de la fuente 2008 engineering structures

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Engineering Structures 30 (2008) 29812990

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Engineering Structures

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

An efficient procedure to obtain exact solutions in random vibration analysis oflinear structures

Enrique de la FuenteEscuela Universitaria de Ingeniera Tcnica Aeronutica, Universidad Politcnica de Madrid, Cardenal Cisneros 3, 28040 Madrid, Spain

a r t i c l e i n f o

Article history:

Received 18 July 2006Received in revised form

20 December 2007

Accepted 9 April 2008

Available online 20 May 2008

Keywords:

Random vibration

RMS stress

Finite elements

Structural analysis

Von Mises stress

a b s t r a c t

A new method for performing random vibration analysis of linear structures is presented in this paper

The method results from direct application of a well known result of Linear Systems Theory and allowscomputation of RMS values of any number of structural outputs: displacements, stresses, element forcesaccelerations, etc. RMS of Von Mises stresses can be also obtained. In all cases, the results produced are

exactfor whitenoise excitation inputsand canbe as accurate as desired forexcitations with arbitraryPSDlaws. These results can be postprocessed by standard Finite Element packages in much the same manneras if they had been obtained in conventional static analysis.

2008 Elsevier Ltd. All rights reserved

1. Introduction

RANDOM vibration analysis is a common task in structuralanalysis.In the space field, forinstance, random vibrationis usuallyneeded to simulate acoustic loading or loading due to atmosphereturbulence during the launch of spacecraft. In the aeronauticalfield, random analysis is required to assess the response of aircraft

structures to continuous turbulence (i.e., turbulence defined byits Power Spectral Density). Civil engineering structures are oftenexcited by random forces such as earthquakes or air turbulence.

The way in which the structural analysis packages deal withthe problem of random vibration starts with the calculation ata number of frequencies of the complex transfer function of

the output variables requested by the user, (i.e. stresses, forces,accelerations, etc.), followed by computation of the individualizedPSD of each output, and finally by performing a numerical

integration of the resulting PSD curves. The final result of thisprocess is the steady state RMS value of the corresponding output.This calculation process presents some shortcomings well known

to the users:

In general, RMS outputs have to be requested one by one(i.e., no instruction such as RMSSTRESS= ALL is available).This inconvenient is important when dealing with large Finite

Element Models. A significant computational effort is needed for each individual

result if this is required to high accuracy. For the solution

E-mail address:[email protected].

obtained to be accurate enough, the PSD of each structuraoutput must be computed at a large number of frequencies.This

usually results, when dealing with large Finite Element Modelsin long CPU times and eventually problems with disk spacestorage.

It is not possible to visualize RMS results (stresses for instance)incolor orcontour plots asit is normallymadein staticor modaanalysis. Postprocessing of RMS results is obviously essential tohave a clear idea of how internal forces/stresses run along thestructure.

For ductile materials, such as those used in engineeringstructures, the equivalent Von Mises stress is an adequatemeasure of stress magnitude.For the case of random vibrationthe RMS value of the Von Mises stress would therefore berequired. However, this equivalent stress being a non linearfunction of the stress components, cannot be obtained simplyfrom their individual RMS values.

The method presented in this paper is intended to overcomeall of these shortcomings. The method is derived from direcapplication of a known result of the Theory of Linear Systems, butthat is howevernot handled frequentlyby structural analysts sinceitsmain useis in LinearSystems Control Theory. Itsmain advantagewith respect to ghe standard sequence is that exactRMS results(stresses, forces, accelerations and so on), are obtained using aprocess that is much cheaper (in teRMS of computational effort)than the standard one outlined above. Since fewer mathematicaoperations are required for each result, the obtention ofall RMSstructural outputs presents no special difficulties and can bepostprocessed graphically much in the same way as if they wouldhave been obtained in a conventional static analysis.

0141-0296/$ see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.04.015
http://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstructmailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.engstruct.2008.04.015http://dx.doi.org/10.1016/j.engstruct.2008.04.015mailto:[email protected]://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstruct


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2982 E. de la Fuente / Engineering Structures 30 (2008) 29812990

Nomenclature

Symbols

A, B, C, D State space matricesM Mass matrix

K Stiffness MatrixE Eigenvector matrix Eigenvalue matrix

X Lyapunov MatrixW PSD Matrix Angular natural frequencies matrix

Z Modal damping matrix Modal damping Angular frequency Element stress

f Element forceu,u,u Nodal displacement, velocity, accelerationE Expectance operator./ Term by term division

1 Square matrix filled with onesI Identity matrixz Any structural output

Subindexes

a Structural degrees of freedomh Modal coordinatesl Degrees of freedom where direct forces are appliedr Degrees of freedom to which enforced base acceler-

ation is inputs State coordinates of the structuren State coordinates of the PSD shaping systemS State coordinates of the compound systemw Force/acceleration excitation

k kth structural output (displacement, stress, etc.)VM Von Mises (stress)i Imaginary unit,ith element (depending on context)

