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总览 评价 唐有绮 1，* ， 陈立群 2，3， （ 1、 上海应用技术学院机械工程学院，上海 201418； 2、 上海大学力学系，上海 200444； 3、 上海市应用数学和力学研究所，上海 200072； ） 摘要： 本文研究了内共振条件下并发生一阶主共振时面内平动黏弹性板的
唐有绮1，*， 陈立群2，3，
（
1、上海应用技术学院机械工程学院，上海 201418； 2、上海大学力学系，上海 200444； 3、上海市应用数学和力学研究所，上海 200072； ）
摘要：
本文研究了内共振条件下并发生一阶主共振时面内平动黏弹性板的横向非线性动力性行为。板的黏弹性本构采用kelvin模型。根据广义哈密尔顿义原理导出了系统的控制方程和相应的边界条件。应用直接多尺度方法建立了系统的可解性条件，然后根据Routh-hurvitz判据确定了幅频响应的稳定性。近似解析分析表明，幅频响应与二阶主共振时不同，仅有双模态解。考察了面内平动速度、黏弹性系数和激励幅值对幅频响应的影响。最后，应用微分求积法数值求解原始方程，得到了一阶主共振和内共振联合激励时具有跳跃现象的幅频响应。对比两种方法的结果表明：数值计算结果验证了近似解析方法的结论。
关键词：
一般力学与力学基础；面内平动板；黏弹性；受迫振动；内共振
Tang Youqi1,*， Chen Liqun2,
（
1、School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai 201418； 2、Department of Mechanics, Shanghai University, Shanghai 200444； ）
Abstract：
In this paper, nonlinear forced vibrations of in-plane translating viscoelastic plates subjected to plane stresses are analytically and numerically investigated on the steady-state responses in external and internal resonances. The plate's material obeys the Kelvin model in which the material time derivative is used. A nonlinear partial-differential equation with four edges simple supports at both ends governing the transverse motion is derived from the generalized Hamilton principle. The method of multiple scales is directly applied to establish the solvability conditions in the primary resonance and the 3:1 internal resonance. The Routh-Hurvitz criterion is used to determine the stabilities of the steady-state responses. The effects of the viscosity coefficient, the excitation amplitude, and the in-plane translating speed on the steady-state responses are examined. The differential quadrature scheme is developed to solve the nonlinear governing equations numerically. The numerical calculations confirm the approximate analytical results regarding the solutions of the steady-state responses.
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