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Journal of Sound and Vibration

Journal of Sound and Vibration 331 (2012) 167–179

0022-46

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jsvi

Stochastic free vibration of orthotropic plates using generalizedpolynomial chaos expansion

K. Sepahvand a,b,n, S. Marburg b, H.-J. Hardtke c

a Department of Mechanical Engineering, Persian Gulf University, Iranb Institute of Mechanics, Universitat der Bundeswehr Munchen, Germanyc Institute of Solid Mechanics, Technische Universitat Dresden, Germany

a r t i c l e i n f o

Article history:

Received 16 September 2010

Received in revised form

15 June 2011

Accepted 14 August 2011

Handling Editor: K. Wordenare represented by truncated polynomial chaos expansions with arbitrary random basis.

Available online 21 September 2011

0X/$ - see front matter & 2011 Elsevier Ltd.

016/j.jsv.2011.08.012

esponding author.

ail address: sepahvand@unibw.de (K. Sepahva

a b s t r a c t

This paper presents the theory and application of the generalized polynomial chaos

expansion for the stochastic free vibration of orthotropic plates. Specifically, the

stochastic analysis of orthotropic plates under the uncertainties in elasticity moduli is

investigated. The uncertain moduli, eigen-frequencies and eigen-modes of the plates

The expansions are substituted in the governing differential equations to calculating the

polynomial chaos coefficients of the eigen-frequencies and the eigen-modes. Distribu-

tion functions of the uncertain moduli are derived from experimental data where the

Pearson model is used to identify the type of density functions. This realization then is

employed to construct random orthogonal basis for each uncertain parameter. Because

of available experimental modal analysis data, this paper provides a useful practical

example on the efficacy of polynomial chaos where the statistical moments and the

probability distributions of modal responses are compared with experimental results.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Orthotropic plates are used in a lot of industrial areas. Due to the natural structural complexity, assigning thedeterministic values for mechanical properties of these elements is not often possible. That means the mechanicalproperties of the plates include some degrees of uncertainties and deterministic simulations lead to the nominal solution.The wide use of these structures requires more accurate investigating to including these uncertainties in simulation.Characterization of the stochastic response due to these uncertainties by sampling methods, e.g. Monte Carlo (MC)simulations, reliability analysis, and perturbation approaches has long interested to researchers. These methods inconjunction with finite element method (FEM) have been widely used by researchers to quantify the structural responseuncertainty [1–3]. The methods are based on the generation of functions or random samples each of which approximatelyfulfills the requirements of uncertain parameters or system response. Owing the fact that the accuracy of samplingmethods is a function of the number of realizations, they are very time-consuming in terms of computation. Inperturbation methods, the random fields are expanded via Taylor series around the mean values [4]. Neumann expansionhas been used by some researchers instead of Taylor series for manipulation of the stochastic system properties andresponse [5]. The main inherent limitation of these methods is that the uncertainties cannot be large, i.e. the deviations ofstructure response cannot be too large compared with the nominal values [6].

All rights reserved.

nd).

K. Sepahvand et al. / Journal of Sound and Vibration 331 (2012) 167–179168

Recently, the development of non-sampling stochastic techniques appears to be the most widely used. The method is todiscretize directly a random quantity in random space. Karhunen–Lo�eve (K–L) expansion is a quite general method of alongthis line utilized for the Gaussian random quantities [7]. Due to the fact that only a few terms are necessary to involve themost of system randomness in response, it can be used in many practical stochastic systems where the uncertainquantities vary smoothly with the time and space [8,9]. However, the covariance function must be known a priori.To overcome this drawback, the generalized polynomial chaos (gPC) expansion can be used [10]. The polynomial basis todiscretize the random space are chosen from the hyper-geometric polynomials of Askey scheme and the underlyingrandom quantities are not restricted to the Gaussian random variables. The stochastic simulations based on the gPCtechniques have been widely employed in various engineering applications, cf. [11–16]. They used mostly the gPC methodto representing the propagation of parametric uncertainty through the response of linear and nonlinear problems wherethe considered uncertain parameters are random variables with standard Probability Density Function (PDF). A generalreview of the methods in structural dynamics has recently reported by Schueller et al. [17].

Limited literature is available on using the above mentioned stochastic methods in orthotropic structures with randommaterial properties. The reported works mainly focused on reliability analysis where uncertain quantities are simplyrepresented by a single random variable and the spatial variability is not taken into account [18–24]. Spectral stochasticFEM has been successfully applied to stochastic simulation of orthotropic structures where the K–L and gPC expansions areused to represent the spectral stochastic nodal displacements in terms of standard normal random variables [25–29]. Themethod has been also applied for stochastic simulation of various vibration problems, see [11,12,30,31] and referencestherein. However, there is no reported work concern to free vibration of orthotropic plates involving uncertain moduli.Furthermore, in the reported works the uncertain parameters have been mostly considered as Gaussian random variableswhich may be not in coincidence with the reality.

