Stochastic Averaging of Oscillators Excite by Colore Gaussian Processes 3 R. Valery Roy Department of Mechanical Engineering, University of Delaware, Newark, Delaware 976. Abstract: The metho of stochastic averaging has been evelope an applie in the past mainly base on Stratonovich-Khasminskii theorem. We examine in this paper the application of this metho in the case of arbitrary colore Gaussian excitations which can be consiere as output of multiimensional linear lters to white Gaussian noise. The metho use is base on a perturbation theoretic approach of the Fokker-Planck-Kolmogorov equation which governs the response probability ensity function. First, for oscillators with linear elastic forces an non-parametric excitation, it is shown that, to leaing orer of perturbation, the results obtaine matches those erive by application of Stratonovich-Khasminskii theorem in the case of broa-ban excitation. Then, more general results are erive for nearly amiltonian systems perturbe by parametric excitations of uncorrelate colore noises. t is shown that the state probability ensity function is governe by a reuce equation in the \slow" amiltonian variable only, which epens on a number of parameters characterizing the colore noise excitations. Several examples are given for illustration. As a preliminary to these theoretical evelopments, the problem of etermining the eigenfunctions an eigenvalues of the Fokker-Planck operator is aresse for a general class of linear multiimensional systems.. ntrouction n the quest for approximate solution techniques for ranom vibration problems, the metho of stochastic averaging has proven to be a powerful analytic tool. First propose by Stratonovich [] in 963, various mathematical justications, erivations an extensions have appeare in the literature. See for example Khasminskii [2] (966), Roberts [3] (978), hu [4] (983), Lin [5] (986), hu an Lin [6] (99), Re-orse an Spanos [7] (992). Comprehensive reviews attesting to the success of stochastic averaging have been written by Roberts an Spanos [8] (986), an by hu [9] (988). The metho is applicable to systems with light amping forces an excite by broa-ban, quasi-white ranom processes. The stanar stochastic 3 nternational Journal of Non-Linear Mechanics, 29, pp. 463-475, 994.
averaging metho consists of rst transforming the original equations of motion into equations for slowly varying processes an rapily varying processes. Equations for the \slow" variables only can then be obtaine by subsequent averaging, using the Stratonovich-Khasminskii limit theorem; a corresponing Fokker-Planck- Kolmogorov (FPK) equation can be written for the joint probability ensity function (p..f.) of the \slow" variables. ence, the \fast variables" are eliminate. For systems with a single egree of freeom, the approach generally leas to a more tractable one-imensional problem in the energy variable. Another version of stochastic averaging is base on a perturbation theoretic approach to the Fokker-Planck- Kolmogorov equation. A perturbation solution of the bounary value problem governing the state p..f. is sought in the asymptotic limit of small friction forces an small noise excitations. For vanishing value of the perturbation parameter, the system is conservative. Such a perturbation scheme has been use by Blankenship an Papanicolaou [](978), orsthemke an Lefever [] (98), Klosek-Dygas, Matkowsky, an Schuss [2, 3] (988). The purpose of this paper is two-fol: rst to introuce this technique to the engineering mechanics community as an alternative to Stratonovich-Khasminskii theorem, an secon to generalize the results of Klosek-Dygas, Matkowsky, an Schuss [2, 3] to the case of colore Gaussian noise excitations which can be treate as outputs of multiimensional linear lters to white noise. Since the temporal evolution of the system is non-markovian in the case of non-white noise excitation, the response of the combine systemplus-lter must be consiere. Central to the erivations of this paper is the expansion of each perturbation coecient along the eigenfunctions of the Fokker-Planck operator associate with the linear lter. ence in x2, we aress the problem of etermining these eigenfunctions an the corresponing eigenvalues. t is shown that for general multiimensional linear