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Prediction of the response of non-linear oscillators under stochastic parametric and external excitations
Int. J Non-Lmcor Mechanic. Rintcd in Great Bntan
Vol. 22. No 2, pp. ISI-160,
0021&7462,‘87 93.a) c 00
1987
C? 1987 Pergamon Journals Ltd.
PREDICTION OF THE RESPONSE OF NON-LINEAR OSCILLATORS UNDER STOCHASTIC PARAMETRIC AND EXTERNAL EXCITATIONS G. E. YOUNG and R. J. CHANG School
of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK
74078, U.S.A. (Received 3 February
1986; received for publication
14 October 1986)
Abstract-The method of equivalent external excitation is derived to predict the stationary variances of the states of non-linear oscillators subjected to both stochastic parametric and external excitations. The oscillator is interpreted as one which is excited solely by an external zero-mean stochastic process. The Fokker-Planck-Kolmogorov equation is then applied to solve for the density functions and match the stationary variances of the states. Four examples which include polynomial, non-polynomial, and Duffing type non-linear oscillators are used to illustrate this approach. The validity of the present approach is compared with some exact solutions and with Monte Carlo simulations.
1. INTRODUCTION
The prediction of the response of non-linear systems subjected to stochastic parametric and external excitations is a central problem in the field of random vibration. In contrast to the problem of the stationary response of non-linear oscillators subjected to external stochastic excitation which has been widely studied [l], the stationary response of nonlinear oscillators under both stochastic parametric and external excitations has been studied only for few non-linear oscillators [2,3]. Actually, in the prediction of the response of these oscillators (very few non-linear oscillators can be solved exactly by using the FokkerPlanck-Kolmogorov (FPK) equation) even the response of a simple non-linear oscillator such as a Duffing system can be obtained only by certain approximate methods under restricted assumptions. Several approximate approaches which have been applied to the problems of non-linear oscillators subjected to external stochastic excitation such as the Gaussian statistical linearization method, perturbation method [l], Gram-Charlier expansion method [4], and the approximate method to solve the FPK equation [l] are based on the implicit or explicit assumption that the stationary probability densities of the states of the non-linear oscillators can be approximated as jointly Gaussian. Unfortunately, this assumption implies that the non-linear oscillators are excited by weak noise intensity and/or the oscillators can be classified as weak non-linear oscillators. For the stochastic parametrically and externally excited non-linear oscillators, the Stratonovich method [S] can be applied to solve for the distributions of amplitude and phase processes, although it is restricted to lightly damped oscillators. Recently, Wu and Lin applied the cumulantneglect closure method [3,7] to solve for the response of stochastic parametrically excited oscillators. This method is very general; however, it will become rather difficult when the higher order cumulants need to be retained. For example, the retaining of the cumulants up to the fourth order used to solve a second-order non-linear oscillator usually requires the solution of ten simultaneous non-linear algebraic equations if one can not eliminate certain variables by trivial substitution. Furthermore, it is formidable to apply the cumulantneglect method when the non-linearity is not of a polynomial form [a]. In this paper, a new approximate method using equivalent external excitation has been developed to predict the stationary response of non-linear oscillators subjected to both stochastic parametric and external excitations when the response of the corresponding systems subjected to external stochastic excitation can be solved by using the FPK equation. Implementing this method allows the straightforward extension of existing solutions of the response of non-linear oscillators subjected to external stochastic excitation to predict the response of non-linear oscillators subjected to both stochastic parametric and external excitations. 151
G. E.
152 2. DERIVATION
and R. J.
YOUNG
OF THE EQUIVALENT
CHANG
EXTERNAL
EXCITATION
METHOD
The derivation of the equivalent external excitation method is illustrated by using the following oscillator. Consider a second-order linear oscillator with stochastic parametric and external excitations 1 + (50 + (‘)a + (po + p’)x = w’
(1)
where co and cl0 are some constants. t’,~’ and w’ are independent zero-mean Gaussian white noise processes with covariances E[5’(t)t’(s)] = Zq&(t-s), E[p’(r)p’(s)] = 2q, ,6(t-s), and E[w’(r)w’(s)] = 2q,,6(r-s), respectively. On introducing x1 = x and x2 = 1, the state equation with diffusional correction term is
x2 --VOX,
-
(Co
-
dr +
1
4221x2
1
0
0
0 (2)
-X,
1
-x2
where dw, = p’dr, dw, = t’dr, and dwj = w’dr and wI, w2, and wj are Wiener processes with independent increments. The propagation of moments can be derived by using Ito’s differential rule [7] ml0
=
m0l
=
mol --Om10
fin,,
=
2m,,
ti 11
-
m02
kit,2
=
2C-fi0mll
-
-
((0
pomzo
-
+ -
422)mol
(q22
(50
-
-
50)mi
1
422)mo2
+
q1 lm20 + 422m02 + q331
(3)
where mij is defined as the expected value of xix{ with i,j = 0,1,2. If one rewrites (1) as .il+(oi+pox=
-/ix-<‘i+W
(4)
and interprets the noise excitation term on the right hand side as an equivalent external excitation with intensity 2qllmzo + 2q,,m,, + 2q,,, then the equivalent state equation with diffusional correction term can be expressed in the form X2
-pox,
1 [I dr+
- (50 -
422)X2
0
(5)
dw
1
where ti is a Gaussian white noise with variance E[ti(r)G(s)] = (2ql,m20 + 2q,,m,, + 2q3J8(r - s). By inspection, (5) maintains the same equations of the propagation of moments up to the second order as (3). However, the oscillator with stochastic parametric and external excitations now becomes an equivalent oscillator subjected to external stochastic excitation. Accordingly, the well-known FPK equation can be applied to solve for the joint probability density function of x1 and x2 without any restrictions on the intensity of the equivalent external noise. The stationary FPK equation to (5) is given by c71 dP -XZQPOXlG+
JP
450 -
q22)xzP zx2
+
1,2Q
d2P_
0
ax:
where Q = 2qllmzo + 2q2,mo2 + 2q,, and P is the stationary joint probability density of x1 and x2. The solution of (6) is obtained as
f’(-u,,.x2) = NexpC-(tO - q22)(poxf + d/Q1
(7)
153
Prediction of the response of non-linear oscillators
with to - qz2 > 0, for convergent solution, and N is a normalized constant. It follows that the stationary probability densities of xi and x2 are expressed as
PC4 = JP~(S~ - qz2)/nQ expC-(to- qz2hx~lQ1 Pbd = ~~expC-(Co
-
(8)
q&lQl.