In this paper, the gPC method is applied to the stochastic analysis of orthotropic plates with uncertain E-moduli. Thegoverning free vibration equation is decomposed for calculation of the eigen-frequencies and the eigen-modes. In eachcase, the stochastic Galerkin technique is used to project the governing equations into a set of deterministic equations.Then the deterministic equations are solved to calculate the deterministic gPC coefficients/functions of system responses.Actually, this work presents significant contributions: application of the gPC expansion in vibration of orthotropic plates,derivation of a closed form formulation for stochastic analysis of these plates including uncertain moduli similarly to thedeterministic form, and using experimental data instead of simulated data for validation the results. Another contributionis using experimental data to identify the probability density function (PDF) of uncertain parameters where thestraightforward Pearson model is used for identification of the PDF type only by using the third and fourth statisticalmoments. Because of available experimental modal analysis data, this paper provides a useful practical example on theefficacy of gPC where the statistical moments and the probability distributions of modal responses are compared withexperimental results.

In the next section we present the basic theory of the gPC and its application for representation of uncertain parameters.The gPC formulation for stochastic free vibration of orthotropic plates is derived in Section 3. The identification of uncertainparameter PDFs from experimental data is demonstrated in Section 4. In Section 5, we compare the numerical results to thatobtained from experimental data. The conclusions are discussed in the last section of paper.

2. Polynomial chaos theory

The polynomial chaos expansion is an effective tool for solving stochastic systems. It was first introduced as thehomogeneous chaos by Wiener [32]. The basic idea is to project the random variables of problem onto a stochastic spacespanned by a set of complete orthogonal polynomials. Let ðO,F ,PÞ be a probability space, in which O is a sample space, F isa s-algebra on O, and P is a probability measure on O. Any uncertain parameter X : O�!R with finite variance, i.e.X 2 L2ðOÞ, can be represented as [7,10]

X ¼ x0C0þX1

i1 ¼ 1

xi1C1ðxi1 ÞþX1

i1 ¼ 1

Xi1

i2 ¼ 1

xi1 i2C2ðxi1 ,xi2 ÞþX1

i1 ¼ 1

Xi1

i2 ¼ 1

Xi2

i3 ¼ 1

xi1i2i3C3ðxi1 ,xi2 ,xi3 Þþ � � � (1)

or in a more compact form

X ¼X1i ¼ 0

xiCiðnÞ: (2)

It is called the gPC expansion of X . The orthogonal polynomials Ci are functions of the multidimensional standard randomvector n¼ fx1,x2, . . . ,xng

T . The standard random variables are functions defined on particular sample spaces, i.e. xi 2 Oi,i¼ 1;2, . . . ,n. Depending on the type of the uncertain parameter, a set of ðfxig,CiÞ can be selected. Accordingly, Gaussianvariables are best approximated by Hermite polynomials. Laguerre polynomials account for the best approximation of aGamma distributed variable whereas Jacobi polynomials should be used for Beta distributions, etc. [10,33]. Thedeterministic unknown coefficients, xi, are denoted as the weights of random space discretization. Furthermore, inpractice, it is not useful to retain infinite summations and a truncated expansion of order N is used [34].

K. Sepahvand et al. / Journal of Sound and Vibration 331 (2012) 167–179 169

The random base functions have orthogonality with respect to the inner product on L2ðOÞ, i.e.

E½Ci,Cj� ¼ E½C2i �dij, (3)

in which dij represents the Kronecker delta and E denotes the expectation value with respect to the probability space. Thisproperty can be used to calculate the truncated gPC representation of X by projecting onto the orthogonal basis:

xi ¼1

/C2i S

ZO/X ,CkðnÞS dmðnÞ, i,k¼ 0;1,2, . . . (4)

where dmðnÞ is the probability measure corresponding to the random space O. If all random variables are independent andcontinuous, then dmðnÞ can be stated as

dmðnÞ ¼ r1ðx1Þr2ðx2Þ . . .rnðxnÞ dx1 dx2 . . .dxn, (5)

and riðxiÞ is PDF of xi. Successively, Eq. (4) is expanded as

xi ¼1

/C2i S

ZO1

ZO2

. . .

ZOn

XCkðnÞr1ðx1Þr2ðx2Þ . . .rnðxnÞ dx1 dx2 . . .dxn: (6)

Eq. (6) represents the polynomial chaos transform of X which minimizes the least-square error on the space spanned by Ck.In this way, the determination of uncertain parameter X is shifted to the calculation of deterministic coefficients xi. Thesecoefficients completely characterize the identification of uncertain parameter X . Clearly, the accuracy of this characterization ishardly depended on the number of these coefficients and selection of orthogonal basis, cf. [10]. The gPC method can beemployed equally for representation of random processes by separating the spatiotemporal and random dependencies, i.e.

