Spectral Approach to Equivalent Statistical

Paola3

Oi

Mario

and

Failla2;

Giuseppe

F.ASCE1;

Spanos,

O.

Pol

Spectral Approach to Equivalent Statistical Quadratization and Cubicization Methods for Nonlinear Oscillators

the

by

and

T,

solved

be

interval

can

long

which adequately

order,

an

cubic

over

or series

quadratic

Fourier

either

of trigonometric

a

in

nonlinearity expanded

is

polynomial

a response

with series

system Volterra

equivalent

The

an

via

method.

series

obtained Volterra

are

Abstract: Random vibrations of nonlinear systemssubjectedto Gaussianinput are investigatedby a technique based on statistical quadratization,and cubicization. In this context, and dependingon the natureof the given nonlinearity, statisticsof the stationaryresponse

exactexpressionsare derived for the Fourier coefficientsof the second-and third-order responsein terms of the Fourier coefficientsof the first-order, Gaussianresponse.By using these expressions,statistics of the responseare determinedusing the statistics of the Fourier coefficients of the first-order response,which can be readily computed since these coefficients are independentzero-meanGaussian variables. In this manner, a significant reduction of the computational cost is achieved. as compared to alternative formulations of quadratizationand cubicization methods where rather prohibitive multifold integrals in the frequency domain must be determined. lliustrative examples demonstratethe reliability of the proposed technique by comparison with data from pertinent Monte Carlo

.

simulations.

DOl: 10.1061/(ASCE)0733-9399(2003) 129:1(31) vib~~_on;

tribution. Clearly, theseeffects cannot be describedproperly by a linearization solution that reflects, multiplicatively with the transfer function of the equivalent linear system, only the frequency rangeof the excitation spectrum,and is obviously Gaussian(Roberts and Spanos 1990). Also, quadratization and cubicization methodsdo exhibit significant additional advantagesas compared to other approximatetechniques,as equivalent linearization with random parameters(Bouc 1994; Bellizzi and Bouc 1996) whose range of validity

con-

process.

is restricted to weakly damped systems, and Cran-

1984;

Un

and

(Wu

closure

cu-

the

in statistical

a

nonlinearities phenomena,

symmetric-type

interaction

handle

To fluid-structure

1992). with

text

1991,

Introduction In the investigation of random vibrations of nonlinear dynamic systemssubjectedto Gaussianinput, statisticalquadratizationand cubicization methodshave received considerableattention in recent decades.80m as natural extension of the well-established statisticallinearizationmethod,a statisticalquadratizationmethod has been developedby Spanosand Donley for asymmetric-type nonlinearities (Donley and Spanos 1990; Spanos and Donley

Gaussian

equations

systems;

moment

Nonlinear

of

Oscillators;

techniques

keywords:

Random

~~atabase

has been subsequentlyused by Kareem and Zhao (1994). In the last few years, various formulations of these methods have been

applied to a number of random vibration problems with considerablesuccess.Specifically,it hasbeen shown that quadratization and cubicization procedurescan capture super- and subharmonic frequency componentsin the responsespectrum.as well as significant deviationsfrom normality in the responseprobability disIL. B. Ryon Chair in Engineering. George R. Brown School of Engineering. Rice Univ.. P.O. Box 1892. Houston. TX 77251. E-mail: [email protected]

2Doctor of Philosophy. Dipt. di Ingegneria Stnatturale e Geotecnica, Univ. degli Studi di Palermo. Viale delle Scienze. 90128 Palenno. Italy. E-mail: [email protected] 3Professor. Dip(. di Ingegneria Stnatturale e Geotecnica, Univ. degli Studi di Palenno. Viale delle Scienze, 90128 Palenno, Italy. E-mail: [email protected] Note. Associate Editor: George Deodatis. Discussion open until June 1,2003. Separate discussions must be submitted for individual papers. To extend the closing date by one month. a written request must be filed with ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on April II. 2002; approved on April 11.2002. This paper is part of the Joumal 01 Engineering Mee1lDlli£s.

nonlin-

polynomial

to

apply

which

1997)

Lin

and

Cai

dal11985;

bicization method, as suggested by Donley and Spanos (1990),

earities only. Further, note that alternative nonlinearizationtechniques basedon potential systemsdo not yield estimatesof the responsespectralproperties (Cai and Un 1988; Cai et al. 1992). The basic concept of quadratization and cubicization methods

is to obtain statistics of an arbitrary nonlinear system from an equivalent systemwith polynomial nonlinearitiesup to quadratic or cubic order, which is solved by using the Volterra series method (Schetzen1980; Worden and Tomlinson 2001). According to this meth-ot-l

Jo }reaccurately the peak of the distribution in terms of both skewness and kurtosis coefficient. involving a lower computational effort as compared to Approach III.