The next step is to solve (8) by matching the variances of xi and x1 as follows:
s 30
m20
=
Pb,)x:dx,
= Q/2(50 - qzzho
-02
m02
Finally, 2q22m02
=
(9)
P0m20.
the solutions of mzo and mo2 are given by substituting 2q,, and solving the algebraic equation (9) to yield
Q = 2q,,m20
+
+
m20
=
mo2ho= qJ3/(50~o - qll
-
2qz2po).
(10)
Specifically, when q1 1 = poqtz, then mzo = q33/c1d50 - h2). If one defines a = q3&l19 b = ((ro/qz2) + 1)/2, it follows that the stationary variance of xi is r/2(/? - 2). Dimentberg [2] has solved the exact stationary probability densities of xi and x2 to (1) using the FPK equation
P(xl) =
feg!$!$J,,+xf)-‘@-
112)
(11)
where I(.) is the gamma function. From (1 l), the variance of xi can be integrated to obtain a/2(/I - 2). The variance of xi is exactly the same as the results obtained by using the equivalent external excitation method. Also, for the linear oscillator (1) the stationary variance of xi can be derived from (3) by setting all derivative terms equal to zero. Again, one derives mzo = mo2/po = a/2(/3 - 2). In summary, the present method can be applied by the following procedure. (1) Replace the stochastic parametrically and externally excited non-linear oscillator by an equivalent one with diffusional correction term which is excited solely by external excitation. The variance of the equivalent external excitation equals the variance of the original external noise plus an equivalent variance which is attributed to the parametrically excited terms. The equivalent variance is obtained by taking the summation of the expected values of the square of the parametrically excited terms multiplied by the corresponding variances of the excited coefficients. (2) Obtain the stationary probability density functions of the states of the equivalent non-linear oscillator from Step 1 by using the Fokker-Planck-Kolmogorov equation. (3) Match the expected value of the square of each of the parametrically excited terms in the oscillator by taking the expected value of those terms through the probability density functions of the states obtained from Step 2. (4) Solve the simultaneous algebraic or integral equations by iterative methods. This step is exactly the same as the procedure when one applies the statistical linearization method [8]. (5) Find the stationary variances of the states of the oscillator by substituting the probability density functions which have been derived from Steps 1 to 4.
3. APPLICATIONS
The following examples follow the same noise notations and definitions as given above. The first example selected is a non-polynomial type non-linear oscillator which can not be solved by using the cumulant-neglect method.
G. E.
154
and R. J. CHANC
YOUNG
Example 1 -d
A second-order non-linear < x < d is given by [6]
oscillator with non-linear
spring F(x) = C tan(xx/2d) for
f + &)a + F(x) = 6.
(12)
The stationary probability density of x is obtained through the FPK equation as P(x) = N exp[(8d5&/n2q,Jln(cos(rcx/2d))].
(13)
If the non-linear oscillator with parametric and external noise excitations is in the form 2 + &?;- +
(1 + /.f’)F(x) = w’
(14)
one can derive the stationary variance of x by the following procedure. Steps 1 and 2. Substitute 2q,, with Q = 2q,, + 2q11E[F(x)2] then, P(x) = N exp[(8d&,C/nQ)ln(cos(nx/2d))].
(15)
Step 3. Match the expected value of F(x)~ by writing
E[F(x)~]
= Q Tqzq” IL
= NC 2/~,tan2(~)exp[(~)ln(cos$)]dx.
(16)
Step 4. Since (16) is an integral equation with unknown Q, the value of Q can be obtained by using numerical iteration. Step 5. Substituting Q into (15), one obtains the probability density function of x. Thus, the stationary variance of x is derived by using P(x) and integrating
s d
m20
=
x2P(x)dx.
