Natural Frequency Statistics of an Uncertain Plate

Transcription

1 THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MECHANICAL AND MANUFACTURING ENGINEERING Natural Frequecy Statistics of a Ucertai Plate Surasak Patarabuditkul Bachelor of Egieerig (Mechatroic) Jue 009 Supervisor Dr Natha Kikaid

2 Abstract This thesis is cocered about desigig a ucertai dyamic system to observe the effect of ucertaity to the system which has perfect symmetry. The steel circular plate was selected ad desiged to be able to stimulate a set of differet cofiguratio of ucertaity. The choice of equipmet ad procedure ad frequecy rage of iterest has bee selected. The qualitative measure of the will be the atural frequecy statistics usig two statistical measures; the statistical overlap factor ad the probability desity fuctio of the spacig betwee successive atural frequecies. The statistical overlap factor is the variatio i a atural frequecy from its mea value measured across the set of system with differet ucertaity. The probability desity fuctio of atural frequecy spacig was applied to each idividual to observe the Rayleigh distributio where it is a feature of the uiversality exhibited by structures with ucertai. The effect will be moitored to see if it is greater or lower as the frequecy is gettig higher. The outcome was cocluded that the effect of material stiffess permutatio usig alumiium bar was eglectable or very low. The uexpected effect of egative mass which holes iitially drilled to fix the alumiium bar to the steel plate was very domiat. The study also icorporated simulatio software to geerate the perfect system without ay egative mass ad other material variatio. The two resulted were compared, simulatio result verified the eglectable effect of alumiium bar. The simulatio study was cotiued to detect at what stiffess the system will be radom eough to produce Rayleigh distributio. ii

3 Statemet of Origiality I hereby declare that this documet cotais o materials previously submitted for ay degree or other award at UNSW or ay other istitutio. It is my belief that all work i this thesis is etirely my ow work, except where stated ad otherwise refereced, is my ow origial work. I coset for this thesis to be available for academics ad studets withi the Uiversity for loa ad database access. Sigature Date iii

4 Ackowledgemet Firstly ad importatly, I would like to thaks Dr Natha Kikaid my supervisor who gave me a chace to do this thesis. I am also appreciatig his guidace, kidess ad expertise throughout the project wheever I eeded. Secodly, I thak Mr Russell Overhall for his help with settig up the experimetal compoets of my thesis. His practical assistaces ad commets from his experiece were valuable. I eed to thaks my family members for their support ad motivatio to ecourage me to complete this thesis. I also specially metio Aruothai Ekkhosit with her help i ANSYS ad support, always. Lastly I caot forget to thaks my frieds Charles Sithu, Day Pradhaa ad Eric Poo for their commets ad helps. Thak you iv

5 Table of cotet Abstract...ii Statemet of Origiality...iii Ackowledgemets...iv Table of Cotets...v List of Figures...viii List of Tables...xii List of Symbols...xiii Chapter 1 Itroductio Literature Review...3 Chapter Vibratio Basics Modal Testig Measuremet Method Excitatio Fast Fourier Trasform...16 v

6 .3 Frequecy Respose Fuctio Plate Theory...1 Chapter 3 Statistical Measure Rayleigh Distributio Fuctio Probability Plot Statistical Overlap Factor...3 Chapter 4 Experimetal Desig ad Setup The Plate Alumiium Bar experimetal Setup ad Techiques Used Supportig Method Excitatio Double Hit Widowig Strikig Locatio Locatio of Accelerometer...4 vi

7 4..6 Coherece Measuremet Aalysig Summary of Equipmets Equipmet Setup Radomisatio the locatio of alumiium bars...48 Chapter 5 Results ad Discussios Experimetal Results Simulatio Results...53 Chapter 6 Coclusio Future Work...69 Referece...70 Appedix...70 vii

8 List of Figure Figure.1 Sigle degree of freedom system Figure. Free body diagram of sigle degree of freedom system Figure 3.1 Examples of Rayleigh Distributio Figure 3. Rayleigh distributio geerated from sample mea ad its histogram Figure 3.3 MATLAB radomly sampled data from Rayleigh distributio Figure 3.4 Rayleigh Probability Plot Figure 3.5 expoetial distributio Figure 3.6 Rayleigh probability plot of expoetial distributio Figure 4.1 Schematic diagram of circular plate Figure 4. - Steel Plate (Left) ad CAD drawig (Right) Figure 4.3 Alumiium Bar of.5 mm (Left) ad 1 mm (Right) Figure 4.4 Suspeded plate Figure 4.5 Hammer Force Spectrum Figure Sample Coherece Figure 4.7 Sample of frequecy respose Figure 4.8 Nyquist plot viii

9 Figure Equipmet setup Figure 4.10 umber assigmet Figure 5.1 Bare plate probability plot ad histogram Figure 5. - Sample of experimetal result probability plot ad histogram that ted toward expoetial distributio Figure Sample of experimetal result probability plot ad histogram that ted toward Rayleigh distributio Figure sample of experimetal result probability plot ad histogram that ted toward Rayleigh distributio betwee 00Hz to 3000Hz Figure Sample of experimetal result probability plot ad histogram that ted toward Rayleigh distributio betwee 3000Hz to 6000Hz Figure bars Plate model Figure bars Plate model Figure bars Plate model Figure Sample of 10 pieces model simulatio result probability plot ad histogram betwee 00Hz to 6000Hz Figure Sample of 30 pieces model simulatio result probability plot ad histogram betwee 00Hz to 6000Hz ix

10 Figure Sample of 50 pieces model simulatio result probability plot ad histogram betwee 00Hz to 6000Hz Figure Sample of 10 pieces model simulatio result probability plot ad histogram betwee 3000Hz to 6000Hz Figure Sample of 30 pieces model simulatio result probability plot ad histogram betwee 3000Hz to 6000Hz Figure Sample of 50 pieces model simulatio result probability plot ad histogram betwee 3000Hz to 6000Hz Figure Sample of 10 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz Figure Sample of 30 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz Figure Sample of 50 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz Figure Sample of 10 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz Figure Sample of 30 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz x

11 Figure Sample of 50 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz Figure 5.1 Statistical Overlap Factor of simulatio results of 10 bars model xi

12 List of Table Table.1 - Type of Frequecy Respose Fuctio Table 4.1 Summary of equipmet Table 5.1 Sample of atural frequecy pairs Table 5. Compariso of atural frequecies xii

13 List of symbol b Parameter of Rayleigh Probability desity fuctio Mea of sample N(i) G U(i) S Ordered statistic media Iverse of the cumulative distributio fuctio Uiform ordered statistic media Statistical overlap factor Stadard deviatio λ² Frequecy parameter Laplacia operator W Positio co-ordiate fuctio W m, Plate displacemet D Flexural rigidity H 1 FRF estimator Phase agle m Mass c Dampig coefficiet xiii

