Reliability analysis of nonlinear vibro-impact systems with both randomly fluctuating restoring and damping terms

Commun Nonlinear Sci Numer Simulat 82 (2019) 105087

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Research paper

Reliability analysis of nonlinear vibro-impact systems with both randomly fluctuating restoring and damping terms Zhicong Ren, Wei Xu∗, Shuo Zhang Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, PR China

a r t i c l e

i n f o

Article history: Received 23 July 2019 Revised 10 October 2019 Accepted 25 October 2019

Keywords: Vibro-impact Modified PI Reliability Random restoring and damping terms

a b s t r a c t In this paper, the reliability analysis of vibro-impact system with both randomly fluctuating restoring and damping terms is studied by a modified path integration (PI) method. Specifically, the Ivanov nonsmooth transformation technique is adopted to transform the vibro-impact system into the system without barrier, then the modified PI application on vibro-impact systems is applied, which is based on the Gaussian closure method and the localized approximation of moment function. According to the first passage theory, the reliability function, the first passage probability density function (PDF) and the mean first passage time are numerically calculated. In the framework of our numerical results, the influences of different random restoring terms, random damping terms and impact conditions on the system’s reliability are discussed. The modified PI results are compared with the Monte Carlo Simulation (MCS) results, which shows that the proposed PI method can not only provide sufficiently accurate results to observe the weak influence of parameters, but also has obvious advantages in computational efficiency. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Vibro-impact system is a type of common dynamic systems with elastic or inelastic collisions. It widely exists in mechanical and structural engineering, such as, oil pipeline collision and constraints, ship rolling with one side iceberg barrier and so on [1–3]. The collision brings about an instantaneous change of velocity, such that the system becomes non-smooth. Non-smooth dynamic systems are always hard to handle and have a deep influence on the dynamic mechanical behaviors. Due to the fact that there are many unavoidable random perturbations in the real motion, many scholars have studied the dynamical response phenomena of vibro-impact systems under random excitations. Different kinds of noise excitations are studied in the vibro-impact systems, which include external Poisson white noises [4,5], correlated Gaussian white noises [6], external and parametric Gaussian white noises [7], parametric Poisson white noises [8], colored noises [9] and so on. Ivanov and Zhuravlev non-smooth transformation techniques are commonly used to transform the original equation into a new system without impact [10,11]. After that, the stochastic averaging method [12–14], the perturbation method [15] and the exponential-polynomial closure method [16] were presented for analyzing the stationary PDF solutions of vibro-impact systems. Except for stationary PDF solutions, the shannon entropy and the largest Lyapunov exponent were also calculated to present a global paramtetric study of the vibro-impact system [17]. Recently, the fractional derivative element is considered in random vibro-impact system, whose mean-square response amplitude is derived by an averaging based method [18]. ∗

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

https://doi.org/10.1016/j.cnsns.2019.105087 1007-5704/© 2019 Elsevier B.V. All rights reserved.

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Fig. 1. Schematic diagram of vibro-impact system subject to multiplicative and additive Gaussian white noises.

In view of above previous studies, the stationary response of vibro-impact system was the main concern. However, in some cases, vibro-impact system may be unstable. The vibro-impact system may reach an unstable equilibrium point on the other side of the collision and moves towards infinity after an enough long time. There is no stationary PDF solution in this situation. Thus it’s necessary to do some research about the reliability of unstable vibro-impact system. Reliability analysis always requires precisely capturing the transient PDF evolution of system, which is much more difficult to handle than the stationary PDF approximations. The reliability function and the probability structure of the first passage time are the most commonly used indicators of system’s stability [19–22]. It has been studied by a great number of scholars with different kinds of analytical and numerical methods, which include stochastic averaging method [23–26], finite element method [27], generalized cell mapping method [28,29], path integration method [30–32] and so on. The reliability of linear system with non-classical inelastic impact was analyzed by applying the backward Kolmogorov equation and Pontryagin equation [33]. Nevertheless, the research about the reliability of vibro-impact system is seldom reported, and no one has studied the reliability analysis of the vibro-impact system with both random varying damping and restoring terms. Motivated by above discussion, a recently proposed PI method [34] is extended to the analysis of the first passage type reliability of nonlinear vibro-impact system with both randomly fluctuating restoring and damping terms. Research methods consist of three steps. Firstly, transform the vibro-impact system into a continuous system by Ivanov non-smooth transformation. Secondly, calculate the PDF evolution of vibro-impact system under the security domain by using the modified PI method. Finally, analyze the reliability of vibro-impact system by calculating multiple reliability indicators, which include the reliability function, the first passage probability density function and the mean first passage time. MCSs are used to verify the accuracy of our PI method. Different intensity of additive noises and different impact conditions are considered to show the parameters’ influence on the system’s reliability. 2. Vibro-impact system with both randomly fluctuating restoring and damping terms Consider a nonlinear vibro-impact system with both randomly fluctuating restoring and damping terms