Superindexes

T Matrix transpose Conjugate transponse Inverse of the conjugate transpose

The paper is structured as follows. In Section 2the standard

process of obtaining RMS values of structural outputs is briefly

reviewed, thus highlightingthe difficultiesmentioned above.Then,

in Section 3 the new procedure is recalled from Linear System

Theory and the sequence of mathematical operations needed is

identified. This sequence of operations is applied first to the case of

general linear systems subjected to white noise and then extended

to the case where systems are subjected to arbitrary PSD input

profiles. Afterwards, in Section 4 the procedure is applied to

structural-like systems. Section9 illustrates the procedure with

several examples. Some intermediate calculations are left as an

Appendix to ease reading the paper.

2. Motivation

The conventional procedure to obtain the steady state root

mean square value of some structural output of a structure

subjected to random forces is recalled below. First, the dynamicequilibrium equations of the structure are written in modal

coordinates,1

uh+2Zhhhhuh+2hhuh= M1hh Tahpawhere Mhh,Zhh and hh are the generalized mass, modal damping

and natural frequencies diagonal matrices, uh are the modal

coordinates, ah is the modal matrix and pa is the applied load

vector. The complex frequency response of the modal coordinates

uhis

uh= Hhh() M1hh Tahpawhere

Hhh()=2Ihh+2hh+i2Zhhhh

1is the modal complex frequency response matrix, a diagonal

matrix as well. The Power Spectral Density matrix of the modal

coordinatesuhis

Whh()=Hhh() M1hh TahWaa()ahM1hh Hhh()

whereWaa()is the PSD matrix of the random forces pa.

Letzkbe some structural output, for instance, the displacement

of a node, a stress component or element force of some finite

element, etc. In general, this output can be written as a linear

combination of the modal displacements uhin the form

zk= ckhuhwhereckhis a row vector with its columns representing the modal

values of the kth structural output.The PSD matrix of the structural

outputzkis given by

Wkk()= ckhWhh() cTkh= ckhHhh() M1hh TahWaa()ah M1hh Hhh() cTkh

and finally the root mean square value of the structural output zk

zk,RMS=

0Wkk() d

1/2.

Taking into account that Hhh() is diagonal and that the PSD

matrix Waa is symmetric, the above expression involves a total ofa(a+1)

2 h(h+1)

2 different integrals (Ref. [9]) for each structural

output required. For the general case, these integrals must be

evaluated numerically. It is apparent that when many outputs are

requested and/or high numerical integration accuracy is desired,

the computational effort to perform that number of numerical

integrations may soon become prohibitive.

Ref. [9] presents a procedure to tackle these integrals and

proposes an exact solution for piecewise polynomial variation withthe frequency of the elements of the PSD matrix Waa(). In this

paper we will follow a completely different approach which we

estimate is easier to implement and which also can provide simple

closed form exact solutionsfor the RMS value of structural outputs.

The procedure that will be presented does not need in fact any

integration at all and is based in a well known result of Linear

Systems Theory. The next section recalls an important result of

Linear System Theorywhich is the base of the proposed procedure.

Later, in Section 4, the procedure will be particularized to the

structural problem.

1 To ease the reading of what follows, each matrix will have two subindexes,indicating the dimensions of the matrix.


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E. de la Fuente / Engineering Structures 30 (2008) 29812990 2983

Fig. 1. The PSD shaping system will model a white noise input to a variable PSD profile.

3. Response of a linear time invariant system to random inputs

3.1. White noise random input

Refs. [13] give a fairly complete and rigorous derivation of thetheory of response of linear systems to random excitation inputs.

Here a brief summary of the main results will be presented. In itsmost general form, a Linear Time Invariant (LTI) system is defined

by a system of first order linear differential equations in the form

xs= Assxs+Bsww (1)z=Czsxsin wherexsis the state vector, wis the input vector,zis the output

vector, and Ass, Bsw and Czs are constant matrices of appropriatedimensions.

Now, let w be a white noise randomprocess with PSDdefined by

a constant, symmetric and positive semidefinite matrix Www. Thenit can be shown that the steady state variance matrix of the statevector is given by

E

xsx

Ts

=Xss

where where E stands for the expectation operator and Xss is the

unique symmetric solution of the linear Lyapunov equation

AssXss+XssATss+BsuWuuBTsu=0. (2)Furthermore, the steady state variance matrix of the output vector

zis given by

EzzT

=z2 = CzsXssCTzs. (3)Finally, the root mean square value of the output elements ofzistherefore

zRMS=

diag

CzsXssCT

zs

. (4)

3.1.1. Solution of the Lyapunov equation

Although the matrix Eq.(2)is a linear function of the elements

of the matrix Xss and could therefore be rewritten as a systemof linear equations in its elements, much powerful and robusttechniques do exist for finding the solution. The interested reader

can see the details in[13].

