Abstract:

The main objective of this paper is to investigate the influence of inertia of nonlinear springs on the dispersion behavior of discrete monoatomic chains with lumped and distributed masses. The developed model can represent the wave propagation problem in a non-homogeneous material consisting of heavy inclusions embedded in a matrix. The inclusions are idealized by lumped masses, and the matrix between adjacent inclusions is modeled by a nonlinear spring with distributed masses. Additionally, the model is capable of depicting the wave propagation in bi-material bars, wherein the first material is represented by a rigid particle and the second one is represented by a nonlinear spring with distributed masses. The discrete model of the nonlinear monoatomic chain with lumped and distributed masses is first considered, and a closed-form expression of the dispersion relation is obtained by the second-order Lindstedt-Poincare method (LPM). Next, a continuum model for the nonlinear monoatomic chain is derived directly from its discrete lattice model by a suitable continualization technique. The subsequent use of the second-order method of multiple scales (MMS) facilitates the derivation of the corresponding nonlinear dispersion relation in a closed form. The novelties of the present study consist of (ⅰ) considering the inertia of nonlinear springs on the dispersion behavior of the discrete mass-spring chains; (ⅱ) developing the second-order LPM for the wave propagation in the discrete chains; and (ⅲ) deriving a continuum model for the nonlinear monoatomic chains with lumped and distributed masses. Finally, a parametric study is conducted to examine the effects of the design parameters and the distributed spring mass on the nonlinear dispersion relations and phase velocities obtained from both the discrete and continuum models. These parameters include the ratio of the spring mass to the lumped mass, the nonlinear stiffness coefficient of the spring, and the wave amplitude.

Key words: nonlinear mass-spring chain, discrete model, continuum model, Lindstedt-Poincare method (LPM), method of multiple scales (MMS), dispersion, phase velocity