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Random Vibration Analysis of Higher-Order Nonlinear Beams and

Composite Plates with Applications of ARMA ModelsYunkai Lu

ABSTRACT

In this work, the random vibration of higher-order nonlinear beams and composite plates

subjected to stochastic loading is studied. The fourth-order nonlinear beam equation is

examined to study the effect of rotary inertia and shear deformation on the root mean

square values of displacement response. A new linearly coupled equivalent linearization

method is proposed and compared with the widely used traditional equivalent

linearization method. The new method is proven to yield closer predictions to the

numerical simulation results of the nonlinear beam vibration. A systematical

investigation of the nonlinear random vibration of composite plates is conducted in which

effects of nonlinearity, choices of different plate theories (the first order shear

deformation plate theory and the classical plate theory), and temperature gradient on the

plate statistical transverse response are addressed. Attention is paid to calculate the

R.M.S. values of stress components since they directly affect the fatigue life of the

structure. A statistical data reconstruction technique named ARMA modeling and its

applications in random vibration data analysis are discussed. The model is applied to the

simulation data of nonlinear beams. It is shown that good estimations of both the

nonlinear frequencies and the power spectral densities are given by the technique.

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Acknowledgment

I would like to thank my advisor, Professor Thangjitham, for his guidance and support all

through my work.

I would like to thank all my committee members, Professor Case, Professor Hyer,

Professor Lu, and Professor Ragab, for the time they took to attend my exams and

defense as well as their advice on my dissertation.

My thanks also go to Professor. Kraige, Professor. Hendricks, and the ESM department,

for their financial support during my Ph.D. study.

Last but not least, I want to thank my parents, Lu, Zhicheng and Zhao, Guiying. I would

not have been able to make it through all the difficulties and hard times in my life without

their unconditional love and support all the time.

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Table of Contents

Chapter 1. Introduction ....................................................................................................... 1

Chapter 2. Literature Review .............................................................................................. 4

Chapter 3. Random Vibration of Geometrically Nonlinear Beams .................................. 14

3.1 Solutions to the Nonlinear Random Vibration of Isotropic Beams ........................ 14

3.2 Effect of Inertia of Rotation and Shear Deformation ............................................. 23

3.3 Numerical Results ................................................................................................... 31

Chapter 4. Nonlinear Random Vibration of Composite Plates ......................................... 54

4.1 Governing Equations .............................................................................................. 54

4.2 Stochastic Response of Linear System ................................................................... 604.3 Stochastic Response of Nonlinear System .............................................................. 63

4.4 Temperature Effects on Random Vibrations of Composite Plate ........................... 68

4.5 Comparison between FSDT and CPT ..................................................................... 72

4.6 R.M.S. Stresses Calculation .................................................................................... 73

Chapter 5. ARMA Model and Its Applications in Random Vibration Data Analysis ...... 79

5.1 Introduction ............................................................................................................. 79

5.2 Theoretical Background .......................................................................................... 80

5.3 Applications of ARMA Model in Identifying, Re-generating, and Extending the

Random Vibration Data ................................................................................................ 83

5.4 Comparison between PSD Curve from ARMA Model and Newlands Approach 88

Chapter 6. Future Work .................................................................................................... 96

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6.1 Durability of Structures Subjected to Random Loading ......................................... 96

6.2 Future Work ............................................................................................................ 99

Reference ........................................................................................................................ 101

Appendix A ..................................................................................................................... 116

A.1 Selected eigenfunctions ....................................................................................... 116

2. Solution procedure for a 22 linearly-coupled system ........................................... 117

Appendix B: Derivation of Plate Equations.................................................................... 121

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List of Figures

Figure 3.1 A beam under pressure .................................................................................... 15Figure 3.2 Correlation between displacement and acceleration for a typical linear beam

(data size: 214

) ........................................................................................................... 27Figure 3.3 Correlation between displacement and acceleration for a typical nonlinear

beam (data size: 214

).................................................................................................. 28Figure 3.4 Two types of loads used in the simulation ...................................................... 33Figure 3.5 A typical stationary Gaussian random process (time domain) ........................ 33Figure 3.6 Histogram of the random process in Figure 3.5 .............................................. 34Figure 3.7 PSDs of the two types of loads used in the simulation.................................... 34Figure 3.8 Displacement R.M.S. of a uniformly loaded F-SS beam vs. different ............ 35Figure 3.9 Mode 1 displacement R.M.S. of a half-uniformly loaded F-SS ...................... 35Figure 3.10 Mode 2 displacement R.M.S. of a half-uniformly loaded F-SS .................... 36Figure 3.11 Displacement R.M.S. (summation of first two modes) of a uniformly loaded

SS-SS beam vs. different random loading PSD levels ............................................. 36Figure 3.12 Coupling effect on mode 1 for a uniformly loaded F-SS beam .................... 38Figure 3.13 Coupling effect on mode 2 for a uniformly loaded F-SS beam .................... 38Figure 3. 14 Coupling effect on mode 1 for a half uniformly loaded F-SS beam ............ 39 Figure 3. 15 Coupling effect on mode 2 for a half uniformly loaded F-SS beam ............ 39 Figure 3.16 Typical mode 1 displacement response (corresponding to data in Table 3.1)

................................................................................................................................... 41Figure 3.17 Typical mode 2 displacement response (corresponding to data in Table 3.1)

................................................................................................................................... 41Figure 3.18 Typical FFT of mode1 displacement response .............................................. 42

Figure 3.19 Typical FFT of mode 2 displacement response ............................................. 42Figure 3.20 Histogram of mode 1 displacement response ................................................ 43Figure 3.21 Histogram of mode 2 displacement response ................................................ 43Figure 3.22 Comparison among different approaches of mode 1 response ...................... 45Figure 3.23 Comparison among different approaches of mode 2 response ...................... 45

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Figure 3.24 Comparison among different approaches of mode 1 response ...................... 46Figure 3. 25 Comparison among different approaches of mode 2 response ..................... 46Figure 3. 26 Comparison among different approaches of mode 1 response ..................... 47Figure 3. 27 Comparison among different approaches of mode 2 response ..................... 47Figure 3. 28 Comparison among different approaches of mode 1 response ..................... 48Figure 3.29 Comparison among different approaches of mode 2 response ...................... 48Figure 3. 30 Comparison among different approaches of mode 1 and 2 responses ......... 49 Figure 3. 31 Mode 1 R.M.S. response of a SS-SS beam subjected to uniform load ........ 51 Figure 3. 32 Mode 2 R.M.S. response of a SS-SS beam subjected to uniform load ........ 52 Figure 3. 33 Mode 1 R.M.S. response of a F-SS beam subjected to half-uniform load ... 52Figure 3. 34 Mode 2 R.M.S. response of a F-SS beam subjected to half-uniform load ... 53

