1. Oscillation Analysis of Some Hybrid Dynamical Systems of Transmission Pipelines Olena Mul ( jointly with Volodymyr Kravchenko) Ternopil Ivan Pul'uj National Technical University Ternopil, Ukraine ESF-EMS-ERCOM/INI Mathematical Conference "Highly Oscillatory Problems From Theory to Applications“, Isaac Newton Institute for Mathematical Sciences, Cambridge, September 12-17, 2010

2. Cambridge, Sept.16, 20102 1. THE PROBLEMS It is necessary to investigate possible oscillations in the transmission pipelines, which always have negative influence on systems functioning. The main problems are: to determine conditions of oscillations self-excitation; to determine frequencies and amplitudes of possible oscillations; to find ways how to decrease negative influence of oscillations or even to avoid them.

3. Cambridge, Sept.16, 20103 2. SOME REFERENCES K. Ya. Kuhta, V. P. Kravchenko, and V. A. Krasnoshapka, The qualitative theory of controllable dynamical systems with continuous-discrete parameters, ''Naukova Dumka'', Kiev, 1986 (Russian). K. Ya. Kuhta, V. P. Kravchenko, and V. A. Krasnoshapka, The qualitative theory of controllable dynamical systems with continuous-discrete parameters, ''Naukova Dumka'', Kiev, 1986 (Russian). A. Samoilenko and R. Petryshyn, Multifrequency oscillations of nonlinear systems, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (2004), no. 567. A. Samoilenko and R. Petryshyn, Multifrequency oscillations of nonlinear systems, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (2004), no. 567. O. Mul and V. Kravchenko, Investigations of Vibrations in the Complex Dynamical Systems of Transmission Pipelines, "Interface and Transport Dynamics. Computational Modelling", Lecture Notes in Computational Science and Engineering, Springer-Verlag Berlin Heidelberg (2003), no.32, 295-300. O. Mul and V. Kravchenko, Investigations of Vibrations in the Complex Dynamical Systems of Transmission Pipelines, "Interface and Transport Dynamics. Computational Modelling", Lecture Notes in Computational Science and Engineering, Springer-Verlag Berlin Heidelberg (2003), no.32, 295-300. O.V. Mul,, D.F.M. Torres, Analysis of Vibrations in Large Flexible Hybrid Systems, Nonlinear Analysis, Elsevier, vol. 63, 2005, USA, pp. 350-363. O.V. Mul,, D.F.M. Torres, Analysis of Vibrations in Large Flexible Hybrid Systems, Nonlinear Analysis, Elsevier, vol. 63, 2005, USA, pp. 350-363.

4. Cambridge, Sept.16, 20104 3. THE SCHEME OF TRANSMISSION PIPELINES X M  m L ) t ) t, L ( u ( F 

5. Cambridge, Sept.16, 20105 4.1. THE MATHEMATICAL MODEL

6. Cambridge, Sept.16, 20106 4.2. THE DIMENSIONLESS MATHEMATICAL MODEL

7. Cambridge, Sept.16, 20107 5. PERTURBATION THEORY METHOD Perturbation theory comprises mathematical methods to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. Applicability: if the problem can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem. The desired solution will be expressed in terms of a power series in some "small" parameter that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, and further terms describe the deviation in the solution, due to the deviation from the initial problem.

8. Cambridge, Sept.16, 20108 5.1. THE IDEA

9. Cambridge, Sept.16, 20109 5.1. THE IDEA

10. Cambridge, Sept.16, 201010 5.2. THE FIRST APPROXIMATION

11. Cambridge, Sept.16, 201011 5.3. THE FIRST TECHNIQUE

12. Cambridge, Sept.16, 201012 5.3. THE FIRST TECHNIQUE

13. Cambridge, Sept.16, 201013 5.3. THE FIRST TECHNIQUE

14. Cambridge, Sept.16, 201014 5.4. THE SECOND TECHNIQUE

15. Cambridge, Sept.16, 201015 5.4. THE SECOND TECHNIQUE

16. Cambridge, Sept.16, 201016 5.4. THE SECOND TECHNIQUE

17. Cambridge, Sept.16, 201017 5.4. THE SECOND TECHNIQUE

18. Cambridge, Sept.16, 201018 5.4. THE SECOND TECHNIQUE

19. Cambridge, Sept.16, 201019 5.4. THE SECOND TECHNIQUE

20. Cambridge, Sept.16, 201020 5.4. THE SECOND TECHNIQUE

21. Cambridge, Sept.16, 201021 6. THE NUMERICAL APPROACH

22. Cambridge, Sept.16, 201022 6. THE NUMERICAL APPROACH

23. Cambridge, Sept.16, 201023 6. THE NUMERICAL APPROACH

24. Cambridge, Sept.16, 201024 6. THE NUMERICAL APPROACH

25. Cambridge, Sept.16, 201025 6. THE NUMERICAL APPROACH

26. Cambridge, Sept.16, 201026 6. THE NUMERICAL APPROACH

27. Cambridge, Sept.16, 201027 6. THE NUMERICAL APPROACH

28. Cambridge, Sept.16, 201028 7. SOME RESULTS Rigidly attached With Lanchester damper

29. Cambridge, Sept.16, 201029 0246810 0 1 2 3 4 5 6 7 8 R 1 А L EV 1 А, E 1 1 1 А А А  7. SOME RESULTS

30. Cambridge, Sept.16, 201030 7. SOME RESULTS 0,00,20,40,60,81,0 0 1 2 3 4 3 1 A 

31. Cambridge, Sept.16, 201031 7. SOME RESULTS

32. Cambridge, Sept.16, 201032 CONCLUSIONS