Periodic Solutions and the Stability in a Non-linear Parametric Excitation System

Abstract

Phenomena on the interaction between self-excited and parametric vibrations in the flow-induced vibration problems are described by a differential equation of Van der Pol - Mathieu type with the non-linear restoring force as a mathematical model. Periodic solutions in the regions of parametric resonances of the first and the second orders are approximated by the sum of two frequency components. These approximated solutions have high accuracy. The stability criterion of periodic solutions established by approximately obtaining the regions of instability in Hill's equation for small disturbances from periodic solutions.