基于Bessel和Meijer-G函数的楔形和锥形悬臂梁振动分析  

Vibration analysis of wedge and cone cantilever beams based on Bessel and Meijer-G functions

作　　者：周坤涛 杨涛[1] 葛根[1] 郝淑英[3] 张琪昌[4] ZHOU Kuntao;YANG Tao;GE Gen;HAO Shuying;ZHANG Qichang(School of Mechanical Engineering,Tiangong University,Tianjin 300387,China;Engineering Training Center,Tianjin University of Technology,Tianjin 300384,China;School of Mechanical Engineering,Tianjin University of Technology,Tianjin 300384,China;School of Mechanical Engineering,Tianjin University,Tianjin 300072,China)

机构地区：[1]天津工业大学机械工程学院,天津300387 [2]天津理工大学工程训练中心,天津300384 [3]天津理工大学机械工程学院,天津300384 [4]天津大学机械工程学院,天津300072

出　　处：《振动与冲击》2022年第4期253-261,共9页Journal of Vibration and Shock

基　　金：国家自然科学基金(12072234,11872044,12072233);天津市自然科学基金(20JCYBJC00510)。

摘　　要：基于欧拉-伯努利梁理论,利用Lagrange法建立了楔形和锥形截面梁在外激作用下的非线性微分方程。提出了一种基于Bessel函数和Meijer-G函数线性组合的无需迭代及近似截断的振型函数,且该振型函数不依赖于楔形和锥形变截面梁的弯曲振动的运动方程是否为标准的Bessel形式,该方法能快速求解线性基频和模态函数。随后将该方法得到的模态函数代入变截面悬臂梁非线性振动的控制方程中,得到了常微分方程的弯曲非线性系数及惯性非线性系数,最后利用多尺度法研究主共振下的幅频响应。结果表明,该方法得到的线性基频及非线性幅频响应曲线与已有文献结果高度吻合,充分验证了用该方法求解的振型函数具有很高的精确性,该方法可为楔形或锥形变截面悬臂梁模态函数解析解提供新思路。In this paper,a nonlinear differential equation model of wedge and cone cantilever beams under external excitation was investigated by the Lagrange method based on the Euler-Bernoulli beam theory.A new type of modal function without iteration and approximate truncation was proposed based on the linear combination of Bessel and Meijer-G functions and it does not depend on whether the equation of motion of the wedge and cone beam in flexural vibration is a standard Bessel form,which can quickly solve linear fundamental frequency and mode shape function.Subsequently,by substituting the modal function obtained in this paper into the governing equation of the vibrating tapered cantilever,the curvature nonlinear coefficient and the inertia nonlinear coefficient were obtained.Finally,the amplitude-frequency response of the nonlinear primary resonance under a given vibration mode was determined using the method of multiple scales.The results show that the linear fundamental frequency and nonlinear amplitude-frequency response curves obtained by the method in this paper are highly consistent with the results of existing literature.

关 键 词：欧拉-伯努利梁 Lagrange法 BESSEL函数 Meijer-G函数 非线性振动 

分 类 号：TH113[机械工程—机械设计及理论]