 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4810 |
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| A.G. Petrov and M.M. Shunderyuk, "On Nonlinear Vibrations of a Heavy Mass Point on a Spring," Mech. Solids. 45 (2), 176-186 (2010) |
| Year |
2010 |
Volume |
45 |
Number |
2 |
Pages |
176-186 |
| DOI |
10.3103/S0025654410020032 |
| Title |
On Nonlinear Vibrations of a Heavy Mass Point on a Spring |
| Author(s) |
A.G. Petrov (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str. 1, Moscow, 119526 Russia, petrov@ipmnet.ru)
M.M. Shunderyuk (Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudny, Moscow District, 141700 Russia, everesk@gmail.com) |
| Abstract |
We present a complete study of small nonlinear vibrations of a swinging spring with a nonlinear dependence of the spring tension on its elongation. We use the Hamiltonian normal form method. The Hamiltonian normal form profitably differs from the general normal form of differential equations, because it has an additional integral. To reduce the Hamiltonian to normal form, we use the invariant normalization method, which significantly reduces the computations. The normal form asymptotics are obtained by successively calculating the quadratures in the same way for both resonance and nonresonance cases. The solutions of Hamiltonian equations in normal form showed that the periodic change of vibrations from vertical to horizontal modes and vice versa occurs only in the case of 1:1 and 2:1 resonances. In the case of 2:1 resonance, this effect manifests itself in the quadratic terms of the equation, and in the case of 1:1 resonance, it manifests itself if the cubic terms are taken into account. In all other cases, both in the case of resonance and without any resonance, the vibrations occur at two constant frequencies, which slightly differ from the linear approximation frequencies. In the case of 2:1 resonance, we found the maximum frequency detuning at which the effect of the energy pumping from one vibration mode to another disappears. 1:1 resonance is physically possible only for a spring with a negative cubic additional term in the strain law. |
| Keywords |
mechanics, spring pendulum, normal form, invariant normalization, resonance vibrations |
| References |
| 1. | A. H. Nayfeh,
Perturbation Methods
(Wiley, New York, 1973; Mir, Moscow, 1976). |
| 2. | V. M. Starzhinskii,
Applied Methods of Nonlinear Vibrations
(Nauka, Moscow, 1977)
[in Russian]. |
| 3. | V. N. Bogaevskii and A. Ya. Povzner,
Algebraic Methods in Nonlinear Perturbation Theory
(Nauka, Moscow, 1987; Springer-Verlag, New York, 1991). |
| 4. | A. P. Markeyev, "Non-Linear Vibrations of a Hamiltonian System with 2:1 Resonance," Prikl. Mat. Mekh. 63 (5), 757-769 (1999) [J. Appl. Math. Mech. (Engl. Transl.) 63 (5),
715-726 (1999)]. |
| 5. | M. N. Zaripov and A. G. Petrov, "Nonlinear Vibrations of a Swinging Spring," Dokl. Ross. Akad. Nauk 399 (3),
347-352 (2004) [Dokl. Phys. (Engl. Transl.) 49 (11),
691-696 (2004)]. |
| 6. | A. G. Petrov,
"Nonlinear Vibrations of a Swinging Spring at Resonance,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 5, 18-28 (2006)
[Mech. Solids (Engl. Transl.)
41 (5), 13-22 (2006)]. |
| 7. | V. Ph. Zhuravlev,
Foundations of Theoretical Mechanics
(Nauka, Fizmatlit, Moscow, 1997)
[in Russian]. |
| 8. | V. Ph. Zhuravlev,
"Invariant Normalization of Non-Autonomous Hamiltonian Systems,"
Prikl. Mat. Mekh.
66 (3), 356-365 (2002)
[J. Appl. Math. Mech. (Engl. Transl.)
66 (3), 347-355 (2002)]. |
| 9. | V. Ph. Zhuravlev and D. M. Klimov,
Applied Methods in Vibration Theory
(Nauka, Moscow, 1988)
[in Russian]. |
| 10. | A. D. Bryno,
"Analytic Form of Differential Equations. II,"
Trudy Moskov. Mat. Obshch.
26, 199-239 (1972)
[Trans. Moscow Math. Soc. (Engl. Transl.)
26, 199-239 (1972)]. |
| 11. | G. D. Birkhoff,
Dynamical Systems
(AMS, Providence, 1927; Gostekhizdat, Moscow-Leningrad, 1941). |
| 12. | T. M. Cherry,
"On the Solution of Hamiltonian Systems of Linear Differential Equations in the Neighborhood of a Singular Point,"
Proc. London Math. Soc. Ser. 2,
27 (1), 151-170 (1928). |
| 13. | F. G. Gustavson,
"On Constructing Formal Integrals of a Hamiltonian Systems near an Equilibrium Point,"
Astron. J.
71 (8), 670-686 (1966). |
|
| Received |
11 April 2008 |
| Link to Fulltext |
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