Stochastic stationary response of a variable-mass system with mass disturbance described by Poisson white noise

Physica A 473 (2017) 122–134

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Stochastic stationary response of a variable-mass system with mass disturbance described by Poisson white noise Yan Qiao, Wei Xu ∗ , Wantao Jia, Qun Han Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, PR China

highlights • • • •

The stationary responses of nonlinear stochastic variable-mass system are studied. The mass disturbance has marked influence on system responses. Differences between two kinds of mass-disturbance on system responses are explored. The theoretical analyses are verified by numerical results.

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Article history: Received 11 May 2016 Received in revised form 22 December 2016 Available online 5 January 2017 Keywords: Variable mass Poisson white noise Stochastic averaging Stochastic response

abstract Variable-mass systems have received widespread attention and show prominent significance with the explosive development of micro- and nanotechnologies, so there is a growing need to study the influences of mass disturbances on systems. This paper is devoted to investigating the stochastic response of a variable-mass system subject to weakly random excitation, in which the mass disturbance is modeled as a Poisson white noise. Firstly, the original system is approximately replaced by the associated conservative system with small disturbance based on the Taylor expansion technique. Then the stationary response of the approximate system is obtained by applying the stochastic averaging method. At last, a representative variable-mass oscillator is worked out to illustrate the effectiveness of the analytical solution by comparing with Monte Carlo simulation. The relative change of mean-square displacement is used to measure the influences of mass disturbance on system responses. Results reveal that the stochastic responses are more sensitive to mass disturbance for some system parameters. It is also found that the influences of Poisson white noise as the mass disturbance on system responses are significantly different from that of Gaussian white noise of the same intensity. © 2017 Elsevier B.V. All rights reserved.

1. Introduction In stochastic dynamics, there were considerable efforts devoted to the study of invariable-mass systems while the variable-mass systems have not drawn enough attention. In fact, in natural phenomena and engineering technology domains, it is unreasonable for some systems that the mass is considered to be constant [1,2]. For instance, comets loose part of mass due to the gravitational interaction when traveling around the other stars [3,4]. The critical problem-oriented models in engineering fields include the motion of rockets which change mass due to fuel burning [5,6], the motion of jet planes [7], the fluid–structure interaction [8], etc. For those systems with big mass, the expelled or captured mass is relatively

∗

Corresponding author. E-mail address: [email protected] (W. Xu).

http://dx.doi.org/10.1016/j.physa.2017.01.039 0378-4371/© 2017 Elsevier B.V. All rights reserved.

Y. Qiao et al. / Physica A 473 (2017) 122–134

123

small and has obvious regularity. Accordingly, mass variation can be described by continuous functions. By far, researches mainly focus on dynamical analyses of deterministic systems with specified mass variation [9], especially with monotone slow-time mass variation [10]. However, in some cases, the mass variation cannot be described with continuous functions due to the existence of unavoidable stochastic mass disturbances [11,12]. In recent years, Microelectromechanical system (MEMS) has attracted enormous attention due to its advantages of small volume, light weight, low energy consumption and intelligence, etc. [13], which has great application prospects and has got brilliant achievements, e.g., miniature motor, micro gyroscope, miniature photoelectric sensors and so on [14,15]. In contrast to the macro mechanical–electrical systems, the scale and mass of MEMS are so small that the adsorption and desorption will significantly change its mass, thereby influencing system dynamic behaviors. For instance, the probe in atomic force microscopy is a crucial component to detect the topographical information of microstructures. Stochastic mass disturbance inevitably reduces its detection precision [16]. In addition, an interesting application of the properties induced by mass variation is detecting the mass of specific molecular [17,18]. Therefore, it is essential to investigate dynamical behaviors of system with mass disturbance in assessing system performance and improving the detection precision. One point worth considering is that the mass variation of MEMS is no longer a small amount when compared with the original mass, which cannot be regarded as a continuous function but discontinuous wave over time. Moreover, due to the random characteristics of adsorption and desorption, mass disturbance should be described by stochastic process. It is worth noting that the differential equation of motion for stochastic variable-mass systems contain random items related to acceleration, which is different from traditional differential equation in essence. Thus those traditional analysis methods cannot be directly utilized to stochastic variable-mass systems. Proper treatment on the mass disturbance is the key to solve this problem. Recently, the stochastic averaging for quasi-integrable Hamiltonian systems with variable mass and mass disturbances described by Gaussian white noise was proposed by Wang [19]. Zhong et al. investigated the stochastic resonance phenomenon in a fractional oscillator with random mass modeled as a trichotomous noise [20]. To the best of our knowledge, Poisson white noise, as a random pulse, can be considered to simulate the mass disturbance in MEMS. However, the stochastic dynamical behaviors of variable-mass systems with mass disturbance described by Poisson white noise have not been reported yet. Since the stochastic averaging method of energy envelope was first proposed by Landa and Stratonovich [21], which has been extensively generalized and applied to investigate the stochastic response, stability and bifurcation of single-degreeof-freedom (SDOF) nonlinear systems with light dampings and weakly random excitations [22–27]. Subsequently, based on the Hamiltonian formulation [28], Zhu and his co-workers have further generalized the stochastic averaging method. Especially, in recent years, this stochastic averaging method is developed to study the stochastic response and stability of the nonlinear systems under the excitation of non-Gaussian random processes, such as Poisson white noise [29–32]. Motivated by the above findings, this paper aims to study the stochastic response of a variable-mass system under the external and parametric excitations of Gaussian white noise and small mass disturbance described by Poisson white noise. The paper is organized as follows. A variable-mass system is introduced and replaced by approximate system in Section 2. In Section 3, stochastic averaging method and perturbation method are successively applied to obtain the statistics of stationary response of the approximate system. In Section 4, an example is given in detail to illustrate the validity of the present method. Furthermore, the effects of mass disturbance on stationary responses are explored adequately and the comparison with Gaussian white noise mass perturbation are carried out. Finally, the paper ends with some conclusions in Section 5. 2. System description Consider a SDOF variable-mass system with mass disturbance modeled as Poisson white noise. The motion is governed by the following equation [19]: mx¨ + ε 2 c (x, x˙ )˙x + u(x) = ε f (x, x˙ )ξe (t ),