14 k Stiffess F Force A Acceleratio V Velocity X Liear displacemet Pa Pascal (N/m ) xiv

15 1. Itroductio Vibratios are a part of huma everyday life i oe form or aother. Vibratios have both a great effect ad devastatig effect o huma society. Sice aciet times humas have experieced the impact of vibratio both mamade ad atural. A earthquake is a example of devastatig cosequece of vibratio. However, beefits from applicatio of vibratio are diverse i both scietific ad o-scietific fields. Music was a early applicatio of vibratio, striged music istrumet appears o the wall of Egyptia tomb which ca be dated back to about 3000 B.C. [1]. Today, vibratio aalysis cotributes toward may fields such as mechaical ad civil egieerig ad medicie. Vibratios, however, ca be passively geerated ad ormally are udesired. Rotatig imbalace i rotatioal machiery such as gearboxes or motors ca create this uwated vibratio. Oscillatio that is a cosequece from operatio of machie ca create excessive oise or fatigue i a machie ad result i catastrophic failure. These factors, good ad bad, have motivated humas to study vibratios to better uderstad, beefit ad prevet their udesirable cosequeces. From a egieerig poit of view vibratios are a importat factor i the desig phase due to udesirables such as oise ad catastrophic factors such as vibratio iduced fatigue ad wear. Oe of aspect of vibratio aalysis that has bee of great iterest is a mathematical model for a particular system. The aalysis of the respose of a vibratioal system relies o costructig a mathematical model or a equatio of motio i order to calculate atural frequecies ad mode shapes. For a complex system that is difficult to model, 1

16 approximatios use various modelig techiques such as fiite elemet aalysis (FEA). Fiite elemet aalysis caot accurately predict respose of a system with iheret radomess but rather a aalysis would yield a approximatio of the respose of the esemble member whose properties most closely match that of the fiite elemet model []. Every egieerig structure ad machie possesses ucertaity. The ucertaity cocered i this thesis is the variatio i material properties such as mass ad stiffess. The respose of ay system with ucertaity will differ from the respose of a system model. The level of variatio of the respose will deped o the degree of ucertaity. At low frequecies, the effect is ofte egligible, but as the frequecy icreases, the effect becomes more apparet. I this thesis, a steel circular plate will be the system of iterest. The ucertaity will be itroduced by fixig a umber of alumiium bars o to the plate. As a rage of similar ucertai systems will be studied, a series of ucertai plate ca be achieved by rearragig the orietatio of each alumiium bar. The study of atural frequecy of a ucertai plate will be usig the statistic to observe the effect of ucertaity o the system at a rage of frequecy. The two mai statistic measuremets are the statistical overlap factor ad the probability desity fuctio of the spacig betwee successive atural frequecies.

17 1.1 Literature Review I 1973, Che ad Soroka [3] study impulse respose of a simple dyamic system. The equatio of motio is modelled with atural frequecy is beig radom variable. The atural frequecy modelled as a sum of mea frequecy ad product of perturbatio parameter ad perturbatio with zero mea. They solve the ordiary differetial equatio with radom coefficiet by Samuels ad Erige. The result was plotted i term of magitude of stadard deviatio. They remarked the amplitude of the stadard deviatio to icrease as the mea frequecy icrease ad dampeed out after it reached certai level. Furthermore, the value of stadard deviatio dimiished as the mea atural frequecy of the system icreased. The result of plottig idicates that a system at low frequecy, the ucertaity i atural frequecy ca be igored i predictig the system respose where as the system with high atural frequecy the effect of ucertaity to the frequecy respose is sigificat. Leissa ad Narita [4] determied a rage of atural frequecy of simply supported circular plate. The motivatio was from the extesive ivestigatio of free ad clamped boudary coditio but ot simply support. The results show iclude Poisso s ratio from 0 to 0.5 i 0.1 steps. The preseted work covers all value of odal diameters ad odal circles s ad which plus s is less tha or equal to 10. The mode shape preseted will be use as referece to determie the locatio of the accelerometer ad excitatio poit. 3

18 S.McWilliam et al [5]. ivestigated the atural frequecy of rigs attached with radom mass. The iterested arragemets were (a) radom harmoic variatios i the mass per uit legth aroud the circumferece of the rig (b) the attachmet of radom poit masses at radom locatios o the rig (c) the attachmet of radom poit masses at uiformly spaced positios o the rig The frequecy spacig s were foud by aalytical expressio rather tha by experimetal results. The Rayleigh-Ritz method was used i the aalysis with assumptio that the mode shape of the imperfect rig was the same as the perfect rig. The results have show that for case (a) the frequecy spacig distributio demostrated Rayleigh distributio. The remarkable observatio for case (b) was the distributio tedig to Rayleigh as the degree of radomess was icreased, i this case the umber of poit mass icreased. Case (c) the distributio was depedig o the particular modes that take ito accout ad umber of poit mass added to the system. I a perfectly axisymmetric body the vibratio modes for a give odal cofiguratio occur i degeerate pairs which (a) have equal atural frequecies, (b) are spatially orthogoal ad (c) have idetermiate agular locatio aroud the axis of symmetry [6]. If ay imperfectio were to be itroduced ito such perfect system, the iitially equal atural frequecy pair would be split. It was foud that the magitude of the frequecy split was a fuctio of the magitude of the imperfectio ad the agular locatio of the modes which determied by the spatial distributio of the imperfectio. 4

19 For a isotropic circular plate with costat thickess ad free edges, its fudametal vibratio mode took the form of a twistig mode with odal diameters. Wag [7] preseted that with certai modificatio, such a fudametal modes with odal diameters become axissymetric mode shape. The proposed method were the addig outer rim to the circular plate with material with lager Youg s modulus or usig the same material but larger i thickess. Li ad Lim[8] studied the rectagular plate with arbitrary mass ad stiffess modificatio. Plate receptaces were derived based o mode superpositio ad the used to calculate the receptaces of the modified plate. Natural frequecies ad mode shapes of the modified plate were idetified by aalysig the receptace data based o the cocept of modal aalysis. They proposed a method which proved to be effective ad accurate to predict atural frequecies ad mode shapes of such a plate metioed above. There were 4 scearios cosidered (a) mass (b) mass ad stiffess (c) mass, stiffess ad dampig (d) mass, stiffess ad dampig at both traslatio ad rotatioal. Here, the result is to be cocered rather the methodology provided. The outcomes were preseted for the first 7 atural frequecies of the dyamic system. The otable result is the variatio i atural frequecies from origial system was most affected by the mass modificatio aloe rather other combiatio of differet modificatio. 5