x¨ + b [1 + γ2 ξ2 (t )] x˙ + k1 [1 + γ3 ξ3 (t )] x + k3 x3 = γ1 ξ1 (t )

(1)

x˙ + = −r x˙ −

(2)

x = ximp

where x is displacement and overhead dot means differentiation with respect to time t. x˙ − and x˙ + are the velocity before and after impact, respectively. ximp is the position of the impact barrier. The restitution coefficient r indicates the rate of energy loss upon impact. When r = 1, the system’s energy does no change during collisions, which means elastic impacts occur. Otherwise, when r < 1, the impact brings about loss of energy, which means inelastic impacts happen. ξ 1 (t), ξ 2 (t) and ξ 3 (t) are independent Gaussian white noises whose intensities are γ 1 , γ 2 , γ 3 , respectively. The consideration of random varying restoring and damping terms relies on the fact that this terms may exhibit some degree of random fluctuation in certain types of mechanical structures [16,35,36]. Thus it’s meaningful to discuss the influence of random varying restoring and damping terms on the system’s reliability. From a mathematical perspective, the Gaussian white noises are borderless whenever its’ intensities are large or small. So the restoring and damping terms always have the probability to be negative due to the large negative values of Gaussian white noises. However, the additive noise intensities γ 2 and γ 3 are usually of order ε. The parameter ε is small so that the additive Gaussian white noises only result in small random fluctuations of restoring and damping terms numerically. So, from a view of numerically analysis and the physical meaning of the system, the case that the restoring and damping terms are always positive is discussed. The corresponding simplified schematic diagram is shown in Fig. 1. The unstable nonlinear oscillator is selected as a commonly used soften duffing oscillator, where k1 > 0, k3 < 0. When the displacement x passes through the unstable equiva-

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lent point ± −k1 /k3 , the restoring force and the motion are in the same direction, the system loses stability, so the security √ domain of Eq. (1) is {(x,x˙ )|x|< −k1 /k3 }. 3. Ivanov non-smooth transformation The Ivanov non-smooth transformation is applied to combine Eqs. (1) and (2) into one continuous equation. Only in this way the modified PI method could be successfully adapted. Herein the variables (x, x˙ ) are mapped into the (U, V) space following the transformation

x − ximp = U sgn(U ),

x˙ = SV sgn(U )

(3)

where S = 1 − ksgn(UV ), k = (1 − r )/(1 + r ). Then the governing equations of variables (U, V) can be expressed in the following first-order form

U˙ = SV     V˙ = −bV − k1 S−1 U +ximp sgn(U ) − k3 S−1 U +ximp sgn(U ) 3





+ S−1 ξ1 (t ) − V bγ2 ξ2 (t ) − S−1 U +ximp sgn(U ) k1 γ3 ξ3 (t )

(4)

The equivalence of equation before and after the Ivanov transformation is simply proofed in Appendix A. In fact, there is no unique inverse transformation from the variables (x, x˙ ) to the variables (U, V), as one point in Eq. (1) always corresponds to two points in Eq. (4). However, so long as the PDF of Eq. (4) is calculated, the PDF of Eq. (1) can be derived according to variables’ relationship (3). A more convenient strategy in reliability analysis is that the first passage problem can be solved directly based on Eq. (4), as the original security domain in Eq. (1) can be corresponded to the one of non-smooth transformed equation. 4. Numerical implementation of the modified PI method The Eq. (4) is of the Stratonovich form. By adding the Wong–Zakai correction term, the following Ito stochastic differential equation is obtained

