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ANALYSIS OF DYNAMICS OF SOME DISCRETE SYSTEMS

Investigating vibrations in dynamical systems to determine stability conditions, amplitudes, and control methods to avoid negative impacts.

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ANALYSIS OF DYNAMICS OF SOME DISCRETE SYSTEMS

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  1. ANALYSIS OF DYNAMICS OF SOME DISCRETE SYSTEMS Olena Mul (1), Ilona Dzenite (2), Volodymyr Kravchenko (3) (1) Ternopil National Ivan Pul’uj Technical University Ruska str., 56, 46001 Ternopil, Ukraine (2) Riga Technical University, Kalku str. 1, Riga LV-1658, Latvia (3)Physico-Technological Institute of Metals and Alloys, Acad. Vernadsky Avenue, 34/1, 03680 Kiev, Ukraine CODY Autumn in Warsaw November 18, 2010

  2. 1. THE PROBLEM To investigate possible vibrations in dynamical systems of machine units, which always have negative influence on systems functioning. We will try: • to determine conditions of stability of different stationary modes as well as transient to them; • to determine amplitudes of possible vibrations; • to find ways how to control vibrations or even to avoid them. CODY Autumn in Warsaw November 18, 2010

  3. 2. SOME REFERENCES • [1] V.L.Vets, M.Z. Kolovskyj, and A.E. Kochura, Dynamics of controlled machine units,“Nauka”, Moscow, 1984, 364 p.(Russian). • [2]V.A. Krasnoshapka, Dynamics of the machine unit subject to nonlinearity of the friction forces moment, Mashynovedenije (1973), no.4, pp. 36-41 (Russian). • [3] F.K. Ivanchenko and V.A. Krasnoshapka, Dynamics of metallurgical machines,“Metallurgiya”, Moscow, 1983, 293 p. (Russian). • [4] N.N. Bogolyubov and Yu.A. Mitropolskii, Asymptotical methods in the theory of nonlinear vibrations, “Nauka”, Moscow, 1974, 503 p.(Russian). MR 0374550 CODY Autumn in Warsaw November 18, 2010

  4. 2. SOME REFERENCES • [5] V.A. Svetlitsky, Engineering vibration analysis: worked problems. 1, Foundations of Engineering Mechanics, Springer-Verlag, Berlin, 2004, 316 p. MR 2081767 • [6]A. Samoilenko and R. Petryshyn, Multifrequency oscillations of nonlinear systems, Mathematics and its Applications, vol. 567, Kluwer Acad. Publ. Group, Dordrecht, 2004. MR 2091174 • [7]N.N. Ivashchenko, Automatic control, Izdat. “Mashinostroenie”, Moscow, 1978, 735 p. (Russian). • [8] M.G. Chishkin, V.I. Klyuchev, and A.C. Sondler, Theory of automatized electric drive,“Energija”, Moscow, 1979, 614 p.(Russian). CODY Autumn in Warsaw November 18, 2010

  5. 3. THE ELECTROMECHANICAL ELASTIC PHYSICAL MODEL OF THE MACHINE UNIT WITHA DIRECT-CURRENT MOTOR CODY Autumn in Warsaw November 18, 2010

  6. 4. THE MECHANICALMODELOF THE MACHINE UNIT WITHA DIRECT-CURRENT MOTOR OF SEPARATE EXCITATION CODY Autumn in Warsaw November 18, 2010

  7. 5. THE MATHEMATICAL MODEL CODY Autumn in Warsaw November 18, 2010

  8. 5. THE MATHEMATICAL MODEL CODY Autumn in Warsaw November 18, 2010

  9. 6. THE AVERAGING METHOD6.1. BASIC APPROACH CODY Autumn in Warsaw November 18, 2010