¯ + ε g ξm (t ), m=m

(1)

¯ subjected to small disturbance ξm (t ); m ¯ and g are positive constants; in which variable mass m is described by mean mass m ε is a small parameter; c (x, x˙ ) is a differentiable function denoting the damping coefficient; f (x, x˙ ) is an infinite differentiable function representing the amplitude of Gaussian white noise excitation ξe (t ) which satisfies ⟨ξe (t )⟩ = 0,

⟨ξe (t )ξe (t + τ )⟩ = 2Dδ(τ ).

(2)

The mass disturbance ξm (t ) is Poisson white noise with zero mean, which can be viewed as the formal derivative of the following compound Poisson process C (t ): C (t ) =

N (t ) 

Yi U (t − Ti ),

(3)

i=1

ξm (t ) =

N (t )  i=1

Yi δ(t − Ti ) =

dC (t ) dt

,

(4)

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Y. Qiao et al. / Physica A 473 (2017) 122–134

where N (t ) denotes Poisson counting process with mean arrival rate λ > 0; U (•) is a unit step function; {Yi , i = 1, 2, . . .} are independent identically Gaussian distributed random variables representing the impulse amplitudes, which are independent of the pulse arrival time Ti ; δ(•) is Dirac delta function. The r-order correlation function of the Poisson white noises is of the form [30] Rr [ξm (t1 ), . . . ξm (tr )] = λE [Y r ]δ(t2 − t1 ) · · · δ(tr − t1 ),

(5)

the mean value of dC (t ) is E [(dC (t )) ] = λE [Y ]dt , r = 1, 2, 3, . . . , ∞. Note that dC (t ) is of the same order as dt for r = 1, 2, 3, . . . , ∞. Besides, ξm (t ) is supposed to be independent with random excitation ξe (t ). Let X1 = x and X2 = mx˙ , the original system (1) is equivalent to the following first-order equations: r

X˙ 1 =

X2

=

m

r

X2

¯ + ε g ξm (t ) m

r

r

,

X˙ 2 = −u(X1 ) − ε 2 c (X1 , X2 )

X2

¯ + ε g ξm (t ) m

(6)

+ εf ξe (t ),

where X1 and X2 are the displacement and momentum, respectively. As seen in Eq. (6), mass disturbance ξm (t ) appears in the denominator, this phenomenon is the most prominent distinction between the variable-mass system and traditional invariable-mass system. Hence the traditional analysis methods cannot be applied directly. It is noted that the mass ¯ we can get disturbance ε g ξm (t ) is relative small, so using Taylor expansion technique to the first equation of Eq. (6) at m, the following approximate formula [19]: X2

¯ + εg ξm (t ) m

≈

X2

¯ m

−

ε X2 g ξm (t ) + ···. ¯2 m

(7)

To keep certain precision, it is necessary to truncate the expansion to obtain the closed form. Substituting Eq. (7) into Eq. (6) and neglecting all terms of ε 3 and higher yield the following approximate system: X˙ 1 =

X2

¯ m

− εg

1 X2

¯ m ¯ m

ξm (t ),

X˙ 2 = −u(X1 ) − ε 2 c (X1 , X2 )

X2

¯ m

(8)

+ ε f (X1 , X2 )ξe (t ).

