Vibro-impact system, as an important type of non-smooth system, exhibits intricately nonlinear characteristics. Inevitably, the vibro-impact system will encounter random excitations, but the conventional methods ar7e not eligible for simultaneous determination of its transient responses and reliabilities. Commonly, existing methods of applying non-smooth transformation tend to ignore the essential non-smooth characteristics of vibro-impact system. To this end, this paper proposes a unified framework based on direct probability integral method (DPIM) to simultaneously determine stochastic dynamic responses and reliabilities of unilateral vibro-impact systems under combined harmonic and random excitation without non-smooth transformation, and captures their complicated dynamical behaviors. Firstly, the impact velocity dependent coefficient of restitution is introduced to establish the motion equation of vibro-impact system. Secondly, the probability density integral equation (PDIE) for the unilateral vibro-impact system is derived from the perspective of probability conservation. Then, the PDIE and governing differential equation of the system is solved in a decoupled and efficient way. Moreover, the first-passage reliability is assessed by introducing extreme value mapping of the stochastic dynamic response. Numerical results of three typical examples using the proposed framework are compared with those using Monte Carlo simulation (MCS), quasi-MCS and from the reference, which highlights the advantages of DPIM in computing the stochastic responses and reliabilities of vibro-impact system under random excitations and random parameters. The stationary probability density functions exhibit periodic fluctuations under combined harmonic and stochastic excitation. Specially, the noise intensity and frequency of harmonic excitation pose the great influence on the reliabilities of systems.

中文翻译：

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联合激励下单侧振动-冲击系统的随机动力学分析


振动冲击系统作为一种重要的非光滑系统，表现出复杂的非线性特性。振动冲击系统不可避免地会遇到随机激励，但传统方法 ar7e 无法同时确定其瞬态响应和可靠性。通常，现有的应用非光滑变换的方法往往忽略了振动冲击系统的基本非光滑特性。为此，本文提出了一种基于直接概率积分法 （DPIM） 的统一框架，以同时确定单侧振动冲击系统在无非平滑变换的谐波和随机联合激励下的随机动力学响应和可靠性，并捕获其复杂的动力学行为。首先，引入冲击速度相关恢复系数，建立振动-冲击系统的运动方程;其次，从概率守恒的角度推导了单侧振动冲击系统的概率密度积分方程 （PDIE）。然后，以解耦和高效的方式求解系统的 PDIE 和控制微分方程。此外，通过引入随机动态响应的极值映射来评估首次通过的可靠性。将采用所提框架的三个典型实例的数值结果与使用蒙特卡洛模拟 （MCS）、准 MCS 和参考文献的数值结果进行了比较，突出了 DPIM 在计算随机激励和随机参数下振动冲击系统的随机响应和可靠性方面的优势。稳态概率密度函数在谐波和随机联合激励下表现出周期性波动。 特别是，谐波激励的噪声强度和频率对系统的可靠性影响很大。