DYNAMIC VIBRATION ABSORBER
A Major Project report submitted to Rajiv Gandhi Proudyogiki Vishwavidyalaya, Vishwavidyalaya, Bhopal towards partial fulfillment of the degree of Bachelor of Engineering 2012
Guided by :
Submitted by :
Dr. R.K. Porwal Department of Mechanical Engineering
Akshay Pratap Singh Deepak Sharma Jyoti Chouhan Kalpana Shankhwar Kuldeep Thakur Pragati Javre Vanya Pande
(AB 46005) (AB 46012) (AB 46022) (AB 46023) (AB 46024) (AB 46032) (AB 46056)
Department of Mechanical Engineering Shri Govindram Seksaria Institute of Technology and Science 23, Park Road, Indore (M.P.)
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SHRI G.S. INSTITUTE OF TECHNOLOGY & SCIENCE INDORE, M.P.
_____________________ ________________________________ _____________________ _____________________ ______________________ ___________________ ________
RECOMMENDATION _____________________ ________________________________ _____________________ _____________________ ______________________ ___________________ ________
The major project report entitled “ Dynamic Vibration Absorber” submitted by Akshay Pratap Singh, Deepak Sharma, Jyoti Chouhan, Kalpana Shankhwar, Kuldeep Thakur, Pragati Javre and Vanya Pande; students of B.E. Mechanical Final Year (VIII sem) in session 2011-12, towards partial fulfilment of degree of Bachelor of Engineering conducted by Rajiv Gandhi Proudyogiki Vishwavidyalaya, Bhopal, is a satisfactory account of their work.
Dr. R.K. Porwal
Dr. M.L. Jain
Project Guide
Head of Department Mechanical Engineering
Director
SGSITS, Indore
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SHRI G.S. INSTITUTE OF TECHNOLOGY & SCIENCE INDORE, M.P.
_____________________ ________________________________ _____________________ _____________________ ______________________ ___________________ ________
CERTIFICATE ______________________ _________________________________ ______________________ _____________________ _____________________ __________________ _______
The major project report entitled “ Dynamic Vibration Absorber” submitted by Akshay Pratap Singh, Deepak Sharma, Jyoti Chouhan, Kalpana Shankhwar, Kuldeep Thakur, Pragati Javre and Vanya Pande; students of B.E. Mechanical Final Year (VIII sem) in session 2011-12, towards partial fulfilment of degree of Bachelor of Engineering conducted by Rajiv Gandhi Proudyogiki Vishwavidyalaya, Bhopal, is a satisfactory account of their work.
Internal Examiner
External Examiner
Date: ………………
Date: ………………
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ACKNOWLEDGEMENT
It is a duty to acknowledge all those people who have contributed in any way in the accomplishment of a task; and it is with deep gratitude that we express our thanks here.
First and foremost, we thank our project guide Dr. R.K. Porwal for his complete support throughout the course of the project, from preparation to final testing in both the phases. Without his constant support and guidance, it would not have been possible to wrap up the project.
We would also like to thank Dr. M.L. Jain, our Head of Department and his team for their valuable inputs at the time of the first internal project progress evaluation.
We also thank our honourable director, Dr. Sudhir S. Bhadauria, for his continuous cooperation and support to our department.
Last, but not the least, we thank our parents and friends, who have always encouraged and supported us incessantly. It is because of them that we stand where we are.
Akshay Pratap Singh Deepak Sharma Jyoti Chouhan Kalpana Shankhwar Kuldeep Thakur Pragati Javre Vanya Pande
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ABSTRACT
A Vibration Absorber is a simple yet important device employed in most of the machineries that experience vibrations. During working, a machine may be subjected to undesirable vibrations under various external excitations. To prevent resonance of the machinery and consequent failure due to breakdown and severe damages owing to harmful vibrations, the Dynamic vibration absorber is a necessity. It has the ability to either absorb or transmit the vibrations elsewhere, hence significantly reducing the vibrations of the machine to which it is coupled to, saving it from damage and preventing loss in terms of performance, capital, additional maintenance and working hours.
An undamped Dynamic Vibration Absorber works by transmitting the vibrations to an auxiliary system (usually comprising a spring-mass system) and thus bringing the main system to rest. It is extremely effective for constant speed machineries.
Since our laboratory lacked any apparatus to observe the phenomenon of dynamic vibration absorption, our project mainly dealt with the fabrication of such a system which generates vibrations due unbalance force excitation along with a dynamic vibration absorber and also to functionalize another apparatus subjected to base motion excitation fabricated by our seniors and equip it with dynamic vibration absorber to understand and observe the absorption of harmful vibrations of the concerned main machinery.
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CONTENTS S. No.
Chapter
1
Introduction
1.1 Preamble 1.2 Objective 1.3 Scope 1.4 Organisation of Report 2
Literature Review
2.1 Vibration and its Causes 2.2 Classification of Vibration 2.3 Resonance 2.4 Need to Study Vibration 2.5 Ways of Eliminating Vibrations 3
Problem Definition
3.1 The Problem 3.2 Object 3.3 Applicability of Dynamic Vibration Absorber 4
Methodology
4.1 Problem Review 4.2 Dynamic Vibration Absorber 4.3 Collection of Data and Calculations 4.4 Fabrication Method 4.5 Testing and Modifications
5
Observations and statistics
5.1 Recording of Results and Specifications 5.2 Data Interpretation and Graphs 6
Results and Discussion
6.1 Interpretation of Results 6.2 Discussion and Fields of Application
Page No. 1-2
1 1 1 2 3-12
3 6 7 7 9 13-14
13 13 14
15-47
15 24 33 43 45
48-54
48 51 55-56
55 55
7 8
Scope of future work Appendices
57 58-60
9
References and Bibliography
61-62 0
Chapter 1 INTRODUCTION 1.1
Preamble:
We continuously experience vibration in our day-to-day life: the reason why we can hear is due to the ability of our ear drum to oscillate due to sound waves and transmit those vibrations to our brain via the internal ear. Music instruments work on the principle of vibration of strings or diaphragms. Earthquakes can be felt and recorded because of the massive vibrations that are produced as a consequence. It can be safely concluded that movement of constituent particles is the origination of vibration in a material. Similarly vibrations also exist in machineries. However, vibration in this case causes wear and tear to the system. Also, resonance is induced on approaching the natural frequency of vibration of the system which causes intense damage. Hence, the need arises to eliminate or at least reduce the vibrations in a machine to a minimum. A dynamic vibration absorber is such a device. An undamped dynamic vibration absorber is basically a spring-mass system that transmits the vibration from the main system to an auxiliary system, preventing damage of the main system. Vibration being an important part of our curriculum, our laboratory lacked an apparatus to enable us to observe and study the transmission and/or absorption of vibrations. Our project deals with equipping our laboratory with a simple apparatus that will wi ll enable en able the observation of dynamic vibration absorption and affect a better understanding of the phenomenon.
1.2
Objective:
The primary objective of our project is to fabricate a machine subjected to unbalance force excitation and functionalise a machine subjected to base motion excitation; and to design and employ „Undamped Dynamic Vibration Absorbers‟ on these machines to demonstrate complete absorption of vibrations of machines which run at constant speed.
1.3
Scope:
Scope of our project is limited to the laboratory only. Its sole purpose is to provide a means of observing and appreciating the phenomenon of vibration absorption at a collegiate level.
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1.4
Organisation of Report:
Chapter 1 This chapter includes the preamble of the report that discusses the phenomenon of vibration in our daily life, its detrimental effects on machineries and hence the need to eliminate vibrations in machine systems.
