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The exact steady-state solution of a class of non-linear stochastic systems
hr. I. Non-Linear Mechanics. Vol. 17. No. 3. pp. 137-142. 1982. Printed in Great Britain.
002~7462/82/030137-06$03.00/0 Pergamon Press Ltd.
THE EXACT STEADY-STATE SOLUTION OF A CLASS OF NON-LINEAR STOCHASTIC SYSTEMS T. K. CAUGHEY and FAI MA California Institute of Technology, Pasadena, CA 91125, U.S.A. Abstract-In this paper exact steady state solutions are constructed for a class of non-linear systems subjected to stochastic excitation. The results are then applied to both classical and non-classical oscillator problems. INTRODUCTION
In the past twenty years the response of non-linear dynamical systems subjected to stochastic excitation has been extensively studied. The diffusion processes approach to this problem leads to the Kolmogorov equations, which have up to now been explicitly solved only in a few simple cases. The exact solutions for linear problems can be obtained by a variety of methods [ 1,2], whereas in the non-linear case only some specific systems have so far been exactly solved [3-51. A number of approximate techniques have been developed for the treatment of non-linear stochastic problems, and many of them have been adopted in computers in recent years [6-8]. The use of various approximate methods will become increasingly popular in the next decade as computing costs decrease. The purpose of this paper, however, is to construct the exact steady-state solution of a class of non-linear dynamical systems subjected to stochastic excitation. DERIVATION
OF THE
MAIN RESULTS
During the course of a continuing investigation of the stochastic stability of a class of hypothetical elastic structures, the steady-state response of the following system is desired f+
(
x2+2$-
x2 :2fi)2M
+2$3$x$
= k(t)
(1)
where G(r) is the formal derivative of a Wiener process w(t) with zero mean and E[+(r,)ti(f2)] = 2D6(tl - t2). Since the above equation is not weakly non-linear, many of the well-known approximate techniques do not apply. If we define y = l/2 1*, we can express (1) in the form
(2) By inspection, one energy integral H(x,y) of the conservative oscillator Z+2x3+ti2=0 x2+ 2i2
(3)
is given by H(x,y) = x4 + i4 + x2i2. It can now be seen that the damping factor and the
restoring force function in (2) are related to H by 2x2+8y-L= x*+4y
J&-H,, hY
T. K. CAUGHEY and FAI
138
MA
In order to solve for the steady-state response of (l), we begin with the seemingly more difficult problem of determining the stationary solution of the following equation: f+
(HJ(H)+y o;+!& ) >
w(t)
(4)
Y
where y = (l/2).+*, H(x,y) and j(H) are functions with continuous second-order derivatives and H,H, r 0. In addition, there exists an Ho > 0 such that j(H) 2 0 if H > Ho, and j-'-&O
as H + ~0.
The previous assumptions on H and j will be needed to show that the steady-state solutions of (4) that we soon construct belong to the class of well-behaved solutions [5,9], and for this purpose they are sufficient but not necessary. We can interpret equation (4) either in the sense of Ito or in the sense of Stratonovich [lo], since in the present case the so-called Wong and Zakai correction terms to the drift vector are identically zero. Using the above remark, it may be shown that the assumptions on H and j need only hold on every finite domain [5]. Let p(x,Mlx&) be the transition probability density function of the process (4), satisfying the following Fokker-Planck equation:
ap
T-
--X
.*a
ax + 2
KHyfW) -@li+~]p+D$
with p(x,x,O~xo,xo)= 8(x - x0). If the steady-state
(5)
p(x,.i) = lim p(x,i,t~x,,i,) I-= exists, then it is the positive solution of the linear partial differential equation
-&2+-I?ax
ai
density
KW(H) -
(6)
Instead of invoking an existence theorem, we shall construct an exact solution of (6). We observe that p(x,i) is certainly a solution of (6) if it satisfies
-i$+-+($+o
(7)
Y
(8) Let p(x,i) = p(x,i)H,
then equation (8) becomes _
$PH, ax
If %O, equation (9) upon expansion,
+x
a(~Hd_ ay
o.