Figure 4.1 Free body diagram of a rectangular plate element (without bending moments)

................................................................................................................................... 55Figure 4.2 Free body diagram of a rectangular plate element (bending moments only) .. 55Figure 4.3 RMS values vs. square root of power spectral density .................................... 67Figure 4.4 R.M.S. values vs.T (based on FSDT) ......................................................... 72Figure 4.5 Variations of the ratios between FSDT and CPT R.M.S. values .................... 73Figure 4.6 Contour plot of

xx R.M.S. at middle plane of the first layer of a

( 60 / 60 / 60 / 60 ) laminate .............................................................................. 75Figure 4.7 Contour plot ofxx R.M.S. at middle plane of the first layer of a

(30 / 30 / 30 / 30 ) laminate .............................................................................. 76Figure 4.8 Effect of ply angle on the maximal R.M.S. values of stress components .... 77

Figure 5.1 PSD of the mode 1 of the beam displacement simulation data ....................... 85 Figure 5.2 Displacement from simulation of mode 2 of a F-SS beam ............................. 86Figure 5.3 Displacement generated by ARMA(4, 3) model ............................................. 87Figure 5.4 Displacement generated by ARMA(8, 7) model ............................................. 87Figure 5.5 Displacement generated by ARMA(11, 10) model ......................................... 87Figure 5.6 Example spectrum plots of 8 out of the 32 segments ...................................... 91Figure 5.7 Comparison of PSD curves from ARMA(4,3) model ..................................... 92

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Figure 5.8 PSD of the beam simulation displacement data (Newlands approach) .......... 93Figure 5.9 PSD from ARMA(4,3) model ......................................................................... 94Figure 5.10 PSD from ARMA(8,7) model ....................................................................... 94Figure 5.11 PSD from ARMA(11,10) model ................................................................... 94

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List of Tables

Table 3.1 Response of a beam (mm) with F-SS boundary condition and subjected to half

uniform load (load PSD = 100000 Pa2/Hz, 2 , 10 1 mh b L h= = = ) ......................... 44

Table 3.2 Response of a beam with F-SS boundary condition and subjected to half

uniform load (load PSD = 100000 Pa2/Hz, 2 , 20 1 mh b L h= = = )......................... 44

Table 3.3 Response of a beam (mm) with F-SS boundary condition and subjected to

uniform load (load PSD = 10000 Pa2/Hz, 2 , 12.5 1 mh b L h= = = ) ..................... 44Tabel 3.4 Response of a beam (mm) with F-Fixed boundary condition and subjected to

half uniform load (load PSD = 100000 Pa2/Hz, 2 , 12.5 1 mh b L h= = = ) ............... 44

Table 3.5 Comparison of R.M.S. response of a 2nd

order beam with that of 4th

order beam

(SS-SS boundary condition with uniform load, 2 , 10 1 mh b L h= = = ) .................. 50Table 3.6 Comparison of R.M.S. response of a 2

ndorder beam with that of 4

thorder beam

(SS-SS boundary condition with half-uniform load, 2 , 10 1 mh b L h= = = ) .......... 50Table 3.7 Comparison of R.M.S. response of mode 1 of 2


th

order beam (F-SS boundary condition with half-uniform load,

2 , 10 1 mh b L h= = = ) ............................................................................................. 50Table 3.8 Comparison of R.M.S. response of mode 2 of 2


th

order beam (F-SS boundary condition with half-uniform load,

2 , 10 1 mh b L h= = = ) ............................................................................................. 51

Table 5.1 Estimated ARMA parameters for different order models ............................... 85Table 5.2 Frequencies predicted by selected ARMA models in Table 5.1....................... 85Table 5.3 Estimated displacement R.M.S. of selected ARMA models ............................ 87

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Chapter 1. Introduction

A great deal of work has been done on the response of beams and plates subjected to

deterministic loading conditions. However, in real life the structure may be subjected to a

stochastic type of loading such as earthquakes, wind turbulence, sea wave, acoustic loads,

etc. These loading conditions are commonly observed on dams, nuclear facilities,

offshore structures, aircraft, etc. The main purpose of this dissertation is to present a

systematical study of the stochastic response of geometrically nonlinear beams/plates

under random excitations.

In this work, a literature review of previous research in the area is given first in Chapter

2. Then, in Chapter 3 random vibration of geometrically nonlinear beams is elaborated

where the detailed procedures to solve the random vibration problem and the traditional

uncoupled equivalent linearization technique are discussed. To study the effect of the

rotary inertia and shear deformation on the root mean square (R.M.S.) of the stochastic

beam response, the fourth-order nonlinear beam equation is examined. The results from

the fourth-order beam equation are compared with those from second-order. A new

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coupled equivalent linearization method is proposed which takes into account the effects

of modal interactions between adjacent modes. Numerical results indicate that the new

method yields closer results to simulation data compared with the uncoupled linearization

approach. In Chapter 4 the focus is moved from one-dimensional beam problem to

rectangular composite plates using both nonlinear classical plate theory (CPT) and first

order shear deformation (FSDT) theory. FSDT takes into account of the transverse shear

strain effect. A nonlinear stress-strain relationship in the von Karman sense is considered

in the formulations of the governing equations. The effects of nonlinearity and

temperature on the R.M.S. values of transverse displacement of the plate and the selected

stress components are examined.

A statistical data characterization and reconstruction technique called ARMA modeling

and its applications in random vibration data analysis are demonstrated in Chapter 5. The

auto-regressive moving averaging (ARMA) model was originally developed as a time-

domain modal analysis method. ARMA model is very efficient in reconstructing the

loading condition directly in the time domain. The model is very concise in its

formulation, requiring very few parameters while preserving the stochastic nature and

spectral information of the original signal history. In an ARMA model, the current value

of the system response is expressed as a linear combination of past values of response

plus a pure white noise. The parameters in the model are determined through a trial and

error procedure in order to minimize the residue variance of the noise. Once the

parameters of the model are known, natural frequencies and damping ratios for all the

modes can be obtained from the autoregressive part. However, the order of an ARMA

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model to fit a certain data set is not unique. In addition, the correlation between the

nonlinear simulation data usually requires a higher order model than the linear case does

in order to accurately represent the spectral properties of the original input, especially the

power spectral density (PSD) plot. It is shown that even though a model can give good

estimations of the frequency values, it may not represent the PSD closely. At the end of

this work, issues regarding the durability of structure under random excitations are

addressed. Future areas of research work are discussed.