U˙ = SV   V˙ = −b + b2 γ22 /2 V







3

− k1 S−1 U +ximp sgn(U ) − k3 S−1 U +ximp sgn(U )





+ S−1 ξ1 (t ) − V bγ2 ξ2 (t ) − S−1 U +ximp sgn(U ) k1 γ3 ξ3 (t )

(5)

Generally, the path integral method is based on the following Chapman–Kolmogorov (CK) equation

p(u, v, t ) =





R2



 



q u, v, t u , v , t  p u , v , t  du dv

(6)

There are mainly two kinds of strategies to calculate the CK equation based on PI method. One is to give a discretized version of CK equation and numerically solve it, which has commonly used in many different stochastic systems and also will be applied in this paper. Another one, which is recently proposed and is called the wiener PI method, is based on a special functional integral formulation [37,38]. The principle of wiener PI calculation is only focused on the most probable path. In this way, wiener PI method can even be applied in 10-degree-of-freedom structural systems [39]. Followed by the discretized strategy of PI method, when the PI method is applied, the continuous time is converted to several uniformly small time intervals. The transition PDF q(u, v, t|u , v , t .) is exactly the same in each time intervals, which need to be calculated only once. Then, the PDF evolution can be obtained by numerically calculating CK equation step by step. In this paper, the short-time Gaussian approximation of transition PDF is adopted, which is based on the Gaussian closure method and the localized approximation of moment function. Different from the Dirac delta transition PDF approximation [30], the short-time Gaussian transition PDF approximation takes diffusions of both displacement and velocity into account, which can more reasonably reflects real stochastic motions. In the traditional Gaussian closure method, calculations of moment function always require that all expressions of Stochastic differential equation are polynomials of variables [40]. Only in this way, the differential of low order moments can be represented as an equation of higher order moments. Eq. (4) obviously does not meet this applicable condition, so the localized approximation of moment function is proposed, which is already used to analyze the equation of ship roll motion with iceberg’s impacts [34]. This assumption is based on the fact that the moment of variables can always be handled in a local range when the time step is sufficiently small, then the following approximation of moment function is obtained

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m˙ 10 = sm01 m˙ 01 = −k 1 m10 − k 2 m20 − k 3 m30 − k 0 − b m01 m˙ 20 = 2sm11 m˙ 11 = sm02 − k 1 m20 − k 2 m30 − k 3 m40 − k 0 m10 − b m11 m˙ 02 = −2k 1 m11 − 2k 2 m21 − 2k 3 m31 − 2k 0 m01 − 2b m02



+ s−2 γ12 + m02 b2 γ22 + s−2 k21 m20 + 2ximp |m10 | + x2imp



where mi j = E U iV

 j



γ32

(7)

, s = 1 − ksgn(m10 m01 ). The expressions of k 0 , k 1 k 2 k 3 and b are as follows

⎧  b = b− b2 γ22 /2 ⎪  ⎪ ⎪ ⎨k 1 = k1 s−1 + 3k3 s−1 x2imp k 2 = 3k3 s−1 sgn(m10 )ximp ⎪ ⎪ k = k3 s−1 ⎪   ⎩ 3 k 0 = s−1 sgn(m10 ) k1 ximp + k3 x3imp

(8)

By using Gaussian closure method, the high order moments are approximated by the following function of low moment variables

m30 = 3m10 m20 − 2m310 m40 = 3m220 − 2m410 m21 = m20 m01 + 2m11 m10 − 2m210 m01 m31 = 3m20 m11 − 2m310 m01

(9)

The Eq. (8) can be converted into a closed moment function with Eq. (9). By using Runge–Kutta method with the zerovariance initial condition, first and second order moments after transition are calculated, which determine the Gaussian approximation of short-time transition PDF. 5. First passage type reliability by PI method The reliability of Eq. (1) is considered as the probability that the motion stays in the security domain Ds during a period of time (0, T], which is expressed in the following form

