Parametric resonances in the problem of longitudinal impact on a thin rod

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Parametric resonances in the problem of longitudinal impact on a thin rod. / Belyaev, A. K.; Morozov, N. F.; Tovstik, P. E.; Tovstik, Tatiana Petrovna.

In: Vestnik St. Petersburg University: Mathematics, Vol. 49, No. 1, 01.01.2016, p. 53-67.

Research output: Contribution to journal › Article › peer-review

Author

Belyaev, A. K. ; Morozov, N. F. ; Tovstik, P. E. ; Tovstik, Tatiana Petrovna. / Parametric resonances in the problem of longitudinal impact on a thin rod. In: Vestnik St. Petersburg University: Mathematics. 2016 ; Vol. 49, No. 1. pp. 53-67.

BibTeX

@article{26fff09ae09f4dfc93f0358e16b27f18,

title = "Parametric resonances in the problem of longitudinal impact on a thin rod",

abstract = "The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.",

keywords = "longitudinal impact, parametric resonances, rod, transverse bending",

author = "Belyaev, {A. K.} and Morozov, {N. F.} and Tovstik, {P. E.} and Tovstik, {Tatiana Petrovna}",

year = "2016",

month = jan,

day = "1",

doi = "10.3103/S1063454116010040",

language = "English",

volume = "49",

pages = "53--67",

journal = "Vestnik St. Petersburg University: Mathematics",

issn = "1063-4541",

publisher = "Pleiades Publishing",

number = "1",

}

RIS

TY - JOUR

T1 - Parametric resonances in the problem of longitudinal impact on a thin rod

AU - Belyaev, A. K.

AU - Morozov, N. F.

AU - Tovstik, P. E.

AU - Tovstik, Tatiana Petrovna

PY - 2016/1/1

Y1 - 2016/1/1

N2 - The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.

AB - The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.

KW - longitudinal impact

KW - parametric resonances

KW - rod

KW - transverse bending

UR - http://www.scopus.com/inward/record.url?scp=84979500270&partnerID=8YFLogxK

U2 - 10.3103/S1063454116010040

DO - 10.3103/S1063454116010040

M3 - Article

AN - SCOPUS:84979500270

VL - 49

SP - 53

EP - 67

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -

ID: 9282197