Compared to the original variable-mass system (6), the mass disturbance term does not appear in the denominator of the approximate system (8) but is included in the first equation. The small parameter ε suggests that variable-mass system can be regarded as the associated invariable-mass system disturbed by small mass disturbance. For the approximate system (8), stochastic averaging method can be utilized to study the stochastic stationary response [19]. 3. Stochastic averaging approach The Stratonovich stochastic differential equations (SDE) of system (8) are expressed as X2

dt − ε g

1 X2

dC (t ), ¯ ¯ m ¯ m m  √ X2 dX2 = −u(X1 ) − ε 2 c (X1 , X2 ) dt + ε f (X1 , X2 ) 2DdB(t ). ¯ m dX1 =

(9)

¯ )(X2 /m ¯ ) only depends on momentum X2 . Thus, there is no Wong–Zakai correction term for the Note that the term g (1/m first equation of Eq. (9) from Stratonovich SDE to Itô SDE. System (9) can be modeled as the following Itô SDE [33]: dX1 =

X2

¯ m

dt − ε g

1 X2

¯ m ¯ m



dC (t ),

dX2 = −u(X1 ) − ε 2 c (X1 , X2 )

X2

¯ m

+ ε 2 Df (X1 , X2 )

 √ ∂ f (X1 , X2 ) dt + ε f (X1 , X2 ) 2DdB(t ), ∂ X2

(10)

where B(t ) is standard Wiener processes; the term ε 2 Df (X1 , X2 )∂ f (X1 , X2 )/∂ X2 is known as the Wong–Zakai correction term, which can be spited into a conservation part and a dissipative part. The conservation part can be combined with −u(X1 ) to ¯ to constitute effective form overall conservative force −˜u(X1 ). The dissipative part can be combined with −ε 2 c (X1 , X2 )X2 /m ¯ with b = b(X1 , X2 ) [24]. Then Eq. (10) can be rewritten as damping forces −ε 2 bX2 /m X2

dt − ε g

1 X2

dC (t ), ¯ ¯ m ¯ m m  √ X2 dt + ε f (X1 , X2 ) 2DdB(t ). dX2 = −˜u(X1 ) − ε 2 b(X1 , X2 ) ¯ m dX1 =

(11)

Y. Qiao et al. / Physica A 473 (2017) 122–134

125

The total energy of associated conservative system is H =

1

¯ 2m

X22 + G(X1 ),

(12)

X

where G(X1 ) = 0 1 u˜ (y)dy. By introducing the transformation X1 = X1 ,

¯ ). H = H (X1 , X2 , m

(13)

Eq. (11) can be transformed into the Itô SDE for displacement X1 and total energy H. The Itô SDE for H is derived from Eq. (11) by utilizing differential rules for systems subject to Gaussian white noise [24] and Poisson white noise [29] as follows [19]: dH = ε



2

−b

2H − 2G(X1 )

¯ m

1

+ (f

√

2D)

2

2

1



¯ m

dt + ε f



√

2D

2H − 2G(X1 )

¯ m

dB(t )

 r  ∞  1 ∂r H 1 2H − 2G(X1 ) + × εg dC (t ) . ¯ ¯ r ! ∂ X1r m m r =1

(14)

According to Zeng and Zhu [29], the SDE for the new Markov process H (t ) can be obtained. Since there are infinite terms in the SDE equation, it is necessary to truncate the equation to obtain the closed form of the averaged SDE. Neglecting the terms of higher order than 4th order yield the following averaged Itô SDE for H: dH = (ε 2 U0 + ε 3 U1 + ε 4 U2 )dt + σ (H )dB(t ) + ε V11 dC¯ 1 + (ε V21 + ε 2 V22 )dC¯ 2 + (ε V31 + ε 2 V32 + ε 3 V33 )dC¯ 3 ,

(15)

where

     1 √ 1 2h − 2G(x1 ) ¯ dx1 + (f 2D)2 (2h − 2G(x1 ))/m −b ¯  ¯  m m T ( h) 2    1 ¯ dx1 , (2h − 2G(x1 ))/m + A1,2 (x1 , h) T ( h)      1 ¯ dx1 , U1 = A1,3 (x1 , h) (2h − 2G(x1 ))/m T (h)      1 ¯ dx1 , U2 = A1,4 (x1 , h) (2h − 2G(x1 ))/m T (h)     1 2h − 2G(x1 )  2 2 2 ¯ dx1 , σ ( h) = ε f 2D (2h − 2G(x1 ))/m ¯ m T (h) U0 =

