Journal of Sound and Vibration (1995) 186(5), 846–855

IMPROVING THE EQUIVALENT LINEARIZATION TECHNIQUE FORSTOCHASTIC DUFFING OSCILLATORS

J. L

Wright Laboratory (FIB), Wright–Patterson Air Force Base, Ohio 45433, U.S.A.

(Received 28 February 1994, and in final form 9 December 1994)

1.

The equivalent linearization technique recovers the exact variance of Duffing’s oscillatorwhen the fourth order cumulant is known. Of course, one does not know it a priori; henceit is assumed zero by Gaussian statistics. Therefore, the most natural way to improve theequivalent linearization entails estimating the fourth order cumulant along with variance.This has been embodied in the non-Gaussian closure schemes of Wu and Lin [13] andCrandall [14]. Wu and Lin used full Edgeworth series for their closure, whereas Crandall’sclosure is couched on an abridged Edgeworth series which omits quadratic cumulants in theexpansion coefficients. We espouse the abridged Edgeworth cumulant analysis because itconverges much faster than the full Edgeworth series.

2. ?

In stochastic non-linear dynamics one is usually content with seeking a statisticaldescription of the lowest order moments of mean and covariance. Under such a limited goal,it may often be possible to replace the original nonlinear system by a surrogate linear systemand thereby attempt to replicate the lowest order statistics. This is the intent of the so-calledstatistical linearization [1, 2]. In particular, when the non-linear system under considerationis an oscillator, it is also called statistical equivalent linearization, conforming with sucha terminology already introduced by Krylov and Bogoliubov [3] for the deterministicnon-linear oscillator. We begin by reviewing the well-known result [4] of a damped harmonicoscillator y0+by0+ky0=f(t), where the overhead dot is d/dt, b is the damping coefficient,and k is the linear spring constant. Under the zero-mean Gaussian white noise, the firsttwo moments of f(t) are � f(t)�=0 and � f(t)f(s)�=2Dd(t−s), where � � denotes anensemble average. Note that the factor 2 for the power input D is completely arbitrary,although agreeing with Wang and Uhlenbeck [5]. For a stationary random forcing, theresponse variance is s2

00�y20�=D/bk, which is linearly proportional to D but inversely

proportional to b and k.Instead, we now consider a damped Duffing oscillator under the same random forcing

f(t),

y+by+ky+ay3=f(t), (1)

where a is the strength of a hard spring. In equivalent linearization [6], equation (1)is replaced by a surrogate linear system y+by+Ky=f(t), where K is the equivalent stiffnessyet to be determined. Since this replacement commits an error e=(k−K)y+ay3, the cruxof the equivalent linearization is to minimize the mean square error by d�e2�/dK=0, yieldingK=k+a�y4�/�y2�, which is still a useless expression because of the unknown fourth ordermoment. However, if y is assumed to be Gaussian we have the moment ratio �y4�g /�y2�2

g=3,where the susbscript g emphasizes the Gaussian ensemble average. We then obtain the usualexpression for equivalent stiffness K=k+3a�y2�g .

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0022–460X/95/400846+10 $12.00/0 7 1995 Academic Press Limited

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Although this could have been obtained more directly from the short-cut formulaK=k+a�1y3/1y� of Atalik and Utku [7], we prefer the original argument of Caughey [6]outlined here, because of the physical insight that it provides. Now, inserting K into theinput–output variance relation �y2�g=Db/K of the surrogate linear system, we obtainthe quadratic variance equation

(3a/k)�y2�2g+�y2�g−s2

0=0, (2)

with the root

�y2�g=(k/6a)[z1+12(a/k)s20−1]. (3)

For weak non-linearity (12(a/k)s20�1) we have

�y2�g=s20[1−3(a/k)s2

0+18(a/k)2s40−· · ·],

whereas equation (3) reduces to �y2�g=z13(s0zk/a ) for strong cubic stiffness

(12(a/k)s20�1).

3. ?