systems satisfying the conition of etaile balance, analytical expressions can be foun in terms of multivariate ermite polynomials. Note that the results of x2 are useful in themselves for purposes other than those of x3 an x4, e. g., for approximate solutions of the FPK equation base on a Bubnov-Galerkin scheme. n x3, the metho is rst applie to the class of oscillators with linear restoring force an non-parametric excitation. The unperturbe system correspons to the classical unampe, unforce linear oscillator. The problem is solve by rst transforming the joint process (x; _x) into amplitue an phase processes (a; ). t is shown that, to leaing orer of perturbation, the results obtaine are ientical to the stanar averaging of the amplitue process by use of Stratonovich-Khasminskii theorem: only the value of the excitation power spectrum at the oscillator's natural frequency is taken into account. n x4, we treat the case of nearly amiltonian systems perturbe by weak friction forces an by weak colore parametric noise excitations. We follow here the methoology of [2] by using a irect perturbation of the FPK equation in the (x; y; u) variables without the use of a transformation to slowly varying processes. The state probability ensity function is shown to be governe, to leaing orer of perturbation, by an FPK equation in the amiltonian variable only. This equation epens on parameters characteristic of the colore noise excitations. Several examples are given. n particular, the results of references [2] an [4, 5] are recovere. 2. Preliminaries: The Eigenvalues an Eigenfunctions of the Fokker-Planck Operator for Linear Systems Consier the general stochastic linear system governe by the equation _u(t) = Au(t) + W(t); () where u = (u ; : : : ; u N ) T. The excitation W(t) is an N-imensional, zero-mean, white Gaussian noise such that hw(t + )W T (t)i = B(). B is assume to be an N 2 N symmetric, non-negative matrix of constant coecients b ij. The N 2 N matrix A of constant, real or complex elements a ij is assume to have a complete set of eigenvalues ; : : : ; N with corresponing eigenvectors e ; : : : ; e N. The eigenvalues ( i ) are assume to all have a negative real part: <( i ) < ; i = ; : : : ; N: (2) 2
With these assumptions, Eq.() has a stationary, zero-mean, Gaussian, Markov solution process whose corresponing probability ensity function is given by p s (u) = N exp( 2 ut K u u) (3) where the normalization constant is N = (2) N=2 (et K u ) =2. The matrix K u stans for the stationary response covariance matrix huu T i an is known to be solution of the matrix equation A K u + K u A T = B: (4) We also enote by U the N 2 N matrix whose columns are e ; : : : ; e N an such that U A U = D = iag( ; : : : ; N ): (5) To Eq.() correspons two linear ierential operators, the (forwar) Fokker-Planck operator Lu an its formal ajoint (backwar Kolmogorov operator) L 3 u whose expressions are given by: Lu = @ ui a ij u j + 2 b ij@ 2 u iu j ; (6) L 3 u = a ij u j @ ui + 2 b ij@ 2 u iu j ; (7) where the usual summation convention is use for repeate inices. We seek in this section analytical expressions for the eigenfunctions 9 (u) an 9 3 (u) an the corresponing eigenvalues an of, respectively, Lu an L 3 u ; that is, we seek the solutions of the equations Lu9 = 9 ; L 3 u9 3 = 9 3 : (8) n general, the two sets (9 ) an (9 3 ) are unrelate, except for the conition of biorthogonality u9 (u)9 3 R n (u) = ; if 6= ; (9) in the case of a iscrete spectrum of eigenvalues. There exists however a conition for which the two operators Lu an L 3 u are relate to one another: the conition of etaile balance. A stationary system is sai to satisfy the principle of etaile balance, if the joint probability ensity of the transition from (u; t) to (u ; t + ) is equal to that of the time reverse transition from (u ; t) to (u; t + ). The reverse transition is such that the variable u = (u ; : : : ; u N ) T is transforme into the variable u = ( u ; : : : ; N u N ) T, where i = for an even variable u i (e.g., a isplacement variable), an i = for an o variable u i (e.g., a velocity variable) uner time reversal. enotes the N 2 N iagonal matrix iag( ; : : : ; N ). See Gariner [6] (983) for more etail. Thus, etaile balance is equivalent to, for Markov systems, p(u ; ju; )p s (u) = p(u; ju ; )p s (u ) () an p s (u) = p s (u): () t can be shown [6] that, for the general linear system given by Eq.(), these conitions are equivalent to the Onsager relations given by (AK u ) = (AK u ) T ; (2) B = B: (3) Uner these conitions, if 9 3 is an eigenfunction of L 3 u corresponing to the eigenvalue, then 9(u) = p s (u) 9 3 (u) is an eigenfunction of Lu corresponing to the same eigenvalue. See [6]. Thus, the problem is reuce to that of solving the equation L 3 u9 3 = 9 3. 3