(17)
-d
This example illustrates that the existing solution of the probability density function of oscillators subjected to external noise excitation can be extended to predict the response of the non-linear oscillators subjected to both parametric and external noise excitations by using the equivalent external excitation approach. The second example chosen for the present investigation is a non-linear oscillator considered by Dimentberg [Z]. Example 2
Consider the non-linear oscillator given by 2 + (if0 + 5’)i +
pi.(x2 + a2//&J+ (PO+ p’)x = w’.
Step 1. The equivalent oscillator with diffusional correction 2 + (50 - q&i
(18)
term is given by
+ pi(x2 + a2/&J + pox = w
(19)
with Q = %,,m20 + Q22m02 + %33. Step 2. The stationary FPK equation is expressed as i?P -x2
dx,
+
POX1
a’ dx,
+
L{[(Co-qz2)+p(x:+$)]x2P+ dx2
l,ZQg}=O.
(20)
155
Prediction of the response of non-linear oscillators
From (20), the solution of the stationary probability density function of P is [9]
P(x,,x,)
2PH2 QPO
- 2(50 -h&H
= Nexp
Q
1
with H
(21)
Step 3. The matching conditions can be applied directly at this step; however, it is easier to apply if one considers the propagation of H. The probability density function of H is in the form
p(H)
exp
= N’
-
2(‘oi q22)H _ 321,
(22)
0
Introducing a = pIQpo, b = po(to - q22)/2p, one obtains 2(50 -
N’ = l/
q22)
Q
H-*H2
QPO
where erfi(b,/%) = 1 - erflbfi) is the complementary seen that the probability density functions of x2 and G value of H and using mo2 = pomzo, one derives
mo2
=
E[H]
Hf’WdH =
=
1
dH
error function. From (21), it is are equal. Taking the expected
&;--;&, - b.
(24)
Step 4. mo2 is readily given by the numerical solution of (24). Since the exact solution of m20 can be derived through the FPK equation provided that the noise intensities qI1 and q22 satisfy qll = poq22, one chooses the appropriate values of noise intensities and parameters to compare several approaches. For 2q 11 = 2q2, = 0.1 and p = p. = to = 1.0, (24) becomes e-O.451/Q
m20
JZrrieerfc(0.672/&)
=
- o*475
with Q = 0.2mzo + 2q,,. (25) The cumulant-neglect method up to the fourth order is derived by solving nine simultaneous non-linear algebraic equations and gives [3] 192m:, + 61.2m:, + (2.55 - 28(2q,,))m,,
- 1.5(2q,,)
= 0.
(26)
By applying the Gaussian closure method, one derives 18 - J324 m20
=
+ 3160(2q,,) -158 ’
(27)
The exact solution of m20 is given by [2] _9 5 +
1 m20=z
[
.
(100(2q,,))‘0°(2933)-9.5e-100(2q33)
woo(2q,,)
-
9.5, W2qA)
1
(28)
G. E. YOUNG and R. J. CHANG
156
where I-$,x) is the incomplete gamma function. The comparisons of (25), (26), (27) and (28) with varying sj = 2q,, are obtained by numerical solution and given in Fig. 1. It is observed that the present approach is in close agreement with the exact solution even when the intensity of the external noise is of the same order as that of parametric noise. The next example selected is governed by a cubic non-linear spring oscillator and can be easily extended to a higher order non-linear spring oscillator. By using the cumulantneglect method such an increase will increase the difficulty in derivation because more cumulant terms are required to express higher order moments. However, this is not a concern when using the equivalent external excitation method. Example
3
A non-linear oscillator with cubic non-linearity 1 + ((0 + <‘)i +
is described by
(/lo + $)x3 = w’.
Step 1. The equivalent oscillator with diffusional correction 2 + (&J - q&i with
Q
=
2qllm60
+ +
pox3 =
h&no,
(29) term is written as
w
+
(30)
2h.
Step 2. The probability density functions of states x, and x2 can be derived through the FPK equation to yield [9]
JYx1) =
Ph)
2k”4 r( h
=p(--k,x;),kl
=
(to - 42&o
> ()
2Q
=
(31)
Step 3. The matching conditions for m60 and moz are expressed as
m60
W&:dx,
=
mo2
=
= 0.254VQAS’o
P(x,)x:dx,
= Q/2(<,
- qzd~o)3’2
(32)
- q2d
(33)
0.300r
0.225 -
%-
0.150
iii
0.075
I
o.oool 0.00
I
0.25
0.50
J
0.75
1.00
%
Fig. 1. The prediction of the stationary variance of displacement with varied intensity s3 by equation (25) and several other approaches.
external
excitation
157
Prediction of the response of non-linear oscillators
Step 4. From (32) and (33), substituting Q and eliminating mdO,one derives
Step 5. mzo is derived by substituting P(x,) and integrating as
m20
=
P(xr)x:dx,
= 0.676(mo2/~o)1’2.
(35)
The comparisons of (35) with a 500 run Monte Carlo simulation of (29) by choosing to = 1.0, p. = 5.0 and with varying qa3/q11 and qj3/qz2 are illustrated in Figs 2, 3 and 4. It is seen that these figures show good agreement in the stationary response of mzo between the present method and Monte Carlo simulations. Since the stationary response of an oscillator will be dominated by the parametric noise excitation when the intensity of the external noise is much less than that of the parametric noise, the concept of using equivalent external excitation to approximate the parametric noise excitation as an equivalent external noise excitation should induce more error in the prediction of the stationary response. Thus, the inaccuracy in the prediction of the stationary mzo by using the present approach
Fig. 2. The prediction of the stationary variance of displacement by equation (35) and Monte Carlo simulation using Zq,, = 0.0, Zq,, = 0.1, 2q,, = 0.1, i.e. q3,/qZL = 1.0.
