Nonlinear flexural waves and chaos behavior in finite-deflection Timoshenko beam

Abstract

Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov's method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.

Propagation of flexural waves in inhomogeneous plates exhibiting hysteretic nonlinearity: Nonlinear acoustic black holes

“…This situation is similar to one with plane bulk shear acoustic waves [29,37,38]. The lowest order nonlinearity which is currently under the consideration for flexural motion in homogeneous materials is classical cubic elastic nonlinearity (of odd-type in its symmetry and cubic in its dependence on the wave amplitude) [39][40][41]. The theories of flexural waves, which include the classical elastic nonlinearities only, could be not relevant for the flexural waves propagation in micro-inhomogeneous plates where other types of nonlinearities, i.e., nonclassical nonlinearities, could dominate [28,31].…”

Chaotic vibrations of beams on nonlinear elastic foundations subjected to reciprocating loads

“…They used Galerkin truncation as well as the differential and integral quadrature method to investigate the nonlinear dynamic behavior of the system. Nonlinear flexural waves and chaotic behavior in Timoshenko beams was studied by Zhang and Liu using the method of Jacobi elliptic function expansion [23]. Younesian and Norouzi [24] analyzed forced vibration analysis of spinning disks subjected to transverse forces.…”