MS02.2: Asymptotic Methods

This study is devoted to the analysis of special nonlinear wave phenomena, emerging in an-harmonic, dissipative Klein-Gordon cyclic chains being subjected to a parametric excitation. In particular, we analyze a special type of spatially extended nonlinear waves, manifested by strong amplitude modulation. Special coordinate transformation allows the exact reduction of the time-modulated waves to the manifold of phase locked beat-wave solutions. Analysis of the system dynamics on the manifold fully recovers the mechanism of system synchronization on the special, stationary traveling beat-wave solutions characterized by the harmonically modulated envelop. Importantly, the derived reduced order model captures exactly the transient evolution as well as the steady-state phases of the collective beat-wave response of PD-DpS model, initiated on the invariant manifold. Special attention of our talk will be given to the rather intriguing extensions of the present methodology to the more complex structures such as parametric chains with multi-component excitations exhibiting the intriguing TBW regimes of relaxation type.

In this paper, an initial-boundary value problem for a semi-infinite string is considered. A tuned-mass-damper (TMD) system is attached at one end of the string. The spring in the system is considered to be non-linear. We study the boundary reflection of waves and damping properties. The well-known D’Alembert formula is used throughout this paper to describe the general solution of the wave equation with corresponding initial velocity and displacement. A Multiple Scales Perturbation (MSP) method is used to approximate the solution of the non-linear problem. How waves are reflected depends strongly on the mass, spring and damper coefficients. Future work will focus on determining the accuracy of our approximations.