scholarly journals Stationary Response of Flag-shaped Hysteretic System under Combined Harmonic and White Noise Excitations

2021 ◽  
Author(s):  
L.C. Chen ◽  
Huiying HU ◽  
Shushen Ye
Keyword(s):  
White Noise ◽  
Digital Simulation ◽  
Gaussian White Noise ◽  
Stochastic Averaging ◽  
Harmonic Excitation ◽  
Hysteretic Behavior ◽  
Stationary Response ◽  
Hysteretic System ◽  
Bifurcation Phenomenon ◽  
Stochastic Jump

Abstract The dynamical system containing flag-shaped hysteretic behavior is common in practice. In this paper, the stationary response of flag-shaped hysteretic system excited by harmonic excitation as well as Gaussian white noise is determined with the technique of stochastic averaging. The reliability of the presented approach is demonstrated by relevant digital simulation. The stochastic jump under a certain combination of parameters is found. The stochastic P-bifurcation phenomenon, i.e., the disappearance or appearance of bimodal shape of stationary response, occurs concerning to the variation of system’s parameters. Besides, the response of the system exposed to only harmonic excitation or non-resonance case is also examined for comparison, respectively. The numerical results show that the stationary amplitude response displays typical “soft” system behavior, and the deterministic jump may occur under pure harmonic excitation. Moreover, the non-resonance response is always weaker than that of resonant case.

Response Analysis of van der Pol Vibro-Impact System with Coulomb Friction Under Gaussian White Noise

2018 ◽  
Vol 28 (13) ◽  
pp. 1830043 ◽  
Author(s):  
Meng Su ◽  
Wei Xu ◽  
Guidong Yang
Keyword(s):  
White Noise ◽  
Coulomb Friction ◽  
Gaussian White Noise ◽  
Original System ◽  
Stochastic Averaging ◽  
Response Analysis ◽  
State Variables ◽  
Van Der Pol ◽  
Impact System ◽  
The Impact

In this paper, the stationary response of a van der Pol vibro-impact system with Coulomb friction excited by Gaussian white noise is studied. The Zhuravlev nonsmooth transformation of the state variables is utilized to transform the original system to a new system without the impact term. Then, the stochastic averaging method is applied to the equivalent system to obtain the stationary probability density functions (pdfs). The accuracy of the analytical results obtained from the proposed procedure is verified by those from the Monte Carlo simulation based on the original system. Effects of different damping coefficients, restitution coefficients, amplitudes of friction and noise intensities on the response are discussed. Additionally, stochastic P-bifurcations are explored.

Dynamics Response of a Hysteretic Structure Under Random Excitations

1993 ◽  
Author(s):  
Ismail I. Orabi
Keyword(s):  
White Noise ◽  
Transition Probability ◽  
Digital Simulation ◽  
Gaussian White Noise ◽  
Vertical Acceleration ◽  
Transition Probability Density ◽  
Vertical Excitation ◽  
Nonstationary Response ◽  
Dynamics Response ◽  
Range Of Values

Abstract The response of a hysteretic structure under horizontal and vertical random excitations is considered. The excitations are modeled by segments of stationary and nonstationary Gaussian white noise and filtered white noise processes. The linearization technique is used and the moments equations of the responses are evaluated. The transition probability density of the response is described and the associated second moment equations are derived. The transient and nonstationary response statistics for a range of values of parameters are obtained. A Monte-Carlo digital simulation study is performed. The results are compared with the theoretical findings and good agreements is observed. Particular attention is given to the amplification effects of the vertical acceleration. It is shown that the effect of the vertical excitation is usually insignificant, unless the load coefficient is quite large.

First-Crossing Problem of Weakly Coupled Strongly Nonlinear Oscillators Subject to a Weak Harmonic Excitation and Gaussian White Noises

2018 ◽  
Vol 140 (4) ◽  
Author(s):  
Y. J. Wu ◽  
H. Y. Wang
Keyword(s):  
Harmonic Function ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Digital Simulation ◽  
Stochastic Averaging ◽  
Harmonic Excitation ◽  
Van Der Pol Oscillator ◽  
Strongly Nonlinear ◽  
Set Up ◽  
White Noises

We study first-crossing problem of two-degrees-of-freedom (2DOF) strongly nonlinear mechanical oscillators analytically. The excitation is the combination of a deterministic harmonic function and Gaussian white noises (GWNs). The generalized harmonic function is used to approximate the solutions of the original equations. Four cases are studied in terms of the types of resonance (internal or external or both). For each case, the method of stochastic averaging is used and the stochastically averaged Itô equations are obtained. A backward Kolmogorov (BK) equation is set up to yield the failure probability and a Pontryagin equation is set up to yield average first-crossing time (AFCT). A 2DOF Duffing-van der Pol oscillator is chosen as an illustrative example to demonstrate the effectiveness of the analytical method. Numerically analytical solutions are obtained and validated by digital simulation. It is shown that the proposed method has high efficiency while still maintaining satisfactory accuracy.

