Stable and unstable solutions in auto-parametric resonance zone of a non-holonomic system

Abstract

The aim of the study is to demonstrate a couple of special states which can be encountered at the system of a ball moving in a spherical cavity working as a passive tuned mass damper (TMD) of slender engineering structures. The system includes six degrees of freedom with three non-holonomic constraints being under horizontal additive kinematic excitation. The Appell–Gibbs approach is used to deduce the governing differential system. Uniaxial and biaxial types of kinematic excitation are considered. Among biaxial, a special attention is paid to circular setting. Influence of the rolling and spinning damping in contact of the ball with cavity is discussed. Under uniaxial excitation is the system auto-parametric and posses multiple solutions. The individual response branches can be identified when the excitation frequency is swept up or down with respect to setting up of initial conditions. Among stable branches reveal those with very low and sometimes zero approaching stability level. Although the accessibility of relevant trajectories is often very subtle due to effect of the dynamic stability, these post-critical phenomena accumulate a lot of energy. Hence, they can be very dangerous for TMD and other important engineering systems. Some general recommendations for practice are formulated.