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

For citation:

Alifov A. A. Mixed forced, parametric, and self-oscillations with nonideal energy source and lagging forces. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 5, pp. 739-750. DOI: 10.18500/0869-6632-2021-29-5-739-750

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
534.16
DOI: 
10.18500/0869-6632-2021-29-5-739-750

Mixed forced, parametric, and self-oscillations with nonideal energy source and lagging forces

Autors: 
Alifov Alishir Ali oglu, Blagonravov Mechanical Engineering Research Institute of RAS
Abstract: 

The purpose of this study is to determine the effect of retarded forces in elasticity and damping on the dynamics of mixed forced, parametric, and self-oscillations in a system with limited excitation. A mechanical frictional self-oscillating system driven by a limited-power engine was used as a model. Methods. In this work, to solve the nonlinear differential equations of motion of the system under consideration, the method of direct linearization is used, which differs from the known methods of nonlinear mechanics in ease of use and very low labor and time costs. This is especially important from the point of view of calculations when designing real devices. Results. The characteristic of the friction force that causes self-oscillations, represented by a general polynomial function, is linearized using the method of direct linearization of nonlinearities. Using the same method, solutions of the differential equations of motion of the system are constructed, equations are obtained for determining the nonstationary values of the amplitude, phase of oscillations and the speed of the energy source. Stationary motions are considered, as well as their stability by means of the Routh–Hurwitz criteria. Performed calculations obtained information about the effect of delays on the dynamics of the system. Conclusion. Calculations have shown that delays shift the amplitude curves to the right and left, up and down on the amplitude–frequency plane, change their shape, and affect the stability of motion.