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Chapter 2. Literature Review

Stochastic loads such as wind turbulence, sea wave, and acoustical loads are commonly

observed on aerospace, mechanical and civil structures. Consequently, random vibration

analysis is necessary to understand the behavior of these structures under stochastic

loading. For a general review on the random vibration theory, one can refer to the

references (Crandall and Mark 1973, Bolotin 1984, Nigam 1983, Roberts and Spanos

1990, Newland 1993, Solnes 1997, Wirsching et al. 1995, Lin 1976, and Elishakoff

1983). The exact solutions to nonlinear random vibration problems are only available for

certain special cases. Therefore, approximation technique and numerical solutions were

developed to find the probability density functions of the response of the nonlinear

system. For limited cases, the moments of the response can be obtained via solving the

Fokker-Planck equation (Stratanovich 1963, Stratanovich 1967, Risken 1996, and

Gardiner 2004). A set of ordinary differential equations for the moment characteristics of

response can be obtained after applying a closure technique such as the Gaussian closure

method (Iyengar and Dash 1978). Perturbation method (Nayfeh 1993, and Nayfeh and

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Mook 1995) is the most widely used technique dealing with the nonlinear dynamic

response of systems with nonlinearity. It can also be adapted to solve the nonlinear

random vibration of such systems (Crandall 1963). However, by nature the perturbation

method is only applicable when the nonlinearity is small, which greatly limited its usage

in a wide range of problems. Another method called stochastic averaging technique

(Roberts and Spanos 1990, and Socha and Soong 1991) can also be applied to weakly

nonlinear systems. The equivalent linearization (Caughey 1959a, Caughey 1959b,

Caughey, 1963, Spanos 1981, and Roberts et al. 1990) is the most commonly used

approximation method due to its straight-forward formulation and effectiveness. When

the load is white noise, the equivalent linearization yields the same results as Gaussian

closure method (Er 1998).

The random vibrations of beams have been studied since the 1950s (Eringen 1957,

Bogdanoff and Goldberg 1960, Crandall and Yildiz 1962, Elishakoff and Livshits 1984,

and Elishakoff 1987). An exact probability density function of modal displacements was

found by Herbert (1964, 1965). Among all the approaches, the two mostly used are the

perturbation method and the stochastic linearization technique. Eringen (1957),

Elishakoff (1987) and Elishakoff and Livshits (1984) came up with closed-form solutions

for simply supported beams subjected to random loading in the form of infinite modal

summation. Exact solutions by the Fokker-Planck equation method only exist for some

extreme cases. Even if an exact solution exists, a large amount of multiple integrations

are needed to evaluate the root mean square value of the response, which makes it

computationally prohibitive. Fang et al. (1995) and Elishakoff et al. (1995) proposed an

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improved stochastic linearization method by minimizing the potential energy of the beam

under stationary random excitation. They claimed that the new approach improved the

accuracy of the conventional stochastic linearization method. Different variations of the

improved stochastic linearization technique can be found in the literature (Elishakoff and

Zhang 1991, Elishakoff 1991, Zhang et al. 1990, and Fang and Fang 1991).

Since one part of this dissertation studies the random vibration of composite plates using

classical plate theory (CPT) and first order shear deformation plate theory (FSDT), a

comprehensive introduction of composite material as well as different plate theories can

be found in the work of Reddy (1997, 2004). Before the stochastic response of composite

plate is examined, a brief review of some of the work on dynamic response of plates

using different theories is given as follows. Some of the studies investigated the nonlinear

vibrations of composite plates or functionally graded plate (a special type of composite

plate), in which iteration scheme was used similar to the equivalent linearization in the

random vibration analysis.

Kim and Noda (2002) discussed transient displacement of functionally graded composite

plates due to heat flux by a Greens function approach based on the classical plate theory.

Praveen and Reddy (1998) investigated the static and dynamic responses of functionally

graded ceramicmetal plates by using a plate finite element that accounts for the

transverse shear strains, rotary inertia and moderately large rotations in the von Karman

sense. Reddy (2000) analyzed the static behavior of functionally graded rectangular

plates based on the third-order shear deformation plate theory via finite element

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approach. Theoretical formulation along with Naviers solution and finite element model

for the plate were presented. Woo and Meguid (2001) applied the von Karman theory for

large deformation to obtain the analytical solution for the plates and shell under

transverse mechanical loads and a temperature field. Zenkour (2006) presented a general

formulation for functionally graded composite plates using the generalized shear

deformation theory that did not require a shear correction factor. Cheng and Batra

(2000a) presented results for the buckling and steady state vibrations of a simply

supported functionally graded polygonal plate based on Reddys plate theory. Cheng and

Batra (2000b) also related the deflections of a simply supported functionally graded

polygonal plate given by the first-order shear deformation theory and a third-order shear

deformation theory to that of an equivalent homogeneous Kirchhoff plate. Loy et al.

(1999) studied the vibration of functionally graded cylindrical shells using Loves shell

theory and RayleighRitz method. Liew et al. (2001, 2002a, 2002b) used classical plate

theory and the first order shear deformation theory to present the finite element

formulation for the shape and vibration control of functionally graded plates with

integrated piezoelectric sensors and actuators. He et al. (2001) presented the vibration

control of functionally graded plate with integrated piezoelectric sensors and actuators by

a finite element formulation based on CPT. Huang and Shen (2004) solved the nonlinear

vibration and dynamic response of simply supported functionally graded plates subjected

to a steady heat conduction process through an improved perturbation technique. Woo et

al. (2006) provided an analytical solution in terms of mixed Fourier series for the

nonlinear free vibration behavior of composite plates. The nonlinear coupling effects on

the fundamental frequencies were examined. Liew et al (2006) presented the nonlinear

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vibration analysis for layered cylindrical panels subjected to a temperature gradient due

to steady heat conduction along the panel thickness direction. A nonlinear pre-vibration

analysis was conducted to obtain the thermally induced pre-stresses and deformation.

