Exceptional points in time-varying oscillators with enhanced sensing sensitivity

We explore the exceptional points (EPs) in mass-spring oscillators with time-varying stiffness. The second-order EPs result from the coalescences of the fundamental and harmonic eigenmodes that can be achieved by a time modulation of the spring stiffness. The occurrence of EPs is first demonstrated in a theoretical model by solving the eigenvalue problem with the state-space method. The signature of the EP can be seen in the response spectrum when the system is subjected to an external excitation. The undamped and damped cases are both considered in this mass-spring oscillator, and the impact of the damping ratio and modulation amplitude on the EPs is systematically investigated. Based on an analogy of the equation of motion with the Mathieu differential equation, we discuss the stability of the undamped time-varying system under different modulation parameters and explore its connection to the EP phenomenon. As a typical property of a second-order EP, the square-root dependence of the frequency splitting response to a perturbation is studied by introducing an added mass. In a second part, the above theoretical concepts are established by simulation methods in an elastic solid mass-spring model and an equivalent time-varying stiffness is realized by shunting a piezoelectric patch with switch-controlled external negative capacitance circuits. This work should pave the way for applications of crack or perturbation detection in elastic media and inspire other elastic wave modulation functions in time-varying systems.