1

(16)

(17)

(18)

(19)

and Vkl (k = 1, 2, 3; l = 1, . . . , k) are derived from the following expressions by equating terms of the same order of ε : 1 dt

E

  j ε V11 dC¯ 1 + (εV21 + ε 2 V22 )dC¯ 2 + (εV31 + ε 2 V32 + ε 3 V33 )dC¯ 3 |H = h =

  4    εm ¯ dx1 , Aj,m (x1 , h)/ (2H − 2G(x1 ))/m T (h) m=j

(20)

where C¯ k are independent compound Poisson processes defined by Eq. (3) with the following properties: r = 1, r = 2, 3 · · · , 4 − k + 1, r = 4 − k + 2, 4 − k + 3, . . .

 0 r r ¯ ¯ ¯ E [(dCk ) ] = λk E [Yk ] = Mk,r mk,r Mk,r ≫ ε

(21)

and mk,r < ε 4 .

Specifically, 4 λ¯ 1 E [Y¯14 ]V11 =

 

1



A4,4 (x1 , h)

T ( h)

1

3 3 λ¯ 1 E [Y¯13 ]V11 + λ¯ 2 E [Y¯23 ]V21 =

2 ¯ 2 E [Y¯23 ]V21 V22 = 3λ

1 T (h)

T ( h)

 

  ¯ dx1 , (2h − 2G(x1 ))/m    ¯ dx1 , A3,3 (x1 , h) (2h − 2G(x1 ))/m

(22)

 

A3,4 (x1 , h)

 

 ¯ dx1 , (2h − 2G(x1 ))/m

(23)

(24)

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Y. Qiao et al. / Physica A 473 (2017) 122–134

2 2 2 λ¯ 1 E [Y¯12 ]V11 + λ¯ 2 E [Y¯22 ]V21 + λ¯ 3 E [Y¯32 ]V31 =

¯ 2 E [Y¯22 ]V21 V22 + 2λ¯ 3 E [Y¯32 ]V31 V32 = 2λ

T (h)

 

1

   ¯ dx1 , A2,2 (x1 , h) (2h − 2G(x1 ))/m 

A2,3 (x1 , h)

T ( h)

2 2 λ¯ 2 E [Y¯22 ]V22 + λ¯ 3 E [Y¯32 ](V32 + 2V31 V33 ) =

 

1

1 T (h)

 



A2,4 (x1 , h)

(25)

 ¯ dx1 , (2h − 2G(x1 ))/m

 

(26)

 ¯ dx1 , (2h − 2G(x1 ))/m

(27)

in which,

  r   ∞ 4   1 ∂r H 1 2H − 2G(X1 ) E × ε g dC ( t ) | H = h = ε k A1,k (x1 , h) + O(ε5 ), r ¯ ¯ dt r ! ∂ X m m 1 r =1 k =2    r j  ∞ 4   1 1 ∂r H 1 2H − 2G(X1 ) E × εg dC (t ) | H = h = εk Aj,k (x1 , h) + O(ε 5 ), r ¯ ¯ dt r ! ∂ X m m 1 k=j r =1  1 T (h) = dx1 . √ ¯ (2h − 2G(x1 ))/m 1

(28)

j = 2, 3, 4,

(29)

(30)

The averaged generalized Fokker–Planck–Kolmogorov (GFPK) equation associated with averaged Itô equation (15) is 3 4 ∂ ¯ ∂ 1 ∂2 ¯ 2 p) − 1 ∂ (A¯ 3 p) + 1 ∂ (A¯ 4 p) + O(ε5 ), p=− (A1 p) + ( A ∂t ∂H 2! ∂ H 2 3! ∂ H 3 4! ∂ H 4

(31)