20 Lucus [9] has doe extesive research o dyamic systems with a rage of permutatio. The experimets were coduct to study the effect of the specific type of permutatio, amely stiffess ad mass, o the atural frequecy of the system. The result is studied statically. The measures that were used to measure the result were successive atural frequecy spacig which quatified by histogram ad statistical overlap factor. The part of research has bee thoroughly reviewed which cosidered to be related to this thesis are poit mass permutatio, stiffess permutatio created by sprig ad the combiatio of both. The dyamic system which these variatios exist was a alumiium rectagular plate of size 899 mm i legth, 600 mm i width ad thickess of mm. The experimet were coduct by (a) icrease the umber variatio o the system which were of the same degree of radomess (b) icrease umber variatio applied o the system but keepig the total of degree of radomess costat These rules were both applied to mass variatio, stiffess variatio ad the combiatio of both. The effect of mass ucertaity revealed that the greater the umbers of ucertaity the greater of the effect. The effect was see by the tred of atural frequecy splittig ted towards Rayleigh distributio as the umber of radomess icreased. It was also oted that the effect of mass permutatio icreased as the frequecy icreased, but it would become saturated at a certai frequecy. This was 6

21 poited out by statistical overlap factor graph where it icreased with frequecy but level off at a specific poit with the bigger the umber of radomess the higher overlap factor curve level off. The frequecy at the curve started to become saturated was relatively the same regardless of umber of variatio. The mea spacig also preseted a relatively costat behaviour across the esemble as well as each member. The iertia of each mass was accouted for oe of the cause of variatio. It was cofirmed i the (b) part experimet that the umber of permutatio applied would determie the variatio i system respose. The result idicated that, eve though, the degree of radomess was costat. The histogram of atural frequecy spacig was earest to Rayleigh distributio whe the umber of mass was greatest although each added mass was less. It was further emphasised whe oly 1 mass, whose mass was equal to total mass of differet cofiguratio applied, showed a almost straight lie i statistical overlap factor curve. The local stiffess variatio, however, showed iverse relatioship i statistical overlap factor. The sprig was attached to the plate ad groud. The curve deceased as the frequecy got higher ad teded toward 0. It was iterpreted that the effect of the sprig was gettig lower at high frequecy. Whe the rule (b) applied, the closest match was ot from the most umber of sprigs. 7

22 . Vibratio Basic Vibratio is a motio of a particle that movig back ad forth about equilibrium positio. The motio is cyclic. Vibratio ca be distiguished ito types, free ad force. Free vibratio is whe the system is oscillatig i the absece of exteral force. It will vibrate at its oe or more atural frequecies. The forced vibratio system will vibrate at the frequecy of the applied force. Whe the applied excitig frequecy is coicide with oe of the system atural frequecy, the respose amplitude become large ad, sometime depedig o system, large eough to cause damage to surroudig or the system itself. It is kow as resoace whe the excitig force is at system atural frequecy. Resoace cotributes to the failure of structure such as bridge, turbie ad machie. The vibratig system geerally icludes a mea for storig potetial eergy (e.g. sprig) ad a mea for storig kietic eergy (e.g. mass) [10]. The vibratio of a system ivolves trasferrig back ad forth betwee potetial ad kietic eergy. The system may also be subjected to eergy dissipatig or resistace source kow as dampig. Sice the system is losig eergy durig the motio, free vibratio will decrease ad evetually become at rest. However the forced vibratio ca remai at the excitig frequecy with requiremet of eergy to be supplied. The system with eergy dissipatig source is called damped system. The dampig of the system also iflueces the atural frequecy of system. The umber of idepedet coordiate ecessary to describe the motio of a system is called the degree of freedom of the system [11]. A system with degrees of freedom will have atural frequecy. 8

23 The system ca be further categorised ito liear ad oliear vibratio. Whe the compoets of the system, sprig ad mass for example, have liear behaviour, it is liear vibratio. The priciple of superstitio holds i liear system whereas the oliear does ot. All system, however, ormally behaves oliearly at high amplitude. The most basic cofiguratio of vibratig system is system with oe degree of freedom, which cosist of a block of weight Figure.1 Sigle degree of freedom system Where m is the mass x is the displacemet k is stiffess c is the dampig coefficiet 9

24 F is the force Figure. Free body diagram of sigle degree of freedom system Equatio of motio of the sigle degree of freedom is F mx (.1) F kx cx mx (.) For modal model, the exteral force is ot take ito accout or it is free vibratio therefore m x cx kx 0 (.3) The geeral solutio to a SDOF system i free vibratio is give by a expoetially decayig sie fuctio as follows ( ) t x t Ae si( t ) (.4) d 10

25 Where A is the amplitude t is the time is the phase agle is the dampig ratio is the system atural frequecy d is the damped atural frequecy give by d 1 (.5) If the sigle degree of freedom systems are subject to force such as harmoic force the equatio of motio becomes m x cx kx F si t (.6) 0 Where F 0 is the maximum force amplitude is the frequecy of harmoic force The geeral solutio to equatio (.6) is x( t) F k 0 1 si( t ) (.7) 11

26 From equatio (.7), it ca be see that the maximum displacemet will occur whe excitig frequecy is ear atural frequecy. The maximum displacemet is also iflueced by the dampig of the system, the higher the dampig, the lower the maximum displacemet..1 Modal testig The study of vibratio respose of a structure is to be to uderstad its ature ad be able to cotrol it. There are geeral types of tests, sigal aalysis ad system aalysis [1]. The sigal aalysis is the process of determiig the respose of a system due to ukow excitatio which geerally operatioal force, i aother word whe it is i real operatio. The system aalysis ivolves determiig the iheret properties of the system by stimulatig the system with kow force ad the study it respose. This is geerally made uder cotrolled coditio ad yield more accurate ad detail iformatio tha the first type. It cosists of data acquisitio ad aalysis of data, the data is from testig of structure of iterest where the aalysis yields mathematical descriptio of their dyamic behaviour. It is kow as modal testig. The ultimate result of modal testig is extractio of modal parameter amely; modal frequecy, modal dampig ad modal shape. They are used to costructig mathematical dyamic model. The applicatios of the acquired model are vast but the most commo applicatio is the measure of vibratio mode to verify with theoretical model that has bee develop earlier. This is to esure the accuracy of theoretical model before use. 1

