CODY Autumn in Warsaw November 18, 2010 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, 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, Kiev, Ukraine

CODY Autumn in Warsaw November 18, THE PROBLEM 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, 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 (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

CODY Autumn in Warsaw November 18, 2010  [5] V.A. Svetlitsky, Engineering vibration analysis: worked problems. 1, Foundations of Engineering Mechanics, Springer- Verlag, Berlin, 2004, 316 p. MR  [6] A. Samoilenko and R. Petryshyn, Multifrequency oscillations of nonlinear systems, Mathematics and its Applications, vol. 567, Kluwer Acad. Publ. Group, Dordrecht, MR  [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). 2. SOME REFERENCES

CODY Autumn in Warsaw November 18, THE ELECTROMECHANICAL ELASTIC PHYSICAL MODEL OF THE MACHINE UNIT WITH A DIRECT- CURRENT MOTOR

CODY Autumn in Warsaw November 18, THE MECHANICAL MODEL OF THE MACHINE UNIT WITH A DIRECT-CURRENT MOTOR OF SEPARATE EXCITATION

CODY Autumn in Warsaw November 18, THE MATHEMATICAL MODEL

CODY Autumn in Warsaw November 18, THE MATHEMATICAL MODEL

CODY Autumn in Warsaw November 18, THE AVERAGING METHOD 6.1. BASIC APPROACH

CODY Autumn in Warsaw November 18, TRANSFORMATIONS

CODY Autumn in Warsaw November 18, TRANSFORMATIONS

CODY Autumn in Warsaw November 18, TRANSFORMATIONS

CODY Autumn in Warsaw November 18, TRANSFORMATIONS

CODY Autumn in Warsaw November 18, TRANSFORMATIONS

CODY Autumn in Warsaw November 18, TRANSFORMATIONS

CODY Autumn in Warsaw November 18, AVERAGING

CODY Autumn in Warsaw November 18, AVERAGING

CODY Autumn in Warsaw November 18, AVERAGING

CODY Autumn in Warsaw November 18, AVERAGING

CODY Autumn in Warsaw November 18, STABILITY CONDITIONS

CODY Autumn in Warsaw November 18, STABILITY CONDITIONS

CODY Autumn in Warsaw November 18, SOME RESULTS 7.1. EXAMPLE 1 7. SOME RESULTS 7.1. EXAMPLE 1

CODY Autumn in Warsaw November 18, Graphs of the dependence of stationary amplitudes of one-frequency vibrations , s e A 0 A A T

CODY Autumn in Warsaw November 18, Graphs of the dependence of stationary amplitudes of biharmonic vibrations A 0 2 A, s e A T

CODY Autumn in Warsaw November 18, Graphs of the dependence of the vibration amplitudes on time t A 1 A A t, s

CODY Autumn in Warsaw November 18, Graphs of the dependence of the motor moment M on time t m. M, N t, s

CODY Autumn in Warsaw November 18, Graphs of the dependence of the elastic deformation on time t  12 t, s

CODY Autumn in Warsaw November 18, Graphs of the dependence of the vibrations amplitudes on time t A 2 A A t, s

CODY Autumn in Warsaw November 18, Graphs of the dependence of the motor moment M on time t m. M, N t, s

CODY Autumn in Warsaw November 18, Graphs of the dependence of the elastic deformation on time t  12 t, s

CODY Autumn in Warsaw November 18, EXAMPLE EXAMPLE 2

CODY Autumn in Warsaw November 18, Graph of the dependence of the stationary amplitude of one-frequency vibrations A 0 2, s e T

CODY Autumn in Warsaw November 18, Graphs of the dependence of the vibrations amplitudes on time t A 2 A A t, s

CODY Autumn in Warsaw November 18, Graph of the dependence of the motor moment M on time t x x10 6 2x10 6 m. M, N t, s

CODY Autumn in Warsaw November 18, Graph of the dependence of the elastic deformation on time t  12 t, s

CODY Autumn in Warsaw November 18, 2010 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.