Differential quadrature method with an iteration scheme was employed to find the

nonlinear vibration characteristics of the panel.

Yang and Shen (2001) presented the dynamic response of initially stressed functionally

graded thin plates. Yang and Shen (2002) investigated the free and forced vibration

problems for the shear-deformable functionally graded plate in thermal environment.

Their results indicated that the plates with intermediate material properties did not

necessarily have intermediate dynamic response. Kitipornchai et al. (2004) gave a semi-

analytical solution for the nonlinear vibration of imperfect functionally graded plates

based on higher-order shear deformation theory with temperature dependent material

properties. The sensitivity of the nonlinear vibration characteristics of plates to the initial

geometric imperfection was evaluated. In Yang et al. (2004), a semi-analytical Galerkin-

differential quadrature approach was employed to convert the governing equations into a

linear system of MathieuHill equations. The influences of various parameters such as

material composition and temperature change on the dynamic stability, buckling and

vibration frequencies were demonstrated through parametric studies. The stability of a

functionally graded cylindrical shell subjected to axial harmonic loading was discussed

by Ng et al. (2001). Patel et al. (2005) conducted the finite element analysis for the free

vibration of elliptical composite cylindrical shells based on the high order shear

deformation theory. Sofiyev (2004) and Sofiyev and Schnack (2004) studied the dynamic

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behavior of nonlinear systems (Rice 1999, Chen and Tomlison 1996, Staszewski 1998,

Boukhrist et al. 1999, and Jaksic and Boltezar 2002). However, when the data consists of

signals from all over the frequency range (white noise or narrow-band white noise),

traditional modeling identification methods have difficulties since extra filtering process

has to be conducted to remove the noise from the data so that the harmonic components

can be exposed. To analyze signal like this, the auto-regressive moving average (ARMA)

model is a very powerful tool. It is also called Box-Jenkins models named after the

people who developed it. A detailed introduction can be found in Box et al. (1994) and

Chatfield (1989). In an ARMA model, the current value of the system response is

expressed as a linear combination of past values plus a white noise. Once the parameters

are determined, natural frequencies, damping ratios (if applicable) can be obtained from

the autoregressive part of the model. The typical procedure for fitting an ARMA models

to a time series involves model identification, model fitting, and model validation. It

should also be pointed out that the applications of ARMA model are not restricted to

engineering field. For instance, it has been used in the analysis of financial data such as

stock market changes and other economical issues (Mills, 1990). Tian and Tan (1987)

used ARMA time series model to study the information of heart sounds of normal human

and patients with cardiovascular disorders. A cardiac functional state which was

determined from the ARMA parameters provided valuable information on the initiation

of heart-failure.

Baek et al. (2006) proposed a modeling method of the mass, the damping coefficient and

the stiffness of a cutting system using an autoregressive moving average (ARMA) model

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and a bisection method. Yoon et al. (2004) compared different algorithms in estimating

the structural dynamic between the endmill and workpiece of a cutting system. Baek et al.

(2006) conducted parameter identification on the experimental data of single-degree-of-

freedom system using ARMA model. Wang et al. (2003) evaluated the nonlinear fluid

force for a freely vibrating cylinder over a wide range of Reynolds numbers, mass and

structural damping ratios. Smail and Thomas (1999) compared the accuracy of three

kinds of ARMA methods (recursive, least-squares output error and corrected covariance

matrix method) in parameter identification of certain simulations and experimental data.

Effects of model orders and sampling frequency were studied. Its found that a good

sampling frequency ranged from three to ten times the maximal frequency of interest.

This information was used while running the simulations for the beam and plate in this

dissertation. Carden and Brownjohn (2007) applied the ARMA modeling technique on

the experimental data from the IASCASCE benchmark four-storey frame structure as

well as two bridge structures. A health-monitoring algorithm was examined that

distinguished a structure in a healthy state from that in an unhealthy state. Mattson and

Pandit (2006) used vector autoregressive (ARV) models to capture the predictable

dynamic properties in the experimental response data. The standard deviation of the

autoregressive residual series provided valuable information on the location of damage in

the structures. Sohn and Farrar (2001) combined auto-regressive and auto-regressive with

exogenous inputs techniques and conducted damage diagnosis of a mass-spring system

with eight degrees of freedom.

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Chapter 3. Random Vibration of Geometrically

Nonlinear Beams

In this chapter the fundamentals of random vibration of geometrically nonlinear beams

are elaborated. Issues such as the solution procedures, linearization technique, and effects

of nonlinearity and modal interaction are addressed. A new equivalent coupled

linearization approach is proposed and compared with the traditional equivalent

uncoupled linearization method. The solution to the random vibration of fourth order

beams is obtained with attention paid to the effects of rotary inertia and shear

deformation.

3.1 Solutions to the Nonlinear Random Vibration of Isotropic

Beams

The geometry of a simply supported beam subjected to uniform pressure is shown in

Figure 3.1. The nonlinear equation of motion for the transverse displacement w(x,t) of a

uniform beam (Foda 1999) can be expressed as

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4 2 4 2 4

4 2 2 2 4

2 4 4

2 4 2 2

1

x

w w w E w I wEI c A I

x t t kG x t kG t

w EI w I wN p

x AG x KAG x t

+ + + +

+ =

(3.1)

where , A, E, G, and I represent the density, cross section area, modulus of elasticity,

shear modulus, and moment of inertia of the cross section of the beam, respectively, and

c is the damping factor,

2

0

02

L

x

EA wN N dx

L x

= + represents the axial force, 0N is the

external axial force and assumed to be zero in the following analysis, and is the shear

correction factor.

Figure 3.1 A beam under pressure

The nonlinear equation of motion for the transverse displacement w(x,t) of a isotropic

beam without considering the rotary inertia and shear deformation effects can be

expressed as

4 2 2

4 2 2x

w w w wEI c A N p

x t t x

+ + =

(3.2)

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It is noticed that from Eq. (3.1) to Eq. (3.2), the order of the differential equation drops

from four to two in the time domain.

Following the method of separation of variables, the response of the displacement field

can be expressed as

1

( , ) ( ) ( )

N

n n

n

w x t f x q t

=

= (3.3)

where ( )n

f x represents the n-th eigenfunction which is determined by the boundary

condition and ( )nq t represents the time-dependant part of the n-th modal response.

Selected choices of eigenfunctions for beams with various support conditions are listed in

Appendix A.1.

