ASME. Methods and Gaussian Criterion for Statistical Linearization of. Stochastic ... 1 Presented at the 1988 ASME Winter Annual Meeting as Paper No. WA/DSC-9. ...... Bendat, J. S., and Piersol, A. G., 1971, Random Data: Analysis and.
R. J. Chang Associate Professor, Department of Mechanical Engineering, National Chong Kung University, Tainan, Taiwan 70101
G. E. Young Associate Professor, School of Mechanical and Aerospace Engineering, Oklahoma State University Stillwater, OK 74078 Mem. ASME
Methods and Gaussian Criterion for Statistical Linearization of Stochastic Parametrically and Externally Excited Nonlinear Systems1 The methods of Gaussian linearization along with a new Gaussian Criterion used in the prediction of the stationary output variances of stable nonlinear oscillators subjected to both stochastic parametric and external excitations are presented. The techniques of Gaussian linearization are first derived and the accuracy in the prediction of the stationary output variances is illustrated. The justification of using Gaussian linearization a priori is further investigated by establishing a Gaussian Criterion. The non-Gaussian effects due to system nonlinearities and/or large noise intensities in a Duffing oscillator are also illustrated. The validity of employing the Gaussian Criterion test for assuring accuracy of Gaussian linearization is supported by performing the Chi-square Gaussian goodness-of-fit test.
1
Introduction The techniques of using statistical linearization for the dynamic analysis of nonlinear stochastic systems are very effective. Since Booton (1954) and Caughey (1959) independently extended the equivalent linearization method (Krylov and Bogoliubov, 1943) to statistical linearization, the techniques have been widely used for the dynamic analysis and controller design of stochastic externally excited nonlinear systems (Gelb and Vander Velde, 1968). However, when the nonlinear systems are subjected to both stochastic parametric and external excitations, the applications of statistical linearization have not been fully investigated (Young and Chang, 1987b). In the applications of the linearization approach, the simple and useful jointly Gaussian distribution is usually applied to the evaluation of expectations of certain nonlinear functions of states. Although Gelb concluded that there was little difference in a linearizing gain when either a non-Gaussian distributed signal including uniform and triangular distributed signal or a Gaussian distributed signal was passed through a saturating element (Gelb, 1974), a quantitative measure of the effects of large nonlinearities on Gaussian linearization of external noise excited nonlinear systems has not been estab-
1 Presented at the 1988 ASME Winter Annual Meeting as Paper No. WA/DSC-9. Discussion on this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th Street, New York, N.Y. 10017, and will be accepted until two months after final publication of the paper itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by ASME Applied Mechanics Division, June 23, 1987; final revision, April 5, 1988.
lished. Some studies are only to compare the output variance of the linearization solution with that of the exact solution of some solvable nonlinear oscillators (Crandall, 1980; Hedrick, 1980). When nonlinear systems are subjected to both stochastic parametric and external excitations, the density function of states is usually far from a jointly Gaussian distribution. For example, the stationary probability density function of the states of a simple second-order stochastic parametrically and externally excited linear system is not a jointly Gaussian distribution (Dimentberg, 1982). Thus, in the applications of statistical linearization for stochastic parametrically and externally excited stable nonlinear systems, a non-Gaussian density needs to be applied for the linearization or a jointly Gaussian distribution can still be used if the system parameters and noise intensities can satisfy certain Gaussian criteria. In this paper, the methods of Gaussian linearization of stable nonlinear stochastic parametrically and externally exicted systems are presented along with the derivation of a new Gaussian criterion. The techniques of Gaussian linearization of these systems are first derived. A parametric and external noise excited oscillator with cubic nonlinear spring is then selected to illustrate these applications. The non-Gaussian linearization is illustrated through implementation of a nonGaussian density which is derived by using the concepts of equivalent external excitation (Young and Chang, 1987a). By following these concepts and their extensions, a Gaussian criterion is further proposed. A stochastic parametrically and externally excited Duffing oscillator is selected to illustrate the applications of the Gaussian Criterion for the justification of employing Gaussian linearization a priori. The validity of usMARCH 1989, Vol. 56/179
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ing the Gaussian Criterion for assuring accurate prediction of the stationary output variance of a stable nonlinear stochastic system by employing Gaussian linearization is supported by performing the Chi-square Gaussian goodness-of-fit test (Bendat and Piersol, 1971).
