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A.K. Belyaev, N.F. Morozov, P.E. Tovstik, and T.P. Tovstik, "Beating in the Problem of Longitudinal Impact on a Thin Rod," Mech. Solids. 50 (4), 451-462 (2015)
Year 2015 Volume 50 Number 4 Pages 451-462
DOI 10.3103/S0025654415040111
Title Beating in the Problem of Longitudinal Impact on a Thin Rod
Author(s) A.K. Belyaev (Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr. 61, St. Petersburg, 199178 Russia, vice.ipme@gmail.ru)
N.F. Morozov (Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr. 61, St. Petersburg, 199178 Russia, morozov@nm1016.spb.edu, morozov@math.spb)
P.E. Tovstik (Saint-Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, 199034 Russia, peter.tovstik@mail.ru)
T.P. Tovstik (Saint-Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, 199034 Russia)
Abstract The longitudinal impact on an elastic rod generating a periodic system of longitudinal waves in the rod, is considered. For certain values of the problem parameters in the linear approximation, these waves generate parametric resonances accompanied by an infinite increase in the transverse vibrations amplitude. To obtain the finite values of the amplitudes, a quasilinear system where the influence of transverse vibrations on the longitudinal ones is taken into account was considered. Earlier, this system was solved numerically by the Bubnov-Galerkin method and the beatings accompanied by energy exchange between the longitudinal and transverse vibrations were obtained. Here an approximate analytic solution of this system based on two-scale expansions is constructed. A qualitative analysis is performed. The maximum transverse deflection depending on the loading method is estimated.
Keywords rod, longitudinal loading, transverse vibrations, parametric resonance, beating, two-scale expansion
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Received 06 March 2015
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