Fig. 8 shows in detail the right-hand tail of the displacement

PDF.As it has been alreadymentioned,solutionsby Gramexpansion are not meaningful, although a certain improvement is noted as long as a fourth-order expansion is employed (Proposed/GC4). Nevertheless, the solutions by the Hermite transformationpredict accuratelythe simulation data.Finally, it is noted that the linearization solution is quite inaccurate and is not included in the SaIm scale. Assumenext that the input excitation/(1) in Eq. (63) haszero mean.In this case,taking into accountthe normality of the input and the symmetry of the nonlinearity, a mean value IJ.x=O is expectedfor the systemresponse.Clearly, a quadratizationsoluCharlier

Fig. 6. Eq. (63) with nonuro-meaninput: Approximatedisplacement responseprobability density functions based on Gram-Charlier

Fig. 8. Eq. (63) with nonzero-meaninput: Right-handtail of dis-

expansions

placement response probability density function

38/ JOURNAL OF ENGINEERING MECHANICS/ JANUARY 2003

'i :1

o.a

D.I

0.71

.

III

'Ylili-a,i+a,(i3_(i3»

(67)

The cubicizationcoefficient a, , a, are determinedby solving the

[

] [(li\i4)_(lili)Jl.Z' (li\i2) ]

1.71

I

response

displacement

of

cant differencesbetween the third- and the fourth-order truncation. Further, in this caseno negative values are encounteredin the tails of the distribution, where a good agreementwith the simulation data is seen. Note that the Gaussian linearization solution captures well the peak of the PDF.

the

Fig. 12 describesthe displacementPDF, with a moment-based Hermite transfonnation employed for the displacementresponse (Proposed/MH,Approach II). It is seenthat the results are accurate both in the vicinity of the mean, and in the tails of the distribution.

sixth

the

for

(68)

zero

to

to

up

equal statistics

initial the

with

scheme

iteration,

each

At

iterative

an

by unknowns.

values

][ala, =

Jl.i2 Jl.i' Jl.i' Jl.i'-~

response

system

density

peak

tion reducesto the linearization solution and a cubicization solution is, thus, pursued. Following the approach outlined in the previoussection,an equivalentcubic systemis constructedby the approximation

1.1

Fig. 10. Eq. (63) with zero-meaninput: Detail of low-frequeIx:y spectral

input:One-sided powerspecU"al

response

power

of disp)~ement

in

Fig. 9. Eq. (63) with zero-~ density

,.

order are computed by assuming the following parameter values for the Fourier series representation: N=60,

~(&)=0.15

(69)

(59)-

Eqs.

of

PDF

approximate

the

to

resorting

by (61).

performed

The evaluationof the expectedvalues on the RHS of Eq. (68) is

Figs. 9 and 10 illustrate the PSD of the zero-meandisplacement response obtained by cubicization, linearization, and digital noted

is

uKJdel

agree-

It good

in specific

the

of

are

excitation.

the

of regardless

solutions

range data,

cubicization

frequency

the simulation

proposed the

the

below of with

all

found ment

that

is

simulation.A small low-frequency peak, due to nonlineareffects,

to re-

fails

the

Ap-

excitaclarity.

by

by

the

the

solution describes

Fig. 11. Eq. (63) with zero-meaninput: Approximatedisplacement response probability density functions based on Gram-Charlier ex-

MECHANICS

ENGINEERING

pansions

OF

Fig. II showsthe resultsin tenDSof displacementPDF, with a Gram-Charlier expansion employed for the displacement response (Proposed/GC3, Proposed/GC4). It is seen that the results are quite satisfactory in the vicinity of the ~an, with no signifi-

JOURNAL

of obtained

for

herein

solution

solution

range

it

linearization included

the

cubicization

frequency

although

the the not

is

it

to

that

the

peak,

However, seen

identical

is

Thus,

virtually

is

II

it

inside

low-frequency accurately

PSD

the

Proposed/MH). Moreover,

methods.

proposed

proach

tion.

sponse

capture

GC4,

assumedfor the displacement PDF (Proposed/GC3,Proposed/

/ JANUARY 2003/39

~

~

'M'

a

Fig. 12. Eq. (63) with uro-mean input: Approximatedisplacement responseprobabilitydensityfunctionsbasedon Hermitetransformation

Figs. 13 and 14 illustrate the tails of the dispiacelrent PDF. It

is seenthat,in this case,the solutionsbasedon Gram-Charlier expansion are more accurate than those obtained by Hermite transfonnation.Note also that the linearization solution deviates significantly

from

Concluding

the

Monte

Carlo

simulation

data.