The equivalent linearization system (2) is then rewritten as x+(^0 + nx+(fi0
+ u')x=W'
(9)
where fo = E aiSi
2 Statistical Linearization of Stochastic Parametrically and Externally Excited Nonlinear Systems /'o = E ai fi
Consider a second-order stochastic parametrically and externally excited stable nonlinear system described by * + E (ai + ai)Hi(x,x)
m'VH'W]
=W
= 2S5(t-s)
(1)
where the a ; are constants and the a, and W are mutually independent zero-mean Gaussian white noises with intensities E[ai(t)ai(s)]=2qii S(t-s) and E\W (t) W (s)} = 2q„+i „ + 1 8(t—s)), respectively. The equivalent linearization system of (1) is expressed as (2)
* + E («, + «/) (g/X+fiX) = W
where/,- =/• (m20, m02), gi=gj(m20, m02) and mv is defined as the expected value of x\x{ with x{ = x and x2 = x. The equation difference between the left-hand side of (1) and (2) is given by n
;=i
(10)
=
\=i
Eln'(t)ix'(s)]
=
' = E («, + « , ) [ / / , - (gix+fix)]=
2RS(t-s)
= (t^u^Hts) The stationary output variances of x and x of (9) can be derived from the moment equations with diffusional correction term as (Young and Chang, 1987a) m20 = m02/^0
i=i
£ ) («/ + a,-)e(
(3)
(11a)
/=i
By substituting (10) into (11a) and (lib), two simultaneous algebraic equations with unknown m20 and m02 are then derived to yield
where e, is defined as an error and given by e^Hj-giX-fiX (4) Thus, the techniques of statistical linearization which were originally derived for nonlinear systems subjected to stochastic external excitation are now extended to include the parametric noise excited terms if the error of equation difference between (1) and (2) is defined as e,- rather than e ' , which is an error usually defined for nonlinear systems subjected to stochastic external excitation. By following the usual concepts of statistical linearization (Caughey, 1959; Krylov and Bogoliubov, 1943), /,• and g ; are selected such that the mean-square error of e,- will be minimized. Utilizing the following equations, dE[ej] (.5a) =0 dE[ej]
(5b)
=0
and substituting e ; from (4),/) and g,- are then derived as m01E[X'Hj (x, x)] - mnE[X'Hj (x, x)] Ji ~ -m2n mmmn g;
'
n
e
:
(i2g„gf)6U-s)
m20E[x-H,(x,
x)]-mnE[x'H,(x, mmmm-mh
(6a)
(6b)
By applying Ito's differential rule to (1), m20 is derived as (Young and Chang, 1987a)
// =
Si =
m20 E[jfH,(x,x)] m.02
(7) simplified by
(8a) (8b)
(12a)
/n,„ = 0
( E a f) w2o - ( E Q« ft) m2
-2(E?^)(E«///)
m
20~Qn+\
n+l — 0
(126)
Thus, m20 and m02 are readily obtained by solving (12a) and (12b) with f-, and g,- derived by using the Gaussian or nonGaussian density function in (8a) and (8b), respectively. When the Gaussian (non-Gaussian) density function is applied to evaluate /} and g, from (8a) and (8b), the statistical linearization approach is called the Gaussian (non-Gaussian) linearization approach. The non-Gaussian densities of the states of (9) under certain conditions can be derived through the Fokker-PlanckKolmogorov (FPK) equaton. For R = n0S, Dimentberg (1982) has solved the stationary joint probability density of the states to (9) as p(xux2)
x)]
m20 = 2mn If stationary, (6a) and (6b) are further substituting tnn = 0 to yield E[x-H,(x,x)]
.-(£«,/,)
=
(P-\)ofi W/7 0
1 (a + xl/iiQ+xlY
(13)
wherea = 1. From (13), it is seen that the stationary states of xx and x2 are not independent in spite of mn = 0 from (7). Thus, the highly nonGaussian effects in the linearization of stochastic parametrically and externally excited nonlinear systems need to be considered. One approach of the non-Gaussian linearizations of stochastic parametrically and externally excited nonlinear systems as proposed is implemented by first deriving the non-Gaussian probability density from the concepts of equivalent external excitation (Young and Chang, 1987a). Then, the equivalent linear gains/) and g,- are readily obtained by using the density function in (8a) and (8b), respectively. The applications of the non-Gaussian linearization are il-
180/Vol.56, MARCH 1989
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Transactions of the ASME
mw = NON-GAUSSIAN-^
0.06
^
s
T(3/4) T(l/4)
1 k\/2
(18a)
1
^
(186)
4A:,
then (16) becomes 0.04
y^
