四边固支功能梯度矩形板的主共振分析

摘要

针对四边固支约束的陶瓷-金属材料功能梯度矩形板,在给出非均匀材料的应力应变关系及非线性几何方程基础上,应用虚功原理导出了横向简谐激励力作用下功能梯度板的非线性振动偏微分方程.通过位移函数的设定,利用伽辽金积分法推得了相应的达芬型非线性振动方程.应用多尺度法对非线性系统的主共振问题进行解析求解,得到了稳态运动下的幅频响应方程.基于李雅普诺夫稳定性理论,得到了共振下解的稳定性判别条件.作为算例,给出了不同参数下功能梯度矩形板共振的幅频曲线图和动相平面相轨迹图,讨论了不同参数对系统非线性振动特性的影响.%A ceramic/metal functionally graded rectangular plate was considered. Based on the stress-strain relationship of inhomogeneous materials and nonlinear geometric equations, the nonlinear partial differential equations of functionally graded plate subjected to transverse harmonic excitation force were derived according to the principle of virtual work. For the clamped rectangular plate, with the assumed displacement functions, the Duffing nonlinear vibration equation was obtained by using Galerkin method. Making use of the modified multiscales method, the amplitude-frequency response equation of steady-state motion was obtained. Based on Lyapunov theory, the stability of the resonance solution was analysed. As in some examples, the curres of amplitude versus detuning parameter and phase trajectories in moving phase plane were plotted under different situations. The effects of different parameters were discussed.