To establish a frame of reference, writing equation (1) in the Ito stochastic differentialequations

0dx1

dx21=0 x2

−bx2−kx1−ax311dt+0 0

dW(t)1, (4)

where the increments of the Wiener process W(t) obey �dW(t)�=0 and �dW(t) dW(s)�=2Dd(t−s), we obtain the Fokker–Planck equation by adapting the general form [8, 9] forP(x1, x2):

1P1t

=−1(x2P)

1x1+

1

1x2(bx2+kx1+ax3

1 )P+D12P1x2

2. (5)

Caughey [10] has presented the stationary (1P/1t=0) distribution P=N exp(−bH/D),whereH=kx2

1 /2+ax41 /4+x2

2 /2 is theHamiltonian andN is normalization. Sincex1 andx2 areindependent, we integrate out x2 to obtain the reduced distribution P(x1)=N' exp(−bS/D),which will be used to compute moments �xn

1�fp=fa−a P(x1)xn

1 dx1, where the subscript fpdenotes an average over the Fokker–Planck distribution. Here, S=kx2

1 /2+ax41 /4 is the strain

(potential) energy and N' is another normalization. By the transformation x21=sz2D/ab ,

we find that the moment integral has the form fa0 exp(−zs−1

2s2)s(n−1)/2 ds=G((n+1)/

2) exp(z2/4)U(n/2, z) of equation (19.5.3) of Abramowitz and Stegun [11], A–S for shorthereafter. Here, z=kzb/2Da , and G andU are the gamma and parabolic cylinder functions, respectively. Hence, the moments are�xn

1�fp=(2D/ab)n/4G((n+1)/2)U(n/2, z)/G(12)U(0, z) for even n. In particular, we have

�x21�fp=zD/2abU(1, z)/U(0, z), which can alternately be expressed by D−a−1/2(z)= U(a, z),

as was done in references [12, 13]. By expressing z=z(1/2s20 )(k/a) in terms of s2

0 , we see thatthe weak and strong non-linearities are represented by z:a and z:0. Hence, using A–S’sequation (19.8.1.) for large z, we have �x2

1�fp=s20 [1−3(a/k)s2

0+24(a/k)2s4

0−· · ·] to be compared with the second order coefficient 18 of equivalent

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linearization discussed previously. In contrast, using A–S’s equation (19.3.5) for z=0we obtain

�x21�fp=

2G(3/4)G(1/4) 0s0Xk

a1in the limit as the cubic stiffness dominates. Note here that 2G(3/4)/G(1/4)10·6760 is theamplitude factor that Atalik and Utku [7] have found for a Duffing oscillator without thelinear stiffness term. It is larger than the equivalent linearization value z1

310·5774.Let us now put the Fokker–Planck variance in the form

�x21�fp

s20zk/a

=1

z2

U(1, z)U(0, z)

, (6)

and compute the moment ratio

�x41�fp

(�x21�fp)2=

3U(2, z)U(0, z)U(1, z)2 . (7)

As expected, for large z we have �x41�fp /(�x2

1�fp )2:3, agreeing with the Gaussian ratio. Onthe other hand, at z=0, equation (7) reduces to�x4

1�fp /(�x21�fp )2=1

4(G(14)/G(3

4))212·188,which

is considerably smaller than the Gaussian ratio. This is why the equivalent linearlizationvariance underestimates the Fokker–Planck in the limit of strong cubic non-linearity, forthe amplitude factor is the square root of the reciprocal of the moment ratio. Suppose thatwe repeat the equivalent linerization by using the moment ratio (7) instead. Similar toequation (3), the upshot is that

�y2�fp

s0zk/a=

1

z2

U(1, z)U(0, z) $zz2U(1, z)2+6U(2, z)U(0, z)−zU(1, z)

3U(2, z) %. (8)

Using the identity U(2, z)=23(U(0, z)−zU(1, z)), we find that the square bracket term is

unity, so that equation (8) is indeed identical to equation (6). In words, had the equivalentlinearization been carried out with the actual moment ratio instead of 3, we wouldrecover the exact variance without any approximation. In reality, the moment ratio is notknown a priori because we do not have a distribution function. This therefore suggestsapproximating the variance along with the moment ratio which is quantified by the fourthorder cumulant.

3.

A systematic framework to simultaneously estimate the variance and cumulants isprovided by the hierarchical moment equations which may be obtained from equation (5)by partial integrations. It is, however, simpler to first derive the evolution equation for ascalar function g(x1, x2) along the trajectory of equation (4) by the Ito calculus [8, 9],

�g�=Wx21g1x1w−W(bx2+kx1+ax3

1 )1g1x2w+DW12g

1x22w, (9)

and then write down the moment equations by letting g=xr1xs

2. It is easily checked thatequation (9) gives three second order (r+s=2) and five fourth order (r+s=4) momentequations which are identical to equations (12) and (14) of Wu and Lin [13]. For thestationary (�g�=0) moment solution, there is only one independent moment equation for