Substituting Eq. (3.3) into Eq. (3.2) and applying the Galerkins method by left-

multiplying both sides of Eq. (3.3) by ( )nf x and integrating over the span of 0 to L, the

following equation for the n-th mode is obtained after the orthogonality condition is

applied

, ,

1 1 1

( ) ( ) ( ) ( ) ( ) ( ) ( )

N N N

n n n n n n k k i j i j n

k i j

q t q t q t q t q t q t p t = = =

+ + + =

(3.4)

where

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2

0

''''

02

0

''

0,

2

0

' '

,0

02

0

( )

( ) ( )

( )

( ) ( )

2 ( )

( ) ( )

1( ) ( , ) ( )

( )

n L

n

L

n n nL

n

L

k n

n k L

n

L

i j i j

L

n nL

n

c

A f x dx

EIf x f x dx

A f x dx

EA f x f x dx

AL f x dx

f x f x dx

p t p x t f x dx

A f x dx

=

=

=

=

=

(3.5)

Eq. (3.4) can not be solved analytically due to the existence of nonlinear terms. When the

load ( )n

p t is random in nature, the property of interest is the root mean square (R.M.S.)

of the response, which is defined by

[ ]/ 2 22

/ 2

1( )lim

T

x xT

T

x t dtT

= (3.6)

where x(t) represents a stationary process that has a constant mean value ofx

, and T

stands for the period that is under consideration. It can be seen that 2x

is also the

variance of the processx(t).

In the equivalent uncoupled linearization method, a linearized equation in the following

form is sought

,( ) ( ) ( ) ( )n n n n n n nq t q t q t p t + + = (3.7)

where ,andn n n represent the damping factor and stiffness of the equivalent system.

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It can be seen that in Eq. (3.6) different modes of the beam are totally decoupled for each

mode. However, recall that in Eq. (3.4) all the modes are actually coupled through the

nonlinear terms. So it makes more sense if the equivalent equation is written in the linear

coupled format as below

,

1

( ) ( ) ( ) ( )

N

n n n n k n n

k

q t q t q t p t =

+ + = (3.8)

It should be noted that by setting the non-diagonal stiffness terms, , withn k n k , to

zero, Eq. (3.8) is the same as the representation of the traditional equivalent linearized

equation as shown in Eq. (3.7).

The difference between Eq.(3.8) and (3.4) is

,

1

, ,

1 1 1

( ) ( ) ( ) ( )

( ) ( ) ( )

N

n k k n n n n n

k

N N N

n k k i j i j

k i j

q t q t q t

q t q t q t

=

= = =

= +

(3.9)

The goal is to find the optimal values of , andn k n of the equivalent linearized system

so that the square of the difference between linear and nonlinear systems is minimalized

in the statistical sense. This requires that

2

2

,

[ ] 0

[ ] 0

n

n k

E

E

=

=

(3.10)

whereE[] stands for the mathematical expectation.

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To demonstrate this procedure in Eq.(3.9) and Eq.(3.10), we look at a simpler casea

beam that is simply supported at both ends, in which Eq.(3.4) is simplified to the

following

2

,1 ,1

1

( ) ( ) ( ( )) ( ) ( )

N

n n n n n k k n n

k

q t q t q t q t p t +=

+ + + = (3.11)

where

2 2 4

,1 4

4 4

,1 2 4

0

4

2( ) ( , )sin( )

n

n

n k

L

n

c

A

k n E

L

n EG

L

n xp t p x t

L L

+

=

=

=

=

(3.12)

The equivalent linear system in Eq.(3.8) is used which is listed below again for the sake

of convenience

,

1

( ) ( ) ( ) ( )

N

n n n n k k n

k

q t q t q t p t =

+ + =

The difference between Eqs.(3.12) and (3.8) is

2

, ,1 ,1

1 1

( ) ( ( )) ( ) ( ) ( )

N N

n k k n n k k n n n n

k k

q t q t q t q t += =

= + + (3.13)

Therefore,

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[ ] [ ] [ ]

2

,

2

, ,1 ,1

1 1

2

, ,1 ,1

1 1

0 [ ]

( ) ( ) ( ( )) ( ) ( )

( ) ( ) ( ) ( ) 3 ( ) ( ) ( )

n k

N N

k n k k n n k k n k

k k

N N

n k n k n n k n k k n k

k k

E

E q t q t E q t q t q t

E q t q t E q t q t E q t E q t q t

+

= =

+

= =

=

= +

= +

(3.14)

and

2

2

0 [ ]

( ) ( )

n

n n n

E

E q t

=

=

(3.15)

which leads to

2

, ,1 ,1

1

3 ( )

N

n k n n k k

k

n n

E q t

+

=

= +

=

(3.16)

In the above derivation, the following relationship and definitions are used under the

assumption that both the load and response follow zero-mean Gaussian distributions

(Soong 2004):

3[ ] [ ] 0

[ ] 0 ( )

n n n n

k n

E q q E q q

E q q k n

= =

=

(3.17)

and4 2 2

3 2

[ ] 3 [ ]

[ ] 3 [ ] [ ]

n n

k n k n k

E q E q

E q q E q q E q

=

=(3.18)

Another example is given in Appendix A.2 for a beam fixed on one end and simply

supported at the other. In that case, a complete set of quadratic terms maintains and

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where ( )PS represents the power spectral density of the original load ( , )p x t =

( ) ( )x P t in Eq. (3.1). By definition, the power spectral density is the Fourier transform

of the autocorrelation function ( )R of load ( , )p x t :

( ) ( ) iPS R e d

= (3.22)

In the traditional equivalent linearization method, the correlation between different

modes is not considered due to the fact that the final linearized equations are decoupled.

However, numerical simulations indicate that under certain boundary conditions, there

are strong correlations between the responses of different modes in the nonlinear

problem. The value of correlation factor, ( )n k n k is calculated from the following

relationship (under the assumption that both the load and response have zero mean

Gaussian distribution)

,

2 2 2

[ ]( )

[ ] [ ]

n kn k

n k

E q qn k

E q E q

= (3.23)

Finally, the displacement R.M.S. of nonlinear random vibration of the beam can be

obtained after an iteration scheme that is similar to that of the uncoupled linearization

method:

(1) Taking the linear part of Eq.(3.8) only and calculate the first estimate of 2[ ]nE q and

[ ]n kE q q via Eq.(3.19) to Eq.(3.21) for each of theNmodes.

