excitation;' Proceedings ofthe Course on Rotor Dynamics and Vibration in Turbomachinery, von Karman Institute for Fluid Dynamics, Belgium (September 1992): pp. 21-25.
207
209
Chapter' ,
Dynamic Stiffness and Rotor Behavior
an important result of the rotor mod el that was developed in the last chapter. Vibration was found to be the ratio of the applied dynamic force to the Dynamic Stiffness of th e rotor system. Thus, a change in vibration is cau sed by either a change in the applied force or a change in the Dynamic Stiffness. By understanding how Dynamic Stiffness affects vibration, we can understand why rotor systems behave the way they do . This understanding will lay the foundation for balancing and the malfunction diagnosis of rotating machinery. In this chapter we will explore Dynamic Stiffness in more detail. We will sta rt with a discussion of the physical meaning of the components of Dynamic Stiffness. Then, we will show how the rotor parameters of mass, spring stiffness, damping, and lambda (A) can be extracted from plots of Dynamic Stiffness versus frequency. Concentrating on synchro nous rotor behavior, we will show how Dynamic Stiffness controls rotor response over the entire speed range of a machine and how it is responsible for the phenomenon of resonance. Finally, we will show how changes in Dynamic Stiffness produce changes in vibration in rotating machinery. TH E CON C E P T OF DYNAMIC STIFFNESS IS
What Is Dynamic Stiffness? Physically, Dynamic Stiffness combines the static effects of spring and tangential stiffnesses with the dynamic effects of mass and damping. We will discu ss the physical meaning of this shortly. First, recall th at the nonsynchronous frequency (of both perturbation and vibration), w, is completely independent of th e rotative speed, Th e equation for the generalized, nonsynchronous Dynamic Stiffness is
n.
210
The Static and Dynamic Response of Rotor Systems
KN
= K -Mw 2 + jD (w->..f! )
(11-1)
Thi s can be broken into the nonsynchronous Direct Dynamic Stiffness, (11-2) and the nonsynchronous Quadrature Dynamic Stiffness, (11-3) The Direct Dynamic Stiffness acts along the line of the applied static or dynamic force. The j term indicates that the Quadrature Dynamic Stiffness acts at 90° (in quadrature) to the instantaneous direction of the applied force. When Equation 11-1 is written as K N = K - M w 2 + jDw - jD>" f!
(11-4)
the stiffness terms can be eas ily related to the internal and external rotor system forces. K, the spring stiffnes s of the rotor system, is a combination of the shaft stiffness, the fluid-film bearing stiffness, the bearing support stiffness, and the foun dation stiffness. This term behaves like the stiffness of a simple spring; when a force is applied to the rotor system, the rotor deflects and the spring is compressed, producing an opposing force (Figure 11-1, middle). Simple spring stiffness always acts in a direction opposite to the direction of the applied force; thus, the K in the Dynamic Stiffness is positive, showing that it will oppose the applied force. Positive springs are stabilizing in the sense that the force produced by the spring pushes in a direction back toward the original position. The second term, -Mw 2 , is the mass stiffness. It is a dynamic term that appears because of the inertia of the rotor. Imagine a mass that is vibrating back and forth about the equilibrium position in a simple system (Figure 11-1, bottom). Whenever the mass moves beyond the equilibrium position, the spring force acts to decelerate the mass. However, the inertia of the mass creates an effective force that acts in the direction of motion of the mass, opposite to the spring stiffness force. Thus, the mass stiffness is negative, and acts to reduce the spring stiffness of the system. Negative springs are potentially destabilizing in the sense that the force tends to push an object farther away from the equilibrium position.
Chapt('f 11 Oynam K StilfM" and Roto r ~avio.
."
".
F
."
"",
f~ 11-' . Spnrq ond "'"" Sl Th<' "",pi<- 'P"ngf......, "Y""'" is shown ....th no apph 1oK,,-. "ati< 1oK,,-_ • ~ ~. W1lrn. >Lilli< 10«. ~ .~ IrrOdd"~ "'" 1'0'0I... 'P""'.l "ofIn~.. ';. proO"cfi. 10«:. pr Oportl<>rlal to "'" di,· p lacftl'O'nt thol GPPO""' "'" ~ bc•. _ •
10K.;, appJ;Nl (b
"'"' , ""'."'oct;, oquMlenllO • "O'Q<>t~ " .......... . - lh.
"""'''
_.11..._· ,
211
ruroe
T..... th ird trnn.J0 "" onJi...l.... in t IM' damping and is calIoiliu tlw roIor ~y"'em.liU I1M' oI,lf....-. I.... darnplng !It,!f......~ ' a ~mic oI,ffno-. and ,t only ~ lhe roI or ",",1m"... ha. 1'IOnun> >TIocih.
Thor lao! Ierm. - jD>.!l. ~ from the WlV"ttiaJ fonT of t .... fluid-film 1"""'8"", ..~ in I.... ~"" or...al. The j indiclot.... t hai Ihi. ttvzgmtitIJ " "" ItrU t....." aJ.., ...-t . a! 9tI' w t d il't'O;tic>n of!be .pphed forw >'t'CtOl: RecalI lhaI t .... taflll:"nti.ol . tiff in Ihe a...", chapter (E
~ 1 1 ,2 ~
_
_'i, o:I.oo ....
~
cloogwam. The 00Kt_
... __-
d""'~_I1 _oI
~ .......... ,,\Ill'
.. _ _ """
... the o-.ct
...
n... Jour 5rJfInesi _ ...... ~I~--' .... the~ ....... _ _..,. Il.• The ~
~
~
" " " - ' rho-Ipploed 100<.. _ the Ii the ...........
Dyrwmi<
5,,,,,,,,,,,- _100
,....."'1<' be,,,,,,, ,..... 'OW"'ll b~ _
the ";brltlon teiPO".... _1OI
_
s __
Chapt..r 11
D ~m i<
Slifftw« a r>d ROlo< Ileha vior
The Quaob a tu Dynamic Stiffn...... i< .....pon
Fig_ 1' -) Sync:hronous Dyr\om<: s,,,,"",,, _ d ""l'om. The QuaodooMU'"
Dynom" ~ conW< of on" """ '''''' ond """,""" OS".... gle ~_. like the <'CJlSytI( """""" -'or d""'J'om, 'ho "'91e af the 'Y'""hrcroou. Dynamic
SU'!....., Y@CIor .. the ...measthe""9'" _ the "e.wr'poI d".rnon ....., the . b o' "", ~""".... W'Ctor(higl't
, i.
--,
21l
214
The Static and Dynamic Response of Rotor Systems
The synchronous Dynamic Stiffness is obtained from the nonsynchronous Dynamic Stiffness by setting w = D: K s = K -MDz
+ jD(l-,\)n
(11-5)
The synchronous Dynamic Stiffness is most important for everyday machine applications, because machines vibrate primarily in response to rotating unbalance, a form of synchronous perturbation. The physical meaning of the synchronous Direct Dynamic Stiffness is similar to the meaning of the nonsynchronous Direct Dynamic Stiffness. However, the Quadrature Dynamic Stiffness term in Equation 11-5 is different. Because D is common to both quadrature terms, it is factored out. Figure 11-3 shows a typical synchronous Dynamic Stiffness vector diagram. Note that the Quadrature Dynamic Stiffness now consists of only one term and appears as a single vector. Like the nonsynchronous vector diagram, the angle of the synchronous Dynamic Stiffness vector is the same as the difference between the heavy spot direction and the vibration response vector (high spot). We will discuss the behavior of K s and its relationship to rotor behavior in detail in a later section. Rotor Parameters and Dynamic Stiffness
Both nonsynchronous and synchronous Dynamic Stiffness can be plotted as functions offrequency (Figure 11-4). Dynamic Stiffness is plotted in two separate plots, with direct stiffness above and quadrature stiffness below. Both nonsynchronous (N) and synchronous (S) data are shown. The horizontal axis represents perturbation frequency, w, in cpm for the nonsynchronous case, and rotor speed, D, in rpm for the synchronous case. Rotor speed for the nonsynchronous perturbation is 900 rpm. In the Direct Dynamic Stiffness plot, the nonsynchronous and synchronous cases plot on the same line. Below the Dynamic Stiffness plots are Bode plots of the corresponding rotor responses. The Dynamic Stiffness plots can be used to obtain the rotor parameters. The figure was created using the same model and rotor parameters as in Chapter 10 (Table 10-1), and the key points are marked in red. The Direct Dynamic Stiffness plot (top) is a parabola. At zero frequency, the mass stiffness term is zero, and the Direct Dynamic Stiffness is equal to the spring stiffness, K. The frequency at which the Direct Dynamic Stiffness is zero yields the resonance speed (this will be discussed below). The Quadrature Dynamic Stiffness plots (second from top) are different for nonsynchronous and synchronous perturbation. Both plots are straight lines, but with different slopes and different Y intercepts.
~
S
.<-'''_'1 S
--.
. ,'- '.:-'-' ~
~I
I "'
_ _ D,nomoc
1."'''''''' So""""" ,,
~ P'
".,.
""_ ·rod."n"".
""' """'''' ...... .-..
- . . "II"'d lor ~.......,....
p.or
:so... t
-I .
_
I
- '-'•,-- ....
