generalized linear random vibration analysis using …€¦generalized linear random vibration...

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Generalized Linear Random Vibration Analysis Using Auto-Covariance Orthogonal Decomposition Sameer B. Mulani, * Rakesh K. Kapania, and Karen M. L. Scott Department of Aerospace and Ocean Engineering Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061-0203 Application of a stationary Gaussian Random process to describe a non-deterministic forcing function of a linear vibrating system is well studied and documented. Two al- gorithms: 1) KL Expansion Method, and 2) Collocation Technique for non-stationary and non-Gaussian forcing processes (narrow or broad band, but not a white noise) for lin- ear dynamic systems are developed here. In the KL Expansion Method, the forcing ran- dom process auto-covariance is decomposed using the well-known Karhunen-Loeve(KL) expansion. In the KL expansion, Galerkin projection (the weighted-residual method) and Collocation Technique (discretized covariance matrix) are used to get the eigenvalues and the eigenvectors of the auto-covariance function, numerically. The steady-state and the transient response of a single-degree-of-freedom (SDOF) system for an exponential auto- covariance (Gaussian random process) is obtained using three methods: 1) Analytical, 2) Semi-Analytical and 3) Numerical. In the Semi-Analytical method, the eigenvalues and the eigenvectors of the exponential auto-covariance function are obtained analytically by solving the Fredholm Integral equation of the 2 nd kind and those analytical definitions of the eigenvectors and the eigenvalues are used to obtain the numerical response of the SDOF system. In the case of the steady state response of the SDOF system, the convergence of the standard deviation of the response is shown to be a function of the number of KL expansion terms used in the expansion of the auto-covariance of the forcing function. The transient analysis of the same SDOF system is carried out using an exponentially mod- ulated non-stationary process. Comparison of the proposed methods with respect to the analytical solutions are presented for both the stationary and the non-stationary Gaussian excitations. The steady-state responses as well as the transient responses for non-Gaussian random processes (Uniform, Triangular, and Beta) for the same SDOF system are also presented. Nomenclature λ c Eigenvalues of the excitation covariance matrix λ n Eigenvalues of the auto-covariance function ω n Natural frequency, rad/sec φ c Eigenvectors of the excitation covariance matrix φ n (t) Eigenvectors of the auto-covariance function ξ n Independent identically distributed (iid) random variables with zero mean and variance equal to one ζ Non-dimensional damping coefficient c Damping coefficient, N s/m C FF Excitation covariance matrix obtained by collocation d jk h j basis function participation factor of φ k eigenvector * Post Doctoral Fellow, Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, AIAA Mem- ber. Mitchel Professor, Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, AIAA Associate Fellow. College of Engineering Graduate Teaching Fellow, Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, AIAA Student Member. 1 of 20 American Institute of Aeronautics and Astronautics 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<br>17th 4 - 7 May 2009, Palm Springs, California AIAA 2009-2264 Copyright © 2009 by Sameer Mulani. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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Page 1: Generalized Linear Random Vibration Analysis Using …€¦Generalized Linear Random Vibration Analysis Using Auto-Covariance Orthogonal Decomposition Sameer B. Mulani, Rakesh K. Kapania,

Generalized Linear Random Vibration Analysis Using

Auto-Covariance Orthogonal Decomposition

Sameer B. Mulani,∗ Rakesh K. Kapania, and † Karen M. L. Scott ‡

Department of Aerospace and Ocean Engineering

Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061-0203

Application of a stationary Gaussian Random process to describe a non-deterministicforcing function of a linear vibrating system is well studied and documented. Two al-gorithms: 1) KL Expansion Method, and 2) Collocation Technique for non-stationary andnon-Gaussian forcing processes (narrow or broad band, but not a white noise) for lin-ear dynamic systems are developed here. In the KL Expansion Method, the forcing ran-dom process auto-covariance is decomposed using the well-known Karhunen-Loeve(KL)expansion. In the KL expansion, Galerkin projection (the weighted-residual method) andCollocation Technique (discretized covariance matrix) are used to get the eigenvalues andthe eigenvectors of the auto-covariance function, numerically. The steady-state and thetransient response of a single-degree-of-freedom (SDOF) system for an exponential auto-covariance (Gaussian random process) is obtained using three methods: 1) Analytical, 2)Semi-Analytical and 3) Numerical. In the Semi-Analytical method, the eigenvalues andthe eigenvectors of the exponential auto-covariance function are obtained analytically bysolving the Fredholm Integral equation of the 2nd kind and those analytical definitions ofthe eigenvectors and the eigenvalues are used to obtain the numerical response of the SDOFsystem. In the case of the steady state response of the SDOF system, the convergence ofthe standard deviation of the response is shown to be a function of the number of KLexpansion terms used in the expansion of the auto-covariance of the forcing function. Thetransient analysis of the same SDOF system is carried out using an exponentially mod-ulated non-stationary process. Comparison of the proposed methods with respect to theanalytical solutions are presented for both the stationary and the non-stationary Gaussianexcitations. The steady-state responses as well as the transient responses for non-Gaussianrandom processes (Uniform, Triangular, and Beta) for the same SDOF system are alsopresented.

Nomenclature

λc Eigenvalues of the excitation covariance matrixλn Eigenvalues of the auto-covariance functionωn Natural frequency, rad/secφc Eigenvectors of the excitation covariance matrixφn(t) Eigenvectors of the auto-covariance functionξn Independent identically distributed (iid) random variables with zero mean and variance equal to oneζ Non-dimensional damping coefficientc Damping coefficient, N s/mCFF Excitation covariance matrix obtained by collocationdjk hj basis function participation factor of φk eigenvector∗Post Doctoral Fellow, Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, AIAA Mem-

ber.†Mitchel Professor, Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, AIAA Associate

Fellow.‡College of Engineering Graduate Teaching Fellow, Aerospace and Ocean Engineering, Virginia Polytechnic Institute and

State University, AIAA Student Member.

