Research on the reliability of friction system under combined additive and multiplicative random excitations

Commun Nonlinear Sci Numer Simulat 54 (2018) 1–12

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Research paper

Research on the reliability of friction system under combined additive and multiplicative random excitations Jiaojiao Sun, Wei Xu∗, Zifei Lin School of Natural and Applied Sciences, Northwestern Polytechnical University, Xi’an 710072, PR China

a r t i c l e

i n f o

Article history: Received 7 November 2016 Revised 20 April 2017 Accepted 18 May 2017 Available online 19 May 2017 Key words: Friction system Stochastic excitations Stochastic averaging method First passage failure Reliability

a b s t r a c t In this paper, the reliability of a non-linearly damped friction oscillator under combined additive and multiplicative Gaussian white noise excitations is investigated. The stochastic averaging method, which is usually applied to the research of smooth system, has been extended to the study of the reliability of non-smooth friction system. The results indicate that the reliability of friction system can be improved by Coulomb friction and reduced by random excitations. In particular, the effect of the external random excitation on the reliability is larger than the effect of the parametric random excitation. The validity of the analytical results is verified by the numerical results. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Friction, which is one of the typical non-smooth factors, exists widely in people’s life and engineering practice, such as robot [1,2], clutch, computerized numeric control (CNC) [3], automobile tires, brake pads, dry friction damper [4]. It may dramatically change the dynamic mechanical behaviors, and even induce the structural insecurity. Grasping the dynamic characteristics of friction can help to identify the causes of unwanted behavior such as squeal of car brakes and to reduce the harm induced by the friction force. Therefore, for many years the topic of friction has been received widespread attention [5–9]. Random factors exist widely in the physical system and may markedly affect the dynamic behavior of the physical system. Whereas, there are few researches on the friction system under random excitations, and current works almost focus on the numerical solutions. Sun [10] researched the random response of Coulomb friction system by applying the generalized cell mapping method which is based on the short-time Gaussian approximation. Feng [11] used a two-dimensional mean Poincare map to establish the discrete model of random friction system and studied the stochastic stick-slip. Brouwers [12] investigated the non-linearly damped response of a marine riser subject to random waves. The equivalent nonlinear method was utilized by Tian [13] to study the optimal load resistance of a randomly excited nonlinear electromagnetic energy harvester with Coulomb friction. Reliability is important in many practical applications, such as the comfort of the vehicle bump vibration, the security of the building under the wind load or the impact of the earthquake, and others. It is one of the most essential and difficult problems in random vibration theory. In the structural systems, the reliability problem is usually the first passage problem. To study the reliability, one should solve the diffusion processes related to the response of a stochastic system. However, it is hard to find out the analytical results of diffusion processes, even for the case of one dimension, the known analytical ∗

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

http://dx.doi.org/10.1016/j.cnsns.2017.05.014 1007-5704/© 2017 Elsevier B.V. All rights reserved.

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J. Sun et al. / Commun Nonlinear Sci Numer Simulat 54 (2018) 1–12

