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Stochastic averaging of quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations
International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
Stochastic averaging of quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations Z.L. Huang∗ , W.Q. Zhu Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China Received 15 November 2003; received in revised form 15 December 2003; accepted 2 February 2004
Abstract A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems to combined harmonic and white noise excitations. According to the proposed method, an n + + -dimensional averaged Fokker–Planck–Kolmogorov (FPK) equation governing the transition probability density of n action variables or independent integrals of motion, combinations of angle variables and combinations of angle variables and excitation phase angles can be constructed when the associated Hamiltonian system has internal resonant relations and the system and harmonic excitations have external resonant relations. The averaged FPK equation is solved by using the combination of the 9nite di:erence method and the successive over relaxation method. Two coupled Du;ng–van der Pol oscillators under combined harmonic and white noise excitations is taken as an example to illustrate the application of the proposed procedure and the stochastic jump and its bifurcation as the system parameters change are examined. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Stochastic averaging method; Quasi-integrable Hamiltonian systems; Combined harmonic and white noise excitations
1. Introduction The stochastic averaging method is a powerful approximate technique for the prediction of response, decision of stability and estimation of reliability of linear or non-linear conservative oscillators subject to lightly linear and (or) non-linear dampings and external and (or) parametric random excitations. It has been extensively used in theoretical investigation and engineering application of random vibration [1,2]. Comprehensive reviews attesting the success of the stochastic averaging method have been written by Roberts and Spanos [3] and by Zhu [4,5]. ∗
Corresponding author. Tel./fax: +86-571-879-526-51. E-mail address: [email protected] (Z.L. Huang).
0020-7462/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2004.02.004
The classical stochastic averaging method has two widely used versions. One is often called Stratonovitch stochastic averaging or standard stochastic averaging. It was initially developed by Stratonovitch [6] and later justi9ed mathematically by Khasminskii [7]. It can be applied to multi-degree-of-freedom (MDOF) quasi-linear system, i.e., linear conservative oscillators subject to lightly linear and (or) non-linear dampings and weakly external and (or) parametric excitations of wide-band random processes or combined harmonic and wide-band random excitations. The other version is usually called the stochastic averaging of energy envelope or quasi-conservative averaging. It was initially proposed by Khasminskii [8] and Landa and Stratonovitch [9], rederived by Zhu [10] and Zhu and Lin [11] based on a theorem due to Khasminskii [12].
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This version of stochastic averaging method is applicable to single-degree-of-freedom (SDOF) non-linear conservative oscillators subject to lightly linear and (or) non-linear dampings and weakly external and (or) parametric excitations of Gaussian white noise. In recent years, the stochastic averaging method for quasi-Hamiltonian systems has been developed [13–15]. The method is applicable to MDOF nonintegrable, completely integrable and partially integrable Hamiltonian systems subject to lightly linear and (or) non-linear dampings and external and (or) parametric excitations of Gaussian white noises. The method has been applied to study the stochastic stability, stochastic bifurcation, reliability and optimal non-linear stochastic control of quasi-Hamiltonian systems [16–24]. On the other hand, the stochastic averaging methods for SDOF non-linear oscillators under combined harmonic and white noise excitations and under bounded noise excitation, respectively, have been developed and they are successfully applied to study the stochastic jump and bifurcation of Du;ng oscillator under combined harmonic and white noise excitations or under bounded noise excitation [25,26]. In the present paper, a stochastic averaging method for quasi-integrable Hamiltonian systems subject to combined harmonic and white noise excitations is proposed. The proposed method is applied to study the response of two coupled Du;ng–van der Pol oscillators under combined harmonic and white noise excitations. The stochastic jump and its bifurcation as system parameters change are examined.
2. Quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations Consider a quasi-integrable Hamiltonian system of n DOF governed by the following equations of motion: 9H Q˙ i = ; 9Pi
(1a)
s = 1; : : : ; m;
(1b)
(2)
u; M vM = 1; : : : ; m: Eqs. (1a) and (1b) are equivalent to the following set of Itˆo equations: 9H dt; 9Pi 9H 9H + cij − hik cos k (t) dPi = − 9Qi 9Pj 9giv dt + 1=2 is dBs (t); − Duv gju 9Pj
dQi =
i; j = 1; : : : ; n;
(3a)
k = 1; : : : ; l;
u; M v; M s = 1; : : : ; m;
(3b)
where k (t) = k t + k ; Bs (t) are the standard Wiener processes and T = 2gDgT . The double summation terms on the right-hand side of Eq. (3b) are known as the Wong–Zakai correction terms. These terms usually can be split into two parts: one has the e:ect of modifying the conservative forces and another modifying the damping forces. The 9rst part can be combined with 9H =9Qi to form an overall e:ective conservative forces 9H=9Qi with a new Hamiltonian H = H (Q; P) and with 9H=9Pi = 9H =9Pi . The second part can be combined with − cij 9H =9Pj to constitute e:ective damping forces − cij 9H=9Pj with cij = cij (Q; P). After doing these, Eqs. (3a) and (3b) can be rewritten as 9H dt; 9Pi 9H 9H + cij − hik cos k (t) dt dPi = − 9Qi 9Pj
+ 1=2 gis Ws (t); k = 1; : : : ; l;
E[WuM(t)WvM(t + )] = 2Duv ();
dQi =
9H 9H P˙ i = − − cij + hik cos(k t + k ) 9Qi 9Pj
i; j = 1; : : : ; n;
where Qi and Pi are generalized displacements and momenta, respectively; H = H (Q; P) is twice di:erentiable Hamiltonian; cij = cij (Q; P) are di:erentiable functions representing coe;cients of quasi-linear dampings; hik = hik (Q; P) are amplitudes of harmonic excitations; gis = gis (Q; P) are amplitudes of white noise excitations; is a small positive parameter; Ws (t) are Gaussian white noises in the sense of Stratonovich with correlation functions
+ 1=2 is dBs (t);
(4a)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
i; j = 1; : : : ; n;
3. Averaged equations
k = 1; : : : ; l;
s = 1; : : : ; m:
(4b)
Suppose that the Hamiltonian system associated Eqs. (4a) and (4b) is integrable, i.e., there exists a set of canonical transformations Ii = Ii (Q; P);
"i = "i (Q; P);
i = 1; : : : ; n
3.1. Internal resonance only At 9rst, consider the case where there are internal resonant relations but no external resonance in system (7). System (7) is said to be internal resonant if there exist (1 6 6 n − 1) resonant relations
(5)
Lui !i = 0;
such that the new Hamilton equations are of the following form: 9H (I) 9H (I) "˙i = = !i (I); I˙i = − = 0; 9Ii 9"i
Lui
i = 1; : : : ; n;
(6)
where Ii and !i are action variables and frequencies, respectively; "i are the angle variable conjugated to Ii ; and H (I) is the transformed Hamiltonian independent of "i . The Itˆo di:erential equations for action and angle variables can be obtained from Eqs. (3a) and (3b) by using transformation (5) and Itˆo di:erential rule as follows: 9H 9Ir 9Ir + hik cos k (t) dIr = −cij 9Pj 9Pi 9Pi 9 2 Ir 9Ir + Duv giu gjv dt + 1=2 is dBs (t) 9Pi 9Pj 9Pi 1=2
(2) = [!r + Vr (I; X; t)] dt + 1=2 rs (I; X) dBs (t);
T
(7b) T
(8)
3.2. External resonance only Next, consider the case where there are (1 6 6 l) external resonant relations but no internal resonance in system (7), i.e., Lvr !r + Mkv k = v ; r = 1; : : : ; n;
k = 1; : : : ; l;
v = 1; : : : ; ;
(9)
where Lvr and Mkv are integers and not all zero for a given v. Introducing combinations v of angle variables and phase angles of harmonic excitations v
= Lvr "r + Mkv k (t); r = 1; : : : ; n;
k = 1; : : : ; l;
v = 1; : : : ; ;
(10)
1 (t); : : : ; l (t) are replaced by 1; : : : ; ; +1 (t); : : : ; l (t). The Itˆo di:erential equations for I1 ; : : : ; In ; 1 ; : : : ; ; "1 ; : : : ; "n are obtained from Eqs. (7a) and (7b) by using transformation (10) and Itˆo di:erential rule as follows: (1) (I; ; X) dBs (t); dIr = Ur (I; ; X; !) dt + 1=2 rs (11a)
k = 1; : : : ; l;
u; M v; M s = 1; : : : ; m;
u = 1; : : : ; ;
where are integers and not all zeros for a 9xed u. In this case, the terms containing cos k (t) in Eqs. (7a) and (7b) can be neglected in the 9rst approximation. The averaged Itˆo di:erential equations are essentially the same as those in Ref. [14].
(1) rs (I; X) dBs (t);
= Ur (I; X; t) dt + (7a) 9H 9"r 9"r + hik cos k (t) d"r = !r − cij 9Pj 9Pi 9Pi 92 "r 9"r + Duv giu gjv dt + 1=2 is dBs (t) 9Pi 9Pj 9Pi
r; i; j = 1; : : : ; n;
1423
where I =[I1 ; I2 ; : : : ; In ] ; X =["1 ; "2 ; : : : ; "n ] ; Qi ; Pi on the right-hand sides of Eqs. (7a) and (7b) should be replaced by Ir and "r in terms of Eq. (5). The form and dimension of averaged equations depend on the resonance of the associated Hamiltonian system as discussed in the following section.
d
v
= [ v + Lvr Vr (I; ; X; !)] dt (2) + 1=2 Lvr rs (I; ; X) dBs (t);
d"r = [!r + Vr (I; ; X; !)] dt (2) + 1=2 rs (I; ; X) dBs (t);
(11b)
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r = 1; : : : ; n;
with initial condition
v = 1; : : : ; ;
s = 1; : : : ; m;
(11c)
where = ( 1 ; : : : ;
p = p(I; ; 0 | I0 ; 0 ) = (I − I0 ) ( − 0 ) and boundary conditions 9 9(brv p) (brr2 p) + = 0; −2UM r p + 9Ir2 9 v
),
! = [ +1 ; : : : ; l ]T It can be seen from Eqs. (11a)–(11c) that I1 ; : : : ; In ; 1 ; : : : ; are slowly varying processes while "1 ; : : : ; "n are rapidly varying processes. Applying deterministic averaging with respect to "1 ; : : : ; "n , one obtained the following averaged equations:
p(I;
1
+ 2k1 *; : : : ;
(15)
I ∈ S; (16)
+ 2k *; t | I0 ; 0 )
=p(I; ; t | I0 ; 0 ):
(17)
dIr = UM r (I; ) dt + 1=2 M(1) rs (I; ) dBs (t);
(12a)
Eqs. (16) and (17) imply vanishing probability Oows in n directions at the boundary S, and periodic boundaries with respect to v , respectively.
