 | | 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 |
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| K. V. Avramov, "The analysis of period-doubling bifurcations of the impact oscillator by means of the amplitude surface method," Mech. Solids. 39 (3), 20-26 (2004) |
| Year |
2004 |
Volume |
39 |
Number |
3 |
Pages |
20-26 |
| Title |
The analysis of period-doubling bifurcations of the impact oscillator by means of the amplitude surface method |
| Author(s) |
K. V. Avramov (Kharkov) |
| Abstract |
Impact-oscillation systems have been considered by many authors in various aspects. This is accounted for by the importance of such systems for engineering. The theory of impact-oscillation systems is considered in [1, 2] and the review of the results in this field is given in [3]. Experimental studies of impact oscillators are presented in [4, 5]. The limit cycles and their bifurcations are analyzed in [6]. The families of subharmonic modes and their bifurcations are considered in [9, 10]. The papers [11, 12] are devoted to the utilization of impact-oscillation systems for modeling the dynamics of force transmissions. Although the theory of bifurcations of periodic motions is a fairly well developed area of applied mathematics [13, 14], the bifurcations of periodic motions in impact-oscillation systems in the case where two parameters change virtually have not been analyzed. The dynamics of an impact-oscillation system with two changing parameters was studied in [7, 8]. However, various oscillation modes and their bifurcations have not been considered in these papers. Apparently, this issue is considered in the present paper for the first time.
We propose a method for the analysis of bifurcations in the impact oscillator under the change of two parameters. This method is based on the construction of the amplitude surface that graphically represents bifurcations in the case of two variable parameters. This method allowed us to identify a number of bifurcation points of co-dimension 2. Publications in which the amplitude surfaces are utilized for the analysis of impact-oscillation systems have not been known to the author. Note that bifurcation points of co-dimension 2 have not been investigated previously for such systems. |
| References |
| 1. | V. I. Babitskii, Theory of Impact-Oscillation Systems
[in Russian], Nauka, Moscow, 1978. |
| 2. | V. I. Babitskii and V. L. Krupenin, Oscillations in Strongly Nonlinear Systems [in Russian], Nauka, Moscow, 1985. |
| 3. | S. R. Bishop, "Impact oscillator," Phil. Trans. Roy. Soc. London, Ser. A, Vol. 347, No. 1683, pp. 347-351, 1994. |
| 4. | S. F. Masri, Y. A. Maramy, and J. C. Anderson, "Dynamic response of a beam with a geometric nonlinearity," Trans. ASME. J. Appl. Mech, Vol. 48, No. 2, pp. 404-410, 1981. |
| 5. | F. C. Moon and S. W. Shaw, "Chaotic vibration of a beam wit non-linear boundary conditions," Intern. J. Non-linear Mech., Vol. 18, No. 6, pp. 465-477, 1983. |
| 6. | S. W. Shaw and P. J. Holmes, "A periodically forces piecewise linear oscillator," J. Sound and Vibration, Vol. 90, No. 1, pp. 129-155, 1983. |
| 7. | F. Peterka and J. Vacik, "Transition to chaotic motion in mechanical systems with impacts," J. Sound and Vibration, Vol. 154, No. 1, pp. 95-115,
1992. |
| 8. | F. Peterka and T. Kotera, "Four ways from periodic to chaotic motion in the impact oscillator," Machine Vibration, Vol. 5, No. 2, pp. 71-82, 1996. |
| 9. | M. I. Feigin, "On the generation of a family of subharmonic modes in a piecewise continuous system," PMM [Applied Mathematics and Mechanics], Vol. 38, No. 5, pp. 810-818, 1974. |
| 10. | M. I. Feigin, "On the behavior of dynamical systems near the boundary of the region of existence of periodic motions," PMM [Applied Mathematics and Mechanics], Vol. 41, No. 4, pp. 628-636, 1977. |
| 11. | K. V. Avramov, A. S. Belomyttsev and V. N. Karaban, "Regions of chaotic vibrations in discrete mechanical systems with piecewise linear elastic characteristics," Prikladnaya Mekhanika, Vol. 30, No. 5, pp. 81-88, 1994. |
| 12. | K. V. Avramov and V. N. Karaban, "Specific features of the development of forced vibrations in nonlinear force transmissions," Izv.
Vuzov. Mashinostroenie, No. 4-6, pp. 3-8, 1994. |
| 13. | Yu. I. Neimark and P. S. Landa, Stochastic and Chaotic Oscillations [in Russian], Nauka, Moscow, 1987. |
| 14. | J. M. Guckenheimer and P. Holmes,
Nonlinear Oscillations, Dynamical Systems and Bifurcation
of Vector Fields, Springer, N.Y., 1983. |
| 15. | M. Qriouet and C. Mira, "Fractional harmonic synchronization in the Duffing-Rayleigh differential equation," Intern J. Bifurcat. Chaos, Vol. 4,
No. 2, pp. 411-426, 1994. |
| 16. | P. J. Holmes and Y. K. Lin, "Deterministic stability analysis of a wind loaded structure," Trans. ASME. J. Appl. Mech, Vol. 45, No. 1, pp. 165-169, 1978. |
| 17. | H. Kawakami, "Bifurcation of periodic responses in forces dynamic nonlinear circuits: computation of bifurcation values of the system parameters," IEEE Trans. Circuits and Systems, Vo. 31, No. 3, pp. 248-260, 1984. |
| 18. | R. Allam and C. Mira, ""Crossroad area-disymmetrical spring area-symmetrical spring area" and "double crossroad area-double spring area" transitions," Intern J. Bifurcat. Chaos, Vol. 3, No. 2, pp. 429-435, 1993. |
| 19. | A. S. Belomyttsev and V. N. Karaban, "An algorithm for solving a nonlinear boundary-value problem for ordinary differential equations in a multi-valued region," Zh. Vychisl. Matematiki i Matem. Fiziki, Vol. 27, No. 7,
pp. 1099-1102, 1986. |
|
| Received |
13 December 2001 |
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