Nonlinear mode coupling in a passively isolated mechanical system

Abstract

Recent studies in passively isolated systems have shown that mode coupling is desirable for better vibration suppression, thus refuting the long-standing rule of modal decoupling. However, these studies have ignored the nonlinearities in the isolators. In this work, we consider stiffness nonlinearity from pneumatic isolators and study the nonlinear forced damped vibrations of a passively isolated ultra-precision manufacturing (UPM) machine. Experimental analysis is conducted to guide the mathematical formulation. The system comprises linearly and nonlinearly coupled in-plane horizontal and rotational motion of the UPM machine with quadratic nonlinear stiffness from the isolators. We present an analytical study using the method of multiple scales and the method of harmonic balance for different cases of external resonances, viz., the primary and the secondary resonances (superharmonic and combined resonances) with 1 : 2 internal resonance between the modes. We further validate our analytical findings using direct numerical integration and observe an excellent match. On extending our analysis, we observe the existence of subcritical, supercritical, and s-shaped bifurcation depending on the location of the isolators and the case of external resonance. Also, the saturation and quasi-saturation phenomenon are observed for the case of resonances close to the higher natural frequency and combined resonance, respectively. A parametric study is conducted to examine the effect of different parameters on the dynamics of the system, and consecutively the critical parameters of the system are identified for different cases of external resonance.