Topology-Dependent Excitation Response of Networks of Linear and Nonlinear Oscillators

This paper investigates the near-resonance response to exogenous excitation of a class of networks of coupled linear and nonlinear oscillators with emphasis on the dependence on network topology, distribution of nonlinearities, and damping ratios. The analysis shows a qualitative transition between the behaviors associated with the extreme cases of all linear and all nonlinear oscillators, respectively, even allowing for such a transition under continuous variations in the damping ratios but for fixed topology. Theoretical predictions for arbitrary members of the network class using the multiple-scales perturbation method are validated against numerical results obtained using parameter continuation techniques. The latter include the tracking of families of quasi-periodic invariant tori emanating from saddle-node and Hopf bifurcations of periodic orbits. In networks in the class of interest with special topology, 1:1 and 1:3 internal resonances couple modes of oscillation, and the conditions to suppress the influence of these resonances are explored.

Responses of Linear and Nonlinear Oscillators to Fractional Gaussian Noise With Hurst Index Between 1/2 and 1

The responses of linear and nonlinear oscillators to fractional Gaussian noise (fGn) are studied. First, some preliminary concepts and properties of fractional Brownian motion (fBm) and fGn with Hurst index 1/2<H<1 are introduced. Then, the exact sample solution, correlation function, spectral density, and mean-square value of the response of linear oscillator to fGn are obtained. Based on the sample solution, it is proved that the long-range correlation index of displacement response of linear oscillator is the same as that of excitation fGn, i.e., 2-2H, while the velocity response has no such long-range correlation. An interesting discovery is that the ratio of kinetic energy to total energy decreases as increasing Hurst index H. Finally, for the responses of one and two degrees-of-freedom (DOF) nonlinear oscillators to fGn, the equivalent linearization method is applied to obtain the sample functions, correlation functions and mean-square values of the responses. Plenty of digital simulation results are obtained to support these solutions. It is shown that the approximate solution is effective for weakly nonlinear oscillators and it is feasible to apply the equivalent linearization to study multi-DOF weakly nonlinear oscillators.