Optimal dynamic stabilisation of a linear system by periodic stiffness excitation

A theoretical study on the dynamic stabilisation of an unstable mechanical structure is performed. The system is unstable due to negative damping introduced by, for example, a self-excitation of linearised van der Pol type. A minimum model possessing two degrees of freedom with linear spring and damper elements is considered. Based on the effect of a parametric anti-resonance such a system may be stabilised by introducing a time-periodic stiffness variation. Optimum conditions are derived for achieving damping by parametric excitation. All stiffness elements in the system are considered to be available for stiffness variation. Using the averaging method in combination with Fourier series, general conditions for full vibration suppression are derived for arbitrary locations and phase relations of the stiffness variations. Analytical conditions are presented, showing how the maximum gain in stability depends on the amplitude, the phase, the location and the shape function of the periodic stiffness excitation. It is shown that only four characteristic values determine the optimum stiffness variation. These analytical predictions are verified by a numerical stability analysis of an example system. The results can be applied to tune the efficiency of vibration suppression achieved by a periodic variation of one or more stiffness parameters.