Free localized vibrations of a semi-infinite cylindrical shell

Kaplunov, Julius; Kossovich, Leonid Yu.; Wilde, Maria

doi:10.1121/1.428426

Cited by 40 publications

(60 citation statements)

References 7 publications

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“….. It is worth noting that at α ∼ β 1 2 all the terms in (3.11) are of the same asymptotic order as it is usually assumed in the dynamic semi-membrane theory [24].…”

Section: Dispersion Relationmentioning

confidence: 87%

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Vibrations of an elastic cylindrical shell near the lowest cut-off frequency

2016

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A new asymptotic approximation of the dynamic equations in the 2D classical theory of thin elastic shells is established for a circular cylindrical shell. It governs long wave vibrations in the vicinity of the lowest cut-off frequency. At a fixed circumferential wave number the latter corresponds to the eigen frequency of in-plane vibrations of a thin almost inextensible ring. It is stressed that the well-known semi-membrane theory of cylindrical shells is not suitable for tackling a near-cut-off behaviour. The dispersion relation within the framework of the developed formulation coincides with the asymptotic expansion of the dispersion relation originating from full 2D shell equations. Asymptotic analysis also enables refining the geometric hypotheses underlying various adhoc setups, including the assumption on vanishing of shear and circumferential midsurface deformations used in the semi-membrane theory. The obtained results may be of interest for dynamic modelling of elongated cylindrical thin walled structures, such as carbon nanotubes.

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“….. It is worth noting that at α ∼ β 1 2 all the terms in (3.11) are of the same asymptotic order as it is usually assumed in the dynamic semi-membrane theory [24].…”

Section: Dispersion Relationmentioning

confidence: 87%

“…see [1,24]. On introducing (5.1) into the Hamiltonian (2.2) taking into account formula (2.3) and integrating over the angle ϕ, we get by varying it in W an equation identical to (4.15) to within asymptotically secondary terms in α and β.…”

Section: Geometric Hypothesesmentioning

confidence: 99%

Vibrations of an elastic cylindrical shell near the lowest cut-off frequency

2016

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“…For this problem Kaplunov et al (2000) found an infinite discrete spectrum of edge-localized solutions, which are naturally related to the surface wave traveling along the strip edge. In her later work Wilde (2004) has also been able to use this spectrum in order to provide an empirical formulae for the edge vibration frequencies in the case of strip with free faces.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue of a semi-infinite elastic strip

2006

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A semi-infinite elastic strip, subjected to traction free boundary conditions, is studied in the context of in-plane stationary vibrations. By using normal (RayleighLamb) mode expansion the problem of existence of the strip eigenmode is reformulated in terms of the linear dependence within infinite system of normal modes. The concept of Gram's determinant is used to introduce a generalized criterion of linear dependence, which is valid for infinite systems of modes and complex frequencies. Using this criterion, it is demonstrated numerically that in addition to the edge resonance for the Poisson ratio ν = 0, there exists another value of ν ≈ 0.22475 associated with an undamped resonance. This resonance is best explained physically by the orthogonality between the edge mode and the first Lamé mode. A semi-analytical proof for the existence of the edge resonance is then presented for both described cases with the help of the augmented scattering matrix formalism.

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“…The result generalizes the edge wave which arises in the twodimensional approximate theory of plate extension (e.g. Kaplunov et al 2000).…”

Section: Explicit Solution For the Fundamental Edge Wave In The Case mentioning

confidence: 76%

“…Counterparts of bending and extensional edge waves also appear in the two-dimensional shell theory (e.g. Kaplunov et al 2000).…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional edge waves in plates

Zernov

Kaplunov

2007

Proc. R. Soc. A.

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This paper describes the propagation of three-dimensional symmetric waves localized near the traction-free edge of a semi-infinite elastic plate with either traction-free or fixed faces. For both types of boundary conditions, we present a variational proof of the existence of the low-order edge waves. In addition, for a plate with traction-free faces and zero Poisson ratio, the fundamental edge wave is described by a simple explicit formula, and the first-order edge wave is proved to exist. Qualitative variational predictions are compared with numerical results, which are obtained using expansions in threedimensional Rayleigh-Lamb and shear modes. It is also demonstrated numerically that for any non-zero Poisson ratio in a plate with traction-free faces, the eigenfrequencies related to the first-order wave are complex valued.

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Free localized vibrations of a semi-infinite cylindrical shell

Cited by 40 publications

References 7 publications

Vibrations of an elastic cylindrical shell near the lowest cut-off frequency

Vibrations of an elastic cylindrical shell near the lowest cut-off frequency

Eigenvalue of a semi-infinite elastic strip

Three-dimensional edge waves in plates

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