1. MMC
  2. All Volumes and Issues
  3. Volume 9, Number 1, 2022
  4. On the external and internal resonance phenomena of the elastic bodies with the complex oscillations

On the external and internal resonance phenomena of the elastic bodies with the complex oscillations

MMC.
2022;
Volume 9, Number 1
: pp. 152–158
https://doi.org/10.23939/mmc2022.01.152
Received: March 03, 2021
Revised: December 30, 2021
Accepted: January 03, 2022

Mathematical Modeling and Computing, Vol. 9, No. 1, pp. 152–158 (2022)

Authors:
  1. N. Huzyk
  2. P. Ya. Pukach
  3. B. Sokil
  4. M. Sokil
  5. M. I. Vovk
1
Hetman Petro Sahaidachnyi National Army Academy
2
Lviv Polytechnic National University
3
Hetman Petro Sahaidachnyi National Army Academy
4
Lviv Polytechnic National University
5
Lviv Polytechnic National University

Complex nonlinear oscillations in the elastic bodies are studied using a priori information about the oscillations form and taking into account a refined mathematical model of the second (other) form of oscillations.  Application of existing methods or development of the new ones for the analysis of received non-autonomous boundary value problems is proposed.  The effectiveness of the practical implementation of the discussed methodology significantly increases in cases where the magnitude of the elastic body displacements due to the one form of oscillations is much higher than the other one.  To analyze the problem one can use the well-known tested analytical methods for the systems with the small nonlinearity.  Torsional and bending oscillations of the elastic body are shown as the example.  It is also demonstrated that especially dangerous resonant processes can be caused not only by the external perturbations but also by the internal influence between some forms of oscillations.  The obtained results allow to choose the basic technological and operational parameters of the machine oscillating elements in order to avoid the resonance phenomena.

nonlinear elastic body
asymptotic methods
amplitude
resonance
complex oscillations
  1. Andrukhiv A., Sokil B., Sokil M.  Asymptotic method in the investigation of complex nonlinear oscillations of the elastic bodies.  Ukrainian Journal of Mechanical Engineering and Materials Science.  4 (2),  58–67 (2018).
  2. Boholyubov N. N., Mitropolsky Yu. A.  Asymptotic methods in the theory of nonlinear oscillations.  Moscow, Fizmatlit (1974), (in Russian).
  3. Mitropolsky Yu. A., Moseenkov B. I.  Asymptotic solutions of partial differential equations.  Kyiv, Vyshcha Shkola (1976), (in Russian).
  4. Bayat M., Pakara I., Domairry G.  Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems:  A review.  Latin American Journal of Solids and Structures.  9 (2), 1–93 (2012).
  5. Pukach P. Ya.  On the unboundedness of a solution of the mixed problem for a nonlinear evolution equation at a finite time.  Nonlinear Oscillations.  14 (3), 369–378 (2012).
  6. Cveticanin L.  Period of vibration of axially vibrating truly nonlinear rod.  Journal of Sound and Vibration.  374, 199–210 (2016).
  7. Cveticanin L., Pogány T.  Oscillator with a sum of non-integer order non-linearities.  Journal of Applied Mathematics.  2012, Article ID: 649050 (2012).
  8. Mitropolsky Yu. A., Sokil B. I.  On the application of Ateb-functions to the construction of an asymptotic solution of the perturbed nonlinear Klein-Gordon equation.  Ukrainian Mathematical Journal.  50 (5), 754–760 (1998).
  9. Sokil B.  Construction of asymptotic solutions of certain boundary-value problems for the nonautonomous wave equation.  Journal of Mathematical Sciences.  96 (1), 2878–2882 (1999).
  10. Ogorodnikov P., Svitlytskyi V., Hohol V.  Study of connection between longitudinal and rotary fluctuations of the drill stem.  Oil & gas industry of Ukraine.  2,  6–9 (2014).
  11. Borshch O., Gulyayev V.  Spiral waves in elastic twisted rotating tube rods with internal flows of liquid.  Acoustic Bulletin.  10 (3), 12–18 (2007).
  12. Goroshko A., Royzman V.  Statistical Methods for Providing the Stability of the Solutions of Inverse Problems and Their Application to Decrease Rotor Vibroactivity.  Journal of Machinery Manufacture and Reliability.  44 (3), 232–238 (2015).
  13. Goroshko A.  Purpose permissible unbalance for high-speed rotor.  Vibrations in engineering and technology.  2 (82), 69–76  (2016).
  14. Pirogova N., Taranenko P.  Calculative and Experimental Analysis of Natural and Critical Frequencies and Mode Shapes of High-speed Rotor for Micro Gas Turbine Plant.  Procedia Engineering.  129, 997–1004 (2015).
  15. Kyuho Sim, Lee Yong-Bok, Tae Ho Kim.  Rotordynamic Performance of Shimmed Gas Foil Bearings for Oil-Free Turbochargers.  Journal of Tribology.  134 (3), 031102  (2012).
  16. Do-Kwan Hong, Daesuk Joo, Byung-Chul Woo, Yeon-Ho Jeong, Dae-Hyun Koo, Chan-Woo Ahn, Yun-Hyun Cho.  Performance verification of a high speed motor-generator for a microturbine generator. International Journal of Precision Engineering and Manufacturing.  14 (7), 1237–1244 (2013).
  17. Pavlovsky M., Putyata T.  Theoretical mechanics.  Kyiv, Vyshcha Shkola (1985), (in Ukrainian).