Random Vibration Analysis of Higher-Order Nonlinear Beams and Composite Plates with Applications of ARMA Models

Transcription

1 Radom Vibratio Aalysis of Higher-Order Noliear Beams ad Composite Plates with Applicatios of ARMA Models by Yukai Lu Dissertatio submitted to the Faculty of the Virgiia Polytechic Istitute ad State Uiversity i partial fulfillmet of the requiremets for the degree of Doctor of Philosophy i Egieerig Mechaics Surot Thagjitham, Chair Scott Case Michael Hyer Guo-Qua Lu Saad Ragab October 3rd, 8 Blacksburg, Virgiia Keywords: oliear, higher-order beam, root mea square, ARMA model, modal iteractio, power spectral desity

2 Radom Vibratio Aalysis of Higher-Order Noliear Beams ad Composite Plates with Applicatios of ARMA Models Yukai Lu ABSTRACT I this work, the radom vibratio of higher-order oliear beams ad composite plates subjected to stochastic loadig is studied. The fourth-order oliear beam equatio is examied to study the effect of rotary iertia ad shear deformatio o the root mea square values of displacemet respose. A ew liearly coupled equivalet liearizatio method is proposed ad compared with the widely used traditioal equivalet liearizatio method. The ew method is prove to yield closer predictios to the umerical simulatio results of the oliear beam vibratio. A systematical ivestigatio of the oliear radom vibratio of composite plates is coducted i which effects of oliearity, choices of differet plate theories (the first order shear deformatio plate theory ad the classical plate theory), ad temperature gradiet o the plate statistical trasverse respose are addressed. Attetio is paid to calculate the R.M.S. values of stress compoets sice they directly affect the fatigue life of the structure. A statistical data recostructio techique amed ARMA modelig ad its applicatios i radom vibratio data aalysis are discussed. The model is applied to the simulatio data of oliear beams. It is show that good estimatios of both the oliear frequecies ad the power spectral desities are give by the techique. ii

3 Ackowledgmet I would like to thak my advisor, Professor Thagjitham, for his guidace ad support all through my work. I would like to thak all my committee members, Professor Case, Professor Hyer, Professor Lu, ad Professor Ragab, for the time they took to atted my exams ad defese as well as their advice o my dissertatio. My thaks also go to Professor. Kraige, Professor. Hedricks, ad the ESM departmet, for their fiacial support durig my Ph.D. study. Last but ot least, I wat to thak my parets, Lu, Zhicheg ad Zhao, Guiyig. I would ot have bee able to make it through all the difficulties ad hard times i my life without their ucoditioal love ad support all the time. iii

4 Table of Cotets Chapter 1. Itroductio... 1 Chapter. Literature Review... 4 Chapter 3. Radom Vibratio of Geometrically Noliear Beams Solutios to the Noliear Radom Vibratio of Isotropic Beams Effect of Iertia of Rotatio ad Shear Deformatio Numerical Results Chapter 4. Noliear Radom Vibratio of Composite Plates Goverig Equatios Stochastic Respose of Liear System Stochastic Respose of Noliear System Temperature Effects o Radom Vibratios of Composite Plate Compariso betwee FSDT ad CPT R.M.S. Stresses Calculatio Chapter 5. ARMA Model ad Its Applicatios i Radom Vibratio Data Aalysis Itroductio Theoretical Backgroud Applicatios of ARMA Model i Idetifyig, Re-geeratig, ad Extedig the Radom Vibratio Data Compariso betwee PSD Curve from ARMA Model ad Newlad s Approach 88 Chapter 6. Future Work iv

5 6.1 Durability of Structures Subjected to Radom Loadig Future Work Referece Appedix A A.1 Selected eigefuctios Solutio procedure for a µ liearly-coupled system Appedix B: Derivatio of Plate Equatios v

6 List of Figures Figure 3.1 A beam uder pressure Figure 3. Correlatio betwee displacemet ad acceleratio for a typical liear beam (data size: 14 )... 7 Figure 3.3 Correlatio betwee displacemet ad acceleratio for a typical oliear beam (data size: 14 )... 8 Figure 3.4 Two types of loads used i the simulatio Figure 3.5 A typical statioary Gaussia radom process (time domai) Figure 3.6 Histogram of the radom process i Figure Figure 3.7 PSDs of the two types of loads used i the simulatio Figure 3.8 Displacemet R.M.S. of a uiformly loaded F-SS beam vs. differet Figure 3.9 Mode 1 displacemet R.M.S. of a half-uiformly loaded F-SS Figure 3.1 Mode displacemet R.M.S. of a half-uiformly loaded F-SS Figure 3.11 Displacemet R.M.S. (summatio of first two modes) of a uiformly loaded SS-SS beam vs. differet radom loadig PSD levels Figure 3.1 Couplig effect o mode 1 for a uiformly loaded F-SS beam Figure 3.13 Couplig effect o mode for a uiformly loaded F-SS beam Figure Couplig effect o mode 1 for a half uiformly loaded F-SS beam Figure Couplig effect o mode for a half uiformly loaded F-SS beam Figure 3.16 Typical mode 1 displacemet respose (correspodig to data i Table 3.1) Figure 3.17 Typical mode displacemet respose (correspodig to data i Table 3.1) Figure 3.18 Typical FFT of mode1 displacemet respose... 4 Figure 3.19 Typical FFT of mode displacemet respose... 4 Figure 3. Histogram of mode 1 displacemet respose Figure 3.1 Histogram of mode displacemet respose Figure 3. Compariso amog differet approaches of mode 1 respose Figure 3.3 Compariso amog differet approaches of mode respose vi

7 Figure 3.4 Compariso amog differet approaches of mode 1 respose Figure 3. 5 Compariso amog differet approaches of mode respose Figure 3. 6 Compariso amog differet approaches of mode 1 respose Figure 3. 7 Compariso amog differet approaches of mode respose Figure 3. 8 Compariso amog differet approaches of mode 1 respose Figure 3.9 Compariso amog differet approaches of mode respose Figure 3. 3 Compariso amog differet approaches of mode 1 ad resposes Figure Mode 1 R.M.S. respose of a SS-SS beam subjected to uiform load Figure 3. 3 Mode R.M.S. respose of a SS-SS beam subjected to uiform load... 5 Figure Mode 1 R.M.S. respose of a F-SS beam subjected to half-uiform load... 5 Figure Mode R.M.S. respose of a F-SS beam subjected to half-uiform load Figure 4.1 Free body diagram of a rectagular plate elemet (without bedig momets) Figure 4. Free body diagram of a rectagular plate elemet (bedig momets oly).. 55 Figure 4.3 RMS values vs. square root of power spectral desity Figure 4.4 R.M.S. values vs. T (based o FSDT)... 7 Figure 4.5 Variatios of the ratios betwee FSDT ad CPT R.M.S. values Figure 4.6 Cotour plot of σ xx R.M.S. at middle plae of the first layer of a ( 6 / 6 / 6 / 6 ) lamiate Figure 4.7 Cotour plot of σ xx R.M.S. at middle plae of the first layer of a (3 / 3 / 3 / 3 ) lamiate Figure 4.8 Effect of ply agle θ o the maximal R.M.S. values of stress compoets Figure 5.1 PSD of the mode 1 of the beam displacemet simulatio data Figure 5. Displacemet from simulatio of mode of a F-SS beam Figure 5.3 Displacemet geerated by ARMA(4, 3) model Figure 5.4 Displacemet geerated by ARMA(8, 7) model Figure 5.5 Displacemet geerated by ARMA(11, 1) model Figure 5.6 Example spectrum plots of 8 out of the 3 segmets Figure 5.7 Compariso of PSD curves from ARMA(4,3) model... 9 vii

8 Figure 5.8 PSD of the beam simulatio displacemet data (Newlad s approach) Figure 5.9 PSD from ARMA(4,3) model Figure 5.1 PSD from ARMA(8,7) model Figure 5.11 PSD from ARMA(11,1) model viii

9 List of Tables Table 3.1 Respose of a beam (mm) with F-SS boudary coditio ad subjected to half uiform load (load PSD = 1 Pa /Hz, h = b, L = 1h = 1 m ) Table 3. Respose of a beam with F-SS boudary coditio ad subjected to half uiform load (load PSD = 1 Pa /Hz, h = b, L = h = 1 m ) Table 3.3 Respose of a beam (mm) with F-SS boudary coditio ad subjected to uiform load (load PSD = 1 Pa/Hz, h = b, L = 1.5h = 1 m ) Tabel 3.4 Respose of a beam (mm) with F-Fixed boudary coditio ad subjected to half uiform load (load PSD = 1 Pa /Hz, h = b, L = 1.5h = 1 m ) Table 3.5 Compariso of R.M.S. respose of a d order beam with that of 4 th order beam (SS-SS boudary coditio with uiform load, h = b, L = 1h = 1 m )... 5 Table 3.6 Compariso of R.M.S. respose of a d order beam with that of 4 th order beam (SS-SS boudary coditio with half-uiform load, h = b, L = 1h = 1 m )... 5 Table 3.7 Compariso of R.M.S. respose of mode 1 of d order beam with that of 4 th order beam (F-SS boudary coditio with half-uiform load, h = b, L = 1h = 1 m )... 5 Table 3.8 Compariso of R.M.S. respose of mode of d order beam with that of 4 th order beam (F-SS boudary coditio with half-uiform load, h = b, L = 1h = 1 m ) Table 5.1 Estimated ARMA parameters for differet order models Table 5. Frequecies predicted by selected ARMA models i Table Table 5.3 Estimated displacemet R.M.S. of selected ARMA models ix

10 Chapter 1. Itroductio A great deal of work has bee doe o the respose of beams ad plates subjected to determiistic loadig coditios. However, i real life the structure may be subjected to a stochastic type of loadig such as earthquakes, wid turbulece, sea wave, acoustic loads, etc. These loadig coditios are commoly observed o dams, uclear facilities, offshore structures, aircraft, etc. The mai purpose of this dissertatio is to preset a systematical study of the stochastic respose of geometrically oliear beams/plates uder radom excitatios. I this work, a literature review of previous research i the area is give first i Chapter. The, i Chapter 3 radom vibratio of geometrically oliear beams is elaborated where the detailed procedures to solve the radom vibratio problem ad the traditioal ucoupled equivalet liearizatio techique are discussed. To study the effect of the rotary iertia ad shear deformatio o the root mea square (R.M.S.) of the stochastic beam respose, the fourth-order oliear beam equatio is examied. The results from the fourth-order beam equatio are compared with those from secod-order. A ew 1