...1
_
D(I - 1}!J
-._
. -,-- .--- .• •
~ 'otOl
Cf/ ,.". bi
,
K - .\ff P
--
I\ .I
•
216
The Static and Dynamic Response of RotorSystems
The nonsynchronous Quadrature Dynamic Stiffness (blue) has a negative Y intercept. The absolute value of this intercept is the tangential stiffness, DA[2, of the rotor system. The stiffness then increases and becomes zero when the perturbation frequency, w, is equal to AD. The slope of the line is equal to the damping, D. There is enough information in this plot to obtain D and A. Thus, the direct and quadrature components of the nonsynchronous Dynamic Stiffness can provide all of the rotor parameters of our model: K, M, D, A, and D. The nonsynchronous plots also define the Margin ofStability, the frequency range between the zero values of the Direct Dynamic Stiffness and the nonsynchronous Quadrature Dynamic Stiffness. Rotor stability will be discussed in detail in Chapters 14 and 22. For now, we state that, if both the Direct and Quadrature Dynamic Stiffnesses become zero at the same nonsynchronous frequency, w, then the Dynamic Stiffness of the rotor system will vanish. This zero in the denominator of the rotor response equation would result in a (theoretically) infinite response amplitude, a condition called instability. The Y intercept of the synchronous Quadrature Dynamic Stiffness is at the origin (0,0), and the slope is damping modified by fluid circulation. This is less than the slope of the nonsynchronous stiffness, which is D. Thus, we can define the effective damping (or observed damping), DE' of the synchronous rotor system as (11-6)
Because A is usually a positive number less than 0.5, the effective damping for synchronous behavior is usually less than the actual damping constant, D. We have already mentioned that the tangential stiffness term is negative, which acts to oppose the stabilizing damping stiffness term. One effect of this negative stiffness is to reduce the effective damping of the system. This makes sense if we imagine a rotor in a forward, circular orbit. Physically, the damping force acts to remove energy from the system. Force times velocity equals power, and the damping force produces negative power because the direction of the damping force is opposite the centerline velocity. At the same time, the tangential force acts in the same direction as rotor motion and pushes on the rotor. This is positive power, and this power input to the system partially cancels out the power loss due to the damping force. Thus, because of fluid circulation and the pressure wedge, the effective damping of the system is lower.
PT. EMOMI DOC. ROOM
Synchr...-J$ RotQf s.t>..w:.. prima.-y 'OOUnT of nhratlOn in _or ~~ nn. ;. dow 10 unbaLo......... U~ ~ .o synduonour. ( I .'( ) rnponw in .n rol atin, madl""",:,, a nd ill thor _ 001I1""""" ~ ~ f1K.ou... of lib i mporU~ ..~ will c...oc.. , lnI.. on STTldIJorIOUIl _ bo-N,'ior;
n...
~ .....
rolor""'f'Ol""'"' <'an br d j,ldrd into th -...tM. fd ~o.: boobo- '""""...~ (..t. id1 ..~ ..,II call >pfftI-~ /ftORlI ~• .oncl-U .o1x.. ~ ................ (-h~ "J"""'d.~ In NdI ~ .o d JI.......1 1nm of I.... I)ncilnmo.... 1»nomic Stiff """I rois th .. l'ftoJI'On ... of thO' r ,)-"",,_ In ow d, ~m. ..~ .. i ll ..... lIMo ")· fOIl0U5 r"' 01 "",dd IN I d~ in 1.... t...1 p!<'•. in " t.ich I Ix> IX pon... ~edor. ,_ i, ......,n.d 1».
.....u
:\) 0
( 11-7)
........... "n..n 11>00 "..,-pha>t........... l OC'CU"'- th.. rotah nll unboo4nt:., d.......mic f,,~
II.........,..
O Hl)
Synchronous 8eha Yior
8@olow
Rnonanu '
FijIu... 11-5 """""~ plolo of ap .dlJo nou. Oi""" l .ond Q......dr..tu O)nanl KSl l l f _ t!Opland Ilod .. and pc>la. rloh of th.. IX rOlor V1hral ~m p iet ed by 11>0' rnod<-L .'\1 th.. . iJlhI of th .. d~-n. mic . ti ff 1'1"11 ..... th .. s)Tlch ron .. w. Dyn..m ;c Sl ,ffn~s ''{'("(o, dl&ftla m. fo r Iow·speed na ...,... a nd h ijlh·.p'"<'d .-ond llion... Th.. o .wnl..t;on o r th.. llil't'ct Dyna mic Sh IT n i. cor... s pond. 10 th....... ,.,. spot loca l IOn. 3 15". In t.... Iow·"J""""d •• "tt"' ( ll"""n ~ ........&1 Ihin!'l-" . ... "ppa... nl : 1.
n... rotor h ip> opot: (vibrat IOn nospon.... ' ''':'0' ) is in the ..... d o.....,""' .. , l hO' ''''''''' "f'Ol. TIliI io appr
,n..
"'P""""i n... hIM)" opot and hilth 'i"'l ...... 1d 10 hf. ... ~
218
The Static and Dynamic Response of Rotor Systems
2. The vibration amplitude increases as the square of the rotor speed. 3. The synchronous Quadrature Dynamic Stiffness is close to zero. 4. The Direct Dynamic Stiffness equals K at zero speed. The high spot/heavy spot relationship is an important key to balancing. Part of the balancing problem involves determining the direction of the rotor heavy spot. Either the Bode or the polar plot can be used to do this, although the polar plot is much easier to interpret. At very low speeds, D is small, and the Dynamic Stiffness is dominated by the spring stiffness, K. Because of this, the mass stiffness and quadrature stiffness terms can be neglected, and Equation 11-8 becomes
Below resonance:
r=A L:a = mruD2 L:6 K
(11-9)
Because the mass and quadrature stiffness terms are neglected, there is no phase change, and the response is in the same direction as the applied force , a = 6. Thus, the high spot and the heavy spot are in the same direction. Synchronous BehaviorAt The Balance Resonance Several aspects of the balance resonance region (red) are important: 1.
The amplitude of vibration reaches a peak, and, at the same time,
2. The phase of the response lags the heavy spot by 90°. This occurs in the Bode plot where the phase slope is steepest and in the polar plot close to the maximum amplitude of the polar loop. 3. The Direct Dynamic Stiffness becomes zero. This can be seen on the direct stiffness plot and in the Dynamic Stiffness vector diagram. 4.
The Quadrature Dynamic Stiffness is the only stiffness element available to restrain the rotor.
1>;1-1<11
-- - '.----,....... .... --.-1
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The Static and Dynamic Response of Rotor Systems
The zero crossing of th e direct stiffness is near the speed of the resonance peak. This is typical for a machine with a moderate Synch ronous Amplifi cation Factor. The onl y way the dire ct stiffness can become zero is for the spring stiffness and the mass stiffnes s to can cel each other. Let fl res be the speed at the zero cro ssing of the direct stiffness . Then, for this condition, K -Mfl;:es = 0
(11-10)
wh ich leads to this important relationship:
(11-11)
This expression is referred to as the rotor system balan ce resonance speed, resonance speed, or critical spee d. Because resonance occurs when the perturbation frequency is equal to th e rotor system natural frequency, this expression is also called the natural fr equency. More accurately, it is the undamped natural fr equency , which ignores the effects of damping. For most rotor systems, damping is relatively small, and the damped natural frequ en cy (the natural frequen cy) is close to and a little below the undamped natural frequency. This expression is one ofthe most powerful tools in rotating machinery diagnostics. It shows that the balance resonance speed is determined by the spring st iffness and mass of the rotor system. Changes in the balance resonance can be ca used by many rotor system malfunctions. Changes in spring st iffness are usually responsible for significa nt changes in the resonance speed (mass doe s not usually change). For example, spring stiffness can increase because of a rub or severe misalignment, or it can decrease because of a weakening foundation or a de veloping shaft cra ck. This expression can be used to solve problems such as piping resonance. For exa mple, say a machine has high amplitude piping vibration when the machine is at running speed. One solution to the problem is to move the resonance of the piping away from running speed. This could be done two ways: add mass to the piping system, which would lower the resonance, or add stiffness (bracing), to increase the resonance frequency to above running speed. The last choice is probably the best one, because the resonance will be moved completely away from the operating speed range of the machine. Note that, in the Bode plot, the actual peak of th e resonance occurs at a slightly higher speed th an th at given by Equation 11-11. This is because the rotating unbalance force increases with the squa re of th e rotor speed. As th e
Chapter 11
Dynamic Stiffness and Rotor Behavior
rotor passes D res' the force temporarily increases more rapidly than the Dynamic Stiffness, and so overcomes the natural tendency for the vibration amplitude to decrease. The higher the quadrature stiffness of the system, which flattens the response and slows its decline, the more the peak is shifted to the right. Because this effect is usually small and not important to rotating machinery malfunction diagnosis, we will ignore it. At the resonance, the Direct Dynamic Stiffness is zero , and the rotor response equation becomes
r= A LO = mruDz LD jD(l-)")D
At resonance:
(11-12)
The only remaining stiffness term at resonance is the Quadrature Dynamic Stiffness, and a major component is the effective damping, D(l-)..). Thej in the denominator is equivalent to subtracting 90 thus, at resonance, the high spot lags the heavy spot by 90 The magnitude of the quadrature stiffness at resonance determines the magnitude of the vibration response. Because the quadrature stiffness is in the denominator, a smaller value results in a higher amplitude peak, and vice versa (low quadrature stiffness produces a high, narrow peak and high quadrature stiffness produces a low, broad peak). The quadrature stiffness expression shows that damping, fluid circulation, and rotor speed all playa part in the behavior of the rotor at resonance. The effective damping applies only to synchronous rotor response. For the machinery operator, a primary concern during startup and shutdown is whether or not the machine can get through a resonance without an internal rub, and the effective damping controls the peak vibration amplitude. When a machine is running at steady state, away from a resonance, nonsynchronous dynamic forces may exist in the machine that act to excite the natural frequency associated with the resonance. In this case, the nonsynchronous rules apply, and the full damping is available to limit machine response. 0
;
0
•
Synchronous Behavior Above Resonance The rotor behavior at high speed (blue) shows two important relationships: 1.
The amplitude of vibration approaches a constant, nonzero value.