There is however an important situation for which the solution

can be obtained in closed form. This case is of direct applicabilityfor the problem at hand as will be seen later, and corresponds tothe particular case of that the state matrix Ass is diagonalizable.

The solution can be seen for instance in [4]. We will herein givea solution in closed matrix form.

Let ssbe the (complex, in general) eigenvalues ofAsswritten in

the form of a diagonal matrix and Essthe eigenvector matrix. In thecase of thatAssis diagonalizable we can write

Ass= EssssE1ss .Substituting in Eq.(2),

EssssE1ss Xss+XssEss ssEss+BsuWuuBTsu= 0.

Premultiplying byE1ss and postmultiplying byEss we obtain

ssXss+Xssss+Wss= 0 (5)

where

Xss= E1ss XssEss (6Wss= E1ss BsuWuuBTsuEss .Since the matrix ssis diagonal, the solution of(5)can be obtained

explicitly. In fact, the(i,j)element of the Lyapunov matrixXssisXssij=

Wssij(ss)ii+(ss)jj

that can be put in matrix compact form as

Xss= Wss./ (ss1+1ss) (7where1is a square matrix of ones of appropriate dimensions and

the symbol (./) stands for term by term (Hadamard) division. Note

that ifiand jare any two eigenvalues of matrix Ass, the solutionis uniqueifi +j=0. Aswillbe seen later,thiscondition is alwayfulfilled for the structural problem.

The final solution of the Lyapunov equation is obtained by

substituting Eq.(7)into Eq.(6),that is

Xss= Ess XssEss= Ess

E1ss BWuuB

TEss./ (ss1+1ss)

Ess (8

and the RMS value of the response variables zis

zRMS=

diag

CzsXssCT

zs

2

which is the final result written in explicit matrix closed form.The factor 2 in the above equation comes from the fact that the

RMS value given in (4) is calculated for a range of frequencies fromto+. However, in engineering problems, the integration isperformed only for positive values of frequencies, that is from 0 to

+.

3.2. Non white noise random inputs

For most practical situations, implementation of the procedure

just described is more than enough. Constant PSD input a

all frequencies is frequently used to model random excitation

processes. However, it is still possible to extend the above

procedure to non white noise PSD laws.

From Linear Systems Theory (Ref. [1]) we know that any PSDfunction as a function of frequency can be obtained as accurately asdesired as the output from some linear system excited with unitwhite noise input. Moreover, this system can be selected to be

stable and minimum phase.2 In our framework, we will name this

system as the PSD Shaping System (PSDSS).

That is, suppose that the input w in Eq.(1) is a random input

with PSD given by some predefined function of frequency, PSD ()

The above statement means that it is always possible to find a

linear, stable, minimum phase, time invariant system which wil

produce an output PSD as close as desired to the law PSD () when

subjected to unit white noise spectrum (see Fig. 1).

2 That is with all his poles and zeros having negative real parts.
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The more general form of such a system will be

xn= Annxn+Bnwwwhite_noise (9)wPSD()= Cwnxn+Dwwwwhite_noisewhere wwhite_noise is an input consisting of unitary white noisespectrum andwPSD()is the output with the desired PSD.

For ease of reading, the process of constructing the PSDSS (i.e.,

finding the system matrices A nn, Bnw, Cun and Duw for a given lawPSD ()) is discussed in some detail in the Appendix. In whatfollows, we will assume that these matrices are known.

3.3. Compounded system

The PSDSS given by Eq.(9)will now be combined in series withthe original system (as outlined inFig. 1)described by Eq.(1).Thecomplete system matrices of the whole system can be verified tobe

xS=ASSxS+BSwwwhite_noisez=CzSxS (10)where

xS

= xs xnT is the state vector of the complete system,

ASS=

Ass BswCwn0ns Ann

BSw=

BswDwwBnw

CzS=Czs 0zn

are the system matrices of the complete system. Note that thecomplete system is nowsubjected to white noise input, (see Eq. (10)),and thus, the result described in Section3 can now be applied tothe new system. Should the PSDSS provide exactly the requiredoutput PSD(), the complete system will in turn provide theexact solution to the problem of a system subjected to generalspectrum. In any case, since any output PSD can be approximated

as accurately as desired, the exact solution can also be approachedas accurately as needed.