11 coupled equivalet liearizatio method is proposed which takes ito accout the effects of modal iteractios betwee adjacet modes. Numerical results idicate that the ew method yields closer results to simulatio data compared with the ucoupled liearizatio approach. I Chapter 4 the focus is moved from oe-dimesioal beam problem to rectagular composite plates usig both oliear classical plate theory (CPT) ad first order shear deformatio (FSDT) theory. FSDT takes ito accout of the trasverse shear strai effect. A oliear stress-strai relatioship i the vo Karma sese is cosidered i the formulatios of the goverig equatios. The effects of oliearity ad temperature o the R.M.S. values of trasverse displacemet of the plate ad the selected stress compoets are examied. A statistical data characterizatio ad recostructio techique called ARMA modelig ad its applicatios i radom vibratio data aalysis are demostrated i Chapter 5. The auto-regressive movig averagig (ARMA) model was origially developed as a timedomai modal aalysis method. ARMA model is very efficiet i recostructig the loadig coditio directly i the time domai. The model is very cocise i its formulatio, requirig very few parameters while preservig the stochastic ature ad spectral iformatio of the origial sigal history. I a ARMA model, the curret value of the system respose is expressed as a liear combiatio of past values of respose plus a pure white oise. The parameters i the model are determied through a trial ad error procedure i order to miimize the residue variace of the oise. Oce the parameters of the model are kow, atural frequecies ad dampig ratios for all the modes ca be obtaied from the autoregressive part. However, the order of a ARMA

12 model to fit a certai data set is ot uique. I additio, the correlatio betwee the oliear simulatio data usually requires a higher order model tha the liear case does i order to accurately represet the spectral properties of the origial iput, especially the power spectral desity (PSD) plot. It is show that eve though a model ca give good estimatios of the frequecy values, it may ot represet the PSD closely. At the ed of this work, issues regardig the durability of structure uder radom excitatios are addressed. Future areas of research work are discussed. 3

13 Chapter. Literature Review Stochastic loads such as wid turbulece, sea wave, ad acoustical loads are commoly observed o aerospace, mechaical ad civil structures. Cosequetly, radom vibratio aalysis is ecessary to uderstad the behavior of these structures uder stochastic loadig. For a geeral review o the radom vibratio theory, oe ca refer to the refereces (Cradall ad Mark 1973, Boloti 1984, Nigam 1983, Roberts ad Spaos 199, Newlad 1993, Soles 1997, Wirschig et al. 1995, Li 1976, ad Elishakoff 1983). The exact solutios to oliear radom vibratio problems are oly available for certai special cases. Therefore, approximatio techique ad umerical solutios were developed to fid the probability desity fuctios of the respose of the oliear system. For limited cases, the momets of the respose ca be obtaied via solvig the Fokker-Plack equatio (Strataovich 1963, Strataovich 1967, Riske 1996, ad Gardier 4). A set of ordiary differetial equatios for the momet characteristics of respose ca be obtaied after applyig a closure techique such as the Gaussia closure method (Iyegar ad Dash 1978). Perturbatio method (Nayfeh 1993, ad Nayfeh ad 4

14 Mook 1995) is the most widely used techique dealig with the oliear dyamic respose of systems with oliearity. It ca also be adapted to solve the oliear radom vibratio of such systems (Cradall 1963). However, by ature the perturbatio method is oly applicable whe the oliearity is small, which greatly limited its usage i a wide rage of problems. Aother method called stochastic averagig techique (Roberts ad Spaos 199, ad Socha ad Soog 1991) ca also be applied to weakly oliear systems. The equivalet liearizatio (Caughey 1959a, Caughey 1959b, Caughey, 1963, Spaos 1981, ad Roberts et al. 199) is the most commoly used approximatio method due to its straight-forward formulatio ad effectiveess. Whe the load is white oise, the equivalet liearizatio yields the same results as Gaussia closure method (Er 1998). The radom vibratios of beams have bee studied sice the 195s (Erige 1957, Bogdaoff ad Goldberg 196, Cradall ad Yildiz 196, Elishakoff ad Livshits 1984, ad Elishakoff 1987). A exact probability desity fuctio of modal displacemets was foud by Herbert (1964, 1965). Amog all the approaches, the two mostly used are the perturbatio method ad the stochastic liearizatio techique. Erige (1957), Elishakoff (1987) ad Elishakoff ad Livshits (1984) came up with closed-form solutios for simply supported beams subjected to radom loadig i the form of ifiite modal summatio. Exact solutios by the Fokker-Plack equatio method oly exist for some extreme cases. Eve if a exact solutio exists, a large amout of multiple itegratios are eeded to evaluate the root mea square value of the respose, which makes it computatioally prohibitive. Fag et al. (1995) ad Elishakoff et al. (1995) proposed a 5

15 improved stochastic liearizatio method by miimizig the potetial eergy of the beam uder statioary radom excitatio. They claimed that the ew approach improved the accuracy of the covetioal stochastic liearizatio method. Differet variatios of the improved stochastic liearizatio techique ca be foud i the literature (Elishakoff ad Zhag 1991, Elishakoff 1991, Zhag et al. 199, ad Fag ad Fag 1991). Sice oe part of this dissertatio studies the radom vibratio of composite plates usig classical plate theory (CPT) ad first order shear deformatio plate theory (FSDT), a comprehesive itroductio of composite material as well as differet plate theories ca be foud i the work of Reddy (1997, 4). Before the stochastic respose of composite plate is examied, a brief review of some of the work o dyamic respose of plates usig differet theories is give as follows. Some of the studies ivestigated the oliear vibratios of composite plates or fuctioally graded plate (a special type of composite plate), i which iteratio scheme was used similar to the equivalet liearizatio i the radom vibratio aalysis. Kim ad Noda () discussed trasiet displacemet of fuctioally graded composite plates due to heat flux by a Gree s fuctio approach based o the classical plate theory. Pravee ad Reddy (1998) ivestigated the static ad dyamic resposes of fuctioally graded ceramic metal plates by usig a plate fiite elemet that accouts for the trasverse shear strais, rotary iertia ad moderately large rotatios i the vo Karma sese. Reddy () aalyzed the static behavior of fuctioally graded rectagular plates based o the third-order shear deformatio plate theory via fiite elemet 6

16 approach. Theoretical formulatio alog with Navier s solutio ad fiite elemet model for the plate were preseted. Woo ad Meguid (1) applied the vo Karma theory for large deformatio to obtai the aalytical solutio for the plates ad shell uder trasverse mechaical loads ad a temperature field. Zekour (6) preseted a geeral formulatio for fuctioally graded composite plates usig the geeralized shear deformatio theory that did ot require a shear correctio factor. Cheg ad Batra (a) preseted results for the bucklig ad steady state vibratios of a simply supported fuctioally graded polygoal plate based o Reddy s plate theory. Cheg ad Batra (b) also related the deflectios of a simply supported fuctioally graded polygoal plate give by the first-order shear deformatio theory ad a third-order shear deformatio theory to that of a equivalet homogeeous Kirchhoff plate. Loy et al. (1999) studied the vibratio of fuctioally graded cylidrical shells usig Love s shell theory ad Rayleigh Ritz method. Liew et al. (1, a, b) used classical plate theory ad the first order shear deformatio theory to preset the fiite elemet formulatio for the shape ad vibratio cotrol of fuctioally graded plates with itegrated piezoelectric sesors ad actuators. He et al. (1) preseted the vibratio cotrol of fuctioally graded plate with itegrated piezoelectric sesors ad actuators by a fiite elemet formulatio based o CPT. Huag ad She (4) solved the oliear vibratio ad dyamic respose of simply supported fuctioally graded plates subjected to a steady heat coductio process through a improved perturbatio techique. Woo et al. (6) provided a aalytical solutio i terms of mixed Fourier series for the oliear free vibratio behavior of composite plates. The oliear couplig effects o the fudametal frequecies were examied. Liew et al (6) preseted the oliear 7

17 vibratio aalysis for layered cylidrical paels subjected to a temperature gradiet due to steady heat coductio alog the pael thickess directio. A oliear pre-vibratio aalysis was coducted to obtai the thermally iduced pre-stresses ad deformatio. Differetial quadrature method with a iteratio scheme was employed to fid the oliear vibratio characteristics of the pael. Yag ad She (1) preseted the dyamic respose of iitially stressed fuctioally graded thi plates. Yag ad She () ivestigated the free ad forced vibratio problems for the shear-deformable fuctioally graded plate i thermal eviromet. Their results idicated that the plates with itermediate material properties did ot ecessarily have itermediate dyamic respose. Kitiporchai et al. (4) gave a semiaalytical solutio for the oliear vibratio of imperfect fuctioally graded plates based o higher-order shear deformatio theory with temperature depedet material properties. The sesitivity of the oliear vibratio characteristics of plates to the iitial geometric imperfectio was evaluated. I Yag et al. (4), a semi-aalytical Galerkidifferetial quadrature approach was employed to covert the goverig equatios ito a liear system of Mathieu Hill equatios. The iflueces of various parameters such as material compositio ad temperature chage o the dyamic stability, bucklig ad vibratio frequecies were demostrated through parametric studies. The stability of a fuctioally graded cylidrical shell subjected to axial harmoic loadig was discussed by Ng et al. (1). Patel et al. (5) coducted the fiite elemet aalysis for the free vibratio of elliptical composite cylidrical shells based o the high order shear deformatio theory. Sofiyev (4) ad Sofiyev ad Schack (4) studied the dyamic 8

18 stability of fuctioally graded shells uder a periodic impulsive loadig ad a liearly icreasig dyamic torsioal loadig, respectively. Large amplitude vibratio aalysis of pre-stressed fuctioally graded plates with both the temperature ad piezoelectric effects take ito cosideratio was studied by Yag et al. (3). Oe dimesioal differetial quadrature techique ad Galerki techique was adopted to obtai both liear ad oliear frequecies of selected plates with two opposite edges clamped. Studies cocerig the radom vibratio of composite plates (Choa 1985, Gray et al. 1985, Cederbaum et al ad 1989, Sigh et al. 1989, Abdelaser ad Sigh, 1993, Harichadra ad Naja 1997, ad Kag ad Harichadra 1999) ca be foud i the literature. The mea square respose of the oliear system is the focus of these studies. Typical umerical schemes ivolved were Mote Carlo simulatio, perturbatio method, ad equivalet liearizatio. A comprehesive review ca be foud i Ibrahim (1987) ad Maohar ad Ibrahim (1999). Gray et al. (1985) preseted a aalytical solutio for large amplitude vibratio ad radom respose of a symmetrically lamiated plate. Cederbaum et al. (1988) studied the radom vibratio of symmetric lamiated plates usig a high-order shear deformatio theory. Two cases of radom pressure fields, amely, ideal white oise ad turbulet boudary layer pressure fluctuatio, were cosidered. Numerical results were provided that could serve as referece for the reliability evaluatio of pertiet structures. Worde ad Maso (1998, 1999) used the Volterra series to approximate the frequecy respose fuctio (FRF) of a Duffig oscillator system uder radom excitatio. The composite 9