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each order, for all others are either trivial or redundant to the moment equations of lowerorder. Summarized here are the stationary moment equations of orders 2, 4, 6 and 8:

(a/k)�x41�+�x2

1�−s20=0, (a/k)�x6

1�+�x41�−3s2

0�x21�=0,

(a/k)�x81�+�x6

1�−5s20�x4

1�=0, (a/k)�x101 �+�x8

1�−7s20�x6

1�=0, . . . (10)

It should be pointed out that one could have written down equation (10) directly fromequation (11) of Crandall [14] which has the form s2

0�1C/1x1�=�(x1+(a/k)x31 )C�, where

C=xn1 (n=1, 3, . . .).

Because of the cubic non-linearity, equations (10) are indeterminate in that each containsan unknown moment of higher order. By restricting ourselves to the first of equations (10)and closing it off by �x4

1�g /�x21�2

g=3, we recover equation (2). Hence, the equivalentlinearization is nothing but the so-called Gaussian closure of the first moment equation ofequations (10). Calling this the first order equivalent linearization, let us now consider thefirst two of equations (10) to effect what may be called the second order equivalentlinearization. Then by invoking another ratio �y6�g /�y2�3

g=15, we obtain a cubic equation15(a/k)2�x2

1�3g−(1+3(a/k)s2

0 )�x21�g+s2

0=0. For the weakly non-linear case (a/k:0),this admits the three-term expansion for �x2

1�fp as a solution, thus agreeing with theFokker–Planck variance up to the second order in a/k. In general, one must solvenumerically the cubic equation which will be put in a dimensionless form

X3−((2z2+3)/15)X+z2z/15=0, (11)

where X=�x21�g /s0zk/a . As shown in Figure 1, equation (11) does not have a physically

relevant root for all z. First, in 0EzQ0·4, one of the positive real roots is shown inFigure 1, for which the amplitude factor is X=z1

5 at z=0. Next, there are one negative realroot and a pair of complex conjugate roots in 0·4QzQ1·6; hence no roots are

Figure 1. Comparison of the response variances. – – – –, �x21�fp of Fokker–Planck distribution; ——, �x2

1�g ofthe second order equivalent linearization.

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Figure 2. The percentage error in the equivalent linearization techniques. ——, first order linearization; —w—,second order linearization.

physically relevant. Finally, for zq1·6, the positive root decreasing with z is shown inFigure 1 as the response variance of the second order equivalent linearization. In view ofFigure 2, which compares variance deviations of the first and second order equivalentlinearization from the Fokker–Planck, we see that the second order equivalent linearizationoutperforms the first order for large zq2·8. On the other hand, for small z the second orderequivalent linearization is worse than the first order and, in particular, it fails completelyin the range 0·4QzQ1·6. Just as in the first order equivalent linearization, the second orderhinges critically on the assumed moment ratio. Suppose that we repeat the second orderlinearization by using the actual moment ratio �x6

1�fp /�x21�3

fp=15U(3, z)U(0, z)2/U(1, z)3.We then recover the Fokker–Planck variance (6); hence the second order equivalentlinearization is indeed superfluous.

3. -

The second order equivalent linearization embodied by equation (11) clearly demonstratesthe failure of Gaussian closure, whereby the Gaussian moment ratios are invoked to closeoff equations (10). As noted previously, it is moment ratios that need to be evaluated alongwith the variance, if one were to improve equivalent linearization at any closure level ofequations (10). Since cumulants express the deviation of moment ratios from the Gaussianvalues, the analysis of cumulants would quantify from the Gaussianity departure in asystematic fashion [15, 16]. As a non-Gaussian closure, Crandall [14, 17] and Wu and Lin[13] have already carried out the cumulant analyses for the Duffing oscillator (1) in parallelformulations. Although they have reported the identical response variance for the fourthorder cumulant analysis, the variance estimates are different at the sixth order level ofcumulant analysis. We shall show here that Wu and Lin [13] have based their cumulantanalysis on full Edgeworth series, whereas the analysis of Crandall [14] involves anabridged Edgeworth series to be defined shortly. This is the source of discrepancy in theirsixth order analyses. However, as will be seen later, the abridged Edgeworth series convergesmuch faser than the full Edgeworth series. We, therefore, espouse the use of the former overthe latter.

