Dynamics of a weakly non-linear periodic chain

Abstract

Harmonic wave propagation in an infinite, non-linear periodic chain is investigated. Both hardening and softening types of non-linearity are considered. A perturbation approach is used to obtain both propagation and attenuation constants which are amplitude dependent for such a non-linear system. Only the first-order non-linear effect is retained in the analysis. Special attention is given to the bounding frequencies of the propagation zone. Propagation constants are used to obtain the non-linear natural frequencies and the associated non-linear modes of both finite chains with homogeneous boundary conditions and endless cyclic chains. The computational effort is shown to be independent of the number of elements present in the chain. The interaction of two opposite-going primary waves in semi-infinite or finite chains are seen to generate secondary waves. The non-linear normal modes are found to consist of atmost two linear modes and for some boundary conditions exhibit restricted orthogonality properties. Some explicit numerical results are included to validate the wave-propagation approach for studying free vibration of such non-linear periodic chains.