R(T | p0 ) = Pr ( (x(t ), x˙ (t ) ) ∈ Ds , t ∈ (0, T ]| p0 )

(10)

where R(T|p0 .) is the reliability function and p0 denotes the initial PDF distribution in (x, x˙ ) space. The procedure for using PI method to analyze the reliability of system [30] has been extended to the system under Poisson white noises and Levy noises [41,42]. It’s also interesting to see if this method still yields accurate results in our vibro-impact system. The Path integration method could only handle the PDF evolution of Eq. (4) at discrete time steps. Therefore, the following approximation of reliability function is proposed

R(tN | p0 ) ≈ Pr ( (x(ti ), x˙ (ti ) ) ∈ Ds , i = 0, 1, 2, . . . N | p0 )   = Pr (U (ti ), V (ti ) ) ∈ D¯ s , i = 0, 1, 2, . . . N | p¯ 0

(11)

where ti = it, D¯ s and p¯ 0 is the corresponding security domain and initial PDF distribution in (U, V) space, respectively. According to the Markov property, the reliability function is calculated by PI method with the following iterations

R(tN | p0 ) ≈





D¯ s

...



  N D¯ s





q u(i ) , v(i ) , t u(i−1) , v(i−1) , 0



i=1

· p¯ u(0 ) , v(0 ) , 0 du(0 ) dv(0 ) . . . du(N ) dv(N )

(12)

The time when the motion firstly moves out of the security domain is called the first passage time. The first passage probability density can be obtained by the following equation

pF (t | p0 ) = −

∂ R(t | p0 ) ∂t

(13)

then the mean first passage time can be deduced directly from the reliability function by distribution integral method:

E (T0 ) =



0

 =

∞

0

∞

t · pF (t | p0 )dt t ·−

∂ R(t | p0 ) dt = ∂t



∞ 0

R(t | p0 )dt

(14)

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Fig. 2. Plot of reliability function (a) and first passage PDF (b) of vibro-impact systems with both randomly fluctuating restoring and damping terms. Both modified PI results and Monte Carlo results with different γ 3 are performed.

6. Numerical results In this section, the influences of different parameters on the reliability function of the vibro-impact system are numerically discussed. For all cases, without special statement, the system parameters are given as b = 0.4, k1 = 3.0, k3 = −3/4, γ1 = 1.0, γ2 = 0.1, γ3 = 0.1, ximp = −0.5, r = 0.9. It is assumed that the motion is initially at rest in zeros point (x, x˙ ) = (0, 0), which corresponds to the point ( ± ximp , 0) in (U, V) space. Therefore the initial PDF distribution is considered as the following Dirac delta function

p¯ (u, v, 0 ) =

     1 δ (v ) δ u + ximp + δ u − ximp 2

(15)

The  time step value  of modified PI method is selected as 0.2. The security domain in (U, V) space is set as [ximp − −k1 /k3 , −ximp + −k1 /k3 ] × (−∞, +∞), which is corresponding to the domain [ximp , −k1 /k3 ] × (−∞, +∞) in (x, x˙ ) space. PI method also require a selection of limited integral range in V axis, so the security domain for numerical   integration is approximated to be [ximp − −k1 /k3 , −ximp + −k1 /k3 ] × [Vdown , Vup ]. In this paper, the upper and lower bounds of V axis are selected as Vdown = −5 and Vup = 5. The security domain is divided into 400 × 200 subintervals. In the meanwhile, MCSs with 106 samples are performed in each parameter’s selection, which is used to verify the accuracy of the modified PI results. The time step value of MCS is selected as 0.02. It’s worth mentioning that the calculation of the first passage PDF by MCS requires larger numbers of MCS samples than the calculation of reliability calculation. Therefore, the Monte Carlo simulation in this paper is extremely time-consuming. While the running time of modified

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Fig. 3. Plot of reliability function (a) and first passage PDF (b) of vibro-impact systems with both randomly fluctuating restoring and damping terms. Both modified PI results and Monte Carlo results with different γ 2 are performed.

