Stationary energy probability density of oscillators driven by a random external force

Abstract

We derive rigorous analytical results for the stationary energy probability density function of linear and nonlinear oscillators driven by additive Gaussian noise. Our study focuses on two cases: (i) a harmonic oscillator subjected to Gaussian colored noise with an arbitrary correlation function and (ii) nonlinear oscillators with a general potential driven by Gaussian white noise. We also derive analytical expressions for the stationary moments of the energy and investigate the partition of the mean energy between kinetic and potential energy. To illustrate our general results, we consider specifically the case of exponentially correlated noise for (i) and power-law and bistable potentials for (ii). Our theoretical results are substantiated by Langevin simulations.

Performance measures for single-degree-of-freedom energy harvesters under stochastic excitation

“…with a restoring force that is nonlinear see e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13]. In all of these approaches, a common characteristic is the employment of intensional nonlinearity in the harvester dynamics with an ultimate scope of increasing performance and robustness of the device without changing its size, mass or the amount of its kinetic energy.…”

Design criteria for dissipative devices in coupled oscillators under seismic excitation

“…It is worth to highlight that the expected value of the Total Energy for a linear oscillator is equally distributed between Potential and Kinetic Energy. Given this observation (reported also in Mendez et al), the manifolds in the design parameter space of the minimum values are the same for all possible adopted energy metrics: Elastic, Kinetic, and Total (Elastic + Kinetic). For this reason, here, only Total Energy has been considered as adopted design method.…”

Performance Barriers for Single-Degree-of-Freedom Energy Harvesters Under Generic Stochastic Excitation

“…. For this set of parameters, equation 13coincides with equation (9) and the solution to (13) will be q (t) = a √ bq 0 (bt). Note that for this solution we also have…”

“…with a restoring force that is nonlinear see e.g. [1][2][3][4][5][6][7][8][9]. In all of these approaches, a common characteristic is the employment of intensional nonlinearity in the harvester dynamics with an ultimate scope of increasing performance and robustness of the device without changing its size, mass or the amount of its kinetic energy.…”