G. Falsone & G. Muscolino - WIT Press · using the classical stochastic linearization method ......

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Response ofa beam-stop system under random excitations by an equivalent non-linearization approach G. Falsone & G. Muscolino Abstract An equivalent non-linearization approach has been implemented in order to find the response of a beam-stop system subjected to a boundary excitation modelled as a stochastic process. In particular the original bilinear impact force has been replaced by a cubic one, whose coefficients have been evaluated by stochastically minimising the difference between the original and the equivalent systems. The applications have evidenced an optimum level of accuracy, if compared with the results of the classical stochastic linearization approach. 1 Introduction In heat exchangers and nuclear piping systems, baffle plates and snubbers are used to control large amplitude displacements of tubes due to flow induced or seismic excitations. These long tubes or pipes expand or contract for the effect of a temperature change and hence clearances are incorporated between tubes and snubbers. Even a single tube leak may result in extended and costly power plant outage and repairs and, hence, an early warning ot impeding failure is of crucial importance. For this reason the evaluation of the response of these systems is fundamental. Due to the fact that either the flow induced and the seismic excitations are usually considered as random processes, the tube or pipe systems can be optimally modelled as a beam-stop system with assigned random support movement. In this way the problem is included in the class of linear continuous beams, randomly boundary excited, with an impact force non-linearly depending on the beam deflection. In particular, the impact force depends bilmeady on the Structures under Shock & Impact VI, C.A. Brebbia & N. Jones (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-820-1

Transcript of G. Falsone & G. Muscolino - WIT Press · using the classical stochastic linearization method ......

Response of a beam-stop system under random

excitations by an equivalent non-linearization

approach

G. Falsone & G. Muscolino

Abstract

An equivalent non-linearization approach has been implemented in order to findthe response of a beam-stop system subjected to a boundary excitation modelledas a stochastic process. In particular the original bilinear impact force has beenreplaced by a cubic one, whose coefficients have been evaluated bystochastically minimising the difference between the original and the equivalentsystems. The applications have evidenced an optimum level of accuracy, ifcompared with the results of the classical stochastic linearization approach.

1 Introduction

In heat exchangers and nuclear piping systems, baffle plates and snubbers areused to control large amplitude displacements of tubes due to flow induced orseismic excitations. These long tubes or pipes expand or contract for the effect ofa temperature change and hence clearances are incorporated between tubes andsnubbers. Even a single tube leak may result in extended and costly power plantoutage and repairs and, hence, an early warning ot impeding failure is of crucialimportance. For this reason the evaluation of the response of these systems isfundamental.

Due to the fact that either the flow induced and the seismic excitations areusually considered as random processes, the tube or pipe systems can beoptimally modelled as a beam-stop system with assigned random supportmovement. In this way the problem is included in the class of linear continuousbeams, randomly boundary excited, with an impact force non-linearly dependingon the beam deflection. In particular, the impact force depends bilmeady on the

Structures under Shock & Impact VI, C.A. Brebbia & N. Jones (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-820-1

Structures Under Shock and Impact VI

beam deflection measured in the impact point. Consequently the beam deflectionis a random field, that is non-Gaussian, even if the excitations are Gaussianprocesses; hence response moments and correlation functions of order greaterthan two have to be evaluated in order to characterise adequately the beamdeflection field.

In the literature there are some works concerning this problem, most of themusing the classical stochastic linearization method (for example Bouc [1], Miles[2]); this method allows us to find accurate results only if the level of non-linearity is low, giving always a Gaussian response field. In other works, in orderto improve the linearization, the concept of equivalent linear system with randomcoefficients has been introduced (Bouc & Defilippi [3]). This makes the responsenon-Gaussian, but, as the classical linearization, this approach underestimates theresponse because it replaces the real bilinear spring with an approximated linearone which is stiffen

In the present work the method of stochastic equivalent non-linearization(Roberts & Spanos [4]) is used for the evaluation of the response of a beam-stopsystem excited by boundary Gaussian random processes. It implies thereplacement of the original non-linear system with another non-linear one, whosesolution, accurate even if approximated, is more easily available. The parametersof the equivalent non-linear system are chosen in such a way that the differencerespect to the original system is minimal in a statistical sense. In particular, dueto the form of the non-linearity in the beam-stop systems, that is a bilinear form,the equivalent non-linear system chosen for the analysis has a cubic non-linearity. The coefficients of the cubic polynomial are chosen in order tostochastically minimise the mean square differences between these systems.

The application of the proposed approach to a simple, but significantexample, shows its good applicability and accuracy compared with the resultsobtained by the equivalent linearization.