3.4. Exact solution for variable PSD spectrum

It is still possible to find an explicit exact solution for thecompounded system given in (10). Let nn and Enn be theeigenvalues and eigenvectors of the PSDSS, that is, they satisfy

AnnEnn=Ennnn.The eigenvalues and eigenvectors of the complete system state

matrix ASScan be obtained from those corresponding to the twoseparate systems by

SS= ss 00 nnESS=

Ess Esn0ne Enn

(11)

where the matrixEsnis given by

Esn= EssEsnand the(i,j)element of the matrixEsnis given by

Esn

ij=

E1ss BswCwnEnn

ij

(ss)ii(nn)jjor in matrix compact form

Esn

= E1ss BswCwnEnn ./ (ss11nn).

This result can be verified simply by substitution.

Once the eigenvalues and eigenvectors of the complete systemare known, the solution to the Lyapunov equation can be obtained

in thesame way as shown in Section 3.1, that is, by first finding thesolution of

ASSXSS+XSSATSS+BSwWwwBTSw=0 (12)which according with Section3.1.1is given by

XSS= ESSXSSESS (13)where

XSSij= E1SS BSwWwwB

TSwE

SS

(SS)ii+(SS)jj. (14)

The steady state variance of the structural responses is given by

Z=CzSXSSCTzS (15)and the steady state mean square values of the response,

z2 =diag

CzSXSSCT

zS

. (16)

Finally, thesteadystateRMS value of the kth structural response

is given by

zRMS=

diag

CzSXSSCT

zS

2

. (17)

4. Random vibration analysis of a linear structure

Section 3 dealt with the analytical solution of a LTI systemsubjected to random excitations.We will now apply the above

general result for the case of a structure subjected to randomexcitation. We will see that for structural analysis, the resultscan be written in a simple explicit form. As before, we willstart by considering that the excitation is white noise. For thesake of completeness, we will consider a structure subjected

simultaneously to both directly applied random forces as well asto imposed random accelerations at some nodal points.

4.1. Dynamic equilibrium equations in modal coordinates

Consider the dynamic equilibrium equations of an elastic linearstructure, in the absence of damping

Maaua+Kaaua= pa (18)in whichMaa, Kaaare the mass and stiffness matrices respectively,

paare the external nodal forces and ua,uathe nodal displacementsand accelerations respectively. We will assume that the abovesystem of equations has been obtained after having eliminatedall the external and internal constraints and performed all the

required condensations (for instance, Guyan reduction).Nodal displacements uawill now be partitioned in sets uland ur,

the first collecting all the degrees of freedom with applied forcesand the last collecting all the degrees of freedom to which thebase acceleration is imposed. The above system of equations ispartitioned accordingly:

Mll Mlr

MTlr Mrr

ulur

+

Kll Klr

KTlr Krr

ulur

=

plpr

.

Note that both ul and u r areabsolute displacements in that theymight include rigid body modes. Since rigid body modes do notproduce internal stresses, the usual procedureto compute the rigidbody modes is to consider the static equation

Kll KlrKTlr Krr

lr

Irr

= 00
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from where

lr= K1ll Klr.The absolute displacement vector ul is the sum of elastic nodaldisplacementsul (i.e., the nodal displacements withur= 0) andrigid body ones, lrur, that is

ul

=ul

+lrur.

Using modal decomposition, the elastic displacementsul can bewritten in the form

ul= lhuhwhere uh are the modal coordinates. The original displacementvectoruacan now be written as

ua

ulur

=

lh lr

0rh Irr

uhur

.

Incorporating this transformation into Eq.(18),we arrive atMhh Lhr

LThr Mrr

uhur

+

Khh 0

0 Krr

uhur

=

TlhplLhrur

pr

(19)

where

Mhh= TlhMlllh Khh= TlhKlllh=Mhh2hhMrr= TlrMlllr+TlrMlr+MTlrlr+Mrr Krr= TlrKlllr+KrrLhr= Tlh(Mlllr+Mlr).

Now, from the first group of equations in(19),

Mhhuh+Khhuh=

Tlh, Lhr

plur

and introducing the modal damping matrix Zhh

uh+2Zhhhhuh+2hhuh=M1hh

Tlh, Lhr

plur

.

4.2. Expression of the structural outputs

We will suppose that the user is interested in obtaining someor all of the RMS nodal displacements, element forces, elementstresses, absolute accelerations, etc. Since the element forces andelement stresses depend only on elastic displacementsulthey canalways be written in the form

f

=

ClCf l

ul=

ChCf h

uh

where the matrices Ch = Cllh and Cf h = Cf llh are the modalstressesand modal element forces respectively, that is, the stressesand forces corresponding to a displacement field given by lh.

On the other hand, the absolute accelerations deserve a little

more attention. These are given by

ul= lhuh+lrur= lh

M1hh Lhrur2Zhhhhuh2hhuh

+ lrur

= lh2hhuhlh2ZhhhhuhM1ll Mlrursince, if all normal modes are taken (h=l), then lhM1hh Tlh= M1ll .