2. The high spot lags the heavy spot by 180
0 •
221
222
The Static and Dynamic Response of Rotor Systems
At high speed, the mass stiffness term dominates the Dynamic Stiffness. The [22 term becomes so large that the other stiffness te rm s ca n be neglected. The n, th e rotor response equat ion becomes
mr. [22 L 8 r = A L O' = _ =ll_ _- M [2 2
(11-13)
The spe ed terms can cel, and th e equat ion reduces to
Above resonance:
mr r=A LO'= _ _ ll L 8 M
(11-14)
Thus, at speeds well above th e balance resonance, the amplitude of vibration is constant and independent of rotor speed. It can be shown that the zero-to-peak amplitude in this equation is equal to the distance from the geometric center of the system to the mass cente r of the system. Thus, above resonance the rotor syste m rotates about its mass center. Th e minus sign indicates that the high spot d irection, 0', is opposite th e heavy spot direction, 8. In othe r words, the high spot lags th e hea vy spot by 180 °. How Changes In Dynamic Stiffness Affect Vibration When the Dynamic Stiffness of a machine cha nges, the vibration of that machine will change. Dynamic Stiffness contains all of the rotor parameters, M, D, K, .\ and [2, and each of th ese parameters may affect th e vibration, depending on what region th e machine is operating in. For a machine operating at steady state, the most likely parameter to change is the spring stiffness, K. This is because K is a combination of so many stiffness elements: shaft stiffness, fluid-film bearing or sea l stiffness, bearing support st iffness, and foundation stiffness. Changes in any of these components can ch ange K, and change the vibration in the low-speed or resonance regions. The change can be in amplitude, phase, or both. Interestingly enough, an increa se in K does not always cause vibration to de creas e, or vice versa. What actually happens will depend on wh ere the machine is operating relative to the balance resonance. The Bode plots of Figure 11-6 show the vibration response of a rotor syst em to changes in K. In all plots, th e original rotor response cur ves are green. On the left, K has increased, shifting th e resonance to a high er speed (red curves). On th e right, K has decreased, shift ing the resonance to a lower speed. In each plot,
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Chapter 11
Dynamic Stiffnessand Rotor Behavior
ing condition is shown in red . Th e IX and 2X pump frequencies are shown. In the amplitude plot, the 2X amplitude changes (colored dots) as the system resonance frequency passes through twice operating speed. The amplitude peaks and then declines, ju st as the APHT data shows. The Bode phase plot is not as good a fit to the APHT plot data, but it does predict that the 2X phase lag will increase as the rotor system natural frequency decreases.
Summary Dynamic Stiffness consists of th e static spring and tangential st iffnesses of the rotor system combined with the dynamic effects of mass and damping. Two general types of Dynamic Stiffness exist. Nonsynchronous Dynamic Stiffness, the most general form, controls the rotor response to an applied dynamic force at any frequency, independent of rotor speed. Synchronous Dynamic Stiffness, a special case of nonsynchronous Dynamic Stiffness, controls the rotor response to a synchronous (IX) forc e, such as unbalance. Dynamic Stiffness can be separated into Direct Dynamic Stiffness, which acts along the line of the applied force vector, and Quadrature Dynamic Stiffness, which acts along a line oriented at 90 to the applied force vector. The comp onents of Dynamic Stiffness are related to the forces that act on the rotor and the inertia of the rotor itself. Dyn amic Stiffness contains spring stiffness, ma ss stiffness, damping stiffness, and tangential stiffness. Synchronous Dynamic Stiffness terms are associated with the rotor response in three speed ranges: below, at, and above a balance resonance. 0
1.
At speeds well below a resonance, spring stiffness dominates, and the rotor high spot is in phase with the heavy sp ot. Vibration amplitude increases as the square of the rotor speed.
2.
At the resonance speed, Dire ct Dynamic Stiffness goes to zero, a nd only Quadrature Dynamic Stiffness remains. Rotor amplitude peaks, and the high sp ot lags the he avy spot by 90 0
•
3.
At speeds well above a resonance, mass stiffness dominates, the vibration amplitude becomes constant, and the high sp ot lags th e heavy spot by 180 The rotor system turns about its mass center. 0
•
Changes in Dynamic Stiffness produce changes in the amplitude or phase (or both) of vibration. How vibration changes depends on where the rotor system is operating relative to a resonance.
225
227
Chapter 12
Modes of Vibration
provides a good description of basic rotor beh avior. In Chapter 11, we used the model to understand the basic pr inciples of synchronous rotor behavior below, at , and above a resonance. Because the model has onl y one mass , it is limited t o describing a system with only one , lateral, natural frequency, one forward resonance, and no gyro scopic effects. While our model has single, lump ed parameters of mass, st iffness, damping, and lambda ().), real machines have continuous distributions of parameters (and often several sources of )., from different bearings and seals) , and larger and higher-speed machines often exhibit several resonances during startup and shutdown. These distributed systems are theoretically capable of an infinite number of resonances. In practice, we are primarily interested in onl y the lowest few resonances that the machine will encounter on the way up to or down from operating speed or that exist at some integer multiple of running speed. When a rotor system encounters a resonance, the system vibration will be amplified. For large , distributed systems, the vibration amplitude and phase will be different at different axial positions along the rotor. Also, the machine casing will participate in the vibration in some complicated way. The total system vibration will affect rotor-to-stator clearances along the rotor, possibly leading to internal rubs on seals or blade tips. Th is complicated, vibration deflection shape of the rotor system is com monly called the mode shape of the system. The mode shape describes the axial distribution of vibration amplitude and phase along the rotor system, and it changes with rotor speed. It is a function of the ma ss, stiffness, damping, and ). TH E ROTOR MOD E L WE DEVELOPED IN CHAPTER 10
228
The Static and Dynamic Response of Rotor Systems
distribution along the roto r, combined with the distribution of unbalance along the rotor. In this chapter we will discuss of the concept of natural frequencies and free vibration mode shapes and show that the forced mode shape is the sum of several free vibration mode shapes, each of which is excited to varying degrees by th e unbalance distribution of the system. We will show how rotor system mode shapes are influenced by the relative stiffness of the shaft and bearings, and we will introduce the concept of modal parameters. Finally, we will discuss some different techniques for estimating the mode shape of a rotor system using vibration data. Mode Shapes All mechanical systems have natural frequenci es of vibration. These natural frequencies can be excited by momentarily disturbing th e system from its equilibrium position. If the system is underdamped, it will vibrate at one or more natural frequencies until the initial input energy decays away. Because the system is not forced continuously, this kind of vibration is called free vibration. Our rotor model is capa ble of only one natural frequency, wll ' which, for low damping, is approximately
(12-1)
As the number of masses in the system increases, so do the number of natural frequencies. A natural frequency is also called a mode, or natural mode of the system. Each natural frequency is associated with a particular vibration pattern, called a mode shape , and each mode shape is independent of all the other mode shapes of the system. Figure 12-1 shows an example of the mode shapes of a simple, two mass.Iinear system. Because the system has two lumped masses, it is capable of two independent free vibration modes. The lowest mode of this system is an in-pha se mode (left, blue), where both masses move in the same direction at the same time. Note that the amplitudes of the motion are different. The second mode (red), occurs at a higher frequency and is an out-ofphase mode, where the ma sses move in opposite directions. The frequencies of the modes and the relative amplitudes of vibration of the masses depend on the valu es of the mass and stiffness elements of the system. Our l -Complex-Degree-of-Freedom (l -CDOF) rotor model ha s only one moving element, the rotor. We can extend that concept to view the rotor and
Chap,.... 12
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The Static and Dynamic Response of Rotor Systems
Rotor systems combine the characteristics of a continuous system (the rotor itself) and a lumped system, where the rotor and casing can behave like large lumped masses. In addition, lightweight, relatively flexible machine casings can deform in their own set of mode shapes. Because of this complexity, the term mode shape can be somewhat confusing. It is fairly easy to imagine a rotating, flexible rotor that deflects in some complicated, three-dimensional shape like a piece of wet spaghetti. We will refer to this as a rotor mode shape. However, vibration modes involve the simultaneous vibration of all coupled components in the system: rotor, casing, foundation, attached piping, etc. The rotor and casing vibrate at the same time because the rotor and casing are coupled to each other through the bearing stiffness and damping. The casing can transmit vibration to the rest of the extended system, and it can transmit vibration to the rotor. The term mode shape can refer to the complicated pattern of rotor, casing, foundation, and piping system vibration. We will refer to this as the system mode shape. The rotor mode shape is part of the overall system mode shape, and each rotor and casing mode shape is associated with a natural frequency that includes the entire system. When a rotor vibrates, it moves (precesses) in an orbit. The precession of the rotor produces a set of rotating reaction forces in the bearings. These forces are transmitted to the casing, and the casing responds dynamically to the rotating forces. Thus, the casing can move in an orbit in response to the rotor vibration, and rotor and casing mode shapes are a complicated, axially distributed set of rotor and casing orbits. Rotor mode shapes are strongly influenced by the ratio of the shaft bending stiffness to the combination of bearing, casing mass, and casing support stiffnesses. If the stiffness ratio is low (relatively high support stiffness), then the bearings and casing will strongly constrain the rotor motion at the bearings, and most of the motion of vibration will occur through bending of the shaft. Rotors that experience bending modes, such as those in aeroderivative gas turbines and boiler feed pumps, are called flexible rotors. On the other hand, if the stiffness ratio is high (relatively high rotor stiffness), then the rotor is likely to exhibit rigid behavior, and the rotor natural modes will be rigid body modes. In this case, bending of the rotor will be relatively small compared to the motion of the rotor in the bearings (although there will be some rotor bending). A large electric motor rotor supported in fluid-film bearings is an example of a rigid rotor. Figure 12-4 shows typical lowest rotor mode shapes for cases of high and low stiffness ratios. A high ratio of rotor stiffness to support stiffness tends to produce rigid body modes (left), while a low ratio tends to produce flexible rotor bending modes (right). The first modes (top) are essentially in phase, as shown by the Keyphasor dots on the orbit, while the second modes are out of phase.