Chapter 2 This chapter discusses the theory behind vibration, its causes, its effect with historical examples, as well as vibration absorption and the basic concepts and formulae related to it with their derivations.
Chapter 3 This chapter deals with the problem definition where the problem regarding an undamped vibration absorber has been defined; the objective stated and discussed; and the various applications of the system have been stated.
Chapter 4 This chapter has discussed in length and detail, the step by step procedure followed during the working on the project beginning from selection of materials to fabrication, from testing to observations to modifications incorporated in the design for a better performance of the system.
Chapter 5 This chapter includes the observations and inferences taken from the practical performed on the two setups with and without coupling of undamped dynamic vibration absorber. Specifications and speed range of the machines in which vibration absorber is effective are also defined.
Chapter 6 This chapter covers the final results interpreted as the verification of the phenomenon of absorption of vibrations using dynamic vibration absorber and its practical applications.
Chapter 7 This chapter gives light on the possibilities of future work to explore new applications of Dynamic Vibration Absorber. Towards the very end, a list of various references that were studied and followed is provided.
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Chapter 2 LITERATURE REVIEW
2.1
Vibration and its Causes:
The term ‘Vibration’ refers refers to repeated oscillations of a body about its mean equilibrium position due to disturbance. These oscillations may be be periodic such as the motion of a pendulum or random random such as the movement of a tire on a gravel road. Further a periodic vibration may be harmonic or non-harmonic in nature.
2.1.1 Vibration The essential condition to have vibration in a body is that, it must possess ‘mass’ and and ‘elasticity’ . From energy conservation point of view, in a vibration motion there is a continuous conversion of „kinetic energy into potential energy‟ or „potential energy into kinetic energy‟ of the body. Mass is a necessary parameter for kinetic energy and elasticity is responsible for storing the elastic energy of deformation (potential energy) in the system to bring it back to mean equilibrium position. A simple spring-mass vibrating system is shown in fig. 2.1
Fig. 2.1 [1]
Where, m = mass ; k = stiffness of spring ; X = maximum displacement of mass from mean equilibrium position (amplitude).
= time period for one oscillation.
When periodic motion of vibrating mass is such that its acceleration is proportional to displacement & always directed to mean equilibrium position, it is called ‘Simple harmonic motion’ (SHM). (SHM). 3
SHM can be represented by a displacement sine wave as shown in fig. 2.2
X Ɵ
x
Fig. 2.2
In simple harmonic motion parameters are given by: Displacement Velocity Acceleration
̇ ̈ ;
;
Fig. 2.3 SHM parameters
4
Mostly, vibration in a machine or in a structure is „undesirable’ as it causes loss of performance, discomfort, danger and destruction. For example:
Building oscillation caused by earthquake and wind flow.
Collapse of bridges due to wind induced vibration.
Oscillations endured by a passenger as a car rides over a bumpy road.
Excessive mechanical vibrations in an engine, etc.
But vibration is occasionally ‘desirable’ for some constructive purposes also. Such as we can hear because our ear drum can oscillate, our heart beat is also an oscillation. Some examples of constructive applications of vibrations are :
In vibratory conveyers and soil compactors.
In musical instruments for generation of sound.
Mechanical shakers for mixing of things.
Vibratory sieves for sorting objects by size, etc.
2.1.2 Common Causes of Vibration [2] Vibration can result from a number of conditions, acting alone or in combination. Sometimes vibration problems might be caused by auxiliary equipment, not just the primary equipment. Some of the main reasons are given here :
Imbalance: A “heavy spot” in a rotating component will cause vibration when the unbalanced weight rotates around the machine‟s axis, creating a centrifugal force. Imbalance can be caused by manufacturing defects (machining errors, casting flaws) or maintenance issues (deformed or dirty fan blades, missing balance weights). As machine speed increases the effects of imbalance become greater. Imbalance can severely reduce bearing life as well as cause undue machine vibration. Misalignment or Shaft Run-out: Vibration can result when machine shafts are out of line. Angular misalignment occurs when the axes of (for example) a motor and pump are not parallel. When the axes are parallel but not exactly aligned, the condition is known as parallel misalignment. Misalignment can be caused during assembly or develop over time, due to thermal expansion, components shifting or improper reassembly after maintenance. The resulting vibration can be radial or axial (in line with the axis of the machine) or both. Wear: As components such as ball or roller bearings, drive belts or gears become worn, they might cause vibration. When a roller bearing race becomes pitted, for instance, the bearing rollers will cause a vibration each time they travel over the 5
damaged area. A gear tooth that is heavily chipped or worn, or a drive belt that is breaking down, can also produce vibration.
Looseness: Vibration that might otherwise go unnoticed can become obvious and destructive if the component that is vibrating has loose bearings or is loosely attached to its mounts. Such looseness might or might not be caused by the underlying vibration. Whatever its cause, looseness can allow any vibration present to cause damage, such as further bearing wear, wear and fatigue in equipment mounts and other components. Wind Induced Vibrations: In case of sky scraper buildings and large bridges, wind blowing around can cause them to vibrate. After a certain speed of wind, the eddy frequency of wind induced disturbance nearly reaches the natural frequency of the structure resulting in violent vibrations and sometimes structural failure.
2.2 Classification of Vibration:
Vibration
linear
free
damped
single dof
non-linear
forced
same as descendants of "free"
undamped
multi dof
continuous
same as descendants of "linear"
same as descendants of "damped"
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2.3
Resonance:
‘Resonance’ is a phenomenon of continuous build up of higher and higher amplitude of vibration. It is destructive form of vibration, when amplitude of oscillations is so high that failure of the machine or structure takes place due to these uncontrolled vibrations. To understand the „resonance‟ consider spring and mass as energy storage elements. Here mass stores ‘kinetic energy’ and the spring stores ‘potential energy’. When mass-spring system has no external force acting on it they transfer energy back and forth at a rate equal to their natural frequency. In other words, if energy is to be efficiently pumped into both the mass and spring the energy source needs to feed the energy in at a rate equal to the natural frequency. Periodic excitation optimally transfers to the system the energy of vibration and stores it there. Because of this repeated storage and additional energy input the system swings ever more strongly, until its load limit is exceeded. Applying a force to the mass and spring is similar to pushing a child on swing; you need to push the swing at the correct moment if you want it to get higher and higher. As in the case of the swing, the force applied does not necessarily have to be high to get large motions; the pushes just need to keep adding energy into the system.
2.4
Need to Study Vibrations:
It is not difficult to observe that vibrations exist everywhere in our daily life. We speak because of larynges can vibrate, we hear because our ear drum can vibrate. When we ride on a motorbike or in a car it shakes partly due to uneven road and partly due to vibrations from engine. Speakers and musical instruments generate sound due to vibration. Human discomfort and fatigue is another important consideration which requires engineers to study human response to vibrations. In mechanics and construction a resonance disaster describes the destruction of a building or a technical mechanism by induced vibrations at the system's resonance frequency, which causes it to oscillate. Usually the best way to decrease such destructive incidents is to consider the possible effects of vibration on the system at its designing stage and accordingly modify the parent design of system. On the other hand one field of interest may be to explore new ideas that use vibration for some constructive and useful applications. Some important features that make study of vibrations essential are:
To verify that vibrations do not exceed the material fatigue limit hence to reduce excessive deflection and failure of machines and structures. To verify that physical activities do not harm or cause any discomfort to human body by controlling unwanted noise and uncomfortable motion. To improve performance efficiency of a certain machinery or process by suppressing vibrations. To make diagnosis of machinery by condition monitoring, fault diagnosis and prognosis. 7
To plan maintenance on machines by making decisions of maintenance activity schedule. To construct or verify computer models of structures in a view to dampen or isolate vibration sources by system parameter identification and model updating. Vibration is studied to stimulate earthquake for geological research and conduct studies in design of important structures such as nuclear reactor.
