ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Kovriguine D. A., Potapov A. I. Nonlinear wave dynamics of one-dimensional elastic systems. Part I. Method оf coupled normal waves . Izvestiya VUZ. Applied Nonlinear Dynamics, 1996, vol. 4, iss. 2, pp. 72-81.

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 1996no2p072.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Review of Actual Problems of Nonlinear Dynamics
Article type: 
Article
UDC: 
534.1

Nonlinear wave dynamics of one-dimensional elastic systems. Part I. Method оf coupled normal waves

Autors: 
Kovriguine Dmitrij Anatolevich, Blagonravov Mechanical Engineering Research Institute of RAS
Potapov Alexandr Ivanovich, Lobachevsky State University of Nizhny Novgorod
Abstract: 

In present paper the theoretical wave approach is developed for investigation of nonlinear dynamics of one-dimensional elastic systems with a strong dispersion. A technique is proposed to reduce the physical field equations to those for nonlinearity coupled normal waves. Then the equations of coupled normal waves are reduced to the evolution equations, which describe either multiwave resonant interaction or wave self-modulation or group synchronism between short and long waves, depending on a problem formulation.

Key words: 
-
Acknowledgments: 
The work was supported by the International Science Foundation (grant N R9B000).
Reference: 
  1. Potapov АI. Nonlinear vibroacoustics of thin-walled structures. In: Wave Dynamics of Machines. М.: Nauka; 1991. P. 110.
  2. Potapov АI. Nonlinear wave processes in elastic media with spatial dispersion. Doctoral Thesis. Gorky: Gorky State University; 1987. 340 p.
  3. Engelbrekht YuK, Nigul UК. Nonlinear Deformation Waves. М.: Nauka; 1981. 256 p.
  4. Bagdoev АG. Wave Propagation in Solid Media. Yerevan; 1981. 308 p.
  5. Nakoryakov VE, Pokusaev BG, Shreiber IR. Wave Dynamics of Gas-Vapor-Liquid Media. M.: Energoatomizdat; 1990. 248 p.
  6. Potapov АI. Nonlinear Deformation Waves in Rods and Plates. Gorky: Gorky University Publishing; 1985. 108 p.
  7. Nayfeh АH. Introduction to Perturbation Techniques. N.Y.: Wiley; 1981. 536 p.
  8. Pelinovskii EN, Fridman VE, Engelbrekht YuК. Nonlinear Evolutionary Equations. Tallinn: Valgus; 1984. 154 p.
  9. Rabinovich MI, Trubetskov DI. Introduction to the Theory of Oscillations and Waves. М.: Nauka; 1984. 432 p.
  10. Zhuravlev VF, Klimov DМ. Applied Methods of Oscillation Theory. M.: Nauka; 1988. 328 p.
  11. Kubenko VD, Kovalchuk PS, Krasnopolskaya TS. Nonlinear Interaction of Forms of Bending Oscillations of Cylindrical Shells. Kiev: Naukova dumka; 1984. 220 p.
  12. Erofeyev VI, Potapov Al. Nonlinear wave processes in elastic media with inner structure. In: Bar’yaktar VG, editor. Nonlinear world. Vol. 2. Singapore: World Scientific; 1990. P. 1197.
  13. Pouget J, Maugin GA. Nonlinear dynamics of oriented elastic solids. Pt. I. Basic equations. J. of Elas. 1991;22(2-3):135-155. DOI: 10.1007/BF00041108.
  14. Novikov AA. Application of the method of coupled waves to an analysis of nonresonance interacti. Radiophys. Quantum Electron. 1976;19(2):225-227. DOI: 10.1007/BF01038531.
  15. Kovrigin DA, Novikov AA, Potapov AN. Nonlinear interactions of normal waves in one-dimensional elastic systems. In: Wave Problems of Mechanics. Nizhny Novgorod; 1991. P.115.
  16. Dodd RK, Eilbeck JC, Gibbon JD, Morris HC. Solitons and Nonlinear Wave Equations. N.Y.: Academic Press; 1982. 630 p.
  17. Zaslavskii GM, Sagdeev RZ. Introduction to Nonlinear Physics. From Pendulum to Turbulence and Chaos. М.: Nauka; 1988. 368 p.
  18. Arnold VI. Loss of auto-oscillation stability near resonances. In: Gaponov-Grekhov АV, editor. Nonlinear Waves. M.: Nauka; 1979. P. 116.
  19. Lvov VS. Nonlinear Spin Waves. М.: Nauka; 1987. 272 p.
  20. Gaponov AV, Ostrovskii LA, Rabinovich МI. One-dimensional waves in disperse nonlinear systems. Rdiophys. Quantum Electron. 1970;13(2):121-161. DOI: 10.1007/BF01031524.
  21. Kovrigin DA, Potapov АI. Nonlinear resonance interactions in a thin ring. Sov. Phys. Doklady. 1989;305(4):803-807.
  22. Phillips OM. Wave interactions - the evolution of an idea. J. Fluid Mech. 1981;106:215-227. DOI:10.1017/S0022112081001572
  23. Bloembergen N. Nonlinear Optics. Reading: Addison-Wesley; 1992. 229 p.
  24. Kaup DJ, Reiman А, Bers А. Space-time evolution of nonlinear three-wave interactions. Interaction in а homogeneous medium. Rev. Mod. Phys. 1979;51(2):275-309. DOI: 10.1103/RevModPhys.51.275.
  25. Kovrigin DА. Nonlinear Resonance Interactions of Waves in Elastic Elements of Structures. PhD Thesis. Nizhny Novgorod: Nizhny Novgorod State University; 1992.
  26. Ginzburg VL. Propagation of Electromagnetic Waves in Plasma. М.: Nauka; 1967. 684 p.
  27. Goncharov VV. On the influence of high-order nonlinearities on three-wave processes. Izvestiya of the Academy of Sciences of the USSR. Atmospheric and Oceanic Physics. 1981;17(1):88.
  28. Volmir АS. Nonlinear Dynamics of Plates and Shells. М.: Nauka; 1972. 432 p.
  29. Shalashov GM. Cross-modulation of acoustic waves on cubic nonlinearity of solids. Acoust. J. 1984;30(3):386-390.
  30. Kovriguine DA, Potapov АI. Identification of large mechanical structures in terms of triad cells. In: Abs JUTAM Symposium on Identification of Mechanical Systems. University of Wuppertal. 1993. P.29.
Received: 
10.12.1995
Accepted: 
20.02.1996
Published: 
21.07.1996
  • 308 reads