0.16
i
0.0
2.5
5.0
7.5
10.0
TIME
Fig. 3. The prediction of the stationary variance of displacement by equation (35) and Monte Carlo simulation using 2q,, = 0.0, Zq,, = 0.5, Zq,, = 0.1, i.e. q3,/qz1 = 0.2.
158
G. E. YOUNG and R. J. CHANG
0.00.
0.0
2.5
5.0
I
7.5
10.0
TIME
Fig. 4. The prediction of the stationary variance of displacement by equation (35) and Carlo simulation using 2q,, = 5.0,Zq,, = 0.0,Zq,, = 0.5,i.e. q,Jq,, = 0.1.
Monte
is expected when the intensity of the damping noise is higher than that of the external noise as shown in Fig. 3. However, the present approach still predicts very accurate stationary response of mzO even when the intensity of the spring noise is ten times that of the external noise as shown in Fig. 4. The last example chosen for discussion is the Duffing oscillator. Example 4
A Duffing oscillator with parametric and external noise excitations is expressed as i + ((0 + 5’)i + x + (po +
$)x3 = w’.
Step 1. The equivalent oscillator with diffusional correction
i +(
Step 2. The probability
Q = &?,lm60
(36)
term is
+ x + pox3 = w + %mo2
(37)
+ %.
density functions of states x, and x2 are given by [l]
WI) =
+ k$)]
exp -k(x; [
+ 9
)I
10
s -P P(x2) =
exp[ -P(xT
J$xp(-
kxi), k =
dx,
50-
422
>
()
Q
’
(38)
Steps 3 and 4. The matching condition of mo2 is used to yield
mo2 =
&,b:dx, = Q/2(1;, - q2d
By using the matching condition of m60 and substituting k and Q, one derives
(39)
Prediction
of the response
of non-linear
oscillators
159
Step 5. The stationary mzo is obtained by
[exp[
-(i&)(x:+%$)]x:dxi (411
m20 = [exp[
-(&)(x:
+ q)]dxi
*
The comparison of (41) with a 500 run Monte Carlo simulation of (36) by choosing to = 1.0, p. = 5.0 and 2q,, = 5.0,q22 = 0.0,2q,, = 1.0is shown in Fig. 5. It is seen that the accurate prediction of the stationary variance of mto is obtained by using the present approach. When the intensity of the external noise is at least the same order as that of the spring noise, say qaJ/qll= 1.0, better results are obtained as can be expected. 4. CONCLUSIONS
A new approach by using equivalent external excitation to predict the stationary variances of the states of non-linear oscillators subjected to both stochastic parametric and external excitations has been introduced in this paper. By using the concept of equivalent external excitation, one can easily extend the existing solution of the response of non-linear oscillators subjected to external excitation to the problem of parametric and external excitations. Since the present approach is to reformulate the stochastic parametric excitation to an equivalent external excitation and apply the FPK equation, this approach can be applied to non-polynomial non-linearities and strong noise excitations in non-linear oscillators. The validity of the prediction of stationary variance by using the present approach has been compared with some exact solutions and Monte Carlo simulations. The results show that better prediction is obtained when the intensity of the external excitation is at least the same order as that of the parametric excitation. When the parametric excitation enters through the spring term, it was shown through an example that the present approach is valid even when the intensity of the spring noise is ten times that of the external excitation. By using the present approach, the computational effort is almost the same as the statistical linearization method and the restricted assumption that the probability densities of the stationary states are jointly Gaussian need not be invoked. Here, the only assumption of the equivalent external excitation approach is based on the validity of using the equivalent external excitation to maintain the same propagation of moments up to the second order. Acknowledgement-This work was supported and is gratefully acknowledged.
by the National
Science
Foundation
under
grant
MEA-8307509
0.16
i
Fig. 5. The prediction of the stationary variance of displacement by equation Carlo simulation using 2q ,, = 5.0, 2q,, = 0.0, 2q,, = 1.0, i.e. q3,/qll
(41) and Monte = 0.2.
160
G. E. YOUNG and R. J. CHANG
REFERENCES 1. N. C. N&am, Inrroduction to Random Vibrations. The MIT Press, Cambridge, MA (1983). 2. M. F. Dimentberg An exact solution to a certain non-linear random vibration problem. Jnt. J. Non-linear Mech. 17, 231-236 (1982). 3. Y. F. Wu and Y. K. Lin, Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations. fnt. J. Non-lineur kech. 19, 349-362 (1984). 4. S. A. Assaf and L. D. Zirkle, Approximate analysis of non-linear stochastic systems. fnt. J. Control 23, 477492 (1976). 5. R. L. Stratonovich, Topics in the Theory of Random Noise, Vols I and II. Gordon & Breach, New York (1963). 6. S. C. Liu, Solutions of Fokker-Planck equation with applications in non-linear random vibration. The BeL/ Sys. tech. J. 48, pt2, 2031-2051 (1969). 7. T. T. Soong, Random Diflerentia[ Equations in Science and Engineering. Academic Press, New York (1973). 8. J. K. Hedrick Non-linear
and A. V. Arslan,
System
Analysis
9. T. K. Caughey, Non-linear Press. New York (1971).
Non-linear
analysis of rail vehicle forced lateral response and stability. In Vol. 2. A.S.M.E., New York (1980). of random vibrations. In Advances in Applied Mechanics, Vol. 11.Academic
and Synthesis,
theory
Vol. 22. No 2, pp. ISI-160,
0021&7462,‘87 93.a) c 00
1987
C? 1987 Pergamon Journals Ltd.
