Internal resonance of vibrational modes in single-walled carbon nanotubes

Shi, M.X.; Li, Qingming; Huang, YG

doi:10.1098/rspa.2009.0147

Cited by 13 publications

(13 citation statements)

References 23 publications

(44 reference statements)

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“…In [9], parametric-resonance-induced mode transformation in single-walled nano-tubes, modeled as continuous elastic shells, is analyzed through a Mathieu equation and multiple instability tongues are obtained. In [10] that analysis is improved by the use of a non-local elasticity model. In [11] parametric-resonanceinduced mode-switching is analyzed on the basis of a continuum shallow shell model.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of hidden modes in a system with internal symmetry

Perchikov

2016

Journal of Sound and Vibration

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We consider a discrete dynamical system with internal degrees of freedom (DOF). Due to the symmetry between the internal DOFs, certain internal modes cannot be excited by external forcing (in a case of linear interactions) and thus are considered "hidden". If such a system is weakly asymmetric, the internal modes remain approximately "hidden" from the external excitation, given that small damping is taken into account. However, already in the case of weak cubic nonlinearity, these hidden modes can be excited, even as the exact symmetry is preserved. This excitation occurs through parametric resonance. Floquet analysis reveals instability patterns for the explored modes. To perform this analysis with the required accuracy, we suggest a special method for obtaining the Fourier series of the unperturbed solution for the nonlinear normal mode. This method does not require explicit integration of the arising quadratures. Instead, it employs expansion of the solution at the stage of the implicit quadrature in terms of Chebyshev polynomials. The emerging implicit equations are solved by using a fixed-point iteration scheme. Poincar\'{e} sections help to clarify the correspondence between the loss of stability of the modes and the global structure of the dynamical flow. In particular, the conditions for intensive energy exchange in the system are characterized

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Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of hidden modes in a system with internal symmetry

Perchikov

2016

Journal of Sound and Vibration

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“…This intermittent transformation between RBM and CFM may be related to the 2:1 internal resonance, which has been observed in other non-linear systems (Nayfeh and Arafat, 2006). The existence of this automated internal nonlinear resonance in SWCNTs has been further demonstrated using molecular dynamics (MD) and local and non-local continuum models (Li and Shi, 2008;Shi et al, 2009). …”

Section: Introductionmentioning

confidence: 89%

Atomic-scale finite element analysis of vibration mode transformation in carbon nanorings and single-walled carbon nanotubes

Shi¹,

Li²,

Liu³

et al. 2009

International Journal of Solids and Structures

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a b s t r a c tAtomic-scale finite element analyses show that 2:1 internal resonance mechanism exists in a range of single-walled carbon nanorings (10-60). When an initial radial breathing mode (RBM) vibration with sufficiently high velocity is imposed to a nanoring, circumferential flexural modes (CFMs) can be excited after a period of RBM-dominated vibration. Then, mode transformations between RBM and the excited CFMs can be observed in the subsequent vibration process. When single-walled carbon nanorings are assembled to make double-or triple-walled carbon nanorings, the 2:1 internal resonance may change to 1:1 internal resonance in a specific ring due to the strong interactions between these nanorings. Furthermore, mode transformations between RBM and the excited CFMs can become unstable in a specific ring if the excited CFMs in neighbouring layer rings are not symmetrically matching between each other. 2:1 internal resonance is also shown in selected armchair single-walled carbon nanotubes except in a special case (armchair (9, 9)), in which 1:1 internal resonance occurs.

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“…Recently, it has been revealed that vibration mode transformation is a common phenomenon that may happen in engineering structures at very different length scales governed by 2:1 or similar internal resonance mechanism. For example, strain growth in explosion containment vessels is related to the vibration mode transformation (Zhu et al, 1997, Duffey and Romero, 2003 and the intermittent mode transformation in single-walled carbon nanotubes was predicted in (Li and Shi, 2008, Shi et al, 2009a,2009b, Shi et al, 2009c. Since many natural, bio-and engineering structures have similar geometries to cylindrical and spherical shells, the vibration mode transformation may be realised in all these structures.…”

Section: Introductionmentioning

confidence: 99%

Instability in the transformation between extensional and flexural modes in thin-walled cylindrical shells

Shi

2011

European Journal of Mechanics - A/Solids

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Abstract:With the inclusion of all terms up to the third-order in the membrane strain to consider the geometric nonlinearity in deformation, a third-order implicit model and a fourth-order explicit model for the vibration transformation between extensional and flexural modes in thin-walled cylindrical shells are established and solved numerically. Numerical instability is observed in numerical solutions based on the explicit model. It is found that such numerical instability does not result from the accumulated numerical errors in the numerical integration process, but from the neglecting of higher order terms in the formulation of the problem. With the inclusion of all terms up to the fourth-order in the strain energy, the explicit model can predict a stable vibration history with periodic mode transformations between the two modes. The same 2:1 internal resonance of the vibration mode transformation is predicted by both models. As flexural stress growth is concerned, the two models matches very well during the first group of peaks but lag in phase gradually appears in later group of stress peaks based on the implicit model. This is understood to be resulting from the limited number of terms included in the series expansions based on the explicit model, which allows the most likely excited flexural mode get a larger share of energy transferred from the principle mode and as a result flexural stress arrives at peaks earlier based on the explicit model.

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Internal resonance of vibrational modes in single-walled carbon nanotubes

Cited by 13 publications

References 23 publications

Nonlinear dynamics of hidden modes in a system with internal symmetry

Nonlinear dynamics of hidden modes in a system with internal symmetry

Atomic-scale finite element analysis of vibration mode transformation in carbon nanorings and single-walled carbon nanotubes

Instability in the transformation between extensional and flexural modes in thin-walled cylindrical shells

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