The solution of the ajoint eigenvalue problem is simplie by using the change of variable u = Uv which iagonalizes the matrix A. The equation L 3 u9 3 (u) = 9 3 (u) becomes U li a ij U jk v k @ vl 9 3 + 2 U li b ij U kj @2 v kv l 9 3 = 9 3 (4) that is, using U A U = D, k v k @ vk 9 3 + 2 ~ b kl @ 2 v kv l 9 3 = 9 3 (5) where the ~ b ij 's are the elements of the symmetric matrix e B = U BU T. The solutions of Eq.(5) are foun by examining the set of multivariate ermite polynomials of the variable v as ene by Appell an Kampe e Feriet (see Erelyi [7], 953): m;:::;mn (v) = () m exp( 2 vt Cv)@ m v m ;:::;v m N exp( N 2 vt Cv) (6) G m;:::;mn (v) = () m exp( 2 vt Cv)@ m w m ;:::;w m N exp( N 2 vt Cv) (7) where w = Cv an C is a positive enite, symmetric N 2 N matrix. Each polynomial G m;:::;mn (v) of orer m = m + + m N is known to satisfy the following system of partial ierential equations (with no summation on the inex k): v k @ vk G l= kl @ 2 v kv l G = m k G; k = ; : : : ; N (8) where the ij are the elements of C. ence, G m;:::;m N (v) also satises the following linear combination of these equations: k v k @ vk G k kl @ 2 v kv l G = m k k G: (9) Thus, one can reaily see that G m;:::;m N (v) will satisfy equation (5) if k kl + l lk = ~ b kl, that is, if the matrix C is chosen to satisfy D C + C D = e B (2) whose unique solution is C = U T K u U: (2) Thus, C is the covariance matrix K v of the process v = U u whose normalize p..f. is given by 9 (u) = N exp( 2 ut K u u) = N exp( 2 vt K v v) = N exp( 2 wt K v w): (22) with w = K v v = K v U u. Therefore, the eigenfunctions 9(u) are given by with corresponing eigenvalues given by 9 m;:::;m N (u) = 9 (u) 9 3 m ;:::;m N (u) = () m @ m w m ;:::;w m N N 9 (u) (23) m;:::;m N = m + + m N N : (24) For m =, 9 (u) is the stationary p..f. p s (u) corresponing to the eigenvalue =. For m =, the eigenfunctions are reaily foun to be 2 6 4 9 (u) 9 2 (u). 9 N (u) 3 7 5 = r w exp( 2 wt K v w) = K v w9 (u) = U u9 (u): (25) 4
Since the set of ermite polynomials forms a complete set, the transition probability ensity of the process u(t) can be expane in the form p(u; ju ) = X m =;:::;mn = c m;:::;m N e m;:::;mn 9 m;:::;m N (u)9 3 m ;:::;m N (u ); ( ) (26) The coecients c m;:::;mn are etermine by taking into account the initial conition lim! p(u; ju ) = (u u ) an the biorthogonality conition (9), leaing to: c m;:::;m N = u 9 m;:::;m N (u)9 3 m ;:::;m N (u) : (27) Then, various statistical response functions can be etermine, such as the response stationary correlation matrix R u () = hu(t)u T (t + )i (for ): R u () = where the vector r m;:::;m N = u X u u u T p(u; ju ) p s (u ) m =;:::;mn = c m;:::;m N r m;:::;m N r T m ;:::;m N exp( m;:::;m N ) (28) is ene as r m;:::;m N = u u 9 3 m ;:::;m N (u)9 (u) = u u 9 m;:::;m N (u): (29) The class of linear systems satisfying the conition of etaile balance can be shown to contain the following linear lters of arbitrary orer N: u (N) + a N u (N) + + a 3 u (2) + a 2 u () + a u = W (t) (3) where u (i) = i u=t i, an the constant coecients (a i ) are such that Eq.(3) amits a stationary solution. This equation can be set in the state-space form () by ening u = (u; _u; : : : ; u (N) ) T, with A an B such that, with hw (t)w (t + )i = (), a ij = 8 < : ; for j = i + a j ; for i = N ; otherwise b ij = ; for i = j = N ; otherwise Note that u, u, u (4), : : : are even variables, whereas _u, u (3), : : : are o variables: = iag(; ; : : : ; () N+ ). t can then be verie that the conitions of etaile balance (2) an (3) are satise, thus generalizing the results of Atkinson [8] (973) for the eigenfunctions corresponing to a lter of orer 2. 3. Stochastic Averaging of Oscillators with Non-Parametric Colore Excitation We seek here an asymptotic solution of the probability ensity function of the nonlinear oscillator (3) x(t) + f(x(t); _x(t)) +! 