Research paperResearch on the reliability of friction system under combined additive and multiplicative random excitations
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], [6], [7], [8], [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 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], [15], [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], [18], [19], [20], [21], [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.
Section snippets
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, where 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 represents the signum function.
ξ1(t), ξ2(t) are independent
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 . According to Eq. (5), H(t) is a
Numerical results
In this paper, set the critical energy 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 . Then the data are saved with 1 × 106 different trajectories under the same initial energy. Set the parameters , , , , , , unless there is a special
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
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11472212 and 11672233) and Innovation and Creative Seed Fund of Northwestern Polytechnical University (Grant No. Z2016161).
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2020, Communications in Nonlinear Science and Numerical SimulationCitation Excerpt :It is difficult to capture the closed-form solution of its PDF, but the numerical results can be obtained by solving the Kolmogorov-Feller equation or the corresponding integral equation, Chapman-Kolmogorov equation using numerical methods, e.g., the finite element method [6,7], the spectral finite difference method [8], the stochastic averaging method [9], the complex fractional moments method [10], the generalized cell mapping method [11], and the path integral method [12-15], etc. Further, if the first-passage reliability in terms of a certain specified threshold is of interest, an absorbing boundary condition can be imposed on the differential or integral equation that the PDF satisfies, namely, the probability density evolution equation, to yield the remaining probability density [16–18]. However, if the threshold is time variant, say due to the deterioration of materials in the long service life [19], or the reliability in terms of different thresholds is of concern, such treatment is not convenient.