Wðx,t; nÞ �XN

i ¼ 0

wiðx,tÞCiðnÞ, (7)

in which wiðx,tÞ are deterministic functions of spatial space and time. If W is available, then these coefficient functions aredetermined as xi in Eq. (6).

3. Stochastic free vibration of orthotropic plates

In this section we derive the general gPC formulation for stochastic free vibration of orthotropic plates with all freeboundary conditions. The stochastic analysis is performed for uncertainties in elasticity moduli, i.e. Ex,Ey and Gxy. Theeigen-frequencies and eigen-modes are considered as random variables and random processes, respectively. Starting fromthe general deterministic governing equation of free vibration for orthotropic rectangular plates as [35]

D11@4w

@x4þ2ðD12þ2D66Þ

@4w

@x2@y2þD22

@4w

@y4¼�rh

@2w

@t2, (8)

we employ the random vector n¼ fx1,x2,x3gT to include the parameter uncertainties in the governing equation. For such

conditions Eq. (8) can be easily rewritten as

DðnÞr2r

2wðx,y,t; nÞ ¼�rh@2wðx,y,t;nÞ

@t2, (9)

where

DðnÞ ¼Dðx1,x2,x3Þ ¼ ½D11ðx1Þ D12ðx1,x2Þþ2D66ðx3Þ D22ðx2Þ�, (10)

D11ðx1Þ ¼Exðx1Þh

3

12ð1�nxynyxÞ, D22ðx2Þ ¼

Eyðx2Þh3

12ð1�nxynyxÞ, (11)

D12ðx1,x2Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinxynyxExðx1ÞEyðx2Þ

ph3

12ð1�nxynyxÞ, D66ðx3Þ ¼

Gxyðx3Þh3

12: (12)

Let us assuming that the uncertain E-moduli can be represented by following gPC expansions:

Exðx1Þ ¼XNx

i ¼ 0

exiC1iðx1Þ, Eyðx2Þ ¼

XNy

j ¼ 0

eyjC2jðx2Þ, Gxyðx3Þ ¼

XNg

k ¼ 0

gxykC3kðx3Þ: (13)

In which the deterministic coefficients, exi,eyj

and gxykare derived from Eq. (6). The polynomial basis C1,C2 and C3 are

individual basis, i.e. polynomials which are used to discretize the random spaces corresponding to the uncertain E-moduli.We attempt to determine the stochastic eigen-modes and eigen-frequencies considering the plate stochastic transversedeflection function wðx,y,t; nÞ as

wmnðx,y,t; nÞ ¼Fmnðx,y;nÞe�2pLmnðnÞt (14)

K. Sepahvand et al. / Journal of Sound and Vibration 331 (2012) 167–179170

where Fmnðx,y; nÞ is the stochastic eigen-mode, LmnðnÞ is the stochastic eigen-frequency and m and n are the number ofnodes along x and y directions, respectively. Then Eq. (9) yields

D11ðx1Þ@4Fmn

@x4þ2 D12ðx1,x2Þþ2D66ðx3Þ� � @4Fmn

@x2@y2þD22ðx2Þ

@4Fmn

@y4¼�rhL2

mnFmn: (15)

The eigen-modes and eigen-frequencies are approximated by the gPC expansions as

Fmnðx,y; nÞ ¼XNf

l ¼ 0

fmnlðx,yÞClðnÞ, L2

mnðnÞ ¼XNl

q ¼ 0

lmnqCqðnÞ: (16)

The polynomial basis C is derived by using Hilbert space properties as C¼C1 �C2 �C3. The unknown deterministicfunctions fmnl

ðx,yÞ and unknown coefficients lmnlare calculated by substituting the gPC in the corresponding equations.

The procedure is discussed in subsequent sections.

3.1. Determination of the gPC coefficients for eigen-frequencies

The stochastic eigen-frequencies of orthotropic plates are derived by substituting Eq. (14) into (9) which leads to

L2mnðnÞ ¼

D4mnðnÞp2

4rha4,

D4mnðnÞ ¼D11ðx1ÞB4

mþD22ðx2ÞB4n

a4

b4þ2

a2

b2ðD12ðx1,x2ÞFmFnþ2D66ðx3ÞJmJ nÞ: (17)

The constants Bi, F i and J i are indices determined according to the boundary conditions, cf. [36]. By substitution the gPCfor the uncertain parameters and the frequency, we have

XNl

q ¼ 0

lmnqCqðnÞ ¼p2

4rha4

8>>>><>>>>:

h3

12ð1�nxynyxÞB4

m

XNx

i ¼ 0

exiC1iðx1ÞþB4

n

a4

b4

XNy

j ¼ 0

eyjC2jðx2Þ

24

35

þ2a2

b2

266664

ffiffiffiffiffiffiffiffiffiffiffiffiffinxynyxp

h3

12ð1�nxynyxÞFmF n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNx

i ¼ 0

exiC1iðx1Þ

XNy

j ¼ 0

eyjC2jðx2Þ

vuut|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

non2linear term

þ2h3

12JmJ n

XNg

k ¼ 0

gxykC3kðx3Þ

377775

9>>>>=>>>>;: (18)

