Parametrically Excited Stability of Periodically Supported Beams Under Longitudinal Harmonic Excitations
Parametrically Excited Stability of Periodically Supported Beams Under Longitudinal Harmonic Excitations
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DOI：
10.1142/s0219455419500950
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发表时间：
2019-08
影响因子：
3.6
通讯作者：
Z. Ying;Y. Ni;Lyu Fan
中科院分区：
文献类型：
--
作者：
Z. Ying;Y. Ni;Lyu Fan
基于Floquet定理、Fourier级数和矩阵特征值分析，提出了求解周期支承梁在一般简谐激励下多模态耦合振动稳定性问题的直接特征值分析方法。考虑了横向周期支承对梁在纵向周期激励下参数激励稳定性的影响。研究了纵向简谐激励下横向间隔支承梁的参激振动的动力稳定性。给出了简谐激励下间隔支承梁的运动偏微分方程，并利用Galerkin方法将其转化为时变周期参数的常微分方程，从而描述了梁的多模态耦合参激振动。基于Floquet定理，方程的基本解表示为周期分量和指数分量的乘积。将周期分量和周期参数展开为傅里叶级数，得到了直接判定参激稳定性的矩阵特征方程。通过对不稳定区域的数值计算，说明了简谐激励下空间支承梁的参激振动的动力稳定性。探讨了周期支承和激励参数对参激稳定性的影响。多模态耦合振动梁的参数激励稳定性可以通过周期支承得到改善。所发展的分析方法适用于更一般的周期参数梁在各种简谐激励下的多模态耦合振动。
A direct eigenvalue analysis approach for solving the stability problem of periodically supported beams with multi-mode coupling vibration under general harmonic excitations is developed based on the Floquet theorem, Fourier series and matrix eigenvalue analysis. The transverse periodic supports are considered for improving the parametrically excited stability of beams under longitudinal periodic excitations. The dynamic stability of parametrically excited vibration of the beam with transverse spaced supports under longitudinal harmonic excitations is studied. The partial differential equation of motion of the beam with spaced supports under harmonic excitations is given and converted into ordinary differential equations with time-varying periodic parameters using the Galerkin method, which describe the parametrically excited vibration of the beam with coupled multiple modes. The fundamental solution to the equations is expressed as the product of periodic and exponential components based on the Floquet theorem. The periodic component and periodic parameters are expanded into Fourier series, and the matrix eigenvalue equation is obtained which is used for directly determining the parametrically excited stability. The dynamic stability of parametrically excited vibration of the beam with spaced supports under harmonic excitations is illustrated by numerical results on unstable regions. The influence of the periodic supports and excitation parameters on the parametrically excited stability is explored. The parametrically excited stability of the beam with multi-mode coupling vibration can be improved by the periodic supports. The developed analysis method is applicable to more general period-parametric beams with multi-mode coupling vibration under various harmonic excitations.