Chaotic behavior of a curved carbon nanotube under harmonic excitation

2009 ◽  
Vol 42 (3) ◽  
pp. 1860-1867 ◽  
Author(s):  
Fathi N. Mayoof ◽  
Muhammad A. Hawwa
Keyword(s):  
Carbon Nanotube ◽  
Chaotic Behavior ◽  
Harmonic Excitation

TRANSITION TO CHAOS IN A CURVED CARBON NANOTUBE UNDER HARMONIC EXCITATION

2012 ◽  
Vol 26 (32) ◽  
pp. 1250210 ◽  
Author(s):  
YIXIANG GENG ◽  
LIXIANG ZHANG
Keyword(s):  
Carbon Nanotube ◽  
Chaotic Behavior ◽  
Numerical Study ◽  
Melnikov Method ◽  
Equation Of Motion ◽  
Harmonic Excitation ◽  
Periodic Attractor ◽  
Runge Kutta Method ◽  
Transition To Chaos ◽  
Melnikov Criterion

The chaotic behavior of a carbon nanotube with waviness along its axis is investigated. The equation of motion involves a quadratic and cubic terms due to the curved geometry and the mid-plane stretching. Melnikov method is applied for the system, and Melnikov criterion for global homoclinic bifurcations is obtained analytically. The numerical solution of the system using a fourth-order-Runge–Kutta method confirms the analytical predictions and shows that the transition from regular to chaotic motion is often associated with increasing the energy of an oscillator. Moreover, a detailed numerical study of the periodic attractor in the period window is also carried out.

Subharmonic Bifurcations and Transition to Chaos in a Pipe Conveying Fluid under Harmonic Excitation

2013 ◽  
Vol 444-445 ◽  
pp. 791-795
Author(s):  
Yi Xiang Geng ◽  
Han Ze Liu
Keyword(s):  
Chaotic Behavior ◽  
Melnikov Method ◽  
Harmonic Excitation ◽  
Phase Portraits ◽  
Pipe Conveying Fluid ◽  
Bifurcation Diagrams ◽  
Dynamical Behaviors ◽  
Transition To Chaos ◽  
Subharmonic Bifurcations ◽  
Conveying Fluid

The subharmonic and chaotic behavior of a two end-fixed fluid conveying pipe whose base is subjected to a harmonic excitation are investigated. Melnikov method is applied for the system, and Melnikov criterions for subharmonic and homoclinic bifurcations are obtained analytically. The numerical simulations (including bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincare map) confirm the analytical predictions and exhibit the complicated dynamical behaviors.

Optimally Designed Nonlinear Feedback Excitation for High-Sensitivity Sensing

2009 ◽  
Author(s):  
Andrew R. Sloboda ◽  
Bogdan I. Epureanu
Keyword(s):  
Chaotic Behavior ◽  
High Sensitivity ◽  
Harmonic Excitation ◽  
Pattern Search ◽  
Control Parameters ◽  
Nonlinear Feedback ◽  
Linear Vibration ◽  
Optimal Excitation ◽  
Multiple Sensor ◽  
Sensitivity Vector

Employing sensitivity vector field (SVF) analysis in active micro-sensors can increase both their sensitivity and their ability to differentiate between changes in multiple sensor parameters. However, since SVF analysis is based on quantifying attractor deformations in state space, maximizing its effectiveness depends on selecting a sensor excitation that generates an attractor having suitable deformation with respect to the parameter(s) of interest. This paper addresses issues surrounding such system excitation design for a simple, linear vibration-based sensor having a combination of harmonic and nonlinear feedback excitation. In order to reframe the search for an optimal excitation as a search for a set of optimal control parameters, the excitation is considered to be of a specified form with a set of adjustable control parameters. Determining how to adjust the excitation parameters so as to maximize the magnitude of the resulting sensitivity vectors is then the formal goal. Using a pattern search method that avoids difficulties caused by bifurcations, we show that improved excitation can be designed reliably and efficiently. We also show that for short trajectory evolution times (suitable for “large” sensor perturbations) limit cycle behavior generates the best SVFs while for longer evolution times (suitable for “small” sensor perturbations) chaotic behavior may be more useful. Other issues discussed include the relative importance of various controller terms and the significance of harmonic excitation phase when generating sensitivity vectors.

