Dynamics of a piecewise non-linear system subject to dual harmonic excitation using parametric continuation

Abstract

The dynamic behavior of piecewise linear and/or non-linear vibratory systems subject to a single harmonic excitation has been recently analyzed by using a one- or multi-term harmonic balance (or Galerkin’s) method, piecewise linear techniques, analog simulation and direct numerical integration (digital simulation). Instead, this paper proposes to utilize the technique of parametric continuationto study the steady state response and global dynamics of a two-degree-of-freedom piecewise non-linear system with backlash or multi-valued springs and impact damping. The physical system is under the influence of a mean load and is subject to a dual harmonic external excitation, where the second frequency is either twice or three times the fundamental excitation frequency. The effects of various parameters such as impact damping, mean and alternating loads, the piecewise stiffness ratios and the relative phase between the excitation harmonics on the system response are thoroughly examined. System or excitation parameter regimes that exhibit sub-harmonic, super-harmonic quasi-periodic and chaotic solutions are obtained efficiently and systematically by using the proposed scheme. This investigation also clarifies the differences between viscous and non-linear impact damping models, which have been proposed earlier in the literature for studying clearance non-linearities. Limited experimental data validate our modelling strategy. Finally, selected predictions from this technique match very well with a multi-term harmonic balance method and with direct numerical integration, wherever applicable.