It is the gPC representation of the stochastic eigen-frequencies. Knowing the gPC expansions of uncertain parameters,Eq. (18) gives the gPC coefficients lmnq . This can be achieved by employing stochastic Galerkin projection. However,Galerkin projection requires no non-linear terms. We substitute this nonlinear term by the approximated form offfiffiffiffiffiffiffiffiffiffi

ExEy

qþ

1

2ffiffiffiffiffiffiffiffiffiffiExEy

q XNx

i ¼ 1

exiC1iðx1Þ

XNy

j ¼ 1

eyjC2jðx2Þ, (19)

where Ex and Ey are the mean values of elasticity moduli. Galerkin projection of above equation can be carried out bymultiply the both sides of Eq. (18) by the test functions fCsg, s¼ 0;1, . . . ,Nl, and using the orthogonality properties ofpolynomial basis. Then we have

lmns ¼p2

4rha4/C2s S

h3

12ð1�nxynyxÞB4

m

XNx

i ¼ 0

exiCisþB4

n

a4

b4

XNy

j ¼ 0

eyjCjs

24

35

8<:

þ2a2

b2

ffiffiffiffiffiffiffiffiffiffiffiffiffinxynyxp

h3

12ð1�nxynyxÞFmF n

ffiffiffiffiffiffiffiffiffiffiExEy

qþ

1

2ffiffiffiffiffiffiffiffiffiffiExEy

q XNx

i ¼ 1

XNy

j ¼ 1

exieyj

Cijs

0B@

1CA

264þ2

h3

12JmJ n

XNg

k ¼ 0

gxykCks

#), s¼ 0;1, . . . ,Nl (20)

where Cij...v ¼/C1iC2j

. . .CvS. Setting

D11 ¼h3

12ð1�nxynyxÞ/C2s S

XNx

i ¼ 0

exiCis, (21)

D22 ¼h3

12ð1�nxynyxÞ/C2s S

XNy

j ¼ 0

eyjCjs, (22)

K. Sepahvand et al. / Journal of Sound and Vibration 331 (2012) 167–179 171

D12 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffinxynyxp

h3

12ð1�nxynyxÞ/C2s S

ffiffiffiffiffiffiffiffiffiffiExEy

qþ

1

2ffiffiffiffiffiffiffiffiffiffiExEy

q XNx

i ¼ 1

XNy

j ¼ 1

exieyj

Cijs

0B@

1CA, (23)

D66 ¼h3

12/C2s S

XNg

k ¼ 0

gxykCks, (24)

Eq. (20) can be rewritten as

lmn ¼p2

4rha4D11B4

mþD22B4n

a4

b4þ2

a2

b2ðD12FmF nþ2D66JmJ nÞ

� �, (25)

where lmn ¼ flmn0,lmn0

, . . . ,lmnNlgT is the vector of gPC coefficients for Lmn. Eq. (25) introduces a general deterministic

form to determining the gPC coefficients of stochastic eigen-frequency for orthotropic plates. The vector coefficients Dij areconstant and are calculated once during the numerical procedure. Furthermore, it is important to take into account thateven the eigen-frequencies are positive values but the gPC coefficients may be positive or negative.

3.2. Determination of the gPC coefficients for eigen-modes

The coefficients of the gPC for the eigen-modes fmnlðx,yÞ in Eq. (16) are calculated by substituting the expansions for

uncertain parameters and eigen-pairs in Eq. (15). This leads to

h3

12ð1�nxynyxÞ

XNx

i ¼ 0

exiC1iðx1Þ

XNf

l ¼ 0

@4fmnl

@x4ClðnÞþ

XNy

j ¼ 0

eyjC2jðx2Þ

XNf

l ¼ 0

@4fmnl

@y4ClðnÞ

24

35

þh3

3

XNg

k ¼ 0

gxykC3kðx3Þ

XNf

l ¼ 0

@4fmnl

@x2@y2ClðnÞþ

ffiffiffiffiffiffiffiffiffiffiffiffiffinxynyxp

h3

6ð1�nxynyxÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNx

i ¼ 0

exiC1iðx1Þ

XNy

j ¼ 0

eyjC2jðx2Þ

vuut|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

non�linear term

�XNf

l ¼ 0

@4fmnl

@x2@y2ClðnÞ ¼ �rh

XNl

q ¼ 0

lmnqCqðnÞXNf

l ¼ 0

fmnlClðnÞ: (26)

