非圆截面杆中的非线性扭转波及其行波解

NONLINEAR TORSIONAL WAVE IN A NON-CIRCULAR CROSS-SECTIONAL ROD AND THE CORRESPONDING TRAVLING WAVE SOLUTIONS

研究了非圆截面杆中非线性扭转波的传播特性.由于非圆截面杆的扭转运动会伴随有横截面的翘曲,这种翘曲运动将引起扭转波的弥散.如果同时考虑有限扭转变形和翘曲弥散的共同作用,将会得到非线性扭转波的方程.在相平面上,对非线性扭转波动方程进行定性分析,结果表明,在一定条件下方程存在同宿轨道或异宿轨道,分别相应于方程的孤波解或冲击波解.本文利用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三类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.这些条件与定性分析的结果相一致.

Propagation properties of the nonlinear torsional wave in a non-circular cross-sectional rod are investigated in this paper. Torsional motion of a non-circular cross-sectional rod may accompany with the warping of the cross-section. The warping motion will cause dispersion of the torsional waves. When both the finite torsional deformation and the warping dispersion are considered simultaneously, a new nonlinear torsional wave equation can be established. Results from the qualitative analysis on the phase plane show that a homoclinic orbit or a heterclinic orbit of the nonlinear torsional wave equation exists under certain conditions, which corresponds to the solitary wave or the shock wave solution, respectively. Using the Jacobi elliptic function expansion method, three kinds of exact periodic solutions of the equation are obtained. The necessary conditions for the existence of these solutions are given, too, which are consistent with results obtained from the qualitative analysis.