19 FRF for a two-degree-of-freedom system with cubic o-liearity uder a white Gaussia excitatio was computed. Dahlberg (1999) studied the modal couplig effects by examiig the respose of a simply supported beam subjected to a statioary radom loadig. Harichadra ad Naja (1997) used equivalet liearizatio i cojuctio with the fiite elemet method to perform o-liear radom vibratio aalysis of lamiated composite plates. A series represetatio of the o-liear shear stress-strai law was selected i the fiite elemet formulatio. However, trasverse shear deformatio was eglected i their aalysis. Kag ad Harichadra (1999) preseted a radom vibratio aalysis techique for lamiated fiber reiforced plastic plates via fiite elemet approach i which the material oliearity was expressed by a approximate fifth-order polyomial. Kitiporchai et al. (6) studied the vibratio of fuctioally graded plates exhibitig radomess i thermo-elastic properties of the costituet materials. A mea-cetered first-order perturbatio techique was adopted to obtai the secod-order statistics of vibratio frequecies. So far the discussio has bee focused o estimatig the system respose while the parameters of the origial systems are kow. O the other had, sometimes dyamic testig is coducted o the structures ad respose data is gathered from the system s dyamical respose such as displacemet, velocity, or acceleratio. Oe would like to kow the properties pertiet to the structure, i.e., atural frequecies, dampig etc. There are several methods to idetify the parameters of the system from the dyamical 1

20 behavior of oliear systems (Rice 1999, Che ad Tomliso 1996, Staszewski 1998, Boukhrist et al. 1999, ad Jaksic ad Boltezar ). However, whe the data cosists of sigals from all over the frequecy rage (white oise or arrow-bad white oise), traditioal modelig idetificatio methods have difficulties sice extra filterig process has to be coducted to remove the oise from the data so that the harmoic compoets ca be exposed. To aalyze sigal like this, the auto-regressive movig average (ARMA) model is a very powerful tool. It is also called Box-Jekis models amed after the people who developed it. A detailed itroductio ca be foud i Box et al. (1994) ad Chatfield (1989). I a ARMA model, the curret value of the system respose is expressed as a liear combiatio of past values plus a white oise. Oce the parameters are determied, atural frequecies, dampig ratios (if applicable) ca be obtaied from the autoregressive part of the model. The typical procedure for fittig a ARMA models to a time series ivolves model idetificatio, model fittig, ad model validatio. It should also be poited out that the applicatios of ARMA model are ot restricted to egieerig field. For istace, it has bee used i the aalysis of fiacial data such as stock market chages ad other ecoomical issues (Mills, 199). Tia ad Ta (1987) used ARMA time series model to study the iformatio of heart souds of ormal huma ad patiets with cardiovascular disorders. A cardiac fuctioal state which was determied from the ARMA parameters provided valuable iformatio o the iitiatio of heart-failure. Baek et al. (6) proposed a modelig method of the mass, the dampig coefficiet ad the stiffess of a cuttig system usig a autoregressive movig average (ARMA) model 11

21 ad a bisectio method. Yoo et al. (4) compared differet algorithms i estimatig the structural dyamic betwee the edmill ad workpiece of a cuttig system. Baek et al. (6) coducted parameter idetificatio o the experimetal data of sigle-degree-offreedom system usig ARMA model. Wag et al. (3) evaluated the oliear fluid force for a freely vibratig cylider over a wide rage of Reyolds umbers, mass ad structural dampig ratios. Smail ad Thomas (1999) compared the accuracy of three kids of ARMA methods (recursive, least-squares output error ad corrected covariace matrix method) i parameter idetificatio of certai simulatios ad experimetal data. Effects of model orders ad samplig frequecy were studied. It s foud that a good samplig frequecy raged from three to te times the maximal frequecy of iterest. This iformatio was used while ruig the simulatios for the beam ad plate i this dissertatio. Carde ad Browjoh (7) applied the ARMA modelig techique o the experimetal data from the IASC ASCE bechmark four-storey frame structure as well as two bridge structures. A health-moitorig algorithm was examied that distiguished a structure i a healthy state from that i a uhealthy state. Mattso ad Padit (6) used vector autoregressive (ARV) models to capture the predictable dyamic properties i the experimetal respose data. The stadard deviatio of the autoregressive residual series provided valuable iformatio o the locatio of damage i the structures. Soh ad Farrar (1) combied auto-regressive ad auto-regressive with exogeous iputs techiques ad coducted damage diagosis of a mass-sprig system with eight degrees of freedom. 1

22 Gautier et al. (1995) proposed a method of idetifyig the modal parameters of structures based o fidig the optimal value of the oise variace to correct the covariace matrix. The method was tested o several dyamic systems ad its advatage over time domai idetificatio methods was demostrated. Popescu ad Demetriu (199) aalyzed the acceleratio record of a earthquake groud motio data with parametric ARMA model. Mobarakeh et al. () used a time-varyig ARMA(,1) model to simulate several earthquakes recorded i Ira ad Mexico. Power spectral desity of iteral carotid arterial doppler sigals (Ubeyli ad Guler 4) was estimated by classical (fast Fourier trasform) ad model-based (autoregressive, movig average, ad ARMA) methods. Compariso was made amog the differet approaches ad it was foud that the autoregressive ad ARMA methods gave the better predictio of power spectral desity fuctios as well as the shapes soograms tha the fast Fourier trasform did. The compariso betwee ARMA modelig ad some traditioal methods ca also be foud i work by other researchers (Kaluzyski 1987, Vaitkus et al. 1988, Guler et al. 1995, Guler et al. 1996, ad David et al. 1997). 13

23 Chapter 3. Radom Vibratio of Geometrically Noliear Beams I this chapter the fudametals of radom vibratio of geometrically oliear beams are elaborated. Issues such as the solutio procedures, liearizatio techique, ad effects of oliearity ad modal iteractio are addressed. A ew equivalet coupled liearizatio approach is proposed ad compared with the traditioal equivalet ucoupled liearizatio method. The solutio to the radom vibratio of fourth order beams is obtaied with attetio paid to the effects of rotary iertia ad shear deformatio. 3.1 Solutios to the Noliear Radom Vibratio of Isotropic Beams The geometry of a simply supported beam subjected to uiform pressure is show i Figure 3.1. The oliear equatio of motio for the trasverse displacemet w(x,t) of a uiform beam (Foda 1999) ca be expressed as 14

24 4 4 4 w w w E w ρ I w EI + c + ρ A ρi x t t kg x t kg t 4 4 w EI w ρi w N x + 4 = x κ AG x KAG x t p (3.1) where ρ, A, E, G, ad I represet the desity, cross sectio area, modulus of elasticity, shear modulus, ad momet of iertia of the cross sectio of the beam, respectively, ad c is the dampig factor, L EA w Nx = N + dx represets the axial force, N is the L x exteral axial force ad assumed to be zero i the followig aalysis, ad κ is the shear correctio factor. Figure 3.1 A beam uder pressure The oliear equatio of motio for the trasverse displacemet w(x,t) of a isotropic beam without cosiderig the rotary iertia ad shear deformatio effects ca be expressed as 4 w w w w EI + c + ρ A N 4 x = p x t t x (3.) 15

25 It is oticed that from Eq. (3.1) to Eq. (3.), the order of the differetial equatio drops from four to two i the time domai. Followig the method of separatio of variables, the respose of the displacemet field ca be expressed as N w( x, t) = f ( x) q ( t) (3.3) = 1 where f ( x ) represets the -th eigefuctio which is determied by the boudary coditio ad q( t ) represets the time-depedat part of the -th modal respose. Selected choices of eigefuctios for beams with various support coditios are listed i Appedix A.1. Substitutig Eq. (3.3) ito Eq. (3.) ad applyig the Galerki s method by leftmultiplyig both sides of Eq. (3.3) by f ( x ) ad itegratig over the spa of to L, the followig equatio for the -th mode is obtaied after the orthogoality coditio is applied N N N q ( t) + βq ( t) + αq( t) + α, kqk ( t) γi, jqi ( t) qj ( t) = p( t) k = 1 i= 1 j = 1 (3.4) where 16

26 α β = α γ L '''' = L ρ A f ( x) dx, k ρ A f ( x) dx L '' k L ρ ( ) EA f ( x) f ( x) dx = AL f x dx L ' ' i, j = i j f ( x) f ( x) dx L c EI f ( x) f ( x) dx L 1 p ( t) = p( x, t) f( x) dx L ρ A f ( x) dx (3.5) Eq. (3.4) ca ot be solved aalytically due to the existece of oliear terms. Whe the load p( t ) is radom i ature, the property of iterest is the root mea square (R.M.S.) of the respose, which is defied by 1 σ = x lim µ T T [ x( t) ] / T T / x dt (3.6) where x(t) represets a statioary process that has a costat mea value of µ x, ad T stads for the period that is uder cosideratio. It ca be see that σ x is also the variace of the process x(t). I the equivalet ucoupled liearizatio method, a liearized equatio i the followig form is sought q ( t) + β q ( t) + α q ( t) = p ( t) (3.7), where β ad α, represet the dampig factor ad stiffess of the equivalet system. 17

27 It ca be see that i Eq. (3.6) differet modes of the beam are totally decoupled for each mode. However, recall that i Eq. (3.4) all the modes are actually coupled through the oliear terms. So it makes more sese if the equivalet equatio is writte i the liear coupled format as below N, k k = 1 q ( t) + β q ( t) + α q ( t) = p ( t) (3.8) It should be oted that by settig the o-diagoal stiffess terms, α, k with k, to zero, Eq. (3.8) is the same as the represetatio of the traditioal equivalet liearized equatio as show i Eq. (3.7). The differece betwee Eq.(3.8) ad (3.4) is N k = 1 Λ = α q ( t) + ( β β ) q ( t) α q ( t), k k N N N α, kqk ( t) γ i, jqi ( t) q j ( t) k = 1 i= 1 j= 1 (3.9) The goal is to fid the optimal values of α, ad β of the equivalet liearized system k so that the square of the differece betwee liear ad oliear systems is miimalized i the statistical sese. This requires that E β α, k E [ Λ ] = [ Λ ] = (3.1) where E[ ] stads for the mathematical expectatio. 18