The non-linear term is substituted with the approximation in Eq. (19) and Galerkin projection is employed using theorthogonal basis Cp, then

h3

12ð1�nxynyxÞ

XNf

l ¼ 0

XNx

i ¼ 0

exiCilp

@4fmnl

@x4þXNf

l ¼ 0

XNy

j ¼ 0

eyjCjlp

@4fmnl

@y4

0@

1A

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffinxynyxp

h3

12ð1�nxynyxÞ

ffiffiffiffiffiffiffiffiffiffiExEy

qCpp

@4fmnp

@x2@y2þ

1

2ffiffiffiffiffiffiffiffiffiffiExEy

q XNf

l ¼ 0

XNx

i ¼ 1

XNy

j ¼ 1

exieyi

Cijlp

@4fmnl

@x2@y2

0B@

1CA

þh3

3

XNf

l ¼ 0

XNg

k ¼ 0

gxykCklp

@4fmnl

@x2y2¼�rh

XNf

l ¼ 0

XNl

q ¼ 0

lmnq Cqlpfmnl, p¼ 0;1, . . . ,Nf, (27)

where

Cij...v ¼/C1i,C2j

, . . . ,CvS: (28)

Eq. (27) can be written more concisely to have a similar schematic form as the deterministic case

D11@4fnm

@x4þ2ðD12þ2D66Þ

@4fnm

@x2@y2þD22

@4fnm

@y4¼ rhX2fnm, (29)

in which

fnm ¼ ffnm0,fnm1

, . . . ,fnmpgT , (30)

and

D11 ¼h3

12ð1�nxynyxÞ

PNx

i ¼ 0 exiCi00

PNx

i ¼ 0 exiCi10 . . .

PNx

i ¼ 0 exiCiNf0PNx

i ¼ 0 exiCi01

PNx

i ¼ 0 exiCi11 . . .

PNx

i ¼ 0 exiCiNf1

^ . . . . . . ^PNx

i ¼ 0 exiCi0p

PNx

i ¼ 0 exiCi1p . . .

PNx

i ¼ 0 exiCiNfp

2666664

3777775, (31)

K. Sepahvand et al. / Journal of Sound and Vibration 331 (2012) 167–179172

D22 ¼h3

12ð1�nxynyxÞ

PNy

j ¼ 0 eyjCj00

PNy

j ¼ 0 eyjCj10 . . .

PNy

j ¼ 0 eyjCjNf0PNy

j ¼ 0 eyjCj01

PNy

j ¼ 0 eyjCj11 . . .

PNy

j ¼ 0 eyjCjNf1

^ . . . . . . ^PNy

j ¼ 0 eyjCj0p

PNy

j ¼ 0 eyjCj1p . . .

PNy

j ¼ 0 eyjCjNfp

26666664

37777775, (32)

D12 ¼

c11þa1 c12 . . . c1p

c21 c22þa1 . . . c2p

^ . . . . . . ^

cp1 cp2 . . . cppþa1

266664

377775, (33)

D66 ¼h3

12

PNg

k ¼ 0 gxykCk00

PNg

k ¼ 0 gxykCk10 . . .

PNg

k ¼ 0 gxykCkNf0PNg

k ¼ 0 gxykCk01

PNg

k ¼ 0 gxykCk11 . . .

PNg

k ¼ 0 gxykCkNf1

^ . . . . . . ^PNg

k ¼ 0 gxykCk0p

PNg

k ¼ 0 gxykCk1p . . .

PNg

k ¼ 0 gxykCkNfp

2666664

3777775, (34)

X2¼

PNlq ¼ 0 lmnq Cq00

PNlq ¼ 0 lmnq Cq10 . . .

PNlq ¼ 0 lmnq CqNf0PNl

q ¼ 0 lmnq Cq01PNl

q ¼ 0 lmnq Cq11 . . .PNl

q ¼ 0 lmnq CqNf1

^ . . . . . . ^PNlq ¼ 0 lmnq Cq0p

PNlq ¼ 0 lmnq Cq1p . . .

PNlq ¼ 0 lmnq CqNfp

2666664

3777775: (35)

The elements of matrix D12 in Eq. (33) are calculated as

clp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffinxynyxp

h3

24ffiffiffiffiffiffiffiffiffiffiExEy

qð1�nxynyxÞ

XNx

i ¼ 1

XNy

j ¼ 1

exieyj

Cijlp, (36)

a1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffinxynyxp

h3

12ð1�nxynyxÞ

ffiffiffiffiffiffiffiffiffiffiExEy

qCpp: (37)

Eq. (29) is a system of ðNfþ1Þ deterministic coupled fourth order homogeneous partial differential equations (PDE) withthe ðNfþ1Þ unknowns fnml

ðx,yÞ. In this way, the stochastic PDE in (15) is transferred to the set of deterministic PDEs andthe solution of stochastic PDE in Eq. (15) is reduced to the calculation of deterministic functions, fmnl

, at any point of theplate for any eigen-frequency Lmn. The solution can be performed by the various methods. Here, we use the semi-analytical procedure based on the admissible beam functions. The procedure is based on the approximation functions offmnl

required for the Ritz method for orthotropic plate. The choice is restricted to the eigen-functions satisfied freeboundary conditions of the beam element. Assuming that the variables in fmnl