PREDICTION OF THE RESPONSE OF NON-LINEAR OSCILLATORS UNDER STOCHASTIC PARAMETRIC AND EXTERNAL EXCITATIONS G. E. YOUNG and R. J. CHANG School
of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK
74078, U.S.A. (Received 3 February
1986; received for publication
14 October 1986)
Abstract-The method of equivalent external excitation is derived to predict the stationary variances of the states of non-linear oscillators subjected to both stochastic parametric and external excitations. The oscillator is interpreted as one which is excited solely by an external zero-mean stochastic process. The Fokker-Planck-Kolmogorov equation is then applied to solve for the density functions and match the stationary variances of the states. Four examples which include polynomial, non-polynomial, and Duffing type non-linear oscillators are used to illustrate this approach. The validity of the present approach is compared with some exact solutions and with Monte Carlo simulations.
1. INTRODUCTION
The prediction of the response of non-linear systems subjected to stochastic parametric and external excitations is a central problem in the field of random vibration. In contrast to the problem of the stationary response of non-linear oscillators subjected to external stochastic excitation which has been widely studied [l], the stationary response of nonlinear oscillators under both stochastic parametric and external excitations has been studied only for few non-linear oscillators [2,3]. Actually, in the prediction of the response of these oscillators (very few non-linear oscillators can be solved exactly by using the FokkerPlanck-Kolmogorov (FPK) equation) even the response of a simple non-linear oscillator such as a Duffing system can be obtained only by certain approximate methods under restricted assumptions. Several approximate approaches which have been applied to the problems of non-linear oscillators subjected to external stochastic excitation such as the Gaussian statistical linearization method, perturbation method [l], Gram-Charlier expansion method [4], and the approximate method to solve the FPK equation [l] are based on the implicit or explicit assumption that the stationary probability densities of the states of the non-linear oscillators can be approximated as jointly Gaussian. Unfortunately, this assumption implies that the non-linear oscillators are excited by weak noise intensity and/or the oscillators can be classified as weak non-linear oscillators. For the stochastic parametrically and externally excited non-linear oscillators, the Stratonovich method [S] can be applied to solve for the distributions of amplitude and phase processes, although it is restricted to lightly damped oscillators. Recently, Wu and Lin applied the cumulantneglect closure method [3,7] to solve for the response of stochastic parametrically excited oscillators. This method is very general; however, it will become rather difficult when the higher order cumulants need to be retained. For example, the retaining of the cumulants up to the fourth order used to solve a second-order non-linear oscillator usually requires the solution of ten simultaneous non-linear algebraic equations if one can not eliminate certain variables by trivial substitution. Furthermore, it is formidable to apply the cumulantneglect method when the non-linearity is not of a polynomial form [a]. In this paper, a new approximate method using equivalent external excitation has been developed to predict the stationary response of non-linear oscillators subjected to both stochastic parametric and external excitations when the response of the corresponding systems subjected to external stochastic excitation can be solved by using the FPK equation. Implementing this method allows the straightforward extension of existing solutions of the response of non-linear oscillators subjected to external stochastic excitation to predict the response of non-linear oscillators subjected to both stochastic parametric and external excitations. 151
G. E.
152 2. DERIVATION
and R. J.
