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 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 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.

Stationary Solution of Duffing Oscillator Driven by Additive and Multiplicative Colored Noise Excitations

2017 ◽  
Vol 139 (2) ◽  
Author(s):  
Siu-Siu Guo ◽  
Qingxuan Shi
Keyword(s):  
White Noise ◽  
Colored Noise ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Approximate Solutions ◽  
Intensity Level ◽  
First Order ◽  
Fpk Equation ◽  
Polynomial Closure ◽  
Mcs Method

A bistable Duffing oscillator subjected to additive and multiplicative Ornstein–Uhlenbeck (OU) colored excitations is examined. It is modeled through a set of four first-order stochastic differential equations by representing the OU excitations as filtered Gaussian white noise excitations. Enlargement in the state-space vector leads to four-dimensional (4D) Fokker–Planck–Kolmogorov (FPK) equation. The exponential-polynomial closure (EPC) method, proposed previously for the case of white noise excitations, is further improved and developed to solve colored noise case, resulting in much more polynomial terms included in the approximate solution. Numerical results show that approximate solutions from the EPC method compare well with the predictions obtained via Monte Carlo simulation (MCS) method. Investigation is also carried out to examine the influence of intensity level on the probability distribution solutions of system responses.

Noise Influenced Responses of Elastic Cantilevers With Nonlinear Tip Force Interactions

2011 ◽  
Author(s):  
Ishita Chakraborty ◽  
Balakumar Balachandran
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Harmonic Excitation ◽  
Attractive Force ◽  
Surface Band ◽  
Micro Scale ◽  
Cantilever Structure ◽  
Macro Scale ◽  
Band Limited

In this article, the effects of noise on a base-excited cantilever structure with nonlinear tip force interactions are studied by using experimental, numerical, and analytical methods. The focus of the study is on the enhancement of the cantilever response, when Gaussian white noise is added to the harmonic base input. The experimental arrangement consists of a base-excited elastic cantilever with a magnet attached to its free end. An attractive force is generated at the cantilever tip magnet through another magnet of opposite polarity, which is fixed to a translatory stage. The second magnet is covered by a thin compliant material, with which the tip magnet makes intermittent contact when the cantilever is subjected to a base excitation. For a purely harmonic excitation, it is observed that the tip magnet of the cantilever sticks to the base magnet due to the attractive force. Starting from a situation where the cantilever tip is sticking to the surface, band-limited white Gaussian noise is added to the excitation and the strength of noise is gradually increased. The cantilever tip resumes its periodic motion when the strength of added noise reaches a sufficient signal to noise ratio. This phenomenon is explored by using a reduced-order numerical model and an analytical framework involving the application of a moment-evolution approximation derived from the associated Fokker Planck equation for the system. Since the macro-scale experimental system qualitatively replicates the micro-scale attractive-repulsive force interaction experienced by an atomic force microscope cantilever operated in tapping mode, this study sheds light on the possible use of white noise to control the sticking of such micro-scale cantilevers with sample surfaces.

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