Analytic Solutions of Nonlinear Wave and Structure Interactions

The purpose of this study is to obtain an analytic solution for the nonlinear problem of incident waves acting on a structure system with one degree of freedom. The analytic solution up to the second-order is pursued. The problem considered is a piston type structure containing mass, damper, and spring sitting at one end of a wave channel, and is subjected to acting of periodic incident waves. The nonlinear problem with structure and waves interaction is rewritten into a first-order (linear) problem and a second-order problem by using Taylor series expansion on moving boundaries together with perturbation expressions. In the second-order solution for the wave problem, the nonhomogeneous boundary value problem is divided into a Stokes wave problem and a wavemaker problem. The second-order solutions contain time-independent parts in addition to time-dependent solutions. Similar to the linear problem the interaction between waves and structure are resolved by using kinematic and dynamic conditions on the interface boundary. Using the present theory nonlinear characteristics involved in the interaction between waves and the structure are studied. For very rigid structure systems total reflection waves are obtained that confirm accuracy of the present derivations. With decreasing structural stiffness the induced reflection waves have increasing phase shift that can cancel incident waves, and can cause lesser wave forces on the structure and lesser structural motions. With structural damping the structural response caused by incident waves can be decreased, and resonant results can be decreased. Using the present theory nonlinear waveforms in front of the structure and the nonlinear structure motions are simulated.