基于路径积分法的悬索非线性随机振动响应分析

Stochastic response analysis of a nonlinear cable with path integration

随机荷载激励下悬索过大的动力响应将影响其正常使用与安全,对其响应概率密度函数的求解与分析是评估悬索随机动力响应的重要途径之一。针对悬索在高斯白噪声激励下的随机振动模态响应,利用基于Gauss-Legendre积分和短时高斯转移概率密度假定的路径积分法,研究了模态振动响应的概率密度函数的平稳数值解与非平稳数值解,并进一步开展了参数研究,揭示了不同参数影响下概率密度函数的分布规律。将路径积分法所得的平稳解和非平稳解,分别与FPK方程的精确平稳解、等效线性化法所得平稳解及蒙特卡罗模拟非平稳解进行对比,结果表明,路径积分法所得的概率密度函数解分别与精确平稳解及蒙特卡罗模拟非平稳解符合良好。对于平稳响应,由于位移二次非线性项的存在,位移概率密度函数分布呈非对称分布形式,但速度概率密度函数并不受其影响,仍服从对称分布;非平稳响应概率密度函数初始时刻峰值较大,且在初始阶段峰值是随着时间不断变化的,波动较明显,随着时间推移逐渐平稳。研究结果对于悬索非平稳响应研究具有重要的工程意义。

The overlarge response of a cable under random excitation may significantly affect its usage and safety, and the analysis of its response probability density function is one of the important ways to evaluate the stochastic dynamic response of the cable. Regarding the random vibration modal response of the cable under Gaussian white noise, stationary and non-stationary numerical solutions of the probability density function of the modal vibration response are studied by using a path integration method. The method is based on the Gauss-Legendre integral rule and the short-time Gaussian transition probability density assumption. Furthermore, the parametric study is carried out to reveal the distribution law of the probability density function with different parameters. The stationary and non-stationary solutions, obtained by the path integration method, are compared with the stationary solution of the FPK equation, the stationary solution of the equivalent linearization method and the Monte Carlo simulation, respectively. The results show that the solutions of the probability density function obtained with the path integration method agree well with the exact stationary solution and the Monte Carlo simulation. For the stationary response, the presence of the quadratic term of displacement leads the probability density function distribution of displacement to be in an asymmetric shape. However, the probability density function of velocity still has a symmetric distribution, which is not affected by nonlinear coefficient. In the non-stationary case, the peak of the non-stationary probability density function solution continuously changes with large fluctuations in the early stage, and then it gradually reaches the stationary state. Therefore, the study on the non-stationary response of a cable is very significant in the field of engineering.