(9)
becomes
.-H$f+H$=O
(10)
The exact slcady-stutc
solution
of non-linear
stochastic
which may now be solved by the method of characteristics of (10) is
systems
[l 11. The general solution
J? = 4(H) and p = t$(H)H, where do is an arbitrary 00, equation (8) implies
function.
(11)
Since p and its first derivatives
vanish as Ix]+ ]rij -*
(HYm) - Z)ip +!$=
0.
(12)
Y
Substituting
139
(11) into (12), we have
iH;[g+f(H)$] =0. It foilows by integration
where A is a normalizing
(13)
that
constant.
Hence the steady state density is given by
It is easy to see that the above function satisfies all the requirements for a probability density and it satisfies (6). From the exponential nature of p in (14), we can readily check that the solution (14) belongs to the class of well-behaved stationary solutions [5,9]. Since it has been shown that the well-behaved solution of a stationary Fokker-Planck equation is unique [93, we arrive at the following result. Theorem. The solution (14) is the unique steady-state solution of (4). Generalization of the above theorem is possible in several directions. Consider, for example the non-linear coupled system ii +
(
Hy,f(H)-
Hyiy’ Qi,+$
H
Yl >
&t)
(15)
Yi
where i=1,2,...,n,yi=
ix!, E[d Wi(t)
dwi(t)] = 246, dt, and H(x,, . . -xm Yl9 * - - ,Y”)
is an energy integral of the following system of conservative
oscillators
&+x0
(16)
HYi for i = 1,2, . . . ,n. In addition, it is required that
HYi= G(xl,x2,. . . ,x,,) for
all i.
and A-r=/_l/lexp(-rf(e)dC)G(
)dxI...dx,dir...dlS,.
T. K. CAUGHEY
140
and
FM
MA
By applying the same constructive technique we have presented conclude that the steady state response of (15) is p(x,, . . . ,xn, i,, . . . ,;i,) = A exp (-161f(nd5)G(x,,x,,...*.). Returning to equation (l), if we identify steady state probability density is given by
(17)
H = x’+4y2 + 2x2y and f = 1 then the
PW) = A exp (- x4 - ? - xV)(~~ + 2i.2) where A is the normalizing
to system (IS), we
(18)
constant.
EXAMPLESANDFURTHERDISCUSSION We shall apply the theorem established in the last section to some random oscillator problems, mentioning possible extensions when appropriate. 1 The stationary
Example
response
of a+&t+x=
b+(t)
(19)
is p(x,x) = A exp [- (p/2D)(i2 + x2)], where A is the normalizing H = ii2 +:x2 and f = (/3/D).
constant. In this case
Example 2 The steady state solution of
(20)
where (/3/D) > 2, is given by
(21)
Example 3 If H is of the form
H = ably where y = $ i2. Then with A a normalizing
+ B(x)
(22)
constant,
~(x,i)=Aexp(-(BID)lo~f(Z)dZ).(~) is the steady state solution of f + @(x)f(H)i
+p
‘(x) + fa’(x)? a(x)
= k(C)
(23)
The exact steady-state solution of non-linear stochastic systems
141
Example 4 If H is of the form H =
+I - In(l + rl31) + y* OX I g(rl) dri
tdl= ‘&Y);
Y = 5,
(24)
G(x) = J,’ g(q).
Thus p(x,f) =cA exp t- (i?/D)lil- (/3ID)‘G(x)] is the steady state probabihty density function for 2 + psgn i +
(1 +&)g(x)
= Hi(t).
(251
En vefupe sta tisticf
Let us define an envelope process n(t) by H(y,x) = H(O,o). The probability density of the process a(t) is, after some manipulation, given by p,(a)
=4$l
ja(4’)p(x,i)
=
dx dl
AH,(O,a) exp (-~‘lOrl~(S)d~)T(a)
where A is a constant
(26)
is the period of the deterministic
oscillator.
(27) The expected number of zero crossings per unit time is m
v; = =
I I
ip(O,i) di
0
m
0
where v(a) = (l/T(a))
pAa)v(a) da
is the frequency of the oscillator (27).
In this paper a class of random systems has been considered, and exact steady state solutions derived. The result has been applied to some classical osciflator problems. If the steady state density of an oscillator has been calculated, it may be possible to obtain the approximate non-stationary response by perturbation techniques 191.This subject will be taken up in a future paper. Acknowtedgemcnf-We would like to thank Mr. S. Tanveer for a helpful discussion.
