DESIGN OF AN ADAPTIVE DYNAMIC VIBRATION ABSORBER

Transcription

1 SUB CRUCE LUMEN DESIGN OF AN ADAPTIVE DYNAMIC VIBRATION ABSORBER Christopher Ting-Kong Department of Mechanica Engineering The University of Adeaide South Austraia 5005 Thesis submitted for the degree of Master of Engineering Science on the 1 st Apri, i

2 Contents DESIGN OF AN ADAPTIVE DYNAMIC VIBRATION ABSORBER TABLE OF CONTENTS Abstract Statement of Originaity Acknowedgements viii i CHAPTER 1. INTRODUCTION AND LITERATURE REVIEW Introduction 1. Literature Review 1..1 Eisting technoogy and prior research 1.. Summary of contribution to current knowedge addressed by this thesis Strategies taken here to achieve the objectives Anaysis of Dynamic Vibration Absorbers Anaysis of vibration in a base structure simpy supported beam Governing Equations 1 CHAPTER. DYNAMIC VIBRATION ABSORBER USING ENCLOSED AIR 15.1 Introduction 15. First Prototype 16. Second prototype 18.4 Eperimenta Resuts.4.1 Frequency range of second prototype.4. Identified probems with the second absorber ii

3 Contents.4. Comparison of eperimenta with theoretica vaues Identified probems with prototype using rubber diaphragm 6.5 Incorporating the use of an Auminium diaphragm 8.6 Impedance of the absorber 1.7 Performance of the absorber.8 Summary 5 CHAPTER. DUAL-MASS CANTILEVERED DYNAMIC VIBRATION ABSORBER 6.1 Introduction 6. Initia theoretica anaysis of cantievered absorber using discrete system anaysis 8. Theoretica anaysis of the cantievered absorber using continuous system theory 41.4 Finite eement anaysis of the absorber 50.5 Eperimenta Setup Introduction Resonance frequency eperimenta measurement of absorber (aone) 5.5. Eperimenta resuts Eperimenta setup with the absorber on the beam 56.6 Contro System Controer Setup Tuning agorithm 6.6. The inear transducer 65 iii

4 Contents.7 Eperimenta resuts for the case of the absorber mounted on the beam Performance of the absorber on a sinusoiday ecited simpy supported beam Variation in natura frequency of the absorber due to change in effective rod ength Speed of adaptation 70.8 Summary 7 CHAPTER 4. SUMMARY AND CONCLUSIONS Summary of Dynamic Absorber using encosed air 7 4. Summary of Dua Cantievered Mass Absorber Future work Future work on the air-spring absorber Future work on the dua cantievered mass absorber Concusions 77 Appendices 78 References 94 iv

5 Notation LIST OF FIGURES Figure 1.1 Representation of an Adaptive Vibration Absorber Figure 1. Resonance curves for a Dynamic Vibration Absorber Figure 1. Changing the natura frequency of a beam Figure.1 Initia concept of the absorber Figure. Variation in stiffness when changing the height of the encosed voume Figure. Design of the second prototype Figure.4 Transfer function of the second prototype Figure.5 Identified probem with the absorber mass Figure.6 Comparison of eperimenta and theoretica stiffness vaues Figure.7 Variation in natura frequency with auminium diaphragm Figure.8 Variation between eperimenta and theoretica stiffness vaues for Auminium diaphragm Figure.9 Harmonic response curve showing bandwidth and haf power points Figure.10 Eperimenta setup for auminium absorber Figure.11 Frequency response of auminium diaphragm dynamic vibration absorber Figure.1 The first modes of the cantievered absorber Figure. Dimensions used for discrete system anaysis Figure.4 Dimensions used for continuous system anaysis Figure.5 Shear and moment sign conventions Figure.6 Boundary conditions for atera vibration in a beam Figure.7 A meshed mode of the Dynamic Absorber Figure.8 Measuring resonance frequency of the absorber aone Figure.9 Transfer function of the absorber v

6 Notation Figure.10 Positioning for the number of turns made by the motor Figure.11 Eperimenta setup for the absorber attached on the beam Figure.1 Fow of contro signas Figure.1 Fow diagram for contro signas in tuning agorithm Figure.14 Location of the inear transducer Figure.15 Variation of natura frequency with dispacement votage Figure.16 Transfer function before and after absorber instaation Figure.17 Comparison of anti-resonances produced due to change in absorber stiffness Figure.18 Time signa for absorber to adapt to a changing ecitation signa Figure.19 Time signa for absorber to track changing ecitation frequencies in succession vi

7 Notation Notation m a = mass of absorber or secondary structure (kg) m b = mass of base or primary structure (kg) ω a = natura frequency of absorber (rad/s) ω b, ω n = natura frequency of base (rad/s) ω 11, ω = first and second modes of the composite system (rad/s) p = ecitation or operating frequency (rad/s) E = moduus of easticity (Pa) I = second moment of inertia (m 4 ) ρ = density (kg/m ) H a = impedance of absorber (N/m/s) H b = impedance of base (N/m/s) u b = maimum dispacement of base with absorber attached (m) u F = maimum dispacement of base without absorber attached (m) vii

8 DESIGN OF AN ADAPTIVE DYNAMIC VIBRATION ABSORBER ABSTRACT The aim of this thesis is to investigate the use of a Dynamic Vibration Absorber to contro vibration in a beam. Traditiona means of vibration contro have invoved the use of passive and more recenty, active methods. This study is different in that it invoves an adaptive component in the design of vibration absorber using two nove designs for the adaptive mechanism. The first design incorporates the use of an encosed air voume to provide the variabe stiffness component in the absorber. By adjusting the voume of compressibe air within the absorber, the stiffness characteristics of the absorber can be atered, enabing the device to adapt to changing vibration frequencies. Work here incudes a theoretica investigation of the device. Foowing this, two prototypes are constructed and tested, the second of which is the refined mode used for further testing. The second design incorporates the use of two concentrated masses cantievered from two rods. The adaptive soution is achieved by moving the two masses aong the ength of the rod, producing a changing natura frequency for the absorber device. An anaytica mode of this device is deveoped as we as a finite eement mode. Resuts from both are compared to those obtained eperimentay. Finay, a tuning agorithm is derived for the second absorber, and a contro system constructed to make the dynamic vibration absorber adaptive. Eperiments are undertaken to determine the effectiveness of the absorber on the beam subject to changing ecitation frequencies. The outcome of this research is that an Adaptive Vibration Absorber has been constructed with a computer interface such that the device can be used on ine. viii

9 Statement of Originaity STATEMENT OF ORIGINALITY This thesis does not contain any materia which has been accepted for the award of any degree or dipoma in this university or other tertiary institution. Uness where stated or referenced, this work contains no materia which has been previousy pubished or written by other researchers. Permission is given by the author to the reevant bodies within the University of Adeaide, for purposes of photocopying and reproduction, and its avaiabiity for oan. Signed:.. Date: i

10 Chapter 1. Introduction ACKNOWLEDGMENTS The author woud ike to thank Dr. Scott D Snyder for his supervision and input, and his wiingness to support the ideas forwarded in the research. Thanks must aso go to Dr. Coin H. Hansen for his interest and comments in the discussion of the work. The study woud never have been possibe without the taents and suggestions from George Osborne, Sivio De Ieso, Aan Mitter and Derek Frankin. The author woud aso ike to thank Ron Jager, Craig Price and Macom Bethune for a their work in the construction of the absorbers. Finay, a big thank you to a the administration staff in the department of mechanica engineering, and my feow cubice peers, and aso in particuar, Benny B. Cazzoato, for much of his inteigent input.