PI only depends on the subdivision degree of integral space and time step value, the modified PI method has significant advantage in efficiency when dealing with this problem.

6.1. Case 1: Reliability with different randomness of restoring term In this case, different randomness of restoring term is considered by changing parameter γ 3 . Fig. 2 shows a comparison on the reliability function and the first passage PDF with different γ 3 using modified PI and MCS methods. From Fig. 2(a), it can be seen that the reliability presents smaller decay rate by smaller γ 3 , while presents larger decay rate by larger γ 3 . This results mean the system with larger randomness of restoring term has poorer reliability. By observing the first passage PDF with different γ 3 in Fig. 2(b), one can conclude that larger γ 3 make the density value more concentrated at the peak of first passage PDF.

6.2. Case 2: Reliability with different randomness of damping term Next, different randomness of damping term is considered by changing parameter γ 2 . In Fig. 3, comparison on the reliability function and the first passage PDF with different γ 2 is represented. Similar to the case 1, it is shown that when γ 2 increases, the decay rate of reliability increases, which means system’s stability is more likely to lost.

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Fig. 4. Plot of reliability function (a) and first passage PDF (b) of vibro-impact systems with both randomly fluctuating restoring and damping terms. Both modified PI results and Monte Carlo results with different restitution coefficients r are performed.

However, through the comparison of Figs. 2 and 3, we can see that the change of reliability function and first passage PDF with different γ 2 is relatively small compared with the change in Case 1, which indicates that the effect of damping term’s randomness on the system’s reliability is weaker than the effect of restoring term’s randomness.

6.3. Case 3: Reliability with different impact conditions In the third case, the influence of different impact conditions on the system’s reliability is considered. Two indicators determine the impact condition: the impact position and the value of restitution coefficient. Therefore the reliability analysis with different ximp and r is performed in Figs. 4 and 5, respectively. It is shown in Fig. 4 that the reliability of vibro-impact system with lower restitution coefficient presents smaller decay rate, and the peak value of the first passage PDF obviously increases when the value of restitution coefficient is further away from unit 1. Fig. 5 presents the first passage PDF and the change of the mean first passage time with different impact positions. A number of impact positions are selected in this comparison, which are −1.5, −1.4, . . . , −0.2, −0.1. As shown in Fig. 5, there are only small changes of first passage PDF with different impact positions. The value of restitution coefficient denotes the energy losing rate during inelastic impact, so it is indicated that the main influence of impact on the reliability is due to the dissipation of system’s energy. Particular attention is paid to this small first passage changes with different impact position. It can be observed in Fig. 5(a) that the peak of the first passage PDF gets the minimum value at about ximp = −0.5, which corresponds to the

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Fig. 5. Plot of first passage PDF (a) and mean first passage time (b) of vibro-impact systems with both randomly fluctuating restoring and damping terms. Both modified PI results and Monte Carlo results with different impact position ximp are performed.

maximum value of the mean first passage time in Fig. 5(b). Therefore, it’s believed that there exists an optimal collision position to make the system reach the most stable state. From Figs. 2–5, we can see that the modified PI results coincide extremely well with the MCS results. This good agreement is made whether at the beginning or at the end of the reliability function and the first passage PDF. Besides, as shown in Fig. 5(b), the proposed PI approach could also provide reliable estimation on the mean first passage time. In view of above discussion, the level of modified PI accuracy is adequate to observe the weak influence of parameters on the reliability.

