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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Euler, Method for Determining the Curves with the Maximum or Minimum Property (GTTI, Moscow–Leningrad, 1934) [in Russian].
W. J. Hytchinson and B. Budiansky, “Dynamic Buckling Estimates,” AIAA J. 4 (3), 527–530 (1966).
W. G. Knauss and K. Ravi-Chandar, “Some Basic Problems in Stress Wave Dominated Fracture,” Int. J. Fract. 27, 127–143 (1985).
N. F. Morozov and Yu. V. Petrov, Dynamics of Fracture (Izdat. SPbGU, St. Petersburg, 1997; Springer, Berlin-Heidelberg-New York, 2000).
V. A. Bratov, N. F. Morozov, and Yu. V. Petrov, Dynamic Strength of Continuum (St. Petersburg Univ. Press, St. Petersbirg, 2009).
V. V. Bolotin, Dynamic Stability of Elastic Systems (Nauka, Moscow, 1956) [in Russian].
A. S. Vol’mir, “Stability of Compressed Rods under Dynamic Loading,” Stroit. Mekh. Rashch. Sooruzh., No. 1, 6–9 (1960).
M. A. Lavrentiev and A. Yu. Ishlinskii, “Dynamic Buckling Modes of Elastic Systems,” Dokl. Akad. Nauk SSSR 64 (6), 776–782 (1949).
Ya. G. Panovko and I. I. Gubanova, Stability and Vibrations of Elastic Systems (Consultant Bureau, New York 1965; Nauka, Moscow, 1967).
A. S. Vol’mir, Stability of Elastic Systems (GITTL, Moscow, 1962) [in Russian].
V. V. Bolotin, Transverse Vibrations and Critical Velocities, Vols. 1 and 2 (Izdat. ANSSSR, Moscow, 1951, 1953) [in Russian].
N. F. Morozov and P. E. Tovstik, “Dynamics of a Rod on Longitudinal Impact,” Vestnik St. Peterzburg. Univ. Ser. I. Mat. Mekh. Astr. No. 2, 105–111 (2009).
A. K. Belyaev, D. N. Il’in, and N. F. Morozov, “Dynamic Approach to the Ishlinsky-Lavrent’ev Problem,” Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 28–33 (2013) [Mech. Solids (Engl. Transl.) 48 (5), 504–508 (2013)].
N. F. Morozov and P. E. Tovstik, “Dynamics of a Rod on Short-Time Longitudinal Impact,” Vestnik St. Peterzburg. Univ. Ser. I. Mat. Mekh. Astr. No. 3, 131–141 (2013).
N. F. Morozov and P. E. Tovstik, “Transverse Rod Vibrations under a Short-Term Longitudinal Impact,” Dokl. Ross. Akad. Nauk 452 (1), 37–41 (2013) [Dokl. Phys. (Engl. Transl.) 58 (9), 387–391 (2013)].
N. F. Morozov and P. E. Tovstik, “The Rod Dynamics under Longitudinal Impact,” in Book of Abstracts of International Conference on Nonlinear Dynamics in Engineering: Modeling, Analysis, and Applications, August 21–23, 2013, Aberdeen, UK, Ed. by J. Ing, Y. Liu, E. Pavlovskaya, A. Postnikov, and M. Wiercigroch, (Aberdeen, 2013) p. 73.
A. K. Belyaev, N. F. Morozov, and P. E. Tovstik, “On Static and Dynamic Instability of Thin Rods,” in Proc. 7 All-Russia Conf. “Mechanics of Deformable Solid” (Izdat. YuFY, Rostov-on-Don, 2013), pp. 80–84 [in Russian].
N. F. Morozov, P. E. Tovstik, and P. E. Tovstik, “Statics and Dynamics of a Rod under Axial Compression,” in ICNAAM 2014, AIP Conference Proc. (2014).
N. F. Morozov, P. E. Tovstik, and P. E. Tovstik, “Statics and Dynamics of a Rod under Longitudinal Loading,” Vestnik Yuzhno-UralUniv. Ser. Mat. Model. Progr. 7 (1), 76–89 (2014).
N. F. Morozov, P. E. Tovstik, and P. E. Tovstik, “Again on the Ishlinskii-Lavrentyev Problem,” Dokl. Ross. Akad. Nauk 455 (4), 412–415 (2014) [Dokl. Phys. (Engl. Transl.) 59 (4), 189–192 (2014)].
N. F. Morozov and P. E. Tovstik, “Dynamic Loss of Stability of a Rod under Longitudinal Load Lower Than the Eulerian Load,” Dokl. Ross. Akad. Nauk 453 (3), 282–285 (2013) [Dokl. Phys. (Engl. Transl.) 58 (11), 510–513 (2013)].
N. F. Morozov, P. E. Tovstik, and P. E. Tovstik, “Stability of a Rod under the Long-Term Axial Compression,” Probl. Prochn. Plastichn. 77 (1), 40–48 (2014).
A. K. Belyaev, N. F. Morozov, P. E. Tovstik, and P. E. Tovstik, “Statics and Dynamics of a Rod in Longitudinal Compression,” in 7th Polyakhov Readings, Theses (St. Petersburg, 2015), p. 9 [in Russian].
A. A. Ilyukhin, Spatial Problems of Nonlinear Theory of Elastic Rods (Naukova Dumka, Kiev, 1979) [in Russian].
V. A. Palmov, Vibrations of Elastoplastic Bodies (Nauka, Moscow, 1976) [in Russian].
A. M. Lyapunov, General Problem of Stability of Motion (GosTekhIzdat., Moscow, 1950) [in Russian].
V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients and Their Applications (Nauka, Moscow, 1972) [in Russian].
N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in Theory of Nonlinear Vibrations (Nauka, Moscow, 1969) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.K. Belyaev, N.F. Morozov, P.E. Tovstik, T.P. Tovstik, 2015, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2015, No. 4, pp. 104–117.
About this article
Cite this article
Belyaev, A.K., Morozov, N.F., Tovstik, P.E. et al. Beating in the problem of longitudinal impact on a thin rod. Mech. Solids 50, 451–462 (2015). https://doi.org/10.3103/S0025654415040111
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654415040111