Failure of Tacoma Bridge (1940) [3]
Collapse of Nimitz freeway in an earthquake, 1989 [5]
Some of the accidents happened in past resulting from mechanical failure due to vibration are following :
Failure of Tacoma Narrows Bridge on 7 Nov 1940 due to wind induced vibrations. [3] Broughton Suspension Bridge collapsed on 12 April 1831 reportedly owing to a mechanical resonance induced by troops marching over the bridge in step . [4]
Collapse of Nimitz freeway in an earthquake in Oakland, 1989. [5]
Collapse of -Königs Wusterhausen Central Tower. [5]
Resonance of the-Millenium Bridge. [5]
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2.5
Ways of Reducing/Eliminating Vibrations : [6]
Vibration control implies limiting the vibratory amplitudes within permissible limits by suppressing cause of vibration. In system designing point of view the best way to control vibrations would be rectification of design/manufacturing processes to eliminate possible causes of vibration. In a machine or structure when vibration amplitudes are beyond permissible limits, vibration control can be achieved by following methods :
Avoiding resonance. Balancing or control of excitation forces. Adequate damping. Vibration isolation. Absorption of vibration using vibration absorber.
2.5.1 Avoiding Resonance The natural frequency of a system is defined as the frequency at which it freely keeps on vibrating without any external energy input. When frequency of any external energy input to the system matches with the natural frequency of the system then the amplitude of vibration goes on increasing. Therefore if the frequency of external excitation is kept away from system‟s natural frequency by choosing system parameters such that its natural frequency lies far away from
k
m
F sin ωt
Fig. 2.4 Avoiding resonance
frequency of excitation force or system is operated at lower or higher side of its natural frequency then vibration amplitude will not build up. This is called „detuning‟.
2.5.2 Balancing or Control of Excitation Forces In rotating type of machines like turbines, motors, I.C. engines centrifugal force due to unbalance is a common cause of harmonic vibration. Practically all newly machined parts are non symmetrical due to blow holes in castings, uneven number and position of bolt holes, parts fitted off-centre, machined diameters eccentric to the bearing locations etc. The vibration due to unbalance can be controlled by
9
incorporating a balancing mass opposite to unbalance mass which nullifies the effect of centrifugal force and thus controls undesirable vibration of machine.
Unbalance mass „U‟ is counter balanced by mass „C‟
Counteracting mass „C‟ is provided for balancing of single cylinder engine
Fig. 2.5, Balancing
2.5.3 Adequate Damping The dissipation of energy of system‟s vibration is known as ‘damping’ . Damping effect is basically an inertia property of system elements. Damping reduces the amplitude of vibrations in a system. It may be due to the friction between moving surfaces, intermolecular resistance of material in deformation or it may also be due to change in physical state of any element. The elements which are used to absorb the energy of vibration are called damping elements and they are widely used to control vibration within permissible limits. Damping effect is linearly related to the velocity of the oscillations. This restriction leads to a „linear differential equation‟ of motion, and a simple analytic solution. Passive, semi-active, and active control methods can be used for vibration damping. The traditional „ passive‟ damping methods include the use of visco -elastic, viscous, and friction dampers, as well as tuned mass dampers for vibration. ‘Active’ damping involves the use of actuators (e.g. motors) along with sensors and controllers (analog or digital) to produce an actuation with the right timing to counteract the resonant oscillation.
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c
m
m Active fluidic damper
Representation of damped system
Viscous damping
Dam ed door closer Fig. 2.6 damping
2.5.4 Vibration Isolation Vibration isolation seeks to reduce the vibration level in one or several selected areas. The idea is to hinder the spread of vibrations along the path from the source to the receiver. A vibration isolation problem is often schematically described by division into substructures: a source structure which is coupled to a receiver structure. Vibration isolation is yet another substructure incorporated between the two structures. The objective of vibration isolation is to reduce the vibrations in some specific portion of the receiver structure. It can be provided in a machine by use of metallic springs, elastomeric mounts, resilient pads and inertia blocks.
Vibration isolated foundation with floor plate
Inertia block for vibration isolation
Fig. 2.7 Vibration Isolation 11
2.5.5 Absorption of Vibration Using Vibration Absorber A „Dynamic vibration absorber‟ (DVA) is a device that consists of an auxiliary mass attached to a vibrating system, in order to reduce the vibration amplitude of the system. The auxiliary mass is attached to the system by means of a spring, usually in parallel combination with a damper. Its mass is relatively small in comparison with the mass of the vibrating system. The basic principle of operation of a DVA is to generate vibration out-of-phase (opposite to) with the vibrating system, thereby applying a counteracting force, resulting in reduction of the vibration amplitude of the vibrating system.
c
Fig. 2.8, Dynamic vibration absorber [7]
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Chapter 3 PROBLEM DEFINITION
3.1
Problem Statement:
Many machines run at constant speed and they may be subjected to external excitation force or motion (base excitation) due to unbalance force, misalignment or looseness because of some manufacturing defect or worn out parts in long operational period. When machine‟s operating speed matches with frequency of excitation resonant vibrations will be produced. This external energy input restricts the speed range of operation of the machine, as it can‟t be operated at a speed which corresponds to machine‟s natural frequency.
3.2
Object:
To fabricate and functionalize machines, one of which is subjected to unbalance force excitation and the other one is subjected to base motion excitation and to design & employ „Undamped dynamic vibration absorbers‟ on these machines to demonstrate complete absorption of vibrations of machines which run at constant speed. To operate the machine at a speed which coincides with its natural frequency or near to it, without affecting its performance an „undamped dynamic vibration absorber‟ can be coupled with the machine. Tuned undamped dynamic vibration absorbers are extremely effective for constant speed machines but they can lose their effectiveness outside a certain range of speed of the machine. The coupled system will have two new natural frequencies which may fall outside the desirable speed range of operation of the machine.
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3.3
Applicability of Dynamic Vibration Absorber: [8]
Dynamic vibration absorbers can be employed in machines with or without the use of damping elements; and accordingly, they are classified as „Damped dynamic vibration absorbers‟ (DDVA) and „Undamped dynamic vibration absorbers‟ (UDVA).
UDVAs are mostly applicable without dampers in constant speed machines, such as turbine in a power plant, compressor, electric motor, generator etc. DDVAs find their application mostly in variable speed machines, such as pumps, engines etc. to suppress their vibrations, but they can also be effectively employed with constant speed machines to achieve a wider range of speed in which machine‟s vibrations can be absorbed. In case of DDVAs, as the viscous damping is increased, wider speed range of vibration absorption is obtained but along with it amount of absorption reduces. Recently DDVAs employed for defence mechanism against earthquakes. Much work has been directed towards the use of DVAs attached to building structures to counter seismic movements and wind forces. In recent studies, interest has also been focused on the use of feedback and feed forward control systems, and the synthesis of DVAs for multiple-degree-of-freedom systems.