11 Chapter 1. Introduction CHAPTER 1. INTRODUCTION AND LITERATURE REVIEW 1.1 INTRODUCTION When a structure is undergoing some form of vibration, there are a number of ways in which this vibration can be controed. Passive contro invoves some form of structura augmentation or redesign, often incuding the use of springs and dampers, that eads to a reduction in the vibration. Active contro augments the structure with sensors, actuators and some form of eectronic contro system, which specificay aim to reduce the measured vibration eves. This research investigates the use of Adaptive Tuned Dynamic Vibration Absorbers (ATDVA) to contro vibration in a structure. A dynamic vibration absorber (DVA) is essentiay a secondary mass, attached to an origina system via a spring and damper. The natura frequency of the DVA is tuned such that it coincides with the frequency of unwanted vibration in the origina system. This resuts in a subcomponent of the tota structure adding a arge input impedance to the primary structure, thus absorbing the inertia energy transferred from the primary structure. Active DVAs are characterised by a secondary mass, mounted to a vibratory primary system, that is directy connected to an actuator and eectronic contro system. However, Adaptive Tuned refers to the abiity of the DVA to change its interna properties to refect changes in the vibratory environment. The use of DVAs is advantageous because of their mobiity, and their abiity to be incorporated into the structure after the design stage. This often aows for the absorber to be mounted without affecting the mounting between the structure and the supporting foundation, therefore providing a rigid mount with minima movement. Other advantages of DVAs over active attenuation techniques incude the abiity to guarantee stabiity, ow cost, ow power, and simpicity of impementation. 1

12 Chapter 1. Introduction This research wi aso investigate the effectiveness of ATDVA, especiay as an aternative to active vibration contro. DVAs can be constructed using a variety of different mechanisms and orientations; this study proposes two different arrangements. The first chapter reviews the theoretica mode of the DVA, and concepts reating to the testing of the devices. A review of past research in this area is aso undertaken. The second chapter detais work undertaken on the first DVA arrangement used in this study. This absorber is based on the idea of using an encosed air voume to provide a variabe stiffness component. The third chapter detais a second arrangement of an adaptive dynamic vibration absorber, one based on the concept of a variabe ength cantievered beam. The fourth chapter summarises the work which has been competed in this study and discusses possibe areas of improvement for both absorbers.

13 Chapter 1. Introduction 1. LITERATURE REVIEW 1..1 Eisting Technoogy and Prior Research Active Dynamic Vibration Absorbers Dynamic Vibration Absorbers were first invented in 1909 (Den Hartog, 1956). Work on DVAs was undertaken rigorousy during the deveopment of heicopter rotor bades after 196 (Fanney, 196, Jones, 1971), and more recenty, for the defence mechanism against earthquakes. Much work has been directed towards the use of DVAs attached to buiding structures, to counter seismic movements and wind forces. In recent studies, interest has aso been focused on the use of feedback and feedforward contro systems, and the synthesis of DVAs for mutipe-degree-of-freedom systems. Yoshida K. et a. (1988) eamined the combined use of both feedforward and feedback contro to attenuate both stationary and non-stationary (random) inputs, with an active DVA. Yoshida used a inear motor controed digitay using optima feedback and feedforward inks, to actuate the absorbing mass. DVAs have the advantage in that the controer is abe to guarantee a cosed-oop stabiity, even with minima knowedge of the controed system (Lee-Gauser, 1995). Harris and Crede (1961) provide a description of a number of different DVA arrangements, incuding penduum and inear spring-mass arrangements. Description is aso given of a tube form, in which a fuid ros from one tank to another, often used for antiro in ships. Absorber systems can incude rotationa or inear motion, or both. As this study focuses on the design of a suitabe and controabe DVA, it is favourabe to restrict the research to singe-degree of freedom inear systems.

14 Chapter 1. Introduction Hunt (1979) outines many appications where ADVAs have been usefu. In the area of heicopter rotor bade deveopment, hydrauic Active Dynamic Absorbers have found use. Here, a doubeacting hydrauic cyinder is paced between the rotor and the fuseage, the rotor being the absorber mass. Two feedback signas are taken from the rotor and the fuseage through an ampifier, which sends a contro signa to a servo vave. This vave then contros the hydrauic actuator. Active Vibration Absorbers have aso been used for suppressing vibration in beams which have mutipe resonant frequencies. Passive dynamic absorbers can be effective ony for a singe resonance frequency, and may not be effective at other resonant frequencies or for modes where a node is present at the ocation of the absorber. A force generator deveoped by Rockwe (1965) aso acting as the absorber mass can be mounted on the beam. A sensor mounted on the other side of the beam detects the motion of the beam and sends a feedback signa to the generator, which in turn reacts against the motion of the vibration. Another fied is in the contro of tower structures. Korenev et a. (1995) detais a number of different designs of dynamic absorbers for structura appications. 4

15 Chapter 1. Introduction Adaptive-Passive Dynamic Vibration Absorbers Numerous appications invoving active contro of Dynamic Absorbers have deat with actuating the absorber mass directy. This study is different in the way that the active component is impemented. Here, the actuator wi be controing the stiffness of the spring, thus controing the natura frequency of the absorber system (the secondary system). This method of contro has the potentia to be ess energy consuming, as the power required to adjust the spring variabe is epected to be ess than the power required to actuate the mass directy. This method is best described as an adaptive-passive approach to vibration contro. Adaptive-passive methods invove the use of passive eements which can be optimay tuned to perform over a certain frequency range. Unike a fuy active system, this form of contro cannot add negative damping to an eisting system (ie. the poes of the system cannot migrate across the imaginary ais). Thus a system which is aready stabe can not be made unstabe through the use of this method. Adaptive Hemhotz resonators, described by Lamancusa (1987), are an eampe of where adaptive-passive methods have been used for narrowband appications. These resonators are abe to contro sound within a certain frequency range, over a number of different speeds, by varying the resonator neck dimensions or cavity voume or both. In the area of broadband appications, Graf et a (1987) epains the use of adaptive-passive methods to vary the stiffness and damping of an engine mount. In the area of structura contro, Ryan et a (1994) designed an adaptive-passive vibration absorber using a variabe spring as the adaptive component. The stiffness was controed using a spring inserted through a siding pate, which coud then be moved to ater the effective number of cois in the spring, which in turn affected the overa stiffness of the spring. The pate was moved according to the feedback signa inked an acceerometer, mounted on the primary vibratory system. 5

16 Chapter 1. Introduction The physica imitation of damping present in a systems of dynamic absorbers, and the imitations of the eectrica properties in the equipment, resuted in some residua motion present in the vibratory structure. However, the resuts from this study indicated that the use of adaptive-passive vibration absorbers is feasibe in the contro of vibration. Sef-adapting absorbers have been previousy used in the vibration contro of unbaanced rotating shafts. These come in the form of centrifuga absorbers, which can vary their penduum ange in accordance with the anguar speed. This resuts in the stiffness of the absorber increasing at a rate square to the anguar speed, which means the eigenfrequency increases directy with the speed of the rotating shaft. 6

17 Chapter 1. Introduction 1.. Summary of contribution to current knowedge addressed by this thesis The main objective of this work is to eamine the use of two different arrangements as adaptive dynamic vibration absorbers. Adaptive dynamic vibration absorbers are essentiay a subset of those in active contro (for vibration and structura/acoustic contro), with particuar emphasis on tona disturbances. Where they can be used, DVAs have a distinct advantage in guaranteed stabiity. This study wi aim to deveop a dynamic vibration absorber with variabe stiffness that is: 1. Practica (construction from common materias, portabiity, and easy to use). Has a broad operating frequency range. Low Cost 4. Low power consumption Firsty, the effectiveness of using encosed air as a means of providing the stiffness for an adaptivetuning DVA wi be investigated. Secondy, the usefuness of using cantievered beams with concentrated end-masses wi aso be investigated as a second mechanism. A design procedure for the tuning agorithm for the absorber wi be derived. 7

18 Chapter 1. Introduction 1.. Strategies taken here to achieve the objectives 1. Determine appropriate absorber sizes by eperimentation with an eisting vibrating structure assemby; investigate the potentia of various absorber modes, and devices which wi enabe adjustment of the absorber stiffness. Determine the frequency range over which the absorber wi be useabe.. Buid a prototype mode of an absorber with a variabe encosed air voume and test its effectiveness for a singe resonance frequency. An anaytica procedure wi be used to design a DVA resonating at a ow frequency (ess than 50) Hz. It is epected that the seaing the air voume wi be critica, as this wi determine whether sufficient stiffness can be provided without pressurising the air itsef. Minima damping is often recommended for an absorber, as ecessive damping wi imit the singe-frequency attenuation in echange for epanded bandwidth. This may be a probem with an air device.. Test the effectiveness of the prototype absorber within a given frequency range. Determine the sensitivity of the absorber to variations in the ecitation frequency. 4. Construct an actuator that can contro the stiffness of a mechanica spring component, and derive agorithms for the contro system. Undertake eperimenta tests and compare the resuts with theoretica data. 5. Verify the usefuness of this second absorber arrangement, by subjecting it to successive ecitation signas of changing frequency. 8

19 Chapter 1. Introduction 1..4 Anaysis of the Dynamic Vibration Absorber The anaytica mode of the absorber wi be assumed to be a spring-mass system (athough minima damping wi aways be present in the rea form). The overa system is represented in figure 1.1. Controer Acceerometer m a F(t) Secondary System k a m s Primary System k s c s Figure 1.1 Representation of an Adaptive Vibration Absorber The range of frequencies over which the study is conducted has been chosen to be ω n [0,100] Hz. This range has been seected because of the vast number of rea vibration probems, which occur within it, particuary in reation to probems associated with rotating machinery. It is aso a range where the use vibration absorbers may be preferabe to other passive treatments, such as constrained ayer damping, owing to size and performance requirements. 9