the coefficients of Eq. (31) can be derived from Eq. (14) as follows: A¯ 1 =

=

A¯ 2 =

=

A¯ 3 =

= A¯ 4 =

=

 ¯ dx1 (2h − 2G(x1 ))/m T (h) dt      2h − 2G(x1 ) 1 √ 2h − 2G(x1 ) ε2 2 1 −b(x1 , x2 ) + (f 2D) dx1 ¯ ¯ ¯ T (h) m 2 m m      ε 2 λ E [Y 2 ] g 2 2h − 2G(x1 ) ∂ 2 h 2h − 2G(x1 ) dx1 + ¯2 ¯ ¯ m m 2!T (h) m ∂ x21  2 4    ε 4 λ E [Y 4 ] g 4 ∂ h 2h − 2G(x1 ) 2h − 2G(x1 ) + dx1 , ¯4 ¯ ¯ 4!T (h) m m m ∂ x41      1 1 2 ¯ dx1 E (dH |H = h) (2h − 2G(x1 ))/m T (h)   dt    2ε 2 2h − 2G(x1 ) 2h − 2G(x1 ) dx1 Df 2 ¯ ¯ m m T (h)   2    ε 2 λ E [Y 2 ] g 2 2h − 2G(x1 ) ∂ h 2h − 2G(x1 ) + dx1 ¯2 ¯ ¯ T ( h) m m ∂ x1 m    2  2 2    ε 4 λ E [Y 4 ] g 4 2h − 2G(x1 ) ∂ h 2h − 2G(x1 ) + dx1 ¯4 ¯ ¯ 4T (h) m m m ∂ x21   2   2ε 4 λE [Y 4 ] g 4 2h − 2G(x1 ) ∂ h ∂ 3h 2h − 2G(x1 ) + dx1 , ¯4 ¯ ¯ 3! m m ∂ x1 ∂ x31 m      1 1 3 ¯ dx1 E (dH |H = h) (2h − 2G(x1 ))/m T (h) dt        2 3ε 4 λE [Y 4 ] g 4 2h − 2G(x1 ) ∂ h 2 ∂ 2h 2h − 2G(x1 ) dx1 , ¯4 ¯ ¯ 2T (h) m m ∂ x1 m ∂ x21      1 1 3 ¯ dx1 E (dH |H = h) (2h − 2G(x1 ))/m T (h) dt         4 2 ε 4 λ E [Y 4 ] g 4 2h − 2G(x1 ) ∂h 2h − 2G(x1 ) dx1 , ¯4 ¯ ¯ T ( h) m m ∂ x1 m 1

 

1

 

E (dH |H = h)

(32)

(33)

(34)

(35)

Y. Qiao et al. / Physica A 473 (2017) 122–134

127

where setting A¯ 1 = ε 2 a1 + ε 4 a2 ,

A¯ 2 = ε 2 b1 + ε 4 b2 ,

A¯ 3 = ε 4 c1 ,

A¯ 4 = ε 4 d1 .

(36)

The boundary conditions of averaged GFPK equation (31) are p(h)|h=0 = finite,

∂ np = 0, h→∞ ∂ hn

lim p(h) = 0,

lim

h→∞

n = 1, 2, . . . .

(37)

The stationary probability density function (PDF) of averaged GFPK equation (31) can be acquired by utilizing the perturbation method. Assume the stationary solution has the following form p(h) = p0 (h) + ε 2 p1 (h) + ε 4 p2 (h) + · · · .

(38)

Substituting Eqs. (36) and (38) into Eq. (31) and grouping terms of same order of small parameter ε yield the following equations:

∂ 1 ∂2 (a1 (h)p0 (h)) + (b1 (h)p0 (h)) = 0; ∂h 2 ! ∂ h2 ∂ 1 ∂2 ε 4 : − (a1 (h)p1 (h)) + (b1 (h)p1 (h)) ∂h 2 ! ∂ h2 2 ∂ 1 ∂ 1 ∂3 1 ∂4 = (a2 (h)p0 (h)) − ( b ( h ) p ( h )) + ( c ( h ) p ( h )) − (d1 (h)p0 (h)); 2 0 1 0 ∂h 2! ∂ h2 3! ∂ h3 4 ! ∂ h4 .... ε2 : −

(39)

The terms pi (h), i = 1, . . . ∞, in Eq. (39) can be obtained step by step. Then the corresponding joint PDF of the displacement and momentum can be gained as follows: p(x1 , x2 ) = p(h)/T (h)|h=h(x1 ,x2 ) .

(40)

The corresponding marginal PDFs of variable x1 , x2 can be calculated as p(x1 ) = p(x2 ) =



+∞



−∞ +∞

p(x1 , x2 )dx2 , (41) p(x1 , x2 )dx1 . −∞

4. An example To illustrate the validity of proposed analytic technique, consider a variable-mass oscillator with Poisson white noise mass-disturbance, the equation of the oscillator is of the form [34,35]

¯ + g ξm (t ))¨x + c x˙ + ax + bx3 = f ξe (t ), (m

(42)

where damping coefficient c, linear stiffness coefficient a and non-linear stiffness coefficient b are all constants. ξe (t ) is Gaussian white noise with intensity 2D; ξm (t ) is Poisson white noise with zero mean and with the normal distribution of impulse strength. 4.1. Effects of Poisson mass-disturbance on stationary response According to Eq. (8), the approximate system is expressed as X˙ 1 =

X2

¯ m

−g

X2

¯2 m

ξm (t ),

X˙ 2 = −aX1 − bX13 − c

X2

(43)

+ f ξe (t ).