28 To demostrate this procedure i Eq.(3.9) ad Eq.(3.1), we look at a simpler case a beam that is simply supported at both eds, i which Eq.(3.4) is simplified to the followig N ( ) + β ( ) + ( α,1 + α,1 + k k ( )) ( ) = ( ) q t q t q t q t p t (3.11) k = 1 where α c β = Aρ α 4 k π E = 4ρL, π κ EG = ρ L,1+ k 4 L π x p( t) = p( x, t)si( ) L L (3.1) The equivalet liear system i Eq.(3.8) is used which is listed below agai for the sake of coveiece N, k k k = 1 q ( t) + β q ( t) + α q ( t) = p ( t) The differece betwee Eqs.(3.1) ad (3.8) is N N α, kqk ( t) ( α,1 α,1 kqk ( t)) q( t) ( + β β) q( t) k = 1 k = 1 Λ = + + (3.13) Therefore, 19

29 α = E[ Λ ], k N N = E qk ( t) α, kqk ( t) E ( α,1 + α,1+ kqk ( t)) q( t) qk ( t) k = 1 k = 1 N N = α, ke q t qk t α,1 E q t qk t + α,1+ ke qk t E q t qk t k = 1 k = 1 [ ( ) ( )] [ ( ) ( )] 3 ( ) [ ( ) ( )] (3.14) ad β = E[ Λ ] = β β ( ) E q ( t) (3.15) which leads to N, k,1 3,1 ke + qk ( t) α = α + α β = β k = 1 (3.16) I the above derivatio, the followig relatioship ad defiitios are used uder the assumptio that both the load ad respose follow zero-mea Gaussia distributios (Soog 4): E q q 3 [ ] = E[ qq] = E[ q q ] = ( k ) k (3.17) ad E q E q 4 [ ] = 3 [ ] E q q = E q q E q 3 [ k ] 3 [ k ] [ k ] (3.18) Aother example is give i Appedix A. for a beam fixed o oe ed ad simply supported at the other. I that case, a complete set of quadratic terms maitais ad

30 makes the derivatio much more tedious. But the idea stays the same. The details are ot discussed here but show i the Appedix. Because of the couplig terms i Eq.(3.14), ot oly the E q for each mode eeds to be [ ] estimated, but also the cross momet terms such as E[ q q ] ( k). From the frequecy k domai aalysis, E q ad E[ q q ] ( k) for a liear system are determied by [ ] k E q G ω G ω S ω dω * [ ] = ( ) ( ) ( ) E q q G ω G ω S ω dω * [ k ] = ( ) k ( ) k ( ) (3.19) where G (ω ) is the frequecy respose fuctio ad G ( ) is its complex cojugate. Sk * ω ( ω ) represets the correspodig power spectral desity associated with the -th ad k-th excitatio p( t ) ad pk ( t ) i the modal equatios. For a liear system govered by Eq. (3.7), G (ω ) takes the form G ( ω) = 1 α ω + i β ω (, ) (3.) ad S ( ω ) are defied by k L L η( x) f( x) dx η( x) fk ( x) dx k ( ω) = SP ( ω) L L ( ρ A) f ( x) dx fk ( x) dx S (3.1) 1

31 where SP( ω ) represets the power spectral desity of the origial load p( x, t ) = η ( x) P( t) i Eq. (3.1). By defiitio, the power spectral desity is the Fourier trasform of the autocorrelatio fuctio R( τ ) of load p( x, t ) : iωτ S ( ω) = R( τ ) e dτ (3.) P I the traditioal equivalet liearizatio method, the correlatio betwee differet modes is ot cosidered due to the fact that the fial liearized equatios are decoupled. However, umerical simulatios idicate that uder certai boudary coditios, there are strog correlatios betwee the resposes of differet modes i the oliear problem. The value of correlatio factor ρ, k ( k) is calculated from the followig relatioship (uder the assumptio that both the load ad respose have zero mea Gaussia distributio) E[ q q ] ρ, k = ( k) (3.3) E q E q k [ ] [ k ] Fially, the displacemet R.M.S. of oliear radom vibratio of the beam ca be obtaied after a iteratio scheme that is similar to that of the ucoupled liearizatio method: (1) Takig the liear part of Eq.(3.8) oly ad calculate the first estimate of E q ad [ ] E[ q q ] via Eq.(3.19) to Eq.(3.1) for each of the N modes. k

32 () The values of E q ad E[ q q ] are the substituted ito relatioships such as [ ] k Eq.(3.16) or Eq.(A..1) to fid ew estimates of parameter α, ad β. (3) The α, ad β obtaied i step () are substituted ito Eq.(3.19) to Eq.(3.1) agai k k to obtai a ew estimate of E q. [ ] (4). Steps ()-(3) are repeated util a certai coverge criterio is achieved after i-th iteratios for all the E q cosidered, i.e., [ ] ( E[ q ]) ( E[ q ]) i ( E[ q ]) i i 1 < ε ( = 1,, 3... N ) where ε represet the desired accuracy ad usually take to be 1% or less. 3. Effect of Iertia of Rotatio ad Shear Deformatio I the previous sectio the oliear radom vibratio of the secod order beam is studied. I that study, the terms associated with the rotary iertia ad shear deformatio are eglected. I this sectio, i order to study how those terms affect the root mea square respose of the beam subjected to radom loadig, these effects are icluded i the goverig equatio. This results i a fourth-order differetial equatio i the time domai. The oliear equatio of motio for the trasverse displacemet w(x,t) of a isotropic beam is expressed i Eq.(3.1) 3

33 4 4 4 w w w E w ρ I w EI + c + ρ A ρi x t t kg x t kg t 4 4 w EI w ρi w N x + 4 = x KAG x KAG x t p For a beam simply supported at both eds, w(, t) = w( L, t) =. Assume the solutio for the beam is the summatio of the first N modes where N represets the total umber of modes cosidered. N π x w = si( ) q( t) (3.4) L Substitutig Eq.(3.4) ito Eq.(3.1) ad applyig the Galerki s method, the followig equatio for the -th mode is obtaied after some legthy maipulatio N + α,1 + α,1 + k k + α,+ N q ( t) ( q ) q ( t) q ( t) k = 1 N ( α,3 + N α,3+ N + kqk ( t)) q ( t) p( t) + + = k = 1 (3.5) where α α α α α AG E + G,1 = +, k + 1 4,+ N κ π ( κ ) ρi ρl 4 k π E = 4ρL cκ G = ρ I 4 4 π κ EG = ρ L,3+ N 4 4 k π E( π EI + κ AGL ) = 4I ρ L,3+ N + k 6 L κg π x p ( t) = p( x, t)si( ) Lρ I L (3.6) 4

34 It should be oted that the shear deformatio effect ad rotary iertia effect are embedded i terms such as α,1 ad α, + N i Eq.(3.6). Notice that i Eq.(3.6) each mode is coupled with the remaiig N-1 modes. The oliear couplig makes it impossible to apply the frequecy domai aalysis to obtai the root mea square of the displacemet respose. Now resortig to the equivalet liearizatio techique, a equivalet liearized goverig equatio for each mode i the followig form is sought: q ( t) + α q ( t) + β q ( t) + γ q ( t) = p ( t) (3.7) The differece betwee Eq.(3.5) ad (3.7) is N,1,1 + k k,+ N Λ = ( α α α q ( t)) q ( t) + ( β α ) q ( t) k = 1 N ( γ α,3 + N α,3+ N + kqk ( t)) q( t) + k = 1 (3.8) The goal is to fid α, β, ad γ so that E α E β E γ [ Λ ] = [ Λ ] = [ Λ ] = (3.9) where E[ ] stads for the mathematical expectatio. 5

35 Because of the couplig terms i Eq.(3.5), ot oly the [ ] E q, E q, ad E[ q q ] for [ ] each mode eed to be estimated, but also the cross terms such as E[ q q ] ( k). They are obtaied from the frequecy domai aalysis as explaied i the followig. Recall that for a liear system, the mea square respose is obtaied by Eq.(3.19) k * σ = G ( ω) G ( ω) S p( ω) dω where G ( ω ) is the frequecy respose fuctio ad G ( ) is its complex cojugate. * ω For a liear system govered by Eq.(3.7), G (ω ) takes the form 1 G ( ω) = ω α ω γ β ω 4 e e e + + i (3.3) Furthermore, via radom vibratio theory, followig formulas [ ] E q ad [ ] E q ca be calculated from the 1 E q S dω [ ] = ( ) ω 4 e e e ( ω αω + γ ) + ( βω ) 4 ω [ ] = ( ) ω E q S dω 4 e e e ( ω αω + γ ) + ( βω ) (3.31) where S ( ω ) represets the power spectral desity of the excitatio p( t ). The result for E[ q q ], o the other had, ca be obtaied from the autocorrelatio betwee q ad q : 6

36 ρ q, q = E( q q ) E( q ) E( q ) ( ) ( ) ( ) ( ) E q E q E q E q (3.3) Sice the correlatio betwee the displacemet ad acceleratio is ukow, we seek help from simulatio results. After geeratig 1 series of data with legth of 14, statistical evaluatio of the correlatio factor was coducted based o Eq.(3.3). Numerical simulatios were ru for differet beams with differet boudary ad loadig coditios. It was foud out that the value of ρ fell ito a cosistet rage of -.88 to q q -.8. For the purpose of simplicity, the value of -.88 was used i the aalytical aalysis. Evetually, the value of E[ q q ] is calculated from the followig relatioship (uder the assumptio that both the respose ad associated acceleratio have zero mea) E q q ρ E q E q (3.33) [ ] = q, [ ] [ ] q. Acceleratio, 1 6 <m/s Displacemet, m Figure 3. Correlatio betwee displacemet ad acceleratio for a typical liear beam (data size: 14 ) 7

37 .8 Acceleratio, 1 6 <m /s Displacemet, m Figure 3.3 Correlatio betwee displacemet ad acceleratio for a typical oliear beam (data size: 14 ) Solvig Eq.(3.9) simultaeously yields the equivalet system parameters α, β ad γ. Durig this process, the followig statistical properties are applied uder the assumptio that both the load ad respose follow the zero-mea Normal distributio. E q q E q E q q 3 [ ] = 3 [ ] [ ] E[ q q ] = E[ q ] E[ q ] + E[ q q ] E q q 3 [ ] = E[ qq] = (3.34) E[ q q ] = E[ q q ] = E[ q q ] = E[ q q ] = ( k ) k k k 8

