基于蒙特卡洛法的Euler-Bernoulli梁基频和振型求解方法  

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

作　　者：祝磊[1,2] 张建勋 孙海林 Zhu Lei;Zhang Jianxun;Sun Hailin(Beijing Advanced Innovation Center for Future Urban Design,Beijing University of Civil Engineering and Architecture,Beijing 100044,China;School of Civil and Transportation Engineering,Beijing University of Civil Engineering and Architecture,Beijing 100044,China;China Architecture Design and Research Group,Beijing 100044,China)

机构地区：[1]北京建筑大学北京未来城市设计高精尖创新中心,北京100044 [2]北京建筑大学土木与交通工程学院,北京100044 [3]中国建筑设计研究院有限公司,北京100044

出　　处：《Journal of Southeast University(English Edition)》2024年第2期203-209,共7页东南大学学报（英文版）

基　　金：The National Natural Science Foundation of China (No.51778035)。

摘　　要：将Rayleigh法和蒙特卡洛法相结合,在Euler-Bernoulli梁理论假设下求解了均匀梁、变截面梁和附带集中质量的变截面梁自由振动问题.对原本连续的梁结构模型进行离散化处理,利用蒙特卡洛法给出梁结构的假设振型.将假设得到的梁结构振型函数代入Rayleigh法,多次计算过程中,将历次基频所得值与计算所得最小值进行比较,根据其相对误差判断是否满足收敛条件,进而求得基频及对应的振型.结果表明,不同计算模型中基频最大误差不超过10%,能够满足工程需求,且精度和时间的控制参数调整灵活,使用者可根据自身需要自行调节.该方法理论简明,适用范围广泛,能够快速准确地求解诸多类型的梁结构基频和振型.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.

关 键 词：EULER-BERNOULLI梁 基频 蒙特卡洛法 数值解 

分 类 号：O302[理学—力学]