27 .1.1 Measuremet method The quality of the obtaied data depeds o the acquisitio method to esure that data aalysis will yield accurate iformatio. There are 3 areas i which eed particular attetio to make sure the quality of measure data is high [13] The mechaical aspect of the supportig ad excitig structure There are ormally 3 optios; free or urestraied, grouded ad i situ. With free coditio, the structure will exhibit rigid body modes which are determied completely by its mass ad iertia properties ad will be o bedig or flexig at all. It is ormally settig the test structure to be completely free i real life. Free is usually achieved by susped the structure to be tested with very soft sprig or elastic bad. The poit of susped should be put as close as possible to odal poit of the mode i iterest. If possible, the susped directio is put ormal to primary directio of vibratio. The structure of iterest that is grouded will be rigidly cramped. It is crucial to provide the foudatio that is sufficietly rigid. The cofirmatio is to check the mobility of the support over the rage of testig frequecy ad compare whether it is relatively low whe compare to the structure at the poit of cotact. It is usually that the free support will be the best optio of supportig type as o extraordiary attetio requires to verify the groud support achieves grouded coditio but i case where structure is large, the free support is almost impossible. 13

28 The coditio of the ormal operatio of test structure ca ifluece the type of support chose, if it is ormally grouded the supportig obviously should be grouded which is close to those of the operatioal structure. I situ meas the structure will be coected to some other structure or compoet o-rigidly The correct trasducer of the quatities to be measured The trasducer is essetial part as it is the compoet that will iput ad measure the data such as force, acceleratio ad velocity. The choices of trasducer deped o the rage of data to be read ad the type of structure that udergo testig. The respose measuremet of motio parameter, force ad acceleratio, ofte use piezoelectric accelerometer due to it umber of advatage. It has good liearity, low weight, broad dyamic rage, wide frequecy rage, low trasverse sesitivity ad applicable for umber of moutig method. Iput force measuremet is usually doe by piezoelectric force trasducer Oe problem eeds to be take ito accout is the ifluece of the trasducer to the test structure. Icorrect use of trasducer ca cotribute to major error. Usig a large trasducer to measure respose o a flexible plate will create local loadig i the area ca lead to iaccurate respose. This will also result i icrease i stiffess or dampig which will lower the measured atural frequecy of the system. Therefore, where possible, a light trasducer is preferred especially whe testig i high frequecy. 14

29 The trasducer moutig method is importat as trasducer itself ad eed to be carefully selected. There are rage of moutigs ca be chose from. The available are steel stud, bee wax, cemet stud, thi tape, thick tape ad maget. For optimal performace the steel stud is the first choice but it ot, however, always coveiece ad applicable i small or flexible structure. The more coveiece way yet still givig good result is applyig bee way at the base of trasducer ad firmly pressed oto structure. Although the arrower frequecy rage but without modificatio of the structure as i steel stud case, it is oe of popular choice of moutig techique The sigal processig which is appropriate to the type of test The choice of fuctio used i the processig eed to be correspoded to the type of test beig performed.1. Excitatio There are excitatio types; cotactig ad o-cotactig [13]. The cotactig exciter will be attached to test structure ad remai there throughout the vibratio period. The secod type of excitatio, the exciter ca always be out of cotact or i cotact for a short period of time to excite the test structure. Examples of attached exciters are electromagetic shaker, electrohydraulic shaker ad eccetric rotatig masses. Examples of o-attached exciters are hammers ad suspeded cable to produce sap back. 15

30 . Fast Fourier Trasform Fourier trasform is iitially used to covert a cotiuous fuctio that is fuctios which are defied at all values of the time t, from time domai ito frequecy domai. Digital sigal processig ivolves discrete sigals which are from the sample of iterested sigal rather tha cotiuous sigals. The coversio process is the eeded what is called Discrete Fourier Trasform or DFT, it is the modificatio of Fourier Trasform. The output or the result of the DFT is a set of sie ad cosie coefficiets. Recostructio of the origial wave sigal is doe superimpose the product of each obtaied coefficiets with appropriate sie or cosie waves with appropriate frequecy. The sie or cosies are the frequecy compoets of the origial wave where the coefficiet is the amplitude cotets. The spectrum is complex quatity. DFT is a straight forward calculatio but it, however, requires umerous steps. It eeds to perform N calculatio for DTF where N is the umber of sample [14]. Numerical algorithm developed by Cooley ad Tukey[15] called Fast Fourier Trasform allows DFT to perform with sigificat reductio i calculatio steps. This is preseted i A algorithm for the machie calculatio of complex Fourier series published i The process of determiig the amplitudes of all the frequecy compoets of a sigal is kow as spectrum aalysis. The frequecy versus amplitude is called the spectrum of the sigal. The term that will come across is autospectrum ad cross spectrum. The autospectum is obtaied by multiplyig a spectrum by its complex cojugate ad by 16

31 averagig a umber of idepedet products while cross spectrum is from the multiplicatio of oe spectrum ad complex cojugate of differet spectrum. The cross spectrum is complex where it phase is the phase shift betwee iput ad output..3 Frequecy Respose Fuctio Frequecy respose fuctio is a trasfer fuctio H() give by [16] H( ) X ( ) F( ) (.8) Where F () is iput force spectrum X () is output spectrum It is complex fuctio which cotais both magitude ad phase. The output spectrum ca be of displacemet, velocity or acceleratio, which reflect the ame of frequecy respose fuctio. 17

32 Displacemet / Force Velocity / Force Acceleratio / Force Admittace Compliace Receptace Mobility Accelerace Iertace Table.1 Type of Frequecy Respose Fuctio The frequecy respose will be applied to sigle degree of freedom as a example From equatio of motio F mx cx kx F m x c m x k m x (.9) c m (.10) k m (.11) Where is udamped atural frequecy is dampig ratio 18

33 19 Substitute equatio (.10) ad (.11) ito equatio (.9) x x x k F (.1) The fial Fourier trasformatio is give by [5] ) ( 1 ) ( ) ( j k F X (.13) This has magitude ) ( 1 ) ( ) ( k F X (.14) Ad phase arcta (.15) The similar aalysis ca be doe usig Fourier trasform to obtai mobility ad accelerace Mobility is ) ( 1 ) ( ) ( j j k F V (.16) The magitude is

34 0 ) ( 1 ) ( ) ( k F V (.17) Ad the phase is arcta (.18) Accelerace has the form ) ( 1 ) ( ) ( j k F A (.19) Where its magitude is ) ( 1 ) ( ) ( k F A (.0) Ad phase is arcta (.1) Sice the measuremet process experiece oise which might come from surroudig eviromet or istrumet itself. Egieerig measuremet ofte ivolves averagig process which the same i this case. The averagig miimise the error causig by oise i