38 Ad, the followig defiitios are used E E E 1 = E q [ ] = E q [ ] = E[ q q ] 3 (3.35) Now Eq.(3.9) ca be writte i the followig explicit form: N = E[ Λ ] = E ( α α,1 α,1 + kqk ( t)) q ( t) α k = 1 N + E q ( t)( γ α,3 + N α,3+ N + kqk ( t)) q ( t) k = 1 + ( β α ) E q ( t) q ( t) [ ],+ N N = ( α α,1 ) E q ( t) α,1+ ke qk ( t) E q ( t) k = 1 [ ] ( ) [ ] + α, + 1 E q( t) q( t) + γ α,3+ N E q( t) q( t) N E[ q ( t) q ( t) ] ( α,3 N ke + + qk ( t) ) + α,3 N E + + q ( t) k = 1 N = ( α α,1 ) E α,1 + kek1 E + α, + 1E3 + ( γ α,3+ N ) E 3 k = 1 N E3 ( α,3+ N + kek1 ) + α,3+ N + E1 (3.36) k = 1 β = E[ Λ ] = ( β α, + N ) E q ( t) (3.37) 9

39 γ = E[ Λ ] N = ( γ α,3 N ) E + q ( t) ( α,3+ N + ke qk ( t) ) k = 1 ( ) [ ] + α,3 N E q ( t) + + E q ( t) + α α,1 E q( t) q( t) N k = 1 [ ] α [ ] ( α,1 + ke qk ( t) ) E q ( t) q( t), + 1E q ( t) E q( t) q( t) N ( γ α,3+ N ) E 1 ( α,3+ N + kek1 ) + α,3+ N + E1 E 1 k = 1 = N,1 3,1+ k k1 3, k = 1 ( ) + α α E ( α E ) E α E E (3.38) From Eq.(3.3)-Eq.(3.34), we have N,1,1+ k k1 k = 1 α = ( α + α E ) β = α,+ N N, + 1 3,3 + N,3+ + N 1,3+ N + k k1 k = 1 γ = α E + α + α E + α E (3.39) It is recalled that based o the aalysis i the past, the dampig term i the equivalet system would stay the same because there is o oliear term i the dampig coefficiet i Eq.(3.5). The results i Eq.(3.39) also verifies that coclusio. Fially, the root mea square of oliear radom vibratio of the fourth-order beam ca be obtaied after a iteratio scheme that is described as follows 3

40 (1) Takig the liear part of Eq.(3.5) oly ad calculate the first estimate of [ ] E q, E q, ad E[ q q ] via Eqs.(3.3) to (3.33) for each of the N modes. [ ] () The values of [ ] E q, E q, ad E[ q q ] are the substituted ito Eq.(3.39) to fid [ ] ew estimates of parameter α, β ad γ. (3) The α, β ad γ obtaied i step () are substituted ito Eq.(3.31) agai to obtai a ew estimate of [ ] E q. (4). Steps ()-(3) are repeated util a certai coverge criterio is achieved after k-th iteratios for all the [ ] E q cosidered, i.e., ( E[ q ]) ( E[ q ]) k ( E[ q ]) k k 1 < ε ( = 1,, 3... N ) where ε represet the desired accuracy ad usually take to be 1% or less. 3.3 Numerical Results Numerical studies are coducted usig the procedures discussed above. The beams used i the study have the same cross sectio aspect ratio (height/width = ) but differet legth/thickess ratios. The legth of the beam is fixed at L = 1 m for the purpose of simplicity, ad the beam is made from material that has a modulus of elasticity E = 7 GPa, ad a desity ρ = 3 kg/m 3. A dampig factor c = 1 N s/m is used. 31

41 Two types of loads are cosidered: a uiformly distributed pressure load over the whole spa of the beam ad a half-uiformly distributed pressure load over the left half spa of the beam, as show i Figure 3.4. Both loads have the spectrum of that of a bad-limited white oise. Figure 3.5 shows the load history durig a spa of four secods. The power spectral desity of these two loads is plotted i Figure 3.7. Oe of the key reasos to choose these two loads is that they represet the symmetric ad usymmetrical type of loadig, respectively. As a result, if the boudary coditios are also symmetric, oly the odd modes will be excited uder symmetric loadig coditio, otherwise all the modes will be excited. Results from the umerical study aim at addressig the followig issues: differece betwee the liear ad oliear radom vibratio aalysis advatage of the equivalet coupled liearizatio method over the equivalet ucoupled liearizatio method, if there is ay effects of rotary iertia ad shear deformatio o the respose of the beam uder radom excitatio impact of differet boudary or loadig coditios o the respose of the beam 3

42 Figure 3.4 Two types of loads used i the simulatio 3 15 Load, N Time, sec Figure 3.5 A typical statioary Gaussia radom process (time domai) 33

43 .16 Relative frequecy Load, N Figure 3.6 Histogram of the radom process i Figure 3.5 load/p, 1/Hz Frequecy/ωc, Hz Figure 3.7 PSDs of the two types of loads used i the simulatio ( ω c :cut-off frequecy) The results i Figures show the differece betwee the liear ad oliear system root mea square values at differet white oise spectral desity levels for a beam with differet cross sectios (F: fixed; SS: simply-supported) ad boudary coditios. It is observed that there is sigificat differece betwee the liear ad oliear R.M.S. 34

44 values. The oliear terms play a very importat role at relatively high spectral desity levels. Displacemet R.M.S., mm h = b, L = 1h = 1 m liear ucoupled coupled Square Root of Load PSD Level, Pa/ Hz Figure 3.8 Displacemet R.M.S. of a uiformly loaded F-SS beam vs. differet radom loadig PSD levels 4 Displacemet R.M.S., mm h = b, L= 1 h = 1 m liear ucoupled coupled Square Root of Load PSD Level, Pa/ Hz Figure 3.9 Mode 1 displacemet R.M.S. of a half-uiformly loaded F-SS beam vs. differet radom loadig PSD levels 35

45 1 Displacemet R.M.S., mm h = b, L= 1 h = 1 m liear ucoupled coupled Square Root of Load PSD Level, Pa/ Hz Figure 3.1 Mode displacemet R.M.S. of a half-uiformly loaded F-SS beam vs. differet radom loadig PSD levels 3 Displacemet R.M.S., mm 1 h = b, L = h = 1 m liear oliear ucoupled Square Root of Load PSD Level, Pa/ Hz Figure 3.11 Displacemet R.M.S. (summatio of first two modes) of a uiformly loaded SS-SS beam vs. differet radom loadig PSD levels It should be oted that for the case i Figure 3.11 the oliear coupled ad ucoupled methods give really close results. Ad the two curves almost overlap each other. 36

46 Therefore, oly the oliear coupled results are show ad referred to as oliear for the purpose of simplicity. The modal iteractio effect is examied. The effects of boudary coditio as well as the loadig coditio o modal iteractios are demostrated by the results i Figures 3.1 to I Figure 3.1 where the beam is subjected to a uiform load, the differece betwee the two approaches is egligible. Although the results from the two approaches start to separate i Figure 3.13 as the load icreases, the relative R.M.S. magitude of the mode is still very small compared to that of mode 1 i this case, so it s ot goig to chage the overall respose of the beam much. I Figure 3.14 ad Figure 3.15, however, for the same beam with the same boudary coditio, sigificat differece is observed betwee the two methods while the uiform load is replaced by the half-uiform load. Also i Figure 3.14 ad 3.15, differet legth/thickess ratios are used. The sleder beam that has a larger legth/thickess ratio demostrates larger differece betwee the two approaches. Geerally speakig, because of the fact that the atural frequecies of the beams are well separated (refer to Appedix A.1), large differece betwee the coupled ad ucoupled method is ot expected. But i systems where there exist two or more atural frequecies that are very close to each other, modal couplig would have a big impact o the results. Also, whe the load is ot uiform ad the beam is subjected to boudary coditio that is ot simply supported, the liear coupled approach usually yields more accurate results. I these scearios, the liear coupled method is highly recommeded over the ucoupled (equivalet liearizatio) techique. 37

47 6 Displacemet R.M.S, mm h = b, L = 1h = 1 m coupled ucoupled Square Root of Load PSD Level, Pa/ Hz Figure 3.1 Couplig effect o mode 1 for a uiformly loaded F-SS beam 5 Displacemet R.M.S, mm h = b, L = 1h = 1 m coupled ucoupled Square Root of Load PSD Level, Pa/ Hz Figure 3.13 Couplig effect o mode for a uiformly loaded F-SS beam 38

48 13 Displacemet R.M.S, mm coupled, L/h = coupled, L/h = 1 ucoupled, L/h = ucoupled, L/h = Square Root of Load PSD Level, Pa/ Hz Figure Couplig effect o mode 1 for a half uiformly loaded F-SS beam ( h = b, L = 1 m ) Displacemet R.M.S, mm ucoupled, L/h = ucoupled, L/h = 1 coupled, L/h = coupled, L/h = Square Root of Load PSD Level, Pa/ Hz Figure Couplig effect o mode for a half uiformly loaded F-SS beam ( h = b, L = 1 m ) I order to test the accuracy of results obtaied from the liear-coupled or equivalet liearizatio method, simulatios of selected beams are coducted. The simulatio is ru 39

49 twety times with a sample size of 14 ad the mea values are listed i Table 3.1 to Table 3.4 alog with the aalytical predictios. The time step is chose to be 1/ ω ( ω : cut-off frequecy) secod to make sure the secod mode frequecy is well covered. I other words, the Nyquist frequecy coditio is satisfied. c c Figure 3.16 ad Figure 3.17 show the displacemet resposes of first two modes of a fixed-simply supported beam uder half-uiform load (refer to Table 3.1 for beam geometry ad loadig/boudary coditios) durig a four-secod spa. I Figure 3.18 to Figure 3.19, the FFT (fast Fourier trasform) plots of the resposes of the two modes are displayed. The peaks i the figures idicate that there are iheret harmoic compoets i the respose. The peak positios i these two figures idicate that the mode 1 has sigificat ifluece o the mode respose. O the other had, the ifluece of mode two o mode 1 is egligible eve though the R.M.S. values of the first two modes are o the same order (as is show i Table 3.1). The ormalized histogram of the first two modes of oe oliear beam respose is show i Figure 3. ad 3.1. The correspodig theoretical probability desity fuctios (PDF) of ormal distributio based o zero mea ad calculated R.M.S. value are also preseted i dashed lies. Recall that a assumptio is made i the very begiig of this study that the respose for a liear system subjected to ormally distributed load also follows ormal distributio. Although for oliear system the shape of the histogram may ot follow that of a ormal distributio perfectly, the area 4