2 o x(t) = p u(t) (32) excite by the colore ranom process u(t), in the limit! of weak noise an weak nonlinearity. The ranom process u(t) is assume to be a stationary, physically realizable Gaussian process, with zero-mean. More specically, it is assume that a Markov realization of this process can be constructe, that is, u(t) is a component of an N-imensional stationary iusion Markov process u(t) = (u = u; u 2 ; : : : ; u N ) T governe by Eq.(). The conition of stability (2), an the conitions of etaile balance (2) an (3) are assume satise, so that the complete sets of eigenvalues an eigenfunctions of the corresponing operator Lu are 5
given by Eq.(24) an Eq.(23) respectively. As in x2, 9 (u) enotes the stationary p..f. of u, an correspons to the eigenvalue =. We also enote by R u the correlation function of u(t), that is, the (; ) element of the correlation matrix Ru of u(t). As a rst step, the joint response process (x; _x) is transforme into (a; ) accoring to the relationships x(t) = a(t) cos (t); _x(t) =! o a(t) sin (t): (33) The equations of motion for the amplitue a(t) an phase (t) processes are then given by _a(t) = p f(a cos ;! o a sin ) sin u(t) sin ; (34)! o! o _(t) =! o +! o a f(a cos ;! oa sin )cos p u(t) cos : (35)! o a n the limit!, the process a(t) is clearly a slow variable, whereas the process (t) is a fast variable. The amplitue of the response will not change signicantly over a time interval of the orer O(). To investigate the response of the system over a time interval of the orer O( ), the time variable is scale as t = =, where is a slow time scale. The extene process (a; ; u) is a Markov process, whose probability ensity p (a; ; u; ), assume to exist, is solution of the following FPK equation where the operators L, L an L 2 are given by an Lp = L + p L + L 2 p = (36) L =! o @ + Lu; (37) L = u! o sin @ a + u! o a @ cos (38) L 2 =! o @ a (f sin )! o a @ (f cos ) @ : (39) A solution of this equation is sought in terms of a perturbation expansion of the parameter in the following form p (a; ; u) = p (a; ; u) + p p (a; ; u) + p 2 (a; ; u) + (4) where, for simplicity, the epenence upon the variable is hereon omitte. nserting (4) into (36) an ientifying terms of equal power of =2 leas to the sequence of equations : L p = ; (4) p : L p = L p ; (42) i=2 : L p i = L p i L 2 p i2 ; i = 2; 3; : : : (43) Each term of expansion (4) is require to be 2-perioic in the variable, integrable over a 2 [; [, an assume to satisfy natural bounary conitions for the variable u. The normalization conition of p is equivalent to 2 a u p (a; ; u) = ; (44) an to, for i, 2 a u p i (a; ; u) = : (45) 6
n general, a linear equation of the type L p = q, such as Eq.(42) or Eq.(43), will amit a solution p if its inhomogeneous part q is orthogonal to all elements of the null space ker L 3 of the ajoint of L. nee, for all q 3 2 ker L 3, that is, such that L3 q3 =, the following solvability conition is obtaine: a u q 3 q = a u q 3 L p = a u pl 3 q3 = (46) Thus, to each equation (42) an (43) correspons a solvability conition as given by Eq.(46). Since the operator L 3 =! o@ + L 3 u is the sum of an operator acting on the variable only, an of the operator L 3 u acting on the variable u only, the elements q 3 of ker L 3 are sought in a series expansion along the eigenfunctions 9 3 m ;:::;m N (u) of L 3 u in the following way q 3 (a; ; u) = X m =;:::;mn = q 3 m ;:::;m N (a; )9 3 m ;:::;m N (u): (47) Then, each coecient q 3 m ;:::;m N is solution of Since each eigenvalue satises < m;:::;mn non-zero coecient in expansion (47) is (! o @ + m;:::;m N )q 3 m ;:::;m N = (48) <, except for = corresponing to 9 3 (u) =, the only perioic q 3 (a; ) = q 3 (a): (49) Therefore, the elements of ker L 3 are arbitrary functions of the amplitue a only. This implies that the solvability conition (46) of equation L p = q takes the expression 2 2 u q(a; ; u) = : (5) which imposes that the average of q over the variables, an u is zero. We then procee to n the leaing term p of perturbation expansion (4). p (a; ; u) is solution of Eq.(4) an can be sought in a series expansion of the eigenfunctions of the operator Lu, that is, Then, each term p () m ;:::;m N p (a; ; u) = is solution of X m =;:::;mn = p () = p () 9 + k= m ;:::;m N (a; )9 m;:::;m N (u) p () k 9 k + k;l= p () kl 9 kl + (5) (! o @ + m;:::;m N )p () m ;:::;m N (a; ) = ; (52) an, upon integration over the variable, is a non-zero, perioic function of if an only if m = corresponing to =. t follows that the solution of Eq.(4) is inepenent of the phase variable an takes the form p (a; ; u) = p (a)9 (u); (53) where p (a) remains to be foun from subsequent solvability conitions. To orer O( ), the amplitue process a(t) is uncouple from the phase process (t). At orer p, the term p is solution of the equation (! o @ + Lu)p = L p = u! o 9 (u)(p =a @ a p ) sin ; (54) 7