35 readig. The approach is to use the priciple of least square method which resultig i FRF estimator * F X F F H * (.) This estimator is called H1 ad is equal to ratio of cross spectrum of respose ad force over autospectrum of force. H 1 ( G ) G FX FF ( ) ( ) (.3).4 Plate Theory The study of plate vibratio was ispired by the successful of Euler s membrae equatio. Euler [17] derived equatios for the vibratio of rectagular membraes, but however it was oly correct for the uiform tesio case. He cosidered the rectagular membrae as a superstitio of two set of strigs laid i perpedicular directio. With the Germa physicist, Chladi, is accouted for the first to have a serious study of plate vibratio. The experimet was o observig the odal patter of vibratio o rectagular plate. Sad was spread o to iterested plate where it was let free to vibrate, after the plate came to rest the sad accumulated alog odal lie. The first umerical method was developed by Germai, where she proposed the differetial equatio of trasverse deformatio of plate by meas of calculus of variatio. 1

36 It, however, was corrected by Lagrage for eglectig the strai eergy due to warpig of the plate midplae [18]. The plate theory was the further improve by Kirchhoff which icluded bedig ad stretchig effect. Kirchhoff plate theory is referred as Classical Plate Theory. Mildli later proposed that the effect of rotatory iertia ad trasverse shear strai was sigificat ad caot be omitted as i the Classical Plate Theory. The modified plate equatio of motio is kow as Shear Deformable Plate Theory. Recet refiemet o plate theory was proposed by Reddy, his third order shear deformatio theory give zero shear stress coditio at free surface is satisfied..4.1 Fudametal Equatio of Classical Plate Theory The displacemet w of plate give by Leissa [19] is 4 w D w 0 (.4) t Where D is the flexural rigidity give by 3 Eh D (.5) 1(1 v ) E is Youg s modulus h is the plate thickess

37 v is Poisso s ratio ρ is mass desity per uit area t is time 4 is the Laplacia operator Whe the free vibratio is assumed, the motio is expressed by w W cos t (.6) ω is circular frequecy i radia per uit time W is a fuctio oly of the positio coordiate Substitute equatio (.6) ito (.4) ( 4 4 k ) W 0 (.7) 4 k (.8) D ( 4 4 k ) W 0 ca be rewritte as ( k )( k ) W 0 The solutios to liear differetial equatio above are W k W 0 W k W 0 (.9) 3

38 4 I the case of plate supported by a massless elastic medium equatio (.4) become 0 4 t w Kw w D (.30) K is stiffess of the foudatio i uit of force per uit legth of defectio per uit area of cotact Assumig the deflectio form (.6) ad substitutig ito equatio (.30) D K k 4 (.31) Circular Plate The Laplacia operator expressed i circular coordiate is 1 1 r r r r (.3) Whe Fourier compoet i θ are assumed r W r W r W )si ( )cos ( ), ( * 1 0 (.33) Substitute equatio (.33) ito equatio (.9) 0 ) ( W k r dr dw r dr W d 0 ) ( 1 W k r dr dw r dr W d (.34)

39 Ad two idetical equatio for equatio havig solutios * W equatios (.34) are recogised as form of Bessel s W 1 A J ( kr) BY ( kr) W CI ( kr) DK ( kr) (.35) J ad Y are the Bessel fuctio of the first ad secod kids I ad K are modified Bessel fuctio of the first ad secod kids coefficiet A B C D determie the mode shape ad solved for from the boudary coditio The geeral solutio to equatio (.7) i polar coordiate is W ( r, ) 1 [ A * J 0 [ A J ( kr) B Y ( kr) C I * * ( kr) B Y ( kr) C I ( kr) D * ( kr) D K K ( kr)]si ( kr)]cos (.36) Whe the origi of a polar coordiate system is take to coicide with the cetre of the solid circular plate equatio (.36) with assumptio, Y (kr) ad K (kr) are discarded to avoid ifiite deflectio ad stresses at r=0, the boudary coditio possess symmetry with respected to oe or more diameter of the circle the the terms ivolvig si is ot eeded become W [ A J ( kr) C I ( kr)] cos (.37) 5

40 ca take o all values from 0 to The subscriptio will correspod to umber of odal diameters 6

41 3. Statistic Measure The statistic descriptors iterested i this study of a structure with radom properties are the probability desity fuctio of atural frequecy spacig ad statistical overlap factor. Set atural frequecy spacig across esemble of sample are arraged i icreasig order the fitted ito a probability desity fuctio. To measure how well a set of data fit ito a distributio fuctio, probability plot is employed. 3.1 Rayleigh Distributio fuctio Rayleigh probability desity fuctio is give by [0] x b e x b (3.1) Where b is the parameter of Rayleigh Probability desity fuctio defied by where is the mea of sample Figure 3.1 illustrates sample Rayleigh distributios, both are Rayleigh distributio but are of differet parameter. The wider the distributio, the higher the parameter which it idicates the higher mea of data. As Rayleigh distributio is determied by the parameter which related to mea of data, the mea of the atural frequecy spacig obtaied experimetally will be used to derive the parameter to geerate Rayleigh distributio ad compare how well it fit i the distributio. 7

42 Figure 3. shows a Rayleigh distributio geerated by a sample mea which data were sampled radomly ad its histogram. Figure 3.1 Examples of Rayleigh Distributio Figure 3. Rayleigh distributio geerated from sample mea ad its histogram 8

43 3. Probability plot Probability plot provides a simple effective meas of determie whether a sample of data belog to specified probability distributio [1]. If the ordered samples are plotted agaist the expected values of the order statistic of stadardised distributio, the the result should be liear. If the test data is ot from the assumed distributio, the liear patter will ot be observed. The equatio is give by N( i) G( U( i)) (3.) Where N(i) are ordered statistic media G is the iverse of the cumulative distributio fuctio U(i) are the uiform ordered statistic media defied by U( 1) 1U( ) i U( i) (3.3) (3.4) U( ) (3.5) The iverse cumulative distributio fuctio is give by [] s c l(1 U( i)) (3.6) The iterested data will be plotted agaist ordered statistic medias N (i), N (i) will form horizo axis where data i arraged icreasig order will form vertical axis. 9

44 Figure 3.3 MATLAB radomly sampled data from Rayleigh distributio Figure 3.4 Rayleigh Probability Plot 30

45 The 100 sample umbers take from Rayleigh distributio geerated by MATLAB are plotted i figure 3.3, the red lie i figure idicate the Rayleigh distributio created by usig the mea of the samples. The Rayleigh probability plot of the same samples is show i figure 3.4, agai the red lie is a bechmark of perfect Rayleigh distributio. Figure 3.5 shows expoetial distributio which was geerated by MATLAB, its Rayleigh probability plot is i figure 3.6. The result is obviously does ot follow the red straight lie, a suggestio for perfect Rayleigh distributio, it ca be cocluded that the data were ot from Rayleigh distributio which is iitially kow. The data is, however, expected to produce a straight lie result i expoetial probability plot. Probability plot is a useful statistical techique to quickly check if the set of data are from assumed distributio. Figure 3.5 expoetial distributio 31