YOUNG
OF THE EQUIVALENT
CHANG
EXTERNAL
EXCITATION
METHOD
The derivation of the equivalent external excitation method is illustrated by using the following oscillator. Consider a second-order linear oscillator with stochastic parametric and external excitations 1 + (50 + (‘)a + (po + p’)x = w’
(1)
where co and cl0 are some constants. t’,~’ and w’ are independent zero-mean Gaussian white noise processes with covariances E[5’(t)t’(s)] = Zq&(t-s), E[p’(r)p’(s)] = 2q, ,6(t-s), and E[w’(r)w’(s)] = 2q,,6(r-s), respectively. On introducing x1 = x and x2 = 1, the state equation with diffusional correction term is
x2 --VOX,
-
(Co
-
dr +
1
4221x2
1
0
0
0 (2)
-X,
1
-x2
where dw, = p’dr, dw, = t’dr, and dwj = w’dr and wI, w2, and wj are Wiener processes with independent increments. The propagation of moments can be derived by using Ito’s differential rule [7] ml0
=
m0l
=
mol --Om10
fin,,
=
2m,,
ti 11
-
m02
kit,2
=
2C-fi0mll
-
-
((0
pomzo
-
+ -
422)mol
(q22
(50
-
-
50)mi
1
422)mo2
+
q1 lm20 + 422m02 + q331
(3)
where mij is defined as the expected value of xix{ with i,j = 0,1,2. If one rewrites (1) as .il+(oi+pox=
-/ix-<‘i+W
(4)
and interprets the noise excitation term on the right hand side as an equivalent external excitation with intensity 2qllmzo + 2q,,m,, + 2q,,, then the equivalent state equation with diffusional correction term can be expressed in the form X2
-pox,
1 [I dr+
- (50 -
422)X2
0
(5)
dw
1
where ti is a Gaussian white noise with variance E[ti(r)G(s)] = (2ql,m20 + 2q,,m,, + 2q3J8(r - s). By inspection, (5) maintains the same equations of the propagation of moments up to the second order as (3). However, the oscillator with stochastic parametric and external excitations now becomes an equivalent oscillator subjected to external stochastic excitation. Accordingly, the well-known FPK equation can be applied to solve for the joint probability density function of x1 and x2 without any restrictions on the intensity of the equivalent external noise. The stationary FPK equation to (5) is given by c71 dP -XZQPOXlG+
JP
450 -
q22)xzP zx2
+
1,2Q
d2P_
0
ax:
where Q = 2qllmzo + 2q2,mo2 + 2q,, and P is the stationary joint probability density of x1 and x2. The solution of (6) is obtained as
f’(-u,,.x2) = NexpC-(tO - q22)(poxf + d/Q1
(7)
153
Prediction of the response of non-linear oscillators
with to - qz2 > 0, for convergent solution, and N is a normalized constant. It follows that the stationary probability densities of xi and x2 are expressed as
PC4 = JP~(S~ - qz2)/nQ expC-(to- qz2hx~lQ1 Pbd = ~~expC-(Co
-
(8)
q&lQl.
The next step is to solve (8) by matching the variances of xi and x1 as follows:
s 30
m20
=
Pb,)x:dx,
= Q/2(50 - qzzho
-02
m02
Finally, 2q22m02
=
(9)
P0m20.
the solutions of mzo and mo2 are given by substituting 2q,, and solving the algebraic equation (9) to yield
Q = 2q,,m20
+
+
m20
=
mo2ho= qJ3/(50~o - qll
-
2qz2po).
(10)
Specifically, when q1 1 = poqtz, then mzo = q33/c1d50 - h2). If one defines a = q3&l19 b = ((ro/qz2) + 1)/2, it follows that the stationary variance of xi is r/2(/? - 2). Dimentberg [2] has solved the exact stationary probability densities of xi and x2 to (1) using the FPK equation
P(xl) =
feg!$!$J,,+xf)-‘@-
112)
(11)
where I(.) is the gamma function. From (1 l), the variance of xi can be integrated to obtain a/2(/I - 2). The variance of xi is exactly the same as the results obtained by using the equivalent external excitation method. Also, for the linear oscillator (1) the stationary variance of xi can be derived from (3) by setting all derivative terms equal to zero. Again, one derives mzo = mo2/po = a/2(/3 - 2). In summary, the present method can be applied by the following procedure. (1) Replace the stochastic parametrically and externally excited non-linear oscillator by an equivalent one with diffusional correction term which is excited solely by external excitation. The variance of the equivalent external excitation equals the variance of the original external noise plus an equivalent variance which is attributed to the parametrically excited terms. The equivalent variance is obtained by taking the summation of the expected values of the square of the parametrically excited terms multiplied by the corresponding variances of the excited coefficients. (2) Obtain the stationary probability density functions of the states of the equivalent non-linear oscillator from Step 1 by using the Fokker-Planck-Kolmogorov equation. (3) Match the expected value of the square of each of the parametrically excited terms in the oscillator by taking the expected value of those terms through the probability density functions of the states obtained from Step 2. (4) Solve the simultaneous algebraic or integral equations by iterative methods. This step is exactly the same as the procedure when one applies the statistical linearization method [8]. (5) Find the stationary variances of the states of the oscillator by substituting the probability density functions which have been derived from Steps 1 to 4.
3. APPLICATIONS
The following examples follow the same noise notations and definitions as given above. The first example selected is a non-polynomial type non-linear oscillator which can not be solved by using the cumulant-neglect method.
G. E.
154
and R. J. CHANC
YOUNG
Example 1 -d
A second-order non-linear < x < d is given by [6]
oscillator with non-linear
spring F(x) = C tan(xx/2d) for
f + &)a + F(x) = 6.
(12)
The stationary probability density of x is obtained through the FPK equation as P(x) = N exp[(8d5&/n2q,Jln(cos(rcx/2d))].
(13)
If the non-linear oscillator with parametric and external noise excitations is in the form 2 + &?;- +
(1 + /.f’)F(x) = w’
(14)
one can derive the stationary variance of x by the following procedure. Steps 1 and 2. Substitute 2q,, with Q = 2q,, + 2q11E[F(x)2] then, P(x) = N exp[(8d&,C/nQ)ln(cos(nx/2d))].
(15)
Step 3. Match the expected value of F(x)~ by writing
E[F(x)~]
= Q Tqzq” IL
= NC 2/~,tan2(~)exp[(~)ln(cos$)]dx.
