含末端质量的悬臂梁随机非线性振动的随机平均法

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振动与冲击 ›› 2021, Vol. 40 ›› Issue (13) : 16-22.

论文

解娜娜，葛根

作者信息 +

Stochastic averaging method for stochastic nonlinear vibration of cantilever beam with end mass

XIE Nana, GE Gen

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文章历史 +

摘要

提出了一种可适用于含末端质量的悬臂梁强非线性振动的随机平均法。该方法将原本只能解决仅含有刚度非线性项振子的随机平均法扩展到了能解决既含有刚度非线性又含有惯性非线性的振子。先对含有末端质量的悬臂梁应用凯恩方法进行了建模，然后再基于哈密尔顿函数将振子化为关于瞬态等效振幅和瞬态相位的两个随机微分方程，随后应用随机平均原理将随机微分方程化简为一个关于等效振幅的伊藤方程。并在此基础上得出了在末端质量取不同值时的等效振幅的稳态概率密度以及位移和速度的联合概率密度。数值模拟很好地证明了该理论方法的正确性。

Abstract

Here, a stochastic averaging method was proposed for strongly nonlinear vibration of a cantilever beam with end mass. This method extended the random average method being only able to solve an oscillator with stiffness nonlinearity into the one able to solve oscillators with both stiffness nonlinearity and inertia nonlinearity. A cantilever beam with end mass was modeled using Kane method, and then the oscillator was converted into two stochastic differential equations with respect to transient equivalent amplitude and transient phase based on Hamilton function, and the stochastic differential equations were simplified into an ITO equation with respect to equivalent amplitude by using the stochastic average principle. Furthermore, the steady-state probability density of equivalent amplitude and the joint probability density of displacement and velocity were solved when the end mass having different values. The theoretical correctness of the proposed method was verified with numerical simulation.

导出引用

解娜娜，葛根. 含末端质量的悬臂梁随机非线性振动的随机平均法[J]. 振动与冲击, 2021, 40(13): 16-22

XIE Nana, GE Gen. Stochastic averaging method for stochastic nonlinear vibration of cantilever beam with end mass[J]. Journal of Vibration and Shock, 2021, 40(13): 16-22

参考文献

［1］ZAVODNEY L D, NAYFEH A H. The non-linear response of a slender beam carrying a lumped mass toa principal parametric excitation: theory and experiment［J］. International Journal of Non-Linear Mechanics, 1989, 24(2): 105-125.
［2］HIROSHI Y, YOSHIRO I, NOBUHARU A. Nonlinear analysis of a parametrically excited cantilever beam effect of the tip mass on stationary response(special issue on nonlinear dynamics)［J］. JSME International Journal, 2008, 41(3): 555-562.
［3］NAYFEH A H, PAI P F. Non-linear non-planar parametric responses of an in-extensional beam［J］. International Journal of Non-Linear Mechanics, 1989, 24(2): 139-158.
［4］DWIVEDY S K, KAR R C. Nonlinear response of a parametrically excited system using higher-order method of multiple scales［J］. Nonlinear Dynamics, 1999, 20(2): 115-130.
［5］HAMDAN M N, AL-QAISIA A A, AL-BEDOOR B O. Comparison of analytical techniques for nonlinear vibrations of a parametrically excited cantilever［J］. International Journal of Mechanical Sciences, 2001, 43(6): 1521-1542.
［6］AKBARZADE M, FARSHIDIANFAR A. Nonlinear transversely vibrating beams by the improved energy balance method and the global residue harmonic balance method［J］. Applied Mathematical Modelling, 2017, 45: 393-404.
［7］NIKKAR A, BAGHERI S, SARAVI M. Dynamic model of large amplitude vibration of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia［J］. Latin American Journal of Solids and Structures, 2014, 11(2): 320-329.
［8］FENG Z H, LAN X J, ZHU X D. Explanation on the importance of narrow-band random excitation characters in the response of a cantilever beam［J］. Journal of Sound and Vibration, 2009, 325(4): 923-937.
［9］FENG Z H, ZHU X D, LAN X J. Stochastic jump and bifurcation of a slender cantilever beam carrying a lumped mass under narrow-band principal parametric excitation［J］. International Journal of Non-Linear Mechanics, 2011, 46(10): 1330-1340.
［10］GE G, YAN W K. Cantilever model with curvature nonlinearity and longitudinal inertia excited by lateral basal moments being Gaussian white noise［J］. Journal of Vibroengineering, 2018, 20(1): 677-690.
［11］WU Y, HE J H. Homotopy perturbation method for nonlinear oscillators with coordinate-dependent mass［J］. Results in Physics, 2018, 10: 270-271.
［12］LEV B I, TYMCHYSHYN V B, ZAGORODNY A G. On certain properties of nonlinear oscillator with coordinate-dependent mass［J］. Physics Letters A, 2017, 381(39): 3417-3423.
［13］LANDAU P S, STRATONOVICH R L. Theory of stochastic transitions of various systems between different states［J］. Proceedings of Moscow University, Series, III, Vestinik, MGU, 1962, pp.33-45.
［14］KHAS’MINSKII R Z. On the behavior of a conservative system

with small friction and small random noise［J］. Journal of Applied Mathematics and Mechanics, 1964, 28(5): 1126-1130.
［15］GE G, LI Z P. A modified stochastic averaging method on single-degree-of-freedom strongly nonlinear stochastic vibrations［J］. Chaos Soliton & Fract, 2016, 91: 469-477.
［16］LAURA P A A, POMBO J L, SUSEMIHI E A. A note on the vibrations of a clamped-free beam with a mass at the free end［J］. Journal of Sound and Vibration, 1974, 37(2): 161-168.
［17］HE J H. Preliminary report on the energy balance for nonlinear oscillations［J］. Mechanics Research Communications, 2002, 29(2/3): 107-111.
［18］王栋. 附带有考虑集中质量的转动惯性的梁固有振动分析［J］. 振动与冲击，2010, 29(11): 221-225.
WANG Dong. Vibration analysis of a beam carrying lumped masses with both translational and rotary inertias［J］. Journal of Vibration and Shock, 2010, 29(11): 221-225.

［19］朱位秋.非线性随机动力学与控制——Hamilton理论体系框架［M］. 北京：科学出版社，2003.
［20］GE G, BO Z. Response of a cantilever model with a surface crack under basal white noise excitation［J］. Computers & Mathematics with Applications, 2018, 76(11/12): 2728-2743.
［21］朱海涛，段玲龙. 附加集中质量块悬臂梁路径积分法分析［J］. 武汉理工大学学报，2015, 37(4): 77-82.
ZHU Haitao, DUAN Linglong. Path integration analysis on a cantilever beam with a lumped mass［J］. Journal of Wuhan University of Technology, 2015, 37(4): 77-82.
［22］ADAMU M Y, OGENYI P, TAHIR A G. Analytical solutions of nonlinear oscillator with coordinate-dependent mass and Euler-Lagrange equation using the parameterized homotopy perturbation method［J］. Journal of Low Frequency Noise, Vibration and Active Control, 2019, 38(3/4): 1028-1035.
［23］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): 82-88. 
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): 82-88.
［24］HONEYCUTT R L. Stochastic Runge-Kutta algorithms. I: white noise［J］. Physical Review A: Atomic, Molecular and Optical Physics, 1992, 45(2): 600-603.