含多维随机变量的广义概率密度演化方程解析解:以Euler-Bernoulli梁为例

ANALYTICAL SOLUTIONS OF THE GENERALIZED PROBABILITY DENSITY EVOLUTION EQUATION WITH MULTIDIMENSIONAL RANDOM VARIABLES:THE CASE OF EULER-BERNOULLI BEAM

广义概率密度演化方程的解析解,不仅具有重要理论价值,而且具有校验数值解、进而标定数值算法误差的作用.以Euler-Bernoulli简支梁为例,推导给出了梁受迫振动时跨中位移响应所对应的广义概率密度演化方程解析解.包括非平稳非高斯随机载荷作用下的解(包含2维随机变量)以及同时考虑载荷随机性和结构参数随机性时的解(分别包含2维、4维和5维随机变量).分析结果表明,真实的概率密度演化是一个十分复杂的过程,远不能用简单的概率分布函数加以描述.这一进展,可为概率密度演化理论的进一步深入研究提供一个方面的基础.

The analytical solution of the generalized probability density evolution equation not only holds significant theoretical value,but also serves the purpose of validating numerical solutions and subsequently calibrating the errors in numerical algorithms.However,the analytical solution for this equation is limited to a small number of simple systems under a single random variable.Therefore,taking the Euler-Bernoulli simply supported beam as an example,the analytical solution of the generalized probability density evolution equation corresponding to the mid-span displacement response of the beam under forced vibration is derived.The solutions include those under non-stationary and non-Gaussian random excitations(involving 2-dimensional random variables)as well as those considering both the randomness of excitations and structural parameters(involving 2-dimensional,4-dimensional and 5-dimensional random variables,respectively).The analysis results indicate that the real evolution of probability density is a highly intricate process that cannot be described through simple probability distribution functions.This advancement can provide a foundational aspect for further in-depth research into the theory of probability density evolution.