7. Conclusions In this paper, the reliability analysis for the vibro-impact system with random restoring and damping terms is presented. Specifically, the Ivanov non-smooth transformation and the modified PI method are employed to efficiently calculate the reliability function, the first passage PDF and the mean first passage time. The numerical results show that the reliability of the system strongly depends on the randomness of restoring term, however was weakly influenced by the randomness of damping term. It can be found that smaller randomness of restoring and damping terms, smaller restitution coefficients will lead to stronger reliability of the system. What’s more, through the observation of the first passage PDF and the mean first passage time with different impact positions, we find that there exists an optimal collision position to make the system reach the most stable state. All results are verified by MCS,

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which show extremely high accuracy and significant efficiency of the modified PI method approach. Thus, the proposed PI procedure may deserve further development and application in reliability analysis of other vibro-impact systems. Declaration of Competing Interest The authors declare this manuscript is original and they have no potential conflicts with respect to the research, authorship, and publication of this article. Acknowledgments The authors thank the reviewer for their careful reading and suggestions. The research was supported by the National Natural Science Foundation of China (Grant nos. 11872305, 11532011). Appendix A

Theorem 1. If the variable U and V are governed by the following differential equation

U˙ = SV   V˙ = S−1 sgn(U ) f t, |U |+ximp , SV sgn(U )

(16)

where S = S (U, V )=1 − ksgn(UV ), k = (1 − r )/(1 + r ). Then the variable x=|U | + ximp is governed by the following equation

x¨ = f (t, x, x˙ ) x˙ + = −r x˙ − f or x = ximp

(17)

Proof. Because of Eq. (16), for any continuous parts of x, the following deduction is satisfied:

 

x˙ = U˙  = U˙ sgn(U ) = SV sgn(U )

 





x¨ = U¨  = U¨ sgn(U )= f t, |U |+ximp , SV sgn(U ) = f (t, x, x˙ )

(18)

The discontinuous part of Eq. (16) is when the point (U, V) passes through the V axis, which corresponds to the impact condition of x. There are two cases of intersection: from U > 0 to U < 0 or from U < 0 to U > 0, as shown in Fig. A1. 2 When the point (U, V) moves from U > 0 to U < 0, which mean U˙ < 0, then V = U˙ /S < 0, thus x˙ = SV sgn(U )= 1+ r V before intersection and x˙ = SV sgn(U )= −

2r 1+r V

after it, which satisfy the impact condition x˙ + = −r x˙ − f or x = ximp When the point (U, V) moves from U < 0 to U > 0, which mean U˙ > 0, then V = U˙ /S > 0, thus x˙ = SV sgn(U )= − 2r SV sgn(U )= 1+ rV

before intersection and x˙ = of that, x is governed by Eq. (17).



2 1+r V

after it, which also satisfy the impact condition x˙ + = −r x˙ − f or x = ximp . In view

Fig. A1. The original impact motion (a) and the motion after Ivanov non-smooth transformation (b).

