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Natural frequencies of longitudinal oscillations for a nonuniform variable cross-section rod

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Moscow University Mechanics Bulletin Aims and scope

Abstract

The natural frequencies of longitudinal oscillations of a rod such that its Young’s modulus, the density, and the cross-sectional area are functions of the longitudinal coordinate are analyzed. For solving the corresponding problem, an integral formula is used to represent the general solution to the original Helmholtz equation with variable coefficients in terms of the general solution to the accompanying equation with constant coefficients. Frequency equations are derived in the form of rapidly converging Leibniz series for three types of boundary conditions. For these cases the frequency zerothapproximation equations are given to quickly find the lowest natural frequencies with an adequate accuracy.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119899, Russia

    V. I. Gorbachev

Authors
  1. V. I. Gorbachev
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Correspondence to V. I. Gorbachev.

Additional information

Original Russian Text © V.I. Gorbachev, 2016, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2016, Vol. 71, No. 1, pp. 31–39.

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Gorbachev, V.I. Natural frequencies of longitudinal oscillations for a nonuniform variable cross-section rod. Moscow Univ. Mech. Bull. 71, 7–15 (2016). https://doi.org/10.3103/S0027133016010027

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  • DOI: https://doi.org/10.3103/S0027133016010027

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