2 Preliminary concepts

The system under consideration is the beam represented in Fig.l, excited by themovement z (f)of ™e supports. The differential equation governing the

deflection of the beam can be written as follows:

02 04

where u(x,t) is the relative deflection, EJ,p and A are the elasticity modulus,

moment of inertia, mass density and area of the section, respectively, all assumedconstant along x . The impact force / is a function of the deflection

%o(f) = u(x t) of the beam at x^ and it is defined as follows:

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Structures Under Shock and Impact VI 613

if -oo < w < -e

/(%J = &(%,- g) if g<w, <oo

k being the spring stiffness and e the clearance. In equation (1) S(x -xj is the

Dirac delta function; at last u (i) is the acceleration of the moving supports that,

in the present analysis, is assumed to be a Gaussian white noise process withpower spectral density S .

0 ' v

If the modal expansion:

(3)

is introduced, in such a way that the following ortho-normality conditions aresatisfied:

f~JY V /0 0 "•*

where S.. is the Kronecker symbol and o). is the modal radian frequency, then

the differential equation governing the modal variable q.(t) assumes the

following form:

^ (r) + 2 ^ ^ (/) + (/ (f) = - (%J/(wJ-fi( (r) (5)

P. being the modal participation factor given as:

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614 Structures Under Shock and Impact VI

In equation (5) the viscous modal damping has been introduced by means of thedamping ratio <%. .

If it is assumed that the beam is simply supported, then the normal modes andthe corresponding modal radian frequencies have the following expressions:

and the corresponding modal participation factors are:

=0 if ) = 2,4,...

With the aim to perform the stochastic analysis of the system underconsideration, it is worth noting that the fundamental problem is represented bythe presence of the non-linear function / in the equation (5). This makes non

Gaussian the response and it makes not possible its probabilistic characterisationin a closed form. In the following sections an equivalent non-linearizationapproach, able to give a better approximating response respect to the usuallinearization techniques, will be introduced.

3 Equivalent non-linearization approach

By following the classical linearization approach, the non-linear function /(%J

is replaced by an equivalent linear one k^ in such a way that the difference is

minimised in some statistical sense. But the linear function is not appropriate inorder to approximate /(wj . The choice of a cubic function au^ + bu] is surely

preferable. And this is made here.By following a procedure which is similar to that one used in the stochastic

linearization, the best coefficients a^ and b^ are evaluated by minimising the

mean square root of the difference f(u ) - au^ - bif , that is:

(9)

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Structures Under Shock and Impact VI 615

E[-] being the mean of (•) . In this way, by taking into account that all the

stochastic moments of odd order of the process u (t) are zero, the following

expressions are obtained (Benfratello & Falsone [5]):

^E ] E % -

(10)

Hence the evaluation of the best approximating coefficients requires theknowledge of the even moments of u^ up to the sixth order and the knowledge

of the two means E[/(wJuJ and E[/(%J%q]. These last quantities can be

expressed in function of the moments of u^ only if its probability density

function is assigned. Taking into account that u^ is a non-Gaussian process, an

appropriate expression of this function is given by the following Gram-Charlierexpansion (Crandall [6]):

where C.[wJ are the so-called Hermite-moments of order j of u^ (Muscolino

[7]); they are related to the corresponding quasi-moments b.[u^\ by means of the

simple relationship:

(12)

a^ being the standard deviation of u^. Hence it is not difficult to find the

algebraic relationships between C\[wJ and the stochastic moments up to the

j - th order; they can be found for example in Muscolino [7]. In equation (11)

H .(•) is the 7-th Hermite polynomial of (•) , z is the standardised variable,

that is z = zjo- , and at last /?°,(z) is the Gaussian probability density function

having zero mean and standard deviation cr, .

By taking into account that for the case under examination all the stochasticmoments and Hermite-moments of odd order are zeros and by arresting thesummation in equation (1 1) to 7=6, after some algebra we obtain:

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616

= a i-

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•]*(«„-*KpJz)dz =

^ V *(13a)

, -exp --_ -

-r 6 L O J ._ 1 /-. , «,>5 12 cr- 3cr,, 4

= (J - erj r*2 crQ[wj)+.-exp *

(13b)

4L oJ^ "„ g «„ g ^

r«i-i-f^ + —cr e^+i—"

In this way, having approximated the probability density function of the non-Gaussian process u^ by neglecting the Herniite-moments of greater order than

the sixth one, it is possible to express the best approximating coefficients a^ and

b^ as functions of the even stochastic moments of u^ up to the sixth order, as it

is evidenced by equations (10) and (13).It is important to note that if a closure of the second order is applied, the

corresponding results are coincident with those obtained by the equivalentlinearization.

