梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性，同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应，利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析，在不同的条件下，该方程在相平面上存在同宿轨道或异宿轨道， 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法，对该非线性方程进行 求解，得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解，讨论了这些解存 在的必要条件，这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程，从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.

STUDY ON PROPAGATION OF NONLINEAR FLEXURAL WAVES IN THE BEAMS

By means of Hamilton variational principle, a nonlinear flexural wave equation for beams taking account of the geometric nonlinearity caused by the large deflection and the dispersive effect of rotational inertia in the beams is derived in this paper. Results of qualitative analysis of the nonlinear evolution equation show that for the equation there exists homoclinic or heteroclinic orbits on the phase plane, which correspond to a solitary wave or shock wave solution, respectively. Nonlinear flexural wave equation is solved by the Jacobi elliptic function expansion method. Two kinds of exact periodic solutions of the nonlinear equations are obtained, that is, the shock wave solution and the solitary wave solution. The necessary condition for existence of exact periodic solutions, shock solution and solitary solution is discussed, which is consistent with the qualitative analysis. By using the reductive perturbation method, two kinds of nonlinear SchrSdinger equations are derived from the nonlinear flexural wave equation. Taking into account of large deflection and rotary inertia of beams, the existence of NLS solitary wave in the beams is possible in theory.