This paper proposes a time-delayed feedback control to improve collection performance of the multiple attractors wind-induced vibration energy harvester system. By employing a conversion mechanism for decoupling the electromechanical equations and the stochastic averaging method, the roles of the time-delayed feedback and the system parameters have been systematically examined. In the deterministic case, the time delay can control efficiently the birhythmic properties; thus one can realize a dividing line in the parameter plane that separates the space into two subspaces of generically distinct nature. Further, it is exclusively demonstrated that implementing time-delayed feedback control serves as a very simple but highly efficient scheme to increase the harvested energy from vibrations in presence of noise disturbances. Additionally, the time constant ratio and coupling coefficients can also be properly tuned to optimize power generation.
This paper presents a study on stochastic bifurcations in a time-delayed birhythmic oscillator possessing a bistability mode with coexistence of two stable limit cycles in the deterministic case. Relying on the approximate methods, the stationary probability density function (PDF) of amplitude and joint PDF of displacement and velocity have been exhibited to characterize the qualitative properties of the system. The investigations indicate that the birhythmic region increases firstly and then decreases when time delay is monotonically varied. Further, system parameters and noise level can induce the appearance of stochastic P-bifurcation. Similar bifurcations can be induced by changing the strength of time delay and delayed feedbacks in displacement and velocity. Interestingly, joint PDF will reflect a more complex regime. And the role of the strength of the delayed velocity feedback on stochastic bifurcation is sensitive to the value of time delay. Numerical simulations are carried out for prototype models, which show basic agreement with our theoretical predictions.
This paper focuses on systems of stochastic partial differential equations with impulse effects. We establish an averaging principle such that the solution to the complex original nonlinear impulsive stochastic evolution equations can be approximated by that to the more simplified averaged stochastic evolution equations without impulses. By adopting stochastic analysis theory, semigroup approach and inequality technique, sufficient conditions are formulated and the mean square convergence is proved. This ensures that we can concentrate on the averaged system instead of the original system, thus providing a solution for reduction of complexity.
Noise induced motions are a significant source of uncertainty in the response of micro-electro-mechanical systems (MEMS). This is particularly the case for electrostatic MEMS where electrical and mechanical sources contribute to noise and can result in sudden and drastic loss of stability. This paper investigates the effects of noise processes on the stability of electrostatic MEMS via a lumped-mass model that accounts for uncertainty in mass, mechanical restoring force, bias voltage, and AC voltage amplitude. We evaluated the stationary probability density function (PDF) of the resonator response and its basins of attraction in the presence noise and compared them to that those obtained under deterministic excitations only. We found that the presence of noise was most significant in the vicinity of resonance. Even low noise intensity levels caused stochastic jumps between co-existing orbits away from bifurcation points. Moderate noise intensity levels were found to destroy the basins of attraction of the larger orbits. Higher noise intensity levels were found to destroy the basins of attraction of smaller orbits, dominate the dynamic response, and occasionally lead to pull-in. The probabilities of pull-in of the resonator under different noise intensity level are calculated, which are sensitive to the initial conditions.
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