¯ m The Itô SDE for system (43) is X2

X2

dC (t ), ¯ ¯2 m m  √ X2 dt + f 2DdB(t ). dX2 = −aX1 − bX13 − c ¯ m

dX1 =

dt − g

(44)

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Y. Qiao et al. / Physica A 473 (2017) 122–134

The total energy of associated invariable-mass system is

¯ = H

1 X22

1

1

2

4

+ aX12 + bX14 .

¯ 2 m

(45)

¯ is as follows: Then the Itô SDE for total energy H  ¯ = dH

 −c

+

X2

2 +f

¯ m

1

(a +

2!

3bX12

2

D



¯ m 

) −

√ dt + f

gX2

2

¯2 m

2D

X2

¯ m



dB(t ) + (aX1 + bX13 ) −

dC (t ) + 2

6bX1 3!

 −

gX2

¯2 m

3

gX2



¯2 m

dC (t ) + 3

6b 4!

dC (t )

 −

gX2

4

¯2 m

dC (t )4 .

(46)

Based on the proposed stochastic averaging method in Section 3, using Eqs. (31)–(35) and omitting terms of ε 5 and higher yield the following averaged GFPK equation:

∂ ¯ 1 ∂2 1 ∂3 1 ∂4 ∂ p=− (A1 p) + (A¯ 2 p) − (A¯ 3 p) + (A¯ p), 2 3 ¯ ¯ ¯4 4 ∂t 2! ∂ H 3! ∂ H 4! ∂ H ∂ H¯

(47)

where the coefficients of Eq. (47)are 1

A¯ 1 = −c

¯)+ F (H

f 2D

+ g2

λE [Y 2 ] ¯ 3 T (H¯ ) m



A

 ¯ H¯ − 2mG ¯ (x1 )dx1 (a + 3bx21 ) 2m

¯ ¯ m m −A 4  A   23 b λ E [ Y ] ¯ − 2mG ¯ ¯ 2 m H ( x ) dx1 , + g4 1 ¯7 2m −A

A¯ 2 = f 2

2D

¯ ) + g2 F (H

2λE [Y 4 ]



A

 ¯ H¯ − 2mG ¯ (x1 )dx1 (ax1 + bx31 )2 2m

¯ 3 T (H¯ ) −A m A   3 λE [Y ] ¯ H¯ − 2mG ¯ (x1 ) 2 dx1 , + g4 7 (a + 3bx21 )2 + 8bx1 (ax1 + bx31 ) 2m ¯ T (H¯ ) −A 2m   3 3λE [Y 4 ] A ¯ H¯ − 2mG ¯ (x1 ) 2 dx1 , (ax1 + bx31 )2 (a + 3bx21 ) 2m A¯ 3 = g 4 7 ¯ ¯ T (H ) −A m   3 2λE [Y 4 ] A ¯ H¯ − 2mG ¯ (x1 ) 2 dx1 , A¯ 4 = g 4 (ax1 + bx31 )4 2m 7 ¯ T (H¯ ) −A m ¯ m

(48)

4

(49)

(50)

(51)

where

¯) = 2 T (H



A

−A

¯ m dx1 , ¯ ¯ H − 2mG ¯ (x1 ) 2m



  (a2 + 4bH¯ )1/2 − a /b,   ¯ ) = 2¯ A 2m ¯ H¯ − 2mG ¯ (x1 )dx1 . and F (H T (H ) −A ¯) = A(H

(52)

(53)

The perturbation method is used to solve Eq. (47) to obtain stationary probability density p(h), and other statistics of random responses can be calculated. For instance, the mean-square displacement is E(

X12

)=



+∞

x21 p(x1 )dx1 .

(54)

−∞

To investigate the effects of mass disturbance on system response, the relative change of mean-square displacement is introduced as

δr =

E [X12 ] − E [X12 ]|λE [Y 2 ]=0 E [X12 ]|λE [Y 2 ]=0

,

(55)

in which E [X12 ] and E [X12 ]|λE [Y 2 ]=0 denote the mean-square displacement of variable-mass system and the associated invariable-mass system, respectively. The larger the relative change value δr is, the more distinct the influence of mass disturbance on system response. Monte Carlo simulation (MCS) is used to demonstrate the effectiveness and precision of ¯ = 1.0, a = 1.0, f = 1.0, g = 0.1. Other the proposed technique. Parameters of the system in Eq. (42) are selected as m parameters used in calculation are given in figure captions. The stationary probability densities of displacement, momentum and total energy for the approximate system (43) and associated invariable-mass system are depicted in Fig. 1(a)–(c), respectively. It can be seen obviously that the mass

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Fig. 1. PDFs of displacement X1 , momentum X2 and total energy H for two cases: mass disturbance intensity λE [Y 2 ] = 0; λE [Y 2 ] ̸= 0. (λ = 0.2, 2D = 0.2, b = 0.2, c = 0.01). —, analytical results for approximate system in Eq. (43); •, results from MCS for original system in Eq. (42).