results are limited. Therefore, some scholars have developed many numerical methods, such as finite difference method, finite element method and generalized cell mapping method [14–16]. Based on the stochastic averaging method, the dimension of a stochastic system can be reduced, which makes further research on the reliability be possible. And many scholars have studied the reliability of a system under random excitations by applying the stochastic averaging method [17–22]. Chen [23] used the stochastic averaging method to investigate the first passage failure of quasi integrable-Hamiltonian systems under combined harmonic and white noise excitations. This method was utilized by Li [24] to study the first passage problem for strong nonlinear stochastic dynamical system. Non-smooth factors are ubiquitous in modern engineering structures. And their existence may make the dynamics behavior of a system more complex. Some scholars tried to apply the stochastic averaging method to study the stochastic responses of non-smooth system. Feng [25] and Zhao [26] researched the stochastic responses of vibro-impact system under additive and multiplicative random excitations. Wu [27] investigated the stationary response of multi-degree-of-freedom vibro-impact systems to Poisson white noises. However, the reliability of non-smooth system is rarely investigated. Inspired by previous work, we tried to use the method to investigate the reliability of friction non-smooth system. The reliability of a system can be measured by many indicators. In this paper, the reliability function, the conditional probability density function (PDF) of first passage time and the mean first passage time are chosen. The averaged Itô stochastic differential equation for the energy of friction system is first derived by using the stochastic averaging method. Then, the Backward Kolmogorov (BK) equation for the conditional reliability function and the Generalized Pontryagin (GP) equations for the moments of first passage time are established. Finally, the analytical results, which are obtained by solving the BK equation and GP equations, are comparing with the numerical results to verity the validity of the analytical method. 2. Description of stochastic friction system and average equation Consider the non-linearly damped Coulomb friction system under combined additive and multiplicative Gaussian white noise excitations,

x¨ + ω0 2 x + 2c1 x˙ + c2 x˙ 3 + fk sgn(x˙ ) = ξ1 (t ) + xξ2 (t )

(1)

where x, x˙ , x¨ are displacement, velocity and acceleration, respectively. The dot · represents the differentiation with respect to the time t. ω0 is the nature frequency of the system, c1 , c2 are small parameters. fk is the amplitude of friction and sgn( ) represents the signum function. ξ 1 (t), ξ 2 (t) are independent Gaussian white noise which satisfy the following conditions

E (ξk (t ) ) = 0, k = 1, 2, E (ξk (t )ξl (t + τ ) ) = 2Dkl δ (τ ), k, l = 1, 2. Introducing the transform x1 = x, x2 = x˙ and considering Wong–Zakai approximation, Eq. (1) is equivalent to the following Itô stochastic differential equations

dx1 = x2 dt, dx2 = m(x1 , x2 )dt + σ (x1 , x2 )dW (t ),

(2)

where W(t) is standard Wiener stochastic process,

m(x1 , x2 ) = −ω0 2 x1 − 2c1 x2 − c2 x2 3 − fk sgn(x2 ),

σ 2 (x1 , x2 ) = 2D11 + 2D22 x1 2 . The integral equation of motion of undamped system is as the following equation

H=

1 2 2 1 2 ω0 x1 + x2 . 2 2

(3)

According to Eqs. (2) and (3), the Itô stochastic differential equation governing energy can be obtained by using the Itô formula

 

 



dH = + 2H − 2G(x1 ) f + 2H − 2G(x1 ) +



 + 2H − 2G(x1 )σ x1 , + 2H − 2G(x1 ) dW(t ), 







 1 2 σ x1 , + 2H − 2G ( x1 ) 2

dt (4)

where G(x1 ) is the potential energy,

1 2 2 ω0 x1 , 2 f (x ) = −2c1 x − c2 x3 − fk sgn(x ).

G ( x1 ) =

Using the stochastic averaging method [28,29], one can obtain the averaged Itô stochastic differential equation for total energy of system (1)

¯ (H )dt + σ¯ (H )dW (t ), dH = m

(5)

J. Sun et al. / Commun Nonlinear Sci Numer Simulat 54 (2018) 1–12

where

   2  σ x1 , 2H − 2G ( x1 ) 1 1 ¯ (H ) = −f 2H − 2G ( x1 ) + m  T (H ) −A 2 2H − 2G ( x1 )  ⎤     1 σ 2 x1 , − 2H − 2G ( x1 ) ⎦dx1 , + f − 2H − 2G ( x1 ) +  2 2H − 2G ( x1 ) 

σ¯ 2 (H ) = T (H ) = 2 A=



1 T (H )  A −A





A







A

3

−A



2H − 2G ( x1 )

1 2H − 2G ( x1 )

d x1 =

(6)

      σ 2 x1 , 2H − 2G ( x1 ) + σ 2 x1 , − 2H − 2G ( x1 ) d x1 ,

(7)

2π

(8)

ω0

,

2H /ω0 2 .