20 Chapter 1. Introduction For a singe degree of freedom system, when an absorber with natura frequency ω n is attached to a system, a transmission zero, or antiresonance in the resuting system response is found at this natura frequency. σ/ st µ=0.05 ξ=0 a 6 ξ=0.1 4 M N p ω 11 ω n 1.0 ω 1. ω a /ω 0 6 µ=0.05 ξ=0 4 Composite system with absorber M b ξ=0. N c ω 11 ω n p ω Figure 1. Resonance Curves of the Dynamic Vibration Absorber ω a /ω 0 10

21 Chapter 1. Introduction Iustrated in figure 1. is a typica frequency response of the primary system (a) and the composite system incuding a dynamic absorber (b). The ampitude vaues are the ratio between acceeration eves σ taken from the primary system and the input eves st entering the primary system. When an absorber with a natura frequency ω n is attached to the system, the singe resonance peak of the primary system is repaced by two resonances on either side of the origina resonance; the origina resonance has now been repaced by an antiresonance. This basic physica behavior has great practica potentia. For eampe, if ecitation is provided by machinery operating near frequency ω n, vibration can be reduced by attaching the absorber so that an antiresonance p, is created near that frequency. Observe from curve (b) in the ower pot in figure 1. that the combined primary system and absorber has two resonance peaks, paced in frequency on either side of the absorber-induced antiresonance. The greater the ratio µ of absorber mass to the system mass, the further apart the resonance frequencies ω 11 and ω wi be. Typica vaues for the absorber mass to primary mass ratio, µ, range from 0.1 to 1.0. For smaer vaues of µ, the sensitivity of the absorber to tuning increases. Line (c) shows the effects of damping within the absorber. With damping present, the eve of attenuation is ess at the resonance frequency, p, but the range over which attenuation in the vibration response is provided is now increased. The optima amount of damping present in the absorber system is achieved when the points M and N are at equa ampitudes. A the resonance curves for a dynamic absorber pass through these two points. For damping which is ecessive, however, it is possibe for the absorber to move in phase with the primary structure, thus increasing its ampitude. 11

22 Chapter 1. Introduction 1..5 Anaysis of Vibration in Base Structure - Simpy Supported Beam The eperimenta tests to be discussed ater in this thesis are undertaken on a simpy supported rectanguar cross-section beam. The resonance frequencies of the modes of the beam are adjusted by adjusting the aia position of the supports. This is iustrated in figure 1.. The natura frequencies of the beam are adjusted by moving the aia ocation of the supports. Figure 1. Changing the natura frequency of a beam The purpose of constructing the test rig such that its resonance frequencies can be adjusted is to test the effectiveness of the absorber over a range of frequencies. When the absorber is mounted on the beam, the resonance of the beam wi be adjusted such that the span in which the absorber is operating wi be within the same proimity as the resonant modes of the beam. This procedure wi aso aow for the different absorber prototypes (which are constructed for different operating ranges) to be tested on the same rig. 1

23 Chapter 1. Introduction 1..6 Governing Equations for Base Structure The governing equation for the natura frequency of a simpy supported beam is given by (Thomson): ω = n n = ( n EI ρa ) EI ρa For the rectanguar uniform beam, E = Pa, ρ = 7800 kg/m. The second moment of inertia, I = (bh )/1. The vaue () depends on the boundary conditions of the beam. Typica end conditions are given in the tabe 1.1 beow (Thomson). 4 Tabe 1.1 Beam Configuration ( 1 ) ( ) ( ) Simpy Supported Cantiever Free-Free Camped-Camped Camped-Hinged Hinged-Free For a beam, b = 0.05m, h = 0.05m, = 1.15m (initia setup of the test rig), I = =.6 10 m For a simpy supported beam, () = 9.87 for the fundamenta mode ω n = = 40rad / s 9 4 f = 64Hz For a free-free beam, () =.4 for the fundamenta mode ω n = = 190rad / s f = 06Hz 1

24 Chapter 1. Introduction The vaue obtained from eperimenta procedures shoud ie in between camped-camped and simpy supported vaue, as the end conditions of the actua beam are somewhere between the two. 14

25 Chapter. Dynamic Absorber Using Encosed Air CHAPTER DYNAMIC VIBRATION ABSORBER USING ENCLOSED AIR.1 INTRODUCTION This chapter considers the design of a dynamic vibration absorber with a variabe air voume backing. The foowing work is undertaken: 1. Conceptua design of a first prototype is done to test the physica mechanism of using air as a variabe spring component.. A second prototype is constructed to address the probems found in the first prototype. The amount of variation in stiffness is investigated for this mechanism.. The second prototype is tested, and further refinement is made to this design. 4. Resuts from the tests are anaysed. 15

26 Chapter. Dynamic Absorber Using Encosed Air. FIRST PROTOTYPE The first absorber proposed here can be epained with reference to figure.1. The impetus behind the concept is that for some appications (particuary vehice appications), the absorber must be as ight as possibe. To be of practica use, the design must incorporate a means of adjusting the absorbing frequency; since the absorber mass is often difficut to adjust, the stiffness in the spring must be the variabe component. piston encosed air absorber mass ight spring attachment rod to primary structure Figure.1 - Dynamic Vibration Absorber using encosed air as a spring variabe Referring to figure.1, the absorber consists of a cyindrica she which contains an absorbing mass. This mass is supported on a ight spring which sets the equiibrium dispacement of the mass. Variabe stiffness is provided by an adjustabe encosed air voume above the mass. Note that the air voume is not pressurised. 16

27 Chapter. Dynamic Absorber Using Encosed Air The spring provides cearance for the mass to dispace in both positive and negative directions aong the vertica ais. The piston on the top of the cyinder moves to adjust the air voume; varying this voume wi vary the stiffness component of the absorber resonance. This piston can be connected to a inear actuator which wi be inked to a feedback contro signa. The absorber is fied onto the primary system using a rod protruding from the bottom of the she. To reduce the damping caused by the friction between the moving mass and the interna was, a ubricant wi be used. To prevent the interna mass from jamming inside the cyinder, it is important that the thickness of the mass not be too sma; if this thickness is sma, then the mass can tip to one side during osciations and jam. The overa ength of the absorber can be made smaer by reducing this thickness and inserting a rod through the mass to keep it in ine. However, this wi ony increase the damping due to friction, thus reducing the overa performance of the absorber. As this is the first prototype, carbon stee wi be used to construct the she and absorber mass. Mass considerations wi be taken into account in the second prototype. For the initia testing of this prototype, it was found that the mass of the she was ecessive. This added consideraby to the equivaent mass of the secondary system (the secondary system wi be detaied ater). Thus, the absorber mass was ineffective in controing the vibration the secondary system. In the design of the second prototype, the mass of the she wi be an important factor. 17

28 Chapter. Dynamic Absorber Using Encosed Air. SECOND PROTOTYPE The second prototype was designed with the aim of reducing the overa weight of the absorber. If the mass of the she is heavy, then the mass of the absorbing mass wi have to be increased to ensure an acceptabe toerance eve between ω 11 and ω. The she of the second prototype was buit using Perspe. This aso aows the user to see inside the device whie it is in use. The optima dimension of the encosed air voume was determined using the foowing equation. K = A P V Where K = stiffness provided by the encosed air (N/m) A = cross sectiona area of the cyinder (m) P = pressure inside the encosed voume (N/m ) V = encosed voume (m ) The objective was to contro the parameters r (radius of the cyinder (m)), and h (height of the encosed voume (m)), such that the variation in the stiffness provided by the device is maimum (ie. to maimize the device sensitivity). Arbitrary vaues of the radius are first chosen to gain an idea of the overa behavior of the stiffness. These can be shown in figure.. Variation in Stiffness When Changing the Height of the Encosed Voume radius = 0.05 m Stiffness (N/m) Height of the Encosed Voume (m) stiffness,k Figure. Variation in stiffness when changing the height of the encosed voume 18

29 Chapter. Dynamic Absorber Using Encosed Air It can be seen that as the height of the encosed voume is reduced to 10mm the stiffness increases substantiay to 70 kn/m. If the height of the voume is retained at 100mm then the stiffness reduces to 5kN/m. Note that this is ony the idea case where the air is competey seaed in the voume. In practice, the imperfection in the seas and the diaphragm woud imit this range in stiffness. The design of the second prototype is shown in figure.. 19