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American Institute of Aeronautics and Astronautics

50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br>17th4 - 7 May 2009, Palm Springs, California

AIAA 2009-2264

Copyright © 2009 by Sameer Mulani. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Page 2: Generalized Linear Random Vibration Analysis Using …€¦Generalized Linear Random Vibration Analysis Using Auto-Covariance Orthogonal Decomposition Sameer B. Mulani, Rakesh K. Kapania,

f(t) Band limited excitation, Nhj Karhunen-Loeve eigenvector basis functionk Stiffness, N/mm Mass, kgRFF Auto-covariance of the excitationRXX Auto-covariance of the displacementSFF Power Spectral Density function of the excitationSXX Power Spectral Density function of the displacementx Displacement, mσx (t) Standard deviation of the displacementPDF Probability Density FunctionPSD Power Spectral Density functionSDOF Single-degree-of-freedom system

I. Introduction

The present work is motivated by the design of a fiber-optics pressure and temperature sensor for the usein advanced liquid propulsion systems against fatigue failure. The temperature sensor (no greater than

1/4 inch diameter) will be housed in a cryogenic piping system to measure the pressure and temperature ofdifferent propulsive fuels. Designing such a pressure and temperature sensor is a Fluid-Structure Interaction(FSI) problem where the sensor vibrations and fluid flows are coupled. Flow induced vibrations of thetemperature sensors are traditionally predicted through the use of the ASME Boiler and Pressure Vessel CodeSection III Appendix N-1300 (BPV). BPV code comprises of two main areas of analysis: vortex inducedvibration and turbulence induced vibration. Currently, only stationary and Gaussian random loadingsgenerated by the turbulence in the flow is assumed in BPV code which results in a stationary response of thesensor. But, intermittent flow during the switch-off and switch-on conditions create a non-stationary randomresponse that may be higher than the stationary response obtained using the BVP code. The loading on thesensor may be non-stationary which causes an inherently non-stationary response of the sensor. Apart fromsuch FSI problems of interest to us, gust response of aircraft, and the vibrations experienced by vehicles onrough roads are other examples of non-stationary vibrations.

In most of the linear random vibration analyses, the non-deterministic forcing function is often assumedto be stationary Gaussian random processes to obtain the Power Spectral Density (PSD) functions of theresponse1,2. Thus, the standard deviation of the response or the distribution of the standard deviation inthe frequency domain can be obtained. In practical problems, most of the random processes describingthe excitation are Non-Gaussian; meaning that their marginal Probability Density Functions (PDF) mayhave positive real domain and which might be finitely bounded. As Gaussian excitations are infinite intheir domain, the response of the system would also be infinite, an impossible situation. For finite boundedexcitations, the response would also be finite. However, response of a system to non-stationary and Non-Gaussian random forcing functions has not been studied in great depth.3 The response of the dynamicsystems under non-stationary excitations has been studied 4–6 for simple systems using integral method;meaning that integrations involved in the response calculations are performed using analytical or numericalmethods. For finite degree of freedom (DOF) systems, this is possible, but for large structures, it is computa-tionally prohibitive to solve such integrations. Most of the time, the non-stationary random processes wouldbe an evolutionary process meaning that the non-stationary process can be split into a reference stationaryprocess and a modulating function. If the excitations are non-Gaussian processes, they would generally beMarkov and Poisson processes and their applications are generally limited to finite DOF models6–8. In theauthors’ opinion, there does not exist a general method which can handle linear random vibration problemsthat are excited by a more general random forcing function than the often used stationary Gaussian process.The PSD definition makes sense for the problems subjected to stationary Gaussian process. However, theauto-covariance definition of the response as well as that of the input functions are applicable to an exci-tation defined by a non-stationary random process. So, the auto-covariance is the fundamental property ofthe random processes as opposed to the PSD. In parallel, efforts have also been made to generate rationalnon-Gaussian random processes for practical applications9,10.

While using the auto-covariance definition to express a given excitation, one has to first discretize theauto-covariance function for the response calculation of the system11,12. The Karhunen-Loeve(KL) expansion

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is one of the way of discretization of the auto-covariance function. The KL expansion of the Gaussian randomprocess in the random space (probability space) is similar to Fourier decomposition in the deterministic space,as it decomposes a random process into random variables (Basis random variables) and a set of deterministicorthogonal functions in real space (domain of the process). The basis random variables of KL expansion13

of Gaussian random processes are uncorrelated standard normal variables. The basis random variables,when using the KL expansion, should be independent identically distributed (iid) random variables withzero mean and unit variance. Earlier, Poirion et al presented the KL expansion of non-Gaussian randomprocesses using Monte-Carlo.14 Later, in 2002, Ghanem et al developed the polynomial chaos decompositionfor the non-Gaussian non-stationary processes15. Recently, we and Phoon et al presented the KL expansionof non-Gaussian random processes16,17.

Here, an algorithm is presented which uses the KL expansion for decomposition of the auto-covarianceinto orthogonal components. During the KL expansion of the auto-covariance of the forcing function, thetime duration for analysis is chosen such that it is greater than the non-zero duration of the auto-covariance.Orthogonality of the basis random variables of the KL expansion is used to obtain the standard deviation andthe auto-covariance of the response. This algorithm is applied to a single-degree-of-freedom (SDOF) systemwhich is excited by a random process having the mean equal to 0 and the variance equal to 1. This randomprocess is described by the stationary exponential and the exponentially modulated non-stationary auto-covariance functions. The KL expansion of the exponential auto-covariance is carried out using analyticalfunctions, Galerkin projection method18, and the Collocation Technique12.

