Study on propagation of nonlinear flexural waves in the beams

By means of Hamilton variational principle, a nonlinearflexural wave equation for beams taking account of the geometricnonlinearity caused by the large deflection and the dispersive effectof rotational inertia in the beams is derived in this paper. Results ofqualitative analysis of the nonlinear evolution equation show that for theequation there exists homoclinic or heteroclinic orbits on the phase plane, whichcorrespond to a solitary wave or shock wave solution, respectively. Nonlinearflexural wave equation is solved by the Jacobi elliptic functionexpansion method.Two kinds of exact periodic solutions of the nonlinear equations areobtained, that is, the shock wave solution and the solitary wave solution.The necessary condition for existence of exact periodic solutions, shock solution andsolitary solution is discussed, which is consistent with thequalitative analysis. By using the reductive perturbation method, two kindsof nonlinear Schr\"odinger equations are derived from the nonlinear flexuralwave equation. Taking into account of large deflection and rotaryinertia of beams,the existence of NLS solitary wave in the beams is possible in theory.