(9)

Substituting Eqs. (8) and (9) into Eqs. (6) and (7), the simplified equations can be obtained

¯ (H ) = D11 − m

√ 2 f k 2H

− 2 c1 H +

π

σ¯ 2 (H ) = 2D11 H +

D22 H 2

ω0 2

D22 H

ω0

−

3 c2 H 2 , 2

.

(10) (11)

3. BK equation and GP equation Research on the first passage failure is devoted to studying the probability that the motion stayed at the security domain. In this paper, the total energy is regard as the standard to study the reliability of the system. And we mainly research the reliability function of system (1), the conditional PDF of the first passage time of system (1), and the mean first passage time of system (1). Assume the safe domain of system (1) is an open domain  = [0, Hc ). According to Eq. (5), H(t) is a one-dimensional process. Reliability analysis aims to research the probability that the responses of a system stay in the safe domain or reach the boundary of the safe domain within the time interval [0, t]. Introducing the conditional reliability function

R(t |H0 ) = P {H (s ) ∈ , s ∈ (0, t ]|H (0 ) = H0 ∈ }. According to the diffusion process theory [30], one can obtain that the conditional reliability function R(t|H0 ) satisfies the following BK equation

∂R ∂R 1 2 ∂ 2R ¯ (H0 ) =m + σ¯ (H0 ) , ∂t ∂ H0 2 ∂ H0 2

(12)

with the initial condition

R(0|H0 ) = 1, H0 ∈ ,

(13)

and the boundary conditions

R(t |Hc ) = 0, t ≥ 0,

(14)

R(t |0 ) = f inite, t ≥ 0.

(15)

¯ (H0 ) and σ¯ (H0 ) are the same as in Eq. (5) with H(t) replaced by initial state H (0 ) = H0 . By assuming the In Eq. (12), m lifetime of first passage T , then the probability distribution function FT (t, H0 ) and the conditional PDF pT (t|H0 ) of the first passage time of system (1) can be obtained as follows

FT (t, H0 ) = P {T < t |H (0 ) = H0 ∈ } = 1 − R(t |H0 )

(16)

∂ FT (t, H0 ) ∂ R(t |H0 ) =− . ∂t ∂t

(17)

pT (t |H0 ) =

Assuming lim t n R(t |H0 ) = 0, then the statistical moments of first passage time satisfy the following equation t→∞

μn (H0 ) = E (T n ) = −



∞ 0

tn

∂ R(t |H0 ) dt = n ∂t

 0

∞

t n−1 R(t |H0 )dt.

(18)

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Fig. 1. When H0 = 0.01, the energy and time of system (1) with four different displacement and velocity. (a) x10 =







2H0 , x20 = 0; (b) x10 = 0, x20 =



2H0 ;

(c) x10 = − 2H0 , x20 = 0 ; (d) x10 = 0, x20 = − 2H0 .

Multiplying Eq. (12) by tn , and then integrating over t, the GP equations for the moments of first passage time can be obtained by using Eq. (18)

1 2 d2 μn+1 dμn+1 ¯ (H0 ) σ¯ (H0 ) +m = − ( n + 1 ) μn , n = 0 , 1 , 2 , · · · , 2 dH0 dH0 2

(19)

where μ0 = 1, Eq. (19) is under the boundary conditions

μn+1 (Hc ) = 0,

(20)

μn+1 (0 ) = f inite,

(21)

Especially, μ1 depicts the most well-known mean first passage time. ¯ (H0 ) = 0. In order to solve Eq. (12), a quantitative It can be known from Eqs. (10) and (11) that lim σ¯ (H0 ) = 0 and lim m H0 →0

H0 →0

boundary condition is required to replace qualitative condition (15). According to authors in [21,31,32], when σ¯ (0 ) = 0 and ¯ (0 ) = 0, boundary condition (15) should be replaced by the following equation m