Many times, the Galerkin projection method yields negative eigenvalues of the auto-covariance func-tion and results in numerical inaccuracies during integration while solving Fredholm equation of the 2nd

kind. So, a user has to make sure that the eigenvalues of the auto-covariance are both positive and finite.One can obtain eigenvalues and eigenvectors of the auto-covariance function using the Collocation Tech-nique12. Generally, such approach does not yield negative and infinite eigenvalues. For the exponentialauto-covariance function, the PSD definition is available, so the response PSD and its standard deviationare obtained. The standard deviation of the steady-state (stationary excitation) response of SDOF systemhaving the Gaussian excitation is compared with the response obtained using Fourier Transform (Analyti-cal Technique) for two approaches: 1) Semi-Analytical method, and 2) Numerical methods (KL ExpansionMethod and Collocation Technique). In the ’Semi-Analytical Method’, eigenvalues and eigenvectors of theauto-covariance of the excitation are obtained by solving Fredholm equation of the 2nd kind analyticallyand analytical definitions of the obtained eigenvectors are used to construct the response of the system.The auto-covariances of the response are compared for semi-Analytical and numerical methods. A transientanalysis response of the SDOF system for the exponential auto-covariance (stationary random process) andthe exponentially modulated non-stationary random process are presented. Additionally, the PDFs of thedisplacement for a non-Gaussian stationary excitation (exponential auto-covariance) as well as for a non-Gaussian non-stationary (exponentially modulated) process are obtained, where the excitations are uniform,triangular and beta random processes.

When the KL Expansion Method and the Collocation Technique applied to the SDOF system subjectedto the stationary and the non-stationary random processes as the forcing function, the obtained standarddeviation of the displacement matches very well with the response obtained using analytical methods. TheKL Expansion Method more accurately describes the statistics of the response as compared to the CollocationTechnique but many times suffers from ill-conditioned matrices obtained during the KL expansion of theexcitation auto-covariance. During the transient analysis of the system, the PDF of the displacement changesfrom dirac-delta function to the PDF having some finite standard deviation, the shape of the PDF of thedisplacement depends upon the number of terms used for the expansion of the excitation auto-covarianceduring theKL Expansion Method or the Collocation Technique. For wide-band excitation, the PDF of thedisplacement is almost Gaussian for both methods. But, for short-band excitation or non-stationary response,the PDF of the displacement obtained using the Collocation Technique may differ with the one obtained usingthe KL Expansion Method. Still, the standard deviation obtained using Collocation Technique matches verywell with the obtained standard deviation using the KL Expansion Method.

II. KL Expansion of the Excitation and the Response Calculation

The proposed algorithms can be easily applied to general systems (multi-degree-of-freedom) using com-mercial Finite Element Method (FEM) softwares. For the derivation convenience, understanding and analysis

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perspective, the proposed algorithms are applied to a SDOF system. A time-invariant linear second-ordersystem subjected to a forcing function can be written as :

m x+ c x+ k x = f (t) (1)

where m, c, and k are mass, damping coefficient and stiffness properties of the system. The forcingfunction, f(t) can range from a narrow band to a broad band (but not white noise) random process and canbe a stationary or a non-stationary process. This process may be Gaussian or non-Gaussian and is prescribedby its auto-covariance and the mean value. Using the KL expansion, the random excitation, f(t) is writtenas :

f (t) = µF (t) +n∑i=1

√λnξnφn(t) (2)

where µF (t) is the deterministic part of the forcing function. λn and φn are eigenvalues and eigenvectorsof the auto-covariance function, ξn are random variables having a mean value equal to zero and a varianceequal to 1. Variables λn and φn(t) are obtained by solving the following Fredholm equation of the 2nd kind

∫D

RFF (t1, t2)φn (t1) dt1 = λnφn (t2) (3)

D ≡ [Tmin, Tmax] (4)

L = (Tmax − Tmin) /2 (5)

where RFF (t1, t2) is the auto-covariance function of the force/excitation, and D is the time domain of theforcing function, Tmin and Tmax are the lower and upper bounds of the time duration in which the responseof the system is sought, and L is the half of the time domain, t. The KL expansion decomposes a randomprocess into deterministic orthogonal components and the independent identically distributed (iid) randomvariables. For a Gaussian process, these will be standard normal variables and for non-Gaussian processes,these random variables should be calculated16 using the PDF definition of the force. Equation (3) can besolved analytically for some special auto-covariances and for general covariances, it should be solved using aGalerkin procedure.18

Let the eigenvectors of the auto-covariance be represented as :

φk (t) =n∑j=1

djkhj (t) (6)

where djk are constants and hj are user-defined basis functions.Therefore for such a time domain, the components of hj should be chosen such that they form a complete

basis of the system, but it is not a necessary condition. The djk can be viewed as the hj basis functionparticipation factors of the eigenvector, φk of the excitation auto-covariance. For the steady state responseof the dynamic system, h are chosen such that they represent complete cosine and sine functions over theselected time domain and are given as :

h =

1, cosπ

Lt, sin

π

Lt, . . . , cos

π (n− 1)L

t, sinπ (n− 1)

Lt

(7)

For the transient analysis of the dynamic systems, h are chosen such that hj basis functions include multiplesof both half cosine and half sine functions to represent the response components corresponding to hj basisfunctions with lower frequencies as the system response starts with constant value having 0 frequency. It is

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important to choose such functions to represent transient response with refined basis functions as comparedto basis functions given in Eq. (7) and h are given as :

h =

1, cosπ

2Lt, sin

π

2Lt, . . . , cos

π (n− 1)2L

t, sinπ (n− 1)