The term M1ll Mlrurcauses some trouble because if the matrixMlr(which represents the inertial coupling between the degreesof freedom u l and u r) is not zero, the RMS value of the absoluteaccelerationul would be infinite for a white noise input baseaccelerationur. In practical problems this will never be an issuesince first Mlris more than often null (this is the case of lumpedmasses) and more important, the PSD of the input acceleration will

be zero at high frequencies. This will be the case of variable PSDwhich shall vanish at high frequencies. In this case obviously the

input cannot be white noise. However the contribution of input

base acceleration at high frequencies will be negligible if sufficient

number of modes have been taken for the analysis. Besides, below

the problem of non white noise input is treated. This situation wil

be covered when the results for non white noise inputs are dealt

with. For these reasons, we will ignore the contribution of the termM

1ll Mlrur.

Summing up,the whole setof outputstructural variables, whichwill be calledz, can be written in the form

z

ul(relative)

f

ul(absolute)

=

lh 0Ch 0Cfh 0

lh2hh 2lhZhhhh

uhuh

. (20

4.3. Equations in state space form

To follow the process described in Section3 we first need to

represent the structure as a linear first order differential equation

system, that is

xs

=Assxs

+Bsww (21

z=Czsxswith

x

uhuh

w

plur

z

ul(relative)

f

ul(absolute)

Ass

0hh Ihh

2hh 2Zhhhh . (22aBsw

0hwBhw

Czs

lh 0lhCh 0hCfh 0f h

lh2hh 2lhZhhhh

and

Bhw=M1hh

Tlh, Lhr

. (23

4.4. Expression of the exact solution

It happens that for a structural-like system, the matrices

that appear in the development of Section 3, acquire simple

expressions. For instance, the eigenvalues of the state matrix Asare

ss=hh 0

0 hh

with

hh= hhZhh+i

IhhZ2hh

1/2 (24

are the complex eigenvalues. The state matrix Ass of the structure

is always diagonalizable, since it comes from a symmetric matrix

problem, so that, the analytical solution of the Lyapunov equationgiven in Section2can be applied.


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The matrix of eigenvectors ofAsscan be verified to be

Ess=

Ihh Ihhhh

hh

=

Ihh Ihh

hh(Zhh+i(IhhZ2hh)1/2) hh(Zhhi(IhhZ2hh)1/2)

(25)

and even its inverse can be obtained in explicit form

E1ss =

hhhh1 hh hhhh1hhhh

1hh

hhhh

1

(26)

and simplifies to

E1ss =1

2

IhhiZhh

IhhZ2hh

1/2 i1hh (IhhZhh)1/2Ihh+iZhh

IhhZ2hh

1/2i

1hh (IhhZhh)1/2

. (27)

Moreover, since the firsth rows of matrix Bsw are always zero, thematrixWssin(6)can be written as

Wss

=E

1ss

0hh 0hh0hh BhwWwwB

Thw E

ss

that can easily be verified to be of the form

Wss=

Shh ShhShh Shh

where

Shh=

hhhh

1BhwWwwB

Thw

hhhh

= 1

4

1hh

IhhZ2hh

1/2BhwWwwB

Thw

1hh

IhhZ2hh

1/2(28)

withBhwgiven by(23).According to (7) the solution of the Lyapunov equation acquires

the form

Xss= Wss./ss1ss+1ssss

=

Shh./

hh1hh+1hhhh

Shh./ (hh1hh+1hhhh)Shh./

hh1hh+1hhhh

Shh./

hh1hh+1hhhh

. (29)Note that the matrixXss has a special structure. In fact, it can beeasily seen that the submatrices

Xss11and Xss22are conjugate ofeach other, as is also the case of

Xss12and Xhh21.The complete Lyapunov solution is

Xss= Ess XssEss (30)whereEssis given by(25).

Finally, the root mean square of the structural outputs vector zis

zRMS=

diag

CzsXssCTzs

2

1/2

(31)

withCzsgiven in(22a)If no nodal accelerationsulare requested, the above expressions

simplify even more since in this case the last h columns of matrixCzsare null, so that only the submatrix (Xss)11of the Lyapunov solution

Xss equation is in fact needed. After some algebraic manipulation,this submatrix can be verified to be

(Xss)11=2Re

Shh./ (hh1hh+1hhhh)Shh./hh1hh+1hhhh

where Re()stands for real part, and the RMS value of the output

vector is simply

zRMS=

diag

Czh(Xss)11CT

zh

2

1/2. (32)

In particular, the RMS value of thekth output will be given by

(zkRMS)2 = ckh(Xss)11c

Tkh

2

=ckRe

Shh./ (hh1hh+1hhhh)Shh./

hh1hh+1hhhh

cTk

(33)

whereckhis thekth row of matrixCzh.

