基于Bessel和Meijer-G函数的楔形和锥形悬臂梁振动分析

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摘要本文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，本文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，充分验证了用本文提出的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。

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	周坤涛1
	2
	杨涛1
	葛根1
	郝淑英3
	张琪昌4

关键词 ：欧拉伯努利梁, Lagrange法, Bessel 函数, Meijer-G 函数, 非线性振动

Abstract：In this paper, a nonlinear differential equation model of wedge and cone cantilever beams under external excitation is investigated by the Lagrange method based on the Euler Bernoulli beam theory. A new type of modal function without iteration and approximate truncation is 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, 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 are obtained. Finally, the amplitude-frequency response of the nonlinear primary resonance under a given vibration mode is 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 the existing literature. It can provide new ideas for the exact solution of wedge and cone cantilever beams.

Key words： Euler-Bernoulli beam Lagrange method Bessel function Meijer-G function nonlinear vibration

收稿日期: 2020-09-14 出版日期: 2022-02-28

引用本文:

周坤涛1,2，杨涛1，葛根1，郝淑英3，张琪昌4. 基于Bessel和Meijer-G函数的楔形和锥形悬臂梁振动分析[J]. 振动与冲击, 2022, 41(4): 253-261.
ZHOU Kuntao1,2, YANG Tao1, GE Gen1, HAO Shuying3, ZHANG Qichang4. Vibration analysis of wedge and cone cantilever beams based on Bessel and Meijer-G functions. JOURNAL OF VIBRATION AND SHOCK, 2022, 41(4): 253-261.

链接本文:

http://jvs.sjtu.edu.cn/CN/ 或 http://jvs.sjtu.edu.cn/CN/Y2022/V41/I4/253

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