REFERENCES 1. hi. C. Wang and G. E. Uhlenbcck, On the theory of the Brownian motion II. Rev. mod. Phys. If. 323-342 (1945). 2. A. T. Bharucha-Reid, ffemenfs of the Theory of Markov Processes and Their Applications. McGrawHill, New York (1960). 3. T. K. Caughqr, Nonlinear theory of random vibrations. Aduances in Applied Mechanics, Vol. 2. Academic Press, New York (1971).
142
T. K. CAUGHEY and FAI MA
of dynamical systems under random perturbations. SIAM Rev. 17. 605-640 (1975). T. K. Caughey and Fai Ma, The steady-state response of a class of dynamical systems to stochastic excitation. 1. oppl. Meek (submitted for publication). P-T D. Spanos and W. D. Iwan, On the -existence of uniqueness of solutions generated by equivalent linearization. Znt. Z. Non-Zineur Me& 13. 71-78 (1978). J. P. Johnson and R. A. Scott, Extension of ei‘genfunction expansion solutions of a Fokker-Planck equation 1. First order system. Znt. J. Non-linear Me&. 14, 315-324 (1979). J. P. Johnson and R. A. Scott, Extension of eigenfunction-expansion solutions of a Fokker-Planck equation 2. Second order system. Znt. 1. Non-linear Mech. 15.41-56 (1980). T. K. Caughey and H. J. Payne, On the response of a class of self excited oscillators to stochastic excitation. Znt. J. Non-linear Mech. 2, 125-151 (1967). L. Arnold, Sfochastic Differential Equations: Theory and Applications. Wiley-Interscience. New York (1974). P. R. Garabedian. Partial ZIiflerential Equations. Wiley, New York (1964).
4. D. Ludwig, Persistence
5. 6. 7. 8. 9. IO.
II.
Zusamnenfassung: In dieser Arbeit werden die genauen stationgren Lijsungen fiireine Klasse nichtlinearer Systeme unter statistischer Erregung aufgebaut. Die Ergebnisse werden dann auf klassische und nichtklassische Schwingungsproblemeangewendet.
002~7462/82/030137-06$03.00/0 Pergamon Press Ltd.
THE EXACT STEADY-STATE SOLUTION OF A CLASS OF NON-LINEAR STOCHASTIC SYSTEMS T. K. CAUGHEY and FAI MA California Institute of Technology, Pasadena, CA 91125, U.S.A. Abstract-In this paper exact steady state solutions are constructed for a class of non-linear systems subjected to stochastic excitation. The results are then applied to both classical and non-classical oscillator problems. INTRODUCTION
In the past twenty years the response of non-linear dynamical systems subjected to stochastic excitation has been extensively studied. The diffusion processes approach to this problem leads to the Kolmogorov equations, which have up to now been explicitly solved only in a few simple cases. The exact solutions for linear problems can be obtained by a variety of methods [ 1,2], whereas in the non-linear case only some specific systems have so far been exactly solved [3-51. A number of approximate techniques have been developed for the treatment of non-linear stochastic problems, and many of them have been adopted in computers in recent years [6-8]. The use of various approximate methods will become increasingly popular in the next decade as computing costs decrease. The purpose of this paper, however, is to construct the exact steady-state solution of a class of non-linear dynamical systems subjected to stochastic excitation. DERIVATION
OF THE
MAIN RESULTS
During the course of a continuing investigation of the stochastic stability of a class of hypothetical elastic structures, the steady-state response of the following system is desired f+
(
x2+2$-
x2 :2fi)2M
+2$3$x$
= k(t)
(1)
where G(r) is the formal derivative of a Wiener process w(t) with zero mean and E[+(r,)ti(f2)] = 2D6(tl - t2). Since the above equation is not weakly non-linear, many of the well-known approximate techniques do not apply. If we define y = l/2 1*, we can express (1) in the form
(2) By inspection, one energy integral H(x,y) of the conservative oscillator Z+2x3+ti2=0 x2+ 2i2
(3)
is given by H(x,y) = x4 + i4 + x2i2. It can now be seen that the damping factor and the
restoring force function in (2) are related to H by 2x2+8y-L= x*+4y
J&-H,, hY
T. K. CAUGHEY and FAI
138
MA
In order to solve for the steady-state response of (l), we begin with the seemingly more difficult problem of determining the stationary solution of the following equation: f+
(HJ(H)+y o;+!& ) >
w(t)
(4)
Y
where y = (l/2).+*, H(x,y) and j(H) are functions with continuous second-order derivatives and H,H, r 0. In addition, there exists an Ho > 0 such that j(H) 2 0 if H > Ho, and j-'-&O
as H + ~0.