If on the other hand, the absolute acceleration is required, thewhole Lyapunov matrixXssmust be used and the solution is givenby(31).

5. Summary of the procedure

The procedure proposed in this paper to calculate RMS valuesof structural outputs of a structure subjected to white noise issummarized below.

(1) Compute the mass and damping modal matrices,Mhh,Zhh andthe natural frequencies hh

(2) Compute the elastic and rigid body modes, lhand lr(3) Compute the modal stresses and modal element forces

(matricesChand Cfh respectively)(4) Construct the output matrix Czs from Eq.(22a)and the input

matrixBhwgiven in(23)(5) Compute hhwith Eq.(24)(6) Calculate the solution of the Lyapunov equation using(30):

(a) Costruct the complex eigenvalue matrixEssand its inverseE1ss with(25)and(27)

(b) Compute intermediate matrixShhwith(28)(c) ComputeXssfrom(29)

(7) Compute the RMS values of the outputs from(32).

5.1. Example of application

As a simple illustration, consider the case of a single degree offreedom system. In this simple case, the state matrices are

A=

0 1

2 2

B=

01/m

C= 1 0 .The intermediate matrices are

=+i

1 2

, =

i

1 2

1+ 1= 2+2i

1 2, 1+ 1= 2S= ()1 BhaWaaBTha()

=1

41 2

W

2m2

and finally

zRMS=

ck(Re {S./ (1+1)S./ (1+1)}) cTk= 1

8

W

m23

that is the correct result, according to Ref. [5].

6. Structures subjected to non white noise input

When the input excitation to the structure is not describedby a white noise spectrum, but by a law PSD (), the procedure

described in Section3.2 can be directly applied. The extension isstraightforward and will not be repeated here.
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7. Calculation of RMS Von Mises stresses

The most usual stress measure of a 3D stress state for ductile

materials is the Von Mises equivalent stress. Since this measure

is a non linear function of the stress components and thus of the

displacements, the RMS Von Mises stress cannot be calculated

directly from the individual RMS values of the stress components.

Ref. [12] offers a procedure for calculating the RMS Von Misesstress in random vibration environment. However, within the

framework of the theory presented in this paper, allows for a

specially simple and explicit expression for the RMS Von Mises

stress.

In a general three dimensional stress field the Von Mises

equivalent stress is given by

2VM=2x+ 2y+ 2zxy+ xz+yz

+3 2xy+ 2xz+ 2yzthat can be written (see[12])

2VM=TVwhereVis a symmetric matrix

V=

1 12

12

12

1 12

12

12

1

33

3

and

= x y z xy xz yz .In a random environment, the expected value of the square on

the Von Mises stress is

E

2VM

=2VM= E

TV

. (34)

In particular the Von Mises stress of the kth element and can be

written as2VM

k= E

xTSC

TkSVCkSxS

.

However, since

xTSCTkSVCkSxS=trace

CTkSVCkSxSx

TS

and since

ExSx

T

S=XSS

withXSSthe solution of the Lyapunov Eq.(12),we obtain2VM

k

= ExTSC

TkSVCkSxS

=E

trace

CTkSVCkSxSx

TS

= trace

CTkSVCkSE

xSx

TS

= trace

CTkSVCkSXSS

.

Finally we obtain the RMS value of theVon Mises stress of element

kth

(VM)k,RMS=

2VM

k=

trace CTkSVCkSXSS1/2

.

To the knowledge of the author, this closed form expression forthe RMS Von Mises stress has never been published before.

Fig. 2. Three degrees of freedom system submitted to base accelerationur.

8. Advantages of the proposed procedure

The advantages of the procedure proposed in this paper are thefollowing

(1) The procedure provides the exact solution for structuresexcited by random white noise excitation, and can be as closeto the exact solution as desired for inputs with arbitrary PSD

laws(2) No integration (neither numerical, nor analytical) is needed(3) The solution is given in explicit matrix form and can be

programmed very easily(4) The calculation of RMS Von Mises stresses is also given in

explicit form(5) Theprocedure canbe applied to arbitrarily large Finite Element

Models and can provide as many structural outputs as desired(6) Since an explicit solution is obtained, extensions and further

applications can be devised very easily. For instance,

(a) Contribution of each mode to the RMS value of somestructural output can be easily isolated by simply selectingthe appropriate column of output matrix CzS.

(b) Sensitivity of RMS structural outputs to any structuraparameter can be easily obtained in explicit form

9. Applications

9.1. Application example 1

As a first illustration of the procedure proposed in this paperconsider the three degrees of freedom system shown inFig. 2.Thesystem is excited at the base by random acceleration, white noisewith PSD given by the scalarW. The a degrees of freedom are the

displacementsof masses 1, 2 and3. Themass andstiffness matricesare

Maa=m 0 00 m 0

0 0 m

Kaa=

k k 0k 2k k

0 k k

.