4 Stochastic analysis of the equivalent non-linear system

The differential equation governing the j - th modal coordinate of the

equivalent non-linear system is given by:

(14)

Taking into account only the first N vibration modes of the beam and by

introducing the modal coordinates vector q^ = (g, q^ ••• q^), we can write:

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Structures Under Shock and Impact VI 617

in which E is the diagonal matrix collecting the viscous damping coefficients

Igj&j , Q' is the diagonal matrix collecting the squares of the modal radian

frequencies, *F is the modal vector, P is the participation factors vector and theexponent into square brackets indicates the power made by following the rules ofthe Kronecker algebra (see Appendix).

Having in mind to perform the stochastic analysis of the system representedin equation (15), it is useful to rewrite it by introducing the state variables vector

> obtaining:

z(f ) = Az(f ) + Bz

where:

A= I ^\ e(/ () (J / v ^ /

(17)

Once that the modal equations are written in the form given into equation(16), if the support acceleration u^ is assumed to be a Gaussian white noise, it is

possible to write the differential equations governing the stochastic moments ofeven order of z (the odd ones are zeros) by means of the I to differential calculus(Ito [8]); they have the following form:

(18)

r = 2,4,6

where:

A,. = A,._, (S) I2K*2K + I]'ZL ® A ; A, = A

B = B , (8)1,., ,„ +l!,7'L ®B ; B. = B __ .,-t 2Ax2K 2/T/2A ^ I ^^^

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g I g Structures Under Shock and Impact VI

the symbol (8> indicating the Kronecker product and Q being the natural

numbers matrices defined in the Appendix.It is worth noting that, due to the presence of the quantities a^ and b^,

depending on the stochastic moments up to the sixth order of u^, respectively in

the matrices A and B , the equation (18) is non-linear. Moreover the equation(18), particularised for r = 6, requires the knowledge of the eighth ordermoments. These last quantities can be expressed in function of less order ones bytaking into account a sixth order closure of the probability density function onthe Hermite moments. As a consequence of these two circumstances the equation(18) is non-linear and it can be solved by an iterative procedure.

If we are interested in the stationary response, then the first member ofequation (18) is posed zero and the response stochastic moments can beevaluated as the solution of the corresponding non-linear algebraic equation:

A,.E[z ] + B, [z[" ] +V,.E[z - ]2 = 0 (19b)

that can be solved by an iterative procedure. Once that the modal responsestochastic moments up to the sixth order are evaluated, the correspondingmoments of the beam deflection u(x) at any point x can be evaluated by means

of:

From these quantities the Hermite moments can be obtained and consequentlythe approximated probability density function, too.

If we are interested to characterise the beam deflection field, it is necessary toevaluate the corresponding correlation functions. And this is possible, too. Infact, for example, the second order correlation function is obtained as follows:

Vf/rr^i (21)

Hence the proposed procedure allows us to characterise accurately thedeflection at any fixed point of the beam or all the deflection field from aprobabilistic point of view.

5. Numerical example

The procedure proposed in the previous sections is now applied to a simplysupported beam-stop system. The geometrical and mechanical parameters are:elasticity modulus E = 1.96xlO"(A/m^) , mass density p = 7.8x1

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Structures Under Shock and Impact VI 619

moment of inertia / = l.l9xlQ~*(m*), length L = 6(m), section area

A = 1.37xlO~*(w*), spring-stop stiffness k - 2xltf(Nm~^ and the excitation

w is a white noise process with unitary power spectral density. Various

locations #<, of the stop will be considered. The deflection is measured at x^ and

at another point *, . Only the first three modes are considered in the analysis, that

is N = 3 in equation (3). The first three modal radian frequencies of the linearsystem, obtained by means of the second one of equations (7), are:

= 364.530/W/j (22)

while the modal damping ratios will be considered equal with value £. - 0.05 .

case a)

case b)

case c)

xo=1.5x,=3Xr,=2x,=3Xo=3xri.5

LE

CT-10'2.6139.7352.6486.0863.8464.760

CEcr-icr

2.72310.0012.7576.2534.0194.891

C4

-0.255-0.152-0.271-0.128-0.280-0.131

C6

-0.187-0.102-0.199-0.081-0.206-0.090

MCcr-10'

2.73110.042276462574.0254.899

Q

-0.261-0.161-0.280-0.135-0.285-0139

c.

-0.187-0.111-0.207-0.090-0.212-0.099

In Tab.I the results obtained by the proposed procedure are compared withthose obtained by applying the classical linearization approach and thoseobtained by a Monte Carlo simulation implemented with 100,000 samples. Theresults are reported in terms of standard deviations and Hermite moments offourth and sixth order. In the case a) the stop is located at x^ = 1.5m and the other

measure point is in the middle of the beam. In the case b) the stop is located at*o = 2.0m and the other measure point is in the middle of the beam, too. At last

in the case c) the stop is located in the middle of the beam while the othermeasure point is at %, =1.5m. The analysis of these results evidences the

optimum level of accuracy of the proposed approach and the substantialdifferences with the linearization results, above all in correspondence of the stoplocation where the effects of the non-linearity are stronger. Moreover theyconfirm the fact that the linearization approach underestimates the stochasticresponse, by considering an equivalent linear system which is stiffer than theoriginal one. All these considerations are confirmed by the results reported inFig.2 and Fig.3 where the probability density functions of the deflection at x^

and at jtj are depicted, respectively, with reference to the first case.

