Stability analysis of linear systems with switchable stiffness using the Floquet theory

Abstract

In this paper, a novel approach based on the Floquet theory is applied for the stability analysis of a mass–spring system with switchable stiffness. The Reid model is used to describe the dynamics of this semi-active vibration control problem. The semi-active control is achieved by a spring which commutes between a maximum and minimum stiffness according to a prescribed state-dependent rule and its performance is characterized by a system parameter, which relates to the extreme values of the stiffness. In order to apply the Floquet theorem, the Reid model is written as a linear periodic differential equation by converting the state-dependent rule into a time-periodic control law. The application of the theory allows us to obtain the Floquet multipliers and exponents in terms of the system parameter. The multipliers lie inside the unitary circle showing asymptotic stability, while the exponents are used to solve an optimization problem by applying a sensitivity analysis. Our results are validated by analyzing the Reid model using nonlinear analysis techniques. According to our findings, the present approach provides a useful tool to analyze the vibration control of linear systems with switchable stiffness in a natural and straightforward way, which also gives mathematical tractability for optimization purposes. In addition, this approach can be extended to study the cases of multi-degree-of-freedom systems and forced systems.