ðx,yÞ are separable, the gPC functions areexpressed in term of appropriate admissible functions which satisfy at least boundary conditions. We begin with the Ritzapproximation of the form

fmnlðx,yÞ �

XNx

i ¼ 1

XNy

j ¼ 1

cijlXiðxÞYjðyÞ, Nx,Ny ¼ 1;2,3, . . . , (38)

where Xi(x) and Yj(y) denote any admissible approximation function for the problem. Here we use the admissible beamfunctions as stated in [37]. They are dictated by the essential projected free–free boundary conditions on fmnl

.Consequently, the gPC representation of the eigen-modes is reduced to the determination of the constant coefficientscijl

, i.e.

Fmnðx,y; nÞ �XNf

l ¼ 0

XNx

i ¼ 1

XNy

j ¼ 1

cijlXiðxÞYjðyÞ

24

35ClðnÞ: (39)

The validity and accuracy of solution depend entirely on the choice of the admissible functions. The solution should beperformed for each mode numbers of (m,n) and, accordingly the coefficients cijl

can be represented as the first coefficientcij0

. This approach gives a reasonable accurate coefficients but, clearly, the convergence of approximation is hardly dependson the number of unknown coefficients, cijl

, and number of admissible functions, i.e. Nx and Ny.

4. Construction of uncertain parameters from experimental data

The first step in the gPC representation of stochastic eigen-frequencies and eigen-modes is identifying the PDF ofuncertain parameters. Indeed, PDF estimation is a central problem in uncertainty quantification by probabilistic methods.

K. Sepahvand et al. / Journal of Sound and Vibration 331 (2012) 167–179 173

The most common approach is digitally generating realizations of uncertain parameters from MC simulations. In thispaper, we use experimental data available for E-moduli which are derived from 100 samples of identical plates. Weassume that a collocation of experimental data is observed over a finite subset for each uncertain parameter. In this way,a set of P ¼ ½p1,p2, . . . ,pN�

T is used for each uncertain parameter. Let n¼ fx1,x2,x3g be a R3-valued random vector withprobability measure of dmðnÞ, that is absolutely continuous in respect to the Lebesgue measure. It is assumed that allrandom variables are independent which implying that dmðnÞ ¼

Q3i ¼ 1 riðxiÞdxi, cf. Eq. (5). The goal is to find general

mapping ri such that xixmap;ri P and it is equal in PDF of data set P. The function riðxiÞ is then used to construct the gPC

expansions of uncertain parameters with respect to the chosen probability measure. To identify this mapping, however,we need to know the properties of random variable xi and mapping ri. That means the PDF type must be pre-known byprior. This can be achieved by using a PDF identification procedure. Here, we use general Pearson model as following [38]:

driðxiÞ

dxi¼

xi�a0

b0þb1xiþb2x2i

riðxiÞ: (40)

In which a0 and bi are constants related to the statistical moments of the distribution function. The most importantadvantage of the Pearson model is that the calculations of these constants are depends only on the third and fourthstatistical central moments, skewness and kurtosis, respectively [38]. Depending on the roots of quadratic functionD¼ 4b0b2�b2

1 and values b2, the solution of the differential equation in Eq. (40) yields to various PDF types. For instance, ifD¼0 and b2 ¼ 0, the solution of the Pearson model is riðxiÞ ¼ C expð�x2

i =2Þ which is considered as normal distribution.To empirical identifying the PDF type, we use the diagram of the Pearson system showing various distribution types in

terms of b1 (squared skewness) and b2 (traditional kurtosis), i.e.

b1 ¼m2

3

m32

, b2 ¼m4

m22

, (41)

and mk is the kth central moment about the parameter mean which for a discrete finite data is given by

mk ¼1

Nsk

XN

i ¼ 1

½pi�EfPg�k ¼ 1

N

XN

i ¼ 1

½pi�EfPg�s

� k

, (42)

where s is the standard deviation of finite subset. Pearson family contains many popular PDFs (e.g. Normal, Beta, Gamma)as well as general non-standard type. However, since the estimation of these parameters is unique, the Pearson systemprovides a unique PDF for each uncertain parameter. Furthermore, since the first four central moments of subsets areprecisely evaluated for calculating of moment rations b1 and b2, the Pearson model provides the approximated PDF with amuch higher accuracy in a wide probability range. Numerical details concerning the identification of the PDF type for theelasticity moduli are presented in next section.