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Chapter 4 METHODOLOGY
4.1
Problem Review:
With the wide use of variable frequency drives, it is becoming more difficult to design mechanical systems free from natural frequencies within operating speed range. If such an occurrence is allowed in the field, a resulting resonance condition threatens to significantly impact performance and longevity of the equipment. Since machines are made up of metallic parts, they have mass and elasticity both. Further if a machine contains any rotating of moving member; it is subjected to forces which vary periodically with time. These forces ma y or may not be harmonic in nature and result into „Forced- vibrations‟ in those machines. An electric motor or any other device with a rotor as its working component is called „rotating machine‟. The machine is said to have unbalance when the centre of gravity of the rotor does not coincide with the axis of rotation. Many systems, such as an internal combustion engine; a turbine in a power plant, operate at constant angular speeds. There is always a possibility that the frequency of excitation due to unbalance may match the natural frequency of the machine (main system). In such a case resonance will occur resulting in undesirable and harmful vibrations and loss of performance. This also limits the speed domain of a machine such that „it can‟t be operated at a particular speed which is close to resonant speed though at that particular speed the performance of machine may be high.‟ On the other hand there are a number of practical situations in which the dynamic system is excited due to the motion of the base. A vehicle moving on a wavy road, a locomotive running on a rail track with gaps between the adjacent rails, a panel of measuring instruments subjected to excitation from the vibrating structure etc. In such cases also there is a chance of resonance to occur when the frequency of base excitation matches with the natural frequency of the machine. Therefore some viable solution to this problem is needed, which can eliminate or reduce the harmful and intolerable vibrations of machine so that i t can perform as expected.
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Fig. 4.1 A rotating machine (force excitation problem) [1]
Fig. 4.2 An automobile on a wavy road (base excitation problem) [1]
Referring to fig. 4.1 (force excitation problem), let „M‟ be the total mass of the machine including the rotor and let „m×e‟ represents the amount of unbalance. Assuming that the machine is constrained to move in vertical direction, main system has only 1 degree of freedom (dof). The unbalance mass „m‟ revolves with angular velocity „ω‟ in counter clockwise direction. Then „(M-m)‟ will be the non-rotating mass of the machine. Let „x‟ be the displacement of the non-rotating mass from static equilibrium position. Here „k‟ is the combined stiffness of spring, and „c‟ is the damping coefficient of damper.
Thus displacement of the unbalance mass „m‟ in vertical direction is given by, „x + e sin ωt‟. Writing the equation of motion for above system:
̈ ̈ ̇ ̈ ̇ ω
ω
...(1)
In eq.1 „ ‟ shows the centrifugal force due to rotation of unbalance mass. For a constant speed this excitation force is also constant, writing it „ ‟
̈ ̇
ω
16
nd
This is a 2 order, linear non-homogeneous differential equation of motion with constant coefficient. The complete solution of this differential equation consists of two parts, thus:
̈ ̇
...(2)
The complementary function „ ‟ is the solution of corresponding homogeneous equation:
ω
ω
...(3)
The part of the equation given by eq.3 dies out with time, therefore it is not considered in the solution here. Using the method of undetermined coefficients to find out the particular solution „ the eq.1:
̇ ̈
Let,
ω
Then,
ω
ω
ω
ω
ω
ω
‟ of
...(4)
ω
ω
ω
Substituting in eq. 1, we get:
ω
ω
ω
ω
ω
ω
ω ...(5)
Comparing coefficients of sine and cosine terms on both sides,
ω
ω
ω
ω
Solving for A and B:
; ;
17
Putting in eq.4, we get:
* +
...(6)
Defining:
Eq.6 can be re-written as:
√ √ ω
ω
;
ω
ω
ω
ω
ω
ω
...(7)
...(8)
Where,
√
X is the amplitude of steady state response and is the phase lag of „
excitation force „
‟.
‟ with respect to
Dividing the numerator and denominator on the right hand side of eq.8 :
Where,
√ √
, is the deflection under static load
.
, is the excitation frequency to natural frequency ratio.
Rewriting above equation:
√
...(9)
Here „K‟ is called the „Amplitude ratio‟ or „Amplification factor‟. 18
Now from eq. 7, we can write,
ω
ω
Or,
ω
ω
...(10)
Multiplying and dividing the right hand side by „M‟ in eq. (10) :
ω
ω
ω
ω
ω
ω
...(11)
Therefore from eq. 11, the amplitude of steady state response:
...(12)
From eq.9,
√
...(13)
19
Fig. 4.4 shows the variation of the non-dimensional ratio (MX/me) versus the frequency ratio r. It follows form eq.(13) that as r→0, (MX/me) →0 for all values of .
Fig. 4.3 „Amplitude ratio v/s Frequency ratio‟ [1]
Fig. 4.4 „Non-dimensional ratio (MX/me) v/s Frequency ratio‟ [1]
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As it is clear from above figure, At r =1,
Thus if This is the condition of Resonance
⁄
;
⁄ ;
Now analyzing the problem of Base excitation referring to Fig. 4.2 & 4.5
Fig. 4.5 System subjected to base excitation [1]
In this problem, let „m‟ be the sprung mass (mass of the vehicle), „k‟ being the combined stiffness of the spring, and „c‟ being the damping coefficient of the damper.
Let the base is given a harmonic base excitation motion of the road. Due to this „ ‟ being the motion of main mass „m‟.
due to waviness
There are two aspects of base excitation motion : The absolute motion of the mass. Relative motion between mass and base.
Assuming
; the relative motion between mass and base is given by :
̇ ̇ ̇ ; And
;
Applying Newton‟s second law to the free body diagram of the mass „m‟ we have:
̈ ̇ ̇ 21
Rearranging the equation, we have
̈ ̇ ̇ ̇
...(14)
Assuming the base excitation to be harmonic, of the form:
we have,
Substituting these values, the above equation of motion becomes
Let,
̈ ̇ ̈ ̇ ⁄ √ and
and substituting in above equation, we have
...(15)
Where:
;
And
Which simplifies to
√ ;
and
Substituting for A in above equation, we have
Or
Where,
̈ ̇ √ ̈ ̇ √
...(16)
...(17)
The steady state response may be expressed as:
22
Which on simplification becomes
√ Thus, the steady state amplitude is
√ √
Hence, the motion transmissibility ratio (T.R.) is given by
√ √
...(18)
o i t a R y t i l i b i s s i m s n a r T
Frequency Ratio „r‟ Fig. 4.6 Transmissibility v/s Frequency ratio curve for different values of damping factor [1]
From fig. 4.6, it is clear that when the frequency ratio r = 1; i.e. frequency of external excitation matches with natural frequency of the system ; the motion transmissibility ratio (T.R.) is governed by damping factor .
When = 0, the T.R. approaches to a very large value, which is the undesirable condition of resonance.
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As a solution to such problems of Force excitation and Base excitation, an arrangement can be applied to main machine which absorbs all of its undesirable vibrations particularly at the speed which matches with resonance frequency and also greatly reduces the extent of harmful vibrations of the machine in a particular speed domain. Such a concept is the “Dynamic Vibration Absorber.”
4.2
Dynamic Vibration Absorber:
Traditional treatment methods that involve structural modifications are often time consuming and expensive. One possible solution is an installatio n of a „Dynamic vibration absorber ‟ (DVA). It has certain advantages over other methods of vibration suppression. It is external to the machine structure, so no re-installation of equipment is necessary. Unlike with structural modifications, when the final effect is unknown until mass-elastic properties of the machine components have been modified, a DVA can be designed and tested before installation. It can be adjusted in the lab environment with predictable field results.