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4 2 4 2 4

4 2 2 2 4

2 4 4

2 4 2 2

1

x

w w w E w I wEI c A I

x t t kG x t kG t

w EI w I wN p

x KAG x KAG x t

+ + + +

+ =

For a beam simply supported at both ends, (0, ) ( , ) 0w t w L t = = . Assume the solution for

the beam is the summation of the firstNmodes

sin( ) ( )

N

n

n

n xw q t

L

= (3.24)

whereNrepresents the total number of modes considered.

Substituting Eq.(3.24) into Eq.(3.1) and applying the Galerkins method, the following

equation for the n-th mode is obtained after some lengthy manipulation

2

,1 ,1 ,2

1

2

,3 ,3

1

( ) ( ) ( ) ( )

( ( )) ( ) ( )

N

n n n k k n n N n

k

N

n N n N k k n n

k

q t q q t q t

q t q t p t

+ +

=

+ + +

=

+ + +

+ + =

(3.25)

where

2 2

,1 2

2 2 4

, 1 4

,2 2

4 4

,3 2 4

2 2 4 2 2 2

,3 2 6

20

( )

4

( )

4

2( ) ( , )sin( )

n

n k

n N

n N

n N k

L

n

AG n E G

I L

k n E

L

c G

I

n EG

L

k n E n EI AGL

I L

G n xp t p x t

L I L

+

+

+

+ +

+= +

=

=

=

+=

=

(3.26)

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Because of the coupling terms in Eq.(3.25), not only the 2[ ]nE q ,2[ ]nE q , and [ ]n nE q q for

each mode need to be estimated, but also the cross terms such as [ ] ( )n k

E q q n k . They

are obtained from the frequency domain analysis as explained in the following. Recall

that for a linear system, the mean square response is obtained by Eq.(3.19)

2 *( ) ( ) ( )n n n pG G S d

=

where ( )nG is the frequency response function and )(*

nG is its complex conjugate.

For a linear system governed by Eq.(3.27), )(nG takes the form

4 2

1( )

n e e e

n n n

Gi

= + +

(3.30)

Furthermore, via random vibration theory, 2[ ]nE q and2[ ]nE q can be calculated from the

following formulas

( ) ( )

( ) ( )

2

2 24 2

42

2 24 2

1[ ] ( )

[ ] ( )

n ne e e

n n n

n ne e e

n n n

E q S d

E q S d

= + +

= + +

(3.31)

where ( )nS represents the power spectral density of the excitation ( )np t .

The result for [ ]n n

E q q , on the other hand, can be obtained from the autocorrelation

betweenn

q andn

q :

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, 2 2 2 2

( ) ( ) ( )

( ) ( ) ( ) ( )n n

n n n nq q

n n n n

E q q E q E q

E q E q E q E q

=

(3.32)

Since the correlation between the displacement and acceleration is unknown, we seek

help from simulation results. After generating 10 series of data with length of 214

,

statistical evaluation of the correlation factor was conducted based on Eq.(3.32).

Numerical simulations were run for different beams with different boundary and loading

conditions. It was found out that the value ofn nq q

fell into a consistent range of -0.88 to

-0.80. For the purpose of simplicity, the value of -0.88 was used in the analytical analysis.

Eventually, the value of [ ]n nE q q is calculated from the following relationship (under the

assumption that both the response and associated acceleration have zero mean)

2 2 2

,[ ] [ ] [ ]n nn n q q n nE q q E q E q= (3.33)

-0.8 -0.4 0 0.4 0.8Displacement, m

-0.2

-0.1

0

0.1

0.2

Acceleration,

106 .

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Another important function associated with an ARMA(p, q) process is the power

spectrum density function. This function, given by Eq.(5.13), is useful in representing the

spectral characteristics of the random signal.

2( 1)2

1

2( 1)

1

...( )

2 ...

qi q i

q

pi p i

p

e ef

e e

=

(5.13)

where is the sampling interval.

General speaking, there are infinite choices of (p, q) to fit the collected data by regression

technique. Therefore, based on the nature of the problem, certain criterions must be

satisfied before an ARMA model can be considered qualified. After this screen

process, the optimal fit is usually taken to be the one that is simplest in format, i.e.,

lowest in order. For detailed procedure as well as adequacy check on the fitted models,

one can refer to Pandit and Wu (1993) and Box et. al (1994).

5.3 Applications of ARMA Model in Identifying, Re-generating,

and Extending the Random Vibration Data

In this section, the applications of ARMA models are demonstrated by some numerical

examples. Suppose some random vibration data of limited length is available from either

experiment or simulation. In this case, the displacement of nonlinear random vibration of

a beam with fixed-simply supported boundary condition (refer to Table 3.1) is used as

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input. The beam is subjected to half uniform load. First, different orders of ARMA

models are fitted to the data and the harmonic components within the data are predicted

using Eq. (5.6) to (5.7). Second, once the ARMA(p, q) parameters 1 , , p , 1 ,, q

are obtained, Eq. (5.1) is used to re-generate the displacement response data and this set

of data is compared with the original signal. Third, sincet is the pure noise that follows

normal distribution, it can be generated via random normal number generator once the

variance is found in the previous steps. Substituting thus-obtainedt

along with the

parameters1 , , p , 1 ,, and q into Eq. (5.1), an analytical express is found that

can be used to generate time series of any desired length depending on the special needs

of the situation.

Selected ARMA(m, n) models are fitted to represent the simulated displacement response

of mode 1 of the beam. The scheme by Pandit and Wu (1993) is used to estimate the

parameters of selected ARMA(p , q) models. The results are listed in Table 5.1 and 5.2.

Figure 5.1 shows the corresponding PSD of the response of the simulation data. It is

observed in Figure 5.1 that the response has a dominant harmonic component at about

900 Hz.

Theoretically, the response of a second-order system could be fitted with a second order

ARMA(2, 1) or even AR(2) model. However, since the sampling interval is so small, the

correlations between adjacent data points must be taken into consideration, which may

require a higher order ARMA model. Another possibility is that there are additional

modes that are introduced into the system due to the stochastic nature of the excitation.

