在大扭转变形条件下,本文建立了新的非对称截面扭转和垂向运动耦合的两自由度动力学方程组.该微分方程组描述了截面在大扭转变形时的动力学行为.在忽略方程组中的平方非线性项,保留线性耦合及立方非线性项情况下,采用多尺度法求解了结构在垂向载荷及扭矩均为简谐载荷并发生主共振时的动力学行为.结果显示,当扭矩诱发低频主共振时,系统的立方非线性项呈现硬弹簧性质.当垂向简谐载荷诱发高频主共振时,立方非线性项呈现软弹簧性质.同时由于非线性的影响,结构的振动幅值会随激励的幅值及激励频率的变化而发生跳跃.这是仅考虑小扭转变形的数学模型所不能揭示的.

the two-degree-freedom dynamics equations that describe the non-linear vibrations of non-symmetrical cross-section are established condition in large rotation. The equations are simplified by ignoring square non-linear terms and retaining linear coupling and cubic non-linear terms. The simplified equations are solved by the multiscale method for harmonic vertical load and harmonic torque. The results illustrate that the cubic non-linearities play a role of hard spring when the external torque induce the resonance of low-frequency, and the cubic non-linearities play a role of soft spring when the vertical load induced the resonance of high-frequency. And as a result of the cubic non-linearities the vibration amplitudes may be suddenly jump with the change in excitation amplitude and excitation frequencies. This does not appear in the linear differential equations that is only considered a small torsion.