Probabilistic Solution of a Duffing-Type Energy Harvester System Under Gaussian White Noise

2015 ◽  
Vol 1 (1) ◽  
Author(s):  
H. T. Zhu
Keyword(s):  
White Noise ◽  
Energy Harvester ◽  
Gaussian White Noise ◽  
Joint Probability ◽  
Solution Procedure ◽  
Displacement Velocity ◽  
Strong Nonlinearity ◽  
Probabilistic Solution ◽  
Joint Probability Density ◽  
Electrical Variable

This paper proposes a solution procedure to formulate an approximate joint probability density function (PDF) of a Duffing-type energy harvester system under Gaussian white noise. The joint PDF solution of displacement, velocity, and an electrical variable is governed by the Fokker-Planck (FP) equation. First, the FP equation is reduced to a lower-dimensional FP equation only about displacement and velocity by a state-space-split (SSS) method. The stationary joint PDF of displacement and velocity can be solved exactly. Then, the joint PDF of displacement, velocity, and the electrical variable can be approximated by the product of the obtained exact PDF and the conditional Gaussian PDF of the electrical variable. A parametric study is further conducted to show the effectiveness of the proposed solution procedure. The study considers weak nonlinearity, strong nonlinearity, high excitation level, and a bistable oscillator. Comparison with the simulated results shows that the proposed solution procedure is effective in obtaining the joint PDF of the energy harvester system in the examined examples.

Probabilistic Solution of Vibro-Impact Systems Under Additive Gaussian White Noise

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
H. T. Zhu
Keyword(s):  
White Noise ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Solution Procedure ◽  
Additive Gaussian White Noise ◽  
Stationary Probability Density Function ◽  
Probabilistic Solution ◽  
Zero Offset ◽  
Polynomial Closure

This paper presents a solution procedure for the stationary probability density function (PDF) of the response of vibro-impact systems under additive Gaussian white noise. The constraint is a unilateral zero-offset barrier. The vibro-impact system is first converted into a system without barriers using the Zhuravlev nonsmooth coordinate transformation. The stationary PDF of the converted system is governed by the Fokker–Planck equation which is solved by the exponential-polynomial closure (EPC) method. A vibro-impact Duffing oscillator with either elastic or lightly inelastic impacts is considered in a numerical analysis. Meanwhile, the level of nonlinearity in displacement is also examined in this study as well as the case of negative linear stiffness. Comparison with the simulated results shows that the EPC method can present a satisfactory PDF for displacement and velocity when the polynomial order is taken as 4 in the investigated cases. The tail of the PDF also works well with the simulated result.

Probabilistic Solutions of a Nonlinear Plate Excited by Gaussian White Noise Fully Correlated in Space

2017 ◽  
Vol 17 (09) ◽  
pp. 1750097 ◽  
Author(s):  
G. K. Er ◽  
V. P. Iu
Keyword(s):  
White Noise ◽  
Random Vibration ◽  
Gaussian White Noise ◽  
Computational Effort ◽  
Degree Of Freedom ◽  
Equivalent Linearization ◽  
Nonlinear Random Vibration ◽  
Split Method ◽  
Probabilistic Solution ◽  
Polynomial Closure

This paper addresses the nonlinear random vibration of a rectangular von Kármán plate excited by uniformly distributed Gaussian white noise which is fully correlated in space. The state-space-split method and exponential polynomial closure method are jointly utilized to analyze the probabilistic solutions of the plate. The computational efficiency and numerical accuracy of the methodology for analyzing the nonlinear random vibration of the plate are verified by comparing the computational effort and numerical results with those obtained by Monte Carlo simulation and equivalent linearization, respectively. Meanwhile, the convergence of the probabilistic solution in the sense of Galerkin’s approximation is examined by analyzing the plate modeled as single-degree-of-freedom and multi-degree-of-freedom systems. Some phenomena are discussed after numerically studying the behaviors of probabilistic solutions of the deflection at different locations of the plate.

Internal Resonance Energy Harvesting

2015 ◽  
Vol 82 (3) ◽  
Author(s):  
Li-Qun Chen ◽  
Wen-An Jiang
Keyword(s):  
Energy Harvesting ◽  
White Noise ◽  
Frequency Response ◽  
Internal Resonance ◽  
Energy Harvester ◽  
Multiple Scales ◽  
Gaussian White Noise ◽  
Resonance Energy ◽  
Correlated Noise ◽  
Amplitude Frequency Response

Internal resonance is explored as a possible mechanism to enhance vibration-based energy harvesting. An electromagnetic device with snap-through nonlinearity is proposed as an archetype of an internal resonance energy harvester. Based on the equations governing the vibration measured from a stable equilibrium position, the method of multiple scales is applied to derive the amplitude–frequency response relationships of the displacement and the power in the first primary resonances with the two-to-one internal resonance. The amplitude–frequency response curves have two peaks bending to the left and the right, respectively. The numerical simulations support the analytical results. Then the averaged power is calculated under the Gaussian white noise, the narrow-band noise, the colored noise defined by a second-order filter, and the exponentially correlated noise. The results demonstrate numerically that the internal resonance design produces more power than other designs under the Gaussian white noise and the exponentially correlated noise. Besides, the internal resonance energy harvester can outperform the linear energy harvesters with the same natural frequencies and in the same size under Gaussian white noise.

Probabilistic analysis and ghost-stochastic resonance of a hybrid energy harvester under Gaussian White noise

Meccanica ◽  
2020 ◽  
Vol 55 (9) ◽  
pp. 1679-1691
Author(s):  
G. J. Fezeu ◽  
I. S. Mokem Fokou ◽  
C. Nono Dueyou Buckjohn ◽  
M. Siewe Siewe ◽  
C. Tchawoua
Keyword(s):  
White Noise ◽  
Stochastic Resonance ◽  
Probabilistic Analysis ◽  
Energy Harvester ◽  
Gaussian White Noise ◽  
Hybrid Energy ◽  
Hybrid Energy Harvester

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

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