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620 Structures Under Shock and Impact VI

.-IT 100 —

I ' I ' I-0.01 0 0.01

displacement (m)

.linearizatio

• MC Simulaion

I I 1 I I-0.02 0 0.02 0.04 0.06

displacement (m)

Fig. 3: Probability density function of the deflection at x, in the case a).

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Structures Under Shock and Impact 17 621

6 Conclusions

A beam-stop system boundary excited by a Gaussian white noise process hasbeen analysed by applying an equivalent non-linearization approach. Inparticular the original bilinear impact force has been replaced by a cubic one,whose coefficients have been evaluated by minimising the mean square root ofthe difference between the original impact force and the approximated cubic one.This procedure can be considered as an improvement respect to the classicalstochastic linearization approach for two reasons: 1) the linearization makes thebeam deflection a Gaussian field, while it is non-Gaussian; 2) the linearizationreplaces the original bilinear spring with a linear one, that is stiffer, givingunderestimated response results.

The application of the procedure to a simple, but significant, example hasshown an optimum level of accuracy of the response, in terms of variance, higherorder statistics (Hermite-moments) and probability density function of the beamdeflection at some particular points.

7 References

1. Bouc, R. The power spectral density of a weakly damped strongly nonlinearrandom oscillation and stochastic averaging, Publication du LMA: CoUoqueControle actif Vibro-acoustique et dinamique stochastique, Marseille, ISSn07JO-7JJ6, 127, 1991.

2. Miles, R.N. Spectral response of a bilinear oscillator. Journal of Sound andK/WfzbM, 163 2, 319-326, 1993.

3. Bouc, R. & Defilippi, M. Spectral response of a beam-stop system underrandom excitation, ILJTAM Symposium on "Advances in Nonlinear6Yoc/7<%s//c Mcc/zamcs (W. MzasA <& 3. AYe/?& cfW, Kluiwer AcademicPublishers, 69-78, 1996.

4. Roberts, J.B. & Spanos, P.T.D. /faWo/;? W6n?//o/? fW j/<?/A5/z<Wlinearization, John Wiley and Sons, New York, 1990.

5. Benfratello, S. & Falsone, G. A non-Gaussian approach for the stochasticanalysis of offshore structures, Journal of Engineering Mechanic Division(WSCE), 121, 1173-1180, 1995.

6. Crandall, S.H. Non-Gaussian closure for random vibration of non-linearoscillations, International Journal of Non-Linear Mechanics, 15, 303-313.1980.

7. Muscolino, G. Response of linear and non-linear structural systems underGaussian and non-Gaussian filtered input, in Dynamic motion: chaotic and6'foc/?o,sn'c W?wWour (F. CksczY/fz' ffh'for), CISM Courses and Lecturesn°340, Springer-Verlag, Wien, 1993.

8. Ito. K. On a formula concerning stochasic differential, Nagoya,, 3, 55-65, 1951.

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622

Appendix

Structures Under Shock and Impact VI

111 this appendix some properties of the Kronecker algebra are reported.If M and N are two matrices of order (pxq) and (sxt) respectively, the

Kronecker product M®N is the matrix of order (psxqt), obtained by

multiplying each element of M for all the matrix N in such a way that:

m ,N m ,N ••• m N/•" pt /•"/

where m.. is the generic element of M . The Kronecker product satisfies the

follows rules:

where P and R are other two matrices. The Kronecker power is defined asfollows:

M^ =M®M<8)---(x)M

The matrix Q^ appearing in the equation (19) of this work is the (2Af)* x(2Af)*

matrix given by:

In which E. . is the so-called permutation matrix, which is a matrix of order

ijxij , consisting of jxi arrays of elementary submatrices E^ , of order ixj ;

each of these matrices has all zero elements, except for the (k, /) - th which is

one. Hence E. . has the following form:

Structures under Shock & Impact VI, C.A. Brebbia & N. Jones (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-820-1

Structures Under Shock and Impact 17

Ell -j-,21JL

El; TTI 2,H/

where, for example, E'* is given by:

0 1

0 0

623

E"

•• 0

.. o

0 0 - 0

It is important to note that the permutation matrix exhibits a single one in eachrow and in each column. Consequently it is easy to verify that the matrix Q

exhibits only zeros and natural numbers in such a way that the sum of theelements of each row, or of each column, is always equal to k .

Structures under Shock & Impact VI, C.A. Brebbia & N. Jones (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-820-1