4.2.1 History: The dynamic vibration absorber was invented in 1909 by ‘Hermann Frahm’ (US Patent #989958, issued in 1911), and since then it has been successfully used to suppress wind-induced vibration and seismic response in buildings. Characteristics of DVA were studied in depth by Den Hartog . Work on DVAs was undertaken rigorously during the development of helicopter rotor blades after 1963 (Flannelly, 1963; Jones, 1971), and more recently for the defence mechanism against earthquakes. In the industry, it has been primarily used to suppress vibration caused by a resonance condition in machinery.
4.2.2 Concept: In its simplest form, a DVA is a small vibratory system consisting „a spring and a mass‟ often called an ‘Auxiliary system’ which is coupled to a machine or structure called ‘Main system’ , so as to control its vibrations. The components of the auxiliary system are so selected that its natural frequency is tuned to match the natural frequency of the machine it is installed on. Because of this tuning DVA exerts a force on the main system that is equal and opposite of the excitation force, canceling vibration at the resonant frequency.
24
The concept of DVA can be implemented to both Force excit ation and Base excitation problems successfully. Let us first analyze the concept of DVA pertaining to a rotating machine subjected to a periodic unbalance force of excitation.
Force Excitation: For force excitation problem, a simple DVA and its equivalent system with free body diagram [1] are shown in fig. 4.7
Fig 4.7 (a) Main system coupled with DVA
(b) Free body diagram
In the above system:
= mass of the main system; = mass of the auxiliary system;
= stiffness of spring of main system; = stiffness of spring of auxiliary system; = excitation force;
= displacement of main system; = displacement of auxiliary system;
̈
For analysis, assuming
>
Writing the differential equations of motion:
25
And
̈ ̈ ̈
Rearranging these equations,
...(19) ...(20)
For steady state solution, assuming solutions:
̈ ̈ and
Then,
and
Substituting in eq. 19 and 20:
* + * +
Or
...(21) ...(22)
Solving for A & B
And
Where,
;
* + represents the frequency equation:
…(23)
26
Dividing eq.(23) by
, we get
{ }
In eq.(24), the natural frequency of main system is frequency of the auxiliary system is
...(24)
; and the natural
; therefore this equation can be
re-written as :
[ ] For a tuned system
Further, as
letting the mass ratio
...(25)
,
=
reduces eq.(25) to
...(26)
This is a parabolic equation and the resonance frequencies of the tuned absorber system can be obtained from the roots of this equation.
...(27)
27
Fig 4.8 shows the „effect of mass ratio on the spread of frequency ratio‟, which decides the working range of the DVA and gives two new natural frequencies of the composite system. 1.8 1.6 1.4 ' r ' o 1.2 i t a r 1 y c n0.8 e u q0.6 e r F0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
1.2
Mass ratio µ
Fig. 4.8 „Effect of mass ratio on the spread of frequency ratio‟
As we have seen that, the steady state amplitude of the main system is,
Thus for A = 0; (for complete absorption of vibration of main system)
;
which is the natural frequency of tuned auxiliary system.
Let the amplitude of static deflection of the main system under static load
is,
28
To write the Amplitude ratios of main system and auxiliary system in nondimensional form, dividing numerator and denominators of A and B by :
()
Or
And
...(28)
...(29)
Eq. (28) & (29) represent the amplification factor of vibrations of
the main system and auxiliary system
respectively.
Base Excitation: Now let us analyze the concept of DVA pertaining to a problem of Base excitation. A concept model of base excitation system coupled with DVA is shown in figure below: Let a periodic motion y = Y sin ωt, is applied on the base, due to which the main mass and auxiliary mass start to vibrate with displacements say and respectively at the frequency of forced excitation motion „ω‟.
29
Fig. 4.9
(a) Main system coupled with DVA
In the above system :
(b) Free body diagram
̈ ̈ ̈ = mass of the main system;
= mass of the auxiliary system;
= stiffness of spring of main system;
= stiffness of spring of auxiliary system; = excitation base motion;
y = displacement of the base
= displacement of main system;
= displacement of auxiliary system;
For analysis, let us assume:
and
Writing the differential equations of motion with the help of free body diagrams;
And Rearranging,
...(30)
30
̈ Now let us assume the solution to above equations is:
̈ ̈ * + * + and
Then, And
Substituting these values in above equations, we get:
Or
...(31)
...(32) ...(33)
Above equations are the same as eq. (21) & (22), which have been solved ear lier. Therefore we can conclude that all the solution of force transfer problem and motion transfer problem due to external excitation are sam e. Thus, steady state amplitude of masses „ respectively.
And
‟ and „
‟ are given by A & B
And amplitude ratios of main and auxiliary systems are given by eq. (28) & (29),
And
...(28)
...(29)
31
Now plotting these amplitude ratios with frequency ratios, we get following trends as shown in fig 4.10 (a) and (b).
r
(a) For main system
r
(b) For auxiliary system
Fig. 4.10 Amplitude ratio v/s frequency ratio [1]
32
4.3
Collection of Technical Data and Calculations:
This project work covers two types of machine setups. A machine for demonstration of „absorption of vibrations of a system subjected to force excitation using DVA‟ was fabricated and a machine subjected to base motion excitation for the same purpose was functionalized for “Vibration and noise control laboratory” of Mechanical engineering department.
4.3.1 Setup for machine subjected to Force excitation For fabrication of the Force excitation setup, following components were used
AC/DC electric motor, rated speed 4000 RPM
A disc with arrangement of increasing/decreasing unbalance mass
Auto transformer to control speed of motor
Wooden platform for machine
Wooden base of (36×30) cm for clamping the motor
Main system springs (×4) each of stiffness =
Auxiliary (system) mass and spring
Steel pipes to provide bearing surface to restrict DoFs of wooden base
/4
The working principle of DVA pertaining to force excitation setup has been explained earlier. From fig. 4.8 the effect of mass ratio on the spread of frequency ratios is clear, which gives new natural frequencies of the 2 dof system. Let us first calculate the parameters of the M ain system : Main mass = (mass of motor + mass of wooden base + mass of rotor + mass of clamping accessories.)
= (1.742 + 1.468 + 0.108 + 0.145) kg = 3.463 kg.
Now, stiffness of the main spring can be obtained by plotting „load deflection curve‟ of the spring. This is done by progressively hanging weights on spring and measuring its length (deflection).
33
Load vs deflection curve 3000
2500
2000
) m g ( 1500 d a o l 1000
500
0 58
60
62
64
66
68
70
72
length of spring (mm)
Fig. 4.11
Stiffness of the spring
Since main mass is supported by 4 such springs therefore, Stiffness of main spring
34
Now calculating the natural frequency of M ain system :
Therefore, speed of the motor
At the speed of 500 rpm of motor, main system will be in resonating condition. To keep the main system from harmful vibrations it is to be coupled with the auxiliary system.
35
Now calculating parameters of the Au xi li ary system : Stiffness of the auxiliary spring can be obtained in similar manner by plotting „load – deflection curve‟ for the spring shown in graph below:
Load vs deflection curve 3500
3000
2500
) 2000 m g ( d a o1500 l
1000
500
0 136
138
140
142
144
146
148
length of spring (mm) Fig. 4.12
Stiffness of the spring
Stiffness of auxiliary spring
;
36
For a tuned DAV, natural frequency of the auxiliary system must match the natural frequency of main system.
