工程力学 ›› 2020, Vol. 37 ›› Issue (3): 28-35.doi: 10.6052/j.issn.1000-4750.2019.05.0260

• 基本方法 • 上一篇    下一篇

基于新型振型函数的渐细变截面悬臂梁的自由振动理论与实验研究

周坤涛1,2, 杨涛1, 葛根1   

  1. 1. 天津工业大学机械工程学院, 天津 300399;
    2. 天津理工大学工程训练中心, 天津 300384
  • 收稿日期:2019-05-12 修回日期:2019-09-27 出版日期:2020-03-25 发布日期:2019-10-18
  • 通讯作者: 杨涛(1970-),男,湖北人,教授,博士,博导,主要从事复合材料加工与检测研究(E-mail:yangtao@tjpu.edu.cn). E-mail:yangtao@tjpu.edu.cn
  • 作者简介:周坤涛(1985-),男,湖北人,博士生,主要从事复合材料非线性结构动力学研究(E-mail:kuntaozhou@tjut.edu.cn);葛根(1982-),男,江苏人,副教授,博士,硕导,主要从事非线性振动研究(E-mail:gegenroot@126.com).
  • 基金资助:
    国家自然科学基金项目(11602169,11402168)

THEORETICAL AND EXPERIMENTAL STUDY ON FREE VIBRATION OF CANTILEVER TAPERED BEAM BASE ON NEW MODAL FUNCTION

ZHOU Kun-tao1,2, YANG Tao1, GE Gen1   

  1. 1. School of Mechanical Engineering, Tianjin Polytechnic University, Tianjin 300399, China;
    2. Engineering Training Center, Tianjin University of Technology, Tianjin 300384, China
  • Received:2019-05-12 Revised:2019-09-27 Online:2020-03-25 Published:2019-10-18

摘要: 该文基于超几何函数和Meijer-G函数的线性组合构建了一种新的变截面悬臂梁的模态函数,该振型函数具有实系数、无近似、精度高等优点。该文分两个步骤验证该振型函数的有效性和精确性:第一步,证明该振型中的自由基频及模态函数形状的准确性;第二步,验证该振型函数在研究变截面梁非线性振动时的效果。第一步中,自由基频及归一化后模态函数形状的理论解、有限元解、有限元半解析解及实验的对照结果精度较好。第二步中,将模态函数代入变截面悬臂梁非线性振动的控制方程,得到了伽辽金截断后的常微分方程的弯曲非线性系数及惯性非线性系数,随后用能量平衡法得到了非线性自由振动时的幅频响应,最后用实验验证了该幅频响应。结果显示,激光位移传感器测得梁上的一个靶点的位移-时间历程图和用振型函数加幅频响应的理论解的预测值吻合,说明了该文方法在预测变截面悬臂梁非线性振动时变形情况的准确性。

关键词: 渐细变截面悬臂梁, 振型函数, 非线性, 模态实验, 能量平衡法

Abstract: A new type of modal function which is a linear combination of hypergeometric and Meijer-G functions is proposed for tapered cantilever beam vibration analysis. This modal function has advantages of real coefficients, accuracy and non-approximation. Two steps are carried out to verify the effectiveness and accuracy of this modal function. In the first step, the fundamental frequency as well as the shape of the modal function is validated. In the second step, the modal function is used to perform nonlinear vibration analysis, and its effectiveness is investigated. A comparison is made among the theoretical prediction, finite element method, finite element semi-analytical method and the experimental results, which demonstrates the accuracy of the calculated frequency and the modal shape, respectively. Subsequently, substituting the modal function into the governing equation of the vibrating tapered cantilever after the Galerkin procedure, the curvature nonlinear coefficient and the inertia nonlinear coefficient are obtained. The frequency-amplitude relationship is deduced by the energy balance method which has been examined by the experiment. The results show that the displacement-time relationship of a selected point on the beam which is detected by a laser sensor perfectively coincides with the theoretical prediction which is given by the modal function and the frequency-amplitude relationship. The results show the effectiveness of the presented method in predicting the deformation of the tapered cantilever during nonlinear vibration.

Key words: tapered cantilever beam, modal function, nonlinearity, mode experiment, energy balance method

中图分类号: 

  • TH113
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