2Lt

(8)

Using the definition of the KL expansion of the forcing function as given in Eq. (2), Eq. (1) is written as :

m x+ c x+ k x = µF (t) +n∑i=1

ξi√λi

n∑j=1

djihj (t)

(9)

As, we are interested in both the steady-state and the transient statistics of the SDOF system, the SDOFresponse which has both the steady state response and the transient response for the step input, F = F0

with zero initial displacement and velocity, is given as :

x =F0

k

(1 − e−ζωnt

(cos (ωdt) + sin (ωdt)

ζ√1− ζ2

))(10)

and for the sinusoidal forcing functions, F = F0 cos Ωt, the total response, xc which includes both thesteady-state and the transient response is given as :

xc = xst + xt (11)

xc =F0 cos (Ωt+ ψ)

k

√(1− β2)2 + (2ζβ)2

+Hce−ζωntcos (ωdt+ψc) (12)

where

β =Ωωn

(13)

ψ = tan−1

(−2ζβ1− β2

)(14)

xt is the transient response, xst is the steady-state response, ζ is non-dimensional damping coefficient, β isthe normalized frequency, ωn is the natural frequency of the SDOF system under consideration, and ψ is thephase angle between the forcing function and the displacement; Hc and ψc are obtained by satisfying initialconditions. For zero initial displacement and velocity,

Hc = −xst

√cos (ψ)2 +

(ζ cos (ψ)− β sin (ψ))2

1− ζ2(15)

ψc = − cos−1

(−xst cos (ψ)

Hc

)(16)

If the forcing function is F = F0 sin Ωt, the total response of the SDOF is given as :

xs = xst + xt (17)

xs =F0 sin (Ωt+ ψ)

k

√(1− β2)2 + (2ζβ)2

+Hse−ζωntsin (ωdt+ψs) (18)

For zero initial displacement and velocity, Hs and ψs are given as :

Hs = −xst

√sin (ψ)2 +

(ζ sin (ψ) + β cos (ψ))2

1− ζ2(19)

ψs = − sin−1

(−xst sin (ψ)

Hc

)(20)

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For the derivation convenience, it is assumed that the mean value of the forcing function, µF (t) is zero,the initial displacement and the velocity are deterministic, and assumed to be zero. Even, if the initialdisplacement and the velocity are non-zero, those can be associated with the mean value of the forcingcomponent. If the mean value of the forcing function, µF (t) is a non-zero function of time, then the meanof the non-stationary response will be non-zero and µF (t) will not affect the calculation of the standarddeviation of the response. The displacement response of the SDOF is given as :

x (t) =n∑i=1

ξi√λi

n∑j=1

djimj (t)

(21)

The response functions, mj , corresponding to the basis functions, hj of the excitation are obtained usingEqs. (10), (11) and (17) by substituting F0 = 1. While evaluating these responses, βn and ψn are calculatedusing Eq. (13) and Eq. (14), respectively and Ωn for the steady-state and the transient analysis are definedby Eq. (22) and Eq. (23), respectively.

Ωn =

0,π

L,π

L, . . . ,

(n− 1)πL

,(n− 1)π

L

(22)

Ωn =

0,π

2L,π

2L, . . . ,

(n− 1)π2L

,(n− 1)π

2L

(23)

By comparing Eqs. (9) and (21), it is seen that for complex, multi-degree-of-freedom structures, thestatistics of the response when excited by the stochastic forcing function can be easily obtained. The usercan choose the basis functions, hj given by either Eq. (7) or Eq. (8). Then, the user has to obtain theeigenvalues, λi and dji, the hj basis function participation factors of the eigenvector, φi of the excitationauto-covariance outside of commercial FEM softwares. The response functions, mj , corresponding to thebasis functions, hj of the excitation should be obtained to construct the PDF or statistics of the response ofthe system.

Using the orthogonality of iid basis random variables, ξ, which is given as :

< ξi, ξj >= δij (24)

where 〈 〉 is the mathematical expectation operator, the covariance, σ2x (t) and auto-covariance, RXX (t1, t2)

of the response are given as :

σ2x (t) =

n∑i=1

λi

n∑j=1

djimj (t)

2

(25)

RXX (t1, t2) = < x (t1) , x (t2) > (26)

=n∑i=1

λi

n∑j=1

djimj (t1)

( n∑k=1

dkimk (t2)

)(27)

From Eq. (25), it can be seen that the standard deviation of the response depends upon the eigenvalues of theforcing auto-covariance function, participation factors of the chosen basis functions and the deterministicsystem response of the basis functions. Equation (27) indicates that the auto-covariance of the responsedepends upon the deterministic responses at two different times of the chosen basis functions.

III. Auto-Covariance Decomposition using Collocation Method and theResponse Calculation

At times, a number of eigenvalues of positive semi-definite auto-covariance function become negativeor infinite when obtained using the KL expansion and the Galerkin procedure because of the used basisfunctions. Similarly, at times, the chosen basis functions may not be optimal in terms of calculations meaningthat a minimum number of basis functions can not represent the given auto-covariance11 as accurately ascompared to some other set of basis functions. To avoid negative and infinite eigenvalues of the auto-covariance function, the eigenvalues and the eigenvectors of the given auto-covariance are obtained from the

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covariance matrix, by collocating the auto-covariance function at discrete points. The Collocation Techniqueshould be applied to the given auto-covariance function in such a way that it results into positive-semidefinitecovariance matrix. By forcing a sufficient number of uniform distribution of collocation points, one can obtainpositive-semidefinite covariance matrix for almost all auto-covariance functions. These collocation pointsshould not be too few otherwise, one would loose important information provided by the auto-covariancefunction. Also, if these points are chosen to be too close to each other, the Collocation Technique wouldresult in an ill-conditioned covariance matrix.

