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.

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.

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.

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.

scholarly journals GENERALIZED WIENER PROCESS AND KOLMOGOROV'S EQUATION FOR DIFFUSION INDUCED BY NON-GAUSSIAN NOISE SOURCE

2005 ◽  
Vol 05 (02) ◽  
pp. L267-L274 ◽  
Author(s):  
ALEXANDER DUBKOV ◽  
BERNARDO SPAGNOLO
Keyword(s):  
White Noise ◽  
Wiener Process ◽  
Anomalous Diffusion ◽  
Gaussian Noise ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Non Gaussian ◽  
Generalized Wiener Process ◽  
Fokker Planck

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker–Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov–Feller equation for discontinuous Markovian processes, and the fractional Fokker–Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.

Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order α

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
White Noise ◽  
Fractional Order ◽  
Gaussian White Noise ◽  
Linear Partial Differential Equation ◽  
Planck Equation ◽  
Stochastic Dynamical Systems ◽  
Poissonian Noise ◽  
Liouville Fractional Derivative ◽  
White Noises ◽  
Taylor’S Series

AbstractIn a first stage, the paper deals with the derivation and the solution of the equation of the probability density function of a stochastic system driven simultaneously by a fractional Gaussian white noise and a fractional Poissonian white noise both of the same order. The key is the Taylor’s series of fractional order f(x + h) = E α(hαDx α)f(x) where E α() denotes the Mittag-Leffler function, and D x α is the so-called modified Riemann-Liouville fractional derivative which removes the effects of the non-zero initial value of the function under consideration. The corresponding fractional linear partial differential equation is solved by using a suitable extension of the Lagrange’s technique involving an auxiliary set of fractional differential equations. As an example, one considers a half-oscillator of fractional order driven by a fractional Poissonian noise.

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.

Piezoelectric Performance of PVDF Composites for Transportation Engineering: A Multi-Scale Simulation Study

2020 ◽  
Author(s):  
Xiao-Hong Yin ◽  
Jin-Wen Jian ◽  
Can Yang ◽  
Tian Lei ◽  
Tao Cheng
Keyword(s):  
Vinylidene Fluoride ◽  
Transportation Engineering ◽  
Piezoelectric Composites ◽  
Micro Scale ◽  
Multi Scale ◽  
Piezoelectric Energy ◽  
Cantilever Structure ◽  
Macro Scale ◽  
Poly Vinylidene Fluoride ◽  
Piezoelectric Performance

Abstract In the present work, the poly (vinylidene fluoride) composite filled with the lead zirconium titanate (PVDF/PZT) was numerically investigated focusing on the improvement of piezoelectric performance parameters. With a multi-scale simulation strategy, effects of the PZT fillers’ orientation and length on the electrical outputs of the piezoelectric energy collectors buried in the roads were systematically examined. Specifically, at the micro-scale, based on our previous research results, Comsol Multiphysics connected with Matlab was utilized to create the unit cell of piezoelectric composites. The simulation results showed that parameters of PZT nano-fillers greatly affect the piezoelectric coefficients. For the macro-scale simulation, a road energy collector with innovative symmetrical cantilever structure was designed, with piezoelectric constants obtained at micro-scale simulation as inputs. The correlation between the output voltage of the energy-collector and PZT parameters (i.e., orientation and length) was successfully developed by applying the vehicle’s axle-load. This work provides a way for tailoring the piezoelectric performance of the macro components (i.e., sensors) through adjusting the states of the fillers inside the piezoelectric composites.

Modified Path Integral Solution of Fokker–Planck Equation: Response and Bifurcation of Nonlinear Systems

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
Pankaj Kumar ◽  
S. Narayanan
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Path Integral ◽  
Duffing Oscillator ◽  
Planck Equation ◽  
Harmonic Excitation ◽  
Integration Scheme ◽  
Pi Method ◽  
Non Gaussian ◽  
Fokker Planck

Response of nonlinear systems subjected to harmonic, parametric, and random excitations is of importance in the field of structural dynamics. The transitional probability density function (PDF) of the random response of nonlinear systems under white or colored noise excitation (delta correlated) is governed by both the forward Fokker–Planck (FP) and the backward Kolmogorov equations. This paper presents a new approach for efficient numerical implementation of the path integral (PI) method in the solution of the FP equation for some nonlinear systems subjected to white noise, parametric, and combined harmonic and white noise excitations. The modified PI method is based on a non-Gaussian transition PDF and the Gauss–Legendre integration scheme. The effects of white noise intensity, amplitude, and frequency of harmonic excitation and the level of nonlinearity on stochastic jump and bifurcation behaviors of a hardening Duffing oscillator are also investigated.

EFFECT OF GAUSSIAN WHITE NOISE ON THE DYNAMICAL BEHAVIORS OF AN EXTENDED DUFFING–VAN DER POL OSCILLATOR

2006 ◽  
Vol 16 (09) ◽  
pp. 2587-2600 ◽  
Author(s):  
XIAOLI YANG ◽  
WEI XU ◽  
ZHONGKUI SUN
Keyword(s):  
White Noise ◽  
Lyapunov Exponents ◽  
Random Noise ◽  
Gaussian White Noise ◽  
Harmonic Excitation ◽  
Basins Of Attraction ◽  
Van Der Pol Oscillator ◽  
Necessary Condition ◽  
Van Der Pol ◽  
Dynamical Behaviors

The influence induced by random noise on dynamical behaviors is a classical yet challenging subject. This paper discusses the influence of Gaussian white noise on the dynamics of a self-excited triple well extended Duffing–Van der Pol oscillator already subjected to harmonic excitation. Firstly, the condition for the rise of hom/heteroclinic chaos is derived by random Melnikov's technique under its corresponding mean-square criterion and the result indicates that the threshold amplitude of harmonic excitation is lowered by the appearance of Gaussian white noise. Moreover, the threshold is decreased as the noise intensity increases. Since the Melnikov's criterion is only a necessary condition for the occurrence of chaotic motion, this prediction is tested against numerical simulations of the basins of attraction and the Lyapunov exponents. By vanishing the largest Lyapunov exponents, another criterion for the onset of chaos is obtained which is accorded with the theoretical one. Finally, how the noise effects the structure of periodic or chaotic attractor is investigated by simulating Poincare maps of the original system and rich transition states displayed by the considered extended Duffing–Van der Pol oscillator are observed.

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