= VM v (I; ) dt + 1=2 M(2) vs (I; ) dBs (t);
(12b)
3.3. Both internal and external resonances
d
v
where UM r =
1 (2*)n+l−
2*
0
Lvr VM v = v + (2*)n+l−
2*
0
0
2*
Suppose that there are internal resonant relations and external resonant relations in system (7), i.e., Ur (I; ; X; !) dXd!;
Lur !r = u ;
Lvr !r + Mkv k = v ;
0
2*
Vr (I; ; X; !) dXd!;
u = 1; : : : ; ;
2*
0
r(1) (I; ; X) r(1) (I; ; X) dX; 1 s1 2 s1
(2) bv1 r1 = br1 v1 = M(1) r1 s1 M v1 s1
=
Lvr21
(2*)n
2*
0
r(1) (I; ; X) r(2) (I; ; X) dX; 1 s1 2 s1
(2) bv1 v2 = M(2) v1 s1 M v2 s1
=
Lvr11 Lvr22 (2*)n
v = 1; : : : ; ;
r = 1; : : : ; n;
k = 1; : : : ; m;
(1) br1 r2 = M(1) r1 s1 M r2 s1
1 = (2*)n
(18a)
0
2*
r(2) (I; ; X) r(2) (I; ; X) dX: (13) 1 s1 2 s1
The averaged FPK equation associated with Eqs. (12a) and (12b) is 9(UM r p) 9(VM v p) 1 92 (br1 r2 p) 9p = − − + 9t 9Ir 9 v 2 9Ir1 9Ir2 1 92 (bv1 v2 p) 92 (br1 v1 p) (14) + + 9Ir1 9 v1 2 9 v 1 9 v2
(18b)
where Lur ; Lvr ; Mkv are integers and not all zero for given u or v. Introducing (1 6 6 n − 1) combinations .u of angle variables and (1 6 6 l) combinations v of angle variables and excitation phase angles, .u = Lur "r ; v
= Lvr "r + Mkv k (t);
(19a) (19b)
"1 ; : : : ; "n and 1 (t); : : : ; l (t) are replaced by .1 ; : : : ; . ; "+1 ; : : : ; "n ; 1 ; : : : ; ; +1 (t); : : : ; l (t). The Itˆo di:erential equations for I1 ; : : : ; In ; .1 ; : : : ; . , 1 ; : : : ; ; "+1 ; : : : ; "n are obtained from Eqs. (7a) and (7b) by using transformations (19a) and (19b) and Itˆo di:erential rule as follows: dIr = Ur (I; "; ; X1 ; !) dt (1) + 1=2 rs (I; "; ; X1 ) dBs (t);
(20a)
d.u = [ u + Lur Vr (I; "; ; X1 ; !)] dt (2) + 1=2 Lur rs (I; "; ; X1 ) dBs (t);
(20b)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
d
v
= [ v + Lvr Vr (I; "; ; X1 ; !)] dt
(2) br1 u2 = ˜(1) r1 s1 ˜ u2 s1 =
(2) + 1=2 Lvr rs (I; "; ; X1 ) dBs (t);
(20c) ×
d"u = [!u + Vu (I; "; ; X1 ; !)] dt + 1=2 u(2) s (I; "; ; X1 ) dBs (t); r = 1; : : : ; n;
u = 1; : : : ; ;
v = 1; : : : ; ;
u = + 1; : : : ; n;
(20d)
where " = {.1 ; : : : ; . }; X1 = {"+1 ; : : : ; "n }. It can be see from Eqs. (20a)–(20d) that I1 ; : : : ; In ; .1 ; : : : ; . ; 1 ; : : : ; are slowly varying processes while "+1 ; : : : ; "n ; +1 ; : : : ; l are rapidly varying processes. Applying deterministic averaging with respect to "+1 ; : : : ; "n , one obtained the following averaged equations: dIr = U˜ r (I; "; ) dt + 1=2 ˜(1) rs dBs (t); d.u = V˜ u (I; "; ) dt + 1=2 ˜(2) us dBs (t); v
= W˜ v (I; "; ) dt + 1=2 ˜(3) vs dBs (t);
(21b) (21c)
1 ˜ Ur = (2*)n+l−− ×
2*
0
×
×
2*
2*
0
2*
0
0
Vr (I; "; ; X1 ; !) dX1 d!;
Lrv n+l−− (2*)
2*
0
(1) br1 r2 = ˜(1) r1 s1 ˜ r2 s1 =
×
Ur (I; "; ; X1 ; !) dX1 d!;
Lur n+l−− (2*)
0
W˜ v = v +
2*
0
V˜ u = u +
2*
0
×
Vr (I; "; ; X1 ; !) dX1 d!; 1 (2*)n−
r(1) (I; "; ; X1 ) r(1) (I; "; ; X1 ) dX1 ; 1 s1 2 s1
2*
0
×
2*
0
2*
0
×
0
2*
Lur11 Lur22 (2*)n−
r(2) (I; "; ; X1 ) r(2) (I; "; ; X1 ) dX1 ; 1 s1 2 s1 Lur11 Lvr22 (2*)n−
r(2) (I; "; ; X1 ) r(2) (I; "; ; X1 ) dX1 ; 1 s1 2 s1
(3) bv1 v2 = ˜(3) v1 s1 ˜ v2 s1 =
Lvr22 (2*)n−
r(1) (I; "; ; X1 ) r(2) (I; "; ; X1 ) dX1 ; 1 s1 2 s1
(3) bu1 v2 = ˜(2) u1 s1 ˜ v2 s1 =
×
Lur22 (2*)n−
r(1) (I; "; ; X1 ) r(2) (I; "; ; X1 ) dX1 ; 1 s1 2 s1
(2) bu1 u2 = ˜(2) u1 s1 ˜ u2 s1 =
(21a)
where
2*
(3) br1 v2 = ˜(1) r1 s1 ˜ v2 s1 =
d
Lvr11 Lvr22 (2*)n−
r(2) (I; "; ; X1 ) 1 s1
× r(2) (I; "; ; X1 ) dX1 : 2 s1 The averaged FPK equation associated Eqs. (21a)–(21c) is 9p 9(U˜ r p) 9(V˜ u p) 9(W˜ v p) − − = − 9t 9Ir 9/u 9 v +
1425
(22) with
1 92 (br1 r2 p) 1 92 (bu1 u2 p) + 2 9Ir1 9Ir2 2 9/u1 9/u2
1 92 (bv1 v2 p) 92 (br1 u2 p) 92 (br1 v2 p) + + 2 9 v1 9 v2 9Ir1 9/u2 9Ir1 9 v2 92 (bu1 v2 p) (23) + 9/u1 9 v2 +
with initial condition p = p(I; "; ; 0 | I0 ; "; 0 ) = (I − I0 ) (" − "0 ) ( − 0 )
(24)
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Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
and boundary conditions 9 9(bru p) 9(brv p) −2U˜r p + (brr2 p) + + = 0; 9Ir2 9/u 9 v I ∈ S;
(25)
p(I; /1 + 2k1 *; : : : ; / + 2k *; 1
+ 2k+1 *; : : : ;
+ 2k+ *; t | I0 ; "0 ; 0 )
=p(I; "; ; t | I0 ; "; 0 ):
(26)
Eqs. (25) and (26) imply vanishing probability Oows in n directions at the boundary S and periodic boundaries with respect to /u and v , respectively. Note that original equations of motion, Eqs. (1a) and (1b), are non-autonomous while the averaged equations, Eqs. (12a), (12b) and (21a)–(21c) are autonomous. There is only potential probability Oow in averaged FPK equations (14) and (23). The dimension of averaged Itˆo equations is equal to n plus the number of sum of internal and external resonant relations and it is usually less than 2n. These are the three advantages of the stochastic averaging method proposed in the present paper.
2 X2 + 2 X23 XQ 2 + (20 + 21 X12 + 22 X22 )X˙ 2 + !20
=h2 cos t + W2 (t) + X2 W4 (t);
where ij are coe;cients of linear and non-linear dampings; !i0 ; i ; hi are positive constants representing the frequencies of associated degenerated linear oscillators, the strengths of non-linear sti:ness of the oscillators and the amplitudes of harmonic excitation, respectively. Wi (t) are independent Gaussian white noises in the sense of Stratonovich with intensities 2Di ; ij ; hi ; Di are assumed to be of the same order of . The Itˆo di:erential equations for independent integrals of motion Hi and angle variables "i are of the form dHi = {Pi [ − Pi (i0 + i1 Q12 + i2 Q22 ) + hi cos t] +Dii + D(i+2)(i+2) Qi2 } dt +Pi dBi (t) + Pi Qi dBi+2 (t);
4. Example Consider two non-linearly coupled Du;ng–van der Pol oscillators subject to additive harmonic excitations and both additive and parametric excitations of Gaussian white noises. The equations of motion are of the form 2 XQ 1 + (10 + 11 X12 + 12 X22 )X˙ 1 + !10 X1 + 1 X13 =h1 cos t + W1 (t) + X1 W3 (t);
(27a)
(28)
9"i [ − Pi (i0 + i1 Q12 + i2 Q22 ) d"i = !i + 9Pi
3.4. Remark In the practical application of the proposed averaging method, it is more convenient to replace n action variables I1 ; : : : ; In with n independent integrals of motion, H1 ; : : : ; Hn , in involution because it is di;cult to obtain the action variables Ii in most cases. The Itˆo di:erential equations for H1 ; : : : ; Hn and "1 ; : : : ; "n can be obtained from Eqs. (7a) and (7b) by using transformation Hr = Hr (I) and Itˆo di:erential rule. The averaged Itˆo equations for Hr and , or Hr ; "; can be derived similarly as those in Sections 3.2 and 3.3.