46 Figure 3.6 Rayleigh probability plot of expoetial distributio 3.3 Statistical Overlap Factor Statistical overlap factor is defied by [3] S i i ( i 1 i ) (3.7) Or S (3.8) is the stadard deviatio of a atural frequecy from its mea is the mea frequecy spacig The meaig of statistical overlap factor is if S is small atural frequecies vary by small amout but if S is large, it idicates atural frequecies move sigificatly. It was stated 3

47 that system uder rai-o-the-roof excitatio woud show statioary i mea whe statistical overlap factor is greater tha ad 3 for steady mea behavior of system uder poit forcig[3]. 33

48 4. Experimetal Desig ad Setup 4.1 The plate The studied plate is a steel circular plate. It is of 500 mm i diameter ad 5 mm i thickess. To stimulate variatio or ucertaity i material stiffess i the system, a umber of alumiium bar are fixed o to the plate. As the statistical measure used eed to be doe across a set of similar structure but differet variatio, it would require to maufacture a great umber of similar iterested system. This is ot effective i practical sese poit of view. It, however, would be idetical if oly oe perfect test system is used but a umber of variatios ca be stimulated o this system. The plate is the required to be able to accommodate variatio of locatio of the alumiium bars. The desig, therefore, was based o chagig of the alumiium bar s orietatio. The fasteig method employed was mechaical i which is screwig. The plate was be drilled with 1/8 ich drill (3.175 mm) which allow M3 screw to go through ad tighte with the bars that are previously threaded at both eds. The dimesios of bars are 150 mm i legth, 10 mm i width, there are thickesses which are 1 mm ad.5 mm. The plate will accommodate maximum of 10 alumiium bars, each of the bars have 3 orietatios as i figure 4.1, the straight lies represet possible set up. The holes that have bee drilled are predetermied. The CAD drawig ad real picture of the plate is i Figure

49 Figure 4.1 Schematic diagram of circular plate Figure 4. - Steel Plate (Left) ad CAD drawig (Right) 35

50 Figure 4.3 Alumiium Bar of.5 mm (Left) ad 1 mm (Right) Alumiium bar Alumiium was chose i order to keep the mass as low as possible while providig a cosiderably stiffess. The desity of alumiium is 700 kg per m 3 where steel has desity 7800 kg per m 3, whe all 10 bars of legth 150 mm, width 10 mm ad thickess.5 mm are fixed o the plate the mass added is kg. The mass of the plate of diameter 500 mm ad thickess of 5 mm is kg. The mass of alumiium is or just 1.3 % of the plate. The mass of 1 mm thick bar is accouted for % of the mass of the plate. This is i order to keep the effect of variatio i mass as low as possible to observe oly stiffess permutatio. This was esured by the reviewed literature [9] showig that a total of 10 % mass variatio with 10 poits permutatio did ot have great effect o the system havig low SOF ad expoetial distributio of atural frequecy spacig. Effect of The alumiium bar will create local stiffess which will vary the atural frequecies of the plate. As the bar will be radomly placed o the plate, differet 36

51 orietatio will differetly affects each mode of vibratio some at great degree, some at lesser degree. The local stiffess will ot oly chage the atural frequecies but also the mode shape of the system as well. The bar will distort the mode where they are fixed, the vibratio amplitude would be expected to be higher as stiffess is icreased. The CAD drawig of types of alumiium bars is i figure experimetal set up ad techiques used 4..1 Supportig method The boudary coditio desired is free. The free coditio meas the test structure is completely isolated from groud ad have rigid body mode at 0 frequecy. I practical sese, it is ot possible to achieve truly floatig structure i space, it is eeded to be support i some maer. Hagig structure from elastic bad will i effect cause the rigid body behaviours to shift away from 0 frequecy. If the elastic bad s stiffess is low eough, ad hece the frequecy is much lower tha flexible mode, the effect will ofte be egligible. Elastic cable was used i this experimet to hag the plate vertically from the groud. Before experimet the plate was give motio i vertical directio ad perpedicular directio to the directio of the suspesio to check if is frequecy is low. The plate was vibrated at aroud 1 to Hz, it was the cocluded that the suspesio was valid to perform experimet give adequate free coditio. The setup is i figure

52 Figure 4.4 Suspeded plate 4.. Excitatio Impact excitatio was chose to perform experimet as it is a quick ad require a few setup procedure. Impact hammer cosists of hammer, a force trasducer ad hammer tip. The wave form geerated by this type of excitatio is trasiet. Sice the force is impulse, the eergy level trasfer to the structure is determied by velocity ad mass of hammer. This follows the theory of liear mometum. The measure force is the mass of the impactor behid the piezoelectric of disc of the force trasducer times the acceleratio. The true force, excitig the structure is the total mass of hammer times the acceleratio durig impact. The true force is the measured force times the ratio of total mass over the mass behid the trasducer [4]. The duratio ad shape of the spectrum is determied by the mass ad stiffess of both structure ad hammer. The stiffess of the hammer tip 38

53 determies the frequecy rage whe hammer is used o relatively large test structure. The harder the tip is, the shorter the pulse which meas higher frequecy rage covered by the impact. The tip, however, should be chose as soft as possible for the iterested frequecy rage to prevet the leakage of eergy to frequecy which is outside the rage of cocer. The useful frequecy rage is from 0 Hz to a frequecy poit where spectrum magitude has decayed by 10 0 db. Figure 4.5 Hammer Force Spectrum[4] The experimet was performed at the rage of up to 6000Hz, the frequecy rage is cosidered high whe impact hammer is used. The tip was chose by checkig the hammer spectrum whether it provided eough eergy across the rage. Steel tip was chose because it was decayed by about 18 db across the rage Hammer, although, regarded as the quickest ad coveiece excitatio choice, it is difficult to obtai a cosistet result, sice it is difficult to cotrol the velocity of the 39

54 hammer. The force spectrum caot be bad-limited at lower frequecies which mea that the techique is ot suitable for zoom aalysis. Aother problems that might be raised durig the experimet ad eed to be take ito accout are oise ad leakage. Noise ca be experieced i both iput force ad respose due to log period of record. The secod problem, leakage, ca be preset i respose sigal as a result of a short record time Double hit Double hit is occur whe the struck structure bouced back to the impactig hammer, there will more chace of double hit with icrease hardess of the hammer tip especially steel tip. Force sigal will reveal or more sharp peak ad the spectrum will ot be smooth, if this occurs the record will be disregard. It caot be compesate by usig trasiet widow or ay other widows Widowig Sice the force sigal is usually short whe compared to the duratio of the record time. The force sigal beside the period of impact is oise ad ca be elimiated without affectig the pulse itself usig widow techique. The widow to use is trasiet widow, this takes out ay sigal outside the widow but it is that short oscillatio after pulse is actually part of the pulse ad should ot be disregarded. 40