Fig. 2. PDFs of displacement X1 and momentum X2 for different external excitation intensity 2D. (λ = 0.2, λE [Y 2 ] = 0.1, b = 0.2, c = 0.01). —, analytical results; •, results from MCS.

disturbance did profoundly impact the system response, which can lead to the stationary PDFs with lower peaks and contribute to the system with large amplitude. It can be concluded from Fig. 2(a) and (b) that higher external excitation intensity leads to stronger response of system. In addition, Figs. 1–2 all show that the second-order perturbation solution for approximate system (43) coincide well with MCS for original system (42). This agreement is also observed in the tail regions as shown in Fig. 3(a) and (b) about the logarithmic PDFs of displacement, which suggests the proposed analytical method gives the accurate evaluation of the original system response.

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Fig. 3. Logarithmic PDFs of displacement X1 : (a) for different mass disturbance intensity, 2D = 0.2; (b) for different external excitation intensity, λE [Y 2 ] = 0.1. (λ = 0.2, b = 0.2, c = 0.01). —, analytical results; •, results from MCS.

Fig. 4. Mean-square displacement E [X12 ] and relative change value δr versus mass disturbance intensity λE [Y 2 ]. (λ = 0.2, 2D = 0.2, b = 0.2, c = 0.01). —, analytical results; •, results from MCS.

Fig. 5. Mean-square displacement E [X12 ] and relative change value δr versus external excitation intensity 2D. (λ = 0.2, λE [Y 2 ] = 0.1, b = 0.2, c = 0.01). —, analytical results; •, results from MCS.

Figs. 4–7 indicate the dependence of the mean-square displacement E [X12 ] and the relative change value δr on mass disturbance intensity λE [Y 2 ], external excitation intensity 2D, damping coefficient c and nonlinear stiffness b, respectively. We can observe from Figs. 4(a) and (b) that with the mass disturbance intensity increases, the mean-square displacement and the relative change value increase gradually, which demonstrates that the influence of mass disturbance on system response becomes more and more significant with the enhanced mass disturbance intensity. It also can be seen that the

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Fig. 6. Mean-square displacement E [X12 ] and relative change value δr versus damping coefficient c. (λ = 0.2, λE [Y 2 ] = 0.1, 2D = 0.2, b = 0.2). —, analytical results; •, results from MCS.

Fig. 7. Mean-square displacement E [X12 ] and relative change value δr versus nonlinear stiffness coefficient b. (λ = 0.2, λE [Y 2 ] = 0.1, 2D = 0.2, c = 0.01). —, analytical results; •, results from MCS.

analytical results agree well with those from MCS, especially for small mass-disturbance intensity. Fig. 5 manifests the effects of external excitation intensity on mean-square displacement and the relative change value. Both are increasing with the external excitation intensity, by contrast, the relative change value δr is not sensitive to external excitation intensity. One can see obviously from Fig. 6(b) that the smaller damping coefficient c is, the faster the increasing speed of relative change value is, which means that the system responses are sensitive to mass disturbance for lighter damping. Fig. 7 exhibits the influence of nonlinear stiffness b on mean-square displacement and the relative change value. With the increase of nonlinear stiffness, the system response decreases while the relative change value ascends, which implies random responses are more sensitive to mass disturbance for stronger nonlinear stiffness. 4.2. Differences between two kinds of mass-disturbance on system response In some literatures, for ease of calculation, the Poisson white noise is often replaced by Gaussian white noise of the same intensity. Nevertheless, this approximate procedure still exists limitations and will bring obvious result errors to system response. The stationary probability density of the Poisson white noise with mean arrival rate λ = 0.02 is compared with that of Gaussian white noises of the same intensity in Fig. 8. It is found that the analysis results obtained by using the proposed method are more accurate than the approximate Gaussian solution derived from the assumption that the massdisturbance is Gaussian white noise. In Fig. 9, the distinctions between the Poisson white noise and Gaussian white noise as the mass-disturbance on the mean-square displacement E [X12 ] are discussed. The solid line denotes the mean-square displacement under Poisson white noise mass-disturbance. The dotted line is for the mass-disturbance of Gaussian white noise of the same intensity. It can be seen from Fig. 9(a) that the difference between the two kinds of mass-disturbance on mean-square displacement is becoming more and more apparent as the increase of mass-disturbance intensity, for large intensity, the Poisson solution is relative smaller than Gaussian solution. Fig. 9(b) portrays that for smaller damping coefficient c, mean-square displacement shows a evident difference between the different mass disturbances.

