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基于已建立的弹性地基上不可伸长梁的非线性动力学模型,利用梁的量纲归一化运动方程和多尺度方法求得梁2次超谐共振的幅频响应方程和位移的二次近似解。进而,运用梁的幅频响应曲线对其超谐共振响应特性进行研究,同时分析了弹性地基模型、Winkler参数、外激励幅值、边界条件等对该共振响应的影响效应。结果表明:弹性地基模型中剪切参数的引入增大了梁2次超谐共振响应的幅值和多值区域;弹性地基Winkler参数的增加会抑制系统的共振响应,但同时会增加系统动力响应的软弹簧特性;在外激励幅值较小的情况下,系统共振响应未展现出明显的非线性特征;边界约束对弹性地基剪切参数作用于梁2次超谐共振响应的效应有显著影响,可在一定程度上改变系统响应幅值及多值区域。
Based on the established nonlinear dynamic model of non-extensible beam on elastic foundation, the amplitude-frequency response equation and the quadratic displacement of the second-order superharmonic resonance of the beam are obtained by using the normalized equations of motion of the beam and the multi-scale method Approximate solution. Furthermore, the effect of the super-harmonic resonance response of the beam is studied by using the amplitude-frequency response curve of the beam, and the effect of the elastic foundation model, Winkler parameter, external excitation amplitude and boundary conditions on the resonance response is also analyzed. The results show that the introduction of shear parameters in the elastic foundation model increases the amplitude and multi-value region of the second-order superharmonic resonance response of the beam. The increase of the Winkler parameter of the elastic foundation suppresses the resonance response of the system, but increases the dynamic response The system resonance response does not show obvious nonlinear characteristics under the condition of small external excitation amplitude.The boundary constraints have a significant effect on the effect of the elastic foundation shear parameters on the second-order superharmonic resonance response of the beam, To a certain extent, the system can change the amplitude and multi-valued area.