Nonlinear Normal Modes and Wave Transmission in a Class of Periodic Continuous Systems

We present two methodologies for studying periodic oscillations and wave transmission in periodic continuous systems with strongly nonlinear supporting or coupling stiffnesses. The first methodology is based on a nonsmooth transformation of the spatial variable and eliminates the singularities (generalized functions) from the governing partial differential equation of motion. The resulting smooth partial differential equation is then analyzed asymptotically. This method is used to study localized nonlinear normal modes (NNMs) of an infinite linear string supported by a periodic array of nonlinear stiffnesses. A second methodology is developed to study primary pulse transmission in a periodic system composed of linear layers coupled by means of strongly nonlinear stiffnesses. A piecewise transformation of the time variable is introduced, and the scattering of the primary pulse at the nonlinear stiffnesses is reduced to solving a set of strongly nonlinear first-order ordinary differential equations. Approximate analytical and exact numerical solutions of this set are presented, and the methodology is employed to study primary pulse transmission in a system with clearance nonlinearities.