Nonlinear Forced Vibration Analysis of a Non-Local Carbon Nanotube Carrying Intermediate Mass

2015 ◽  
Author(s):  
Zia Saadatnia ◽  
Ebrahim Esmailzadeh
Keyword(s):  
Carbon Nanotube ◽  
Forced Vibration ◽  
Beam Theory ◽  
Analytical Form ◽  
Harmonic Excitation ◽  
Local Theory ◽  
Mass Detection ◽  
Intermediate Mass ◽  
Non Local ◽  
Forced Vibration Analysis

The aim of this study is to model and investigate the nonlinear transversal vibration of a carbon nanotube carrying an intermediate mass along the structure considering the nonlocal and non-classical theories. Due to the application of the proposed system in sensors, actuators, mass detection units among others, the analysis of forced vibration of such systems is of an important task being considered here. The governing equation of motion is developed by combining the Euler-Bernoulli beam theory and the Eringen non-local theory. The Galerkin approach is employed to obtain the governing differential equation of the system and the transient beam response for the clamped-hinged boundary condition. A strong perturbation method is utilized to solve the equation obtained and the system responses subjected to a harmonic excitation is examined. The steady-state motion is studied and the frequency response in an analytical form is obtained. Finally, results are evaluated for some numerical parameter values and their effect on the frequency responses are presented and fully discussed.

Large-Amplitude Vibrations of Thin Panels

2006 ◽  
Author(s):  
M. Amabili
Keyword(s):  
Boundary Conditions ◽  
Lyapunov Exponents ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Time Integration ◽  
Period Doubling ◽  
Harmonic Excitation ◽  
Different Boundary Conditions ◽  
Model C

Geometrically nonlinear vibrations of circular cylindrical panels with different boundary conditions and subjected to harmonic excitation are numerically investigated. The Donnell's nonlinear strain-displacement relationships are used to describe geometric nonlinearity; in-plane inertia is taken into account. Different boundary conditions are studied and the results are compared; for all of them zero normal displacements at the edges are assumed. In particular, three models are considered in order to investigate the effect of different boundary conditions: Model A for free in-plane displacement orthogonal to the edges, elastic distributed springs tangential to the edges and free rotation; Model B for classical simply supported edges; Model C for fixed edges and distributed rotational springs at the edges. Clamped edges are obtained with the Model C for very high value of the stiffness of rotational springs. The nonlinear equations of motion are obtained by the Lagrange multi-mode approach, and are studied by using the code AUTO based on pseudo-arclength continuation method. Convergence of the solution with the number of generalized coordinates is numerically verified. Complex nonlinear dynamics is also investigated by using bifurcation diagrams from direct time integration and calculation of the Lyapunov exponents and the Lyapunov dimension. Interesting phenomena such as (i) subharmonic response, (ii) period doubling bifurcations, (iii) chaotic behavior and (iv) hyper-chaos with four positive Lyapunov exponents have been observed.

Chaotic Behavior in the Double Pendulum Under Parametric Resonance

2016 ◽  
Author(s):  
Rafael H. Avanço ◽  
Helio A. Navarro ◽  
Airton Nabarrete ◽  
José M. Balthazar ◽  
Angelo Marcelo Tusset
Keyword(s):  
Friction Coefficient ◽  
Parametric Resonance ◽  
Chaotic Behavior ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Phase Portraits ◽  
Double Pendulum ◽  
Bifurcation Diagrams ◽  
Harmonic Motion ◽  
Parametric Pendulum

In literature, the classic parametrically excited pendulum is vastly studied. It consists of a pendulum vertically displaced with a harmonic motion in the support while it oscillates. The chaos in this mechanism may appear depending on the frequency and amplitude of excitation in superharmonic and subharmonic resonance. The double pendulum is also well analyzed in literature, but not under parametric excitation. Therefore, this is the novelty in the present paper. The present analysis considers a double pendulum under a harmonic excitation following the same idea performed previously for a single pendulum. The results are obtained based on methods, such as, phase portraits, Poincaré sections and bifurcation diagrams. The 0–1 tests analyze the presence of chaos while the parameters are varied. The dimensionless parameters take into account the excitation frequency and amplitude as mentioned for the classic parametric pendulum. In this case, we have the particular characteristic that the two pendulums have the same length, the same mass and the same friction coefficient in the joints. The types of motion observed include fixed points, oscillations, rotations and chaos. Results also demonstrated that there was a self-synchronization between these pendulums in ideal excitation.

