Effect of Parameter Configuration on Free Vibration of Networked Harmonic Oscillators with Viscous Damping

Heterogeneous oscillator networks consist of a dynamical function that describes the state of the oscillator (containing several parameters) and a topology that reflects the connections between the oscillators. Revealing the macroscopic dynamics of systems under different configurations of parameters and topology is a topic worthy of discussion and comes with great challenges. In this study, we discuss the effect of different parameter configurations on the damped free vibration of a classical spring oscillator network. On the regular network, we give the analytical expression satisfied by the free vibration of the system under over-damping and critical damping. Furthermore, we discuss the fastest and slowest exponential rates of decay of the system for different parameter conditions. In conjunction with quadratic eigenvalue theory, we extend the above analysis from regular networks to complex networks. We give a general method for calculating the eigenvalue spectrum of the system for arbitrary parameter configurations. In conjunction with the stability of the system and the rate of decay of the exponential rate, we also give numerical simulation results for the second largest and smallest real parts of the eigenvalues of the system. Finally, through the comparative analysis of the simulation results, we relax the matching relationship between the three parameters (mass, damping, and degree) to the matching between two parameters (mass and damping).