Numerical solution method for fundamental frequency and mode shape of Euler-Bernoulli beam based on Monte Carlo method

Abstract

To address the challenge of solving free vibration problems in beams with uniform cross-sections, beams with variable cross-sections, and Euler-Bernoulli beams with concentrated masses, an innovative method combining the Rayleigh method and the Monte Carlo method is introduced. This dual-method strategy offers a novel solution by first discretizing the continuous beam structure model, followed by employing the Monte Carlo method to determine the vibration modes of the beam structure. Subsequently, these identified vibration modes are integrated into the Rayleigh method to calculate the fundamental frequency and vibration modes. The process involves a meticulous comparison with the minimum value obtained during calculations to ensure the satisfaction of the convergence condition. The results show that this combined method achieves a maximum error of 10% or less in predicting the fundamental frequency across different calculation models. This accuracy level is well within acceptable engineering requirements. The control parameters for accuracy and time can be easily adjusted to meet various needs. The method, which is simple in theory and widely applicable, enables the quick and precise determination of fundamental frequencies and vibration modes for diverse beam structures.

Keywords

Euler-Bernoulli beam / fundamental frequency / Monte Carlo method / numerical solution