Stabilization of statically unstable systems by parametric excitation

A linear vibrational system with multiple degrees of freedom subjected to parametric excitation is considered. It is assumed that the system is statically unstable but close to a critical point, the excitation amplitude and damping are small, and the excitation frequency is arbitrary. A new stabilization condition is derived in terms of integrals depending on eigenfrequencies and modes of the undisturbed conservative system and the symmetric excitation matrix. As a special case, an approximation for high-frequency excitation is deduced from this condition. Influence of damping on stabilization region is shown to be very small. Two examples for systems with one and two degrees of freedom are presented. It is shown that stabilization of statically unstable systems is possible for low, medium and high excitation frequencies.