Let CFF represent the excitation covariance matrix obtained by collocation of RFF , the excitation auto-covariance function, and the elements of CFF matrix are given by Eq. (28)

CFF (i, j) = RFF (ti, tj) (28)

Eigenvalues, λc and eigenvectors, φc of CFF are obtained using currently available deterministic algorithmsin commercial softwares like MATLAB and LAPACK and made sure that all λc are positive and finite. Toobtain the dynamic response of the system using the algorithm explained in Section II and the CollocationTechnique, djk coefficients are obtained by choosing the basis functions given in Eq. (8) and using Eq. (29).The chosen basis functions are orthogonal to each other and satisfy Eq. (30).

djk =

Tmax∫Tmin

φckhj dt

Tmax∫Tmin

hjhj dt

(29)

Tmax∫Tmin

hihj dt = 0 if i 6= j (30)

While obtaining djk, the basis functions, hi are discretized with the same collocation points that were usedto obtain the covariance matrix given in Eq. (28). In Eq. (29), the cross product of discretized vectors suchas φck

, hi and hj is carried out before evaluating various integrals. Once djk are obtained, the proceduredescribed in Section II is followed to obtain the statistics of the response using Eqs. (9-27). The flowchart ofthe proposed algorithms, the KL Expansion Method and the Collocation Technique are shown in Fig. (1).

IV. Numerical Examples

To check the accuracy of the proposed algorithms, these algorithms are applied to the SDOF systemwhich is excited by the stationary exponential auto-covariance random forcing function. This SDOF systemis shown in Fig. (2). In this SDOF system, mass of the system, m is 1 kg, stiffness, k is 300 N/m anddamping coefficient, ζ = 0.2 is chosen such that the response of the system converges to the steady-stateresponse within t = 1 sec. The steady-state response of the SDOF system is obtained using four differentapproaches. The convergence study of the standard deviation of the response with respect to number ofterms KL expansion is done. The steady-state response to a non-Gaussian stationary exponential excitationis carried out. The transient analysis of the same SDOF system is obtained using numerical techniques (theKL Expansion Method and the Collocation Technique) and compared to analytical solution. The SDOF’stransient response to a non-stationary exponentially modulated excitation is compared with the solutionobtained by numerical integration of the integral generated using convolution theorem and unit impulseresponse. In addition to this, a non-Gaussian response to an exponentially modulated excitation is alsoobtained.

A. An Exponential Auto-Covariance

The steady-state and the transient response of the SDOF is obtained for an exponential auto-covariance whichis given in Eq. (31). The applied excitation is a stationary process, its steady-state would be stationary butthe transient response would be non-stationary.

RFF (τ) = e−20|τ | (31)

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Choose Basis Functions, hj

Auto-covariance Functionof the Excitation, RFF

Geometry, Mass, Damping, and Force

Definition of the Structure

Determine the Response

Functions, mj

FEM or Analytical KL Expansion andGalerkin Projection

Collocation TechniqueCovariance Matrix, CFF

Eigenvalues, λi andBasis Function Participation

Factor, dji

Response of the System

x (t) =

nXi=1

ξipλi

⎛⎝ nXj=1

djimj (t)

⎞⎠

Figure 1. The Flowchart of the KL Expansion Method and the Collocation Technique

mm

k

c

x(t)

f(t)

Figure 2. Single-Degree-of-Freedom System

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1. Analytical Method

For the particular auto-covariance defined in Eq. (31), the PSD function given in Eq. (32) can be found bytaking the Fourier transform of the auto-covariance defined in Eq. (31).

SFF (ω) =6.366197400 + ω2

(32)

The PSD function of the displacement is shown in Fig. (3) and is given as :

SXX (ω) = |H (ω)|2 SFF (ω) (33)

where H (ω) is the frequency response function and is given as :

|H (ω)|2 =1

k2(

(1− β2)2 + (2ζβ)2) (34)

Covariance of the displacement is calculated by integrating the PSD of the response as given in Eq. (35)

−50 −40 −30 −20 −10 0 10 20 30 40 500

1

2

3

4

5

6

7x 10

−7

Frequency, rad/sec

S xx, m

2 /(ra

d/se

c)

Figure 3. Power Spectral Density Function of the Displacement when Excited by Exponential Auto-Covariance

and the standard deviation of the displacement, σX for this particular SDOF system and excitation is foundto be 3.9307mm.

σ2X =

∫ ∞−∞

SXX (ω) dω (35)

2. Semi-Analytical Method

The exponential auto-covariance described by Eq. (31) is a particular type of the auto-covariance whoseeigenfunctions and eigenvalues can be obtained using analytical methods. The obtained eigenfunctions ofthis auto-covariance are defined by analytical functions which would be accurate as compared to the obtainedby weighted residual method while solving the Fredholm Eq. (3) of the 2nd kind. By differentiating Eq. (3)twice with respect to time and applying appropriate boundary conditions at t = Tmin and t = Tmax,transcendental Eq. (36) and Eq. (39) are obtained. Solutions of Eq. (36) along with Eqs. (37, 38) definesymmetric eigenvalues and eigenfunctions. In the same way, solutions of Eq. (39) along with Eqs. (40, 41)define anti-symmetric eigenvalues and eigenfunctions. Additional information on analytical method used toobtain eigenfunctions and eigenvalues is given.18

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For symmetric eigenfunctions of an exponential auto-covariance :

C − κ tan (κa) = 0 (36)

φn (t) =cos (κnt)√a+ sin (2κna)

2κn

(37)

λn =2C

κ2n + C2

(38)

For anti-symmetric eigenfunctions of an exponential auto-covariance :

κ∗ + C tan (κ∗a) = 0 (39)

φ∗n (t) =sin (κ∗nt)√a+ sin (2κ∗na)

2κ∗n

(40)

λ∗n =2C

κ∗2n + C2(41)

where C is the correlation length parameter of the forcing auto-covariance, a is equal to the half of the timedomain considered in the KL expansion, and κn and κ∗n are the solutions of the transcendental Eq. (36) andEq. (39). The φn and φ∗n are symmetric and antisymmetric eigenfunctions of the exponential auto-covariance,respectively. The λn and λ∗n are the eigenvalues of the same auto-covariance function. Hence, the forcingfunction and the displacement can be written as summation of orthogonal components obtained using theKL expansion and are given in Eq. (42) and Eq. (43), respectively.