(27b)
+ hi cos t] + [Dii + +
2
"i 9Pi2
9 D(i+2)(i+2) Qi2 ]
dt
9"i 9"i dBi (t) + Qi dBi+2 (t); 9Pi 9Pi
where Qi = Xi ;
Pi = X˙ i ;
!2 Pi2 i + i0 Qi2 + Qi4 ; 2 2 4 √ 2 * i ai + b2i !i = √ ; 2 2 K(mM i )
Hi =
* F(Zi ; mM i ); 2K(mM i ) Qi ; Zi = arccos ai
2 !i0 4i Hi 2 ai = 1+ −1 ; 4 i !i0 "i =
b2i
!2 = i0 i
4i Hi 1+ +1 ; 4 !i0
ai mM i = ; 2 ai + b2i
i = 1; 2:
(29)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
ai is the amplitude of oscillator i; K(mM i ) and F(Zi ; mM i ) are the complete elliptic integral and the elliptic integrals of the 9rst kind, respectively. Pi ; Qi ; 9"i =9Pi ; 92 "i =9Pi2 can be expanded into Fourier series and only the 9rst several leading terms needed to be retained in analysis. Two cases are considered in following subsections.
and qMij ; pM ij ; "Mij are the coe;cients of the following Fourier expansions: ∞
Qi = qMij cos(2j − 1)"i ; j=1
Pi =
∞
Consider the case of primary resonance between the harmonic excitation and the 9rst oscillator of system (28) and no internal resonance between the two oscillators. Assume that (30)
where is a detuning parameter. Introducing new variable 5 as de9ned in Eq. (10). Following the procedure in Subsection 3.2, one obtains the averaged Itˆo equations for H1 ; H2 and 5 as follows: dH1 = m1 (H1 ; H2 ; 5) dt + 1k dBk (t);
(31)
m1 = −10 pM s11 − 11 pM q11 − 12 pM s11 qMs21
Qi2 =
∞
qMsij cos(2j − 2)"i ;
j=1
Pi Qi =
∞
pM qij sin 2j"i ;
j=1
9"i = 9Pi
∞
"Mij cos(2j − 1)"i ;
j=1
∞
p 9"i P= "Mij sin 2j"i ; 9Pi
i = 1; 2:
(33)
j=1
da1 = m˜ 1 (a1 ; a2 ; 5) dt + ˜1k d BM k (t);
+D44 qMs21 ;
da2 = m˜ 2 (a1 ; a2 ; 5) dt + ˜2k d BM k (t);
h1 M m3 = − !1 − "11 cos 5; 2 b11 = 1k 1k = 2D11 pM s11 + 2D33 pM q11 ;
d5 = m˜ 3 (a1 ; a2 ; 5) dt + ˜3k d BM k (t);
(34)
where dai 1 d 2 ai m˜ i = mi + bii ; dHi 2 dHi2 Hi =Hi (ai ) 2 da i b˜ii = ˜ik ˜ik = bii ; i = 1; 2; dHi
b22 = 2k 2k = 2D22 pM s21 + 2D44 pM q21 ; ∞
"M21j + D33 2("Mq11 )2 b33 = 3k 3k = D11 j=1
Hi =Hi (ai )
("Mq1j )2 ;
bij = ik jk = 0(i = j):
j=1
Note that ai is related to Hi only as shown in Eq. (29). The Itˆo di:erential equations for a1 ; a2 and 5 can be derived by using Itˆo di:erential rule as follows:
h1 pM 11 sin 5 + D11 + D33 qMs11 ; − 2 m2 = −20 pM s21 − 21 pM s21 qMs11 − 22 pM q21 + D22
j=2
pM sij cos(2j − 2)"i ;
j=1
where Bk (t) are independent unit Wiener processes,
+
∞
∞
d5 = m3 (H1 ; H2 ; 5) dt + 3k dBk (t);
∞
Pi2 =
q 9"i Qi = "Mij cos(2j − 2)"i ; 9Pi
dH2 = m2 (H1 ; H2 ; 5) dt + 2k dBk (t); k = 1; : : : ; 4;
pM ij sin(2j − 1)"i ;
j=1
4.1. External resonance only
− !1 = ;
1427
m˜ 3 = m3 (H1 ; H2 ; 5)|Hi =Hi (ai ) ; (32)
b˜33 = ˜3k ˜3k = b33 |Hi =Hi (ai ) :
(35)
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Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
The averaged FPK equation associated with averaged Itˆo equation (34) is
(36)
It is well known that in a Du;ng oscillator with hardening sti:ness subject to harmonic excitation the phenomenon of sharp jumps in amplitude may occur. Averaged Itˆo equation (34) is reduced to ordinary differential equation when Di = 0. The stationary amplitude response curves of the system can be obtained by letting da1 =dt = da2 =dt = d5=dt = 0 and then are shown in Fig. 1. It is seen from Figs. 1(a) and (b) that the amplitude responses of the two oscillators are triple-valued in certain frequency interval of harmonic excitation. Among the three values of amplitudes, two are stable while the other is unstable. The deterministic jump occurs at the two extreme values of the frequency interval of triple-valued amplitude. A typical response exhibiting stochastic jump phenomena of two coupled Du;ng–van der Pol oscillators under combined harmonic and white noise excitations is shown in Fig. 2. Figs. 2(a) and (b) are the stationary joint probability densities of amplitude and phase of the 9rst oscillator obtained from solving the averaged FPK equation (36) by using 9nite di:erence method and from digital simulation of original system (27), respectively. Figs. 2(c) and (d) are the stationary probability densities of amplitude and phase, respectively, of the 9rst oscillator. Figs. 2(e) and (f) are the sample functions of displacements of the 9rst and second oscillators, respectively, of original system (27) form digital simulation. It is seen from Fig. 2 that the probability densities obtained from the combination of the stochastic averaging method and the 9nite di:erence method are in good agreement with those from digital simulation of original system (27). The stationary joint probability densities are bimodal. It implies that there are two more probable motions in the response of the coupled Du;ng–van der Pol oscillators under combined harmonic and white noise excitations and stochastic jump may occur. This is veri9ed by the sample functions of displacements of the system. The stochastic jump of the coupled Du:-
B
1.2
A
1
C
0.8
D
0.6 0.4 0.2 0
1
1.2
1.4
1.6
(a)
Ω
1.8
2
2.2
2.4
2.2
2.4
2 1.8 1.6 D
1.4
a2
1 92 (b˜22 p) 1 92 (b˜33 p) + + : 2 2 9 2 9a22
1.4
a1
9p 9(m˜ 1 p) 9(m˜ 2 p) 9(m˜ 3 p) 1 92 (b˜11 p) − − =− + 9t 9a1 9a2 9 2 9a21
1.6
1.2
C
1
A
0.8
B
0.6 0.4 0.2 0
(b)
1
1.2
1.4
1.6
1.8
2
Ω
Fig. 1. Amplitude response curves of two coupled Du;ng– van der Pol oscillators under pure harmonic excitation with external resonance only. !10 = 1:0; !20 = 2:0; 10 = 11 = 12 = 21 = 22 = 0:05, 20 = −0:05. (a) First oscillator, (b) second oscillator. A: 1 = 2 = 2:0; h1 = h2 = 0:2; B: 1 = 2 = 2:0; h1 = h2 = 0:25; C: 1 = 2 = 2:0; h1 = h2 = 0:13; D: 1 = 2 = 5:0; h1 = h2 = 0:2.
ing–van der Pol oscillators under harmonic and white noise excitations can be regarded as random spread of deterministic jump of the same oscillators under pure harmonic excitations, as shown by curve A in Fig. 1. However, there are signi9cant di:erence between the deterministic jump and stochastic jump. The deterministic jump occurs only at two extreme values of the frequency interval of triple-valued amplitude and only from upper branch to lower branch of amplitude response curve A at the right extreme value of the frequency interval or from lower branch to upper branch at the left extreme value of the frequency interval while the stochastic jumps may occur back and forth at any frequency within this frequency interval.
6
5
1 0.8 0.6
4
ψ
3
(a)
1
5
0.8 0.6
4
ψ
3
a1
0.4
2 0.2
1
(b)
0 0
0.45
2 1.8
0.4
1.6
0.35
1.4
0.3
1.2
p (ψ)
p (a1)
0.8 0.6 0.4 0.2 0 6
0 0
1.2 1 0.8 0.6 0.4 0.2 0 1.4 1.2
p(a1,ψ )
1
0.2
1
1429
1.2
a1
0.4
2
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1.4 1.2
p(a1,ψ )
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
p(a1,ψ )
p(a1,ψ )
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
1 0.8
0.25 0.2 0.15
0.6 0.4
0.1
0.2
0.05
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
(c)
0
1.6
a1
0
1
2
900
1,000
(d)
3
ψ
4
5
6
7
2
1.5
1.5
1
1 0.5
X2
X1
0.5 0
0 -0.5
-0.5
-1 -1 -1.5 800
(e)
-1.5 900
1,000
1,100
1,200
1,300
-2 800
1,400
time
(f)
1,100
1,200
1,300
1,400
time
Fig. 2. Stationary probability densities and sample functions of the responses of system (27) with external resonance only. !10 = 1:0; !20 =2:0; 10 =11 =12 =21 =22 =0:05, 20 =−0:05, 1 =2 =2:0; h1 =h2 =0:2; =1:5; D1 =D2 =D3 =D4 =0:01. (a) Joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation obtained from stochastic averaging method, (b) joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation obtained from digital simulation of original system (27), (c) probability density of amplitude of the 9rst oscillator, (d) probability density of phase di:erence between the 9rst oscillator and the harmonic excitation, (e) sample function of displacement of the 9rst oscillator, (f) sample function of displacement of the second oscillator, —: from stochastic averaging method; •: from digital simulation of original system (27).
As in the deterministic case, the occurrence of stochastic jump depends on the system parameters, such as the frequency and amplitude of harmonic ex-
citation, the intensities of white noises, the strength of non-linearity and the linear and non-linear damping coe;cients. Since the occurrence of stochastic
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
1 0.8 0.6 a 1 0.4
5 4
ψ
3 2
0.6 0.4 0.2 0
1.4 1.2
6
0
1.4 1.2 1 0.8
5 4 0.6
3
ψ
0.4
2
0
1 0.8
5 4
ψ
(b)
0.6
3 0.4
2
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1.4 1.2
6
a1
0
4
ψ
0
Fig. 3. Stationary joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation of system (27) with external resonance only. The parameters are the same as those in Fig. 2 except = 1:3. (a) From stochastic averaging method, (b) from digital simulation of original system (27).
jump is related to bimodal probability, we call the appearance or disappearance of stochastic jump as the system change the bifurcation of stochastic jump. For = 1:3 and 1.8, the joint stationary probability densities of amplitude and phase of the 9rst oscillator obtained from solving averaged FPK equation (36) by using 9nite di:erence method are shown in Figs. 3(a) and 4(a), respectively, while the results obtained from digital simulation of original system (27) are shown in Figs 3(b) and 4(b), respectively. For 1 = 2 = 5:0, the joint probability densities of amplitude and phase of the 9rst oscillator obtained from solving averaging FPK equation (36) by using 9nite di:erence method and from digital simulation of original system (27) are shown in Figs. 5(a) and (b), respectively. It can be seen from these 9gures that the analytical results agree well with those from digital simulation of original system (27) and the joint probability densities are unimodal and no jump may occur. This is because
1 0.8
5
0.2
1
0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1.4 1.2
6
0
p (a1,ψ)
4 3.5 3 2.5 2 1.5 1 0.5 0
p (a1,ψ)
p (a1,ψ)
(a)
a1
0.2
1
4 3.5 3 2.5 2 1.5 1 0.5 0
p (a1,ψ)
0.8 0.6 0.4 0.2 0
0.8
0.2
1
(a)
1
1
p (a1,ψ)
6
1.2
1.2
p (a1,ψ)
4 3.5 3 2.5 2 1.5 1 0.5 0
4 3.5 3 2.5 2 1.5 1 0.5 0
p (a1,ψ)
p (a1,ψ)
1430
(b)
0.6
3 0.4
2
a1
0.2
1 0
0
Fig. 4. Stationary joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation of system (27) with external resonance only. The parameters are the same as those in Fig. 2 except = 1:8. (a) From stochastic averaging method, (b) from digital simulation of original system (27).
