基于Hertz接触的单自由度碰振系统的随机响应近似闭合解

Approximate closed-form solution to random response of a SDOF vibro-impact system based on Hertz contact theory

碰振系统普遍存在于工程领域中,因此开展碰撞振动的研究具有重要的实际意义。应用随机振动研究的最新成果--迭代加权残值法,求解了高斯白噪声激励下基于Hertz接触的单自由度碰振系统的随机响应近似闭合解。首先,结合概率环流和概率势流的概念,构造系统FPK方程稳态解的近似表达式;然后,运用加权残值法获得近似表达式中的待定系数;最后,应用迭代技术在指定均方误差下获得特定精度的概率密度估计。为了说明该方法的有效性,分别考察了Duffing碰振系统和干摩擦碰振系统,再将蒙特卡罗模拟结果与理论解析解进行对比验证。研究表明,理论解析解与模拟结果吻合的非常好。

Vibro-impact systems exist widely in engineering fields, and studying collision-vibration is of great practical significance. Here, the latest achievement in studying random vibration, i.e., the iterative weighted residual method, was employed to obtain the approximate closed-form solution to random response of a SDOF vibro-impact system based on Hertz contact theory under Gaussian white noise excitation. Firstly, the approximate expression of the steady-state solution to Fokker-Planck-Kolmogorov(FPK) equation was constructed using concepts of the probabilistic circulation and probabilistic potential flow. Then, undetermined coefficients in the approximate expression were obtained using the weighted residual method. Finally, the iterative technique was used to gain the probability density estimation of specific accuracy under specified mean square error. To demonstrate the effectiveness of the proposed method, a Duffing vibro-impact system and a dry friction vibro-impact one were investigated, respectively. The theoretical analytical solutions were compared with the simulation results of Monte Carlo method. It was shown that the theoretical analytical solutions agree well with Monte Carlo simulation results.