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Table 5.1 Estimated ARMA parameters for different order models

ARMA

parameters

ARMA (p, q) model

(2,1) (3,2) (4,3)

1 1.608 1.465 1.241

2 -1.057 -0.946 -1.160

3 -0.0211 0.668

4 -0.692

1 -0.269 -0.527 -0.591

2 -0.254 -0.680

3 -0.310

Table 5.2 Frequencies predicted by selected ARMA models in Table 5.1

ARMA (p, q) model Predicted frequency, Hz

(2,1) 857

(3,2) 917

(4,3) 875

0 400 800 1200 1600 2000Frequency, Hz

0

10

20

30

40

mm

2/Hz

Figure 5.1PSD of the mode 1 of the beam displacement simulation data

The data in Table 5.2 indicates that the selected ARMA models in Table 5.1 all give very

good predictions of the dominant harmonic frequency component embedded in the signal.

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After the ARMA(p, q) parameters 1 , , p , 1 ,, q are obtained as shown in Table

5.1 and the variance of t is found, Eq. (5.1) is used to re-generate a time series based on

the ARMA model. The re-generated data is compared with the original simulation results.

Although ARMA(2,1) model gives a pretty good prediction of the frequency component

embedded in the signal, a look at the re-generated displacement field by the model

reveals that the simulated displacement quickly blows out of proportion, thus suggesting

an instability in the system. This is part of the problem associated with ARMA

applications. The condition of stability (Box et. al 1994) for a certain ARMA model

requires that the norm of the each of the complex root determined by Eq.(5.6) is less than

one. For the ARMA(2,1) parameters listed in Table 5.1, the norm of the two complex

roots is 1.028 which is just over one. That causes the instability of the re-generated data.

Hence, a higher order model is needed in order to produce a stable time series.

0 1 2 3 4Time, sec

-0.15

0

0.15

Displacement,m

Figure 5.2Displacement from simulation of mode 2 of a F-SS beam

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0 1 2 3 4Time, sec

-0.15

0

0.15

Displacement,m

Figure 5.3Displacement generated by ARMA(4, 3) model

0 1 2 3 4Time, sec

-0.15

0

0.15

Displac

ement,m


0 1 2 3 4Time, sec

-0.15

0

0.15

Displace

ment,m


Table 5.3 Estimated displacement R.M.S. of selected ARMA models

Displacement R.M.S., mm

simulation ARMA(4,3) ARMA(8,7) ARMA(11,10)

32.1 30.1 34.4 32.3

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Further numerical examples are shown in Figures 5.2 to 5.5 for the mode 2 displacement

of the same beam. The original simulation result is shown in Figure 5.2. In this case

ARMA(3,2) model turns out to be unstable, so the lowest model used is ARMA(4, 3).

During the process, the estimated variance of the pure white noise part of each model is

used to produce random displacement data. Thats why the generated displacement looks

different each case. It is also seen that the generated displacement looks quite different

from the original data. In the original data, it is obvious that there is a significant

harmonic component. While in Figures 5.2 to 5.5, that harmonic component is shadowed

by the random noise. However, the R.M.S. of all the displacement data are very close, as

is shown in Table 5.3. So from the statistical point of view, they can be considered

equivalent signals. This is a very useful application of ARMA model because it can

extend data serial to unlimited length in the time domain while preserving the spectral

information of the original data. It is also observed that as the order of ARMA model

increases, the R.M.S. of the predicted displacement gets closer to that of the original

signal.

5.4 Comparison between PSD Curve from ARMA Model and

Newlands Approach

There are some long-established methods that estimate the power spectral densities of

digital signals. One of the most widely used techniques was summarized by Newland

(1993). The detailed procedures are not repeated here. Although the guideline is the

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same, there are variations in the executions of the technique. In this section, different

trials are conducted in order to get a smoothed PSD curve.

(1) Divide the original data set (usually 218) into segments in the size of 213 to 215, dothe spectral estimation for each segment following Newlands approach (with

smoothing), then average across all the segments.

(2) Same as (1), except that no smoothing is applied during the spectral estimationfor each segment .

(3)

Same as (1), except that during the spectral estimation for each segment,

triangular window was used instead of the default rectangular window.

(4) Do the FFT with each segment, average across all the segments, then do thespectral estimation based on the averaged FFT data (now having the size of one

segment).

It is found out that if there is enough data in each segment (say 213 or more) and enough

segments (say, 32 or more), all the methods yielded almost identical curves. This result

suggests that one can skip the smoothing process and just use the approach in (4) since

this is most efficient one computationally.

As a quick example, the above spectrum estimation technique is applied to the simulated

transverse response data of a 3m by 2m rectangular composite plate with a thickness of

0.06 m. The plate is subjected to a white noise excitation and has a first natural frequency

of 70.2 Hz. The four-layer laminate ( 45 / 45 / 45 / 45 ) has material properties of

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3

1 2 3 1 12 13 12206.84 GPa, 0.1 , 10.34 GPa, 0.25, and 2070 kg/m .E E E E G G = = = = = = =

The size of the simulation data is 218

which is divided into 32 sections. In Figure 5.6

selected 8 out of the 32 segments are displayed. The averaged PSD curve via Newlands

approach is plotted in Figure 5.7.

The PSD plot given by Eq. (5.13) based on an ARMA(4,3) model is plotted in Figure 5.7

along with the estimated PSD curve from simulation data. Although the curves from the

two approaches dont overlap, especially the height of the peaks, the areas under the

curves that represent the variance of the data are very close to each other, which is what

is expected. Generally Newlands approach gives a very conservative estimation for the

PSD curve. It tends to underestimate the peak values.

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0 40 80 120 160Freqency, Hz

0

200

400

100

300

m

m2/Hz

0 40 80 120 160Freqency, Hz

0

200

400

100

300

m

m2/Hz

0 40 80 120 160Freqency, Hz

0

200

400

100

300

m

m2/Hz

0 40 80 120 160Freqency, Hz

0

200

400

100

300

m

m2/Hz

0 40 80 120 160Freqency, Hz

0

200

400

100

300

m

m2/Hz

0 40 80 120 160Freqency, Hz

0

200

400

100

300

m

m2/Hz

0 40 80 120 160Freqency, Hz

0

200

400

100

300

mm

2/Hz

0 40 80 120 160Freqency, Hz

0

200

400

100

300

mm

2/Hz

R.M.S. is 22.30 mmR.M.S. is 21.05 mm

R.M.S. is 20.48 mmR.M.S. is 21.02 mm

R.M.S. is 21.36 mmR.M.S. is 21.30 mm

R.M.S. is 20.83 mmR.M.S. is 21.67 mm

#9 #10

#11 #12

#14 #15

#16 #17

Figure 5.6 Example spectrum plots of 8 out of the 32 segments

(segments #9-#17, each of which is smoothed)

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0 40 80 120 160 200Frequency, Hz

0

50

100

150

200

mm

2/Hz

ARMA(4,3) model

predictionNewland method

Figure 5.7 Comparison of PSD curves from ARMA(4,3) model

and Newlands approach

Now attention is turned to cases when multiple harmonic components exist in the data.