The previous assumptions on H and j will be needed to show that the steady-state solutions of (4) that we soon construct belong to the class of well-behaved solutions [5,9], and for this purpose they are sufficient but not necessary. We can interpret equation (4) either in the sense of Ito or in the sense of Stratonovich [lo], since in the present case the so-called Wong and Zakai correction terms to the drift vector are identically zero. Using the above remark, it may be shown that the assumptions on H and j need only hold on every finite domain [5]. Let p(x,Mlx&) be the transition probability density function of the process (4), satisfying the following Fokker-Planck equation:
ap
T-
--X
.*a
ax + 2
KHyfW) -@li+~]p+D$
with p(x,x,O~xo,xo)= 8(x - x0). If the steady-state
(5)
p(x,.i) = lim p(x,i,t~x,,i,) I-= exists, then it is the positive solution of the linear partial differential equation
-&2+-I?ax
ai
density
KW(H) -
(6)
Instead of invoking an existence theorem, we shall construct an exact solution of (6). We observe that p(x,i) is certainly a solution of (6) if it satisfies
-i$+-+($+o
(7)
Y
(8) Let p(x,i) = p(x,i)H,
then equation (8) becomes _
$PH, ax
If %O, equation (9) upon expansion,
+x
a(~Hd_ ay
o.
(9)
becomes
.-H$f+H$=O
(10)
The exact slcady-stutc
solution
of non-linear
stochastic
which may now be solved by the method of characteristics of (10) is
systems
[l 11. The general solution
J? = 4(H) and p = t$(H)H, where do is an arbitrary 00, equation (8) implies
function.
(11)
Since p and its first derivatives
vanish as Ix]+ ]rij -*
(HYm) - Z)ip +!$=
0.
(12)
Y
Substituting
139
(11) into (12), we have
iH;[g+f(H)$] =0. It foilows by integration
where A is a normalizing
(13)
that
constant.
Hence the steady state density is given by
It is easy to see that the above function satisfies all the requirements for a probability density and it satisfies (6). From the exponential nature of p in (14), we can readily check that the solution (14) belongs to the class of well-behaved stationary solutions [5,9]. Since it has been shown that the well-behaved solution of a stationary Fokker-Planck equation is unique [93, we arrive at the following result. Theorem. The solution (14) is the unique steady-state solution of (4). Generalization of the above theorem is possible in several directions. Consider, for example the non-linear coupled system ii +
(
Hy,f(H)-
Hyiy’ Qi,+$
H
Yl >
&t)
(15)
Yi
where i=1,2,...,n,yi=
ix!, E[d Wi(t)
dwi(t)] = 246, dt, and H(x,, . . -xm Yl9 * - - ,Y”)
is an energy integral of the following system of conservative
oscillators
&+x0
(16)
HYi for i = 1,2, . . . ,n. In addition, it is required that
HYi= G(xl,x2,. . . ,x,,) for
all i.
and A-r=/_l/lexp(-rf(e)dC)G(
)dxI...dx,dir...dlS,.
T. K. CAUGHEY
140
and
FM
MA
By applying the same constructive technique we have presented conclude that the steady state response of (15) is p(x,, . . . ,xn, i,, . . . ,;i,) = A exp (-161f(nd5)G(x,,x,,...*.). Returning to equation (l), if we identify steady state probability density is given by
(17)
H = x’+4y2 + 2x2y and f = 1 then the
PW) = A exp (- x4 - ? - xV)(~~ + 2i.2) where A is the normalizing
to system (IS), we
(18)
constant.