30 Chapter. Dynamic Absorber Using Encosed Air Hande Pis ton Hoe to equaize pressure during piston movement Copper Screw Air H oes Rubber O Ring 6 mm thick perspe wa 100 h 1 Stee stee ring ring camp camp (to (to hod hod thein the diaphragm). Fat fat screws ( aow (which the aows piston to the c to come within 5mm of the mas come within 5mm of the mass) Diaphragm fange 8 Mass is given by 1.0 kg. mass This incudes must be the 1.0 mass kg of (the mass stee bock is critica, and camped not the he spring ring. must have stiffness k= Spring stiffness if given by k=100n/m Figure. Design of the second prototype 0

31 Chapter. Dynamic Absorber Using Encosed Air Note that with the second prototype, a rubber diaphragm is used to encose the voume of air above the absorber mass. If the air is competey seaed from escaping from the space during adjustment of the voume, then the effort required to move the voume-adjusting piston woud be ecessive. As a resut, a singe O-ring was chosen for the piston sea. With this rather poor sea, the air is sti abe to escape when the piston is moving up or down, as the rate of this vertica movement is sma. Whie the piston is stationary, the pressure created by the vibrating mass is maintained within the encosure. It is recognized that this arrangement wi pace a ow-frequency imit on the operation of the device, as the pressure wi equaize over time (simiar to the equaization of charge in a piezoeectric device pacing a ow-frequency imit on its operation). The equaization effect is achieved incorporating a hoe into the piston (as shown in figure.). However, it is epected that the device wi sti be abe to function in the target span of Hz. 1

32 Chapter. Dynamic Absorber Using Encosed Air.4 EXPERIMENTAL RESULTS.4.1 Frequency Range of Second Prototype The natura frequency of the absorber is controed by adjusting the piston verticay at specified distances. The figure beow shows the change in natura frequency attainabe when the device is attached directy to a shaker. A force transducer is ocated at the point of attachment, and an acceerometer is ocated on the absorbing mass, to provide the transfer function. 0 Ampitude (db re 1v=10N/m/s^) ω n = Hz ω n = 5Hz ω n = 7 Hz Frequency (Hz) 1cm 4cm 8cm Figure.4 Transfer function of second prototype Figure.4 shows that the natura frequency varies over a range from Hz, when the height of the encosure is at 8cm, to 7 Hz, when the height is at 1cm. The absorber mass decreases in ampitude by 6 db as the voume height is decreased from 8cm to 1cm. This is in accordance with the theoretica anaysis. As the voume of the encosure decreases, the amount of compressibe air becomes ess. Hence, with sma vaues the rate of compression per unit dispacement of the piston is much greater when the voume is reduced, and so the dispacement wi aso be reduced.

33 Chapter. Dynamic Absorber Using Encosed Air.4. Identified Probems with the Second Prototype The range in natura frequencies obtained with the absorber device was ess than that predicted in the numerica anaysis. This is ikey due to a number of factors. First, the encosed air voume was assumed to be idea, such that the air was neither escaping nor entering into the encosure whie the mass is osciating. In practise, imperfections in the rubber o-ring and the difficuties in cutting the Perspe to a perfect circe wi aow for some air to move between the encosed voume and the eterna environment. This wi reduce the maimum pressure attainabe when the air is compressed by the moving absorber mass, thus reducing the span of natura frequency when the piston is in different positions. Second, the diaphragm which supports the absorber mass may not be competey uniform after the manufacturing process. The diaphragm used in the device was initiay a fat piece of rubber which was heat treated and curved into a membrane. Because of this, the mass which sits on top may sti osciate eccentricay, possessing a moment about the z-ais. This can be iustrated in figure.5. +M. (a) (b) Figure.5 Identified probem with absorber mass

34 Chapter. Dynamic Absorber Using Encosed Air This probem coud be minimised by manufacturing a rubber membrane via a moud, but this woud increase the cost of the device substantiay. The aternative was to construct a siding rod through the absorber mass, so that the mass woud move aiay aong the y-ais without rotation (figure b). However without constant ubrication, the frictiona damping between the rod and the mass woud affect the performance of the absorber at resonance. Figure 1. shows the decrease in attenuation obtainabe when there is ecessive damping present in the absorber. A third possibiity is that the rubber diaphragm wi actuay distort under the pressure of the compressed air voume, and so effectivey pace an upper imit on the compression reached. 4

35 Chapter. Dynamic Absorber Using Encosed Air.4. Comparison of Eperimenta with Theoretica Vaues The graph beow shows the comparison of the eperimenta vaues for the stiffness of the absorber with the theoretica vaues. The vaues shown are for the encosed air voume, and does not incude the stiffness vaue provided by the supporting spring (k spring = 100 N/m). Varying Stiffness with Changing Encosure Height Stiffness (N/m) radius = m Height (m) Theoretica Eperimenta Figure.6 Comparison of eperimenta and theoretica stiffness vaues Figure.6 shows that the eperimenta data corresponds to the theoretica vaues at piston heights above 6cm. When the piston is beow this distance, the theoretica vaues increase at a rate faster than the eperimenta. This is most ikey because the absorber is unabe to hod the pressure. The theoretica stiffness (N/m) vaue is given by, k = ω nt.m eq where m eq (kg) = absorber mass ( + mass of diaphragm and stee cap) + spring mass /. ω nt (rad/s) = the theoretica natura frequency cacuated for an idea case. 5

36 Chapter. Dynamic Absorber Using Encosed Air And the eperimenta stiffness vaue is given by, k = ω ne.m eq where ω ne (rad/s) = the eperimenta natura frequency ω d = 1 ζ ω d (rad/s) = damped natura frequency ξ = damping coefficient measured from transfer function. 6

37 Chapter. Dynamic Absorber Using Encosed Air.4.4 Identified Probems with Prototype Using Rubber Diaphragm The damping introduced by the diaphragm was found to be ecessive; ς 0.7. This woud resut in the absorber mass osciating in phase with the primary mass. For a tuned and damped absorber, the reaction force is ideay 90 out of phase with the dispacement and the acceeration at the point of interface. Damping coud be decreased by reducing the thickness of the diaphragm. However, this wi imit the maimum attainabe pressure of the encosed air, as a thin diaphragm wi buge under air pressure. Performance of the absorber is sensitive to mounting, as the absorber mass is prone to rotationa motion as we as dispacement. The first attempt of mounting the absorber such that the mass was osciating horizontay resuted in a frequency function producing two peaks. The second of which coud be attributed to the moment action on the mass. The aternative method of mounting the absorber such that the motion was vertica, eiminated the two peaks. In order to provide a sea for the encosed air, the piston acting on the encosed voume has a rubber o-ring around the edges of the piston. Without any form of ubrication, this created ecessive friction whie moving the position of the piston. This point must be taken when designing the contro system for the piston. The motor used to drive the piston must have sufficient power to overcome this friction, if ubrication is not to be used. It was anticipated that the absorber device woud be free of maintenance. There eists a ack of pressure due to distortion of the rubber diaphragm. A thicker diaphragm woud reduce the distortion of the diaphragm when it is under air pressure. 7

38 Chapter. Dynamic Absorber Using Encosed Air.5 INCORPORATING THE USE OF AN ALUMINIUM DIAPHRAGM Figure.4 showed that the span of the natura frequency of the origina absorber air voume was between Hz and 7 Hz. It was necessary to increase this range of frequencies for the absorber to be of practica use. A sheet of auminium curved into a membrane (as shown in Figure C-) was triaed as an aternative materia to rubber for the diaphragm. The thickness of the auminium was 0.5 mm. The operating frequencies of the absorber are increased to a greater span. This is iustrated Ampitude (db) Frequency (Hz) in figure.7. h=100mm h=50mm h=10mm Referring to figure.7, it can be surmised that use of the auminium diaphragm in pace of the rubber diaphragm significanty improves the variabe stiffness of the device. At a piston height of 100mm, the owest natura frequency recorded from the transfer function pot was approimatey 51 Hz. This compares favouraby with the previous vaue of Hz when the rubber diaphragm was 8

39 Chapter. Dynamic Absorber Using Encosed Air used. The maimum natura frequency obtained using the auminium diaphragm was at 75 Hz, with the piston at a height of 10mm. Heights beow this vaue were impractica as the maimum dispacement of the mass was 8mm. The workabe span of the absorber when using the auminium increased from 4 Hz to approimatey 4 Hz, greaty improving the usabiity of the device. Note that arger spans woud be possibe by maintaining a constant pressure above atmospheric. However, this woud increase the running cost of the absorber as pressure vaves and an intake of airfow woud have to be incorporated into the design. Construction costs woud aso increase as better quaity seas woud be required to ensure that the increased pressure is maintained. Varying Stiffness with Changing Encosure Height Stiffness (N/m) radius = m Height (m) Theoretica Eperimenta Figure.8 Variation between eperimenta and theoretica stiffness vaues for Auminium diaphragm Figure.8 shows the increase in stiffness achieved when the rubber diaphragm is repaced by an auminium diaphragm. The range of stiffness is increased from the previous case, and the maimum 9