Auxiliary mass
Now, let us find the spread of frequency ratios which gives two new natural frequencies of the coupled system and it also gives the speed range of working of DVA. Mass ratio
From eq. (26),
Putting the value of:
On solving above eq. it gives two values of new frequency ratios,
37
Upper natural frequency of the system as a whole :
= 70 rad/s
;
Lower natural frequency of the system as a whole:
= 39.16 rad/s
The practical results of absorption of vibrations of the main system subjected to unbalance force excitation by transferring its vibrations to auxiliary system are shown in fig. 5.1 (a) & (b).
Force excitation setup
38
4.3.2 Setup for machine subjected to Base excitation For functionalizing the Base excitation setup, components that were used are –
AC Synchronous motor 2600 RPM
Variable speed drive with cone pulley arrangement
Eccentric cam for generation of base excitation motion
Main (system) mass and spring
Auxiliary (system) mass and spring
The working principle of DVA is explained earlier. From fig. 4.8 the effect of mass ratio on the spread of frequency ratios is clear, which gives new natural frequencies of the 2-DoF s ystems.
Using eq. (26):
For simplicity, taking the mass ratio
;
On solving the above eq. it gives two values of new frequency ratios,
; ;
Limiting speeds achieved through variable speed drive after speed reduction are, Maximum speed = 800 rpm; Minimum speed =
300 rpm;
A combination of mass and spring properties would be suitable for use if natural frequency of the Main system made of them lies in between the above speed range.
39
Taking main spring with stiffness from „load v/s deflection curve‟ as shown in graph:
Load vs deflection curve 1600
1400
1200
1000
) m g ( d 800 a o l 600
400
200
0 50
55
60
65
70
75
80
length of spring (mm) Fig. 4.13
Stiffness of the spring
For main mass, arbitrarily choosing
;
40
Checking for the natural frequency,
Therefore speed
;
which is in the speed range and therefore suitable for use on this setup.
Since we used mass ratio µ =1; therefore the auxiliary mass and spring will also have same properties as that of main system‟s. Thus,
;
;
Now it is clear from fig.8 and eq.28 that when this auxiliary system is attached to the main system, it will completely absorb the vibrations of main system at its natural frequency. The combined natural frequency of the 2-DoF system is now shifted to upper and lower frequencies from the natural frequency of main system. Those new natural frequencies are given by frequency ratios .
Upper natural frequency of the system as a whole:
; 41
Lower natural frequency of the system as a whole:
The practical results of absorption of vibrations of the main system by transferring its vibrations to auxiliary system are shown in fig. 5.2 (a) & (b).
Base Excitation setup
42
4.4
Fabrication Method:
Following steps were followed for fabrication of setup for force excitation and the vibration absorber:
4.4.1 Selection of Motor Selection of a small sized, low hp motor whose speed can be varied by varying voltage supply. The motor that was used has following specifications
Type – AC/DC (universal)
Rated speed = 4000 rpm
Voltage supply = 220 V
Current = 0.5 A
Mass = 1.742 kg
Diameter = 10 cm; length = 20 cm.
4.4.2 Determining the Minimum Motor Speed With the help of auto-transformer available in the laboratory minimum speed of motor was checked by varying supply voltage. Minimum speed of motor was found to be 300 rpm.
4.4.3 Wooden Base and Unbalance Disc A wooden base for clamping the motor and an unbalance disc were purchased in which amount of unbalance can be varied by increasing no. of nuts. Motor was centrally located on wooden base and clamped with nut-bolts.
4.4.4 Purchase of Springs Now any combination of mass and spring would be useful if the natural frequency of the system made of them lies in speed range 300-4000rpm. Since main mass would be including motor, rotor, base etc adopting the speed range of 400-3000 rpm for ease, some springs of certain stiffness wer e purchased.
4.4.5 Determining the Spring Stiffness To check the stiffness of springs, each spring was loaded progressively and change in length was recorded. Now plotting „load v/s deflection curve‟ which is a straight line stiffness was obtained by slope of the line.
Stiffness of one of the main springs = 2373.4 N/m
Stiffness of auxiliary spring = 3270 N/m
43
4.4.6 Calculation of Main Mass Now total main mass was obtained by adding all sprung mass Main mass = (mass of motor + mass of wooden base + mass of rotor + mass of clamping accessories). It was found to be
= 3.463 kg.
4.4.7 Selection of Number of Springs Selection of no. of main springs would perform two tasks. Firstly, it would decide the combined stiffness of main spring in parallel arrangement; and secondly these springs would support the wooden base on which motor has been clamped.
No. of springs used = 4 Combined stiffness of main spring
.
4.4.8 Calculation of Natural Frequency of Main System Natural frequency of the main system was calculated and checked whether it lies in the adopted speed range otherwise no. of springs had to be changed.
Natural frequency
, @ N = 500 rpm.
4.4.9 Arrangement of the Wooden Base A wooden platform was arranged the top plate of which has a circular hole large enough to allow free movement of auxiliary mass across it, and that would support motor base by 4 main springs.
4.4.10 Mounting of Motor Wooden platform and motor base were connected together by 4 main springs using nut-bolts.
4.4.11 Preparation of Auxiliary Mass Since auxiliary spring (stiffness = 3270 N/m) has been already purchased, an auxiliary mass was prepared with a hook welded on its top such that the natural frequency of auxiliary system matches with the natural frequency of main system, making it a tuned vibration absorber.
Material used- Mild steel
Auxiliary mass
4.4.12 Mounting of Unbalance Disc Unbalance disc was tightened on the motor shaft. 44
4.4.13 Arrangement of Auxiliary System Motor base, auxiliary spring and mass were provided with end hooks to couple & decouple main system and auxiliary system at will. This completes the fabrication process of force excitation setup coupled with Dynamic vibration absorber.
4.5
Testing and Modifications:
When both the machines are ready, the following testing procedure was conformed to:
4.5.1 Testing of force excitation setup
Note down the specifications of machine, e.g. stiffness of main spring, main mass (including rotor), unbalance mass etc. and c alculate the natural frequency of main system thus find speed of motor at natural frequency.
Weight the auxiliary mass and find stiffness of auxiliary spring by plotting „load deflection curve‟.
Pick unbalance disk and load some known unbalance mass on it by applying nuts on the bolt provided on disk.
Mount this unbalance disk on motor shaft by screw and connect motor to auto transformer.
Set the auto transformer to zero voltage supply and switch it on.
Now slowly increase the voltage supply from auto transformer so that motor starts to rotate. Speed of motor increases as the supply voltage increases.
Set the voltage supply to a certain value and measure the speed of motor using tachometer.
At some particular speed of motor the vibration amplit ude of main system will be highest. Set the voltage supply to a certain value where maximum amplitude of vibration is observed.
Now machine (main system) is operating at a speed that corresponds to its natural frequency. Couple the auxiliary spring and mass with the main system and observe the absorption of vibrations.
Calculate natural frequency shifts using formula and obtain two new speeds of motor which correspond to 2 new natural frequencies of the 2 dof system (main system coupled with auxiliary system).
45
Set voltage supply to a value that corresponds to speed of motor at new natural frequencies of 2 dof system, observe the vibration amplitude and interpret results by graphs of „vibration amplitude v/s frequency ratio.‟
4.5.2 Precautions
Check for all fasteners are tight, and ensure there is no loose joint anywhere. Before switching on the auto transformer set it to zero voltage supply, then progressively increase supply voltage. Tight the screw which holds unbalance disk on motor shaft firmly, otherwise at higher speeds it may be thrown away. Calibrate auto transformer for motor speed also, since it would be difficult to measure motor speed when it is vibrating.