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We begin with the formal expansion of an arbitrary distribution by the Hermite poly-nomials Hn ,

P(x)=61+sa

n=3

cn

n !Hn 0xs17f0xs1, (12)

where f(x/s)=(1/z2ps) exp{−12(x/s)2} is the Gaussian distribution with zero mean and

variance s2. Using the orthogonality of Hn , one can compute the coefficients cn in terms ofcentral moments mn ; i.e., c3=m3/s3, c4=m4/s4−3, etc., as in equation (6.31) of Kendall andStuart [18]. Expansion (12) with such central-moment coefficients is the Gram–Charlierexpansion. Now, expressing the central moments in terms of cumulants kn (equation (3.38)in reference [18]), the cn are given alternately in kn (equation (6.41) in reference [18]).For a symmetric distribution, all odd cumulants are zero so that the non-vanishingcoefficients are

c4=k4/s4, c6=k6/s6, c8=(k8+35k24 )/s8, c10=(k10+210k6k4)/s10, . . . . (13)

Expansion (12) with coefficients (13) is the Edgeworth series which is formally identicalto the Gram–Charlier expansion, as pointed out by Kendall and Stuart [18]. However,a difference can arise when one considers the Edgeworth series as an asymptotic expansionand rearranges terms as shown in equation (17.7.3) of Cramer [19].

We now compute some lower order even moments of distribution P(x1):

�x21�es=s2, �x4

1�es=k4+3s4, �x61�es=k6+15k4s

2+105s6,

�x21�es=k8+28k6s

2+210k4s4+35k2

4+105s8,

�x101 �es=k10+45k8s

2+210k6k4+630k6s4+3150k4s

6+1575k24s

2+945s10, (14)

where the subscript ‘‘es’’ indicates the average being over the Edgeworth series (12), withequations (13). For the cumulant analysis we introduce equations (14) into equations(10) to derive the corresponding cumulant equations, noting that s2=k2. It is, however,advisable to define dimensionless variables X=s2/s0zk/a, xn=�xn

i �es/(s0zk/a)n/2,and ln=kn /(s0zk/a )n/2, and thereby retain the hierarchical structure of momentequations,

1−z2zx2−x4=0, 3x2−z2zx4−x6=0,

5x4−z2zx6−x8=0, 7x6−z2zx8−x10=0, (15)

where

x2=X, x4=l4+3X2, x6=l6+15l4X+15X3,

x8=l8+28l6X+210l4X2+35l24+105X4,

x10=l10+45l8X+210l6l4+630l6X2+3150l4X3+1575l24X+945X5. (16)

Although equations (15) are just as indeterminate as the original moment equations, thetransformation (16) has rendered the system of cumulant equations more amenable tosuccessive approximations. Alternately, we consider an abridged Edgeworth series withthe coefficients cn=kn /sn by omitting the quadratic k2

4 and k6k4 in equations (13). Thus,

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such quadratic cumulants are also omitted from equations (14) so that equations (15) arenow supplemented by

x8=l8+28l6X+210l4X2+105X4,

x10=l10+45l8X+630l6X2+3150l4X3+945X5, (17)

whereas, x2, x4 and x6 are the same as in equations (16).Let us for short call the cumulant analysis to be of the nth order if cumulants of order

higher than nth order are suppressed for closure approximations.Second order cumulant analysis. For l4=0 the first of equations (15) gives

3X2+z2zX−1=0. Since this is identical to equation (2), it is the usual equivalentlinearization.

Fourth order cumulant analysis. Under l6=0, combining the first two of equations (15),with equations (16) yields

30X3+15dX2+(d2−12)X−d=0, (18)

where d=z2z. Suppose that we redefine d=1/ze. By substituting X=Aze we findthat equation (18) reduces to equation (17) of Wu and Lin [13] and equation (36) ofCrandall [14] when X=s2ze is substituted. Equation (18) has been plotted in terms of e inFigure 1 of Wu and Lin [13] and in Figure 3 of Crandall [17]. In the limiting strong non-linearcase, equation (18) degenerates to 30X2=12 at z=0. Hence, the amplitude factorX=z2

510·6325 is the same as in equation (24) of reference [13] and equation (43) ofreference [14].

