基于超几何函数和梅哲G函数的变截面梁的非线性振动建模及自由振动

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振动与冲击 ›› 2019, Vol. 38 ›› Issue (23) : 77-83.

论文

薄喆，葛根

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Nonlinear dynamic modelling and free vibration for a tapered cantilever beam based on hyper-geometric function and Meijer-G function

BO Zhe,GE Gen

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文章历史 +

摘要

对具有等厚度变宽度的悬臂梁的非线性振动进行了理论研究。提出了基于超几何函数和梅哲G函数的变截面悬臂梁的无需摄动近似的振型函数，在此基础上采用凯恩方程建立了包含几何非线性和惯性非线性的振动方程。该方程的各项系数和现有文献中其他方法建模结果的计算结果相同，而表达式更简洁。得到的线性基频和有限元等其他多种方法结果做了对比，具有非常良好的精度。在考虑了强非线性振动的情况下，用变分法和能量平衡法得到了体系的幅频响应关系，该结果在大振幅条件下精度优于多尺度法的结果。还改进了能量平衡法，使之和数值解更为贴近。得出了判断系统渐硬或渐软特性的有效非线性系数。

Abstract

Here, a tapered cantilever beam’s nonlinear vibration was studied theoretically.The beam’s modal function was based on the hyper-geometric function and Meijer-G one without needing perturbation and approximation.Kane’s equation was used to establish the beam’s vibration equation with geometric and inertial nonlinearities.The calculation results for various coefficients of this equation were the same as those obtained with other modeling methods in literature and their expressions were more concise.The beam’s fundamental natural frequency gained with the proposed method was compared with that obtained with the FE one, Rayleigh-Ritz one and other ones, and it has a very good accuracy.Under strong nonlinear vibration cases, the system’s amplitude-frequency response relation was obtained with the variational method and the energy balance one, the results calculated using this relation are more accurate than those gained with the multi-scale method under the condition of large amplitude.Furthermore, the energy balance method was improved, and the results obtained with the improved one are closer to numerical solution.Finally, the effective nonlinearity coefficients were calculated to judge the system characteristics becoming hard or soft.

导出引用

薄喆，葛根. 基于超几何函数和梅哲G函数的变截面梁的非线性振动建模及自由振动[J]. 振动与冲击, 2019, 38(23): 77-83

BO Zhe,GE Gen. Nonlinear dynamic modelling and free vibration for a tapered cantilever beam based on hyper-geometric function and Meijer-G function[J]. Journal of Vibration and Shock, 2019, 38(23): 77-83

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