Stochastic stability of a piezoelectric vibration energy harvester under a parametric excitation and noise-induced stabilization

Vibration energy harvesters seek to convert the energy of ambient, random vibrations into electrical power, often using piezoelectric transduction. The stochastic dynamics of a piezoelectric harvester subjected to a parametric (state-dependent) excitation has not been comprehensively investigated in the nonequilibrium regime. Motivated by mathematical results that establish the stabilization of dynamic response using noise, we investigate the stochastic stability of a generic harvester in the linear and the monostable nonlinear regimes excited by a multiplicative noise process characterized by both Brownian and Lévy distributions. Stability is characterized in each case by studying the approach of the harvester response towards equilibrium in the time domain, longer term proximity (or divergence) of two solutions starting with nearby initial conditions in the phase plane as well as the Lyapunov exponent. In the linear case, we analytically obtain a lower bound on the magnitude of the noise intensity that guarantees stability. The bound is derived as an inequality in terms of the system parameters. This analytic result is validated numerically using the Euler-Maruyama scheme by: (1) computing the harvester response and its approach to equilibrium in terms of its displacement, velocity and voltage, (2) computing the trajectories of two solutions with nearby initial conditions in the phase plane and (3) the sign of the maximal Lyapunov exponent in terms of energy. We find that noise-induced stabilization occurs consistently for the noise intensities greater than the lower bound in linear and weakly nonlinear harvesters, for both Brownian and Lévy excitation. Instabilities emerge for the noise intensities lower than the bound. The results lead to the interesting conclusion that noise of appropriate strength can induce stabilization and are expected to be useful in the design of energy harvesters. In addition, the results are expected to be significant in the study of phenomena such as noise-induced transport in Brownian rotors where stability is an important aspect of the stochastic dynamics.