(t>.aple' 12
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Chapter 12
Modes of Vibration
Figure 12-5 shows several rotor mode shapes for three common machine configurations. For each group, the modes are shown with the lowest mode at top. The first group shows rigid and bending modes for typical, single overhung rotors, such as a single-stage pump. The rigid and first bending mode are very similar; which form will appear depends on the ratio of shaft stiffness to bearing stiffness. When shaft stiffness is relatively high, the rigid mode will appear. More flexible rotors will show the first and second bending modes. If the rotor disk has a relatively high polar moment of inertia, gyroscopic effects at higher speeds will tend to resist bending in the area of the disk, forcing the disk to the near vertical orientation shown in the figure. The double overhung configuration is capable of two rigid body modes. Here, the order of appearance of the rigid modes depends on the relative strength of the system angular rotor support stiffness versus the lateral bearing stiffness. This is largely determined by the bearing spacing. Systems with relatively closely spaced bearings will have relatively low angular stiffness and will experience a pivotal mode first. When bearings are widely spaced, angular stiffness is high, and the first rigid body mode will be cylindrical. The last group shows typical mode shapes for the most common rotor configuration, where most of the rotor mass is concentrated between the bearing centers. The first mode is a cylindrical or pivotal rigid body mode; again, which occurs first will depend on the bearing spacing, angular/lateral stiffness ratio, and the mass distribution. First and second bending modes follow one or both pivotal modes. Because of the relatively large rotor vibration amplitude in bearings during rigid body modes, damping forces can be very high, and these modes may not be visible on a polar or Bode plot during startup or shutdown. Some points along the rotor mode shape have relatively high vibration, while others have little or no vibration. A location with no significant vibration is called a node, or nodal point (see also Figure 12-4). Locations where the vibration amplitude is maximum are called antinodes. Nodal points are important because the vibration on either side of a nodal point will have a large phase difference, often 180°. If we fail to detect a nodal point, our perception of the mode shape of the system may be incorrect. Antinodes are important because they are regions of high vibration amplitude. Because, for flexible rotor modes, bearings usually constrain nearby rotor vibration, relatively high vibration tends to occur near the midspan of the rotor. This has the potential to produce rubs that can damage seals, blade tips, or impellers. For these reasons, it is desirable to know the forced mode shape of the rotor
235
236
The Static and Dynamic Response of Rotor Systems
Forced Mode Shapes and Multimode Response
Up to this point we've discussed free vibration. However, operating rotor systems are subjected to forc ed vibration. Subsynchronous forcing can be caused by aerodynamic instabilities in compressors (rotating stall) or by fluid-induced instabilities associated with fluid-film bearings or seals. Supersynchronous forcing can be caused by compressor or turbine blades passing a close clearance, or pump vanes passing a cutwater. Rotors with cross-section asymmetries have a bending stiffness that depends on angular orientation. Asymmetry can be by design, such as in generators and motors, or the result of a shaft crack. When such a rotor is subjected to a static radial load when rotating, the asymmetry can produce supersynchronous forcing, most often 2X. Coupling problems can also produce 2X forcing. Rotor-to-stator rub can produce impacting, a special situation where a periodic impulsive force (the rub contact) produces free vibration that decays until the next contact. Rub impacting can produce subsynchronous vibration, but, more often, a rub will produce a mild, once-per-turn impact that modifies the rotor stiffness in the area of the rub and the IX unbalance response of the system. The most common form of excitation of the rotor is rotating unbalance. The axial distribution of rotating unbalance produces an axially distributed force system that is available to excite natural frequencies. This force system has a shape (magnitude and angular orientation as a function of axial position) that is similar in concept to a mode shape. The degree of excitation of any natural mode depends on how well the unbalance distribution fits the particular natural mode shape. A good unbalance fit will produce a relatively large excitation of a natural mode, a balance resonance. A poor unbalance fit will result in little or no excitation of a mode (Figure 12-6), and little or no resonance. This characteristic is often deliberately exploited to balance one mode while not changing the unbalance state of another mode (Chapter 16). The concept of an unbalance force distribution that matches a natural mode can be extended to include all of the excitation sources we have discussed. Any forcing function in the rotor system is available to excite system natural frequencies when the frequencies coincide. As with unbalance, the degree of excitation of a system mode will depend on the magnitude, orientation, and distribution of the source. The discussion that follows, while oriented toward unbalance excitation, applies to all sources of excitation in the system. When the rotor speed equals a natural frequency of the rotor and the unbalance distribution approximately fits that natural mode shape, then that mode will be strongly excited, producing a balance resonance. The forced mode shape of the system will be dominated by that natural mode shape. As the rotor speed
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Chapter 12
Modes of Vibration
Figur e 12-7 shows the pred icted behavior of a steam turbine generator set. Th e unbalance was the same ma gnitude at all four locations (yellow dots), and all ma sses were placed at 0 relative to the me asurement transducer. The Bode plot , generated by a sophisticated rotor modeling program, shows the synchronou s vibration respon se to th is unbalance distribution. The rotor mode shapes at 1500 rpm (green) and 3600 rpm (red ) ar e det ermined by the IX orbit Keyph asor dots at each location . It is interesting that this complicated syst em st ill follows the behavior predicted by our simple rotor model. At 1500 rpm, the machine operates below the first balance resonance, and the phase of the gen erator response (top) is close to the location of the heavy spots . The steam turbine and generator rotor mode shape (green) deflects toward the heavy spots. At 3600 rpm, the ma chine operates between the first and second modes, wh ere the phase of the generator response lags the he avy spot by about 180°.The rotor mode shape (red) shows that both the turbine and generator rotors are deflected away from the heavy spot s. Mode shapes are often three-dimensional; these mode shapes, though, are almost completely in the plane of the paper. 0
Modal Parameters Rotor system behavior involving multiple modes is quite complex. The mathematical expressions necessary to accurately de scribe such behavior are well beyond the scope of thi s book. Instead, we would like to develop a more intuitive approach, which will allow us to extend the simple concepts we have already de veloped to the more complicated multimode rotor behavior we obs erve. We have stated th at the natural frequency of a on e mode system is approximately given by Equation 12-1:
wh ere K is the combination of shaft spring stiffness, fluid -film bearing spring stiffness, and support spring st iffness, and M is the rotor mass. We want to apply this simple expression to the natural frequencies of higher modes of the system. In th e development of the sim ple model in Chapter 10, we ass umed that the rotor parameters of mass, stiffness, damping, and A were con stant. Obviously, to obtain a higher natural frequ ency from this equation,
239
240
The Static and Dynamic Response of Rotor Systems
either K must become larger or M must become smaller. We will show how both of these things happen. In a fluid-film bearing, the spring stiffness, damping, and>. are nonlinear functions of eccentricity ratio. Thus, static radial load (which affects the average eccentricity ratio) and rotor speed (which affects the amplitude of vibration and the dynamic eccentricity ratio) produce changes in K and D, and in >., which change the effective damping of the system. More important than these eccentricity-related effects, though, is the mode shape of the shaft, which directly influences the effective stiffness, damping, and mass of the rotor itself. A simple, vibrating, mechanical system involves the continuous cycling of energy between the potential energy of a spring and the kinetic energy of a moving mass. When the velocity of the mass is zero, all the energy of the system is stored in the compressed spring in the form of potential energy. When the velocity of the mass is maximum, at the equilibrium point, all the energy of the system is stored as kinetic energy of the mass, and the potential energy of the spring is zero. It is the ratio of these energy storage elements, K and M, that determines the natural frequency of the system. When viewed from the side, a rotor can be viewed as a more complicated, vibrating, mechanical system. The energy in the system is traded between the potential energy of shaft deflection (the spring) and the kinetic energy of shaft motion. The mode shape of the shaft influences how much deflection is available for energy storage. From the side, a deflected rotor shaft looks very much like a simple beam. At the top of Figure 12-8, the rotor behaves like a beam that is supported at the ends, which corresponds to a typical, first bending mode. According to beam theory, a beam that is supported in this way will have an effective stiffness in response to a static deflection force applied at midspan that is inversely proportional to the length cubed. In the second case, the beam has the same shape as a typical, s-shaped, bending mode. This is equivalent to the beam having a pinned joint at the midspan nodal point, which prevents any deflection there. The beam is now similar to two beams with one-half the total length. A force applied to the onequarter point will produce a deflection, but the perceived stiffness of the beam will be much larger than for the first mode shape. Thus, the effective stiffness of the rotor is much higher in the second mode than in the first mode. We use the term modal stiffness to describe the effective stiffness of a system in dynamic motion. The modal stiffness of a rotor will be different for each rotor mode. The only rotor mass that is available to store kinetic energy is mass that is available to vibrate. The rotor mass near the center of the beam span (top) can
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On the other hand, flexible rotor modes tend to have nodal points that are located relatively close to bearings. This results in lower vibration amplitudes in the bearings, which produces low vibration velocity and a low damping force. Such modes tend to have lower Quadrature Dynamic Stiffness and higher Synchronous Amplification Factors, with higher vibration at the antinodes. We use the term modal damping to describe the actual damping force available to the system. The modal damping depends on the actual damping of the bearings and seals combined with the mode shape. This qualitative discussion shows how modal stiffness, modal mass, and modal damping depend on mode shape. The rotor modal parameters combine with bearing parameters, which change with eccentricity ratio, to produce overall modal parameters. Thus, each mode of a rotor system can be viewed as having a different set of modal parameters that are associated with each natural frequency. The variations in modal damping will produce a different Synchronous Amplification Factor for each mode. The Measurement of Mode Shape
Measurement of rotor mode shape involves determining the three-dimensional, dynamic deflection shape of the rotor at any point in time. A complete description of instantaneous rotor position would require unfiltered orbits and average shaft centerline position data. However, for the purpose of defining rotor dynamic behavior, we are usually concerned with defining only IX rotor behavior versus speed. For that reason, the discussion that follows is concentrated on the measurement of IX mode shape. In principle, a set of axially spaced, shaft relative transducers can be used to determine the rotor mode shape over the entire operating speed range of the machine. This can be combined with data from casing transducers to establish the system mode shapes. This can done fairly easily with IX polar plots or more accurately with IX orbits. IX polar plots provide the easiest approach to estimating the mode shape. This technique is commonly used in balancing and works best when orbits are circular. However, when the system has a significant degree of anisotropy, which produces elliptical orbits (Chapter 13), polar plots can produce misleading results. The polar plot technique uses data from a single transducer per mea surement plane to display the startup or shutdown data at points along the length of the rotor. The plots have the same full scale range and are positioned in a way that represents the axial location on the machine. Points at the same rotor speed are linked by a curve, which estimates the mode shape.
