Longitudinal vibrations in circular rods: A systematic approach |
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Authors: | NG Stephen KF Lai K Young KT Chan |
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Institution: | 1. Faculty of Engineering and the Environment, Mechanical Engineering, University of Southampton, Highfield, Southampton SO17 1BJ, UK;2. Department of Physics, The Chinese University of Hong Kong, Hong Kong, China;3. Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, China |
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Abstract: | A systematic method is developed for expressing the frequency squared ω2 and the corresponding displacement fields of harmonic waves in a long thin rod as an even power series in qa, where q is the wavenumber along the rod and a is a representative transverse dimension. For longitudinal waves in a circular rod, the evaluation is reduced to algebraic recursion, giving coefficients analytically in terms of Poisson's ratio v, to many orders. The second nontrivial coefficient, corresponding to Rayleigh–Love theory in the present longitudinal case and Timoshenko theory in the flexural case, is thus put on a firm footing without reliance on ad hoc physical assumptions. The results are compared to available exact predictions, and shown to be accurate for moderate values of qa (5% accuracy for qa≤1.5) with just two terms. Improvements based on the Rayleigh quotient guarantee positivity and the correct asymptotic power, and the variational principle further ensures that the accuracy improves monotonically with the order of approximation. With these features, accurate results are obtained for larger qa (5% accuracy for qa≤3), so that results are valid for rods that are by no means thin. Application of these methods to the flexural case has been presented separately. |
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