Equations of motion of a hyperelastic beam under time-varying axial loading are derived via the extended Hamilton’s principle in this work, where the transverse vibration is coupled with the longitudinal vibration, and nonlinear vibrations of the beam in the subcritical buckling regime are investigated. Complex nonlinear boundary conditions of the beam are determined under some geometric constraints. The critical buckling load is first determined through linear bifurcation analysis. Effects of material and geometric parameters on the forced longitudinal vibration of the beam are numerically investigated. Steady harmonic shapes of the beam at different times under harmonic axial loading are determined. The beam is in the barreling deformation state even when the axial load is not in excess of the critical buckling load. The governing equation for the nonlinear transverse vibration of the beam is obtained by decoupling its equations of motion. Natural frequencies of the free linearized transverse vibration of the beam are studied. By applying the eigenfunction expansion method, the governing equation for the nonlinear transverse vibration of the beam transforms to a series of strongly nonlinear ordinary differential equations (ODEs). Two-to-one internal resonance of the beam is studied by the numerical integration method and its phase-plane portraits are obtained. The harmonic balance method and pseudo arc-length method are used to determine steady-state periodic solutions of the beam from the strongly nonlinear ODEs, and amplitude-frequency responses of the beam are determined. Effects of the external mean axial load, excitation amplitude, and damping coefficient on the amplitude-frequency response of the beam are numerically investigated. Combined effects of the external excitation amplitude and frequency on response amplitudes are also investigated.

利用扩展的汉密尔顿原理，推导了超弹性梁在时变轴向载荷下的运动方程，将横向振动与纵向振动耦合，研究了亚临界屈曲状态下梁的非线性振动。在某些几何约束下确定了梁的复杂非线性边界条件。临界屈曲载荷首先通过线性分叉分析确定。数值研究了材料和几何参数对梁受迫纵向振动的影响。确定在谐波轴向载荷下梁在不同时间的稳态谐波形状。即使当轴向载荷不超过临界屈曲载荷时，梁也处于击鼓变形状态。通过将梁的运动方程解耦，可以得到梁的非线性横向振动的控制方程。研究了梁的自由线性化横向振动的固有频率。通过应用本征函数展开法，梁的非线性横向振动的控制方程转化为一系列强非线性常微分方程（ODE）。利用数值积分方法研究了光束的二比一内共振，并获得了其相平面图。利用谐波平衡法和伪弧长法从强非线性ODE确定光束的稳态周期解，并确定光束的幅频响应。外部平均轴向载荷，激励幅度，数值研究了阻尼系数对梁幅频响应的影响。还研究了外部激励幅度和频率对响应幅度的组合影响。