40 Chapter. Dynamic Absorber Using Encosed Air stiffness vaue obtained is greater, when the height of the piston is at 0.01m. The vaues obtained for the auminium diaphragm are sighty higher than the predicted theoretica vaues, due to the auminium providing some of the stiffness, in addition to the supporting spring. 0

41 Chapter. Dynamic Absorber Using Encosed Air.6 IMPEDANCE OF THE ABSORBER The peak absorber impedance is given by the equation: H = a ma m. Q. ω a a a Where H a (N/m /s) = maimum impedance of the absorber. m a (kg) = absorber mass. Q a = quaity factor. ω a (rad/s) = natura frequency of absorber. The quaity factor is measured from the transfer function for the auminium diaphragm graph. The quaity factor is a measure of the damping within the system and is represented by the sharpness of the peak (db bandwidth) at the point of resonance. A ma f db A ma f 1 f n f Figure.9 Harmonic response curve showing bandwidth and haf power points From figure.7, at a resonance of 51.5 Hz, Q db f = f 51.5 = =

42 Chapter. Dynamic Absorber Using Encosed Air For m = 1.15kg, ω a = 51.5 π H a = ma = 71N / m / s The impedance of the absorber eceeds that of the primary structure. This is a requirement if the absorber device is to be effective. At resonance, when the absorber device has an impedance eceeding that of the primary structure, the ratio of the motion of the primary structure before and after absorber instaation can be shown to be (Von Fotow, 1994), u u b F ω a = H H b ( ω ) a ma a a ( ω ) Hb a = m Q ω a a

43 Chapter. Dynamic Absorber Using Encosed Air.7 PERFORMANCE OF THE ABSORBER To test the performance of the absorber, it was mounted onto a simpy supported vibrating beam. This test rig is aso used for the testing of the Dua Cantievered Mass Absorber, and wi be discussed in detai in section.5.4. The absorber was camped onto the beam, and a shaker is attached at a point aong the beam to ecite it, as shown in figure.10. The auminium diaphragm dynamic absorber was then manuay tuned unti its natura frequency coincides with the resonance frequency of the beam. A transfer function is then taken between the point of attachment of the absorber and the point near the ecitation force. Dynamic absorber acceerometer shaker simpe supports force transducer charge ampifier anayser Figure.10 - Eperimenta Setup for Auminium Absorber

44 Chapter. Dynamic Absorber Using Encosed Air The frequency response obtained from the eperiment is shown in figure.11 Frequency Response of Tuned Auminium Absorber on Beam 5.00 Ampitude (db re 1v = 10 N/m) resonance at 5 Hz w11 at 47 Hz w at 60 Hz Frequency (Hz) Without Absorber With Absorber Figure.11 Frequency response of Auminium Diaphragm Dynamic Vibration Absorber The frequency response on the beam shows that when the absorber is tuned to the resonance frequency of 5 Hz, there is an attenuation of 10 db. This eve of attenuation can be further increased if the damping within the absorber is reduced. The two peaks produced (ω 11 and ω ) are at 47 Hz and 60 Hz respectivey. 4

45 Chapter. Dynamic Absorber Using Encosed Air.8 SUMMARY In this chapter, two prototypes for dynamic vibration absorbers have been designed, using air as a spring variabe. The range in stiffness achievabe using this mechanism has been investigated and appropriate dimensions have been chosen for the design of the second absorber. The testing of the absorber has been undertaken and further probems have been identified and addressed: 1. The operating frequency range (which made the absorber impractica when using the rubber diaphragm) has been improved by using the auminium diaphragm incorporated into the design. The buging effect on the rubber diaphragm has been minimized with the auminium diaphragm, thus the maimum attainabe pressure within the encosed voume has been increased.. The overa mass of the absorber has been reduced, by construction of the second prototype from Perspe. This has reduced the equivaent mass required for the absorbing mass to contro the vibration. The absorber has been shown to be effective in obtaining a sma reduction in the vibration of the test rig, when correcty tuned. 5

46 Chapter. Dua Cantievered Mass Adaptive Absorber CHAPTER DUAL CANTILEVERED MASS DYNAMIC VIBRATION ABSORBER.1 INTRODUCTION Studies directed at assessing the effectiveness and practicaity of the first absorber, which used air as a variabe spring, indicated two major probems with the design: 1. A imitation in the range of frequencies over which the device coud be used, imposed by materia imitations (particuary in the case of the rubber diaphragm, which buged under pressure). A high eve of damping inherent in the arrangement As such, a second improved absorber design wi be investigated. Torsiona vibration absorbers have been used for many appications, such as reduction of vibration in aircraft fuseages. An adaptation of this idea wi be treated as a repacement for the initia prototype. Referring to figure.1, a cantievered mass arrangement consisting of two masses (the absorber mass) inked by a beam which is fied at the centre wi be used here as the absorber. The fied centre is attached directy to the vibrating surface. cantievered rod absorber mass 1st mode nd mode Figure.1 The first modes of the cantievered absorber 6

47 Chapter. Dua Cantievered Mass Adaptive Absorber The effective motion of the each mass is that of a cantiever. It is possibe to tune the absorber using some adaptive strategy, by moving the masses aong the beams. This resuts in a change in natura frequency of the device and thus a change in the notch (absorbed) frequency. Theoretica anaysis of the cantievered absorber can be undertaken using the impedance method (Punket, 1958). The motion of a structure can be characterised by its impedance, epressed as the ratio of an eciting force at a particuar frequency to the resuting veocity at another point on the structure at the same frequency. Once the impedance of a structure is known, prediction of the motion of the structure due to other forcing functions is possibe. It is possibe (and possiby more accurate) to design this absorber using finite eement anaysis, undertaken here using the computer package Ansys. With this, the resonance frequencies of the first 5 modes can be found. The Ansys program is incuded in appendi A. Later, the absorber is to be attached to a uniform beam undergoing transverse vibration. The design of the test rig is discussed in section.5.4. The aim is to minimise the oca vibration at a sensor ocation on the beam. 7

48 Chapter. Dua Cantievered Mass Adaptive Absorber. INITIAL THEORETICAL ANALYSIS OF CANTILEVERED ABSORBER USING DISCRETE SYSTEM METHOD As a simpified first anaysis, the absorber system is assumed to be composed of discrete systems. The absorber mass at the end of the rod is assumed to be one system, and the rod itsef is another. If the damping present in the system is negected, Dunkereys equation can be used for anaysis. Dunkereys equation states that for a muti degree-of-freedom system, the inverse of the fundamenta frequency (Seto, 1964), ω 1 1 = ω ω ω = n nn i= 1 ω 1 ii where ω 11,ω ω nn are the corresponding natura frequencies of each component of the system acting aone D beam =0.01m u(,t) D a = 0.08m W beam W a =0.105m w=0.05m L= m Figure. Dimensions used for Discrete System Anaysis For the natura frequency, ω b of a cantievered beam of mass, m 1, EI ω b = 1.7. m 1 8

49 Chapter. Dua Cantievered Mass Adaptive Absorber For the absorber device, two rods are in parae, with the mass attached at =L. As k = ω m, EI k = 1.7 The tota stiffness produced by the rods in parae is, k t = k 1 + k EI k = 1.7 EI = 5.4 EI ω b = 5.4 m For the natura frequency, ω a of a cantievered beam of negigibe mass with a concentrated mass attached at one end, EI ω a = m Using Dunkereys equation, ω 1 1 = ω 1 b 1 + ω a mb ma = + 5.4EI EI mb + 5.4ma = 76.EI 76.EI ω1 = m + 5.4m b a ω = 76.EI m 5.4m b + a 9

50 Chapter. Dua Cantievered Mass Adaptive Absorber For the materia properties of stee, E = 07 GPa, (Moduus of Easticity) I = πr 4 /4 = π(0.005) 4 /4 = m 4. (Second Moment of Area) ρ = 7800 kg/m. (density) m b = π(0.005).(0.105).(7800). = 0.19 kg m a = π(0.04).(0.05).7800 = 1.96 kg L = = 0.1m ω = = 65.0rad / s = 4.Hz This vaue is ower than the vaue obtained for the eperiment (46.5 Hz) at L=0.105m, which is common for such anaysis. 40