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References [1] Jing H, Young M. Random response of a single-degree-of-freedom vibro-impact system with clearance. Earthq Eng Struct D 1990;19(6):789–98. doi:10. 1002/eqe.4290190602. [2] Ibrahim RA, Babitsky VI, Okuma M. Vibro-impact dynamics of ocean systems and related problems. Berlin: Springer-Verlag; 2009. p. 67–78. doi:10. 1007/978- 3- 642- 00629- 6. [3] Ibrahim RA. Recent advances in vibro-impact dynamics and collision of ocean vessels. J Sound Vib 2014;333(23):5900–16. doi:10.1016/j.jsv.2014.02.006. [4] Zhu HT. Stochastic response of a vibro-impact duffing system under external poisson impulses. Nonlinear Dyn 2015;82(1–2):1001–13. doi:10.1007/ s11071-015-2213-z. [5] Wu Y, Zhu WQ. Stationary response of multi-degree-of-freedom vibro-impact systems to poisson white noises. Phys Lett A 2008;372(5):623–30. doi:10. 1016/j.physleta.2007.07.083. [6] Li C, Xu W, Feng JQ, Wang L. Response probability density functions of Duffing-Van der Pol vibro-impact system under correlated Gaussian white noise excitations. Phys A 2013;392(6):1269–79. doi:10.1016/j.physa.2012.11.053. [7] Wang DL, Xu W, Gu XD, Yang YG. Stationary response analysis of vibro-impact system with a unilateral nonzero offset barrier and viscoelastic damping under random excitations. Nonlinear Dyn 2016;86(2). doi:10.1007/s11071- 016- 2931- x. 891–C909 [8] Yang GD, Xu W, Jia WT, He MJ. Random vibrations of Rayleigh vibroimpact oscillator under parametric poisson white noise. Commun Nonlinear Sci Numer Simul 2016;33:19–29. doi:10.1016/j.cnsns.2015.08.003. [9] Liu D, Li JL, Meng Y. Probabilistic response analysis for a class of nonlinear vibro-impact oscillator with bilateral constraints under colored noise excitation. Chaos Soliton Fract 2019;122:179–88. doi:10.1016/j.chaos.2019.03.024. [10] Ivanov AP. Impact oscillations: linear theory of stability and bifurcations. J Sound Vib 1994;178(3):361–78. doi:10.1006/jsvi.1994.1492. [11] Zhuravlev VF. A method for analyzing vibration-impact systems by means of special functions. Mech Solids 1976;11. 23–C27. http://refhub.elsevier. com/S0378-4371(18)31359-1/sb49 [12] Li C. Stochastic response of a vibro-impact system with variable mass. Phys A 2019;516:151–61. doi:10.1016/j.physa.2018.10.021. [13] Gu XD, Zhu WQ. A stochastic averaging method for analyzing vibro-impact systems under Gaussian white noise excitations. J Sound Vib 2014;333(9):2632–42. doi:10.1016/j.jsv.2013.12.027. [14] Chen LC, Qian JM, Zhu HS, Sun JQ. The closed-form stationary probability distribution of the stochastically excited vibro-impact oscillators. J Sound Vib 2019;439:260–70. doi:10.1016/j.jsv.2018.09.061. [15] Yang GD, Wei X, Jia WT, He MJ. Random vibrations of Rayleigh vibroimpact oscillator under parametric poisson white noise. Commun Nonlinear Sci Numer Simul 2016;33:19–29. doi:10.1016/j.cnsns.2015.08.003. [16] Zhu HT. Response of a vibro-impact Duffing system with a randomly varying damping term. Int J NonLin Mech 2014;65:53–62. doi:10.1016/j. ijnonlinmec.2014.05.003. [17] Kumar P, Narayanan S, Gupta S. Stochastic bifurcations in a vibro-impact Duffing C Van der Pol oscillator. Nonlinear Dyn 2016;85(1):439–52. doi:10. 1007/s11071- 016- 2697- 1. [18] Yurchenko D, Burlon A, Paola MD, et al. Approximate analytical mean-square response of an impacting stochastic system oscillator with fractional damping. ASCE-ASME J Risk Uncertainty Engin Syst Part B 2017;3(3):030903. doi:10.1115/1.4036701. [19] Bergman LA, Spencer BF. First passage time for linear systems with stochastic coefficients. Probabilist Eng Mech 1987;2(1):46–53. doi:10.1016/ 0266- 8920(87)90030- 0. [20] Bergman LA, Spencer BF. On the solution of several first passage problems in nonlinear stochastic dynamics. Nonlinear Stochastic Dyn Eng Syst 1988:479–92. doi:10.1007/978- 3- 642- 83334-2_35. [21] Langtangen HP. Numerical solution of first passage problems in random vibrations. SIAM J Sci Comput 1994;15(4):977–96. doi:10.1137/0915059. [22] Grigoriu M. Reliability of linear systems under Poisson white noise. Probabilist Eng Mech 