whose solvability conition (46) is clearly always satise. As was one for p, p is etermine as the eigenfunction expansion p (a; ; u) = X m =;:::;mn = p () = p () 9 + k= m ;:::;m N (a; )9 m;:::;m N (u) p () k 9 k + k;l= p () kl 9 kl + (55) We rst note that the function u 9 (u) can be expresse as a linear combination of eigenfunctions (9 k ) k=;:::;n of orer m = : u 9 (u) = k= k 9 k (u): (56) where the scalar coecient k is in fact the rst element of the vector r k =c k ene in x2 for the orer m =. Then, the coecients p () m ;:::;m N are solution of (! o @ + m;:::;m N )p () m ;:::;m N (a; ) = k! o (p =a @ a p ) sin ; for m = ; ; otherwise (57) The only non-zero perioic solutions are corresponing to m = an =, an p () (a; ) = p() (a) (58) p () k (a; ) = k (p! o 2 + 2 =a @ a p )(cos + k sin ) (59) k! o corresponing to m = an k = k, for k = ; : : : ; N. The coecient p () (a) correspons to the homogeneous solution of Eq.(54), an is so far unetermine. Finally, p takes the expression p (a; ; u) = p () (a)9 (u)(p =a @ a p ) k= k! 2 o + 2 k (cos + k! o sin )9 k (u) (6) R which satises the normality conition (45) if p () (a)a =. The analytical expression for p (a; ) is etermine by application of the solvability conition of equation (43) at the orer, that is, We rst n where 2 2 2 2 F (a) = 2 u (L 2 p + L p ) (a; ; u) = : (6) u L 2 p (a; ; u) = @ p! o @ a (F (a) p ); (62) 2 f(a cos ;! o a sin ) sin : (63) Then, using Eq.(6), an noting that the term p () (a)9 (u) oes not contribute, where 2 2 u L p (a; ; u) = D u @ a ( 4! o 2 a p + @ a p ); (64) D u = k= 2 k k k : (65)! o 2 + 2 k 8
The scalars k enote the following averages of the process u: k = u u 9 k (u); k = ; : : : N: (66) The scalar D u can in fact be expresse in terms of the excitation power spectral ensity by using the expression of the correlation function (28) an the biorthogonality property of eigenfunctions: S u (!) = R u ()e i! = k= 2 k k k! 2 + 2 k ence, D u = S u (! o ). Finally, it is seen that p (a; ) satises the following FPK-type equation @ p = @ a F (a) + S u(! o )! o 4! o 2 a (67) p + S u(! o ) @ 2 4! o 2 aa p : (68) Thus, it is seen that, to orer O( ), the response probability ensity function is governe by an FPK equation ientical to that erive from Stratonovich-Khasminskii theorem for a broa-ban excitation u(t). To this orer, only the value of the excitation spectral ensity at the oscillator's natural frequency is taken into account, as woul be the case in a white noise approximation of u(t). The joint probability ensity of (x; _x) is given by, to orer O( ), p (x; _x; ) = 2! o a p (a; ; ) = 2! o a p (a; ) (69) an, is therefore inepenent of the phase angle. At the next orer of perturbation the term p () (a) woul be etermine from the solvability conition of the equation L p 3 = L p 2 L 2 p after rst solving for p 2 (a; ; u) up to an unknown term p (2) (a). These higher orer perturbation terms woul take further account of the excitation spectrum an yiel phase-epenent contributions. 4. Stochastic Averaging of Nearly-amiltonian Systems Perturbe by Parametric Colore Excitation We consier in this section systems escribe by a amiltonian function (x; y) an perturbe by weak friction forces an weak parametric colore noise, an whose equations of motion are consiere in the form x t = @ @y + f(x; y) + p (x; y) u(t) y t = @ @x + g(x; y) + p 2 (x; y) u(t): (7) We seek an asymptotic expression for the joint probability ensity of (x(t); y(t)) in the limit!. t is assume that the unperturbe trajectories of the system, that is, for =, are perioic functions of time an are ene in the x; y plane by the close contours of constant values of the amiltonian (x; y). The excitation u(t) is chosen as in x3. For small, the probabilistic behavior of the response is not expecte to change signicantly over time intervals of the orer O(). After scaling the time variable as t = =, the probability ensity of the extene process (x; y; u) is governe by Eq.(36) where the operators L, L, an L 2 are now given by @ L = @ x @y @ + @ y + Lu = @ @x @y @ x + @ @x @ y + Lu (7) L = u @ x (x; y) u @ y 2 (x; y) (72) L 2 = @ x f(x; y) @ y g(x; y) @ : (73) 9