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The mode shape is more accur ately determined using IX orbits (Figure 1212), which require two tr ansducers at each measurement plane. Filtered orbits are constructed for each measurement plane, the orbits are plotted at the same scale , and the Keyphasor dots are connected. If the orbits are digitally sampled, and all the waveform samples are synchronized, then individual mode shapes can be constructed for each sample time (individual point) in the sampled waveforms. For example, if 128 X and Y waveform samples were taken for each revolution, then X and Y sample 99 defines a point on the orbit. This point can be linked to sample 99 points on other orbits, defining the rotor mode shape. Th e collection of the mode shapes for all the samples defines the three-dimensional envelope formed by the orbits. Mode Identification Probes Accurate determination of the rotor and casing mode shapes would require a large set of XY shaft relative and casing transducers spaced along the axis of the machine. In practice, such a large set of shaft relative transducers cannot be installed because of physical limitations. It is not physically possible (and not economically acceptable) to install large numbers of shaft relative transducers in extremely high -pressure or high-temperature regions, or where transducers would interfere with process fluid flow paths. Because of these considerations, shaft relative transducers are usually mounted near bearings, where access, tem perature, and interference with the process are not a factor. Thus, mode shapes must be interpolated between a small set of measurement points. Usually, this interpolation includes places far away from the actual measurement points. This is unfortunate, because often our primary objective is to determine the clearance between the rotor and stator at the midspan of the rotor, exactly the area of highest uncertainty. On large , critical machinery, modern management practice dictates the installation of XY shaft relative transducers at each fluid-film bearing. Unfortunately, this set of transducers will not always provide enough information to measure complicated, higher-order mode shapes when nodal points exist in the interior of a machine; often, more than one possible mode shape can fit the data. Additional probes, called mode identification probes, can be installed on both sides of each bearing, to provide more information. The problem often occurs when nodal points exist at a location outside the bearing. Figure 12-13 (top) shows a machine with transducers installed on the inboard sides of the two bearings; the actual mode shape is shown in black. In this situation, a nodal point occurs inboard of the left probe. More than one possible mode shape (middle) could fit the observed data from these probes. When additional probes are installed outboard of the bearings, the additional infor-
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mation helps identify the actual mode shape. For very complex mode shapes, even this probe configuration may not provide enough information to unambiguously define the mode shape. The best technique for mode shape estimation is to use advanced rotor modeling software in combination with vibration measurements. A model of the rotor system is constructed using accurate shaft and rotor disk dimensions and material properties, and the program calculates the theoretical mode shape based on known physical laws. When combined with actual measurement data, the software can automatically optimize bearing parameters to provide an accurate mode shape of the rotor. The time is coming when such software will be able to use real time vibration measurements to provide accurate rotor mode shape information.
Summary A rotor mode shape is the rotor's three-dimensional, dynamic deflection shape, which changes with axial position. A system mode shape includes information about the relative motion of the rotor, casing, piping system, and any other part of the coupled system. The rotor system can exhibit free vibration at one or many natural frequen cies. Each natural frequency, or mode, has its own characteristic mode shape that is different from the mode shapes at other natural frequencies. A free vibration mode can be forced, or excited, by the distributed unbalance of the system. An unbalance distribution will have its own characteristic three-dimensional shape, with both amplitude and phase as a function of axial position. If the unbalance distribution shape is a good fit to the rotor mode shape, then that mode will be strongly excited, producing a balance resonance when the rotor speed is near the natural frequency. Rotor system forced vibration includes contributions from many free vibration modes. Each mode is excited to some extent by the unbalance distribution, and the resulting rotor response is the sum of the contributions of the individual forced modes. Each mode can be characterized by a set of modal parameters. The square root of the ratio of modal mass to modal stiffness determines the natural frequency for that mode. The modal damping determines the Synchronous Amplification Factor for the mode. Modal mass, stiffness, and damping derive from the mode shape of the rotor. Mode shape can be estimated using polar plots from different axial locations or from orbits at these locations. Nodal points can make determination of the mode shape difficult. Mode identification probes are used to provide more information about nodal points near bearings.
249
Chapter 13
Anisotropic Stiffness
is isotropic, produces only circular, IX orbits, and predicts one forward balance resonance. The model also serves as a basis for defining the Synchronous Amplification Factor. An important result of the synchronous rotor model is that, at low speed, the heavy spot and high spot are approximately in phase. This allows us to use polar and Bode plots to identity the angular location of the heavy spot for balancing. This capability depends on the assumptions used in the model, which include single mode behavior and isotropic parameters of mass, stiffness, damping, and lambda. However, in real operating machinery, multi mode behavior, in combination with fluid-film bearings, can sometimes produce a larger than expected phase lag near a bearing. This effect will be discussed in Chapter 16. A more common problem is that IX orbits are elliptical, and the orientation of the ellipse usually changes with speed. Elliptical orbits produce shaft centerline velocity variations that affect the interpretation of phase, leading to a breakdown of the assumed heavy/high spot relationship used in balancing. Also, because of this ellipticity, measurement of vibration amplitude depends on the orientation of the orbit relative to the measurement probe. Many machines that produce highly elliptical orbits have closely spaced resonances that have a similar mode shape, called split resonances. In between these split resonances, over a short speed range, the rotor may even travel in a reverse, unbalance-driven, IX orbit. These effects are a result of anisotropic stiffness in rotor systems. In this chapter, we will discuss how anisotropic stiffness influences rotor system behavior, with an emphasis on measured vibration amplitude and heavy spot location. THE ROTOR MODEL THAT WE DEVELOPED IN CHAPTER 10
250
The Static and Dynamic Response of Rotor Systems
We will start with a discussion of the meaning of anisotropic stiffness and the physical reasons why it is common in machinery. We will then discuss how anisotropic stiffness manifests itself in rotor behavior, show how, for ani sotropic systems, measured vibration amplitude and phase depend on probe mounting orientation, and how thi s behavior can lead to ambiguity as to the location of the heavy spot. Finally, we will present two signal processing techniques that improve the vibration measurements of anisotropic systems: virtual probe rotation and forward and reverse vector transformation.
Anisotropic Stiffness For the purposes of this discussion, a parameter, such as mass, stiffness, damping, or A (lambda), is isotropic if it has the same value when measured in all radial directions. A parameter is anisotropic if it has different values when measured in different radial directions (Figure 13-1). Because mass distributions, shafts, bearings, and support structures are not perfectly symmetric, all rotor parameters exhibit some degree of anisotropy. Uneven mass distributions on rotor casings and support structures contribute to anisotropic modal mass in rotor systems. External piping can cause different observed modal casing mass along the axis of the piping compared to directions perpendicular to the piping. Rotor mass (and stiffness) can also be anisotropic due to shape asymmetry, which is common in electric motors, wind turbines, and generators. However, these asymmetries, because of rotor rotation, typically manifest themselves as higher-order excitation of the rotor system, particularly in the presence of a side load. A stationary observer sees an average value of rotating rotor mass, so we will assume all rotor parameters to be stationary in this sense. Most rotor systems have relatively low Quadrature Dynamic Stiffness. Thus, damping and A anisotropy will be assumed to have a relatively small effect on rotor response, and the tangential stiffness, DAn, will be assumed to be relatively isotropic. However, anisotropic spring stiffness is common in rotating machinery and has a strong effect on rotor system response. Figure 13-2 shows an end view of a typical, horizontally split machine and the stiffness contributions of various components, including the piping, fluid film, and support structure. Note that the XY vibration measurement probes are mounted at ±45° from the vertical, to avoid the split line. This is a common mounting orientation that has important implications for vibration measurement of anisotropic systems. The method of mounting the casing to the foundation can also produce anisotropic stiffness characteristics. Angular stiffness about the long axis of the
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machine can be significantly lower than vertical stiffness; thi s can appear as a relatively weak horizontal stiffness . The stiffness of the casing and support is influenced by the stiffness of the piping system and it s attachments, and, typically, it will be different in the horizontal and vertical directions. Most importantly, a typical rotor is supported in fluid-film bearings. Remember that the rotor model was developed with the assumption that the rotor was operating, fully lubricated, in the center of the bearing. In this region of the bearing, the spring stiffness is essentially isotropic. Lightly loaded, plain cylindrical fluid -film bearings operate at low eccentricity ratios and can have large attitude angles, sometimes reaching 90° or more. Also, externally pressurized (hydrostatic) bearings normally operate in a fully lubricated condition at very low eccentricity ratios; these bearings are essentially isotropic in behavior. However, normally loaded, internally pressurized (hydrodynamic), plain cylindrical bearings operate in a partially lubricated condition at moderately high eccentricity ratios. At high eccentricity ratios, because of the action of th e hydrodynamic fluid wedge, the journal sees anisotropic spring stiffness. (Imagine it much smaller th an the bearing; ifit were sitting in the bottom of the bearing, it couldn't move as freely down as it could move left to right.) The anisotropic stiffness resolves itself into a strong and a weak axis. The strong axis is approximately at the position angle of the rotor, acting in a radial direction, and the weak axis is at 90° in the tangential direction. (Be careful here. We are talking about variations in the sp ring stiffness in the radial and tangential directions, not about tangential stiffness, which we assume to be isotropic.) In a horizontal machine, a properly aligned, gravity-loaded rotor with plain, cylindrical, fluid -film bearings, will operate in the bearing at an attitude angle of a few tens of degrees. For tilting pad bearings, this is typically less than fifteen degrees. Thus, the orientation of the st rong spring stiffness in the radial direction and weak spring stiffness in the tangential direction will be approximately vert ical and horizontal, and the horizontal spring stiffness will be lower than the vertical spring stiffness. In an anisotropic system, the radial spring stiffness distribution can be a complicated function of angle. In this chapter, we will assume that the spring stiffness distribution has an elliptical shape (like that in Figure 13-1) and can be resol ved into strong and weak stiffness axes that are perpendicular to each other. Th e orientation of these axes can be in any direction, but because of th e machine characteristics we have discussed, we will assume they are approximately horizontal (weak) and vertical (strong). We will call the weak spring stiffness K weak and the strong spring stiffness K strong'
Chapter 13 Anisotropic Stiffness
During the following discussion, remember th at this stiffness orientation is an assumption based on gravity loading of the rotor. Some machines may have a radial load vector that points in some other direction (for example, a gearbox), and the rotor may operate with a large position angle. In this case, the stiffness axes may be oriented in some other direction than horizontal and vertical. As we discussed in the last chapter, different modes of vibration have different modal parameters . By extension, the degree of a nisot ropy may change from mode to mode. Thus, a rotor system with a relatively low degree of anisotropic stiffness in the first mode may have a higher degree in the second mode, or vice versa. Anisotropy may also change with axial position. We will primarily discuss rotor response from the perspective of a single mode.