2009;24(3):397–406. doi:10.1016/j.probengmech.2008.12.001. [23] Zhu WQ, Deng ML, Huang ZL. First-passage failure of quasi-integrable hamiltonian systems. J Appl Mech 2002;69(3):274–82. doi:10.1115/1.1460912. [24] Zhu WQ, Wu YJ. First-passage time of duffing oscillator under combined harmonic and white-noise excitations. Nonlinear Dyn 2003;32:291–305. doi:10.1016/j.probengmech.2018.06.003. [25] Sun JJ, Xu W, Lin ZF. Research on the reliability of friction system under combined additive and multiplicative random excitations. Commun Nonlinear Sci Numer Simul 2018;54:1–12. doi:10.1016/j.cnsns.2017.05.014. [26] Li XP, Huan RH, Wei DM. Feedback minimization of the first-passage failure of a hysteretic system under random excitations. Probabilist Eng Mech 2010;25(2):245–8. doi:10.1016/j.probengmech.20 09.12.0 03. [27] Bergman LA, Heinrich JC. On the reliability of the linear oscillator and systems of coupled oscillators. Int J Numer Meth Eng 1982;18(9):1271–95. doi:10.1002/nme.1620180902. [28] Han Q, Xu W, Yue XL, Zhang Y. First-passage time statistics in a bistable system subject to poisson white noise by the generalized cell mapping method. Commun Nonlinear Sci Numer Simul 2015;23(1–3):220–8. doi:10.1016/j.cnsns.2014.11.009. [29] Sun JQ, Hsu CS. First-passage time probability of non-linear stochastic systems by generalized cell mapping method. J Sound Vib 1988;124:233–48. doi:10.1016/s0022-460x(88)80185-8. [30] Iourtchenko D, Mo E, Naess A. Reliability of strongly nonlinear single degree of freedom dynamic systems by the path integration method. J Appl Mech 2008;75(6):1055–62. doi:10.1115/1.2967896. [31] Naess A, Iourtchenko D, Batsevych O. Reliability of systems with randomly varying parameters by the path integration method. Probabilist Eng Mech 2011;26(1):5–9. doi:10.1016/j.probengmech.2010.05.005. [32] Kougioumtzoglou IA, Spanos PD. Stochastic response analysis of the softening duffing oscillator and ship capsizing probability determination via a numerical path integral approach. Probabilist Eng Mech 2014;35:67–74. doi:10.1016/j.probengmech.2013.06.001. [33] Xu M. First-passage failure of linear oscillator with non-classical inelastic impact. Appl Math Model 2018;54:284–97. doi:10.1016/j.apm.2017.09.036. [34] Ren ZC, Xu W, Wang DL. Dynamic and first passage analysis of ship roll motion with inelastic impacts via path integration method. Nonlinear Dyn 2019;97:391–402. doi:10.1007/s11071- 019- 04975- x. [35] Brouwers JJH. Stability of a non-linearly damped second-order system with randomly fluctuating restoring coefficient. Int J Nonlin Mech 1986;21(1):1– 13. doi:10.1016/0 020-7462(86)90 0 08-9. [36] Brouwers JJH. Asymptotic solutions for mathieu instability under random parametric excitation and nonlinear damping. Phys D 2011;240(12):990– 10 0 0. doi:10.1016/j.physd.2011.02.009. [37] Kougioumtzoglou IA, Spanos PD. An analytical wiener path integral technique for non-stationary response determination of nonlinear oscillators. Probabilist Eng Mech 2012;28:125–31. doi:10.1016/j.probengmech.2011.08.022. [38] Psaros AF, Brudastova O, Malara G, Kougioumtzoglou IA. Wiener path integral based response determination of nonlinear systems subject to non-white, non-Gaussian, and non-stationary stochastic excitation. J Sound Vib 2018;433:314–33. doi:10.1016/j.jsv.2018.07.013. [39] Psaros AF, Kougioumtzoglou IA, Petromichelakis I. Sparse representations and compressive sampling for enhancing the computational efficiency of the wiener path integral technique. Mech Syst Signal Pr 2018;111:87–101. doi:10.1016/j.ymssp.2018.03.056. [40] Yu JS, Cai GQ, Lin YK. A new path integration procedure based on Gauss–Legendre scheme. Int J Nonlin Mech 1997;32(4):759–68. doi:10.1016/ S0 020-7462(96)0 0 096-0. [41] Bucher C, Di Paola M. Efficient solution of the first passage problem by path integration for normal and poissonian white noise. Probabilist Eng Mech 2015;41:121–8. doi:10.1016/j.probengmech.2015.06.007. [42] Bucher C, Di Matteo A, Di Paola M. First-passage problem for nonlinear systems under levy white noise through path integral method. Nonlinear Dyn 2016;85(3):1445–56. doi:10.1007/s11071- 016- 2770- 9.