Influence of different delays on mixed types of oscillations under limited excitation

This review summarizes the findings of some published studies that have explored the influence of various delays (elasticity, damping, and self-oscillatory mechanism of excitation) on the dynamics of classes (or types) of mixed oscillations (MO) without and with consideration of the interaction between the oscillating system and the energy source. A general holistic framework was provided for how such delays, both separately and in combination, affect the dynamics of MOs. A unified computational scheme (model) used in the works studied made it easy to understand and compare the results of this influence on different types of MOs. With the account of the interaction with the energy source, the known calculation scheme (or model) of a mechanical frictional self-oscillating system serves as a unified basis for considering all types of MOs. Nonlinear differential equations of motion valid for all types of MOs with their respective solutions were presented, from which the relations for any certain type of MO are derived as special cases. Equations of unsteady motion and relations to calculate the amplitude and phase of stationary oscillations, the velocity of the energy source and the load of the oscillating system on it, as well as the stability conditions of stationary oscillations were given. The results of the calculations carried out to gain insight into the influence of delays on the system dynamics were discussed. Overall, the calculations show that the interaction between the forces with delay and the forces in the energy source is at the core of a variety of phenomena. Different delays in the same system change the shape of the amplitude-frequency curves, shift them, and influence the stability of motion.