Effect of geometry configuration on bursting oscillations of the mechanical oscillator with strong irrational nonlinearities and its FPGA-based implementation

This paper analyzes the dynamic bursting of a mechanical oscillator with two strong irrational restoring forces when the external force changes slowly. The oscillator can exhibit both smooth and discontinuous characteristics depending on its geometry configuration, which is determined by the smoothness parameter. Using the fast-slow dynamics analysis method, we explore the equilibrium stability and bifurcation diagram of the unperturbed system, considering the external periodic excitation as a control parameter for both smooth and discontinuous cases. The bifurcation diagram shows two-fold/fold hysteresis cycles, which disappear as the smoothness parameter increases. Additionally, we examine the effects of the smoothness parameter and amplitude of the external excitation on the system. The study explores bursting oscillations and identifies two normal patterns—"fold/fold" and "node/node". Although there is no significant difference between smooth and discontinuous cases, the amplitude, shape, and time interval between spiking states of bursting oscillation depend on the smoothness parameter and external excitation amplitude. Furthermore, changing the smoothness parameter can create or eliminate bursting oscillations in the SD. To validate the numerical simulation results, the study implements the dynamics of a discontinuous oscillator in Field Programmable Gate Arrays (FPGA), which yields results that align with the numerical simulations.