∂R ∂R ¯ (H0 ) =m , H = 0. ∂t ∂ H0 0

(22)

Similarly, to solve the GP equations, qualitative boundary condition (21) also should be replaced by a quantitative condi¯ (0 ) = 0, the quantitative condition is given as the following equation tion. Based on the condition that σ¯ (0 ) = 0 and m

μ n+1 = −

(n + 1)μn (H0 ) , H0 = 0. ¯ (H0 ) m

(23)

J. Sun et al. / Commun Nonlinear Sci Numer Simulat 54 (2018) 1–12

Fig. 2. When H0 = 0.1, the energy and time of system (1) with four different displacement and velocity. (a) x10 =





5



2H0 , x20 = 0; (b) x10 = 0, x20 =

(c) x10 = − 2H0 , x20 = 0 ; (d) x10 = 0, x20 = − 2H0 .

Fig. 3. Reliability function of system (1). (a) Analytical results; (b) Monte Carlo simulation results.



2H0 ;

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Fig. 4. (a) Conditional reliability function of system (1); (b) conditional PDF of first passage time of system (1). - Analytical results; · Monte Carlo simulation results.

Fig. 5. When f k = 0.02, reliability function of system (1). (a) Analytical results; (b) Monte Carlo simulation results.

Fig. 6. When f k = 0.03, reliability function of system (1). (a) Analytical results; (b) Monte Carlo simulation results.

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Fig. 7. (a) Conditional reliability function of system (1); (b) conditional PDF of first passage time of system (1); (c) mean first passage time of system (1). - Analytical results; · Monte Carlo simulation results.

Fig. 8. When D11 = 0.005, reliability function of system (1). (a) Analytical results; (b) Monte Carlo simulation results.

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Fig. 9. When D11 = 0.02, reliability function of system (1). (a) Analytical results; (b) Monte Carlo simulation results.

Fig. 10. (a) Conditional reliability function of system (1); (b) Conditional PDF of first passage time of system (1); (c) mean first passage time of system (1). - Analytical results; · Monte Carlo simulation results.

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Fig. 11. When D22 = 0.005, reliability function of system (1). (a) Analytical results; (b) Monte Carlo simulation results.

Fig. 12. When D22 = 0.05, reliability function of system (1). (a) Analytical results; (b) Monte Carlo simulation results.

When n = 0, the two order ordinary differential equation governing the mean first passage time can be obtained by Eq. (19), and it satisfies the following boundary condition

μ 1 = −

1 , H0 = 0. ¯ (H0 ) m

(24)

4. Numerical results In this paper, set the critical energy Hc = 0.25 of the system, that is, when total energy H(t) is equal to or larger than the critical energy Hc , the system is damaged. The numerical data of time series are obtained by using second-order Runge– Kutta algorithm [33,34] with a time step of t = 0.005. Then the data are saved with 1 × 106 different trajectories under the same initial energy. Set the parameters ω0 = 1, c1 = 0.005, c2 = 0.01, fk = 0.01, D11 = 0.01, D22 = 0.01, unless there is a special definition. In order to observe the first passage of system (1) directly, the pictures of total energy and time with different displacement and velocity under the same initial energy are given in Figs. 1 and 2. In Figs. 1 and 2, the blue line represents the curve of the energy and time, the green line represents the critical energy, and the red line shows the time when the total energy is equal to or larger than the critical energy for the first time. It indicates that the movement stays at the security domain when the blue line is under the green one. From Figs. 1 and 2, we can see that the first passage occurs with some initial values. In addition, we can also see that the first passage time is different with the same initial energy and that the first passage is more likely to occur with larger initial energy.

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Fig. 13. (a) Conditional reliability function of system (1); (b) conditional PDF of first passage time of system (1); (c) the mean first passage time of system (1). - Analytical results; · Monte Carlo simulation results.