the values of frequencies are well o: the interval of triple-valued amplitude. Similar conclusions can also be drawn for other system parameters. So, the bifurcation of the stochastic jump of the system can be roughly estimated based on the amplitude response curve of the same system under pure harmonic excitation as the system parameter change. 4.2. Both internal and external resonances Consider the case of both primary external resonance between the harmonic excitation and the 9rst oscillator and internal resonance between the two oscillators of system (27). Assume that − !1 = ; !1 − !2 = ;
(37)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
p (a1,ψ)
4 3.5 3 2.5 2 1.5 1 0.5 0 6
5
3 2
(a)
pM 2j qM1j cos 2(j − 1).;
j=2
m∗3 = m3 (H1 ; H2 ; 5) − 12
m∗4 = m4 (H1 ; H2 ; 5) +
0.2
1
∞
∞
j=1
1 0.8 0.6 a1 0.4
4
ψ
m∗2 = m2 (H1 ; H2 ; 5) − 21 p (a1,ψ)
4 3.5 3 2.5 2 1.5 1 0.5 0 1.4 1.2
1431
∞
j=1
0 0
"Mp1j qMs2j sin 2j.;
(21 "Mp2j qMs1j − 12 "Mp1j qMs2j )
4 3.5 3 2.5 2 1.5 1 0.5 0 6
(b)
5
4
ψ
3 0.4
2
1 0.8 0.6 a
p (a1,ψ)
p (a1,ψ)
×sin 2j.; 4 3.5 3 2.5 2 1.5 1 0.5 0 1.4 1.2
b∗11 = b11 ;
b∗22 = b22 ;
b∗ij = 0(i = j); b∗44 =
∞
j=1
1
(D11 ("M1j )2 + D22 ("M2j )2 ) + 2D33 ("Mq11 )2
0.2
1
+2D44 ("Mq21 )2 +
0 0
Fig. 5. Stationary joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation of system (27) with external resonance only. The parameters are the same as those in Fig. 2 except 1 = 2 = 5; = 1:3. (a) From stochastic averaging method, (b) from digital simulation of original system (27).
b∗33 = b33 ;
∞
j=2
[D33 ("Mq1j )2
+D44 ("Mq2j )2 ];
where mi (H1 ; H2 ; 5) and bii are de9ned in Eq. (32). The Itˆo di:erential equations for a1 ; a2 ; 5 and . can also be derived as follows
where and are detuning parameters. Introduce new variables 5 and . as de9ned in Eqs. (19a) and (19b). Following the procedures described in Subsection 3.3, one obtains the averaged Itˆo equations for H1 ; H2 ; 5 and . as follows
da2 = m˜ ∗2 (a1 ; a2 ; 5; .) dt + ˜∗2k d BM k (t);
∗ dH1 = m∗1 (H1 ; H2 ; 5; .) dt + 1k dBk (t);
d. = m˜ ∗4 (a1 ; a2 ; 5; .) dt + ˜∗4k d BM k (t);
∗ dH2 = m∗2 (H1 ; H2 ; 5; .) dt + 2k dBk (t);
where
∗ d5 = m∗3 (H1 ; H2 ; 5; .) dt + 3k dBk (t);
m˜ ∗1 =
∗ d. = m∗4 (H1 ; H2 ; 5; .) dt + 4k dBk (t);
(38)
where m∗1
= m1 (H1 ; H2 ; 5) − 12
∞
j=2
pM 1j qM2j cos 2(j − 1).;
(39)
da1 = m˜ ∗1 (a1 ; a2 ; 5; .) dt + ˜∗1k d BM k (t);
d5 = m˜ ∗3 (a1 ; a2 ; 5; .) dt + ˜∗3k d BM k (t);
da1 ∗ 1 d 2 a1 ∗ m1 + b ; dH1 2 dH12 11 Hi =Hi (ai ) da1 ∗ 1 d 2 a2 ∗ m2 + b ; m˜ ∗2 = dH1 2 dH22 22 Hi =Hi (ai )
m˜ ∗3 = m∗3 (H1 ; H2 ; 5; .)|Hi =Hi (ai ) ; m˜ ∗4 = m∗4 (H1 ; H2 ; 5; .)|Hi =Hi (ai ) ;
(40)
b˜∗22 = ˜∗2k ˜∗2k =
da1 dH1 da2 dH2
2
b∗11
2
; Hi =Hi (ai )
b∗22
3 2.5 2 1.5 1 0.5 0
; Hi =Hi (ai )
6
5
b˜∗33 = ˜∗3k ˜∗3k = b∗33 |Hi =Hi (ai ) ;
ψ
(41)
The averaged FPK equation associated with Eq. (40) is as follows 9p 9(m˜ ∗1 p) 9(m˜ ∗2 p) 9(m˜ ∗3 p) 9(m˜ ∗4 p) =− − − − 9t 9a1 9a2 9 9/
1 92 (b˜∗11 p) 92 (b˜∗22 p) 92 (b˜∗33 p) + + + 2 9 2 9a21 9a22 92 (b˜∗44 p) + 9/2
:
2 0.2
1
(42)
The joint probability densities of amplitude and phase of the 9rst oscillator obtained from solving FPK equation (42) by using 9nite di:erence method and from digital simulation of original system (27) are shown in Figs. 6(a) and (b), respectively. The stationary probability densities of a1 ; 5; ., respectively, are shown in Figs. 7(a)–(c). It can be seen from these 9gures that the analytical solutions agree well with those from digital simulation of original system and no jump may occur in the system. 5. Concluding remarks A stochastic averaging method for quasi-integrable Hamiltonian system under combined harmonic and white noise excitations is proposed. It is shown that the dimension and form of the averaged equations depend upon the number of internal and (or) external resonant relations. It is noted that the stochastic averaging method for quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations developed in the present paper is reduced to
p (a1,ψ)
(i = j):
3
(a)
b˜∗44 = ˜∗4k ˜∗4k = b∗44 (H1 ; H2 ; 5; .)|Hi =Hi (ai ) ; b˜∗ij = ˜∗ik ˜∗jk = 0
1 0.8 0.6 a 1 0.4
4
0 0
3 2.5 2 1.5 1 0.5 0 1.4 1.2
3 2.5 2 1.5 1 0.5 0 6
(b)
3 2.5 2 1.5 1 0.5 0 1.4 1.2
5
1 0.8 0.6 a1 0.4
4
ψ
p (a1,ψ)
b˜∗11 = ˜∗1k ˜∗1k =
p (a1,ψ)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
p (a1,ψ)
1432
3 2 0.2
1 0 0
Fig. 6. Stationary joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation of system (27) with both external and internal resonances. !10 = !20 = 1:0; 1 = 2 = 2:0; = 1:3; h1 = h2 = 0:2, ij = 0:05 (i = 1; 2; j = 0; 2), (a) from Di = 0:01(i = 1; 4); stochastic averaging method, (b) from digital simulation of original system (27).
the stochastic averaging method for quasi-integrable Hamiltonian systems [14] when no external resonance exist in system (1). The proposed procedures have been successfully applied to predict the responses, the stochastic jump and bifurcation of the two coupled Du;ng–van der Pol oscillators under combined harmonic external excitation and additive and parametric white noises excitations. The proposed procedures can be further extended to quasi-integrable Hamiltonian systems under wide band or narrow band random excitations. These will be our future research subjects. Acknowledgements The work reported in the paper was supported by National Natural Science Foundation of China under Key Grant No. 10332030 and Grant No. 10002015, the special Fund for Doctor Programs in Institutions
3.5
0.7
3
0.6
2.5
0.5
2
0.4
p (φ)
p (a1)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
1.5
1433
0.3
1
0.2
0.5
0.1 0
0 0
(a)
0.2
0.4
0.6
0.8
1
1.2
1.4
0
1.6
1
2
3
6
7
(b)
a1
φ
4
5
6
7
0.8 0.7 0.6
p (ψ)
0.5 0.4 0.3 0.2 0.1 0
0
1
2
(c)
3
ψ
4
5
Fig. 7. Stationary probability densities of the responses of system (27) with both external and internal resonances. The parameters are the same as those in Fig. 6. (a) Stationary probability density of amplitude of the 9rst oscillator, (b) stationary probability density of angle variable di:erence between two oscillators, (c) stationary probability density of phase di:erence between the 9rst oscillator and the harmonic excitation, —: from stochastic averaging method; •: from digital simulation of original system (27).
of Higher Learning of China under Grant No. 20020335092 and Zhejiang Provincial Natural Science Foundation under Grant No. 102040.
References [1] Y.K. Lin, G.Q. Cai, Probabilistic Structural Dynamics, Advanced Theory and Applications, McGraw-Hill, New York, 1995. [2] M.F. Dimentberg, Statistical Dynamics of Non-Linear and Time-Varying Systems, Research Studies Press, England, 1988. [3] J.B. Roberts, P.D. Spanos, Stochastic averaging: an approximate method for solving random vibration problems, Internat. J. Non-linear Mech. 21 (1986) 111–134. [4] W.Q. Zhu, Stochastic averaging method in random vibration, ASME Appl. Mech. Rev. 41 (5) (1988) 189–199.
[5] W.Q. Zhu, Recent developments and applications of stochastic averaging method in random vibration, ASME Appl. Mech. Rev. Part 2 49 (10) (1996) s72–s80. [6] R.L. Stratonovitch, in: Topics in the Theory of Random Noise, Vol. 1, Gordon & Breach, New York, 1963. [7] R.Z. Khasminskii, A limit theorem for solution of di:erential equations with random right hand sides, Theory Probab. Appl. 11 (1966) 390–405. [8] R.Z. Khasminskii, On the behavior of a conservative system with friction and small random noise, Appl. Math. Mech. 28 (1964) 1126–1130 (in Russian). [9] P.S. Laudan, R.L. Stratonovich, Theory of stochastic transitions of various systems between di:erent states, Proceedings of the Moscow University, Series III, 1962, pp. 33–45. [10] W.Q. Zhu, Stochastic averaging of energy envelope of nearly Lyapunov system. in: K. Henning (Ed.), Random Vibrations and Reliability, Proceedings IUTAM Symposium, Akademie-Verlag, Berlin, 1983, pp. 347–357. [11] W.Q. Zhu, Y.K. Lin, Stochastic averaging of energy envelop, ASCE J. Eng. Mech. 117 (1991) 1890–1905.