F (t) =n∑i=1

ξi√λiφi (t) (42)

x (t) =n∑i=1

ξi√λiSi (t) (43)

where φi and Si are defined in Eqs. (44-47) and γi and νi are defined in the same way as defined in Eq. (13)and Eq. (14), respectively.

φ1,3,5,...,n = ci cos (ηit) (44)φ2,4,6,...,n = ei sin (ηit) (45)

S1,3,5,...,n =ci cos (ηit+ νi)

k

√(1− σ2

i )2 + (2ζσi)2

(46)

S2,4,6,...,n =ei sin (ηit+ νi)

k

√(1− σ2

i )2 + (2ζσi)2

(47)

Therefore, the covariance, σ2x (t) and the auto-covariance, RXX (t1, t2) of the response are given as

σ2X (t) =

n∑i=1

λiS2i (t) (48)

RXX (t1, t2) =n∑i=1

λiSi (t1)Si (t2) (49)

As, the KL expansion is used to decompose auto-covariance of the forcing random process, the standarddeviation of the displacement shows Gibbs phenomenon and it depends upon the number of terms usedin the KL expansion. The variation of the standard deviation with respect to time is shown in Fig.( 4)with ’Semi-Analytical’ legend for the SDOF system. While calculating the response, 32 eigenfunctions areused to represent input auto-covariance. In the same Fig. (4), standard deviation is plotted and denotedas ’Analytical’. Similarly, the auto-covariance of the displacement is shown in Fig. (6(a)). To remove theeffects of Gibbs phenomenon for the stationary excitation, smoothing of the standard deviation is carriedout by averaging over the time in which the analysis is carried out, and the mean of the standard deviationcomes out to be 3.8228mm.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.01.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

−3

Sta

ndar

d D

evia

tion

of th

e D

ispl

acem

ent (

m),

σ

Time, t (sec)

Analytical

Semi−Analytical

KL (21 Terms)

Collocation (100 Terms)

Figure 4. Exponential Auto-Covariance’s Steady-State Standard Deviation of the Displacement

3. Numerical Methods

For general application of these algorithm to non-stationary and non-Gaussian random forcing randomprocesses, the KL Expansion Method and the Collocation Technique should be used as described in Section IIand Section III, respectively. These algorithms are applied to the SDOF system, described earlier. Thevariation of the standard deviation is shown in Fig. (4). The mean of the standard deviation is 3.88995mmusing the KL Expansion Method and 4.12461 mm using the Collocation Technique. The convergence ofthe standard deviation of the displacement with respect to the number of terms in the KL expansion isshown in Fig. (5). By including more than 17 terms in the KL expansion, the converged steady-statestandard deviation of the displacement is obtained when the SDOF system is excited by the exponentialauto-covariance, defined by Eq. (31). The auto-covariance of the displacement using the KL ExpansionMethod is shown in Fig. (6(b)). The auto-covariance of the displacements is shown in Fig. (7) when obtainedusing the Collocation Technique. During application of the KL Expansion Method, 21 terms are used torepresent the eigenvectors and 100 terms are used during the application of the Collocation Technique.

While describing a stochastic excitation, one needs to define the auto-covariance of the random processand the marginal PDF of the process. From the marginal PDF definitions, the iid basis random variablesshould be defined to obtain the KL expansion of the given random process.16 Using the same approach,16 iidbasis random variables for non-Gaussian random process having zero mean and unit variance are obtainedand are shown in Fig. (8(a)) for Uniform, Triangular, and Beta PDFs along with the input Gaussian marginalPDF. Here, it is assumed that the user has obtained the correct marginal PDF. The purpose of using differentmarginal PDFs for non-Gaussian random process is to show the applicability of the proposed algorithms. Thesteady-state displacement PDFs are obtained using Eq. (21) and the KL Expansion Method for these randomprocesses and are shown in Fig. (8(b)). For all the excitation processes, the steady-state displacement PDFsare almost Gaussian because displacement calculation requires 21 KL expansion terms and 100 collocationterms. Central Limit Theorem states that as the number of the independent distributed variables increasesin the summation, the linear combination of these variables tends to be a Gaussian.

The transient analysis of the same SDOF system having zero initial displacement and velocity and excitedby an exponential auto-covariance is carried out using both the KL Expansion Method and the CollocationTechnique. The same definition of the auto-covariance is used as defined in Eq. (31). Even though the auto-covariance is stationary, the transient response is non-stationary. The transient response obtained usingnumerical techniques is compared with the analytical solution in Fig. (9). The derivation of the analyticalsolution is described in Appendix. The obtained displacement auto-covariance using the KL Expansion

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5 10 15 20 25 302

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4x 10

−3

Number of Karhunen−Loeve Expansion Terms

Sta

ndar

d D

evia

tion

of th

e D

ispl

acem

ent (

m),

σ

Analytical

KL Expansion

Figure 5. Convergence of the Steady-State Standard Deviation of the Displacement when Excited by Expo-nential Auto-Covariance

Method and the Collocation Technique are shown in Fig. (10) and Fig. (11), respectively. Both the auto-covariances match approximately. The transient response is obtained using 6 KL expansion terms and 100collocation terms.