Theldegrees of freedom are the displacements of masses 1 and 2,andthe rdegree of freedom is the displacement of mass 3. The Thus

lr= K1ll Klr= k kk 2k1

0k= 11

and the matrices reduced to set lare

Mll=

m 00 m

Mlr=

00

Kll=

k kk 2k

.

The eigenvalues and eigenvectors of the system are3

ll=

3

2+ 1

2

5

k

m0

0

3

2 1

2

5

k

m

3 The analytical solution was obtained with Maple 9.0.
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lh=12

1

2

5,

1

2+ 1

2

5

1 1

Mhh= m

1

2 1

2

5

2+1 0

0

1

2+ 1

2

5

2+1

Khh= k

52+ 1

25 0

0 5

2 1

2

5

.

The input and output distribution matrices Bhwand Czhare

Bhw=

2 5+3

5

5+

5

5+

5

4

5+

5

Czh= 2m

1 1

3+

55

+1

535

1

and the intermediate matrices

1 1

1 1

+

1 1

1 1

=(

5+1)(+i

(1 2))

5+i

(1 2)

5

5+i

(1 2)

5 (

51)(+i

(1 2))

(35)

1 1

1 1

+

1 1

1 1

=

5+1

5+i

1 2

5i

1 2

51

. (36)

The matrixSis calculated as

S= w2

1 2

51

25+1

28

5+

52

3+

5 1

20

1

20

51

25+1

285+

52

53

and finally the RMS displacement of masses 1 and 2 are

(u1,2)RMS= Wm21+42

2 4

5

2025+116

2

4510

5+122

.


It is now desired to verify the procedure outlined in this paper

against a large Finite Element Model. For this purpose, a FiniteElement Model of a rectangular plate was prepared. The model has

250 000 elements which leads to 250000 6=1500 000 stresses(i.e., three stress components at the plate top fiber, x,y, xyand three at the bottom fiber). The plate is simply supported on

its four edges and subjected to a white noise input accelerationnormal to the plane of the plate and applied at its four edges. The

model was run under MSC/NASTRAN to extract the normal modes,natural frequencies and modal matrices. One hundred modes were

Table 1

Comparison of exact RMS stresses with those obtained by using increasing order

of PSDSS

Exact RMS stresses Percent error

n=4 n=6 n=8 n=10x 0.00228071 2.56 0.94 0.57 0.03y 9.35048700 3.54 1.29 0.98 0.08xy 0.00123717 2.27 1.07 0.82

0.02

requested and used in the analysis The modal matrices Mhh,Zhhas well as the natural frequencies hh, the modal participation

matrix,Lhrand the modal stresses were written to a file for further

processing.

An external program was written in matlab to first read

the matrices previously written by MSC/NASTRAN and after to

perform the set of operations described in Section 5 to calculate

all the 1 500 000 RMS stresses. The elapsed time was 21 minutes

approximately on a Pentium 4 personal computer to 3 GHz.


As a final example, we will study a rectangular plate ofdimensions 5 10 1, the Modulus of Elasticity E= 7 1010and density = 4.86 107 all data given in coherent units. Itwas modeled by 20 000 CQUAD4 elements shell elements and is

excited by an acceleration normal to the plate applied at its four

edges. The PSD of the base acceleration has an PSD spectrum given

bythe curve shown in Fig. A.1. Twohundred modes were requested

ranging from 1 to 92 rad/s and several stress components wereobtained for comparison of results. In fact, the numerical data of

the problem were chosen so as to have all its natural frequencies

well inside the bandwidth of the excitation spectrum.

The plate was first analyzed by the conventional procedure

of numerical integration of the structural responses PSD and

also by the procedure described in this paper. The RMS stress

components at the most loaded element are shown in Table 1for both procedures and for an input filter of order 4, 6, 8 and

10. The numerical integration was performed with a very fine

frequency discretization with more than 30 000 points. Therefore,

the solution given by the numerical integration procedure can be

considered as exact.

With the very simple fourth order input PSDSS, (see system

state space matrices inBox II), the error was less than 4% which

is completely acceptable from a practical point of view. The tenth

order system produced an error of only 0.08%.

10. Conclusions

An efficient method for computation of RMS values of any

structural variable (i.e., displacement, acceleration, stress, internalforce, constraint force and so on) of an structure subjected to

random loads has been presented. The method allows input

random PSD spectrum of any shape.

Main advantages of the described method are:

For structures excited by white noise random loads/accelerat-ions, the RMS values obtained are exact. For PSD laws variablewith frequency, the procedure requires the construction of an

auxiliary shaping system and the results can be as accurate as

desired.

The procedure does not require any numerical integration as isthe case of the standard procedure that Finite Element Solvers

use. Thus the procedure proposed is by far much more efficient,

given that the number of mathematical operations needed islower in orders of magnitude.