  10. 6.2. TRANSFORMATIONS CODY Autumn in Warsaw November 18, 2010

  11. 6.2. TRANSFORMATIONS CODY Autumn in Warsaw November 18, 2010

  12. 6.2. TRANSFORMATIONS CODY Autumn in Warsaw November 18, 2010

  13. 6.2. TRANSFORMATIONS CODY Autumn in Warsaw November 18, 2010

  14. 6.2. TRANSFORMATIONS CODY Autumn in Warsaw November 18, 2010

  15. 6.2. TRANSFORMATIONS CODY Autumn in Warsaw November 18, 2010

  16. 6.3. AVERAGING CODY Autumn in Warsaw November 18, 2010

  17. 6.3. AVERAGING CODY Autumn in Warsaw November 18, 2010

  18. 6.3. AVERAGING CODY Autumn in Warsaw November 18, 2010

  19. 6.3. AVERAGING CODY Autumn in Warsaw November 18, 2010

  20. 6.4. STABILITY CONDITIONS CODY Autumn in Warsaw November 18, 2010

  21. 6.4. STABILITY CONDITIONS CODY Autumn in Warsaw November 18, 2010

  22. 7. SOME RESULTS7.1. EXAMPLE 1 CODY Autumn in Warsaw November 18, 2010

  23. 0.7 0.6 A 1 0 0.5 0.4 A 0.3 0.2 A 0.1 2 0 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 , s T e 7.1.1. Graphs of the dependence of stationary amplitudes of one-frequency vibrations CODY Autumn in Warsaw November 18, 2010

  24. 0.20 0.15 A 1 0 0.10 A 0.05 A 2 0 0.00 0.010 0.015 0.020 0.025 0.030 0.035 T , s e 7.1.2. Graphs of the dependence of stationary amplitudes of biharmonicvibrations CODY Autumn in Warsaw November 18, 2010

  25. 0.6 0.5 0.4 0.3 A A 0.2 1 A 0.1 2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t, s 7.1.3. Graphs of the dependence of the vibration amplitudes on time t CODY Autumn in Warsaw November 18, 2010

  26. M, N . m 600 400 200 0 -200 -400 -600 -800 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t, s 7.1.4. Graphs of the dependence of themotor moment M on time t CODY Autumn in Warsaw November 18, 2010

  27. q 12 0.9 0.6 0.3 0.0 -0.3 -0.6 -0.9 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t, s 7.1.5. Graphs of the dependence of the elastic deformation on time t CODY Autumn in Warsaw November 18, 2010

  28. 0.16 0.14 0.12 0.10 0.08 A A 0.06 2 A 0.04 1 0.02 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t, s 7.1.6. Graphs of the dependence of the vibrations amplitudes on time t CODY Autumn in Warsaw November 18, 2010

  29. . M, N m 50 0 -50 -100 -150 -200 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t, s 7.1.7. Graphs of the dependence of the motor moment M on time t CODY Autumn in Warsaw November 18, 2010

  30. q 12 1.0 0.5 0.0 -0.5 -1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t, s 7.1.8. Graphs of the dependence of the elastic deformation on time t CODY Autumn in Warsaw November 18, 2010

  31. 7.2. EXAMPLE 2 CODY Autumn in Warsaw November 18, 2010

  32. 0.5 0.4 0.3 A 2 0 0.2 0.1 0.0 0.00 0.04 0.08 0.12 0.16 T , s e 7.2.1. Graph of the dependence of the stationary amplitude of one-frequencyvibrations CODY Autumn in Warsaw November 18, 2010

  33. 0.4 0.3 A 2 0.2 A A 0.1 1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t, s 7.2.2. Graphs of the dependence of the vibrations amplitudes on time t CODY Autumn in Warsaw November 18, 2010

  34. . M, N m 6 2x10 6 1x10 0 6 -1x10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t, s 7.2.3. Graph of the dependence of the motor moment M on time t CODY Autumn in Warsaw November 18, 2010

  35. q 12 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t, s 7.2.4. Graph of the dependence of the elastic deformation on time t CODY Autumn in Warsaw November 18, 2010

  36. CONCLUSIONS • Vibrations in controlled machine units with discrete parameters are investigated. The mathematical model is the 5th order ODE's system, for which the averaging method is used. • In such systems both stable one-frequency modes on the first two frequencies and an unstable biharmonic mode may be excited. The biharmonic mode is possible only in a small range of the motor electromagnetic constant. With time, it passes to one of the one-frequency modes depending on initial conditions. • Vibrations amplitudes are significantly dependent on the motor electromagnetic constant, and amplitudes in transient states may be significantly greater than amplitudes of stationary modes. • The feedbacks, which allow to change purposefully the dynamical characteristics of the motor, should be used for decreasing of the negative effect of vibrations on system functioning. CODY Autumn in Warsaw November 18, 2010

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