55 The respose sigal might, as well, suffer error. There are situatios where the error may arise. The first sceario is whe the structure is lightly damped which the record time is shorter tha decayig time. The secod case rose if the test structure is heavily damped i which its respose decay really fast ad ed much faster tha the record time. The first case will itroduce leakage error whereas the secod case will have poor sigal to oise ratio because the measuremet is suffered from oise. This is whe expoetial widow is employed; it ca hadle well both situatios. The widow fuctio is give by w( t) e t Where is time costat The widow will miimise the leakage error i the first case by forcig the respose to decay completely before the ed of record time. This, however, brig i the extra dampig to the system resultig i broader resoace frequecy. It ca be corrected o the post processig stage. It elimiates the remaiig oise i the measuremet i the case where respose dies out before the ed of record time. The selectio of the time costat, (the time required for the amplitude to be reduced by a factor of 1/e), is about oe-fourth the time record legth. 41

56 4..4 Strikig locatio The strikig locatio was chose at where there is o alumiium bar fixed. As the experimet is oly iterested o the locatio of atural frequecies, the strikig locatio could be chage for every set of cofiguratio. The coherece was check if the strikig locatio was o a odal lie. At each set of cofiguratio several place was tested to fid optimum poit where it revealed the most modes Locatio of accelerometer The locatio of trasducer ca be ay poit of iterest provided that it is ot o or close to a ode or more of the structure s modes. It will be difficult to make a effective measuremet. Similar approach to strikig locatio was used, if poor coherece was experieced locatio of accelerometer was chaged but kept costat for each cofiguratio to fid the optimum locatio to record poit Coherece Coherece fuctio is give by [] FX ( ) S FF S FX ( ) ( ) S ( ) XX S () is output spectrum XX 4

57 S () is iput spectrum FF S () is cross spectrum XF ( FX ) is betwee 0 to 1. Coherece is a measure of liear relatioship betwee iput ad output for each frequecy. It will show 1 for o oise measuremet while 0 idicates pure oise measuremet. The coherece that is less tha perfect 1 idicates poor sigal to oise ratio, umber of measuremet errors, structure s oliear behaviour, iadequate frequecy resolutio or a combiatio of more tha 1 error. The use of coherece fuctio o impact testig is limited due to determiistic ature of the sigal coherece fuctio caot show either leakage or oliear behaviour. The source of less tha 1 i coherece fuctio are the sigal to oise ratio is poor at ati resoace, ot strikig the test structure perpedicularly with surface ad impact poit is at or ear odal poit which ofte result i very low coherece. Figure 4.6 shows a sample of good measuremet where coherece reveals 1 everywhere except aroud atiresoace. 43

58 Figure 4.6 Sample Coherece 4..7 Measuremet aalysig As this experimet is oly iterested oly o the locatio of the atural frequecy, o other post-processig is required. Figure 4.7 shows a sample of frequecy respose which each peak represet atural frequecy which ca be read of the display usig program cursor. Figure 4.7 Sample of frequecy respose 44

59 It might be, however, possible that some peak may ot obvious or verificatio was eeded. This is whe Nyquist plot techique was employed. It is the plot of real part agaist imagiary part of frequecy respose fuctio. If it plotted rage just cover the resoace regio, the plot will be circular as i the sample plot of obvious peak at 15 Hz from figure 4.7 is illustrated i figure 4.8. Oly frequecy poits closest to resoace are idetifiable because those away from resoace are very close together [5]. Ay ucertai peak with o-circular Nyquist plot was disregarded from aalysis. Resoace at 15Hz Resoace at 15Hz Figure 4.8 Nyquist plot 45

60 4..8 Summary of equipmets Equipmet Persoal Computer istalled with Pulse software Data logger Name of equipmet HP Laptop Bruel & Kjær Pulse FrotEd Type 3560C Charge Amplifier Bruel & Kjær Type 635 Piezoelectric Accelerometer Bruel & Kjær Type 4393 Impact Hammer Bruel & Kjær Type 80 Force Trasducer o hammer Bruel & Kjær Type 800 Elastic Cable Table Summary of equipmet 46

61 4..9 Equipmet setup Figure Equipmet setup 47

62 Radomisatio the locatio of alumiium bars The orietatio of each of the bar 10 bars were radom usig MATLAB. Each orietatio i each circles represetig the possible orietatio were assiged with a umber 1 to 3 as i the sample circle i figure A set of umber will be radomly geerated by MATLAB fuctio called radit where it geerates a radom iteger with specified row, colum ad iteger rage. The sample full commad is the followig as=radit(1,10,[1,3]); The commad will geerate 1x10 iteger matrix with radom iteger betwee 1 to 3. The geerated matrix was to positio the alumiium bar. The step was repeated if more cofiguratio is required. The sample output is below. as = Figure 4.10 umber assigmet 48

63 5. Result ad Discussio 5.1 Experimetal result 5 samples were measured to use i aalysis phase. Figure 5.1 Bare plate probability plot ad histogram The egative mass was foud to have great ifluece o the system whe compared to perfect solid circular plate. The plate has holes that were made to fix alumiium bars. Each hole has dimesio as described i previous sectio chapter 4, a diameter of (1/8 iches) with drilled through the plate of 5 mm thick. The steel plate has diameter of 500 mm therefore each hole is effectively is a 3.175/500 fractio or percet egative mass of the total mass of the plate. The histogram ad Rayleigh probability plot is i figure 5.1 showig a close match to Rayleigh. The plate was desiged to have 60 holes. It is however, whe perform the experimet there were 10 bars fixed o the plate, the hole were filled with screw, therefore oly 40 49

64 holes were left o the plate but the followig result showed that they are still strog effect o most of respose i which there were little chage o the spacig distributio. The frequecy rage i figure 5. to 5.4 is 00Hz to 6000Hz, 10 of.5 mm thick bars were used Figure 5. - Sample of experimetal result probability plot ad histogram that ted toward expoetial distributio 50

65 Figure Sample of experimetal result probability plot ad histogram that ted toward Rayleigh distributio Sice the plate showed the Rayleigh distributio eve though the alumiium bars were ot attached, the respose will be use as a bechmark to observe ad effect of mass stiffess permutatio, figure 5.1 showed the spacig distributio of the bare plate. 80% of results showed a Rayleigh distributio which i this case meas that the alumiium bar was with o sigificat differet to the match to Rayleigh. This effect ca be cocluded as eglectable or little. It, however, the fact that the 0% of the result showed the shift back toward expoetial. This would be a result from couter effect of the alumiium bar as well as locatio ad orietatio of the bar to couter the effect of egative mass. This similar effect has bee previously observed by Fox [6] which used mass to attach to specific locatio to chage specific atural frequecy so called mode trimmig where he 51