50 uder the ormalized histogram closely matches the area uder the theoretical PDF curve..3 Displacemet, mm Time, sec Figure 3.16 Typical mode 1 displacemet respose (correspodig to data i Table 3.1).15 Displacemet, mm Time, sec Figure 3.17 Typical mode displacemet respose (correspodig to data i Table 3.1) 41

51 FFT of displacemet, mm Frequecy, Hz Figure 3.18 Typical FFT of mode1 displacemet respose (correspodig to data i Table 3.1) FFT of displacemet, mm Frequecy, Hz Figure 3.19 Typical FFT of mode displacemet respose (correspodig to data i Table 3.1) 4

52 .1 Relative frequecy Displacemet, m Figure 3. Histogram of mode 1 displacemet respose (correspodig to data i Table 3.1, sample size: 14 ).1 Relative frequecy Displacemet, m Figure 3.1 Histogram of mode displacemet respose (correspodig to data i Table 3.1, sample size: 14 ) Good agreemet betwee the aalytical approach ad simulatio is foud. For the mode 1 results i Table 3.1 ad Table 3., the aalytical predictios from liear-coupled 43

53 approach are much closer to simulatio results tha the values from ucoupled approach are. More results are demostrated i Figures 3. to 3.9 for two Fixed-SS beams subjected to various load PSD levels. I these figures the beam has a differet cross sectio from those i Table 3.1 ad 3.. It ca be see that for both mode 1 ad mode, the liear coupled method yields closer results to those obtaied from simulatios. It should be oted that the beam examied i all these figures is asymmetrically supported ad loaded. Table 3.1 Respose of a beam (mm) with F-SS boudary coditio ad subjected to half uiform load (load PSD = 1 Pa /Hz, h = b, L = 1h = 1 m ) simulatio liear coupled ucoupled & w/ corre. ucoupled & w/o corre. mode mode Table 3. Respose of a beam with F-SS boudary coditio ad subjected to half uiform load (load PSD = 1 Pa /Hz, h = b, L = h = 1 m ) simulatio liear coupled ucoupled & w/ corre. ucoupled & w/o corre. mode mode Table 3.3 Respose of a beam (mm) with F-SS boudary coditio ad subjected to uiform load (load PSD = 1 Pa/Hz, h = b, L = 1.5h = 1 m ) simulatio liear coupled ucoupled & w/ corre. ucoupled & w/o corre. mode mode Tabel 3.4 Respose of a beam (mm) with F-Fixed boudary coditio ad subjected to half uiform load (load PSD = 1 Pa /Hz, h = b, L = 1.5h = 1 m ) simulatio liear coupled ucoupled & w/ corre. ucoupled & w/o corre. mode mode

54 Displacemet R.M.S., mm simulatio coupled ucoupled Square Root of Load PSD Level, Pa/ Hz Figure 3. Compariso amog differet approaches of mode 1 respose for a half uiformly loaded Fixed-S (L = 1 m, h/b = 1, L/h = 1) Displacemet R.M.S., mm simulatio coupled ucoupled Square Root of Load PSD Level, Pa/ Hz Figure 3.3 Compariso amog differet approaches of mode respose for a half uiformly loaded Fixed-SS beam (L = 1 m, h/b = 1, L/h = 1) 45

55 Displacemet R.M.S., mm simulatio coupled ucoupled Square Root of Load PSD Level, Pa/ Hz Figure 3.4 Compariso amog differet approaches of mode 1 respose for a half uiformly loaded Fixed-SS beam (L = 1 m, h/b = 1, L/h = ) Displacemet R.M.S., mm simulatio coupled ucoupled Square Root of Load PSD Level, Pa/ Hz Figure 3. 5 Compariso amog differet approaches of mode respose for a half uiformly loaded Fixed-SS beam (L = 1 m, h/b = 1, L/h = ) 46

56 Displacemet R.M.S., mm simulatio coupled ucoupled Square Root of Load PSD Level, Pa/ Hz Figure 3. 6 Compariso amog differet approaches of mode 1 respose for a half uiformly loaded Fixed-SS beam (L = 1 m, h/b =, L/h = 1) Displacemet R.M.S., mm simulatio coupled ucoupled Square Root of Load PSD Level, Pa/ Hz Figure 3. 7 Compariso amog differet approaches of mode respose for a half uiformly loaded Fixed-SS beam (L = 1 m, h/b =, L/h = 1) 47

57 Displacemet R.M.S., mm simulatio coupled ucoupled Square Root of Load PSD Level, Pa/ Hz Figure 3. 8 Compariso amog differet approaches of mode 1 respose for a half uiformly loaded Fixed-SS beam (L = 1 m, h/b =, L/h = ) Displacemet R.M.S., mm simulatio coupled ucoupled Square Root of Load PSD Level, Pa/ Hz Figure 3.9 Compariso amog differet approaches of mode respose for a half uiformly loaded Fixed-SS beam (L = 1 m, h/b =, L/h = ) 48

58 O the other had, i Table 3.3 ad Table 3.4, the simulatio, liear-coupled, ad ucoupled cases geerate very close values. I other words, for these two beams the choice of solutio approach does ot really matter, ad oe may just seek the ucoupled method due to its simplicity ad efficiecy. Same coclusio ca be draw for a fixedfixed beam subjected to a half uiform load, as show i Figure 3.3. Notice that either the support coditio or the load is symmetric for the beam discussed i Table 3 ad 4, ad Figure 3.3. Displacemet R.M.S., mm ucoupled coupled mode 1 mode Square Root of Load PSD Level, Pa/ Hz Figure 3. 3 Compariso amog differet approaches of mode 1 ad resposes for a half uiformly loaded Fixed-Fixed beam (L = 1 m, h/b =, L/h = ) Comparisos betwee the results of 4 th order beam ad d order beam are show i Tables 3.5 to e 3.8, ad Figures 3.31 to It ca be see that the differece betwee the two cases is almost egligible eve whe the load is high eough to cause a R.M.S. deflectio about % of the legth of the beam. Furthermore, the iteratio takes much loger i the 4 th order case compared to the d order case. For the cases discussed here 49

59 (refer to data preseted i Table 3.5 to Table 3.8), the effect of shear deformatio ad rotary iertia effects o the radom vibratio of oliear beam is ot sigificat. Table 3.5 Compariso of R.M.S. respose of a d order beam with that of 4 th order beam (SS-SS boudary coditio with uiform load, h = b, L = 1h = 1 m ) Square root of mode 1 R.M.S., mm mode R.M.S., mm load PSD, Pa/Hz 1/ d order 4th order % diff d order 4th order % diff % % % % % % % % % % Table 3.6 Compariso of R.M.S. respose of a d order beam with that of 4 th order beam (SS-SS boudary coditio with half-uiform load, h = b, L = 1h = 1 m ) Square root of mode 1 R.M.S., mm mode R.M.S., mm load PSD, Pa/Hz 1/ d order 4th order % diff d order 4th order % diff % % % % % % % % % % Table 3.7 Compariso of R.M.S. respose of mode 1 of d order beam with that of 4 th order beam (F-SS boudary coditio with half-uiform load, h = b, L = 1h = 1 m ) Square root of mode 1 R.M.S., mm load PSD, Pa/Hz 1/ d order w/o C.P. d order w/ C.P. 4th order

60 Table 3.8 Compariso of R.M.S. respose of mode of d order beam with that of 4 th order beam (F-SS boudary coditio with half-uiform load, h = b, L = 1h = 1 m ) Square root of mode R.M.S., mm load PSD, Pa/Hz 1/ d order w/o C.P. d order w/ C.P. 4th order Oe other iterestig poit to be oticed it that the results i Table 3.7 ad 3.8 idicate the 4 th order approach yields results very close to those from d order equivalet coupled liearizatio method. I situatios where there is sigificat differece betwee the d order equivalet coupled ad ucoupled liearizatio approaches, the results from the 4 th order equivalet liearizatio method matches those from the d order coupled method very well. But oce agai, i terms of computatio time, the d order method is much more efficiet tha the 4 th order oe. Displacemet R.M.S., mm h = b, L = 1h = 1 m d order 4th order Square Root of Load PSD Level, Pa/ Hz Figure Mode 1 R.M.S. respose of a SS-SS beam subjected to uiform load 51

61 Displacemet R.M.S., mm 8 4 h = b, L = 1h = 1 m d order 4th order Square Root of Load PSD Level, Pa/ Hz Figure 3. 3 Mode R.M.S. respose of a SS-SS beam subjected to uiform load Displacemet R.M.S., mm h = b, L = 1h = 1 m 6 d order coupled 4th order d order ucoupled Square Root of Load PSD Level, Pa/ Hz Figure Mode 1 R.M.S. respose of a F-SS beam subjected to half-uiform load 5

62 Displacemet R.M.S., mm h = b, L = 1h = 1 m d order coupled 4th order d order ucoupled Square Root of Load PSD Level, Pa/ Hz Figure Mode R.M.S. respose of a F-SS beam subjected to half-uiform load I summary, a ew coupled liearizatio method is proposed i this chapter. It proves to yield closer results to umerical simulatio data compared with the traditioal equivalet liearizatio method. The radom vibratio of fourth order beams is studied i which the effects of rotary iertia ad shear deformatio are ivestigated. Based o the umerical examples obtaied, those effects have ot bee foud to have sigificat ifluece o the R.M.S. values of displacemet respose. 53

63 Chapter 4. Noliear Radom Vibratio of Composite Plates I this chapter the techiques discussed i Chapter 3 are exteded to solve the oliear radom vibratio problem of composite plates. Classical plate theory (CPT) ad the first order shear deformatio theory (FSDT) i both liear ad oliear forms are used to obtai the R.M.S. respose of the trasverse displacemet. The differece betwee the two theories is demostrated through umerical examples. Geometric oliear effect is examied. Temperature effects o the R.M.S. values of both displacemet ad stress compoets are also studied. 4.1 Goverig Equatios Figure 4.1 (without bedig momets) ad Figure 4. (bedig momets oly) show the free body diagrams of a rectagular plate elemet i geeral loadig coditio. 54

64 Figure 4.1 Free body diagram of a rectagular plate elemet (without bedig momets) Figure 4. Free body diagram of a rectagular plate elemet (bedig momets oly) 55