B. Exponentially Modulated Auto-Covariance

Mostly, a non-stationary excitation occurs during initial time. It is important to find out transient responsewhen excited by the non-stationary excitation because the standard deviation of the displacement duringthe transient period may be more than the steady-state response and many times, the excitation becomesdeterministic after some time of the application of the force. So, the standard deviation during the transientperiod changes from 0 to some value and then settles down to a constant value. To check the accuracy ofdeveloped algorithms, the same SDOF system is excited by non-stationary excitation as given by Eq. (50).

RFF (t1, t2) = e−(t1+t2)e−20|t1−t2| (50)

The standard deviation of the displacement for exponentially modulated auto-covariance is shown in Fig. (12)when obtained using both the KL Expansion Method and the Collocation Technique. In Fig. (12), the ana-lytical solution is also plotted which is obtained by direct integration of Eq. (51). The transient displacementauto-covariance is plotted in Fig. (13) for both methods and they match approximately. Equation (51) isobtained by taking expectation of the product of the responses obtained using the convolution theorem andthe unit impulse response4.

σ2x (t) =

t∫0

s∫0

RFF (r, s) e−(ζωn(2t−r−s)) sin (ωd (t− r)) sin (ωd (t− s))ω2d

drds (51)

The non-Gaussian response is obtained using the same approach,16 iid basis random variables for non-Gaussian random process having zero mean and unit variance are obtained and are shown in Fig. (8(a))for Uniform, Triangular, and Beta PDFs along with the input Gaussian marginal PDF. The transientdisplacement PDFs at t = 0.17 sec are obtained using Eq. (21) and the KL Expansion Method for theserandom processes and are shown in Fig. (14). For all the excitation processes, transient displacement PDFsare non-Gaussian because the displacement calculation requires only 6 KL expansion terms.

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00.2

0.40.6

0.81

00.2

0.40.6

0.81

−2

−1

0

1

2

3

x 10−5

Time, t1 (sec)Time, t

2 (sec)

Aut

o−C

ovar

ianc

e of

the

Dis

plac

emen

t, R xx

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x 10−5

(a) Semi-Analytical Method

0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1

1.5

2

x 10−5

Time, t1 (sec)

Time, t2 (sec)

Aut

o−C

ovar

ianc

e of

the

Dis

plac

emen

t, R xx

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

−5

(b) KL Expansion Method

Figure 6. Auto-Covariance of the Displacement for T = [0 1] sec when Excited by Exponential Auto-Covariance

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0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1

1.5

2

x 10−5

Time, t1 (sec)Time, t

2 (sec)

Aut

o−C

ovar

ianc

e of

the

Dis

plac

emen

t, R xx

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

−5

Figure 7. Auto-Covariance of the Displacement using Collocation Technique for T = [0 1] sec when Excited byExponential Auto-Covariance

−4 −3 −2 −1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Force (N)

PD

F o

f the

Inpu

t For

ce

GaussianUniformTriangularBeta

(a) Input Random Process Marginal PDFs

−0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.0250

10

20

30

40

50

60

70

80

90

100

110

Displacement (m)

PD

F o

f the

Dis

plac

emen

t

Gaussian Input PDFUniform Input PDFTriangular Input PDFBeta Input PDF

(b) Steady-State Displacement PDFs

Figure 8. Exponential Auto-Covariance Random Process Marginal PDFs and Steady-State Displacement PDFs

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−3

Time, t (sec)

Sta

ndar

d D

evia

tion

of th

e D

ispl

acem

ent (

m),

σ

AnalyticalKL (6 Terms)Collocation (100 Terms)

Figure 9. Exponential Auto-Covariance’s Transient Response Standard Deviation of the Displacement

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8−1

0

1

2x 10

−5

Time, t1 (sec)

Time, t2 (sec)

Aut

o−C

ovar

ianc

e of

the

Dis

plac

emen

t, R xx

−6

−4

−2

0

2

4

6

8

10

12

14

x 10−6

Figure 10. Transient Auto-Covariance of the Displacement for T = [0 0.7] sec for Exponential Auto-Covarianceusing the KL Expansion Method

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0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8−1

0

1

2x 10

−5

Time, t1 (sec)

Time, t2 (sec)

Aut

o−C

ovar

ianc

e of

the

Dis

plac

emen

t, R xx

−5

0

5

10

15

x 10−6

Figure 11. Transient Auto-Covariance of the Displacement for T = [0 0.7] sec for Exponential Auto-Covarianceusing the Collocation Technique

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

2

2.5

3

3.5x 10

−3

Time, t (sec)

Sta

ndar

d D

evia

tion

of th

e D

ispl

acem

ent (

m),

σ

AnalyticalKL (6 Terms)Collocation (140 Terms)

Figure 12. Exponentially Modulated Auto-Covariance’s Transient Standard Deviation of the Displacement

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0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8−5

0

5

10

15x 10

−6

Time, t1 (sec)

Time, t2 (sec)

Aut

o−C

ovar

ianc

e of

the

Dis

plac

emen

t, R xx

−4

−2

0

2

4

6

8

10x 10

−6

(a) KL Expansion Method

00.2

0.40.6

0.8

0

0.2

0.4

0.6

0.8−5

0

5

10

15

x 10−6

Time, t1 (sec)

Time, t2 (sec)

Aut

o−C

ovar

ianc

e of

the

Dis

plac

emen

t, R xx

−2

0

2

4

6

8

10

x 10−6

(b) Collocation Technique

Figure 13. Transient Auto-Covariance of the Displacement for T = [0 0.7] sec for Exponentially ModulatedAuto-Covariance