66 trimmed the atural frequecies of a imperfect rig to elimiate certai of the frequecy splits preset. Figure sample of experimetal result probability plot ad histogram that ted toward Rayleigh distributio betwee 00Hz to 3000Hz Figure Sample of experimetal result probability plot ad histogram that ted toward Rayleigh distributio betwee 3000Hz to 6000Hz 5

67 The most results are similar to figure 5.4 ad figure 5.5 which of the result i the rage of 00Hz to 3000Hz ad 3000Hz to 6000Hz respectively. There were o sigificat differet i the plottig rage where the effect was cocluded to be eglectable or little eve at higher frequecy. 5. Simulatio Result As the system which has bee built showed that it was really sesitive to egative mass variatio. Simulatio was employed to study the perfect circular disc behaviour subjected to mass stiffess ucertaity. ANSYS was used to model steel disc from the oe that used i experimetal study but holes were eglect to create a perfect symmetric vibratio. The simulatio revealed vibratio which occurred i pair as expected. The sample of atural frequecy of the plate is i table 5.1. The model was the modified to cotai alumiium bar which boded to the steel plate model. The models which had differet umber of bar was also studied. This was to observe ay chage whe umber of variatio was icreased, there were 10, 30 ad 50 bar models created, 10 differet cofiguratio were made for each type. The volume of material added was costat, the higher umber of bar, the shorter the legth. It thickess ad width were kept costat. The sample of models of 10, 30 ad 50 bars is i figure 5.6, figure 5.7 ad figure 5.8 respectively. 53

68 Mode Number Natural Frequecy Table 5.1 Sample of atural frequecy pairs Figure bars Plate model 54

69 Figure bars Plate model Figure bars Plate model 55

70 The comparisos were made o atural frequecy spacig distributio of a steel circular plate subjected to 10, 30 ad 50 material stiffess permutatio. Material stiffeer used was a thi alumiium bar of Youg s modulus = 70 GPa which has the same dimesio as the oe that used i the practical experimet. The total volume material added was costat. It was first to perform the check if model s atural frequecies varied from the oe of the perfect plate, the result is show i table 5.. The sample of modified plate was a member of the 10 bars model. It ca be see that the atural pairs were separated. No tred o splittig was observed. Most of the splittig was less tha 5 Hz apart eve at higher frequecy (up to 6000Hz). The rest of aalysis was i the rage from 00 Hz to 1000 Hz. Each sample from the simulatio result is show i figure 5.9 to 5.14 which are separated i aalysis rage 00Hz to 6000Hz ad 6000Hz to 1000Hz respectively to idetify ay chage as frequecy icreased. The red lie i probability plot is the fit for perfect Rayleigh distributio where the red lie i histogram shows a Rayleigh distributio calculated usig the samples mea. 56

71 Mode umber Bare plate Modified plate Table 5. Compariso of atural frequecies 57

72 Figure Sample of 10 pieces model simulatio result probability plot ad histogram betwee 00Hz to 6000Hz Figure Sample of 30 pieces model simulatio result probability plot ad histogram betwee 00Hz to 6000Hz 58

73 Figure Sample of 50 pieces model simulatio result probability plot ad histogram betwee 00Hz to 6000Hz The result showed o sigificat differece amog the sample. They showed a strog expoetial distributio. The compariso of bare plate ad permutated plates showed o differet. This meas that the alumiium bar had little or o effect o the system. This cofirmed the validity of experimetal result ad that i overall the alumiium bar has too small effect o steel plate to make the respose radom eough to geerate Rayleigh distributio. The result also revealed that mass of alumiium bar did ot or rarely cotribute to ay effect o the result i experimet. The splittig of atural frequecy pairs were also low eve icreasig the umber of bar. 59

74 The frequecy rage i figures below is betwee 6000Hz to 1000Hz, which were from the same sample simulatio result but the frequecy rage is from 6000Hz to 1000Hz Figure Sample of 10 pieces model simulatio result probability plot ad histogram betwee 3000Hz to 6000Hz Figure Sample of 30 pieces model simulatio result probability plot ad histogram betwee 3000Hz to 6000Hz 60

75 Figure Sample of 50 pieces model simulatio result probability plot ad histogram betwee 3000Hz to 6000Hz There was still o sigificat chage at higher frequecy rage. The distributio remai strog expoetial. It was correspod to the experimetal result where the 80 % of systems esemble member did ot or little chage the system atural frequecy spacig distributio from the origial system. 61

76 The followig sectio the simulatio was used to fid at what material stiffess, the system would respose i a shift i atural frequecy spacig to Rayleigh distributio. A rage of stiffess has bee test ad at Youg modulus of 800 GPa, the splittig distributio started to depart from expoetial distributio. It is oted here that the rest of material properties are of the alumiium which iclude Poiso ratio ad desity. This is to keep other variable costat to observe what the effects of variatios i material stiffess are. Although i reality is merely impossible to have a material stiffess variatio of 800 GPa which is almost of diamod stiffess, Youg s modulus of alumiium is edited to 800 GPa. Figure Sample of 10 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz 6

77 Figure Sample of 30 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz Figure Sample of 50 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz 63

78 The sample histogram showed that at 800 Gpa, the closest match was whe 30 pieces of alumiium bars were added. Whe 50 bars were used, the result shifted back towards expoetial distributio. As the umber of bar icreased, keepig the total material added, the bar would be too small to iterfere each mode which otherwise would vary atural frequecy to be radom eough to geerate Rayleigh distributio. The higher umber of bar would be expected to give a closer match to expoetial distributio. The results were more clear ad poitig to the same coclusio whe the frequecy rage is betwee 6000Hz to 1000Hz which show i figure 5.18 to figure

79 Figure 5.18 ad figure 5.0 show below are the sample atural frequecy spacig of rage betwee 6000Hz to 1000Hz usig the same sample as above 00Hz to 6000Hz Figure Sample of 10 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz Figure Sample of 30 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz 65

80 Figure Sample of 50 pieces model simulatio result which stiffess is edited to 800GPa showig probability plot ad histogram betwee 00Hz to 6000Hz It has bee observed that the higher the frequecy the closer match to Rayleigh distributio. The effect of material stiffess radomess was icreasig as the frequecy got higher. Agai, the result coicided with the lower frequecy rage where 30 bars system was foud to be the closest match. 66