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Fig. 8. PDFs of displacement X1 and momentum X2 for E [Y 2 ] = 20, 2D = 0.2, c = 0.01, b = 0.2.

Fig. 9. Mean-square displacement E [X12 ] from analytical results: —, mass disturbance described by Poisson white noise; - - -, mass disturbance described by Gaussian white noise of the same intensity. In addition to λ = 0.01, the parameters of (a), (b) are the same as in Figs. 4 and 6, respectively.

Fig. 10. Relative change of mean-square displacement δr induced by mass disturbance from analytical results: —, mass disturbance described by Poisson white noise; - - -, mass disturbance described by Gaussian white noise of the same intensity. The parameters are the same as Fig. 9.

The distinctions between the two kinds of mass-disturbance on relative change value δr are also displayed in Fig. 10. Compared with the mean-square displacement, the relative change value is more sensitive to the types of mass-disturbance, which implies influences of those two kinds of mass-disturbance on system response have obvious differences. Specifically, Fig. 10(a) demonstrates the relative change value of Poisson mass-disturbance is smaller than Gaussian mass-disturbance of the same intensity with the increase of mass disturbance intensity. That is to say, the influence of Poisson mass-disturbance

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Fig. 11. Mean-square displacement E [X12 ] and relative change value versus mass disturbance intensity λE [Y 2 ] from analytical results: —, mass disturbance described by Poisson white noise; - - -, mass disturbance described by Gaussian white noise of the same intensity (λ = 0.2, 2D = 0.2, c = 0.01, b = 0.2).

Fig. 12. Mean-square displacement E [X12 ] plotted against mass disturbance intensity λE [Y 2 ] (2D = 0.2, c = 0.01, b = 0.2).

on system response is weaker. Fig. 10(c) indicates that the Poisson solution is quite close to Gaussian solution for relative larger damping coefficient c. Yet, with the decrease of damping coefficient c, there is considerable difference between those two, for instance, the gap approaches to 7.2% for damping coefficient c = 0.005. In this situation, if the approximate Gaussian solution is utilized to replace Poisson solution, it will cause a certain error to the statistics of system response, thereby affecting the mass control of the system. To summarize, with the decrease of mean arrival rate λ, the difference between the two kinds of mass-disturbance on system response is increasing, especially for larger mass disturbance and lighter damping. When the λ = 0.2 is fixed, it can be seen from Fig. 11 that the second-order perturbation solution of mean-square displacement is close to the approximate Gaussian solutions for λE [Y 2 ] < 0.2, yet with the increase of mass disturbance intensity, there will be marked difference between the two. It is also seen from Fig. 12 that, if the intensity of Poisson white noise is a constant and λ changes from 0.02 to 1.0, the mean-square displacement approaches to that for Gaussian white noise mass-disturbance. This confirms the claim that a Poisson white noise tends to a Gaussian white noise if λE [Y 2 ] keeps constant while λ tends to infinity. 5. Conclusions In the present paper, the stochastic averaging method is generalized to investigate the stochastic response of the nonlinear variable-mass system with mass disturbance described by Poisson white noise. Specifically, the original variablemass system is transformed into associated approximate system via Taylor expansion technique. Stochastic averaging method is carried out to derive the averaging Itô equations and associated GFPK equation for the transition probability density of total energy. Subsequently, perturbation method is utilized to solve the reduced GFPK equation to obtain the approximate stationary probability density and other statistics of the system responses. The effectiveness of this generalized stochastic averaging method is validated in an example by comparing with numerical simulation of the original system. In addition, although the proposed analytical method is confined to SDOF variable-mass system, it can be directly generalized to deal with the variable-mass systems with multi-degree-of freedom [29–32].

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The results obtained in this paper suggest that mass disturbance has marked influence on the dynamical behaviors of systems, especially for the systems with lighter damping, larger external excitation, and stronger nonlinear stiffness. The distinctions between the Poisson white noise and Gaussian white noise as the mass disturbance on systems response have been intensively explored. The results show that as mass disturbance, there exist some differences between the two kinds of noise with the same intensity, especially for larger mass-disturbance intensity, lighter damping and smaller mean arrival rate λ. Accordingly, it is significant and necessary to study the influence of mass disturbance which described by Poisson white noise on system response. 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