51 Chapter. Dua Cantievered Mass Adaptive Absorber. THEORETICAL ANALYSIS OF THE CANTILEVERED ABSORBER USING CONTINUOUS SYSTEM THEORY To improve upon the previous simpe anaysis, the foowing is a derivation to cacuate the natura frequencies of the dua cantievered mass device. The device is first assumed to be symmetric about the centre ine y-ais, and modeed as a cantievered cyindrica beam with a uniform mass attached to the end. y U(,t) 0.08m F beam F mass L = 0.1m 0.05m Figure.4 Dimensions used for continuous system anaysis Firsty, the frequency equation is found using the method of separation of variabes (Tse et. a.) The ongitudina dispacement of the mass is assumed to be of the form, u(,t) = φ().q(t) (-1) where φ() is a function invoving ony, and q(t) is a function invoving ony time, t. u V V + d d M M M + d V d Figure.5 Shear and moment sign conventions 41

52 Chapter. Dua Cantievered Mass Adaptive Absorber Appying Newton s aws in the atera direction gives (Mc Caion, 197), V + V V + u d = m. d. t u V m d = ( V + d) + V t For m= ρa= mass/unit ength (kg/m). where ρ = density (kg/m ) and A = cross sectiona area of beam (kg ) u m t V = (-) Taking the moment about a point on the right side of the eement, + M right _ side = 0 M M + M + d Vd = 0 V M = Using this resut in Equation (-) gives, M u + m t = 0 Using the Bernoui-Euer beam theory, the appied moment is reated to the radius of curvature by, u EI = M, (-) Combining Equation (-) and Equation (-) forms the beam equation for atera vibration, u m t = u EI = + t m 4 u. 4 u EI = 0 (-4) This is the equation of motion for the eement, d. 4

53 Chapter. Dua Cantievered Mass Adaptive Absorber If the wave equation, (-1) is substituted into Equation (-4), this gives 4 d q EI d φ + q = 0 4 dt m d u φ u t q t 4 4 φ As = q and = φ 4 4 EI m 4 1 d φ.. = 4 φ d 1. q d q dt (-5) As the right hand side of Equation (-5) is independent of, and the eft hand side is independent of t, then they must both equa a constant, as the equation is true for a vaues of and t. So et, 4 EI 1 d φ.. = w 4 m φ d 4 d φ mω 4 d EI φ = 0 (-6) where ω is a constant Simiary, as 1 d q. = ω q dt d q + ω q = 0 dt (-7) 4

54 Chapter. Dua Cantievered Mass Adaptive Absorber Equations (-6) and (-7) form two inear differentia equations. The soution of each of these can be shown to be, φ( ) = C sin + C and q( t) = C 5 1 sinωt + C 6 cos + C cosωt sinh + C 4 cosh (-8) (-9) So using Equation (-1), ( C sin + C cos + C sinh + C cosh )( C sinωt C cosωt) u(, t) = (-10) by etting m = EI 4 ω This is the genera soution to a beam undergoing atera vibration. To find the frequency equation for the cantievered absorber mass attached to the end of the cyindrica uniform beam, the boundary conditions must be known. For the purpose of this study, the beam is assumed to be rigidy camped to the wa of the auminium bock. (Thus u(,t)=0). The boundary conditions, however, are normay difficut to impement eacty in practica appications. Common boundary conditions for atera vibration in beams are shown in figure.6 (Tse et a,1978). 44

55 Chapter. Dua Cantievered Mass Adaptive Absorber 45 Left Boundary Conditions Beam End Conditions Right Boundary Conditions Fied Free Spring Load 0 = u EI u(,t)=0 u(0,t)=0 m m Inertia Load Simpy Supported = 0 u = 0 u u u u(0,t)=0 u(0,t)=0 0 = u EI 0 = u EI 0 = u EI 0 = u EI 0 = u EI u k u EI t = u k u EI t = u k u EI t = u k u EI t = 0 = u EI t u m u EI = 0 = u EI t u m u EI = component sope u _ = component moment u _ = component shear u _ = Figure.6 Boundary conditions for atera vibration in a beam

56 Chapter. Dua Cantievered Mass Adaptive Absorber So for a beam which has a camped end on the eft, and an inertia oad on the right hand side, the foowing boundary conditions appy. At the fied end (=0), the sope and defection are zero, u(0, t) = 0 u(0, t) = 0 (-11) (-1) At the mass oaded end (=), the moment is zero, M = or u EI 0 = 0 (-1) And the shear force at this end is equivaent to the force acceeration of the mass, V u = m t or u EI u = m t (-14) Using the first boundary condition, and using Equation (-11) in (-8), (0, t) = 0, φ(0) = C cos (0) + C 4 cosh (0) = 0 as cos(0) and cosh(0) 0 C + C 4 = 0 Using the second boundary condition, Equation (-1), (-15a) As then u( ) = ( C u(0, t) = ( C 1 cos C 1 cos (0) + C sin + C cosh + C cosh (0)) = 0 4 sinh ) C 1 + C = 0 (-15b) 46

57 Chapter. Dua Cantievered Mass Adaptive Absorber 47 Thus, using Equations (-15a) and (-15b) in (-8) gives Using the third boundary condition, Equation (-1), Now we assume that q(t) has the form, (-16) ) cosh (cos ) sinh (sin cosh sinh cos sin ) ( C C C C C C φ + = + = ( ) ( ) ( ) ( ) ( ) ( ) ) sinh (sin ) cosh cos ( sinh cosh sin cos ), ( ) cosh (cos ) sinh (sin cosh sinh cos sin ), ( ) sinh sin ( ) cosh (cos sinh cosh sin cos ), ( C C C C C C d d t u and C C C C C C d d t u and C C C C C C d d t u φ φ φ + = + = = = = = + = = = (-17) ( ) 0 ) cosh (cos ) sinh (sin 0, 0 ) cosh (cos ) sinh (sin 1 1 = = C C EI as C C EI t t t u and t t t u and t t u t t q ω φ ω ω ωφ ω φ ω )sin ( ), ( )cos ( ), ( )sin ( ), ( sin ) ( = = = =

58 Chapter. Dua Cantievered Mass Adaptive Absorber 48 From Equation (.16), Using the fourth boundary condition, Equation (-14), (-18) t C C t t u ω ω sin ) cosh (cos ) sinh (sin ), ( 1 + = ( )( ) ( )( ) ( ) ( ) [ ] m EI m m EI EI gives sides both from C by Division C C m C C C EI C C Equation from C m C C C C EI t C C C m C t C C C C EI t t u m t u EI ω ω ω ω ω ω ω ω cosh sinh cosh sin sinh cos cos sin cosh sinh cosh sin sinh cos cos sin ) sinh sinh sin sinh sin (sin ) cosh cosh cos (cos ) sinh )(sin cosh cos ( ) sinh )(sin cosh (cos ) sinh )(sin sinh (sin ) cosh )(cos cosh (cos, ) cosh (cos ) sinh (sin ) sinh (sin ) cosh (cos ) sinh (sin ) cosh (cos ) sinh (sin ) cosh (cos ) sinh (sin ) cosh (cos (.17), ) cosh (cos ) sinh (sin ) sinh (sin ) cosh (cos sin cosh sinh cos sin sin sinh cosh sin cos ), ( ), ( = = = = + = = + =

59 Chapter. Dua Cantievered Mass Adaptive Absorber EI ω m [(sin + cos ) + cos cosh + (cosh sinh ) ] = cos sinh + sin cosh EI ω m [ 1+ cos cosh + 1] = cos sinh + sin cosh EI ω m EI [ + cos cosh ] + cos sinh sin cosh = 0 ( 1+ cos cosh ) + ω m( cos sinh sin cosh ) = 0 (-19) Hence for a beam with the eft end camped and the right end inertiay oaded, the frequency equation is given by Equation (.19). Note that the frequency equation can then be used to produce the moda shapes for beam under vibration. This is done by cacuating the eigenvaues, n for n=1,, n. The eigenvaues can be found by computation agorithms invoving iterative methods. 49

60 Chapter. Dua Cantievered Mass Adaptive Absorber.4 FINITE ELEMENT ANALYSIS OF THE ABSORBER In order to gain an accurate prediction of the modes of the absorber, a numerica anaysis using finite eements was used. This anaysis aows determination of the resonance frequency of each mode, which wi be a function of the ocation of the mass aong the two shafts. Two main types of meshing eements (in Ansys) were used to mode the absorber device. For the hoow square housing in the centre (which attaches to the vibrating surface), she6 was used. The suitabiity of this eement was based on its bending and membrane properties. For the modeing of the two shafts and absorber masses, SOLID45 tetrahedra was used. More detai can be found within the Ansys manuas. The boundary conditions were then programmed by fiing (in a directions) one side of the square housing. The fina mode of the device is shown in figure.7. Note that one of the shafts wi be threaded and the other is smooth. A controer wi be attached to the threaded rod. When it is turned, the end masses wi move in or out. ω n=1 = 45.5 Hz at L=0.1m Figure.7 A meshed mode of the Dynamic Absorber 50