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[12] K.Z. Khasminskii, On the averaging principle for stochastic di:erential Itˆo equation, Kibernatika 4 (1968) 260–279. [13] W.Q. Zhu, Y.Q. Yang, Stochastic averaging of quasi-non-integrable Hamiltonian systems, ASME J. Appl. Mech. 64 (1997) 157–164. [14] W.Q. Zhu, Z.L. Huang, Y.Q. Yang, Stochastic averaging of quasi-integrable Hamiltonian systems, ASME J. Appl. Mech. 64 (1997) 975–984. [15] W.Q. Zhu, Z.L. Huang, Y. Suzuki, Stochastic averaging and Lyapunov exponent of quasi-partially integrable Hamiltonian systems, Internat. J. Non-linear Mech. 37 (2002) 419–437. [16] W.Q. Zhu, Z.L. Huang, Stochastic stability of quasi-nonintegrable Hamiltonian systems, J. Sound Vibration 218 (1998) 769–789. [17] W.Q. Zhu, Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems, Internat. J. Non-linear Mech. 39 (2004) 569–579. [18] W.Q. Zhu, Z.L. Huang, Lyapunov exponent and stochastic stability of quasi-integrable Hamiltonian system, J. Appl. Mech. 66 (1999) 211–217. [19] W.Q. Zhu, Z.L. Huang, Stochastic Hopf bifurcation of quasi-non-integrable Hamiltonian systems, Internat. J. Non-linear Mech. 34 (1999) 437–447.
[20] C.B. Gan, W.Q. Zhu, First-passage failure of quasinon-integrable Hamiltonian systems, Internat. J. Non-linear Mech. 36 (2001) 209–220. [21] W.Q. Zhu, M.L. Deng, Z.L. Huang, First-passage failure of quasi-integrable Hamiltonian systems, ASME J. Appl. Mech. 69 (2002) 274–282. [22] W.Q. Zhu, Z.L. Huang, M.L. Deng, First-passage failure and its feedback minimizations of quasi-partially integrable Hamiltonian systems, Internat. J. Non-linear Mech. 38 (2003) 1133–1148. [23] W.Q. Zhu, Z.G. Ying, T.T. Song, An optimal nonlinear feedback control strategy for randomly excited structural system, Non-linear Dynamics 24 (2001) 31–51. [24] W.Q. Zhu, Z.L. Huang, Feedback stabilization of quasi-integrable Hamiltonian systems, ASME J. Appl. Mech. 70 (2003) 129–136. [25] Z.L. Huang, W.Q. Zhu, Stochastic averaging of strongly nonlinear oscillators under combined harmonic and white noise excitations, J. Sound Vibration 238 (2000) 233–256. [26] Z.L. Huang, W.Q. Zhu, Y.Q. Ni, J.M. Ko, Stochastic averaging of strongly nonlinear oscillators under bonded noise excitations, J. Sound Vibration 254 (2002) 245–267.
Stochastic averaging of quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations Z.L. Huang∗ , W.Q. Zhu Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China Received 15 November 2003; received in revised form 15 December 2003; accepted 2 February 2004
Abstract A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems to combined harmonic and white noise excitations. According to the proposed method, an n + + -dimensional averaged Fokker–Planck–Kolmogorov (FPK) equation governing the transition probability density of n action variables or independent integrals of motion, combinations of angle variables and combinations of angle variables and excitation phase angles can be constructed when the associated Hamiltonian system has internal resonant relations and the system and harmonic excitations have external resonant relations. The averaged FPK equation is solved by using the combination of the 9nite di:erence method and the successive over relaxation method. Two coupled Du;ng–van der Pol oscillators under combined harmonic and white noise excitations is taken as an example to illustrate the application of the proposed procedure and the stochastic jump and its bifurcation as the system parameters change are examined. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Stochastic averaging method; Quasi-integrable Hamiltonian systems; Combined harmonic and white noise excitations
1. Introduction The stochastic averaging method is a powerful approximate technique for the prediction of response, decision of stability and estimation of reliability of linear or non-linear conservative oscillators subject to lightly linear and (or) non-linear dampings and external and (or) parametric random excitations. It has been extensively used in theoretical investigation and engineering application of random vibration [1,2]. Comprehensive reviews attesting the success of the stochastic averaging method have been written by Roberts and Spanos [3] and by Zhu [4,5]. ∗
Corresponding author. Tel./fax: +86-571-879-526-51. E-mail address: [email protected] (Z.L. Huang).
0020-7462/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2004.02.004
The classical stochastic averaging method has two widely used versions. One is often called Stratonovitch stochastic averaging or standard stochastic averaging. It was initially developed by Stratonovitch [6] and later justi9ed mathematically by Khasminskii [7]. It can be applied to multi-degree-of-freedom (MDOF) quasi-linear system, i.e., linear conservative oscillators subject to lightly linear and (or) non-linear dampings and weakly external and (or) parametric excitations of wide-band random processes or combined harmonic and wide-band random excitations. The other version is usually called the stochastic averaging of energy envelope or quasi-conservative averaging. It was initially proposed by Khasminskii [8] and Landa and Stratonovitch [9], rederived by Zhu [10] and Zhu and Lin [11] based on a theorem due to Khasminskii [12].
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Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
This version of stochastic averaging method is applicable to single-degree-of-freedom (SDOF) non-linear conservative oscillators subject to lightly linear and (or) non-linear dampings and weakly external and (or) parametric excitations of Gaussian white noise. In recent years, the stochastic averaging method for quasi-Hamiltonian systems has been developed [13–15]. The method is applicable to MDOF nonintegrable, completely integrable and partially integrable Hamiltonian systems subject to lightly linear and (or) non-linear dampings and external and (or) parametric excitations of Gaussian white noises. The method has been applied to study the stochastic stability, stochastic bifurcation, reliability and optimal non-linear stochastic control of quasi-Hamiltonian systems [16–24]. On the other hand, the stochastic averaging methods for SDOF non-linear oscillators under combined harmonic and white noise excitations and under bounded noise excitation, respectively, have been developed and they are successfully applied to study the stochastic jump and bifurcation of Du;ng oscillator under combined harmonic and white noise excitations or under bounded noise excitation [25,26]. In the present paper, a stochastic averaging method for quasi-integrable Hamiltonian systems subject to combined harmonic and white noise excitations is proposed. The proposed method is applied to study the response of two coupled Du;ng–van der Pol oscillators under combined harmonic and white noise excitations. The stochastic jump and its bifurcation as system parameters change are examined.
2. Quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations Consider a quasi-integrable Hamiltonian system of n DOF governed by the following equations of motion: 9H Q˙ i = ; 9Pi
(1a)
s = 1; : : : ; m;
(1b)
(2)
u; M vM = 1; : : : ; m: Eqs. (1a) and (1b) are equivalent to the following set of Itˆo equations: 9H dt; 9Pi 9H 9H + cij − hik cos k (t) dPi = − 9Qi 9Pj 9giv dt + 1=2 is dBs (t); − Duv gju 9Pj
dQi =
i; j = 1; : : : ; n;
(3a)
k = 1; : : : ; l;
u; M v; M s = 1; : : : ; m;
(3b)
where k (t) = k t + k ; Bs (t) are the standard Wiener processes and T = 2gDgT . The double summation terms on the right-hand side of Eq. (3b) are known as the Wong–Zakai correction terms. These terms usually can be split into two parts: one has the e:ect of modifying the conservative forces and another modifying the damping forces. The 9rst part can be combined with 9H =9Qi to form an overall e:ective conservative forces 9H=9Qi with a new Hamiltonian H = H (Q; P) and with 9H=9Pi = 9H =9Pi . The second part can be combined with − cij 9H =9Pj to constitute e:ective damping forces − cij 9H=9Pj with cij = cij (Q; P). After doing these, Eqs. (3a) and (3b) can be rewritten as 9H dt; 9Pi 9H 9H + cij − hik cos k (t) dt dPi = − 9Qi 9Pj
+ 1=2 gis Ws (t); k = 1; : : : ; l;
E[WuM(t)WvM(t + )] = 2Duv ();
dQi =
9H 9H P˙ i = − − cij + hik cos(k t + k ) 9Qi 9Pj
i; j = 1; : : : ; n;
where Qi and Pi are generalized displacements and momenta, respectively; H = H (Q; P) is twice di:erentiable Hamiltonian; cij = cij (Q; P) are di:erentiable functions representing coe;cients of quasi-linear dampings; hik = hik (Q; P) are amplitudes of harmonic excitations; gis = gis (Q; P) are amplitudes of white noise excitations; is a small positive parameter; Ws (t) are Gaussian white noises in the sense of Stratonovich with correlation functions
+ 1=2 is dBs (t);
(4a)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
i; j = 1; : : : ; n;
3. Averaged equations
k = 1; : : : ; l;
s = 1; : : : ; m:
(4b)
Suppose that the Hamiltonian system associated Eqs. (4a) and (4b) is integrable, i.e., there exists a set of canonical transformations Ii = Ii (Q; P);
"i = "i (Q; P);
i = 1; : : : ; n
3.1. Internal resonance only At 9rst, consider the case where there are internal resonant relations but no external resonance in system (7). System (7) is said to be internal resonant if there exist (1 6 6 n − 1) resonant relations
(5)
Lui !i = 0;
such that the new Hamilton equations are of the following form: 9H (I) 9H (I) "˙i = = !i (I); I˙i = − = 0; 9Ii 9"i
Lui
i = 1; : : : ; n;
(6)
where Ii and !i are action variables and frequencies, respectively; "i are the angle variable conjugated to Ii ; and H (I) is the transformed Hamiltonian independent of "i . The Itˆo di:erential equations for action and angle variables can be obtained from Eqs. (3a) and (3b) by using transformation (5) and Itˆo di:erential rule as follows: 9H 9Ir 9Ir + hik cos k (t) dIr = −cij 9Pj 9Pi 9Pi 9 2 Ir 9Ir + Duv giu gjv dt + 1=2 is dBs (t) 9Pi 9Pj 9Pi 1=2
(2) = [!r + Vr (I; X; t)] dt + 1=2 rs (I; X) dBs (t);
T
(7b) T
(8)
3.2. External resonance only Next, consider the case where there are (1 6 6 l) external resonant relations but no internal resonance in system (7), i.e., Lvr !r + Mkv k = v ; r = 1; : : : ; n;
k = 1; : : : ; l;
v = 1; : : : ; ;
(9)
where Lvr and Mkv are integers and not all zero for a given v. Introducing combinations v of angle variables and phase angles of harmonic excitations v
= Lvr "r + Mkv k (t); r = 1; : : : ; n;
k = 1; : : : ; l;
v = 1; : : : ; ;
(10)
1 (t); : : : ; l (t) are replaced by 1; : : : ; ; +1 (t); : : : ; l (t). The Itˆo di:erential equations for I1 ; : : : ; In ; 1 ; : : : ; ; "1 ; : : : ; "n are obtained from Eqs. (7a) and (7b) by using transformation (10) and Itˆo di:erential rule as follows: (1) (I; ; X) dBs (t); dIr = Ur (I; ; X; !) dt + 1=2 rs (11a)
k = 1; : : : ; l;
u; M v; M s = 1; : : : ; m;
u = 1; : : : ; ;
where are integers and not all zeros for a 9xed u. In this case, the terms containing cos k (t) in Eqs. (7a) and (7b) can be neglected in the 9rst approximation. The averaged Itˆo di:erential equations are essentially the same as those in Ref. [14].