Key words: 
mixed oscillations
forced oscillations
parametric oscillations
Self-oscillations
method
direct linearization
the source of energy
damping
elasticity
delay
Reference: 
  1. Kononenko VO. Vibrating Systems with Limited Power-Supply. London: Iliffe Books; 1969. 236 p.
  2. Krasnopolskaya TS, Shvets AY. Regular and Chaotic Dynamics of Systems with Limited Excitation. Moscow-Izhevsk: SPC «Regular and Chaotic Dynamics», Izhevsk Institute of Computer Science; 2008. 280 p.
  3. Alifov AA, Frolov KV. Interaction of Nonlinear Oscillatory Systems with Energy Sources. New York, Washington, Philadelphia, London: Hemisphere Publishing Corporation; 1990. 327 p.
  4. Alifov AA. About calculation of self-oscillatory system delayed and limited excitation. In: Proceedings of the International Scientific and Technical Conference «Measurement and Quality: Problems, Prospects». Baku, Azerbaijan, 21-23 November 2018. Baku: AzTU; 2018. P. 289–293. 
  5. Bogoliubov NN, Mitropolskii YA. Asymptotic Methods in Theory of Non-linear Oscillations. New York: Gordon and Breach; 1961. 537 p.
  6. Andronov AA, Vitt AA, Khaikin SE. Theory of Oscillators. USA: Pergamon Press; 1966. 848 p.
  7. Moiseyev NN. Asymptotic Methods of Nonlinear Mechanics. Moscow: Nauka; 1981. 400 p. (in Russian).
  8. Zhuravlev VF, Klimov DM. Applied Methods in the Theory of Oscillations. Moscow: Nauka; 1988. 328 p. (in Russian).
  9. Hayashi C. Nonlinear Oscillations in Physical Systems. New York: McGraw-Hill; 1964.
  10. Butenin NV, Neymark YI, Fufayev NA. Introduction to the Theory of Nonlinear Oscillations. Moscow: Nauka; 1976. 384 p. (in Russian).
  11. Alifov AA. Methods of Direct Linearization for Calculation of Nonlinear Systems. Moscow Izhevsk: SPC «Regular and Chaotic Dynamics»; 2015. 74 p. (in Russian).
  12. Alifov AA. Method of the direct linearization of mixed nonlinearities. Journal of Machinery Manufacture and Reliability. 2017;46(2):128–131. DOI: 10.3103/S1052618817020029.
  13. Alifov AA. About direct linearization methods for nonlinearity. In: Hu Z, Petoukhov S, He M, editors. Advances in Artificial Systems for Medicine and Education III. AIMEE 2019. Vol. 1126 of Advances in Intelligent Systems and Computing. Cham: Springer; 2020. P. 105–114. DOI: 10.1007/978-3-030-39162-1_10.
  14. Alifov AA, Farzaliev MG, Dzhafarov EN. Dynamics of a self-oscillatory system with an energy source. Russian Engineering Research. 2018;38(4):260–262. DOI: 10.3103/S1068798X18040032.
  15. Alifov AA. On the calculation by the method of direct linearization of mixed oscillations in a system with limited power-supply. In: Hu Z, Petoukhov S, Dychka I, He M, editors. Advances in Computer Science for Engineering and Education II. ICCSEEA 2019. Vol. 938 of Advances in Intelligent Systems and Computing. Cham: Springer; 2019. P. 23–31. DOI: 10.1007/978-3-030-16621-2_3.
  16. Alifov AA. About application of methods of direct linearization for calculation of interaction of nonlinear oscillatory systems with energy sources. In: Proceedings of the International Symposium of Mechanism and Machine Science (ISMMS-2017). Baku, Azerbaijan, 11–14 September 2017. Baku: AzC IFToMM; 2017. P. 218–221.
  17. Panovko YG. Method of direct linearization in nonlinear problems of torsional oscillations. Scientific Notes of the Latvian State University. 1953;4:73–90 (in Russian).
  18. Loytsyansky LG. Free and forced oscillations in the presence of quadratic and intermediate between linear and quadratic laws of resistance. Bulletin of the Academy of Sciences of the USSR. Engineering Collection. 1954;18:139–148 (in Russian).
  19. Alifov AA. About mixed forced, parametric and self-oscillations by limited excitation and delayed elasticity. PNRPU Mechanics Bulletin. 2020;(3):12–19 (in Russian). DOI: 10.15593/perm.mech/2020.3.02.
  20. Zolotukhin YN, Kotov KY, Maltsev AS, Nesterov AA, Filippov MN. Compensation of transport lag when controlling the trajectory movement of a group of mobile robots. In: Proceedings of the XIII International Conference «Problems of Control and Modeling in Complex Systems» (PCMCS-2011). Samara, Russia, 15–17 June 2011. Samara: Institute for Complex Systems Control Problems; 2011. P. 223–229.
  21. Tretyakova TV, Wildeman VE. Spatio-Temporal Inhomogeneity of the Processes of Inelastic Deformation of Metals. Moscow: FIZMATLIT; 2017. 120 p. (in Russian).
  22. Alifov AA. Calculating mixed forced and self-oscillations for delayed elastic constraint and a limited power energy source. Journal of Machinery Manufacture and Reliability. 2020;49(2): 105–109. DOI: 10.3103/S1052618820020053.
  23. Alifov AA, Farzaliev MG. On the calculation by the method of linearization of the interaction of parametric and self-oscillations at delay and limited excitation. Tomsk State University Journal of Mathematics and Mechanics. 2020;(68):41–52 (in Russian). DOI: 10.17223/19988621/68/4.
  24. Alifov AA. Autoparametric oscillations with delays in elastic and frictional forces. Journal of Machinery Manufacture and Reliability. 2021;50(2):98–104. DOI: 10.3103/S1052618821020023.
  25. Alifov AA. Self-oscillations in delay and limited power of the energy source. Mechanics of Solids. 2019;54(4):607–613. DOI: 10.3103/S0025654419040150.
  26. Klimov DM. On one type of self-excited vibrations in a system with dry friction. Mechanics of Solids. 2003;(3):6–13 (in Russian).
  27. Frolov KV. Selected Works. In 2 Volumes. Moscow: Nauka; 2007 (in Russian).
  28. Panovko YG. Fundamentals of the Applied Theory of Vibrations and Shock. Leningrad: Mashinostroeniye; 1976. 320 p. (in Russian).
  29. Abdiyev FK. Delayed self-oscillations of a system with a non-ideal energy source. Bulletin of the Academy of Sciences of the AzSSR. Series of Physical, Technical and Mathematical Sciences. 1983;(4):134–139 (in Russian).
  30. Rubanik VP, Starik LK. On the stability of self-oscillations of a cutter in the case of an non-ideal energy source. Scientific Works of Universities of the Lithuanian SSR. Vibration Technology. 1971;(2):205–212 (in Russian).
  31. Murashkin LS, Murashkin SL. Applied Nonlinear Mechanics of Machine Tools. Leningrad: Mashinostroeniye; 1977. 192 p. (in Russian).
  32. Ponomarev AS, Sokolov VI, Kozhukhar AA. Transverse self-oscillations of power tables caused by friction forces. Bulletin of the Kharkov Polytechnic Institute. Mechanical Engineering. 1977; 8(130):67–70 (in Russian).
  33. Bronovets MA, Zhuravlev VF. On self-excited vibrations in friction force measurement systems. Mechanics of Solids. 2012;47(3):261–268. DOI: 10.3103/S0025654412030016.
Received: 
25.05.2020
Accepted: 
16.07.2021
Published: 
30.09.2021
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