轴向运动正交各向异性板的超谐波共振及稳定性  

Superharmonic resonance and stability analysis of orthotropic plates with axial motion

作　　者：胡宇达[1,2] 张晓宇 Hu Yuda;Zhang Xiaoyu(School of Civil Engineering and Mechanics,Yanshan University,066004,Qinhuangdao,China;Key Laboratory of Mechanical Reliability for Heavy Equipment and Large Structures of Hebei Province,Yanshan University,066004,Qinhuangdao,China)

机构地区：[1]燕山大学建筑工程与力学学院,秦皇岛066004 [2]燕山大学河北省重型装备与大型结构力学可靠性重点实验室,秦皇岛066004

出　　处：《应用力学学报》2020年第2期674-681,I0015,共9页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金(11472239)。

摘　　要：研究了轴向运动正交各向异性条形薄板在线载荷作用下的超谐波共振问题。通过哈密顿原理导出了几何非线性下正交各向异性条形板的非线性振动方程。运用伽辽金积分法,推得了关于时间变量的量纲归一化非线性振动微分方程组。应用多尺度法求解三阶超谐波共振问题,得到了稳态运动下一阶、二阶、三阶共振形式的共振幅值响应方程。利用Liapunov方法推得不同共振形式稳态解的稳定性判据,并据此分析不同参数对系统稳定性的影响。绘制了振幅特性变化曲线图和与之对应的激发共振多解临界点曲线图,分析系统参数对共振的影响,并预测系统进入非线性共振区域的临界条件。得出激励在特定位置区间时可激发系统的超谐波共振,随着激励幅值的增加,上稳定解支减小,下稳定解支增加,且一阶模态振幅大于二阶、三阶振幅。Study on the superharmonic resonance of orthotropic strip-shaped thin plates under axial linear load is presented.The nonlinear vibration equation of orthotropic strips under geometric nonlinearity is derived by the Hamiltonian principle.Based on the Galerkin method,the non-dimensional nonlinear differential equations with time variables are derived.The multi-scale method is used to solve the first-three order superharmonic resonance differential equations and achieve the common amplitude response equations of the first-three orders resonance forms of steady-state motions.The stability discriminant for the steady solutions of the first-three order modes are deduced by using the Liapunov method,which can be used to analyze the effects of different parameters on the stability of systems.The amplitude characteristic curves and the corresponding critical point curves of the excited resonance are plotted,where the influence of parameters on resonance is analyzed,and the critical conditions of entry of nonlinear resonance regions are predicted.Results show that,with the increase of the tuning parameters,the amplitude of superharmonic resonance increases and the jump phenomenon occurs in the multi-value solutions.In addition,superharmonic resonance can be excited in specific regions of the excitation force position,where,as the excitation amplitude increases,the stable upper solution branch of the resonant amplitudes decreases,however,the lower stable solution branch increases.Moreover,the first-order modal amplitude is larger than the second-order and third-order amplitudes.

关 键 词：超谐波共振 正交各向异性板 轴向运动 线载荷 多尺度法 

分 类 号：O322[理学—一般力学与力学基础] O343.8[理学—力学]