Newlands approach would always work no matter what the data is like. ARMA models

ability to capture those multiple frequency components is demonstrated by the following

example. The simulation data as shown in Figure 5.3 of the mode 2 of a fixed-simply

supported beam is used as the original signal. Figure 5.8 shows the estimated PSD of the

response via Newlands approach. In the previous section, ARMA(4,3) model has been

proven to give an accurate reproduction of the time series. However, the PSD curve of

the ARMA(4,3) model does not represent the spectral characteristic of the original data

well, as is shown in Figure 5.9. First of all, it is not able to pick out the frequency at about

900 Hz. Second, the two frequencies it predicts are 1732 Hz and 2330 Hz, respectively.

Both of them miss the dominant component at around 1800 Hz. This indicates that a

higher order ARMA model is needed if one wants to accurately capture the spectral

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properties of the signal. So in the next steps, ARMA(5,4), ARMA(6,5), ,

ARMA(11,10) models are tested. The details for the intermediate models are not shown

here. When ARMA(11,10) model is selected, it successfully picks out five harmonic

components: 918 Hz, 1733 Hz, 2023 Hz, 2626 Hz, and 3482 Hz. The PSD plot associated

with the ARMA(11,10) model is shown in Figure 5.11. By comparing the curves in

Figures 5.9 and 5.11, it can be found that the PSD curve via ARMA(11,10) model is

almost a perfect match to the one by Newlands approach. Not only does it capture all the

spectral characteristics, but gives a smooth curve with an analytical expression as

compared to the sketchy one estimated from the Newlands approach. Therefore, it can

be concluded that once the order of the ARMA model is high enough, it will accurately

predict the PSD of the signal.

0 1000 2000 3000 4000Frequency, Hz

0

1

2

3

4

mm

2/H

z

Figure 5.8 PSD of the beam simulation displacement data (Newlands approach)

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0 1000 2000 3000 4000Frequency, Hz

0

1

2

3

mm

2/Hz

Figure 5.9 PSD from ARMA(4,3) model

0 1000 2000 3000 4000Frequency, Hz

0

1

2

3

mm

2/Hz


0 1000 2000 3000 4000Frequency, Hz

0

1

2

3

mm

2/Hz


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In summary, the ability of ARMA models in identifying, re-generating, and extending the

random vibration data is demonstrated and verified in this chapter. If one is only

interested in regenerating the displacement response and using it as a random input for

situations such as structural fatigue analysis, relatively lower ARMA models are efficient

enough to get the job done. However, higher order models are required to accurately

capture the spectral characteristics of the random signal. Overall, ARMA models have

great potentials in the analysis and application of random vibration data.

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Chapter 6. Future Work

6.1 Durability of Structures Subjected to Random Loading

So far in this study the dynamic response of beams and plates under stochastic loading

conditions has been examined. One significant type of reliability concern associated with

oscillating loads of random nature is fatigue. As the magnitude and the direction of the

load vary incessantly with time, the material undergoes fluctuating strains that may

eventually cause the structure to fail at a stress well below the yield stress of the

constituent material. The most straightforward way to evaluate fatigue is the S-N curve.

The S-N curve is obtained through experiments in which a material under study is

subjected to a cyclic stress and the number of cycles to failure is counted. Numerous tests

are run at different stress levels and a smooth curve is drawn with the help of some

knowledge of data processing or linear regression technique. Traditional theories in

fatigue include Miners Rule and Paris equation. Miner (1945) proposed that failure

occurs if the following relationship for a material subjected to kdifferent stress levels is

satisfied

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1

k

i

ii

nC

N=

= (6.1)

where C is a constant determined experimentally and ranges from 0.7 to 2.2, andi

n

represents the number of cycles under the i-th stress level andi

N is the total number of

cycles before failure.

The expected fatigue damage after duration of time Tunder random loading is found to

be (Liou et al. 1999)

[ ] 00

( )( )

( )

d

f m

pE D T v T d

N

+ =

(6.2)

where [ ]E means the mathematical expectation, d is Morrows plastic work interaction

exponent, 0v+ represents the number of stress cycle per unit time for a narrow-band

process, ( )p is the PDF (probability density function) of stress amplitudes, m is the

maximum stress magnitude, and ( )f

N is the number of stress peaks to cause failure if a

constant amplitude is applied and can be obtained from the Basquins equation:

'pfN C = (6.3)

wherep and 'C are material-dependant constants.

In Nigam and Narayanan (1994), the estimated fatigue damage after time T under

stationary narrow-band random vibration is expressed by

(0)[ ( )] ( 2 ) 1

2

bb

q

x

N T qE D T

c

+ = +

(6.4)

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2 2 2 2

,

( , ) ( ) ( )mn mnw mn Q Q

m n

W x y S H d

= (6.8)

where (m, n) represents mode number.

Using Eqs. (6.6)-(6.8), the reliability vs. time curve can be obtained for a beam or plate

subjected to a known random excitation. Due to the fact that this is a straightforward

procedure and no material fatigue data is available, no numerical example is given while

the outline of the solution process is summarized here.

6.2 Future Work

The fatigue life analysis mentioned in section 6.1 is one area future work can be devoted.

So far this study has focused on the stationary random process whose statistical

properties do not vary with time. In reality many random excitations are non-stationary

such as earthquake wave. It would be interesting to examine the dynamic response of

structure as time evolves subjected to this type of loading condition. Different approaches

to model the non-stationary process can be found in work of Cederbaum et. al (1992).

On the other hand, although a new linearly coupled linearization method is proposed in

addition to the traditional equivalent linearization to obtain the numerical results in this

study, the limitation of this type of technique is that the load and system response must be

Gaussian. It has been discussed that technically when governing equations of motion of

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the system are nonlinear, the responses can be non-Gaussian even if the loading is.

Hence, in order to pursue more accurate results for multi-degree systems, other

linearization techniques need to be explored for the higher-order beams or plates such as

the Gaussian closure technique, statistical linearization, and statistical non-linearization.

These are all potential fields where future can be conducted.

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