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
MECHANICS
ENGINEERING
OF
JOURNAL
I
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