Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

Zhang, Wei; Wang, Fengxia; Yao, Minghui

doi:10.1007/s11071-005-6435-3

Cited by 91 publications

(33 citation statements)

References 41 publications

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“…A nonlinear, nonplanar beam model considering the coupling of transverse vibrations is constructed to understand the nonlinear dynamics of the cantilevered SWCNT. The nonlinear differential equations in non-dimensional form describing the flexural motions of a nonlinear, nonplanar beam were derived by Crespo da Silva and Glynn [35][36][37] : where the symbol '·' is the derivative with respect to time t, the symbol ' ′ ' is the derivative with respect to x, v (s,t), w (s,t) are the transverse elastic displacements of the beam, c is a damping factor and it is set as zero in this calculation, β y = D ζ /D η , and For a beam with a circular cross section, β y = 1, Eq. (1) can be reduced tö…”

Section: Resultsmentioning

confidence: 99%

Coupling between flexural modes in free vibration of single-walled carbon nanotubes

Liu

Wang

2015

AIP Advances

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The nonlinear thermal vibration behavior of a single-walled carbon nanotube (SWCNT) is investigated by molecular dynamics simulation and a nonlinear, nonplanar beam model. Whirling motion with energy transfer between flexural motions is found in the free vibration of the SWCNT excited by the thermal motion of atoms where the geometric nonlinearity is significant. A nonlinear, nonplanar beam model considering the coupling in two vertical vibrational directions is presented to explain the whirling motion of the SWCNT. Energy in different vibrational modes is not equal even over a time scale of tens of nanoseconds, which is much larger than the period of fundamental natural vibration of the SWCNT at equilibrium state. The energy of different modes becomes equal when the time scale increases to the microsecond range.

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Section: Resultsmentioning

confidence: 99%

Coupling between flexural modes in free vibration of single-walled carbon nanotubes

Liu

Wang

2015

AIP Advances

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“…5. Here also to verify the degree of correctness of the perturbation results, one may compare the response obtained by method of multiple scales with those obtained by numerically solving the Cartesian form of modulations (19,20). Figures 6a, b illustrated the time response and phase portrait, respectively, obtained by numerically solving the Cartesian form of the modulation equations (19) and (20).…”

Section: Subharmonic Resonance Condition: (ω Is Nearly Equal To 3)mentioning

confidence: 92%

“…For both the trivial and nontrivial responses, the system will be stable if real part of all the eigenvalues of the Jacobian matrix (J ) obtained from the reduced equations (19,20) is negative.…”

Section: Stability Analysismentioning

confidence: 99%

“…Yabuno and Nayfeh [18] theoretically obtained the nonlinear normal modes of a parametrically excited cantilever beam by using method of multiple scales. Zhang et al [19] investigated the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. Zhou [20] obtained the exact and analytical expression for eigenfrequencies and mode shapes of a cantilever beam with heavy tip mass with rotary and translation elastic support.…”

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confidence: 99%

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Vibration control of a transversely excited cantilever beam with tip mass

Pratiher

2011

Arch Appl Mech

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The problem of controlling the vibration of a transversely excited cantilever beam with tip mass is analyzed within the framework of the Euler-Bernoulli beam theory. A sinusoidally varying transverse excitation is applied at the left end of the cantilever beam, while a payload is attached to the free end of the beam. An active control of the transverse vibration based on cubic velocity is studied. Here, cubic velocity feedback law is proposed as a devise to suppress the vibration of the system subjected to primary and subharmonic resonance conditions. Method of multiple scales as one of the perturbation technique is used to reduce the second-order temporal equation into a set of two first-order differential equations that govern the time variation of the amplitude and phase of the response. Then the stability and bifurcation of the system is investigated. Frequency-response curves are obtained numerically for primary and subharmonic resonance conditions for different values of controller gain. The numerical results portrayed that a significant amount of vibration reduction can be obtained actively by using a suitable value of controller gain. The response obtained using method of multiple scales is compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. Numerical simulation for amplitude is also obtained by integrating the equation of motion in the frequency range between 1 and 3. The developed results can be extensively used to suppress the vibration of a transversely excited cantilever beam with tip mass or similar systems actively.

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“…A third method, so-called energy-phase method proposed by Haller and Wiggins [18,19], detects the existence of orbits doubly asymptotic to slow manifolds and reveals families of multi-pulse solutions which are not amenable to Melnikov method. The latter two of these methods involve an exchange of energy between the modes of a system so that there are many applications to study the global dynamics of the systems [20][21][22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Multi-pulse jumping orbits and homoclinic trees in motion of a simply supported rectangular metallic plate

Chen

2010

Arch Appl Mech

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The global bifurcations in mode interaction of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation are investigated with the case of the 1:1 internal resonance, the modulation equations representing the evolution of the amplitudes and phases of the interacting normal modes exhibit complex dynamics. The energy-phase method proposed by Haller and Wiggins is employed to analyze the global bifurcations for the rectangular metallic plate. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case in both Hamiltonian and dissipative perturbations, which imply that chaotic motions may occur for this class of systems. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. To illustrate the theoretical predictions, we present visualizations of these complicated structures and numerical evidence of chaotic motions.

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Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

Cited by 91 publications

References 41 publications

Coupling between flexural modes in free vibration of single-walled carbon nanotubes

Coupling between flexural modes in free vibration of single-walled carbon nanotubes

Vibration control of a transversely excited cantilever beam with tip mass

Multi-pulse jumping orbits and homoclinic trees in motion of a simply supported rectangular metallic plate

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