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

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.