4.1. Construction of the gPC for elasticity moduli

To construct the gPC of uncertain elasticity moduli, the experimental modal analysis are performed using 100 identicalplates, cf. Fig. 1. The plate dimensions and the average standard mechanical properties [39] are shown in Table 1. Tominimize the errors, the measurements at each point have been carried out three times and the average has been recorded.The first five eigen-frequencies of the plates are measured and then using the deterministic form of Eq. (17), the E-modulifor each individual plate are calculated, see Fig. 1. The propagation of the moduli are shown in Fig. 2. These variationsconstitute inherent uncertainty in these parameters. The propagation shows that the E-moduli of the plates are differentfrom each other due to the inherent structural uncertainties in the microstructure of the plates. To represent theseuncertainties by gPC, we need to know the PDF type and consequently the random space corresponding to each uncertainparameters. To achieve this goal, the Pearson model parameters, bi, and corresponding values of b1 and b2 for eachuncertain parameter are calculated in Table 2. The results are shown in Fig. 3 on Pearson chart in terms of the variables b1

and b2. In this chart the Pearson model shows that all uncertain parameters are distributed by type I which iscorresponding to Beta PDF. This shows that the uncertain moduli are far from Gaussian random variables withcoordination ðb1,b2Þ ¼ ð0;3Þ. It is shown in Fig. 3 that the coordination point ðb1,b2Þ of shear modulus is very close totype III line which is corresponding to Gamma PDF. This means that a little data disturbance may push the PDF type fromBeta to Gamma for this uncertain parameter. In such conditions, we need to use different orthogonal basis and standardrandom variable to representing the gPC of Gxy. This issue makes our simulation more complicated due to the fact that theorthogonal basis for system outputs, e.g. eigenpairs, must be represented by using 2D expansions. However, as mentionedbefore, the Pearson model provides unique PDF type and this property is the most notable advantage of the model incompression with other PDF identification methods. The identified PDFs for the E-moduli based on the Pearson system areshown in Fig. 4. All PDFs are identified as generalized Beta-distributions and are limited to a specific intervals by theminimum and maximum values of relevant parameters.

Fig. 1. Layout of experimental model analysis.

Table 1Sample dimensions; a: length, b: width and h: thickness; and material properties of the investigated orthotropic plates.

Dimensions (mm) a¼190 b¼100 h¼10

Elasticity moduli (GPa) Ex ¼ 15:20 Ey ¼ 1:303 Gxy ¼ 0:862

Poisson’s ratios nxy ¼ 0:44 nyx ¼ 0:08 –

1.1

1.2

1.3

1.4

1.5

1.6x 1010 Ex

1.1

1.2

1.3

1.4

1.5

1.6x 109 Ey

7.5

8

8.5

9

9.5x 108 Gxy

Fig. 2. Propagation of the experimental E-moduli for orthotropic plates. The x-axis shows the number of samples, here 100.

Table 2Pearson parameters for uncertain E-moduli, cf. Eqs. (40) and (41).

b0 b1 b2 D b1 b2

Ex 1.465 �0.441 �0.155 �2.605 0.296 2.881

Ey 5.808 1.527 �1.6303 32.903 0.170 2.142

Gxy 1.000 �0.304 �0.000 �1.2159 0.368 3.551

K. Sepahvand et al. / Journal of Sound and Vibration 331 (2012) 167–179174

5. Numerical simulation

In this section we apply the above gPC formulation to represent the eigen-modes and eigen-frequencies of orthotropicplates with uncertain E-moduli. In so far as the distribution of the uncertain E-moduli were identified in previous sectionas Beta PDF, the standard random orthogonal Jacobi polynomials, JðxÞ, are used as the gPC basis, i.e. in Eq. (13) we setC1ðx1Þ ¼C2ðx2Þ ¼C3ðx3Þ ¼ JðxÞ where x 2 U½0;1�. Since JðxÞ are optimal basis for the gPC of uncertain parameters withBeta PDF, cf. [10,34], the first order expansions are employed for representation of these parameters and from Eq. (25)the gPC coefficients of the first five modes up to the third order are calculated, see Table 3. The gPC coefficients foreigen-frequencies converge rapidly. Due to the fact that the first mode frequency, L11, depends only on the uncertainties in

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

Impossible area

Type I

Type IV

Ex

Ey

Gxy

β1

β 2

Pearson system

Type VI

Type IIIType V

Fig. 3. The Pearson chart relating the type of PDF for uncertain E-moduli. The coordination of Gaussian distribution with normal PDF is located on ðb1 ,b2Þ ¼ ð0;3Þ.

Fig. 4. The identified PDFs for uncertain elasticity moduli of orthotropic plates.

K. Sepahvand et al. / Journal of Sound and Vibration 331 (2012) 167–179 175

shear modulus, it shows higher convergence rate so that the third order coefficients for this mode can be ignored. The PDFsof eigen-frequencies are shown in Fig. 5 in comparison with the identified PDFs by kernel density estimation (kde) method.It seems that the third order gPC has reasonable accuracy for the representation of the eigen-frequencies in respect to thekde simulations and experimental data.