(1) rs (I; X) dBs (t);
= Ur (I; X; t) dt + (7a) 9H 9"r 9"r + hik cos k (t) d"r = !r − cij 9Pj 9Pi 9Pi 92 "r 9"r + Duv giu gjv dt + 1=2 is dBs (t) 9Pi 9Pj 9Pi
r; i; j = 1; : : : ; n;
1423
where I =[I1 ; I2 ; : : : ; In ] ; X =["1 ; "2 ; : : : ; "n ] ; Qi ; Pi on the right-hand sides of Eqs. (7a) and (7b) should be replaced by Ir and "r in terms of Eq. (5). The form and dimension of averaged equations depend on the resonance of the associated Hamiltonian system as discussed in the following section.
d
v
= [ v + Lvr Vr (I; ; X; !)] dt (2) + 1=2 Lvr rs (I; ; X) dBs (t);
d"r = [!r + Vr (I; ; X; !)] dt (2) + 1=2 rs (I; ; X) dBs (t);
(11b)
1424
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
r = 1; : : : ; n;
with initial condition
v = 1; : : : ; ;
s = 1; : : : ; m;
(11c)
where = ( 1 ; : : : ;
p = p(I; ; 0 | I0 ; 0 ) = (I − I0 ) ( − 0 ) and boundary conditions 9 9(brv p) (brr2 p) + = 0; −2UM r p + 9Ir2 9 v
),
! = [ +1 ; : : : ; l ]T It can be seen from Eqs. (11a)–(11c) that I1 ; : : : ; In ; 1 ; : : : ; are slowly varying processes while "1 ; : : : ; "n are rapidly varying processes. Applying deterministic averaging with respect to "1 ; : : : ; "n , one obtained the following averaged equations:
p(I;
1
+ 2k1 *; : : : ;
(15)
I ∈ S; (16)
+ 2k *; t | I0 ; 0 )
=p(I; ; t | I0 ; 0 ):
(17)
dIr = UM r (I; ) dt + 1=2 M(1) rs (I; ) dBs (t);
(12a)
Eqs. (16) and (17) imply vanishing probability Oows in n directions at the boundary S, and periodic boundaries with respect to v , respectively.
= VM v (I; ) dt + 1=2 M(2) vs (I; ) dBs (t);
(12b)
3.3. Both internal and external resonances
d
v
where UM r =
1 (2*)n+l−
2*
0
Lvr VM v = v + (2*)n+l−
2*
0
0
2*
Suppose that there are internal resonant relations and external resonant relations in system (7), i.e., Ur (I; ; X; !) dXd!;
Lur !r = u ;
Lvr !r + Mkv k = v ;
0
2*
Vr (I; ; X; !) dXd!;
u = 1; : : : ; ;
2*
0
r(1) (I; ; X) r(1) (I; ; X) dX; 1 s1 2 s1
(2) bv1 r1 = br1 v1 = M(1) r1 s1 M v1 s1
=
Lvr21
(2*)n
2*
0
r(1) (I; ; X) r(2) (I; ; X) dX; 1 s1 2 s1
(2) bv1 v2 = M(2) v1 s1 M v2 s1
=
Lvr11 Lvr22 (2*)n
v = 1; : : : ; ;
r = 1; : : : ; n;
k = 1; : : : ; m;
(1) br1 r2 = M(1) r1 s1 M r2 s1
1 = (2*)n
(18a)
0
2*
r(2) (I; ; X) r(2) (I; ; X) dX: (13) 1 s1 2 s1
The averaged FPK equation associated with Eqs. (12a) and (12b) is 9(UM r p) 9(VM v p) 1 92 (br1 r2 p) 9p = − − + 9t 9Ir 9 v 2 9Ir1 9Ir2 1 92 (bv1 v2 p) 92 (br1 v1 p) (14) + + 9Ir1 9 v1 2 9 v 1 9 v2
(18b)
where Lur ; Lvr ; Mkv are integers and not all zero for given u or v. Introducing (1 6 6 n − 1) combinations .u of angle variables and (1 6 6 l) combinations v of angle variables and excitation phase angles, .u = Lur "r ; v
= Lvr "r + Mkv k (t);
(19a) (19b)
"1 ; : : : ; "n and 1 (t); : : : ; l (t) are replaced by .1 ; : : : ; . ; "+1 ; : : : ; "n ; 1 ; : : : ; ; +1 (t); : : : ; l (t). The Itˆo di:erential equations for I1 ; : : : ; In ; .1 ; : : : ; . , 1 ; : : : ; ; "+1 ; : : : ; "n are obtained from Eqs. (7a) and (7b) by using transformations (19a) and (19b) and Itˆo di:erential rule as follows: dIr = Ur (I; "; ; X1 ; !) dt (1) + 1=2 rs (I; "; ; X1 ) dBs (t);
(20a)
d.u = [ u + Lur Vr (I; "; ; X1 ; !)] dt (2) + 1=2 Lur rs (I; "; ; X1 ) dBs (t);
(20b)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
d
v
= [ v + Lvr Vr (I; "; ; X1 ; !)] dt
(2) br1 u2 = ˜(1) r1 s1 ˜ u2 s1 =
(2) + 1=2 Lvr rs (I; "; ; X1 ) dBs (t);
(20c) ×
d"u = [!u + Vu (I; "; ; X1 ; !)] dt + 1=2 u(2) s (I; "; ; X1 ) dBs (t); r = 1; : : : ; n;
u = 1; : : : ; ;
v = 1; : : : ; ;
u = + 1; : : : ; n;
(20d)
where " = {.1 ; : : : ; . }; X1 = {"+1 ; : : : ; "n }. It can be see from Eqs. (20a)–(20d) that I1 ; : : : ; In ; .1 ; : : : ; . ; 1 ; : : : ; are slowly varying processes while "+1 ; : : : ; "n ; +1 ; : : : ; l are rapidly varying processes. Applying deterministic averaging with respect to "+1 ; : : : ; "n , one obtained the following averaged equations: dIr = U˜ r (I; "; ) dt + 1=2 ˜(1) rs dBs (t); d.u = V˜ u (I; "; ) dt + 1=2 ˜(2) us dBs (t); v
= W˜ v (I; "; ) dt + 1=2 ˜(3) vs dBs (t);
(21b) (21c)
1 ˜ Ur = (2*)n+l−− ×
2*
0
×
×
2*
2*
0
2*
0
0
Vr (I; "; ; X1 ; !) dX1 d!;
Lrv n+l−− (2*)
2*
0
(1) br1 r2 = ˜(1) r1 s1 ˜ r2 s1 =
×
Ur (I; "; ; X1 ; !) dX1 d!;
Lur n+l−− (2*)
0
W˜ v = v +
2*
0
V˜ u = u +
2*
0
×
Vr (I; "; ; X1 ; !) dX1 d!; 1 (2*)n−
r(1) (I; "; ; X1 ) r(1) (I; "; ; X1 ) dX1 ; 1 s1 2 s1
2*
0
×
2*
0
2*
0
×
0
2*
Lur11 Lur22 (2*)n−
r(2) (I; "; ; X1 ) r(2) (I; "; ; X1 ) dX1 ; 1 s1 2 s1 Lur11 Lvr22 (2*)n−
r(2) (I; "; ; X1 ) r(2) (I; "; ; X1 ) dX1 ; 1 s1 2 s1
(3) bv1 v2 = ˜(3) v1 s1 ˜ v2 s1 =
Lvr22 (2*)n−
r(1) (I; "; ; X1 ) r(2) (I; "; ; X1 ) dX1 ; 1 s1 2 s1
(3) bu1 v2 = ˜(2) u1 s1 ˜ v2 s1 =
×
Lur22 (2*)n−
r(1) (I; "; ; X1 ) r(2) (I; "; ; X1 ) dX1 ; 1 s1 2 s1
(2) bu1 u2 = ˜(2) u1 s1 ˜ u2 s1 =
(21a)
where
2*
(3) br1 v2 = ˜(1) r1 s1 ˜ v2 s1 =
d
Lvr11 Lvr22 (2*)n−
r(2) (I; "; ; X1 ) 1 s1
× r(2) (I; "; ; X1 ) dX1 : 2 s1 The averaged FPK equation associated Eqs. (21a)–(21c) is 9p 9(U˜ r p) 9(V˜ u p) 9(W˜ v p) − − = − 9t 9Ir 9/u 9 v +
1425
(22) with
1 92 (br1 r2 p) 1 92 (bu1 u2 p) + 2 9Ir1 9Ir2 2 9/u1 9/u2
1 92 (bv1 v2 p) 92 (br1 u2 p) 92 (br1 v2 p) + + 2 9 v1 9 v2 9Ir1 9/u2 9Ir1 9 v2 92 (bu1 v2 p) (23) + 9/u1 9 v2 +
with initial condition p = p(I; "; ; 0 | I0 ; "; 0 ) = (I − I0 ) (" − "0 ) ( − 0 )
(24)
1426
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
and boundary conditions 9 9(bru p) 9(brv p) −2U˜r p + (brr2 p) + + = 0; 9Ir2 9/u 9 v I ∈ S;
(25)
p(I; /1 + 2k1 *; : : : ; / + 2k *; 1
+ 2k+1 *; : : : ;
+ 2k+ *; t | I0 ; "0 ; 0 )
=p(I; "; ; t | I0 ; "; 0 ):
(26)
Eqs. (25) and (26) imply vanishing probability Oows in n directions at the boundary S and periodic boundaries with respect to /u and v , respectively. Note that original equations of motion, Eqs. (1a) and (1b), are non-autonomous while the averaged equations, Eqs. (12a), (12b) and (21a)–(21c) are autonomous. There is only potential probability Oow in averaged FPK equations (14) and (23). The dimension of averaged Itˆo equations is equal to n plus the number of sum of internal and external resonant relations and it is usually less than 2n. These are the three advantages of the stochastic averaging method proposed in the present paper.
2 X2 + 2 X23 XQ 2 + (20 + 21 X12 + 22 X22 )X˙ 2 + !20
=h2 cos t + W2 (t) + X2 W4 (t);
where ij are coe;cients of linear and non-linear dampings; !i0 ; i ; hi are positive constants representing the frequencies of associated degenerated linear oscillators, the strengths of non-linear sti:ness of the oscillators and the amplitudes of harmonic excitation, respectively. Wi (t) are independent Gaussian white noises in the sense of Stratonovich with intensities 2Di ; ij ; hi ; Di are assumed to be of the same order of . The Itˆo di:erential equations for independent integrals of motion Hi and angle variables "i are of the form dHi = {Pi [ − Pi (i0 + i1 Q12 + i2 Q22 ) + hi cos t] +Dii + D(i+2)(i+2) Qi2 } dt +Pi dBi (t) + Pi Qi dBi+2 (t);
4. Example Consider two non-linearly coupled Du;ng–van der Pol oscillators subject to additive harmonic excitations and both additive and parametric excitations of Gaussian white noises. The equations of motion are of the form 2 XQ 1 + (10 + 11 X12 + 12 X22 )X˙ 1 + !10 X1 + 1 X13 =h1 cos t + W1 (t) + X1 W3 (t);
(27a)
(28)
9"i [ − Pi (i0 + i1 Q12 + i2 Q22 ) d"i = !i + 9Pi
3.4. Remark In the practical application of the proposed averaging method, it is more convenient to replace n action variables I1 ; : : : ; In with n independent integrals of motion, H1 ; : : : ; Hn , in involution because it is di;cult to obtain the action variables Ii in most cases. The Itˆo di:erential equations for H1 ; : : : ; Hn and "1 ; : : : ; "n can be obtained from Eqs. (7a) and (7b) by using transformation Hr = Hr (I) and Itˆo di:erential rule. The averaged Itˆo equations for Hr and , or Hr ; "; can be derived similarly as those in Sections 3.2 and 3.3.