61 Chapter. Dua Cantievered Mass Adaptive Absorber.5 EXPERIMENTAL SETUP.5.1 Introduction The eperiments undertaken on the dua cantievered mass absorber are performed in three setups: 1. Eamining the resonance frequencies of the absorber aone when it is directy attached to a shaker.. Eamining the resonance frequency and vibration attenuation achieved by the absorber when it is attached to the simpy supported beam. At this stage the absorber is tested without the contro system.. Eamining the effectiveness of the contro system and rate of adaptation when the absorber is attached to the beam. 51

62 Chapter. Dua Cantievered Mass Adaptive Absorber.5. Resonance frequency eperimenta measurement of absorber (aone) To determine the natura frequency of the absorber (aone), the device was attached directy to a shaker and subjected to a variabe frequency sinusoida input (sine sweep). The transfer function was then taken between the point of attachment and the absorber mass. The arrangement is shown in figure.8. shaker force transducer absorbing mass (protruding out of page) stepper motor acceerometer anayser charge ampifier Figure.8 Measuring the resonance frequency of the absorber aone In the setup the absorber device is acting in a bending motion. That is, it acts as a uniform beam which has a force input in the centre, and is inertiay oaded at both ends. The resut obtained here wi be different from the resut obtained when the absorber is attached to the beam. 5

63 Chapter. Dua Cantievered Mass Adaptive Absorber.5. Eperimenta Resuts The pot of the transfer function obtained from this setup is shown in figure.9. Note that three different curves are given, for the mass ocated at distances of 9mm, 98mm, and 105mm from the edge of the absorber base. (refer to figure.10 for datum point). This eperiment verifies that the natura frequency of the absorber changes when the position of the mass aong the rod changes. However, the vaues for the natura frequencies shown here on figure.9 do not span the fu frequency range possibe with the absorber. The fu frequency range wi be found when the absorber is ater attached to the beam. Comparison of Absorber Natura Frequency for Given Mass Distances Aong the Shaft Hz 4 Hz 47 Hz 75 Hz 78 Hz 8 Hz Ampitude (db) Frequency (Hz) mass at 105mm mass at 98mm mass at 9mm Figure.9 Transfer function of the absorber 5

64 Chapter. Dua Cantievered Mass Adaptive Absorber The absorber is tuned according to the number of steps turned by the stepper motor. The datum for the position is given by 0 mm at the point when the mass is up against the she wa. This is iustrated in figure.10. contro signa =0 Figure.10 Positioning for the number of turns made by the motor The worm gear buit within the device aows the absorber mass to be precisey tuned to the number of steps of the motor. Starting from the datum, at =0, the number of steps is 0. For =9mm, the number of steps required for the motor to turn is 80,000. For =98mm the motor has to turn times, and for 105mm the motor is turned times. The peaks obtained from these vaues of turns is shown in the transfer function pot of figure.9 above. It can be seen that at turns the natura frequency of the absorber is at 47.0 Hz. At turns the natura frequency decreases to 4.0Hz and for turns, the corresponding vaue is 40.0 Hz. These vaues are ower than those obtained when the absorber is fied to the beam (discussed ater). The theoretica anaysis for the absorber when it is undergoing a bending motion is done in section.. As the natura frequencies correspond to the number of steps, it is then possibe to tune the absorber to any frequency within its range, by resetting the mass to its datum position whenever the device is mounted onto a structure. 54

65 Chapter. Dua Cantievered Mass Adaptive Absorber The pot of the transfer function in figure.9 shows that the second mode of the absorber is around 80Hz; ω 80k =75.0Hz, ω 100k =78.0.0Hz, ω 10k =8.0Hz, for turns of 80000, and respectivey. The second mode of vibration of the absorber is a twisting action as shown in figure.1. As this is not a bending motion, this mode is not effective in controing the transverse vibration in the beam. It is possibe that this second mode may contribute to the moda ampitude at frequency ω (as shown in figure 1.) when the absorber is mounted on the beam. However, this wi not affect the attenuation achieved by the absorber at the frequency of ecitation. 55

66 Chapter. Dua Cantievered Mass Adaptive Absorber.5.4 Eperimenta setup with the absorber on the beam In the second setup, the absorber device is acting as two cantievers with a concentrated mass attached to the end of each. Each cantiever acts as a beam which is inertiay oaded at one end and camped at the other. The theoretica anaysis of this has been described previousy in section.. The shaker is attached to the beam such that it is abe to move aong the beam for different point source ocations. This wi assist in the study of the effect of source ocation on the frequency response of the absorber. The absorber itsef is attached to the beam via camps. Simiary, this aows the ocation of the absorber to be changed to produce a favourabe output signa from the acceerometer. It is assumed that the fundamenta mode of vibration contributes the most towards the overa vibration in the beam. The absorber is ocated sighty off-centre on the beam, to aow it to respond to the first and second modes of vibration. However, the second mode is not aimed to be controed by the absorber in this eperiment. 56

67 Chapter. Dua Cantievered Mass Adaptive Absorber The setup is iustrated in figure.11. stepper motor acceerometer shaker simpe supports force transducer absorbing mass charge ampifier anayser bandpass fiters (ater impemented using fiter functions written on PC) Figure.11 - Eperimenta setup for the absorber attached on the beam The input force is measured by a force transducer paced between the shaker stinger and the point of attachment on the beam. The acceeration eve of the absorber mass is measured by an acceerometer paced on top of the mass. This acceerometer is ocated on the horizonta pane tangentia with the top of the absorber mass. This is to ensure that the acceeration recorded by the acceerometer does not incude any ongitudina movement of the mass aong the horizonta ais. 57

68 Chapter. Dua Cantievered Mass Adaptive Absorber The transfer function is cacuated by dividing the input eve from the force transducer over the input eve from the acceerometer, thus giving a ratio F input /a absorber mass. The foowing properties were used for the simpy supported beam. Property Vaue Length, 1.15m Height,h 0.05m Width,b 0.05m Moduus of Easticity, E Pa Density, ρ 7800 kg/m Second Moment of Inertia, I bh /1 The ength of the beam was arbitrariy chosen such that the resonance frequency of the beam woud be near the operating range of the absorber. This vaue was cacuated using Euer equations for beams. The fundamenta frequency for a simpy supported beam is given by, EI ω1 = ( ) for ( ) = ρa For I bh = 1 = = m 4 and ρa = 9.75kg / m ω = = 40rad / s = 64Hz This measured vaue for the beam resonance (at this ength) is epected to be greater than this vaue. 58

69 Chapter. Dua Cantievered Mass Adaptive Absorber.6 CONTROL SYSTEM.6.1 Controer setup The aim of the work described in this thesis is to produce an absorber which is tunabe on-ine, to provide an aternative to the actuators currenty used in active vibration contro and structura / acoustic contro. Adjustment of the resonance frequency of the cantiever beam absorber can be made via adjustment of the ocation of the masses on the shafts. The hardware used to faciitate movement of the absorber mass in and out on the shafts consists of a stepper motor and a CIO-DIO (digita input-output) card. The stepper motor is connected to the main shaft of the mass by a set of worm gears. The digita input-output card is inserted into a standard ISA port on a Pentium PC running at 150 MHz. The PC is used to contro both the controer card and the data acquisition card. As wi be described in detai ater, the software used to contro the card was written in C++ using the Nationa Instruments package, Lab Windows. The fow of contro signas is iustrated in figure.1. 59

70 Chapter. Dua Cantievered Mass Adaptive Absorber Stepper motor controer unit acceerometer 1 (ocated on the absorber mass) acceerometer Input force CIO-DIO (855) card Charge ampifier NI-DAQ card P-150 channe input-output modue Figure.1 Contro Signas 60

71 Chapter. Dua Cantievered Mass Adaptive Absorber For the purposes of eperimenta anaysis of the absorber device (itsef, as opposed to the overa contro strategy), the stepper motor was initiay driven using a simpe program written in C++. The program is incuded in Appendi B. The stepper motor operates by a series of puses or steps. A step is triggered when a signa changing from 0v to 5v is detected by the motor. Thus by sending a series of puses ranging from 0-5v, the motor can be precisey controed in terms of movement through a number of steps. The program first sets the headecima base address of the CIO-DIO card to 1b0. Then according to user input, it sets the direction for the motor to turn by sending a word to certain addresses above its base address. A word consists of 8 bits which can be turned on or off, depending on the operation which is required by the stepper motor. A oop is then set up to generate the required number of puses to turn the motor a given number of steps. There are acceerometers used in the system; one ocated on the absorber mass to determine the natura frequency at which the absorber is tuned, and the second is attached to the beam to measure the ecitation frequency. 61