Six order cumulant analysis. Thus far only the first three of equations (16) have been used,so that the distinction between full and abridged Edgeworth series was unnoticeable. Thisis about to change at this level of analysis. For l8=0 we obtain from the first three ofequations (15) with equations (16):

630X4+420dX3+(63d2−336)X2+d(d2−90)X−(d2−30)=0. (19)

Note that all but the leading coefficient 630 are identical to those in equation (18) of Wuand Lin [13]. (their coefficient 714 is a misprint). For the limiting strong non-linearitywe have 630X4−336X2+30=0 at z=0; hence the amplitude factor X=z(4+Z37/7)/1510·6480 agrees with that of Wu and Lin [13]. However, it differs from the 0·6606 ofCrandall [14]. The simplest way to resolve this discrepancy is to view Crandall’snon-Gaussian closure as based on abridged Edgeworth series. Then, by appending equations(17) instead, we obtain

315X2+210dX3+(28d2−126)X2+d(d2−20)X−(d2+5)=0, (20)

which is essentially equation (33) of Crandall [14]. And, the limiting amplitude factorX=z(3+2Z22/7)/1510·6606, as given in equation (43) of reference [14] is the root of315X4−126X2−5=0.

Eight order cumulant analysis. For the full Edgeworth series analysis, we considerequations (15) and (16) in their entirety under l10=0. Without writing down theensuing quintic equation for X, we present here 1890X4−1260X2+193=0 for thelimiting strong non-linearity. The amplitude factor X=z(1+Z17/210)/310·6543 is onlysightly larger than the corresponding sixth order 0·6480. On the other hand, if equations (17)are appended to equations (15) we obtain 945X4−315X2−51=0. Hence, the amplitudefactor X=z(1+Z103/35)/610·6727 of the abridged Edgeworth series is much close to theFokker–Planck limit (0·6760) than the full Edgeworth value of 0·6543.

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Figure 3. The convergence of variance estimates. —W—, abridged Edgeworth cumulant analysis; —w—, fullEdgeworth cumulant analysis; ––––, Fokker–Planck limit=0·6760.

Encouraged by the convergence of abridged Edgeworth series analysis, we shall carry itout to the tenth and twelfth order. The limiting equation 5775X6−1155X4−671X2−5=0of the tenth order gives X10·6767, and 19305X6−3861X2−92=0 of the twelfth ordergives X10·6772. In Figure 3 we compare the successive variances of the full and abridgedEdgeworth series cumulant analyses. Clearly, the abridged Edgeworth series convergesmuch faster than the full Edgeworth series. In fact, it overshoots the Fokker–Planck limitbeyond the eight order cumulant analysis. Unfortunately, aside from being a generalizationof Crandall’s non-Gaussian closure [14], we have no theoretical justification for theabridging Edgeworth series as was done in this paper. Yet, the rapid approach towardsFokker–Planck variance is a practical justification, echoing the adage ‘‘the test of thepudding is in the taste’’.

In conclusion, we have summarized in Table 1 the successive variance estimates up to thetwelfth order abridged Edgeworth cumulant analysis. Here, the positive/negative percentageerror denotes under/overestimation of the Fokker–Planck value. Also included in Table 1are dimensionless cumulants up to the eight order analysis, from which Edgeworthdistributions are constructed with cn=kn /sn, as shown in Figure 4. Note that the eight orderabridged Edgeworth series captures faithfully the flattened central peak and broadened sidebands of the Fokker–Planck distribution. Clearly, the appearance of slight bimodality isan undesired side effect of the abridged Edgeworth distribution; a price paid for fastconvergence.

T 1

Variance and cumulants of the abridged Edgeworth series analysis

Method X Error (%) k4/s4 k6/s6 k8/s8

Second order 0·5774 14·6 0 0 0Fourth order 0·6325 6·44 −0·5 0 0Sixth order 0·6606 2·28 −0·7084 2·501 0Eighth order 0·6727 0·49 −0·7904 3·485 −12·19Tenth order 0·6767 −0·10 — — —Twelfth order 0·6772 −0·18 — — —Fokker–Planck 0·6760† 0 −0·8120‡ 3·739§ −38·36¶

† 2/G; ‡ (G2/4)−3; § −3G2+30 ¶ −15G2/8+84G2−630, where G=G(1/4)/G(3/4)12·9587.

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Figure 4. The abridged Edgeworth series distribution. ––––, Fokker–Planck distribution; ——, second ordercumulant analysis; —W—, fourth order cumulant analysis; —(—, sixth order cumulant analysis; —w—, eighthorder cumulant analysis.

ACKNOWLEDGMENTS

This work was supported by AFOSR Task 2304N1. The author wishes to thank thereferees for their comments which have led him to carry out higher order cumulant analyses.

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