Crandall85 NonGaussian Closure Nonlinear Vibration

8/10/2019 Crandall85 NonGaussian Closure Nonlinear Vibration

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W?O- 7462/X5 S3.W + .#I

Pergamon Pres Lid

NON GAUSSIAN CLOSURE TECHNIQUES

FOR STATIONARY RANDOM VIBRATION

STEPHENH.

CRANDALL

Massachusetts Institute of Technology. Cambridge, MA 02139, U.S.A.

(Received 30 April 1984; eceiv ed jar puhli carion 3 September 1984)

Abstract-The classical method of statistical linearization when applied to a non-linear oscillator

excited by stationary wide-band random excitation. can be considered as a procedure in which the

unknown parameters in a Gaussian distribution are evaluated by means of moment identities derived

from the dynamic equation of the oscillator. A systematic extension of this procedure is the method of

non-Gaussian closure in which an increasing number of moment identities are used to evaluate

additional parameters in a family of non-Gaussian response distributions. The method is described

and illustrated by means of examples. Attention is given to the choice of representations of non-

Gaussian distributions and to techniques for generating independent moment identities directly from

the differential equation of the non-linear oscillator. Some shortcomings of the method are pointed

out.

INTRODUCTION

The method of non-Gaussian closure for obtaining statistical information about the

response of non-linear systems to white noise was proposed by Dashevskii and Liptser [ I]

and has been studied by several others [2-61. In outline, the method consists of constructing

a non-Gaussian probability distribution with adjustable parameters for the response and

using moment relations derived from the dynamical equations of the system to obtain

differential or algebraic equations for the unknown parameters. When the parameters are

found the resulting probability distribution is used to provide approximate response

statistics. Here the method is discussed within the framework of the problem of finding the

stationary response of a non-linear oscillator. As a result only algebraic equations are

required to fix the unknown parameters.

NON-LINEAR OSCILLATOR

A mechanical oscillator with displacement x(t) and velocity v(t) = 1 is excited by a force per

unit mass f(t). The equation of motion is taken in the normalized form

2 + qi + g(x) =

tp2f(t)

(1)

where 11s the loss factor and g(s) is the non-linear restoring force per unit mass, here assumed

to be an odd function of s, sufficiently well-behaved to ensure a stationary solution to (1)

when f(r) is an ideal white noise random process with autocorrelation function

R,(7) = E[.fw fu + 7)1 = N7).

(2)

The normalization of (1) is such that when g(.u) is replaced by x the stationary mean square

values of both displacement and velocity are unity: i.e.

E [x2] =

1 and

E [c2] = 1.

As a model for a physically realizable stationary broad-band random process with zero

mean it is more appropriate to consider f(t) to be the limit of a band-limited white noise

whose cut-off frequency increases indefinitely than to considerf(t) to be the formal derivative

of a Wiener Process. The white noise model then provides useful estimates for those statistics

of the physically realizable response which are substantially independent of the cut-off

frequency.

RESPONSE STATISTICS

A family of relations between response statistics for (1) can be derived by using the

elementary properties of stationarity and commutativity of operations OD correlation


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S H CRA\DALL

functions._ Let i(.u) be an arbitrary continuously differentiable function of .x. We multiply

each term of

(1)

by $(.u) and average across the ensemble produced by the process,/ (r) to get

E[rC/rj ] + qE[@] + E[l l /g] = r/1,7E[ j ].

(3)

Now it is known [6] that, for the stationary response. .~(t) and r(t) are statistically

independent. Furthermore, the mean values of both .x(t) and r(r) are zero. This implies that

E[ v] = E[(//(.u)] E[v ] = 0.

The first term on the left of (3) is transformed as follows

E[$i;] =

E[ l :(r + r)],=,

=

E[ 2] =

-E[@?].

The mean square velocity here can be evaluated by returning to

( I ).

multiplying each term by

u(t), and averaging across the ensemble to get

Using the statistical independence of

E[uti ] = + E[u] = 0 w e

reduce (5) to

E[c]

+ E[cg(.u)] = q l*E[~y ] .

i i

x and L and the stationarity of r(t) to set

= I

-

E[uf-].