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A=

(2ff+2gg) (2f+ 4gffg+2g) (2f2gg+2g2ff) 2g2f

1 0 0 00 1 0 00 0 1 0

B= 1 0 0 0T C= 2gg g 0 0 D=0Box I.

Ann BnwCan Daw

=

20.9798 13.1689 5.1463 4.1485

13.1689 0.0485 0.1537 0.27725.1463 0.1537 0.6901 3.03534.1485 0.2772 3.0353 2.2182

9.2548

0.4122

0.84641.0202

9.2548 0.4122 0.8464 1.0202 [0.0345]

Box II.

Fig. A.1. Approximation of a two step PSD profile by increasing order systems.

Owing to its efficiency, there is no limitation on the size of theFinite Element Model, nor in the number of structural variablesfor which the RMS values are required.

RMS Von Mises stresses can be obtained. As with any structuraloutput the results are exact (for white noise random forces) andas accurate as desired for general PSD laws.

Appendix. Construction of the PSD shaping system

A.1. PSD given in analytical form

In some cases of practical importance, the PSD is given inanalytical form. In these cases, the synthesis of the PSDSS isimmediate. We will provide two examples.

One of such cases is the Dryden law for gust velocity PSD, whichis of wide use in turbulence modeling for aircraft structures (seeRef.[10]). The PSD law is given by4

()=2wL

1+3L221+L223

4 The original notation is maintained.

where w = RMS gust velocity and L = scale of turbulence. It iseasy to verify that the PSDSS state matrices are

A B

C D

=

2L

1

L2

1 0

1

0

w1

L2(L22

+1)

3L3/2

L [0]

.

Another important and widely used analytical PSD spectrumis the one suggested by Clough and Penzien [11] for describing

ground excitation induced by earthquakes:

W= 2

g+422g2g2 2g

2 +422g2g4

2 2f2 +422f2f

.

Input filter matrices can be verified to be as in Box I.

A.2. PSD law not given in analytical form

Usually, PSD specifications come in tabular form, most of thetime as a series of straight lines in loglog axes. Finding an LTI

system that produces the given PSD profile for a white noiseinput is a basic problem in areas such as System Identification


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Table A.1

PSD specification

Frequency(rad/s) PSD

0.1 0.01

2 4

5 4

10 16

20 16

800 0.01

Filter Design, Signal processing, System Control, etc. There exist a

number of robust and reliable procedures. In particular, the authorhas found that the subroutine fitmagin the Mu-Analysis Toolboxof matlabbehaves quite satisfactorily (see[68]). However, whatthis subroutine approximates in fact is the modulus of the transfer

function instead of the PSD, so that the user must input thesquare root of the required PSD for fitmag to give the desiredresult.

Fig. A.1shows the result of applying fitmagto approximate a

two step PSD law given by the specification ofTable A.1The required PSD curve is shown in the thicker line inFig. A.1.

The Figure also shows the approximation obtained using systems

of orders ranging from 4 to 10.As it can be seen, the curves becomehardly distinguishable from the required profile.

For the system of order 4, the system matrices obtained are

Box IIand they are used in Example9.3.

References

[1] Kwakernaak H, Sivan R. Linear optimal control systems.New York: JohnWileyand Sons, Inc.; 1972.

[2] Skelton RE. Dynamic systems control. Linear systems analysis and synthesis.

John Willey & Sons; 1988.[3] Boyd SP, Barrat CH. Linear controller design. Limits of performance. Prentice

hall information and systems sciences series, 1991.[4] Junkins JL, Kim Y. Introduction to the dynamics and control of flexible

structures. AIAA; 1992.[5] Crandall SH, Mark WD. Random vibration in mechanical systems.NY, London:

Academic Press; 1963.[6] Anonymous, MATLAB control systems toolbox, Users guide. The Mathworks

Inc.[7] Balas GJ, Doyle JC, et al. -analysis and synthesis TOOLBOX for use with

MATLAB. The Mathworks Inc; 1993.[8] Oppenheim AV, Schaffer RW. Digital signal processing. New Jersey: Prentice

Hall; 1975. pp. 513.[9] Chen Mu-Tsang, Ali Ashraf. An efficient and robust integration technique

for applied random vibration analysis. Comput and Structures 1998;66(6):78598.

[10] Hoblit FM. Gust loads on aircraft: Concepts and applications. AIAA educationseries, 1988.

[11] Clough RW, Penzien J. Dynamics of structures. NY: Mc Graw-Hill, Inc.; 1993.

[12] Segalman DJ, Fulcher CWG. et al. An efficient method for calculating RMS vonMises stress in a random vibration environment.SandiaReport, SAND98-0260.UC 705, Feb 1998.

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