(16)
Step 4. Since (16) is an integral equation with unknown Q, the value of Q can be obtained by using numerical iteration. Step 5. Substituting Q into (15), one obtains the probability density function of x. Thus, the stationary variance of x is derived by using P(x) and integrating
s d
m20
=
x2P(x)dx.
(17)
-d
This example illustrates that the existing solution of the probability density function of oscillators subjected to external noise excitation can be extended to predict the response of the non-linear oscillators subjected to both parametric and external noise excitations by using the equivalent external excitation approach. The second example chosen for the present investigation is a non-linear oscillator considered by Dimentberg [Z]. Example 2
Consider the non-linear oscillator given by 2 + (if0 + 5’)i +
pi.(x2 + a2//&J+ (PO+ p’)x = w’.
Step 1. The equivalent oscillator with diffusional correction 2 + (50 - q&i
(18)
term is given by
+ pi(x2 + a2/&J + pox = w
(19)
with Q = %,,m20 + Q22m02 + %33. Step 2. The stationary FPK equation is expressed as i?P -x2
dx,
+
POX1
a’ dx,
+
L{[(Co-qz2)+p(x:+$)]x2P+ dx2
l,ZQg}=O.
(20)
155
Prediction of the response of non-linear oscillators
From (20), the solution of the stationary probability density function of P is [9]
P(x,,x,)
2PH2 QPO
- 2(50 -h&H
= Nexp
Q
1
with H
(21)
Step 3. The matching conditions can be applied directly at this step; however, it is easier to apply if one considers the propagation of H. The probability density function of H is in the form
p(H)
exp
= N’
-
2(‘oi q22)H _ 321,
(22)
0
Introducing a = pIQpo, b = po(to - q22)/2p, one obtains 2(50 -
N’ = l/
q22)
Q
H-*H2
QPO
where erfi(b,/%) = 1 - erflbfi) is the complementary seen that the probability density functions of x2 and G value of H and using mo2 = pomzo, one derives
mo2
=
E[H]
Hf’WdH =
=
1
dH
error function. From (21), it is are equal. Taking the expected
&;--;&, - b.
(24)
Step 4. mo2 is readily given by the numerical solution of (24). Since the exact solution of m20 can be derived through the FPK equation provided that the noise intensities qI1 and q22 satisfy qll = poq22, one chooses the appropriate values of noise intensities and parameters to compare several approaches. For 2q 11 = 2q2, = 0.1 and p = p. = to = 1.0, (24) becomes e-O.451/Q
m20
JZrrieerfc(0.672/&)
=
- o*475
with Q = 0.2mzo + 2q,,. (25) The cumulant-neglect method up to the fourth order is derived by solving nine simultaneous non-linear algebraic equations and gives [3] 192m:, + 61.2m:, + (2.55 - 28(2q,,))m,,
- 1.5(2q,,)
= 0.
(26)
By applying the Gaussian closure method, one derives 18 - J324 m20
=
+ 3160(2q,,) -158 ’
(27)
The exact solution of m20 is given by [2] _9 5 +
1 m20=z
[
.
(100(2q,,))‘0°(2933)-9.5e-100(2q33)
woo(2q,,)
-
9.5, W2qA)
1
(28)
G. E. YOUNG and R. J. CHANG
156
where I-$,x) is the incomplete gamma function. The comparisons of (25), (26), (27) and (28) with varying sj = 2q,, are obtained by numerical solution and given in Fig. 1. It is observed that the present approach is in close agreement with the exact solution even when the intensity of the external noise is of the same order as that of parametric noise. The next example selected is governed by a cubic non-linear spring oscillator and can be easily extended to a higher order non-linear spring oscillator. By using the cumulantneglect method such an increase will increase the difficulty in derivation because more cumulant terms are required to express higher order moments. However, this is not a concern when using the equivalent external excitation method. Example
3
A non-linear oscillator with cubic non-linearity 1 + ((0 + <‘)i +
is described by
(/lo + $)x3 = w’.
Step 1. The equivalent oscillator with diffusional correction 2 + (&J - q&i with
Q
=
2qllm60
+ +
pox3 =
h&no,
(29) term is written as
w
+
(30)
2h.
Step 2. The probability density functions of states x, and x2 can be derived through the FPK equation to yield [9]
JYx1) =
Ph)
2k”4 r( h
=p(--k,x;),kl
=
(to - 42&o
> ()
2Q
=
(31)
Step 3. The matching conditions for m60 and moz are expressed as
m60
W&:dx,
=
mo2
=
= 0.254VQAS’o
P(x,)x:dx,
= Q/2(<,
- qzd~o)3’2
(32)
- q2d
(33)
0.300r
0.225 -
%-
0.150
iii
0.075
I
o.oool 0.00
I
0.25
0.50
J
0.75
1.00
%
Fig. 1. The prediction of the stationary variance of displacement with varied intensity s3 by equation (25) and several other approaches.
external
excitation
157
Prediction of the response of non-linear oscillators
Step 4. From (32) and (33), substituting Q and eliminating mdO,one derives
Step 5. mzo is derived by substituting P(x,) and integrating as
m20
=
P(xr)x:dx,
= 0.676(mo2/~o)1’2.