Table 3

The coefficients of gPC of the first 5 eigen-frequencies, lmns , for the orthotropic plate.

gPC coefficients

lmn0lmn1

lmn2lmn3

L11 675.05 10.2893 �1.1404 0.0272

L20 1332.2 �41.9021 �5.3736 1.2948

L02 1485.7 �62.6373 7.0744 6.5507

L21 1844.6 46.3146 �7.1521 0.4754

L12 1956.2 �56.3187 2.7784 2.0437

620 640 660 680 700 720

0.005

0.015

0.025

0.035

0.045

Λ [Hz]

1150 1200 1250 1300 1350 1400 1450

0.002

0.006

0.01

0.014

Λ [Hz]

1350 1450 1550 1650

1

3

5

7

9x 10

Λ [Hz]

1650 1750 1850 1950

0.005

0.01

0.015

Λ [Hz]

1800 1900 2000 2100 220

1

3

5

7

9x 10

Λ [Hz]

Fig. 5. The gPC representation of PDFs for the first five eigen-frequencies in comparison with the kernel density estimation (kde) method and

experimental data.

K. Sepahvand et al. / Journal of Sound and Vibration 331 (2012) 167–179176

Now, we try to derive the gPC functions of eigen-modes. The admissible beam functions which satisfied the plategeometry boundary conditions are used as basis for determination of the coefficients cijl

in Eq. (38) where a six termsfunction is considered for each eigen-mode. The gPC coefficient functions are shown in Fig. 6 for the first five modes.However, the fifth order gPC coefficients are shown but the high rate convergency of the coefficients shows that the fourthand fifth terms can be ignored and the third order gPC is accurate enough for representation of the stochastic modes. Toinvestigate this convergence, we calculate statistical moments as

Mnmn ¼ E

XNf

l ¼ 0

fmnlðx,yÞClðxÞ

!n24

35, n¼ 1;2, . . . (43)

In whichMnmn is the nth statistical moment of the stochastic eigen-mode Fmnðx,y; xÞ. By substitution the gPC coefficients

from Eq. (38), we have

Mnmnðx,yÞ � E

XNf

l ¼ 0

XNx

i ¼ 1

XNy

j ¼ 1

cijlXiðxÞYjðyÞClðxÞ

0@

1An2

435: (44)

The mean and the variance functions of the first nine eigen-modes derived from the third order gPC are plotted inFig. 7(a) and (c), respectively. The same results are plotted in Fig. 7(b) and (d) from experimental data. The plots show asatisfactory agreement between the results for the first five modes in both cases but large discrepancy in higher ordermodes. This discrepancy is more conspicuous for variance results whereas the result agreement for mean functions is

Fig. 6. The gPC coefficients, fmnl, of the first five stochastic eigen-modes in Eq. (16).

Φ11 Φ02 Φ20

Φ12 Φ21 Φ22

Φ03 Φ13 Φ23

Φ11 Φ02 Φ20

Φ12 Φ21 Φ22

Φ03 Φ13 Φ23

Fig. 7. (a) and (b) show the mean of vibration eigen-modes calculated from the gPC and experimental, respectively. (c) and (d) the same results for the

variance function.

K. Sepahvand et al. / Journal of Sound and Vibration 331 (2012) 167–179 177

acceptable. This disagreement shows higher degrees of uncertainty in higher order modes which cannot be captured bythird order gPC expansions. This happens again for higher order moments as shown in Fig. 8 for the third and fourthstatistical moments. Though the agreement between the results for these moments at the first and second modes arereasonable, but this agreement is lost for other modes. Clearly, because the gPC method converges in the mean-squaresense [7] there is no guarantee for higher order moment convergency calculated by this method.

Fig. 8. (a) and (b) show the third order statistical moment of vibration mode shapes calculated from the gPC and experimental, respectively. (c) and (d)

the same results for fourth order moments. The mode order are as same as shown in Fig. 7.

K. Sepahvand et al. / Journal of Sound and Vibration 331 (2012) 167–179178

6. Conclusions

The theory and application of the gPC method for stochastic free vibration of orthotropic plates with uncertain moduliinvestigated. The gPC employed to uncertainty representation in the E-moduli with pre-identified PDF. A generalformulation for stochastic free vibration of orthotropic plates by the gPC method is presented and it is shown howthe stochastic problem is transferred to a deterministic one. The Pearson model implemented to identifying the PDF of theE-moduli from experimental data. These real unique identified PDFs then are used for definition of the random spacescorresponding to each uncertain parameter. A numerical simulation shows the reliability and accuracy of the method. Theefficacy of the gPC in a practical example has been shown where the statistical moments and the probability distributionsof modal responses compared with experimental results. It is also shown that the first (mean) and the second (variance)statistical moments derived from the gPC method for the eigen-modes have reasonable accuracy in comparison with theexperimental results. However, this accuracy is lost for higher modes as same as for higher order moments. This point maybe denoted as a common drawback of uncertainty quantification by the gPC.

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