Split Resonances A split resonance consists of two balance resonances that have a similar mode sh ape, but are separated in frequency. Split resonances are a direct result of anisotropic spring st iffness. Recall that, for low damping, a balance resonance will occur when the rotor speed, fl, is equal to th e natural frequency of the rotor system,
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peak of the split. At 4000 rpm, the system is well above the split resonance, and the mass st iffness in the model dominates the response. Becau se mass stiffness is isotropic, the orbit becomes nearly circular. (If the next rotor system mode has anisotropic spring stiffness, then orbits will become elliptical when the second mode begins to dominate the rotor response.) The appearance of split resonances can vary from a slightly broadened, single resonance peak to clearly separated resonance peaks. The separation will depend on the degree of anisotropy in th e spring stiffnesses, the amount of Quadrature Dynamic Stiffne ss, and the viewpoint, or angular orientation, of the probe. Low quadrature stiffness will tend to create clearly separated peaks; high quadrature stiffness will tend to smooth the peaks together. The small polar plot loop will also change appearance; clearly separated peaks produce a well defined small loop, while , on a high quadrature stiffness machine, the small loop may only appear as a small bump. As we will see shortly, probes misaligned from the stiffness axes will tend to highlight split resonances, and those aligned with the stiffness axes will tend to obscure the split. Measured Rotor Behavior and Anisotropic Stiffness
There are significant differences in the observed behavior of systems with ani sotropic spring stiffness compared to isotropic systems : elliptical orbits, disagreement between high spot and heavy spot location at low speeds, and the presence of split resonances. Also, in anisotropic systems, data becomes dependent on the orientation of the measurement probes: there are differences in the appearance of Bode and polar plots, in the observed Synchronous Amplification Factors, and in the measured vibration amplitudes. Probe mounting orientation becomes an important factor for anisot ropic systems. Transducers (probes) are usually mounted in mutually perpendicular, XY pairs in each measurement plane of a critical machine. The probes can be mounted at 45° Land 45° R (typical for horizontal machines), 0° and 90° R, or any other desired orientation. These probes generate timebase signals that are combined to produce an orbit (Chapter 5). Each transducer signal can be used to create a separate nX Bode or polar plot of a machine startup or shutdown. There is a tendency to use the vector data from only one probe for analysis. For isotropic behavior, this is acceptable, but isotropic behavior is rare in machinery with fluid-film bearings. For anisotropic behavior, the use of data from only one probe can lead to a serious misunderstanding of machine behavior. Imagine a rotor system that is producing circular, IX orbits. Because of the symmetry, the I X vector from a single probe could be used to reconstruct the original orbit. However, if a rotor system is producing ellipt ical orb its , it is not
Chapter 13
Anisotropic Stiffness
possible to reconstruct the orbit with vector data from a single probe. Vibration vectors are required from an XY pair to reconstruct a IX, elliptical orbit. Similarly, a complete picture of anisotropic, nX rotor behavior versus speed requires data from two probes: two polar or two Bode plots. A single plot will not convey an adequate description of the vibration behavior of an anisotropic machine. Table 13-1 summarizes the important differences in observed behavior between isotropic and anisotropic rotor systems.
Table 13-1. Isotropic versus anisotropic rotor behavior. Isotropic
Anisotropic
Circular 1X orb its
Elliptical 1X orb its, possib ly reverse between splits Resonances can be split Measured response changes with probe orientation Polar plots may not be circular, and a small loop may appear X and Y po lar plots are different Low-speed heavy/high spot are aligned only under special circumstances SAFis viewpoint dependent
No split resonances Measured vibration not viewpoint dependent 1X polar plots look circular
X and Y polar plots look the same Low-speed heavy / high spot are aligned Single mode SAF matches theory
257
258
The Static and Dynamic Responseof Rotor Systems
Because orbits are circular in an isotropic system, measured vibration amplitude and phase behavior will not depend on the probe mounting orientation, and, consequently, polar plots from XY transducers will look identical (Figure 13-5). In isotropic systems, the polar plot of the rotor response through a balance resonance will have a circular shape. Isotropic, single mode rotor system behavior matches the ideal behavior used to define the Synchronous Amplification Factor (SAF). Thus, the SAF measured on a single mode isotropic system will be a good match to theory and will also be independent of probe mounting orientation. However, SAF measurement can become troublesome even for isotropic systems when closely spaced modes can interact and distort the SAF measurement. In anisotropic systems, IX orbits can vary from nearly circular, for mildly anisotropic stiffness, to extremely elliptical, line orbits. In systems with low quadrature stiffness, orbits can also exhibit reverse precession between split resonances. At speeds well below a resonance, rotor response is quasi-static, and the Dynamic Stiffness is dominated by spring stiffness (Chapter 11). The relatively weak spring of the anisotropic stiffness allows more rotor deflection in the direction of the weak stiffness axis; thus, the major axis of the low-speed, elliptical orbit will be approximately aligned with the weak stiffness axis. Anisotropic spring stiffness can produce split resonances. The visibility of the split will depend on the degree of anisotropy of the spring stiffness (the separation of the two resonances), the Quadrature Dynamic Stiffness of the system, and the orientation of the viewing probes. Relatively high quadrature stiffness will tend to broaden resonance amplitude peaks and blend them together, especially if the degree of anisotropy (and separation) is low. For this reason, split resonance peaks may not be clearly separated on polar and Bode plots. If a measurement probe is not aligned with the axis of the elliptical orbit, then a phase measurement anomaly will exist, producing an error in the inferred heavy spot location. The data from an XYpair will disagree on the implied location of the heavy spot. This will be discussed in more detail below. Anisotropic stiffness can produce significant differences in measured IX vibration response (both amplitude and phase) between X and Yprobes in the same plane. This difference is largest when orbits are highly elliptical and disappears when orbits are circular. Figure 13-6 shows the effect of probe mounting angle (viewpoint) on the measurement of vibration amplitude of an elliptical orbit. In the elliptical orbit at left, the Y probe sees a much smaller vibration amplitude than the X probe. In the circular orbit, both probes measure the same amplitude.
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The bottom plots show the same machine response, but measured by a n XY probe pair mounted at 0° a nd 90° R; thus, the probes are no w aligned with the stiffness axes. The onl y difference between the upper a nd the lower plots is the orientation of the probes, yet the two sets of plots are very different. When the probes are aligned with the stiffness axes, the response (high spot) phase at low speed agrees with the location of the heavy spot. The pol ar loops are more similar to each other, and the sm all loop has disappeared. The calculated SAFs are still different from each other, but within the range of values obtained when the probes were at ±45°. The different resonance frequencies have condensed into two, closer to what we would expect from the strong/weak anisotropic stiffnes s model. The heavy spot/high spot anomaly is related to the ellipticity of the orbit and the mounting ori entation of the probes. In I X circular orbits, both the rotor rotation and the orbital precession of the shaft centerline (high spot) have constant angular velocity; thus, the high spot maintains a constant angular rela tionship with the heavy spot. However, in elliptical orbits the centerline velocity ch anges, and the relationship is not constant. At low speed, the velocity variations in elliptical orbits can cause the high spot to go in and out of sync with the heavy spot. They are in sync onl y at the locations of the major and minor axes . If the probes are located at these points, then the phase of the vibration not only identifies the high spot, but also, at low speed, the heavy spot (Franklin and Bently [1]). At low speed the orbit major axis will be approximately aligned with the weak stiffness axis. Thus, if the probes are aligned with the stiffness axes (0° and 90° R in this example), they will also be aligned with the orbit major or minor axis at low speed, and the inferred heavy spot location for each probe will be the same and will be correct. However, if the probes are mounted at some other angle (for example, 45° L and 45° R), then a phase measurement anomaly will exist, which is evident by the fact that the phase measurement for each probe will not locate the high spot in the same location; this would incorrectly indicate that there are two he avy spots When making phase measurements with a single probe, we tend to make the unconscious assumption that the high/heavy spot relationship is constant, but this is only true of a isotropic system with circular orbits. Imagine th e machinery diagnostician who views only the polar plot for the Yprobe at the top of Figure 13-8. The conclusion might be that this ma chine's vibration wa s not high enough to worry ab out. A very different perspective appears when both th e X and Y plots are viewed at the same time, and when the probes are aligned with the stiffness axes !
Chapter 13
Anisotropic Stiffness
Thus, we arrive at some important findings for systems with anisotropic stiffness: 1.
At low speed, the high spot direction will point toward the heavy spot only if the measurement probes are aligned with the lowspeed orbit axes (which, at low speed, are aligned with the spring stiffness axes).