(27b)
+ hi cos t] + [Dii + +
2
"i 9Pi2
9 D(i+2)(i+2) Qi2 ]
dt
9"i 9"i dBi (t) + Qi dBi+2 (t); 9Pi 9Pi
where Qi = Xi ;
Pi = X˙ i ;
!2 Pi2 i + i0 Qi2 + Qi4 ; 2 2 4 √ 2 * i ai + b2i !i = √ ; 2 2 K(mM i )
Hi =
* F(Zi ; mM i ); 2K(mM i ) Qi ; Zi = arccos ai
2 !i0 4i Hi 2 ai = 1+ −1 ; 4 i !i0 "i =
b2i
!2 = i0 i
4i Hi 1+ +1 ; 4 !i0
ai mM i = ; 2 ai + b2i
i = 1; 2:
(29)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
ai is the amplitude of oscillator i; K(mM i ) and F(Zi ; mM i ) are the complete elliptic integral and the elliptic integrals of the 9rst kind, respectively. Pi ; Qi ; 9"i =9Pi ; 92 "i =9Pi2 can be expanded into Fourier series and only the 9rst several leading terms needed to be retained in analysis. Two cases are considered in following subsections.
and qMij ; pM ij ; "Mij are the coe;cients of the following Fourier expansions: ∞
Qi = qMij cos(2j − 1)"i ; j=1
Pi =
∞
Consider the case of primary resonance between the harmonic excitation and the 9rst oscillator of system (28) and no internal resonance between the two oscillators. Assume that (30)
where is a detuning parameter. Introducing new variable 5 as de9ned in Eq. (10). Following the procedure in Subsection 3.2, one obtains the averaged Itˆo equations for H1 ; H2 and 5 as follows: dH1 = m1 (H1 ; H2 ; 5) dt + 1k dBk (t);
(31)
m1 = −10 pM s11 − 11 pM q11 − 12 pM s11 qMs21
Qi2 =
∞
qMsij cos(2j − 2)"i ;
j=1
Pi Qi =
∞
pM qij sin 2j"i ;
j=1
9"i = 9Pi
∞
"Mij cos(2j − 1)"i ;
j=1
∞
p 9"i P= "Mij sin 2j"i ; 9Pi
i = 1; 2:
(33)
j=1
da1 = m˜ 1 (a1 ; a2 ; 5) dt + ˜1k d BM k (t);
+D44 qMs21 ;
da2 = m˜ 2 (a1 ; a2 ; 5) dt + ˜2k d BM k (t);
h1 M m3 = − !1 − "11 cos 5; 2 b11 = 1k 1k = 2D11 pM s11 + 2D33 pM q11 ;
d5 = m˜ 3 (a1 ; a2 ; 5) dt + ˜3k d BM k (t);
(34)
where dai 1 d 2 ai m˜ i = mi + bii ; dHi 2 dHi2 Hi =Hi (ai ) 2 da i b˜ii = ˜ik ˜ik = bii ; i = 1; 2; dHi
b22 = 2k 2k = 2D22 pM s21 + 2D44 pM q21 ; ∞
"M21j + D33 2("Mq11 )2 b33 = 3k 3k = D11 j=1
Hi =Hi (ai )
("Mq1j )2 ;
bij = ik jk = 0(i = j):
j=1
Note that ai is related to Hi only as shown in Eq. (29). The Itˆo di:erential equations for a1 ; a2 and 5 can be derived by using Itˆo di:erential rule as follows:
h1 pM 11 sin 5 + D11 + D33 qMs11 ; − 2 m2 = −20 pM s21 − 21 pM s21 qMs11 − 22 pM q21 + D22
j=2
pM sij cos(2j − 2)"i ;
j=1
where Bk (t) are independent unit Wiener processes,
+
∞
∞
d5 = m3 (H1 ; H2 ; 5) dt + 3k dBk (t);
∞
Pi2 =
q 9"i Qi = "Mij cos(2j − 2)"i ; 9Pi
dH2 = m2 (H1 ; H2 ; 5) dt + 2k dBk (t); k = 1; : : : ; 4;
pM ij sin(2j − 1)"i ;
j=1
4.1. External resonance only
− !1 = ;
1427
m˜ 3 = m3 (H1 ; H2 ; 5)|Hi =Hi (ai ) ; (32)
b˜33 = ˜3k ˜3k = b33 |Hi =Hi (ai ) :
(35)
1428
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
The averaged FPK equation associated with averaged Itˆo equation (34) is
(36)
It is well known that in a Du;ng oscillator with hardening sti:ness subject to harmonic excitation the phenomenon of sharp jumps in amplitude may occur. Averaged Itˆo equation (34) is reduced to ordinary differential equation when Di = 0. The stationary amplitude response curves of the system can be obtained by letting da1 =dt = da2 =dt = d5=dt = 0 and then are shown in Fig. 1. It is seen from Figs. 1(a) and (b) that the amplitude responses of the two oscillators are triple-valued in certain frequency interval of harmonic excitation. Among the three values of amplitudes, two are stable while the other is unstable. The deterministic jump occurs at the two extreme values of the frequency interval of triple-valued amplitude. A typical response exhibiting stochastic jump phenomena of two coupled Du;ng–van der Pol oscillators under combined harmonic and white noise excitations is shown in Fig. 2. Figs. 2(a) and (b) are the stationary joint probability densities of amplitude and phase of the 9rst oscillator obtained from solving the averaged FPK equation (36) by using 9nite di:erence method and from digital simulation of original system (27), respectively. Figs. 2(c) and (d) are the stationary probability densities of amplitude and phase, respectively, of the 9rst oscillator. Figs. 2(e) and (f) are the sample functions of displacements of the 9rst and second oscillators, respectively, of original system (27) form digital simulation. It is seen from Fig. 2 that the probability densities obtained from the combination of the stochastic averaging method and the 9nite di:erence method are in good agreement with those from digital simulation of original system (27). The stationary joint probability densities are bimodal. It implies that there are two more probable motions in the response of the coupled Du;ng–van der Pol oscillators under combined harmonic and white noise excitations and stochastic jump may occur. This is veri9ed by the sample functions of displacements of the system. The stochastic jump of the coupled Du:-
B
1.2
A
1
C
0.8
D
0.6 0.4 0.2 0
1
1.2
1.4
1.6
(a)
Ω
1.8
2
2.2
2.4
2.2
2.4
2 1.8 1.6 D
1.4
a2
1 92 (b˜22 p) 1 92 (b˜33 p) + + : 2 2 9 2 9a22
1.4
a1
9p 9(m˜ 1 p) 9(m˜ 2 p) 9(m˜ 3 p) 1 92 (b˜11 p) − − =− + 9t 9a1 9a2 9 2 9a21
1.6
1.2
C
1
A
0.8
B
0.6 0.4 0.2 0
(b)
1
1.2
1.4
1.6
1.8
2
Ω
Fig. 1. Amplitude response curves of two coupled Du;ng– van der Pol oscillators under pure harmonic excitation with external resonance only. !10 = 1:0; !20 = 2:0; 10 = 11 = 12 = 21 = 22 = 0:05, 20 = −0:05. (a) First oscillator, (b) second oscillator. A: 1 = 2 = 2:0; h1 = h2 = 0:2; B: 1 = 2 = 2:0; h1 = h2 = 0:25; C: 1 = 2 = 2:0; h1 = h2 = 0:13; D: 1 = 2 = 5:0; h1 = h2 = 0:2.
ing–van der Pol oscillators under harmonic and white noise excitations can be regarded as random spread of deterministic jump of the same oscillators under pure harmonic excitations, as shown by curve A in Fig. 1. However, there are signi9cant di:erence between the deterministic jump and stochastic jump. The deterministic jump occurs only at two extreme values of the frequency interval of triple-valued amplitude and only from upper branch to lower branch of amplitude response curve A at the right extreme value of the frequency interval or from lower branch to upper branch at the left extreme value of the frequency interval while the stochastic jumps may occur back and forth at any frequency within this frequency interval.
6
5
1 0.8 0.6
4
ψ
3
(a)
1
5
0.8 0.6
4
ψ
3
a1
0.4
2 0.2
1
(b)
0 0
0.45
2 1.8
0.4
1.6
0.35
1.4
0.3
1.2
p (ψ)
p (a1)
0.8 0.6 0.4 0.2 0 6
0 0
1.2 1 0.8 0.6 0.4 0.2 0 1.4 1.2
p(a1,ψ )
1
0.2
1
1429
1.2
a1
0.4
2
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1.4 1.2
p(a1,ψ )
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
p(a1,ψ )
p(a1,ψ )
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
1 0.8
0.25 0.2 0.15
0.6 0.4
0.1
0.2
0.05
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
(c)
0
1.6
a1
0
1
2
900
1,000
(d)
3
ψ
4
5
6
7
2
1.5
1.5
1
1 0.5
X2
X1
0.5 0
0 -0.5
-0.5
-1 -1 -1.5 800
(e)
-1.5 900
1,000
1,100
1,200
1,300
-2 800
1,400
time
(f)
1,100
1,200
1,300
1,400
time
Fig. 2. Stationary probability densities and sample functions of the responses of system (27) with external resonance only. !10 = 1:0; !20 =2:0; 10 =11 =12 =21 =22 =0:05, 20 =−0:05, 1 =2 =2:0; h1 =h2 =0:2; =1:5; D1 =D2 =D3 =D4 =0:01. (a) Joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation obtained from stochastic averaging method, (b) joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation obtained from digital simulation of original system (27), (c) probability density of amplitude of the 9rst oscillator, (d) probability density of phase di:erence between the 9rst oscillator and the harmonic excitation, (e) sample function of displacement of the 9rst oscillator, (f) sample function of displacement of the second oscillator, —: from stochastic averaging method; •: from digital simulation of original system (27).
