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.

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.

Effects of noise on periodic orbits of the logistic map

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Feng-guo Li
Keyword(s):  
Periodic Orbit ◽  
White Noise ◽  
Probability Density ◽  
Periodic Orbits ◽  
Gaussian Noise ◽  
Dynamical Behavior ◽  
Gaussian White Noise ◽  
Logistic Map ◽  
Period Doubling ◽  
Colored Gaussian Noise

AbstractNoise can induce an inverse period-doubling transition and chaos. The effects of noise on each periodic orbit of three different period sequences are investigated for the logistic map. It is found that the dynamical behavior of each orbit, induced by an uncorrelated Gaussian white noise, is different in the mergence transition. For an orbit of the period-six sequence, the maximum of the probability density in the presence of noise is greater than that in the absence of noise. It is also found that, under the same intensity of noise, the effects of uncorrelated Gaussian white noise and exponentially correlated colored (Gaussian) noise on the period-four sequence are different.

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.

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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