2. Measured vibration amplitudes will seldom equal the major axis of the orbit, because the orbit, typically, is not aligned with the measurement axes. 3. SAF measurements will be different, depending on the degree of anisotropy of the system and the probe orientation, and results using different calculation methods will differ from each other. 4. When XY probe data is available, polar and Bode plots should always be viewed in pairs. Anisotropic stiffness is common in machinery. Because of the way many horizontal machines with fluid-film bearings are constructed, stiffness axes tend to be near vertical (strong) and horizontal (weak). Since probes are often mounted at ±45° to avoid split lines , the amplitude and phase measurement anomalies we have discussed are common. While it is often not physically possible to mount probes at 0° and 90° R, other factors in the machine (such as process loads or misalignment) may cause the shaft position angle to be different from what we expect, causing the stiffness axes to be oriented at some other angle. Ideally, we would like to adjust our view of the rotor response to any angle we choose and to have another method for accurate determination of the heavy spot location. There are two methods we will discuss: Virtual Probe Rotation, and transformation to Forward and Reverse components. Virtual Probe Rotation
In an ideal world, we would like to be able to install a set of probes at any arbitrary angle. While we cannot always do that physically (or economically), we can take a pair of XY probes and rotate them mathematically to any angle we choose. We do this by modifying the original data set as a function of the angle we want to rotate the probes, creating data from a set of virtual probes. For each sample speed in a database, the original pair of vibration vectors are trans-
265
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Figure 13-10 is an example from a gas turbine generator set, showing IX shutdown data from the outboard bearing on the generator. This machine is also horizontal, with fluid-film bearings, and has probes mounted at ±45° from the vertical. The physical XY polar plots (top) display an overall pattern very similar to the steam turbine data in Figure 13-9. The virtual probes (bottom) have been rotated 28 against rotation from the physical probes. The heavy spot locations now show good agreement, and the axes of the low-speed, IX, elliptical orbit are in line with the polar plot axes, similar to the previous example. The virtual probes are now aligned with the spring stiffness axes, which are controlled by the fluid-film bearing. When XY polar plots agree on the location of the heavy spot, the probes are aligned with the major and minor axes of the low-speed orbit, and the polar plot and orbit axes are aligned with the anisotropic spring stiffness axes of the machine. These two examples, combined with the predicted behavior shown in Figure 13-8, provide strong evidence that the primary source of anisotropic stiffness in a typical machine is the fluid-film bearing. When balancing is what you need to accomplish, there is a much more efficient method of locating the heavy spot: the forward response vector. 0
Forward and Reverse Vectors
In Chapter 8 we showed that filtered orbits can be represented as the sum of a forward and reverse rotating vector and that the amplitudes of these vectors appear as frequency lines in a full spectrum plot. The full spectrum is created using a transform of X and Y data that preserves the phase information. When applied to startup or shutdown data, forward and reverse Bode and polar plots can be created. Research [1] has shown that the forward response maintains the correct heavy spot/high spot relationship with the dynamic unbalanceforce acting on the rotor. This means that, at low speed, the forward response vector will identify the direction of the heavy spot. When the system spring stiffness is completely isotropic, the reverse response disappears. This agrees with our understanding of orbit shape and full spectrum. A forward, IX circular orbit will have a full spectrum consisting of only a single, forward frequency component at the IX frequency. The shape of the orbit can be described by this rotating vector. Anisotropic rotor systems produce elliptical orbits and, by extension, forward and reverse vibration vectors that are the components of the elliptical orbits. The magnitude of the reverse component is related to the degree of ellipticity of the orbit.
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In Figure 13-11, the data used to generate the polar plots (top) has been transformed to show the forward and reverse response polar plots (bottom). The bottom plots are labeled with direction of precession, not rotation. The phase markings on each plot are in degrees lag relative to the direction of precession (forward or reverse), using a coordinate system based on the X probe. Because the forward and reverse responses are derived from the data from two probes, they are completely independent of the mounting orientation of the physical probes. The low-speed part of the forward plot (left) points toward the heavy spot, and this direction agrees with that found using virtual probe rotation in Figure 13-10. For any speed, the sum of the forward and reverse vectors, which contain both amplitude and phase information, will completely reconstruct the original orb it.
Summary Anisotropic spring stiffness results when the support stiffness seen by the rotor mass is not the same in all radial directions. It is common in horizontal machines with fluid-film bearings. While bearing support, casing, and foundation asymmetries can contribute to anisotropic stiffness, the primary source appears to be the unequal fluid-film spring stiffness in the radial and tangential directions when the rotor operates off center in the bearing. Isotropic systems produce circular, IX-filtered orbits; anisotropic stiffness produces elliptical, IX orbits. Because of orbit ellipticity, measured IX vibration amplitude will depend on the angular mounting location of the measurement probe. Also, the high spot, as it moves along the orbit, does not maintain a fixed relationship with the heavy spot; the phase doesn't always accurately locate the heavy spot. Vector data from an XY probe pair observing an anisotropic system produces pairs of IX polar and Bode plots that do not look alike. There are differences in measured amplitude, SAF, and phase when the probes are not aligned with the stiffness axes. The polar plots may show small loops that are not due to structural resonances. Virtual Probe Rotation transforms the IX vector data from a physical pair of XY probes to what it would look like from a set of virtual probes mounted at any desired angle. If the virtual probes are aligned with the anisotropic stiffness axes of the system, then the low-speed phase data will point toward the heavy spot. Forward vibration vectors from XY probe data eliminate the phase measurement anomaly produced by elliptical orbits. Therefore, the low-speed part of a forward polar plot will point toward the heavy spot, a benefit when balancing.
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This is a more efficient and accurate method than probe rotation for determining heavy spot location. The forward vectors are related to the isotropic stiffness behavior of the system and are calculated using the same transform used for a full spectrum. References 1. Franklin,
w., and Bently, D. E., "Balancing Nonsymmetrically Supported Rotors Using Complex Variable Filtering;' Proceedings ofthe Twenty-First Annual Meeting, Vibration Institute, Willowbrook, Illinois (June 1997): pp . 67-72.
273
Chapter 14
Rotor Stability Analysis: The Root Locus
UNTIL NOW, WE HAVE PRIMARILY DISCUSSED various aspects offorced vibration in rotor systems. The model we developed in Chapter 10 was solved to determine the steady state behavior of a rotor system. Steady state behavior describes how a rotor system responds to a continuous perturbation over a long period of time. However, all vibrating systems also exhibit transient behavior, which describes how they respond to brief disturbances over relatively short time spans. An example of transient behavior is the motion of a pendulum after being displaced from its rest po sition. Transient vibration always involves the free vibration of a system at one or more natural frequencies. This differs from steady state vibration, which depends on the presence of a continuous forcing function and (for linear systems) takes place at a frequency equal to the frequency of the perturbation; for example, IX vibration response due to unbalance. At any time, a complete description of the vibration of a system will include the sum of both steady state and transient vibration. Transient disturbances in rotor systems are usually small, but they can occasionally become significant. Examples of small disturbances are the periodic impulses caused by blade passage across a small gap, the forces due to meshing gear teeth, or disturbances due to turbulent fluid flow. Rub impact is an example of a larger disturbance. All disturbances excite the free vibration of a rotor at one or more natural frequencies. In stable rotor systems, transient vibration dies out over time as the damping force gradually removes the energy associated with the free vibration of the system. However, an unstable rotor system can respond to a disturbance with a dramatic increase in vibration, causing vibration levels to exceed allowable lim-
274
The Static and Dynamic Response of Rotor Systems
its. Thus, the analysis of the stability of rotor systems involves the analysis of transient vibration. When an instability does appear, it is important to be able to recognize it and to know how to eliminate it. In this chapter we will develop a powerful analytical tool, root locus, that can be used to reveal many general aspects of rotor behavior and help analyze rotor stability problems in particular. This chapter will present some basic analytical tools and concentrate on the data presentation of the root locus plot. See Chapter 22 for a discussion of the underlying physical causes and the diagnostic symptoms of fluid-induced instability. We will use the simple rotor model we developed in Chapter 10 to explore the transient behavior of rotor systems. Our rotor model has a tangential stiffness term that mimics the effect of rotor interaction with a surrounding, circulating fluid. This fluid circulation can trigger instability in the rotor system. Though our discussion will concentrate on this fluid-induced instability, the basic analytical principles can be extended to any other type of instability. We will start with a discussion of stability of both linear and nonlinear systems, followed by a transient analysis of our linear model. We will obtain results from this model, called roots or eigenvalues, that describe the free vibration of the system versus time, and we will show how these results can be used to determine the speed at which a rotor system goes unstable. We will show how the free vibration behavior of a rotor system changes with rotor speed and how the information can be displayed in a convenient form on a special plot, called a root locus plot. We will show how to extract a large amount of useful information from this plot. We will show how the root locus plot is related to (and superior to) the logarithmic decrement, which is commonly used to express the results of stability analysis. We will also compare the root locus plot to the Campbell diagram, which is used to show natural frequency relationships in rotor systems. Finally, we will show how to use the root locus plot to perform stability analysis of rotating machinery.