65 The equatios of equilibrium (without i-plae loadig, dampig terms ot show for the purpose of simplicity) ca be obtaied by the priciple of virtual work. The detailed derivatio ca be foud i Appedix B. T T N N xx xy N N xx xy u φ = I + I 1 x y x y t t T T N xy N yy N xy N yy v ψ = I + I 1 x y x y t t V V x y w w T w T w + + N xx + N xy + N xx + N xy x y x x y x y w w T w T w + N xy + N yy + N xy + N yy = I y x y y y w t M M x y x y t t T T M xx xy M xx xy φ u Vx = I + I 1 x y x y t t T T M xy M yy M xy M yy ψ v Vy = I + I 1 (4.1) where I = h / i h / i ρ z dz (i = 1,, 3), ad the force resultats N s ad momet resultats M s ca be obtaied from the relatioship ad w x N xx A11 A1 A16 ε xx α xx T B11 B1 B16 w N yy A1 A A 6 ε yy α yy T B1 B B = + 6 y N xy A16 A6 A 66 γ xy α xy T B16 B6 B 66 w x y M xx M = yy M xy w x B11 B1 B16 ε xx α xx T D11 D1 D16 w B1 B B 6 ε yy α yy T D1 D D + 6 y B16 B6 B 66 γ xy α xy T D16 D6 D 66 w x y (4.) 56

66 w + ψ Vx A44 A45 y = K V y A45 A 55 w + φ x (4.3) where the oliear strais i the vo Karma sese are expressed as u 1 w + x x ε xx v 1 w ε yy = + y y γ xy u v w w + + y x x y (4.4) The defiitios for stiffess costats A ij, v, w, ψ, ad φ ca be foud i Appedix B. B ij, D ij ad the five displacemet fuctios u, Substitutio of Eqs. (4.)-(4.4) ito Eq. (4.1) yields the goverig equatios of motio i terms of the displacemet fuctios for the first order shear deformatio theory (FSDT): (o i-plae or thermal load, dampig terms ot show for the purpose of simplicity) u w w v w w u v w w w w A A A x x x y x y y x y x y x y y x y φ ψ φ ψ u φ + B11 + B 1 + B66 + I = + I 1 x x y y y x t t (4.5) 57

67 u w w v w w u v w w w w A A A x y x x y y y y y x x x y x y x φ ψ φ ψ v ψ + B1 + B + B 66 + I = + I 1 x y y x y x t t (4.6) w ψ w φ u w w v w w KA44 + KA A A y y x x x x x y x y y x u v w w w w φ ψ φ ψ w + A B B 1 + B66 + y x y x y y x y x x y y y x x u w w v w w u v w w w w + A1 + + A + A x y x x y y y y y x x x y x y x φ ψ φ ψ w u 1 w w + B1 + B + B 66 + A x y y x y x y x x x v 1 w w φ w A1 B11 A u v w w φ ψ w B66 + y y y x x x y x y y x x y u 1 w w v 1 w w φ ψ w w + A1 + + A + + B1 + B + q = I (4.7) x x x y y y x y y t u w w v w w u v w w w w B B B x x x x y y x y y y x y x y x y w u D φ 11 D ψ 1 D φ ψ 66 A 55 I φ κ + φ = + I 1 x y x y y x x t t (4.8) 58

68 u w w v w w u v w w w w B B B x y x x y y y y x y x y x x x y w v D φ 1 D ψ D φ ψ 66 κ A 44 ψ I ψ = + I 1 x y y x y x y t t (4.9) where κ is the shear correctio factor ad take to be 5/6 i the followig discussio, ad q is the trasverse loadig. It should be oted that the dampig terms are ot displayed i the above equatios. I the umerical examples i this chapter, proportioal dampig is assumed so that there are o coupled dampig terms i the modal coordiates. The liear couterparts to Eqs. (4.5)-(4.9) take the form u w w v w w u v w w w w A A A x x x y x y y x y x y x y y x y = I u + I t 1 φ t (4.1) u w w v w w u v w w w w A A A x y x x y y y y y x x x y x y x = I v + I t 1 ψ t (4.11) w ψ w φ w KA44 + KA q I + = y y x x t (4.1) w u D φ 11 D ψ 1 D φ ψ 66 A 55 I φ κ + φ = + I 1 x y x y y x x t t (4.13) w v D φ 1 D ψ D φ ψ 66 κ A 44 ψ I ψ = + I 1 x y y x y x y t t (4.14) 59

69 4. Stochastic Respose of Liear System Next the liear system of equatios (Eqs. (4.1)-(4.14)) is used to demostrate the procedures to calculate the root mea square values of the displacemet compoets. I order to study the respose of the plate uder radom loadig, the atural frequecies of the plate eed to be kow first. So as a first step, the solutio to the goverig equatios is assumed to take the followig form: u( x, y, t) = U cos( λ x)si( µ y) e iωt v( x, y, t) = V si( λ x)cos( µ y) e iωt w λ µ ) iωt ( x, y, t) = W si( x)si( y e (4.15) φ ( x, y, t) = Φ cos( λx)si( µ y) e iωt ψ ( x, y, t) = Ψ si( λx)cos( µ y) e iωt where U, V, W, Φ, ad Ψ are ukow costats to be determied. For a plate simply supported o both x ad y directios, mπ λ =, a π µ =, with a, b represetig the legth b ad width of the plate, respectively, ad (m, ) stads for the mode umber. Notice that Eq.(4.15) satisfies the simple-supported boudary coditios 6

70 v w = ψ = N xx = M = at x =, a ad = xx u w = φ = N yy = M = at y =, b (4.16) = yy After substitutig Eq. (4.15) ito Eqs. (4.1)-(4.14) ad rearragig terms, oe obtais ([L]-ω [I] ) U V W Φ Ψ = (4.17) where [L] is fuctio of A ij, B ij, D ij, λ ad µ (expressio ot show here), ad [I] is a matrix composed of I i defied by I = h / i h / i ρ z dz (i = 1,, 3). The atural frequecies of the plates are obtaied by settig the determiat of ([L]- ω [I] ) to zero. Det( [L]-ω [I] ) = (4.18) Next from the theory of radom vibratio, the R.M.S. of the trasverse displacemet of the composite plate subjected to a excitatio with a spectral desity of S F x, y, x, y, ) is foud to be ( 1 1 ω kπ x lπ y mπ x π y * w kl m klm k, l m, a b a b = (4.19) σ si si si si G ( ω) G ( ω) S ( ω) dω * kl ω where (k, l) ad (m, ) represet the mode umbers, G ( ) is the cojugate of the frequecy respose fuctio G (ω ) that takes the form of kl 61

71 G kl 1 ( ω) = (4.) ω ω + iξ ω ω ad S (ω ) is the geeralized power spectral desity defied by klm kl kl kl S klm kπ x lπ y mπ x π y η( x, y)si si dxdy ( x, y) si dxdy A A ( ) a b η ω = a b S ( ) F ω kπ x lπ y kπ x lπ y I si si dxdy si si dxdy A a b A a b (4.1) where SF ( ω ) is the power spectral desity fuctio of load q( x, y, t) = η( x, y) F( t). For white oise excitatio, we have S F ( x y x y ω = S where S = costat (4.) 1, 1,,, ) So, oce the atural frequecies are solved from Eq. (4.18), Eq. (4.19)-Eq.(4.) ca be used to obtai the R.M.S. of the plate displacemet uder radom excitatio. It has bee foud that whe the atural frequecies of a plate are well separated, ad for the case of light dampig, oly autocorrelatios terms eed to be cosidered i Eq. (4.19). As a result, Eq. (4.19) ca be simplified to mπx πy * σ w = (si ) (si ) Gm ( ω) Gm ( ω) S mm ( ω) dω (4.3) a b m, 6

72 4.3 Stochastic Respose of Noliear System Usig the first order shear deformatio theory (FSDT), for a plate simply supported o all edges we assume mx y u( x, y, t) = U m( t) cos( x)si( y) a b m, mx y v( x, y, t) = Vm ( t)si( x) cos( y) a b m, mx y w( x, y, t) = Wm ( t)si( x)si( y) a b m, mx y φ( x, y, t) = Φm ( t)cos( x)si( y) a b m, mx y ψ ( x, y, t) = Ψ m ( t)si( x) cos( y) a b m, (4.4) Substitutig Eq.(4.4) ito Eq.(4.1) ad applyig the Galerki method, the followig coupled oliear system of equatios i the matrix form are obtaied: [ ] [ M ]{ X } + [ C]{ X } + K { X} + { } = { Q} (4.5) where {X} is the traspose of the vector { U 11 ( t) V 11 ( t) W 11 ( t) Φ 11 ( t) Ψ 11 ( t) Φ m( t) ( t)} Ψ, } cotais both quadratic ad third-order terms of the displacemet m { NL fuctios ad their cross-products. The previous procedure to obtai the R.M.S. of the liear system does ot apply to Eq. (4.5) due to the existece of oliear terms { NL }. L NL 63

73 Followig the method of equivalet liearizatio, it is assumed that the oliear system i Eq. (4.5) ca be coverted ito a equivalet liear system as show i Eq. (4.6). [ M ]{ X } + [ C]{ X } + [ K ]{ X} = { Q} (4.6) eq The differece betwee Eq.(4.5) ad Eq.(4.6) is { e} = [ K ]{ X} [ K ]{ X} { } (4.7) eq L NL Uder the assumptio that both the excitatio ad respose are Gaussia, the error i Eq.(4.7) is miimize i the statistical sese by requirig T E[{ e}{ e} ] k ij ij = (4.8) T where the subscript ij represets the ij-th elemet of the correspodig [{ e }{ e} ] or [ K] eq matrix. The symbol E [ ] i Eq.(4.8) stads for the mathematical expectatio. It is oted that the solutio to each elemet k ij of matrix [ K] eq is a fuctio of all the E W (m, = 1,,, N, if N modes are take ito accout). The umber of ukow [ m] costats to be solved i matrix [ K] eq is N. Eq. (4.8) provides just eough equatios to solve for those ukows. However, the first few modes usually domiate i terms of cotributio to the R.M.S. values. 64

