Longitudinal vibrations of a Rayleigh-Bishop rod

Fedotov, Igor; Polyanin, Andrei D.; Shatalov, M. Y.; Tenkam, H.M.

doi:10.1134/s1028335810120062

Cited by 14 publications

(19 citation statements)

References 3 publications

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“…Similar results for pseudo-hyperbolic operators have been exposed to only some extent by Fedotov and Volevich [3]. In spite of the absence of a general theory of solvability of pseudo-hyperbolic problems, in certain cases it is possible to determine an analytical solution of a mixed problem such as the Rayleigh-Bishop equation (even for variable coefficients) [4][5][6]. The idea of Theorem 4 (and partly of Theorem 5) is that hyperbolicity (or pseudo-hyperbolicity) can be reformulated in terms of Cauchy's problem.…”

Section: Resultsmentioning

confidence: 85%

Hyperbolic and pseudo-hyperbolic equations in the theory of vibration

2016

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Longitudinal vibration of bars is usually considered in mathematical physics in terms of a classical model described by the wave equation under the assumption that the bar is thin and relatively long. More general theories have been formulated taking into consideration the effect of the lateral motion of a relatively thick bar (beam). The mathematical formulation of these models includes higher-order derivatives in the equation of motion. Rayleigh derived the simplest generalization of the classical model in 1894, by including the effects of lateral motion and neglecting the shear stress. Bishop obtained the next generalization of the theory in 1952. The Rayleigh-Bishop model is described by a fourth-order partial differential equation not containing the fourth-order time derivative. He took into account the effects of shear stress. Both Rayleigh's and Bishop's theories consider lateral displacement being proportional to the longitudinal strain. The Bishop model was generalized by Mindlin and Herrmann. They considered the lateral displacement proportional to an independent function of time and longitudinal coordinate. This result is formulated as a system of two differential equations of second order, which could be replaced by a single equation of fourth order resolved with respect to the highest order time derivative. To obtain a more general class of equations, the longitudinal and lateral displacements are expressed in the form of a power series expansion in the lateral coordinate. In this paper, all of the above-mentioned equations are considered in the framework of a general theory of hyperbolic equations, with the aim of classifying the equations into general groups. The solvability of the corresponding problems is also discussed.

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Section: Resultsmentioning

confidence: 85%

Hyperbolic and pseudo-hyperbolic equations in the theory of vibration

2016

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“…All numerical results and graphs were generated using Mathematica © . Fedotov et al [5] showed that the eigenfunctions corresponding to the Sturm-Liouville problem (11)-(13) for a Rayleigh-Bishop rod with conical cross sections can be given in terms of generalised hypergeometric functions.…”

Section: Numerical Example: Eigenvalues and Eigenfunctionsmentioning

confidence: 99%

“…Inclusion of these effects when deriving the equation of motion results in the so called Rayleigh-Bishop equation [7], a linear fourth order partial differential equation not resolved with respect to the highest order time derivative. It has been shown [5,9] that the Rayleigh-Bishop model improves on estimations made by the classical wave equation. The Rayleigh-Bishop model makes it possible to analyse longitudinal wave propagation in solid rods that are relatively thick (the maximum radius is comparable with the length of the rod) due to the inclusion of transverse effects in the model.…”

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confidence: 97%

Longitudinal vibrations of a cylindrical rod based on the Rayleigh-Bishop theory

2014

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A one dimensional model describing the longitudinal vibrations of a cylindrical rod composed of multiple sections with different diameters and made of different materials is derived. The hypothesis that lateral displacements and axial shear stresses in the bar are zero is rejected. The lateral displacements are assumed to be proportional to the axial strain and the Poisson ratio, resulting in the so-called Rayleigh-Bishop equation as opposed to the classical one dimensional wave equation. The Rayleigh-Bishop equation is a fourth order partial differential equation, not resolved with respect to the highest order time derivative. A system of partial differential equations, as well as the corresponding boundary conditions and continuity conditions at the junctions between the sections, are derived using Hamilton's variational principle. An analytical solution is obtained in terms of a Green function. A numerical example is provided for a rod consisting of two cylindrical sections, in which the first five eigenvalues and corresponding eigenfunctions are presented.

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“…By using the above procedure and Equations (20), (29), (30) and (31), the Fourier coefficient A j can be found as follows:…”

Section: Stokes' Transformationmentioning

confidence: 99%

“…Longitudinal vibrations of conical and cylindrical rods have been presented by Fedeto et al. [29] and Marais et al. [30] based on the Rayleigh‐Bishop theory.…”

Section: Introductionmentioning

confidence: 99%

Axial dynamic analysis of a Bishop nanorod with arbitrary boundary conditions

2020

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In the present work, axial vibrational behavior of nanorods with different boundary conditions is researched. Bishop's rod theory is implemented to simulate the axial deflection. Size‐dependency is captured by using Eringen's nonlocal elasticity theory. Based on nonlocal deformable boundary conditions and Stokes’ transformation, a system of linear equations is derived and then constructed an Eigen value problem. Several numerical examples are presented to investigate the significance of various parameters such as geometric parameters, vibrational modes, various values of nonlocal parameter and axial spring parameters on the axial frequencies of nanorods. The numerical examples indicate that the deformable boundary conditions and small scale parameter have considerable effects on the axial vibration response.

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Longitudinal vibrations of a Rayleigh-Bishop rod

Cited by 14 publications

References 3 publications

Hyperbolic and pseudo-hyperbolic equations in the theory of vibration

Hyperbolic and pseudo-hyperbolic equations in the theory of vibration

Longitudinal vibrations of a cylindrical rod based on the Rayleigh-Bishop theory

Axial dynamic analysis of a Bishop nanorod with arbitrary boundary conditions

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