(35)
The comparisons of (35) with a 500 run Monte Carlo simulation of (29) by choosing to = 1.0, p. = 5.0 and with varying qa3/q11 and qj3/qz2 are illustrated in Figs 2, 3 and 4. It is seen that these figures show good agreement in the stationary response of mzo between the present method and Monte Carlo simulations. Since the stationary response of an oscillator will be dominated by the parametric noise excitation when the intensity of the external noise is much less than that of the parametric noise, the concept of using equivalent external excitation to approximate the parametric noise excitation as an equivalent external noise excitation should induce more error in the prediction of the stationary response. Thus, the inaccuracy in the prediction of the stationary mzo by using the present approach
Fig. 2. The prediction of the stationary variance of displacement by equation (35) and Monte Carlo simulation using Zq,, = 0.0, Zq,, = 0.1, 2q,, = 0.1, i.e. q3,/qZL = 1.0.
0.16
i
0.0
2.5
5.0
7.5
10.0
TIME
Fig. 3. The prediction of the stationary variance of displacement by equation (35) and Monte Carlo simulation using 2q,, = 0.0, Zq,, = 0.5, Zq,, = 0.1, i.e. q3,/qz1 = 0.2.
158
G. E. YOUNG and R. J. CHANG
0.00.
0.0
2.5
5.0
I
7.5
10.0
TIME
Fig. 4. The prediction of the stationary variance of displacement by equation (35) and Carlo simulation using 2q,, = 5.0,Zq,, = 0.0,Zq,, = 0.5,i.e. q,Jq,, = 0.1.
Monte
is expected when the intensity of the damping noise is higher than that of the external noise as shown in Fig. 3. However, the present approach still predicts very accurate stationary response of mzO even when the intensity of the spring noise is ten times that of the external noise as shown in Fig. 4. The last example chosen for discussion is the Duffing oscillator. Example 4
A Duffing oscillator with parametric and external noise excitations is expressed as i + ((0 + 5’)i + x + (po +
$)x3 = w’.
Step 1. The equivalent oscillator with diffusional correction
i +(
Step 2. The probability
Q = &?,lm60
(36)
term is
+ x + pox3 = w + %mo2
(37)
+ %.
density functions of states x, and x2 are given by [l]
WI) =
+ k$)]
exp -k(x; [
+ 9
)I
10
s -P P(x2) =
exp[ -P(xT
J$xp(-
kxi), k =
dx,
50-
422
>
()
Q
’
(38)
Steps 3 and 4. The matching condition of mo2 is used to yield
mo2 =
&,b:dx, = Q/2(1;, - q2d
By using the matching condition of m60 and substituting k and Q, one derives
(39)
Prediction
of the response
of non-linear
oscillators
159
Step 5. The stationary mzo is obtained by
[exp[
-(i&)(x:+%$)]x:dxi (411
m20 = [exp[
-(&)(x:
+ q)]dxi
*
The comparison of (41) with a 500 run Monte Carlo simulation of (36) by choosing to = 1.0, p. = 5.0 and 2q,, = 5.0,q22 = 0.0,2q,, = 1.0is shown in Fig. 5. It is seen that the accurate prediction of the stationary variance of mto is obtained by using the present approach. When the intensity of the external noise is at least the same order as that of the spring noise, say qaJ/qll= 1.0, better results are obtained as can be expected. 4. CONCLUSIONS
A new approach by using equivalent external excitation to predict the stationary variances of the states of non-linear oscillators subjected to both stochastic parametric and external excitations has been introduced in this paper. By using the concept of equivalent external excitation, one can easily extend the existing solution of the response of non-linear oscillators subjected to external excitation to the problem of parametric and external excitations. Since the present approach is to reformulate the stochastic parametric excitation to an equivalent external excitation and apply the FPK equation, this approach can be applied to non-polynomial non-linearities and strong noise excitations in non-linear oscillators. The validity of the prediction of stationary variance by using the present approach has been compared with some exact solutions and Monte Carlo simulations. The results show that better prediction is obtained when the intensity of the external excitation is at least the same order as that of the parametric excitation. When the parametric excitation enters through the spring term, it was shown through an example that the present approach is valid even when the intensity of the spring noise is ten times that of the external excitation. By using the present approach, the computational effort is almost the same as the statistical linearization method and the restricted assumption that the probability densities of the stationary states are jointly Gaussian need not be invoked. Here, the only assumption of the equivalent external excitation approach is based on the validity of using the equivalent external excitation to maintain the same propagation of moments up to the second order. Acknowledgement-This work was supported and is gratefully acknowledged.
by the National
Science
Foundation
under
grant
MEA-8307509
0.16
i
Fig. 5. The prediction of the stationary variance of displacement by equation Carlo simulation using 2q ,, = 5.0, 2q,, = 0.0, 2q,, = 1.0, i.e. q3,/qll
(41) and Monte = 0.2.
160
G. E. YOUNG and R. J. CHANG
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and A. V. Arslan,
System
Analysis
9. T. K. Caughey, Non-linear Press. New York (1971).
Non-linear
analysis of rail vehicle forced lateral response and stability. In Vol. 2. A.S.M.E., New York (1980). of random vibrations. In Advances in Applied Mechanics, Vol. 11.Academic
and Synthesis,
theory