What is Stability? Stability is a broad term that can be interpreted in different ways. A good, general definition of stability is that a mechanical system is stable if, when it is disturbed from its equilibrium condition, it eventually returns to that equilibrium condition. A system is unstable if, when it is disturbed, it tends to move away from the original equilibrium condition. We can think of a stable system as one that is easy to control and behaves in a predictable manner. A stable system mayor may not vibrate; either way it behaves as we expect it to. An unstable system will behave in a way that is unpre-
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276
The Static and Dynamic Response of Rotor Systems
A ball in equilibrium at the top of an convex surface (Figure 14-1, right), is an example of a nonvibrating, unstable system. If it is disturbed, the ball will move away from the equilibrium position and, because of the shape of the surface, will never return. Many mechanical systems vibrate when disturbed from equilibrium. A stable, linear (in the sense of our linear rotor model) system with low damping, when disturbed from equilibrium, will vibrate with decreasing amplitude. The vibration amplitude of an unstable, linear system will increase forever. This is impossible in a real, physical system. In real systems, either nonlinear effects come into play that prevent the system vibration from exceeding a certain level, or the system destroys itself. Because of these effects, unstable, nonlinear systems may eventually reach a new, stable, operating condition that is different from the original one. Rotor systems usually vibrate in a steady state condition (for example, a circular, IX orbit) about a static equilibrium position. While moving in the orbit, a stable machine is in dynamic equilibrium; if it is subjected to some temporary disturbing force, it will eventually return to the original, dynamic equilibrium position. However, when a rotor system with a surrounding, circulating fluid (for example, in a fluid-film bearing or seal) is at or above a particular speed, any disturbance will trigger a fluid-induced instability. This speed is called the Threshold ofInstability. The machine will temporarily become unstable, and a subsynchronous vibration (at a rotor system natural frequency) will begin and rapidly increase in amplitude. As the vibration amplitude begins to increase, the machine will still behave in a linear way (that is, the spring stiffness is approximately constant). However, as the dynamic rotor position nears the surface of a fluid-film bearing or seal, the spring stiffness will increase dramatically (a nonlinearity), producing a stabilizing restraining force. The rotor will evolve into a new, stable operating condition, characterized by higher amplitude, forward, subsynchronous vibration (Figure 14-2). According to our general definition of stability, the machine is stable in this new orbit. The unstable orbit in the figure shows a mixture of subsynchronous and IX vibration; the measurement probe was located near the midspan of the rotor some distance from the source of the instability, a fluid-film bearing. High amplitude vibration is very undesirable and potentially damaging; thus, even though, technically, the new operating condition is stable, the machine may be unstable from a practical point of view. Thus, a practical definition of instability is an undesireable level ofsubsynchronous vibration. Fluid-induced instability is an example of a self-excited vibration, where the energy of the circulating fluid is converted into the energy of vibration. Like all
Chapter 14 Rotor Stability Analysis:The Root Locus
Figure 14-2. Stab ility and pract ical instability in a rotor syst em. Rotor systems usually operate in dynam ic equilibrium about a st at ic equilibrium positio n (for example, a , X orbit, left).When disturbe d, a st able machine w ill eventually return to the original, dynami c equ ilibrium condition . Above t he Threshold of Instability, any disturbance will trigger what is called fluid -induced instability: the rotor will evolve into a new, st able dynam ic equ ili brium , dominated by high er ampl itude, subsynchronous vibration (right ).The orbit show s a mixtu re of , X and 0.48X vibration. According to ou r gen eral definition of st ability, the machine is stable in this new orb it, but the higher amplitude vibration may be undesirable and potentially damaging; thus, the machine is unstable from a practical point of view.
Stable
o
Unstable
G
self-excited vibration, the frequency of the instability vibration occurs at a natural frequency of the system, with a mode shape associated with that natural frequency. The amplitude of the mea sured vibration is affected by three conditions: how mu ch the vibration increases before the system restabilizes, the mode shape of the vibration, and where the measurement probe is in relationship to the source and the mode shape. The highest amplitudes of the subsynchronous instability vibration will be physically located at the antinodes for the mode shape associated with the instability natural frequency. This could be, and often is, some distance away from the source of the instability. Accurate stability analysis of rotor systems requires models based on nonlinear mathematics, but these models are difficult to solve analytically. This is why our rotor model was developed using a line ar model. However, rotor behavior below, at , and immediately aft er crossing the Threshold of Instability is essentially linear in behavior, and we can use the linear model to understand the basic mechanism that leads to instability. We will see that adjustment of the linear model can explain the nonlinear effects that prevent the vibration from growing forever.
277
278
The Static and Dynamic Response of Rotor Systems
Stability and Dynamic Stiffness Im agine a rotor th at has been disturbed fro m a static eq uilibrium po sit ion. The spring force , -Kr, tries to push the rotor back toward t he equilibrium po sitio n (see Chapter 10). However, the tangential force, +j DAS?r, pushes the rotor in a direction 90° from the di spl acement and preven ts t he roto r from returning directly to equilibrium. The tangential force is proportional to the rotor speed, S?, becoming st ro nger as rotor speed increases, and it ac ts ag ainst the stabilizin g d amping force; ultim at ely it can destabilize the ro tor system. Becau se the tan gential force is the effect of the fluid circulating aro und th e rotor, we call this form of instability fluid -induced instability. This destabilizing effec t is related to a loss of Dyn amic Stiffness in the rotor system . We have shown t ha t rotor vibration is th e ra tio of the applied forc e to the Dynamic Stiffness of t he syste m . The Dynamic Stiffness acts to restrain the motion in a way that is sim ilar to a spring (Chapter 11). If the Dynamic Stiffn ess were to disappear, there would be no constraint to rotor motion, and, when disturbed , the rotor would move away from the equilibrium position forever. Thi s m eets the general defini tion of in stability. Thus, we ca n see that when th e Dyn amic Stiffness becom es zero, t he rotor system becom es un stable. Th e nonsynchronous Dyn amic Stiffness is
(14-1)
Dyn amic Stiffness is a complex quantity, contain ing both direct and quadrature parts. For the Dynamic Stiffness to equal zero, both th e direct and quadrature parts must be zero, simultaneously. Thus, the Dir ect Dyn amic Stiffness is zero ,
K -Mw2=0
(14-2)
and the Quadrature Dynamic Stiffness is zero:
;D(W-AS?)=O
(14-3)
Chapter 14
Rotor Stability Analysis: The Root Locus
For this last expression to be t rue , th e te rm in parentheses must be to equa l zer o:
w ->'S2= 0
(14-4)
Because Equations 14-2 and 14-4 are equal to zero, th ey are equa l to each other. We can find th e rotor speed that satisfies this system of equations. Eliminating w, and solving for [2 , we find the Threshold ofInstability, nt/I' to be
Threshold of Instab ility
(14-5)
This is the rotor spee d at or abo ve whi ch the rotor system will be un stable. This exp ression is a very powerful diagnostic tool, and it is the key to understanding how to prevent and curefluid-indu ced instability problem s in rotor systems. This expression combines the Fluid Circumferential Average Velocity Rati o, >., with th e undamped natural frequency of th e rotor system,
(14-6)
For plain , cylindrical, hydrodynamic bearings (internally pressurized bearings) that become fully lubricated, >. is typi cally less than 0.5. Thus, th e reciprocal of >. in Equation 14-5 is typ ically greater than 2. This tells us that a typical rot or system mu st op erate at more than twice a natural freque ncy to trigger fluid-induced instability. For systems with multiple modes, the rotor syste m will encounter the lowest natural frequency first. Instability is almost always associated with the lowest mode of a rotor system. To ensure rotor stability, all that is required is to rais e nth above the highest opera tin g speed of th e rotor. This can be done by reducing fluid circulation, whi ch decreases >., or by increasin g the spring sti ffness , K. Externally pressurized (hydrostatic) bearings, tilting pad bearings, and other bear ings of noncircular geometry accomplish on e of both of these objectives. This will be discus sed in more detail in Chapter 22. For now, we will move on to another form of stability analysis th at can be extended to more complex rotor syste ms.
279
280
The Static and Dynamic Response of Rotor Systems
Stability Analysis
Stability analysis requires the development of a mathematical model of the rotor system. The level of required detail in the model depends on the complexity of the machine being considered and the likely instability mechanism. Models can range from simple, lumped mass systems, to more complicated, multiple lumped mass systems, to very complicated, finite element models. Here, for clarity of understanding, we will use the same rotor model that was developed in Chapter 10. We start with the equation of motion of the rotor model,
Mr+Dr+(K - jD>'[l)r=O
(14-7)
where the perturbation force on the right side is zero, because we want to concentrate on the free vibration behavior of the rotor. This form of the equation of motion is called the homogeneous equation of the system. We will assume a solution to this equation that is similar to the one we used in Chapter 10: r=Re st
(14-8)
where R is an arbitrary, constant displacement vector. We know that the position will change in some way with time, t, but, since we don't know how it will change, we will use a general variable, s. To solve Equation 14-7 we need expressions for the velocity and acceleration: r= slie" r=s2Re st
(14-9)
Substituting these expressions into 14-7, and collecting terms, we obtain
[MS2 +Ds+(K - jD>'[l)]Rest =0
(14-10)
For this expression to be true, either the term in square brackets must be zero or the initial displacement, R, must be zero. If R is zero, then the system is
Chapter 14
Rotor Stability Analysis:The Root Locus
resting at equilibrium, which is a valid, but not very interesting case. We want to examine the case where R is not zero, which requires that
Ms 2 +Ds+K - jD>'fl=O
(14-11)
This is an important relationship known as the characteristic equation of the system. It is a quadratic (second order) polynomial in s. The values of s that satisfy this equation are called the roots of the equation. When both >. and fl are nonzero, solution of this equation will yield two complex roots of the form s] = 1'] + j Wd S2 = 1'2 - jWd
(14-12)
where 1'] (Greek lower case gamma), 1'2' and wd are complicated functions of M, D, K, >., and fl (see Appendix 6). I' is called the growth/decay rate and has units of lis; w d is the damped natural frequency and has units ofrad/s. The meaning of these terms will be discussed shortly. The roots are also known as the characteristic values, eigenvalues, and, in control theory, poles of the system. If we substitute these two solutions into Equation 14-8, we obtain two expressions:
r] = (R]el,t )e jwdt r2 = (R2el ,t )e- jwdt
(14-13)
where the complex arguments of the exponential function have been separated into amplitude and frequency components. R] and R 2 are constant vectors that depend on the conditions at the beginning of free vibration. The complete, free vibration response of the rotor is given by the sum of r ] and r» (14-14) where r] and r 2 are a pair of forward and reverse rotating vectors whose frequency of rotation is the damped natural frequency, w d ' Because the frequency of r I is positive, it represents forward precession. The frequency of r 2 is negative and represents reverse precession at the same frequency. The amplitudes of r] and r 2 are given by the expressions in parentheses. At time to' the initial amplitudes, as a result of a disturbance, are R I and R 2 • Once
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