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−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.020

20

40

60

80

100

120

140

PD

F o

f the

Dis

plac

emen

t

Displacement (m)

Gaussian Input PDFUniform Input PDFTriangular Input PDFBeta Input PDF

Figure 14. Exponentially Modulated Auto-Covariance’s Transient Standard Deviation of the Displacement

V. Conclusion and Future Work

The KL Expansion method and the Collocation Technique are proposed to calculate the response of thedynamic system (second-order) subjected to stationary or non-stationary random processes which might beGaussian or non-Gaussian. These random processes must be described by the auto-covariance functions andgenerally those are available. The excitation random processes are descritized into orthogonal componentsand those are used to obtain the statistics of the response. If the auto-covariance function correlation lengthis small meaning that input random process is wide-band, then expansion of the random process has manyKL expansion or collocation terms. During the transient analysis, the PDF of the displacement evolvesfrom zero standard deviation to some constant standard deviation. During the evolution of the PDF of thedisplacement, many times the standard deviation of the displacement might be more than the stationarystandard deviation. Even during the evolution of the PDF, the PDF changes its density function, it startswith dirac − delta at time t = 0 sec and may become or tend to become Gaussian if excitation is wide-band (number of terms required during the auto-covariance expansion). The KL Expansion Method moreaccurately describes the statistics of the dynamic response as compared to the Collocation Technique butmany-times suffers from ill-conditioned matrices that are obtained during the KL expansion of the auto-covariance function. The Collocation Technique generally over-predicts the system response as comparedto the KL Expansion Method. During the transient analysis, the standard deviation predicted by both themethods are same but the PDF function will differ drastically because the Collocation Technique requires toomany terms as compared to the KL Expansion Method. So the Collocation Technique predicted PDF wouldbe Gaussian in almost all cases. Still the Collocation Technique method is good to predict the standarddeviation or the auto-covariance of the response, but not for predicting PDF function description. Theproposed algorithms require deterministic responses to obtain stochastic response of the dynamic systemwhen excited by a random process. For general application, these algorithms will be next applied to acontinuous system using commercial FEA software.

Acknowledgments

The authors would like to thank Luna Innovations for a partial support of this work. We would also liketo thank the Institute for Critical Technology and Applied Science (ICTAS) for use of their facilities andcontinued support.

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Appendix: Non-stationary Response of the SDOF system subjected to anExponential Auto-covariance

The variance of SDOF system, σ2x (t), when excited by stationary auto-covariance with zero initial dis-

placement and velocity is given as4 :

σ2x (t) =

∞∫−∞

SFF (ω)dω

(ω2n − ω2)2 + (2ζωnω)2

[1 + e−2ζωnt

1 +

ωnωdζ sin 2ωdt

−eζωnt

(2 cosωdt+

2ωnωd

ζ sinωdt)cosωt− eζωnt

2ωωdζ sinωdt sinωt

+(ζωn)2 − ω2

d + ω2

ω2d

sin2 ωdt

](52)

Equation (52) requires the definition of power spectral density function of the auto-covariance. The PSD ofexponential auto-covariance function can be defined :

SFF (ω) =C1

α2 + ω2(53)

where C1 and α are constants, which define exponential auto-covariance’s distribution. The transient dis-placement variance of the SDOF system when excited by an exponential auto-covariance is given as :

σ2x (t) =

[1 + e−2ζωnt

(1 +

ωnωdζ sin (2ωdt)

)]I1 − 2

[e−2ζωnt

(cosωdt+

ωnωdζ sinωdt

)]I2

− 2[e−2ζωnt

ωdsinωdt

]I3 +

e−2ζωnt

ω2d

sin2 ωdt[(

(ζωn)2 − ω2d

)I1 + I4

](54)

where I1, I2, A, I3, and I4 are given as :

I1 =π (α+ 2ζωn)C1

2ζω3nα (α2 + ω2

n + 2ζωnα)

I2 = πAC1

[e−αt

α− e−ζωnt

ζ√

1− ζ2

(√1− ζ2 cosωdt− ζ sinωdt

)

+

(α2 + 2ω2

n

(1− 2ζ2

)) (√1− ζ2 cosωdt+ ζ sinωdt

)2ω3

nζ√

1− ζ2

A =1

(α2 + ω2n)2 − (2αζωn)2

I3 = πAC1

[e−αt − e−ζωnt

2ζ√

1− ζ2

((2ζ√

1− ζ2)

cosωdt−

(2ζ2 − 1 +

α2 + 2ω2n

(1− 2ζ2

)ω2n

)sinωdt

)]

I4 =πC1

2ζωn (α2 + 2αζωn + ω2n)

(55)

References

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4Benaroya, H. and Han, S. M., Probability Models in Engineering and Science, CRC Press, Taylor & Francis Group, 2005.5Spanos, P. D., Tezcan, J., and Tratskas, P., “Stochastic Processes Evolutionary Spectrum Estimation via Harmonic

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2007.13Loeve, M., Probability Theory, Springer-Verlag, 1977.14Poirion, F. and Soize, C., “Monte Carlo Construction of Karhunen Loeve Expansion for Non-Gaussian Random Fields,”

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Stochastic Processes,” Journal of Engineering Mechanics, Vol. 128, No. 2,, 2002, pp. 190–201.16Mulani, S. B., Kapania, R. K., and Walters, R. W., “Karhunen-Loeve Expansion of Non-Gaussian Random Process,”

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18Ghanem, R. G. and Spanos, P. D., Stochastic Finite Elements: A Spectral Approach, Dover Publications Inc., reviseded., 1991.

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