CHAOTIC RESPONSE ANALYSIS OF SINGLE-WALLED CARBON NANOTUBE DUE TO SURFACE DEVIATIONS

NANO ◽  
2012 ◽  
Vol 07 (02) ◽  
pp. 1250008 ◽  
Author(s):  
ANAND Y. JOSHI ◽  
SATISH C. SHARMA ◽  
S. P. HARSHA
Keyword(s):  
Carbon Nanotube ◽  
Chaotic Behavior ◽  
Response Analysis ◽  
Central Plane ◽  
System Behavior ◽  
Frequency Spectra ◽  
Single Walled Carbon Nanotube ◽  
Single Walled Carbon ◽  
Mass Sensors ◽  
Surface Deviations

Nonlinear vibrational behavior of a single-walled carbon nanotube based mass sensors is considered. The modeling involves stretching of the mid plane and damping. The equation of motion involves two nonlinear terms due to the curved geometry and the stretching of the central plane due to the bridged boundary conditions. The manifestation of instability and chaos in the dynamic response is observed. The regions of periodic, sub-harmonic and chaotic behavior are clearly seen to be dependent on added mass and the surface deviations. Poincaré maps and frequency spectra are used to explicate and demonstrate the miscellany of the system behavior.

CHAOTIC BEHAVIOR AND ITS CONTROL IN THE SINGLE-WALL CARBON NANOTUBE

2014 ◽  
Vol 28 (03) ◽  
pp. 1450005 ◽  
Author(s):  
JIANSHU FANG
Keyword(s):  
Carbon Nanotube ◽  
Theoretical Analysis ◽  
Chaotic Behavior ◽  
Perturbation Technique ◽  
Sufficient Conditions ◽  
Single Wall Carbon Nanotube ◽  
Single Wall Carbon ◽  
System Parameters ◽  
Direct Perturbation ◽  
Necessary And Sufficient

Using the direct perturbation technique, we obtained a general perturbed solution of single-wall carbon nanotube (SWNT). We also provided two necessary and sufficient conditions for boundedness of the perturbed solution. Theoretical analysis results and the corresponding numerical results showed that the perturbed solution of the SWNT was unbounded in generally, which indicated the chaotic behavior of the system. Moreover, we were able to control the chaos of the carbon nanotube by adjusting the system parameters.

Chaos of a Single-Walled Carbon Nanotube Resulting from Periodic Parameter Perturbation

2021 ◽  
Vol 31 (09) ◽  
pp. 2150130
Author(s):  
Zhen Wang ◽  
Weipeng Hu
Keyword(s):  
Carbon Nanotube ◽  
Chaotic Behavior ◽  
Parameter Perturbation ◽  
External Excitation ◽  
Subharmonic Solution ◽  
Homoclinic Loop ◽  
Single Walled Carbon Nanotube ◽  
System Parameters ◽  
Single Walled Carbon ◽  
Excitation Parameters

Carbon nanotubes (CNTs) are used in various nano-electromechanical systems (NEMS), and the parameters (including the system parameters and the excitation parameters) may result in chaos in these systems. Thus, understanding the mechanism of the chaos arising from NEMS is vital for CNT’s applications. Motivated by this need, the chaotic properties of a single-walled carbon nanotube system resulting from parametric excitation and external excitation are investigated in this paper. The criteria for the existence of the chaotic behavior in the system with periodic and quasi-periodic perturbations are obtained by the homoclinic Melnikov and the second-order average methods. Furthermore, in order to show the connection between periodic motion and complex behavior, the subharmonic periodic solutions, inside and outside the homoclinic loop, are analyzed. The global structure and the saddle-node bifurcation of the unperturbed averaged system are also considered. Finally, the Poincaré section and the transversal intersection of the unstable and stable manifolds are presented to verify the occurrence of chaos or subharmonic solution. The simulation results confirm the correctness of the theoretical analysis.

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