As in the deterministic case, the occurrence of stochastic jump depends on the system parameters, such as the frequency and amplitude of harmonic ex-
citation, the intensities of white noises, the strength of non-linearity and the linear and non-linear damping coe;cients. Since the occurrence of stochastic
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
1 0.8 0.6 a 1 0.4
5 4
ψ
3 2
0.6 0.4 0.2 0
1.4 1.2
6
0
1.4 1.2 1 0.8
5 4 0.6
3
ψ
0.4
2
0
1 0.8
5 4
ψ
(b)
0.6
3 0.4
2
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1.4 1.2
6
a1
0
4
ψ
0
Fig. 3. Stationary joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation of system (27) with external resonance only. The parameters are the same as those in Fig. 2 except = 1:3. (a) From stochastic averaging method, (b) from digital simulation of original system (27).
jump is related to bimodal probability, we call the appearance or disappearance of stochastic jump as the system change the bifurcation of stochastic jump. For = 1:3 and 1.8, the joint stationary probability densities of amplitude and phase of the 9rst oscillator obtained from solving averaged FPK equation (36) by using 9nite di:erence method are shown in Figs. 3(a) and 4(a), respectively, while the results obtained from digital simulation of original system (27) are shown in Figs 3(b) and 4(b), respectively. For 1 = 2 = 5:0, the joint probability densities of amplitude and phase of the 9rst oscillator obtained from solving averaging FPK equation (36) by using 9nite di:erence method and from digital simulation of original system (27) are shown in Figs. 5(a) and (b), respectively. It can be seen from these 9gures that the analytical results agree well with those from digital simulation of original system (27) and the joint probability densities are unimodal and no jump may occur. This is because
1 0.8
5
0.2
1
0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1.4 1.2
6
0
p (a1,ψ)
4 3.5 3 2.5 2 1.5 1 0.5 0
p (a1,ψ)
p (a1,ψ)
(a)
a1
0.2
1
4 3.5 3 2.5 2 1.5 1 0.5 0
p (a1,ψ)
0.8 0.6 0.4 0.2 0
0.8
0.2
1
(a)
1
1
p (a1,ψ)
6
1.2
1.2
p (a1,ψ)
4 3.5 3 2.5 2 1.5 1 0.5 0
4 3.5 3 2.5 2 1.5 1 0.5 0
p (a1,ψ)
p (a1,ψ)
1430
(b)
0.6
3 0.4
2
a1
0.2
1 0
0
Fig. 4. Stationary joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation of system (27) with external resonance only. The parameters are the same as those in Fig. 2 except = 1:8. (a) From stochastic averaging method, (b) from digital simulation of original system (27).
the values of frequencies are well o: the interval of triple-valued amplitude. Similar conclusions can also be drawn for other system parameters. So, the bifurcation of the stochastic jump of the system can be roughly estimated based on the amplitude response curve of the same system under pure harmonic excitation as the system parameter change. 4.2. Both internal and external resonances Consider the case of both primary external resonance between the harmonic excitation and the 9rst oscillator and internal resonance between the two oscillators of system (27). Assume that − !1 = ; !1 − !2 = ;
(37)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
p (a1,ψ)
4 3.5 3 2.5 2 1.5 1 0.5 0 6
5
3 2
(a)
pM 2j qM1j cos 2(j − 1).;
j=2
m∗3 = m3 (H1 ; H2 ; 5) − 12
m∗4 = m4 (H1 ; H2 ; 5) +
0.2
1
∞
∞
j=1
1 0.8 0.6 a1 0.4
4
ψ
m∗2 = m2 (H1 ; H2 ; 5) − 21 p (a1,ψ)
4 3.5 3 2.5 2 1.5 1 0.5 0 1.4 1.2
1431
∞
j=1
0 0
"Mp1j qMs2j sin 2j.;
(21 "Mp2j qMs1j − 12 "Mp1j qMs2j )
4 3.5 3 2.5 2 1.5 1 0.5 0 6
(b)
5
4
ψ
3 0.4
2
1 0.8 0.6 a
p (a1,ψ)
p (a1,ψ)
×sin 2j.; 4 3.5 3 2.5 2 1.5 1 0.5 0 1.4 1.2
b∗11 = b11 ;
b∗22 = b22 ;
b∗ij = 0(i = j); b∗44 =
∞
j=1
1
(D11 ("M1j )2 + D22 ("M2j )2 ) + 2D33 ("Mq11 )2
0.2
1
+2D44 ("Mq21 )2 +
0 0
Fig. 5. Stationary joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation of system (27) with external resonance only. The parameters are the same as those in Fig. 2 except 1 = 2 = 5; = 1:3. (a) From stochastic averaging method, (b) from digital simulation of original system (27).
b∗33 = b33 ;
∞
j=2
[D33 ("Mq1j )2
+D44 ("Mq2j )2 ];
where mi (H1 ; H2 ; 5) and bii are de9ned in Eq. (32). The Itˆo di:erential equations for a1 ; a2 ; 5 and . can also be derived as follows
where and are detuning parameters. Introduce new variables 5 and . as de9ned in Eqs. (19a) and (19b). Following the procedures described in Subsection 3.3, one obtains the averaged Itˆo equations for H1 ; H2 ; 5 and . as follows
da2 = m˜ ∗2 (a1 ; a2 ; 5; .) dt + ˜∗2k d BM k (t);
∗ dH1 = m∗1 (H1 ; H2 ; 5; .) dt + 1k dBk (t);
d. = m˜ ∗4 (a1 ; a2 ; 5; .) dt + ˜∗4k d BM k (t);
∗ dH2 = m∗2 (H1 ; H2 ; 5; .) dt + 2k dBk (t);
where
∗ d5 = m∗3 (H1 ; H2 ; 5; .) dt + 3k dBk (t);
m˜ ∗1 =
∗ d. = m∗4 (H1 ; H2 ; 5; .) dt + 4k dBk (t);
(38)
where m∗1
= m1 (H1 ; H2 ; 5) − 12
∞
j=2
pM 1j qM2j cos 2(j − 1).;
(39)
da1 = m˜ ∗1 (a1 ; a2 ; 5; .) dt + ˜∗1k d BM k (t);
d5 = m˜ ∗3 (a1 ; a2 ; 5; .) dt + ˜∗3k d BM k (t);
da1 ∗ 1 d 2 a1 ∗ m1 + b ; dH1 2 dH12 11 Hi =Hi (ai ) da1 ∗ 1 d 2 a2 ∗ m2 + b ; m˜ ∗2 = dH1 2 dH22 22 Hi =Hi (ai )
m˜ ∗3 = m∗3 (H1 ; H2 ; 5; .)|Hi =Hi (ai ) ; m˜ ∗4 = m∗4 (H1 ; H2 ; 5; .)|Hi =Hi (ai ) ;
(40)
b˜∗22 = ˜∗2k ˜∗2k =
da1 dH1 da2 dH2
2
b∗11
2
; Hi =Hi (ai )
b∗22
3 2.5 2 1.5 1 0.5 0
; Hi =Hi (ai )
6
5
b˜∗33 = ˜∗3k ˜∗3k = b∗33 |Hi =Hi (ai ) ;
ψ
(41)
The averaged FPK equation associated with Eq. (40) is as follows 9p 9(m˜ ∗1 p) 9(m˜ ∗2 p) 9(m˜ ∗3 p) 9(m˜ ∗4 p) =− − − − 9t 9a1 9a2 9 9/
1 92 (b˜∗11 p) 92 (b˜∗22 p) 92 (b˜∗33 p) + + + 2 9 2 9a21 9a22 92 (b˜∗44 p) + 9/2
:
2 0.2
1
(42)
The joint probability densities of amplitude and phase of the 9rst oscillator obtained from solving FPK equation (42) by using 9nite di:erence method and from digital simulation of original system (27) are shown in Figs. 6(a) and (b), respectively. The stationary probability densities of a1 ; 5; ., respectively, are shown in Figs. 7(a)–(c). It can be seen from these 9gures that the analytical solutions agree well with those from digital simulation of original system and no jump may occur in the system. 5. Concluding remarks A stochastic averaging method for quasi-integrable Hamiltonian system under combined harmonic and white noise excitations is proposed. It is shown that the dimension and form of the averaged equations depend upon the number of internal and (or) external resonant relations. It is noted that the stochastic averaging method for quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations developed in the present paper is reduced to
p (a1,ψ)
(i = j):
3
(a)
b˜∗44 = ˜∗4k ˜∗4k = b∗44 (H1 ; H2 ; 5; .)|Hi =Hi (ai ) ; b˜∗ij = ˜∗ik ˜∗jk = 0
1 0.8 0.6 a 1 0.4
4
0 0
3 2.5 2 1.5 1 0.5 0 1.4 1.2
3 2.5 2 1.5 1 0.5 0 6
(b)
3 2.5 2 1.5 1 0.5 0 1.4 1.2
5
1 0.8 0.6 a1 0.4
4
ψ
p (a1,ψ)
b˜∗11 = ˜∗1k ˜∗1k =
p (a1,ψ)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
p (a1,ψ)
1432
3 2 0.2
1 0 0
Fig. 6. Stationary joint probability density of amplitude and phase di:erence between the 9rst oscillator and the harmonic excitation of system (27) with both external and internal resonances. !10 = !20 = 1:0; 1 = 2 = 2:0; = 1:3; h1 = h2 = 0:2, ij = 0:05 (i = 1; 2; j = 0; 2), (a) from Di = 0:01(i = 1; 4); stochastic averaging method, (b) from digital simulation of original system (27).
the stochastic averaging method for quasi-integrable Hamiltonian systems [14] when no external resonance exist in system (1). The proposed procedures have been successfully applied to predict the responses, the stochastic jump and bifurcation of the two coupled Du;ng–van der Pol oscillators under combined harmonic external excitation and additive and parametric white noises excitations. The proposed procedures can be further extended to quasi-integrable Hamiltonian systems under wide band or narrow band random excitations. These will be our future research subjects. Acknowledgements The work reported in the paper was supported by National Natural Science Foundation of China under Key Grant No. 10332030 and Grant No. 10002015, the special Fund for Doctor Programs in Institutions
3.5
0.7
3
0.6
2.5
0.5
2
0.4
p (φ)
p (a1)
Z.L. Huang, W.Q. Zhu / International Journal of Non-Linear Mechanics 39 (2004) 1421 – 1434
1.5
1433
0.3
1
0.2
0.5
0.1 0
0 0
(a)
0.2
0.4
0.6
0.8
1
1.2
1.4
0
1.6
1
2
3
6
7
(b)
a1
φ
4
5
6
7
0.8 0.7 0.6
p (ψ)
0.5 0.4 0.3 0.2 0.1 0
0
1
2
(c)
3
ψ
4
5
Fig. 7. Stationary probability densities of the responses of system (27) with both external and internal resonances. The parameters are the same as those in Fig. 6. (a) Stationary probability density of amplitude of the 9rst oscillator, (b) stationary probability density of angle variable di:erence between two oscillators, (c) stationary probability density of phase di:erence between the 9rst oscillator and the harmonic excitation, —: from stochastic averaging method; •: from digital simulation of original system (27).
of Higher Learning of China under Grant No. 20020335092 and Zhejiang Provincial Natural Science Foundation under Grant No. 102040.
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