Theory of free and forced vibrations of a rigid rod based on the Rayleigh model

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

doi:10.1134/s1028335807110080

Cited by 24 publications

(14 citation statements)

References 2 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%

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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%

“…This equation can be considered as one-mode model equation since it contains only one unknown function u and it is described in detail by Fedotov et al in [4]. Here η is Poisson's coefficient, and I is the axial moment of inertia.…”

Section: The Wave Equation Is the Simplest One-mode Model The Longitmentioning

confidence: 98%

Hyperbolic and pseudo-hyperbolic equations in the theory of vibration

2016

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“…It is possible to show [3] that an alternative representation of the solution can be obtained by using the second orthogonality condition (20).…”

Section: Solution Of the Bvp And The Green Functionmentioning

confidence: 99%

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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“…Longitudinal waves are broadly used for the purposes of non-destructive evaluation of materials and for generation and sensing of acoustic vibration [1] of surrounding. Many mathematical models describing longitudinal wave propagation in solids have been derived in order to analyse the effects of different materials and geometries on vibration characteristics without the need for costly experimental studies.…”

Section: Introductionmentioning

confidence: 99%

Young's modulus of BF wood material by longitudinal vibration

Phadke¹,

Shrivastava

Mishra

et al. 2014

J. Phys.: Conf. Ser.

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Abstract. All engineered structures are designed and built with consideration of resisting the same fundamental forces of tension, compression, shear, bending and torsion. Structural design is a balance of these internal and external forces. So, it is interesting to calculate the Young's moduli of Borassus Flabellifier BF wood are quite important from the application point of view. The ultrasonic waves are closely related with the elastic and inelastic properties of the materials. In the present study, we measured longitudinal wave ultrasonic velocities in BF wood material by longitudinal vibration method. After measuring ultrasonic velocity in BF wood material, we calculated Young's modulus of Borassus Flabellifier BF wood material. We used ultrasonic interferometer for measuring longitudinal wave ultrasonic velocity in BF wood material made by Mittal Enterprises, New Delhi, India in our laboratory. Borassus Flabellifier BF wood material was collected from Dhar district of Madhya Pradesh, India.

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Theory of free and forced vibrations of a rigid rod based on the Rayleigh model

Cited by 24 publications

References 2 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

Young's modulus of BF wood material by longitudinal vibration

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