Again, we seek an expression for the leaing term p (x; y; u) (omitting the -epenence) of the perturbation expansion (4), solution of the equation L p =. After expaning p on the complete set of eigenfunctions 9 m;:::;m N (u) of Lu, we n that each coecient is solution of the linear partial ierential equation @ @y @ x + @ @x @ y + m;:::;m N p () m ;:::;m N = ; (74) which, accoring to the metho of characteristics, is equivalent to the system of equations x y ( = @ @y = @ @x m;:::;m N )p () m ;:::;m N = : (75) The rst two equations ene the contours of constant -values, an their solutions correspon to the unperturbe trajectories, an hence are perioic functions x(; ), y(; ) of the variable. Clearly, the only perioic function p () m ;:::;m N solution of Eq.(75) correspons to the eigenvalue =, an hence, is inepenent of the variable. Thus, the solution of L p = must be of the form p (x; y; u) = p ((x; y)) 9 (u); (76) where p () remains to be etermine from subsequent solvability conitions. ere, plays the role of a slow variable, an that of the fast variable. At the next orer, the solvability conition of equation L p = L p can be state as x y uq 3 L p = uq 3 L p = (77) for all q 3 such that L3 q3 =. We have use the fact that the Jacobian of the transformation from the (x; y) variables to the (; ) variables is equal to. Furthermore, since the elements q 3 are arbitrary functions of the variable only, this conition implies that ul p = ; (78) @ @y @ x + @ @x @ y + m;:::;m N p () m ;:::;m N = where R the contour integral is evaluate on the curves of constant. This conition is always satise since uu 9 (u) =. An expression for p is then sought as an expansion on the set of eigenfunctions (9 m;:::;mn (u)), leaing to the following equation for the expansion coecients: k (@ x ( p ) + @ y ( 2 p )) ; for m = ; (79) ; otherwise which is equivalent to, on each characteristic curve (x(; ), y(; )) (the unperturbe trajectories of the system): m ;:::;m N p () m ;:::;m N = k (@ x + @ y 2 )p k ( @ x + 2 @ y )@ p ; for m = ; ; otherwise ere, the constants k, (k = ; 2; : : : ; N), are ene as in x3. The only non-zero, perioic solutions correspons to m = m + + m N = an. For m =, the solution is function of only, p () (), an remains to be etermine. For m =, Eq.(8) can be viewe as that of a linear system with a perioic input, an whose response is perioic only at steay state. Thus, the perioic solution of Eq.(8) is given by the following (8)
convolution integrals (k = ; 2; : : : ; N): where p () k (; ) = p () k (; ) @ p J k (; ); (8) k (; ) = k e k (@ x e + @ y e 2 ); (82) J k (; ) = k e k (e g @x + e 2 g @y ); (83) e = (x(; ); y(; )) ; (84) e 2 = 2 (x(; ); y(; )) ; (85) g@ x = @ x (x(; ); y(; )) ; (86) g@ y = @ y (x(; ); y(; )) : (87) Finally, the solvability conition of the next orer equation L p 2 = L 2 p L p is state as u (L 2 p + L p ) = ; (88) which leas successively to, noting that the unetermine term p ( )() oes not contribute, with an nally to with u L 2 p = @ p () F k () = G k () = (f @ x + g @ y ) @ p ; (89) ul fp () k (; )9 k (u)g = k k @ fp () F k ()g (9) ( @ x + 2 @ y ) e k (@ x e + @ y e 2 ) (9) ul f(@ p )J k (; )9 k (u)g = k k @ fg k ()@ p g ; (92) ( @ x + 2 @ y ) e k (e g @x + e 2 g @y ): (93) ence, this solvability conition leas to the following FPK equation for the joint probability ensity p (x; y) = p ((x; y)) T () @ p = @ fp (f 3 () + F ())g + @ fg()@ p g (94) where T () (the perio of the unperturbe trajectories), f 3 (), F (), an G() are given by f 3 () = F () = G() = T () = (95) (f @ x + g @ y ) (96) k= k= k k F k () (97) k k G k (): (98)