72 Chapter. Dua Cantievered Mass Adaptive Absorber.6. Tuning Agorithm Vibration absorbers are typicay tuned in one of two ways; either to a moda eigenfrequency which is some cause of concern, or to a disturbing frequency, which may vary over the whoe spectrum of the primary structure. In the first case, uness the primary structure undergoes some change in its interna properties, due to time or temperature effects, then there woud normay be no need for an adaptive absorber. In this work, the tuning aws wi be written to track the ecitation or disturbance frequency; this wi have more practica use. In designing the tuning agorithm for the absorber device, it is important to note the areas of deay in which absorber adaptation are normay reated (Von Fotow, 1994). Firsty, there is the ogic deay. This is the time required for the feedback contro system, in this case the computer program, to process any acquired data and give a response to the incoming information. Athough a P-150 computer is used as the main hardware for the fow of contro signa, it has to perform the simutaneous processing of the incoming and outgoing information. It is possibe that this wi contribute significanty to the overa deay of the contro system. With the presence of noise in the incoming signa, this deay wi be further increased, as more time wi be required for averaging, fitering and further iterative processes within the agorithm. Secondy, there is the actuation deay. This is the time taken for the absorber device to change its interna properties so that it is adapted to the changing environment. In this case, this is the time required for the absorber mass to move to a specified position which wi change its eigenfrequency to match the ecitation frequency. It is anticipated that this wi be the main area of deay, as this is the compromise which has to be made for a device which has a fine-tuning abiity. 6

73 Chapter. Dua Cantievered Mass Adaptive Absorber Finay, there is the dynamic deay. This is the time required for the device to change its state of vibration once it is tuned to those conditions of the primary structure. The dynamic deay is proportiona to the quaity, Q, of the absorber, and is often represented by a time constant (Von Fotow, 1994), τ dynamic Qa = ( periods) where τ dynamic = dynamic deay ep ressed as an eponentia π Q a = quaity factor of absorber It can be seen that the dynamic deay wi decrease if the damping coefficient is increased. However, this resuts in residua motion of the primary structure even when the absorber is perfecty tuned. Because of this, it is usua to decrease ony the first two forms of deay, and maintain the quaity of the absorber at a high vaue. Dynamic deay is normay considered negigibe compared to the ogic and actuation deays. The dynamic deay aso incudes the time required for the primary structure to respond to changes in the absorber. The program written to contro the input and output signas can be seen in Appendi B. It is written under the Windows NT4 environment using the Nationa Instruments package CVI (C for Virtua Instrumentation). Because this program has numerous buit in functions, the use of ow-eve Data Acquisition ibraries coud be incorporated directy into the program. The overa process for the feedback agorithm can be shown in figure.1. 6

74 Chapter. Dua Cantievered Mass Adaptive Absorber Input contro variabes (channe 1&, sampe rates, input gains). Input buffer size and initiaise buffers for data coection. Open graphica interface Initiaise Data acquisition card and Controer card. Begin data acquisition from channe 1 (dispacement) and input votage vaues into buffer 1. Begin data acquisition from channe (tracking frequency) and input votage vaues into buffer Scae vaues contained in buffer to engineering units. Perform FFT on buffer and cacuate maimum vaue and array ocation contained in buffer, and store as variabe. Perform averaging on Buffer 1 and store as variabe. Perform oop n number of times depending on amount of noise present in incoming signa Pot dispacement votage on graphica interface. Pot frequency response on separate graph. Trace dispacement votage to natura frequency of absorber Compare frequency of absorber to tracking frequency If tracking frequency of absorber is greater than absorber frequency by 1Hz then send p number of positive puses to contro card If tracking frequency of absorber is ess than absorber frequency by 1 Hz then send p number of negative puses to contro card Figure.1 Fow diagram for contro signas in tuning agorithm 64

75 Chapter. Dua Cantievered Mass Adaptive Absorber.6. The Linear Transducer The position of the absorber mass at any point in time is known by incorporating a inear transducer into the design of the device. This inear transducer consists of a rotationa pot which gives out a votage variation depending on the number of turns it has competed. A torsiona spring is buit into the transducer so that it wi recoi whenever the absorber mass is moving inwards aong the rod. The pot is connected to one of the masses via a string which is gued to the surface. The arrangement can be shown in figure.14. inear transducer stepper motor absorber mass connect to controer card connect to transducer ampifier Figure.14 - Location of the Linear Transducer The output of the transducer is passed through a power ampifier and the signa is converted to a DC votage and then into the NI-DAQ card. 65

76 Chapter. Dua Cantievered Mass Adaptive Absorber Figure.15 shows the reationship between the votage produced from the inear transducer and the natura frequency of the absorber. Variation of Natura Frequency with Dispacement Votage Frequency (Hz) Votage (v) Natura Frequency (Hz) Figure.15 - Variation of Natura Frequency with Dispacement Votage Figure.15 shows that there is a inear reationship between the votage output from the transducer and the natura frequency of the dynamic absorber. It is because of this inear reationship that a proportiona controer can be written into the tuning agorithm. The target frequency is measured from the sensor, which tracks the ecitation frequency. From this, a precacuated votage is known, which is the target vaue for the output votage from the inear transducer. The stepper motor is then actuated to move the masses in either direction, unti the vaues for both the cacuated and output votage coincide. The sensitivity of the absorber can be adjusted by defining the toerance between the cacuated and output votage. When the difference between the two votages is greater than the toerance, then the motor is actuated. 66

77 Chapter. Dua Cantievered Mass Adaptive Absorber.7 EXPERIMENTAL RESULTS FOR THE CASE OF THE ABSORBER MOUNTED ON THE BEAM.7.1 Performance of the absorber on a sinusoiday ecited simpy supported beam Figure.16 shows the transfer function obtained from the second setup in section w11 = 47.5 Hz Beam Resonance at 77.5 Hz w = 95.0 Hz Ampitude (db re 1v) antiresonace at 75.0 Hz Without Absorber Frequency (Hz) With Absorber Figure.16 Transfer function before and after absorber instaation Figure.16 shows the fundamenta frequency of the beam at 77Hz, without the absorber mounted. This vaue is higher than the cacuated vaue using the Euer equation, due to the discrepancies in the beam terminations. The boundary conditions achieved at both ends of the beam in the eperiment suggests a range between a simpy supported termination (f n=1 =64 Hz) and free-free (f n=1 =06 Hz). The scae is caibrated for 1 db reative to 1 N/m/s. 67

78 Chapter. Dua Cantievered Mass Adaptive Absorber When the absorber is mounted on the beam and sef-tuned to remove the peak, it can be seen that an antiresonance is produced near the frequency at which the resonance previousy occurred. This resuts in the ampitude of response at the target frequency being attenuated by approimatey 45dB. This indicates that the dua cantievered mass absorber, when correcty tuned has the capacity to substantiay reduce the vibration in the beam. 68

79 Chapter. Dua Cantievered Mass Adaptive Absorber.7. Variation in natura frequency of the absorber due to change in effective rod ength Iustrated in figure.17 is the change in frequency responses of the absorber / beam system as the absorber mass is moved aong the rod. The transfer function is taken between the source and at a point aong the beam as shown in figure.1 using acceerometer.the antiresonance produced had a range from 45 Hz up to approimatey 90 Hz. At ower haf of this range the antiresonances were sharper. This is due to the damping in the absorber being frequency dependent. When the mass is positioned 0mm from the base, the frequency of the absorber is at 88 Hz. At distances of 70 mm and 110 mm the frequencies are 65 Hz and 45 Hz respectivey Ampitude (db) Hz at 110mm 65 Hz at 70mm 88 Hz at 0mm -50 Frequency (Hz) Figure.17 - Comparison of anti-resonances produced due to change in absorber 69

80 Chapter. Dua Cantievered Mass Adaptive Absorber.7. Speed of Adaptation For tests in the frequency range of interest here, attachment of the absorber to the vibrating beam resuted in an attenuation of approimatey 0dB. (This was when the absorber was ocated in a position sighty off centre). The time varying signa from the sensor to measure the ampitude of the vibrating beam taken from an acceerometer ocated near the point of attachment can be seen in figure.18. Figure.18 Time signa for absorber to adapt to a changing ecitation signa Figure.18 shows the time required for the absorber to adapt to an ecitation signa. In this case, the absorber is initiay tuned to a frequency of 46 Hz. This frequency is chosen arbitrariy as it is in the ower end of the operating span of the absorber. The ecitation frequency is then increased to 7 Hz and at t=7sec, the controer program is run. The distortion which can be seen starting at t=7 sec is due to the sma vibration caused by the stepper motor itsef. This eterna noise does not interfere with the tuning agorithm since the resonance caused by the motor is at a frequency ω e = 50Hz (reca that there is a band-pass fiter on the sensing signa). 70