4.5.3 Modifications When the machine was run first it was found that vibrations due to unbalancing force were occurring in horizontal plane also. It was due to the fact that motor was centrally clamped to wooden base and plane of rotation of unbalance mass was located at some distance from clamping, therefore unbalance force was generating a moment on wooden base resulting in horizontal oscillations of it. Since we have adopted single degree of freedom system, main system has to be made to vibrate in one direction only (1-DoF). To ensure vibrations of the main system would be in vertical plane only, some bearing surfaces were provided in the form of motion blocking rods clamped to wooden platform surrounding motor base and preventing vibration in horizontal plane.
46
4.5.4 Testing of Base excitation setup
4.5.5
Note down the specifications of machine, e.g. minimum and maximum speed available. Ensure for the variable speed drive that belt is tight enough on cones to avoid mutual slipping. Weight main and auxiliary mass. Also obtain stiffness of main and auxiliary spring by plotting load v/s deflection curve. Calculate natural frequency of the main system (mass & spring) and corresponding speed of rotation after reduction through variable drive. Connect motor to power supply and note down the speed available after reduction of speed through variable drive at the suspension end of cam motion generator. Suspend main spring & mass at the suspension end of cam motion generator, and vary the frequency of excitation (speed) by shifting of belt on variable drive and observe the vibration amplitude of main system. Set the speed of drive at which the vibration amplitude of main system is maximum. At this speed the machine is operating at its natural frequency. Now couple the main system with auxiliary system (mass & spring) and observe vibrations of the system as a whole. Calculate natural frequency shifts using formula. Vary the frequency of excitation (speed) to lower and upper side and observe the vibration amplitude. Interpret results by graphs of „vibration amplitude v/s frequency ratio.‟
Precautions
Clean cone surfaces to avoid slipping of belt on them.
Tight the transmission belt so that it may not slip during operation.
Do not shift the belt when machine is not running.
47
Chapter 5 OBSERVATIONS & STATISTICS
5.1
Recording of Test Results and Specifications:
To demonstrate the vibration absorption of main systems subjected to unbalance force excitation and base motion excitation respectively, when these systems were tested and coupled with undamped dynamic vibration absorbers the experimental results obtained from both machines are tabulated in following tables.
5.1.1 Specifications for Force excitation setup Frequency ratio is defined earlier as ratio of operation excitation frequency to the natural frequency of the system. During testing of the machine following specifications were concluded;
Natural frequency of the main system
52.36 rad/s. @ 500 rpm.
Maximum speed range of motor = 400-3000 rpm
Range of frequency ratio „r‟ = 0 to 3
Increment in frequency ratio = + 0.1
Mass ratio µ = 0.35
Natural frequency shifts : r = 0.7 and 1.3
Main spring stiffness
Auxiliary spring stiffness
Auxiliary mass
= 9493.6 N/m = 3270 N/m
= 1.2 kg
5.1.2 Specifications for Base motion excitation setup During testing of the machine following specifications were concluded ;
Natural frequency of the main system
55.80 rad/s. @ 533 rpm.
Maximum speed range of variable speed drive = 800 rpm
Range of frequency ratio „r‟ = 0 to 3
Increment in frequency ratio = + 0.1
Mass ratio µ = 1
Natural frequency shifts : r = 0.6 and 1.6
Main mass
= 350 gm
Main spring stiffness
= 1090 N/m 48
Frequency ratio 'r'
Amplitude ratio of main system 'A/Xst'
Amplitude ratio of auxoliary system 'B/Xst'
0
1
1
0.1
1.0137
1.024
0.2
1.0577
1.1018
0.3
1.1424
1.2553
0.4
1.2931
1.5394
0.5
1.5789
2.1053
0.6
2.2567
3.5261
0.7
5.7562
11.287
0.8
3.8136
10.593
0.9
0.76799
4.042
1
0
2.8571
1.1
0.55351
2.6357
1.2
1.4175
3.2216
1.3
5.9792
8.6655
1.4
4.0747
4.2445
1.5
1.6129
1.2903
1.6
1.0146
0.65036
1.7
0.73811
0.39053
1.8
0.57678
0.25749
1.9
0.47039
0.18023
2
0.39474
0.13158
2.1
0.33814
0.099161
2.2
0.29422
0.076619
2.3
0.25917
0.060413
2.4
0.2306
0.048446
2.5
0.2069
0.039409
2.6
0.18694
0.032455
2.7
0.16994
0.027018
2.8
0.15531
0.022706
2.9
0.1426
0.019244
3
0.13147
0.016434
Table 5.1, Matlab simulation “Response of Main system & auxiliary system coupled together against Frequency ratio” when subjected to Unbalance force excitation.
49
Frequency ratio 'r'
Amplitude ratio of main system 'A/Xst'
Amplitude ratio of auxiliary system 'B/Xst'
0
1
1
0.1
1.0205
1.0308
0.2
1.0889
1.1343
0.3
1.2329
1.3548
0.4
1.5396
1.8328
0.5
2.4
3.2
0.6
12.903
20.161
0.7
2.2184
4.3497
0.8
0.70533
1.9592
0.9
0.24551
1.2922
1
0
1
1.1
0.18012
0.85771
1.2
0.35302
0.80231
1.3
0.56842
0.82379
1.4
0.9245
0.96302
1.5
1.8182
1.4545
1.6
12.342
7.9114
1.7
2.7709
1.4661
1.8
1.2601
0.56256
1.9
0.81509
0.3123
2
0.6
0.2
2.1
0.47242
0.13854
2.2
0.38766
0.10095
2.3
0.32713
0.076254
2.4
0.2817
0.05918
2.5
0.24633
0.046921
2.6
0.21804
0.037854
2.7
0.19489
0.030985
2.8
0.17563
0.025677
2.9
0.15936
0.021506
3
0.14545
0.018182
Table 5.2, Matlab simulation “Response of Main system & auxiliary system coupled together against Frequency ratio” when subjected to Base motion excitation.
50
5.2
Data Interpretation and Graphs:
Now to interpret the results obtained from testing of both machines, system‟s response as a function of frequency ratio are plotted in the following figures.
5.2.1 Force excitation setup Fig. 5.1 shows the transmission of vibration to main system and its absorption by the auxiliary system having tolerable amplitude of vibrations having mass ratio µ=0.35, for unbalance force excitation setup.