Noise and Contact in Dynamic AFM Operations

2011 ◽  
Author(s):  
Ishita Chakraborty ◽  
Balakumar Balachandran
Keyword(s):  
White Noise ◽  
Evolution Equations ◽  
Stochastic Dynamics ◽  
Gaussian White Noise ◽  
Excitation Frequency ◽  
Period Doubling ◽  
Harmonic Excitation ◽  
Dynamic Mode ◽  
Grazing Impacts ◽  
Representative System

In this article, the authors study the effects of Gaussian white noise on the dynamics of an atomic force microscope (AFM) cantilever operating in a dynamic mode by using a combination of numerical and analytical efforts. As a representative system, a combination of Si cantilever and HOPG sample is used. The focus of this study is on understanding the stochastic dynamics of a micro-cantilever, when the excitation frequencies are away from the first natural frequency of the system. In the previous efforts of the authors, period-doubling bifurcations close to grazing impacts have been reported for micro-cantilevers when the excitation frequency is in between the first and the second natural frequencies of the system. In the present study, it is observed that the addition of Gaussian white noise along with a harmonic excitation produces a near-grazing contact, when there was previously no contact between the tip and the sample with only the harmonic excitation. Moment evolution equations derived from a Fokker-Planck system are used to obtain numerical results, which support the statement that the addition of noise facilitates contact between the tip and the sample.

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.

scholarly journals Homoclinic Bifurcation and Chaos in a Noise-InducedΦ6Potential

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Gen Ge ◽  
Zhi Wen Zhu ◽  
Jia Xu
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Nonlinear Damping ◽  
Harmonic Excitation ◽  
Simple Zero ◽  
Necessary Condition ◽  
Bounded Noise ◽  
Basin Boundaries ◽  
Zero Points ◽  
Safe Basin

The present paper focuses on the noise-induced chaos in aΦ6oscillator with nonlinear damping. Based on the stochastic Melnikov approach, simple zero points of the stochastic Melnikov integral theoretically mean the necessary condition causing noise-induced chaotic responses in the system. To quantify the noise-induced chaos, the Poincare maps and fractal basin boundaries are constructed to show how the system's motions change from a periodic way to chaos or from random motions to random chaos as the amplitude of the noise increases. Three cases are considered in simulating the system; that is, the system is excited only by the harmonic excitation, by both the harmonic and the Gaussian white noise excitations, or by both the bounded noise and the Gaussian white noise excitations. The results show that chaotic attractor is diffused by the noises. The larger the noise intensity is, the more diffused attractor it results in. And the boundary of the safe basin can also be fractal if the system is excited by the noises. The erosion of the safe basin can be aggravated when the frequency disturbing parameter of the bounded noise or the amplitude of the Gaussian white noise excitation is increased.

Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations

2013 ◽  
Vol 81 (4) ◽  
Author(s):  
Wantao Jia ◽  
Weiqiu Zhu ◽  
Yong Xu ◽  
Weiyan Liu
Keyword(s):  
White Noise ◽  
Probability Density ◽  
Hamiltonian Systems ◽  
Transition Probability ◽  
Digital Simulation ◽  
Relaxation Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Transition Probability Density ◽  
Poisson White Noise

A stochastic averaging method for quasi-integrable and resonant Hamiltonian systems subject to combined Gaussian and Poisson white noise excitations is proposed. The case of resonance with α resonant relations is considered. An (n + α)-dimensional averaged Generalized Fokker–Plank–Kolmogorov (GFPK) equation for the transition probability density of n action variables and α combinations of phase angles is derived from the stochastic integrodifferential equations (SIDEs) of original quasi-integrable and resonant Hamiltonian systems by using the jump-diffusion chain rule. The reduced GFPK equation is solved by using finite difference method and the successive over relaxation method to obtain the stationary probability density of the system. An example of two nonlinearly damped oscillators under combined Gaussian and Poisson white noise excitations is given to illustrate the proposed method. The good agreement between the analytical results and those from digital simulation shows the validity of the proposed method.

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