Remarks

An efficient solution scbetre bas been presentedin context with statistical quadratizationand cubicization solutions for nonlinear

Fig. 14. Eq. (63) with zero-meaninput: Right-handtail of displacement response probability density function

oscillator responseto Gaussian excitation. The Volterra series method, which is ordinarily used to treat the auxiliary equivalent with polynomial nonlinearity, hasbeenreformulated.Specifically, the convolution

integrals

representation

of the system

response

hasbeenreplacedby a truncatedFourier seriesexpansionover an adequatelylong time interval T. Consistently with this representation, statisticsof the responsehave beencomputedby ensemble averaging sample values, which are given as time averageover the interval T. Exact solutions, already developedby the writers (Spanoset aI. 2000), have been used to expressthe Fourier coefficients of the second-and third-order oscillator responseas functions of the Fourier coefficients of the first-order response.Then, by using these solutions and properties of integrals of harmonic functions, statisticsof the responsehavebeendeterminedin terms of the statistics of the Fourier coefficients of the first-order response.It has been found that statistical propertiesof thesecoefficients, which are independent zero-mean Gaussian variables, can be used to reduce significantly the computational effort involved in computing high-order statistics. Examples of application have shown reliability of the proposedsolution schemefor oscillators with both nonlinear damping and stiffnesstenDS. Although SDOF systems only have been consideredin this paper, an extension to multidimensional systemscan be readily developed by resorting to the same concepts.Additional work, however, is warranted to test the versatility of the method for a broad class of engineeringproblems. Specifically, attention must be given to the question of selecting a high enough cutoff frequency for the spectrum of the excitation so that the superharmonics induced by the nonlinear behavior of the system,and not capturedby the representationof Eq. (40), can be neglected.

&

the proposed

I

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ENGINEERING

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Fig. 13. Eq. (63) with zero-mean input: Left-hand tail of displacement response probability density function

40

JANUARY2003

solution

by

statistics

response

of

evaluation

the

discusses

Appendix

This

Appendix

sche~. For simplicity,a cubicnonlinearity

in the stiffness term only is considered in the equivalent system given by Eq. (8)

1+2t(llo.i+(II~+a.li+a..(i3-(i3»=

8\3)=': (TiCl)'(t)sin(CI)lt)dt

(70)

excitation, with

narrow-band

TJo

and the input /(1) is taken to be a

/(I)-lA.j

two-sided PSD S/1'j

~I), U~I), Y (IS)

and

/

MECHANICS

11aefme.Eq. (82) may be simplified in the f2)2+( and

(~1>2)( Mechanics

~VI2)(

IH(

Offshore

~1)2)+(~1)~2.)

platfonns,

offshore

Li,X.M.(1998). Stochastic

Computa-

Vol. L, ASME. New York. Kareem. A., Zbao, J., and Tognarelli, M. A. (1995). "Surge response statisticsof tension leg platfonns under wind and wave loads: a statistical quadratization approach." Probab. Eng. Mech., 10(4), 225240. response

Eq. (86).

Mech.,

Eng.

J.

cubicization."

statistical

by 1056-1068.

121(10),

References

platforms

tional Mechanics Publications, Southampton,U.K. Li, X. M., Quek, S. T., and Koh, C. G. (1995). "Stcx:hasticresponseof offshore

involved in

~1)2X

of

termsof thestatisticsof U~I), accordingto Eq.(44).

It is noted that, if the standardconvolution integral representation in Eq. (26) is adopted,one single, one double, and three improper triple integrals must be solved numerically, with a significant increaseof computationaleffort as comparedto the one

Proc.,

Note that. in deriving Eq. (86), the statistics of Vii) have been expressed in

12 Re[H(wl)]«

~l)r+

+

U~I»3)

~1)2)(~1)2)+(~1)2)1)+

~2)12(

IH(

~1»3+

~I)r(

+27(IH(

12 Re[H(wV]«

36(IH(

~1)~+2