Referring to table 5.1 and fig. 5.1 following conclusions can be stated regarding Force excitation setup: When main system is not coupled with the auxiliary system, resonance occurs at a motor speed of 500 rpm which corresponds to the natural frequency of the system at r = 1. On coupling the main system with the auxiliary system, vibration amplitude „A‟ of main system reduces to zero at the same speed (r =1), and auxiliary system vibrates with maximum vibration amplitude ratio of 2.857. On varying the operation speed of the machine i.e. changing the frequency ratio „r‟ on higher or lower sides vibration transmissibility of main system increases and it reaches a value twice of steady state amplitude at frequency ratios r = 0.85 & 1.25. The coupled system (2 DoF system) has two new natural frequencies at which vibration amplitudes exceed tolerable limits. These natural frequencies corresponds to frequency ratios r = 0.7 & 1.3
51
Amplitude ratio vs Frequency ratio curve for 'Main mass' 5
4
3
t s X / A
2
1
0
0
0.5
1 1.5 2 Frequency Ratio 'r'
2.5
3
Amplitude ratio vs Frequency ratio curve for 'Auxiliary mass' 8 7 6 5 t s X / 4 B
3 2 1 0
0
0.5
1 1.5 2 Frequency Ratio 'r'
2.5
3
Fig. 5.1 System Response v/s Frequency Ratio Curves for Force Excitation Setup 52
5.2.2 Base motion excitation setup Fig. 5.2 shows the transmission of vibration to main system and its absorption by the auxiliary system having tolerable amplitude of vibrations having mass ratio µ=1, for base motion excitation setup. Referring to table 5.2 and fig. 5.2 following conclusions can be stated regarding Base excitation setup:
When main system is not coupled with the auxiliary system, resonance occurs at a motor speed of 533 rpm which corresponds to the natural frequency of the system at r = 1. On coupling the main system with the auxiliary system, vibration amplitude „A‟ of main system reduces to zero at the same speed (r =1), and auxiliary system vibrates with maximum vibration amplitude ratio of 1. On varying the operation speed of the machine i.e. changing the frequency ratio „r‟ on higher or lower sides vibration transmissibility of main system increases and it reaches a value twice of steady state amplitude at frequency ratios r = 0.68 & 1.52 The coupled system (2 DoF system) has two new natural frequencies at which vibration amplitudes exceed tolerable limits. These natural frequencies corresponds to frequency ratios r = 0.6 & 1.6 Since we have taken a higher mass ratio in base excitation setup, therefore we get a wider range of frequency ratio in which undamped dynamic vibration absorber works effectively.
53
Amplitude ratio vs Frequency ratio curve for 'Main mass' 10
8
t s X / A
6
4
2
0
0
0.5
1 1.5 2 Frequency Ratio 'r'
2.5
3
Amplitude ratio vs Frequency ratio curve for 'Auxiliary mass' 7 6 5 t 4 s X / B
3 2 1 0
0
0.5
1 1.5 2 Frequency Ratio 'r'
2.5
3
Fig. 5.2 System Response v/s Frequency Ratio Curves for Base Excitation Setup 54
Chapter 6 RESULTS AND DISCUSSION
6.1
Interpretation of Results:
After performing tests on both the machines and taking out the experimental observations following results were obtained:
Undamped dynamic vibration absorber, when coupled to main system in „tuned‟ condition completely absorbs the vibration of main system at its natural frequency. At an operating speed at lower or higher side of zero amplitude condition (r = 1), vibration amplitude of the combined system increases and finally meets resonance condition at lower & higher natural frequencies of the system as a whole. New resonance frequencies of the combined system subjected to force excitation
New resonance frequencies of the combined system subjected to base excitation
6.2
Undamped dynamic vibration absorbers are extremely effective for constant speed machineries in a certain speed range. The spread of this working speed range depends on the mass ratio of the system. As in the testing of force excitation setup, working range of DVA is r = 0.82 to 1.25, for mass ratio µ = 0.35. Whereas in testing of base excitation setup, working range of DVA was found to be r = 0.68 to 1.52 Auxiliary mass vibrates with low amplitudes which are in tolerable limits absorbing undesirable vibrations form main system. At an operating frequency away from natural frequency lower or higher side vibration transmission problem is insignificant.
Discussion & Fields of Application:
As the working principle of Undamped dynamic vibration absorber suggests, it can be applied to a machine or structure which is subjected to external excitation at a constant frequency. The experimental results obtained from the machines on which work is done verify the phenomenon of absorption of vibrations using DVA. Therefore the undamped dynamic vibration absorber can be successfully employed in turbines & compressors of a power plant, motor of flour mill, on bridges when tuned to its natural frequency so that
55
when excitation frequency due to passing of vehicles or troops matches with natural frequency of the bridge so it should not collapse. It should be noted that the use of DVA with machinery sh ould not be viewed only as a fix to a vibration problem. In some cases, when a machine structure has to be tall with a high centre of gravity, a DVA can be designed to be built into a machine, very much like it would be installed in a tall building. For example, heavy motors designed for vertical installation often have their fundamental natural frequency (often called reed frequency) just slightly above the operating speed. If such a motor is installed above a pump on a pedestal, the system natural frequency can get dangerously close to the operating range, causing a resonance. A DVA could be incorporated into the motor structure and tuned appropriately to prevent resonance vibration. For this purpose an element of the motor structure, such as a fan cover, may serve as an absorber mass if it is mounted to the motor by elastic springs.
56
Chapter 7 SCOPE OF FUTURE WORK
This project work covers the utility of „Undamped Dynamic Vibration Absorber‟ (UDVA) in its simplest form which comprises of an auxiliary mass and a spring. UDVA can be of some other forms also which depends on its application. Such as in a vertical motor it can be a fan or flywheel connected through elastic spring, it can also be in the form of an auxiliary mass with enclosed air or a beam of metallic strip as variable spring element. However, with incorporation of a damper in auxiliary system a better attenuation of vibration in a wider range of excitation frequency can be achieved by selecting a damping element with suitable damping coefficient. DVA‟s can be designed for tall buildings and structures also in suitable form which will keep safe those structures from earthquake and wind induced vibrations.
57
APPENDIX
1. Matlab program for „Response of Main system‟ in Force excitation setup :
>> r=0:0.1:3 >> x=(1-r.^2); >> y=[(r.^4)-(2.35*(r.^2))+1]; >> a=(x./y) >> b=(a.^2); >> c=sqrt(b) >> plot (r,c) >> grid
2. Matlab program for „Response of Auxiliary system‟ in Force Excitation setup
>> r=0:0.1:3; >> x=[(r.^4)-(2.35*(r.^2))+1]; >> y=(1./x) >> m=(y.^2); >> n=sqrt(m) >> plot (r,n) >> grid
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3. Matlab program for „Response of Main system‟ in Base excitation setup :
>> r=0:0.1:3; >> x=(1-(r.^2)); >> y=[(r.^4)-(3*(r.^2))+1]; >> m=(x./y) >> a=(m.^2); >> b=sqrt(a) >> plot (r,b) >> grid
4. Matlab program for „Response of Auxiliary system‟ in Base excitation setup :
>> r=0:0.1:3; >> x=[(r.^4)-(3*(r.^2))+1]; >> y=(1./x); >> a=(y.^2) >> b=sqrt(a) >> plot (r,b) >> grid
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REFERENCES
1. http://www. Googlebooks.com/Mechanical Vibrations and Noise Engineering by A. G. Ambekar. 2. http://www.reliableplant.com/Read/24117/introduction-machinery-vibration 3. http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_(1940)#Film_of_collapse 4. http://en.wikipedia.org/wiki/Broughton_Suspension_Bridge 5. http://en.wikipedia.org/wiki/Resonance_disaster#Resonance_disaster 6. Methods of Vibration control, Paper presentation iitd. 7. http://www.deicon.com/vib_categ.html 8. http://www.pump-zone.com/instrumentation/controls/dynamic-vibration -absorber 9. http://www.iitr.ac.in/outreach/web/CIRCIS/PG/NVH/Design_for_NVH_Lab_Experi ment.pdf 10. http://en.wikipedia.org/wiki/Damping 11. http://www.iitr.ac.in/outreach/web/CIRCIS/UG/FSV/Vibration%20Isolation.pdf 12. http://www.universal-balancing.com/en/balancing-information/what-is-balancing 13. http://alexandria.tue.nl/repository/books/571783.pdf 14. http://www.acs.psu.edu/drussell/Demos/absorber/DynamicAbsorber.html 15. http://digital.library.adelaide.edu.au/dspace/bitstream/2440/37922/1/02whole.pdf
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