74 I summary, the solutio procedure to obtai the R.M.S. of the oliear system trasverse displacemet respose is as follows: (1) By modal aalysis, the liear system (by settig { NL} = i Eq. (4.5)) ca be decoupled ito the followig N 5 equatios through modal trasformatio { X} = [ u]{ q} u( j,3) q mj + ξωmjq mj + ωmjq = Fj ( t) (4.9) M where [u] is the trasformatio matrix composed of modal vectors, j = 1,,, 5N (N is the umber of modes cosidered), u( j, 3) represets the elemets o the third colum of the matrix [ u ], fuctio. mj M mj represets the j-th modal mass, ad F j (t) represets the modal load From radom vibratio theory, the R.M.S. value of modal coordiate q mj ca be determied from u ( j,3) E[ q ] H ( ω) H ( ω) S( ω) dω (4.3) * mj = j j M mj where S( ω ) is spectral desity of the load F j (t) ad H ( ω ) is the frequecy respose fuctio correspodig to the j-th modal coordiate j 1 H j ( ω) = ω ω + iζ ω ω mj mj mj (4.31) where ω ad ζ represet the j-th frequecy withi the (m, ) mode ad mj mj correspodig dampig ratio, respectively. 65

75 Fially, the mea square value of the origial displacemet compoet [ m] E W ca be calculated from: 5 E[ Wm ] = um(3, j) E[ qmj ] (4.3) j= 1 (). This value [ m] E W is the substituted ito Eq.(4.8) to obtai a ew equivalet matrix [ K eq ]. (3). [ K eq ] is the substituted ito Eq.(4.6) to fid a ew estimate of [ m] E W. (4). Steps ()-(3) are repeated util a certai coverge criterio is achieved after k-th iteratio for all the [ m] E W cosidered, i.e., ( E[ Wm ]) ( E[ Wm ]) k ( E[ Wm ]) k k 1 < ε where ε represets the pre-set error ad usually take to be 1% or less. The above theory is applied i the followig umerical example. Cosider a symmetric cross-ply boro-epoxy lamiate ( / 9 / 9 / ) with a total thickess h, legth a ad width b, ad with the followig lamia orthotropic material properties (Reddy, 1997): E E E E υ G G 3 1 = 6.84 GPa, = 3 =.1 1, 1 =.5, 1 = 13 = 1.34 GPa, = 7 kg/m, ρ α =.5 1 / K, ad α = 8 1 / K. The plate is subjected to differet levels of white oise excitatios. 66

76 18 a = b = 5h = 1 m, DT = Displacemet RMS, mm Liear Noliear Power Spectral Desity, Pa/ Hz Figure 4.3 RMS values vs. square root of power spectral desity Figure 4.3 shows the differece betwee the liear ad oliear displacemet R.M.S. values as a fuctio of white oise excitatio power spectral desity (PSD) levels. It is see that the oliear terms play a very importat role at relatively high PSD levels. Recall that similar coclusio is draw i Chapter 3 o the oliear radom vibratio of beams. 67

77 4.4 Temperature Effects o Radom Vibratios of Composite Plate I this sectio the effect of temperature o the radom vibratio properties of composite plate is examied. The existece of the temperature field exerts its ifluece o the vibratio characteristics of the composite plate by affectig the effective fudametal frequecies of it. The first order shear deformatio plate theory (FSDT) (Reddy 4) icludig the temperature force resultat terms is expressed by Eqs. (4.33)-(4.37) i which the superscripts T represet the temperature-itroduced force or momet resultat terms. u w w v w w u v w w w w A A A x x x y x y y x y x y x y y x y T T φ ψ φ ψ N N xx xy u φ + B11 + B 1 + B66 + I + + = + I 1 x x y y y x x y t t (4.33) u w w v w w u v w w w w A A A x y x x y y y y y x x x y x y x T T φ ψ φ ψ N xy N yy v ψ + B1 + B + B 66 + I + + = + I 1 (4.34) x y y x y x x y t t 68

78 w ψ w φ u w w v w w KA44 + KA A A y y x x x x x y x y y x u v w w w w φ ψ φ ψ w + A B B 1 + B66 + y x y x y y x y x x y y y x x u w w v w w u v w w w w + A1 + + A + A x y x x y y y y y x x x y x y x φ ψ φ ψ w u 1 w w + B1 + B + B 66 + A x y y x y x y x x x v 1 w w φ w A1 B11 A u v w w φ ψ w B66 + y y y x x x y x y y x x y u 1 w w v 1 w w φ ψ w + A1 ( + ) + A ( + ) + B1 + B x x x y y y x y y (4.35) T w T w T w T w w + N xx + N xy + N xy + N yy + q = I x x y y x y t u w w v w w u v w w w w B B B x x x x y y x y y y x y x y x y w u D φ 11 D ψ 1 D φ ψ 66 A 55 I φ κ + φ = + I 1 (4.36) x y x y y x x t t u w w v w w u v w w w w B B B x y x x y y y y x y x y x x x y w v D φ 1 D ψ D φ ψ 66 κ A 44 ψ I ψ = + I 1 (4.37) x y y x y x y t t As a example, a special case is examied where the plate is subjected to a temperature gradiet varyig i the thickess directio oly. The composite plate has the same 69

79 material properties as listed i Sectio 4.3. The temperatures at the top surfaces is icreased to a certai value ad the temperature at the bottom surface icreased/decreased to aother value, which creates a steady-state heat trasfer problem alog the thickess directio of the plate govered by the followig equatio withi each layer d dt ( z) ki ( z) dz = dz (4.38) where k i is the thermal coductivity of the i-th layer of the composite plate. The boudary coditios cosidered here are T = T t at T = T b at h z = ad T ( z ) ( ) i h z = (4.39) = T zi 1 at each iterface where the subscript b ad t refer to the top ad bottom surface, respectively. The thermal force resultats itroduced by the varyig temperature field ca be obtaied from N Q Q Q α T ( x, y, z) N = N = Q Q Q T x y z dz (4.4) T { } T xx N xx z T k+ 1 yy 1 6 yy (,, ) z α k T k 1 N = xy Q16 Q6 Q 66 α xy T ( x, y, z) k where α ij is the thermal expasio coefficiet i the ij-th directio of the composite plate, ad Qij s are defied i Appedix B. 7

80 Whe the material properties are idepedet of temperature, solvig Eqs. (4.38)-(4.4) together yields the thermal force resultats which ca the be substituted ito the plate goverig equatios i Eqs. (4.33)-(4.37). Followig the procedures discussed i Sectio 4.3, the R.M.S. resposes of trasverse displacemet of a composite plate with differet legth/thickess ratios are displayed i Figure 4.4. Figure 4.4 idicates that the temperature field affects the vibratio characteristics of the composite plate by affectig the fudametal frequecies ad hece the R.M.S. respose of the plate displacemet. It is observed that the R.M.S. values icrease mootoically with that of the temperature gradiet. The reaso is that the thermally itroduced i-plae compressive force resultat decreases the equivalet stiffess of the system, which i tur leads to a decrease i the equivalet frequecy of the plate. For the case a/h = 4, o data is available beyod T = 19 K due to plate bucklig. Oe kows the plate buckles because the iteratio process discussed i Sectio 4.3 o loger coverges. It oscillates betwee two values, displayig a sap-through type of bucklig. So i this case, oliear radom vibratio actually provides aother way to fid the buckig temperature gradiet for the composite plate uder study. 71

81 9 Displacemet RMS, mm a h = a h = 3 Plate Buckles a h = 4 a = b = 1 m Temperature Chage DT, K Figure 4.4 R.M.S. values vs. T (based o FSDT) 4.5 Compariso betwee FSDT ad CPT So far the first order shear deformatio theory (FSDT) has bee used throughout the umerical examples. The key differece betwee the FSDT ad classical plate theory (CPT) is that the former takes ito accout the trasverse shear strais. I other words, the trasverse ormals are o loger perpedicular to the mid-plae of the plate after deformatio. I this sectio both theories are applied to study the R.M.S. respose of the same composite plate ad the results are compared i Figure 4.5 7

82 I Figure 4.5 the calculated R.M.S. values uder differet temperature gradiet ( T) ad plate aspect ratios are show. As the plate becomes thicker, the differece of the R.M.S. resposes based o the two theories becomes more sigificat. However, for relatively high a/h ratios (> 4), the two theories yield almost the same results. Agai, o data is available beyod a/h = 3 for the FSDT case ( T = K ) due to plate bucklig. 1. a = b, h =. m Displacemet RMS Ratio DT = (FSDT) CPT DT = K (FSDT) Plate Buckles a/h Figure 4.5 Variatios of the ratios betwee FSDT ad CPT R.M.S. values vs. plate legth to thickess ratio a/h 4.6 R.M.S. Stresses Calculatio So far i this chapter attetio has bee paid to the R.M.S. of displacemet respose. Sice the stress is a fuctio of the displacemet, the stochastic ature of the respose dictates that stress withi the plate also varies with time. The R.M.S. values of stress 73

83 compoets are very useful umbers whe it comes to the fatigue aalysis of structures. So i this sectio, the calculatio of R.M.S. values of stress compoets at a give locatio (x, y, z) withi the plate is briefly discussed. First, recall that the defiitios of the oliear strais ad stresses are: u 1 w + α xx T x x φ v 1 w x ε xx + α yy T ψ ε y y yy y γ u v w w xy = + + α z xy T + φ ψ γ y x x y + yz y x γ w xz + ψ y w + φ x (4.41) σ xx Q11 Q1 Q 16 ε xx σ yy = Q1 Q Q6 ε (4.4) yy τ Q xy 16 Q6 Q 66 γ xy Based o the above defiitios, the value of E[ σ ij ] is ot oly fuctio of the locatio coordiate (x, y, z), but also E[ U ( t ) ], E[ V ( t ) ], E[ W ( t ) ], E[ Φ ( t ) ], ad E[ Ψ ( t ) ] m m (ote: uder the assumptio that both the load ad respose are zero-mea ad follow ormal distributio, expectatio of cross products of displacemet compoets such as U V ], E[ Φ ( t) Ψ ( t) ], E[ U ( t) W ( t) ], ad E[ V ( t) W ( t) ] are zeroes.) E[ ( t) ( t) m m m m m which are calculated from radom vibratio aalysis. This is a straightforward but tedious process due to the fact that so may terms are ivolved. But it ca be efficietly m m m m m m 74

84 carried out usig software such as Mathematica. As a example, cotour plots of selective R.M.S. stress compoets for a 3 m by m composite plate with a thickess of.6 m are displayed i Figure The plate is subjected to uiform radom pressure loadig with a PSD level of Pa /Hz. Figure 4.6 Cotour plot of σ xx R.M.S. at middle plae of the first layer of a ( 6 / 6 / 6 / 6 ) lamiate 75

85 Figure 4.7 Cotour plot of σ xx R.M.S. at middle plae of the first layer of a (3 / 3 / 3 /3 ) lamiate 76