All these expressions are compute from contour integrals over the noise-free an friction-free trajectory escribe by the functions (x(; ), y(; )), at xe value of the variable. Equation (94) can be easily generalize in the case of multiple, uncorrelate, parametric colore noise excitations: each noise excitation will contribute inepenently to the (noise-inuce) rift F () an iusion G() functions. We illustrate the calculation of Eq.(94) with the example of a general, linear parametric oscillator governe by the equation x + 2! o ( + p 3 (t)) _x +! 2 o( + p 2 (t))x = p (t); (99) where, 2, an 3 are taken to be mutually uncorrelate, colore, Gaussian noise excitations. This oscillator has been investigate by Ariaratnam an Tam [9] (979). The unperturbe trajectories are given by the functions x(; ) = =! o (2) =2 cos! o ; y(; ) = (2) =2 sin! o () of perio T () = T = 2=! o, an where = y 2 =2 +! 2 o x2 =2. Then, using Eq.(67) for the excitation power spectrum, we n f 3 () = 2! o T F () = ; G () = T 2 N X k= F 2 () = ; G 2 () =!2 4 T 2 F 3 () = 2 2! 2 o T N X k= G 3 () = 2! 2 o T 2 N X k= 2 3;k 3;k 3;k 2 ;k ;k ;k = T! o 2 + 2 ;k 2 S (! o ) k= 2 2;k 2;k 2;k =!2 4! o 2 + 2 ;k 4 T S 2 (2! o ) 2 () = 2 2! 2 o T S 3 () 3;k 3;k 4 3;k + 2 3;k 4! 2 o + 2 3;k! = 2! 2 o T (2S 3 () + S 3 (2! o )) 2 where F i an G i correspon to noise i, i =, 2, an 3 an have etermine accoring to Eqs.(97) an (98). Using these expressions in Eq.(94), the stationary ensity is reaily etermine: p (x; y) = where N is a normalization constant, an where N (c + c 2 (y 2 +! 2 o x2 )) c3=c2 (2) c = S (! o ); c 2 =!2 4 S 2 (2! o ) + 2! 2 o(2s 3 () + S 3 (2! o )); c 3 = 2! o ( +! o S 3 ()): (3) This solution will exist only if c 3 =c 2 >, that is, if 2! o >!2 4 S 2 (2! o ) + 2! 2 o S 3 (2! o ): (4) The results of references [2, 3] are recovere in the case of nonlinear oscillators governe by the equation x + f(x; _x) _x + U (x) = p (x; _x) u(t); (5) in which case, becomes the energy variable E = y 2 =2 + U(x) with y = _x. Note that in references [2, 3] the mean rst-passage time from energy level E to E c, characteristic of the system reliability, is shown to be governe by an \average" equation by aopting a similar perturbation scheme. n the case of non-parametric excitation, that is, where (x; _x) =, the functions F (E) an G(E) of Eq.(36) can be simplie further, after 2
expaning the perioic velocity response y(e; ) of the unperturbe system, that is, of 2 x= 2 + U (x) = in Fourier series of perio T (E) = 2=!(E): y(e; ) = _x(e; ) = X y n (E) exp(i n!(e)): (6) n= Then, by substituting this series expansion in Eq.(9) an Eq.(93) we obtain, with =, 2 =, E y = y, an for k = ; 2; : : :; N: F k (E) = ; (7) an ence, F (E) =, an G k () = E = T (E) X y(e; ) e k y(e; ) X n= k k G(E) = T (E) y n (E)y n (E) n= k + in!(e) k= = T (E) 2 jy (E)j 2 S u () + y n (E)y n (E) k + in!(e) : (8) X n= jy n (E)j 2 S u (n!(e))! ; (9) where S u (!) is the excitation power spectrum, an jy n j 2 = y n y n. ence, the results of Roberts [4, 5] are recovere. Similar expressions can be obtaine in the case of parametric excitations. For example, for the oscillator x + f(x; _x) _x + U (x) = p ( (t) + x 2 (t) + _x 3 (t)) ; () where, 2, an 3 are three inepenent Gaussian colore noise excitations, the corresponing functions F i an G i, i = ; 2; 3, are given by F (E) = ; () G (E) = T (E) 2 jy (E)j 2 S () + X n= jy n (E)j 2 S (n!(e))! ; (2) an F 2 (E) = ; (3) G 2 (E) = T (E) 2 jz (E)j 2 S 2 () + X n= jz n (E)j 2 S 2 (n!(e))! ; (4) F 3 (E) = 2 T (E) w (E)S 3 (); (5) G 3 (E) = T (E) 2 jw (E)j 2 S 3 () + X n= jw n (E)j 2 S 3 (n!(e))! : (6) n these expressions, y n, z n, an w n are the Fourier coecients of y(e; ), x(e; )y(e; ), an y 2 (E; ) respectively, corresponing to the unperturbe system. 3
Conclusions Moeling excitations by colore noises is justie by the fact that real life uctuations of the environment have a non-vanishing correlation time an a non-white power spectrum. ence, the results of x3 an x4 are useful to stuy the eect of nite correlation time of the excitation on the system's behavior. We have foun that, to leaing orer of perturbation, the state probability ensity function is governe by a reuce equation in the \slow" variable only, that is, the amplitue variable for the harmonic oscillator, or the amiltonian variable for general amiltonian systems. n some cases, we have foun that the results obtaine are ientical to those erive in the past by application of Stratonovich-Khasminskii theorem. n such cases, the assumption of etaile balance for the excitation process u was only neee to the extent that the existence of the eigenfunctions an eigenvalues is then guarantee, but the nal results such as given by Eq. (68) o not explicitly epen on the etaile knowlege of the eigenfunctions an eigenvalues of the noise process. owever, for the more general case of amiltonian systems perturbe by parametric noise, the result of Eq.