Longitudinal Vibration of Isotropic Solid Rods: From Classical to Modern Theories

Shatalov, M. Y.; Marais, Julian; Fedotov, Igor; Tenkam, Michel Djouosseu

doi:10.5772/15662

Cited by 19 publications

(19 citation statements)

References 8 publications

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“…1 Referring to Fig. 1 and noting that if u is the axial displacement at a distance x from the origin, the kinetic and potential energies of the bar Tbar and Vbar are respectively given by [7,21]…”

Section: Dynamic Stiffness Matrix Of a Rayleigh-love Barmentioning

confidence: 99%

“…In these earlier works, when the axial stiffnesses were incorporated into the bending stiffnesses to construct the dynamic stiffness matrix of an individual element, only classical theory for longitudinal free vibration of bars which ignores the transverse inertia effect was used. This is generally justified, particularly in the low and probably in the medium frequency range, but for high frequency vibration, the so-called Rayleigh-Love theory [20,21] which accounts for the effects of transverse inertia during longitudinal vibration and the Timoshenko theory [17] which accounts for the effects of shear deformation and rotatory inertia during bending vibration need to be considered. This is particularly important when applying the widely accepted SEA technique for which the high frequency vibration problem must be modelled properly [1][2].…”

Section: Introductionmentioning

confidence: 99%

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An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

Banerjee

Ananthapuvirajah

2019

International Journal of Mechanical Sciences

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An exact dynamic stiffness matrix for a beam is developed by integrating the Rayleigh-Love theory for longitudinal vibration into the Timoshenko theory for bending vibration. In the formulation, the Rayleigh-Love theory accounted for the transverse inertia in longitudinal vibration whereas the Timoshenko beam theory accounted for the effects of shear deformation and rotating inertia in bending vibration. The dynamic stiffness matrix is developed by solving the governing differential equations of motion in free vibration of a Rayleigh-Love bar and a Timoshenko beam and then imposing the boundary conditions for displacements and forces.Next the two dynamic stiffness theories are combined using a unified notation. The ensuing dynamic stiffness matrix is subsequently used for free vibration analysis of uniform and stepped bars as well as frameworks through the application of the Wittrick-Williams algorithm as solution technique. Illustrative examples are given to demonstrate the usefulness of the theory and some of the computed results are compared with published ones. The paper closes with some concluding remarks.

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Section: Dynamic Stiffness Matrix Of a Rayleigh-love Barmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

Banerjee

Ananthapuvirajah

2019

International Journal of Mechanical Sciences

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“…2. It is found that the present nonlocal nanorod model can be degenerated to the classical rod model (Shatalov et al 2011) without wave dispersion, and the nonlocal rod model proposed by Narendar and Gopalakrishnan (2010) where the nonlocal scale effect is introduced. Furthermore, the present model is also compared with Li's model including lateral inertia and shear effects (Li et al 2017), which suggests the nonlocal Rayleigh-Bishop model can be obtained from the present model when the elastic medium is neglected.…”

Section: Nonlocal Nanorod Model Verificationmentioning

confidence: 99%

Effect of uncertainty in material properties on wave propagation characteristics of nanorod embedded in elastic medium

Liu

2017

Int J Mech Mater Des

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The effect of uncertainty in material properties on wave propagation characteristics of nanorod embedded in an elastic medium is investigated by developing a nonlocal nanorod model with uncertainties. Considering limited experimental data, uncertain-but-bounded variables are employed to quantify the uncertain material properties in this paper. According to the nonlocal elasticity theory, the governing equations are derived by applying the Hamilton's principle. An iterative algorithm based interval analysis method is presented to evaluate the lower and upper bounds of the wave dispersion curves. Simultaneously, the presented method is verified by comparing with Monte-Carlo simulation. Furthermore, combined effects of material uncertainties and various parameters such as nonlocal scale, elastic medium and lateral inertia on wave dispersion characteristics of nanorod are studied in detail. Numerical results not only make further understanding of wave propagation characteristics of nanostructures with uncertain material properties, but also provide significant guidance for the reliability and robust design of the next generation of nanodevices.

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“…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.…”

mentioning

confidence: 97%

“…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. Although the applicable range of vibration (or excitation) frequencies is still limited, analysis of higher frequency vibrations has also been justified [9].…”

mentioning

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 Vibration of Isotropic Solid Rods: From Classical to Modern Theories

Cited by 19 publications

References 8 publications

An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

Effect of uncertainty in material properties on wave propagation characteristics of nanorod embedded in elastic medium

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

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