On The Rayleigh-Love Rod Accreting In Both Length And
Cross-Sectional Area: Forced And Damped Vibrations
On The Rayleigh-Love Rod Accreting In Both Length And Cross-Sectional Area: Forced And Damped Vibrations
M.L.G. Lekalakala1, M. Shatalov2, I. Fedotov3, S.V. Joubert4
of Mathematics, Vaal University of Technology, P.O. Box 1889, Secunda, 2302,
2,3,4Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria, South Africa.
In this paper an elastic cylindrical rod that is subjected to forced and damped vibrations is considered. The rod is assumed to be isotropic. The applied external force of excitation is assumed to be harmonic, and the damping force is that of Kelvin-Voigt. The longitudinally vibrating rod is fixed at the left end and free at the other end. The rod is assumed to be accreting in length and cross-sectional area as it vibrates. The problem arising and the dynamics of the vibrating rod are described and investigated within the Rayleigh-Love theories of the rod. A partial differential equation describing the longitudinal displacement of the rod is formulated. The formulated partial differential equation, together with the corresponding boundary conditions as per the configuration of the rod, is solved numerically using the Galerkin-Kantorovich method. The frequency of vibration of the harmonic exciting force is kept constant in this investigation.
It is shown that in this periodically forced viscoelastic damped vibration, all the modes of vibration are subjected to the resonance behaviour within a proper time interval, depending on the length of the accreting rod.
Hyperbolic Models Arising in the Theory of Longitudinal Vibration of Elastic Bars
1I. Fedotov, 1J. Marais, 1,2M. Shatalov and 1H.M. Tenkam
1Department of Mathematics and Statistics,
Tshwane University of Technology
Private Bag X6680, Pretoria 0001
firstname.lastname@example.org, email@example.com, firstname.lastname@example.org.
Council of Scientific and Industrial Research (CSIR)
P.O. Box 395, Pretoria, 0001
In this paper a unified approach to the
derivation of families of one
dimensional hyperbolic differential equations and boundary conditions describing
the longitudinal vibration of elastic bars is outlined. The longitudinal and
lateral displacements are expressed in the form of a power series expansion in
the lateral coordinate. Equations of motion and boundary conditions are derived
using Hamilton's variational principle. Most of the well known models in this
field fall within the frames of the proposed theory, including the classical
model, and the more elaborated models proposed by by Rayleigh, Love, Bishop,
Mindlin, Herrmann and McNiven. The exact solution is presented for the
Mindlin-Herrmann case in terms of Green functions. Finally, deductions regarding
the accuracy of the models are made by comparison with the exact
Pochhammer-Chree solution for an isotropic cylinder.
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