(94) oes require the assumption of etaile balance in orer to etermine in this equation certain parameters characterizing the ranom excitation. But we surmise that even in this case the results woul be ientical to those obtaine by using Stratonovich-Khasminskii theorem, an shoul only be vali for broa-ban excitations, that is, when the ratio of the relaxation time of the system to the correlation time of the ranom excitation is much larger than one. A rawback of the perturbation scheme is that it oes not impose any restrictions on the shape of the excitation power spectrum. We observe though that narrow-ban excitations corresponing to one or more eigenvalues m;:::;mn close to the natural frequency of the system (near resonance) woul generate nearly singular terms in the present erivations. Thus other asymptotic methos shoul be evelope in the case of narrow-ban ranom excitations. Acknowlegements Many thanks to one of the reviewers for his careful review. References [] Stratonovich, R. L., Topics in the Theory of Ranom Noise, Goron an Breach, New York, (963). [2] Khasminskii, R.., \A Limit Theorem for the Solutions of Dierential Equations with Ranom Right- an Sies", Theory of Probability an Applications, Vol., No. 3, pp. 39-45, (966). [3] Roberts, J. B., \The Energy Envelope of a Ranomly Excite Non-Linear Oscillator", Journal of Soun an Vibration, Vol. 6, pp. 77-85, (978). [4] hu, W. Q., \Stochastic Averaging of the Energy Envelope of Nearly Lyapunov Systems", Proceeings of the UTAM Symposium on Ranom Vibrations an Reliability, (Frankfurt/Oer, G. D. R., 982), (K. ennig, E.), pp. 374-357, Akaemie, Berlin, (983). [5] Lin, Y. K., \Some observations on the Stochastic Averaging Metho", Probabilistic Engineering Mechanics, Vol., No., pp. 23-27, (986). [6] hu, W. Q., an Lin, Y. K., \On Stochastic Averaging of Energy Envelope", Journal of Engineering Mechanics, ASCE, Vol. 7, No. 8, pp. 89-95, (99). [7] Re-orse, J. R., an Spanos, P. D., \Generalization to Stochastic Averaging in Ranom Vibration", nternational Journal of Non-Linear Mechanics, Vol. 27, No., pp. 85-, (992). [8] Roberts, J. B., an Spanos, P. D., \Stochastic Averaging: An Approximate Metho for Solving Ranom Vibration Problems", nternational Journal of Non-Linear Mechanics, Vol. 2, No. 2, pp. -34, (986). 4
[9] hu, W. Q., \Stochastic Averaging in Ranom Vibration", Applie Mechanics Reviews, Vol. 4, No. 5, 89-99, (988). [] Blankenship, G., an Papanicolaou, G. C., \Stability an Control of Stochastic Systems with Wie-Ban Noise Disturbances", Siam J. Appl. Math., Vol. 34, No. 3, pp. 437-476, (978). [] orsthemke, W., an Lefever R., \A Perturbation Expansion for External Wie Ban Markovian Noise: Application to Transitions nuce by Ornstein-Uhlenbeck Noise",. Physik B, Vol. 4, pp. 24-247, (98). [2] Klosek-Dygas, M. M., Matkowsky, B. J., an Schuss,., \Stochastic Stability of Nonlinear Oscillators", Siam J. Appl. Math., Vol. 48, No. 5, pp. 5-27, (988). [3] Klosek-Dygas, M. M., Matkowsky, B. J., an Schuss,., \Colore Noise in Dynamical Systems", Siam J. Appl. Math., 48, No. 2, pp. 425-44, (988). [4] Roberts, J. B., \A Stochastic Theory for Nonlinear Ship Rolling in rregular Seas", J. Ship Research, Vol. 26, No. 4, pp. 229-245, (982). [5] Roberts, J. B., \Energy Methos for Non-Linear Systems with Non-white Excitation", Proceeings of the UTAM Symposium on Ranom Vibrations an Reliability, (Frankfurt/Oer, G. D. R., 982), (K. ennig, E.), pp. 285-294, Akaemie, Berlin, (983). [6] Gariner, C. W., anbook of Stochastic Methos, Springer-Verlag, Berlin, (983). [7] Erelyi, A., igher Transcenental Functions, Vol., McGraw-ill, New York, (953). [8] Atkinson, J. D., \Eigenfunction Expansions for Ranomly Excite Non-Linear Systems", Journal of Soun an Vibration, Vol. 3, No. 2, pp. 53-72, (973). [9] Ariaratnam, S. T., an Tam, D. S. F., \Ranom Vibration an Stability of a Linear Parametrically Excite Oscillator", AMM, Vol. 59, pp. 79-84, (979). 5