In order to study the reliability of system (1) further, the conditional reliability function, the conditional PDF of first passage time and the mean first passage time should be obtained. However, it is difficult to solve the BK equation analytically. Here, the conditional reliability function is calculated by using the implicit finite-difference method of Crank–Nicolson type which has been used in Ref. [14]. Friction force can hinder the movement of a system, and its existence makes the original smooth system into a nonsmooth system and also makes the movement more complex. In this section, we mainly consider the influence of friction force on the conditional reliability function of system (1), the conditional PDF of first passage time of system (1) and the mean first passage time of system (1). Fig. 3 shows the picture of the reliability function of system (1), in which (a) represents analysis results and (b) represents Monte Carlo simulation results. From Fig. 3, it can be seen that the reliability function is monotonously decreasing function of the initial energy, which is consistent with the conclusion drawn from Figs. 1 and 2. In addition, the reliability function becomes smaller with the increase of time, that is, when the time increases, the reliability reduces. Compared with Figs. 3(a) and (b), it can be found that the analytical results are in agreement with the numerical results. Fig. 4(a) shows that the conditional reliability function decreases more rapidly with larger initial value. It can be seen from Fig. 4(b) when the initial energy is close to the critical energy, the conditional PDF curve of first passage time increases rapidly and then drop sharply in the vicinity of t = 0. It can be found from Figs. 3, 5 and 6 that when the amplitude of friction increases, the reliability function also increases, that is, the reliability of system (1) can be improved by friction force. This conclusion can also be drawn from Fig. 7(a). As fk is increased, it can be seen in Fig. 7(b) that the peak of the conditional PDF curve become lower which implies the probability of first passage within a short time is smaller. It is seen from Fig. 7(c) that as the amplitude of friction increases, the mean first passage time increases, which is agree with the conclusions that the reliability of the system improve and that the probability of first passage within a short time is smaller.

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The existence of random excitations makes the range of random motion of a system expanded. Intuitively, we can draw the conclusion that the probability that the responses of a system stay at the security field will be smaller with the increase of the intensity of random excitations. Compared with Figs. 3, 8 and 9, it can be seen that the reliability function decreases when the intensity of external random excitation increases. Namely, the reliability of system (1) can be reduced by the external random excitation, which can also be drawn from Fig. 10(a). It can be seen from Fig. 10(b) that as the intensity of external random excitation increases, the peak of the conditional PDF curve become higher. That is, the probability of first passage within a short time is larger. Fig. 10(c) shows that the mean first passage time decreases with the increase of the intensity of external random excitation, which is agree with the conclusions that the reliability reduces and that the probability of first passage within a short time is larger. It can be also seen that the decrease of the mean first passage time when the intensity of external random excitation changes from D11 = 0.005 to D11 = 0.01 is larger than the decrease when the intensity changes from D11 = 0.01 to D11 = 0.02. Compared with Figs. 3, 11 and 12, it can be seen that the effect of the parametric random excitation on the reliability of system (1) is similar to the case of the external random excitation. In addition, it also has its own characteristic. From Fig. 13, it can be seen that the reliability function, the mean first passage time and the conditional PDF of first passage time change more slowly when the intensity of the parametric random excitation increases from D22 = 0.005 to D22 = 0.05. From Figs. 7 and 10, it can be seen that the effect of the parametric random excitation on the reliability of system (1) is smaller than the external random excitation. 5. Conclusion The reliability of a lightly non-linearly damped friction oscillator under combined additive and multiplicative Gaussian white noise excitations is investigated. The BK equation for the conditional reliability function and the GP equations for the moments of first passage time have been established based on the stochastic averaging method. These equations have been solved numerically by using finite difference method and Runge-Kutta method, then the analytical results have been verified by the results obtained from Monte Carlo simulation. The results show that the reliability can be raised by the friction force, while reduced by the random excitations. Particularly, the effect of the external random excitation on the reliability is larger than the parametric random excitation. 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