Explicit solution for a vibrating bar with viscous boundaries and internal damper |
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Authors: | Vojin Jovanovic Sergiy Koshkin |
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Affiliation: | (1) Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt;(2) Faculty of Industrial Education, Helwan University, Cairo, Egypt |
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Abstract: | We investigate longitudinal vibrations of a bar subjected to viscous boundary conditions at each end and an internal damper
at an arbitrary point along the bar’s length. The system is described by four independent parameters and exhibits a variety
of behaviors including rigid motion, super stability/instability and zero damping. The solution is obtained by applying the
Laplace transform to the equation of motion and computing the Green’s function of the transformed problem. This leads to an
unconventional eigenvalue-like problem with the spectral variable in the boundary conditions. The eigenmodes of the problem
are necessarily complex-valued and are not orthogonal in the usual inner product. Nonetheless, in generic cases we obtain
an explicit eigenmode expansion for the response of the bar to initial conditions and external force. For some special values
of parameters the system of eigenmodes may become incomplete, or no non-trivial eigenmodes may exist at all. We thoroughly
analyze physical and mathematical reasons for this behavior and explicitly identify the corresponding parameter values. In
particular, when no eigenmodes exist, we obtain closed form solutions. Theoretical analysis is complemented by numerical simulations,
and analytic solutions are compared to computations using finite elements. |
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Keywords: | |
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