Intermittent transformation between radial breathing and flexural vibration modes in a single-walled carbon nanotube

Li, Qingming; Shi, M.X.

doi:10.1098/rspa.2007.0253

Cited by 13 publications

(16 citation statements)

References 25 publications

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“…The nonlinear differential equations in Li & Shi (2008) are further extended to include non-local effects described by equation (3.1), as shown in appendix A, which can be reduced to Mathieu equations to describe the coupling between the RBM and inextensional flexural modes (Goodier & McIvor 1964;Li & Shi 2008). The two non-dimensional parameters in the Mathieu stability diagram with the consideration of non-local effects are …”

Section: Non-local Elastic Shell Modelmentioning

confidence: 99%

“…On the other hand, figure 8 also demonstrates the necessity to include the nonlocal effects in the continuum model for the correct prediction of 2 : 1 internal resonance. In Li & Shi (2008), where a local continuum model is employed using the old thickness parameter 0.066 nm (Yakobson et al 1996), mode transformation between the RBM and CBM-4 was predicted. If the excitation power intensity is further reduced, CBM-5 will be involved under 1 : 1 internal resonance mechanism, as shown in figure 10.…”

Section: Non-local Elastic Shell Modelmentioning

confidence: 99%

“…Lower frequency modes than RBMs in Raman spectra have received much less attention in spectrum analysis. It has been shown that the intermittent transformation between the RBM and a lower frequency circumferential flexural mode (CFM) may happen when the amplitude of initial RBM vibration is greater than a critical value for the given geometry of single-walled carbon nanotubes (SWCNTs; Li & Shi 2008). This intermittent transformation between the RBM and CFM may be related to the 2 : 1 internal resonance, which has been observed in many nonlinear systems (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…The other term c n refers to inextentional flexural deformation, which corresponds to the deformation caused by CFM vibration. Using the same governing equations in Li & Shi (2008), the unknown coefficient functions a 0 and c n in equations (A 1) and (A 2) can be obtained by means of the Lagrange equation as…”

mentioning

confidence: 99%

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Internal resonance of vibrational modes in single-walled carbon nanotubes

Shi

Huang

2009

Proc. R. Soc. A.

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We show, by molecular dynamics simulations, that 2 : 1 internal resonance may occur between a radial breathing mode (RBM) and a circumferential flexural mode (CFM) in single-walled carbon nanotubes (SWCNTs). When the RBM vibration amplitude is greater than a critical value, automatic transformations between the RBM and CFM with approximately half RBM-frequency are observed. This discovery in the discrete SWCNT atom assembly is similar to the 2 : 1 internal resonance mechanism observed in continuum shells. A non-local continuum shell model is employed to determine the critical conditions for the occurrence of observed 2 : 1 internal resonance between the RBM and CFMs based on two non-dimensional parameters and the Mathieu stability diagram.

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Section: Non-local Elastic Shell Modelmentioning

confidence: 99%

Section: Non-local Elastic Shell Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Internal resonance of vibrational modes in single-walled carbon nanotubes

Shi

Huang

2009

Proc. R. Soc. A.

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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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Nonstationary Resonant Dynamics of Oscillatory Chains and Nanostructures

Manevitch¹,

Kovaleva

Смірнов³

et al. 2018

Foundations of Engineering Mechanics

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Intermittent transformation between radial breathing and flexural vibration modes in a single-walled carbon nanotube

Cited by 13 publications

References 25 publications

Internal resonance of vibrational modes in single-walled carbon nanotubes

Internal resonance of vibrational modes in single-walled carbon nanotubes

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

Nonstationary Resonant Dynamics of Oscillatory Chains and Nanostructures

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