(6)

When (6) is inserted in (4) and the result substituted in (3) we obtain

which contains two cross-correlations between the excitationf(r) and the response quantities

v(t) and $(_~(t)}. N

ow the response of the oscillator (1) is determined by the past history of

excitation. This means that for an excitation with short correlation time (broad-band

process) these cross-correlations will be small. In fact. whenf(t) is a band-limited white noise

whose cut-off frequency increases without limit we have

lim E [Idif]= 0

lim E[tf ] = ql i2.

(8)

These results are carefully derived in the Appendix of [6]. A simple heuristic argument

follows.

To evaluate the expectations in (8) as the correlation time of the excitation approaches

zero it is sufficient to use response representations which are only valid for very short times.

Now for very short times the response of the oscillator (1) is determined entirely by the inertia

term and is independent of the damping and stiffness. For example, the unit impulse response

k,(t) for the velocity starts with a step function of amplitude q1 2 and the unit impulse

response

k,(t)

for the displacement starts with a ramp whose initial slope is v~.

independently of g(.u). Thus, for short time intervals t - t, the velocity response of (1) can be

approximated by


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Non-Gaussian closure techniques for stationary random vibration 3

When the correlation time of,f(t) is short enough,f(t) will be uncorrelated with the response

at the earlier time to and thus the cross-correlation between the excitation and the response

velocity reduces to

s

-I

=

U@R,(We.

0

As the excitation autocorrelation function approaches the limit (2) the integral (10)

approaches the value h,(O + ) = q I2 which verifies the second of (8). Note that the eveness of

the autocorrelation function during the limiting process accounts for the fact that only unit

area under (2) is employed.

In a similar manner it can be shown that E[x(t)f(r)] = h,(O+) = 0. Finally, since a

continuously differential function $(x(t)}

can be approximated by A + Bx over a sufficiently

short interval t - to it follows tha.t E [t,b(x)f(r)] = 0 in the limit as_f(r) approaches the white

noise of (2). This verifies the first of (8). When the limiting values of (8) are inserted in (7) we

obtain the following important relation between response statistics for the oscillator (1)

E[ WMx l = E[ W dxl .

(11)

Here

x(r)

is the stationary response of (1) to the white noise of (2) and Ii/(x) is an arbitrary

differentiable function of x.

RELATION TO STATISTICAL LINEARIZATION

In the well-known method of statistical linearization [7 J, the non-linear oscillator (1) is

replaced by the equivalent linear oscillator described by

f + vi + IX =

/2f(r)

(12)

where iL s fixed by requiring the mean square of the equation difference 4 = Lx - g(x) to be

minimum. From

dE [42]/aA = 0

there follows

AE[x] = E[xg(x)]

(13)

which can be interpreted as a relation between response statistics for the non-linear oscillator

(I ). The left side of (13) can be evaluated by calculating the mean square response of the

equivalent linear oscillator (12). IJsing [8] we find

E[x] = f

(14)

which when inserted in (13) produces

E[xg(x)] = 1.

(15)

The identity (15) provides a constraint on the probability distribution of x. In the usual

application of statistical linearization [8] the distribution is assumed to be Gaussian with

zero mean and unknown variance (14). Application of the constraint (15) leads to a non-

linear equation for the determination of the variance.

The link between this procedure and the general family (11) of relations between response

statistics is provided by the fact that the identity (15) is just the special case of

(11)

for which

I,I?(s) s simply x itself. The derivation of (11) shows that the identity (15) holds exactly for the

nonlinear system: not just for the equivalent linear system.


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S H. CRA~DALL

NON-GAUSSIAN CLOSURE

The method of non-Gaussian closure can be viewed as an extension of the statistical

linearization procedure in that a probability distribution with II unknown parameters is

selected and n independent constraints. obtained from ( 11) by selecting II different functions

$;(s), are used to fix unknown parameters. If the non-Gaussian distribution is expected to be

close to Gaussian a convenient choice for an even probability density function is the Gram-

Charlier expansion

pls) =

e-- ?a2

iI

hca)

2

1 + i SH,,(.Y,)

,[=A n

1

(16)

where the summation is over even integer values of II and where H,,I< Jstands for the Hermits

polynomials

H, = 1

HI =

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Crandall85 NonGaussian Closure Nonlinear Vibration