EXAMPLESANDFURTHERDISCUSSION We shall apply the theorem established in the last section to some random oscillator problems, mentioning possible extensions when appropriate. 1 The stationary
Example
response
of a+&t+x=
b+(t)
(19)
is p(x,x) = A exp [- (p/2D)(i2 + x2)], where A is the normalizing H = ii2 +:x2 and f = (/3/D).
constant. In this case
Example 2 The steady state solution of
(20)
where (/3/D) > 2, is given by
(21)
Example 3 If H is of the form
H = ably where y = $ i2. Then with A a normalizing
+ B(x)
(22)
constant,
~(x,i)=Aexp(-(BID)lo~f(Z)dZ).(~) is the steady state solution of f + @(x)f(H)i
+p
‘(x) + fa’(x)? a(x)
= k(C)
(23)
The exact steady-state solution of non-linear stochastic systems
141
Example 4 If H is of the form H =
+I - In(l + rl31) + y* OX I g(rl) dri
tdl= ‘&Y);
Y = 5,
(24)
G(x) = J,’ g(q).
Thus p(x,f) =cA exp t- (i?/D)lil- (/3ID)‘G(x)] is the steady state probabihty density function for 2 + psgn i +
(1 +&)g(x)
= Hi(t).
(251
En vefupe sta tisticf
Let us define an envelope process n(t) by H(y,x) = H(O,o). The probability density of the process a(t) is, after some manipulation, given by p,(a)
=4$l
ja(4’)p(x,i)
=
dx dl
AH,(O,a) exp (-~‘lOrl~(S)d~)T(a)
where A is a constant
(26)
is the period of the deterministic
oscillator.
(27) The expected number of zero crossings per unit time is m
v; = =
I I
ip(O,i) di
0
m
0
where v(a) = (l/T(a))
pAa)v(a) da
is the frequency of the oscillator (27).
In this paper a class of random systems has been considered, and exact steady state solutions derived. The result has been applied to some classical osciflator problems. If the steady state density of an oscillator has been calculated, it may be possible to obtain the approximate non-stationary response by perturbation techniques 191.This subject will be taken up in a future paper. Acknowtedgemcnf-We would like to thank Mr. S. Tanveer for a helpful discussion.
REFERENCES 1. hi. C. Wang and G. E. Uhlenbcck, On the theory of the Brownian motion II. Rev. mod. Phys. If. 323-342 (1945). 2. A. T. Bharucha-Reid, ffemenfs of the Theory of Markov Processes and Their Applications. McGrawHill, New York (1960). 3. T. K. Caughqr, Nonlinear theory of random vibrations. Aduances in Applied Mechanics, Vol. 2. Academic Press, New York (1971).
142
T. K. CAUGHEY and FAI MA
of dynamical systems under random perturbations. SIAM Rev. 17. 605-640 (1975). T. K. Caughey and Fai Ma, The steady-state response of a class of dynamical systems to stochastic excitation. 1. oppl. Meek (submitted for publication). P-T D. Spanos and W. D. Iwan, On the -existence of uniqueness of solutions generated by equivalent linearization. Znt. Z. Non-Zineur Me& 13. 71-78 (1978). J. P. Johnson and R. A. Scott, Extension of ei‘genfunction expansion solutions of a Fokker-Planck equation 1. First order system. Znt. J. Non-linear Me&. 14, 315-324 (1979). J. P. Johnson and R. A. Scott, Extension of eigenfunction-expansion solutions of a Fokker-Planck equation 2. Second order system. Znt. 1. Non-linear Mech. 15.41-56 (1980). T. K. Caughey and H. J. Payne, On the response of a class of self excited oscillators to stochastic excitation. Znt. J. Non-linear Mech. 2, 125-151 (1967). L. Arnold, Sfochastic Differential Equations: Theory and Applications. Wiley-Interscience. New York (1974). P. R. Garabedian. Partial ZIiflerential Equations. Wiley, New York (1964).
4. D. Ludwig, Persistence
5. 6. 7. 8. 9. IO.
II.
Zusamnenfassung: In dieser Arbeit werden die genauen stationgren Lijsungen fiireine Klasse nichtlinearer Systeme unter statistischer Erregung aufgebaut. Die Ergebnisse werden dann auf klassische und nichtklassische Schwingungsproblemeangewendet.