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UDK 534.112

DOI: 10.15507/0236-2910.028.201804.472-485

The Problem of Damping the Transverse Oscillations on a Longitudinally Moving String

Leonid A. Muravey
Professor, Chair of Computer Mathematics, Moscow Aviation Institute (National Research University) (3 Orshanskaya St., Moscow 121552, Russia), D.Sc. (Physics and Mathematics), ResearcherID: U-2857-2018, ORCID: https://orcid.org/0000-0001-5432-8843, This email address is being protected from spambots. You need JavaScript enabled to view it.

Victor M. Petrov
Associate Professor, Chair of Computer Mathematics, Moscow Aviation Institute (National Research University) (3 Orshanskaya St., Moscow 121552, Russia), Ph.D. (Physics and Mathematics), ResearcherID: U-2845-2018, ORCID: https://orcid.org/0000-0003-3382-7496, This email address is being protected from spambots. You need JavaScript enabled to view it.

Alexandr M. Romanenkov
Associate Professor, Chair of Computer Mathematics, Moscow Aviation Institute (National Research University) (3 Orshanskaya St., Moscow 121552, Russia), Ph.D. (Engineering), ResearcherID: T-3538-2018, ORCID: https://orcid.org/0000-0002-0700-8465, This email address is being protected from spambots. You need JavaScript enabled to view it.

Introduction. The problem under consideration is relevant to production processes associated with the longitudinal movement of materials, for example, for producing paper webs. For these processes transverse disturbances, which in the vertical section are described by the hyperbolic equation of a longitudinally moving string, are extremely undesirable. That gives the problem of damping these oscillations within a finite time.
Materials and Methods. To solve the problem of damping the oscillations, the authors suggest reducing it to the trigonometric problem of the moments at an arbitrary time interval. When considering moving materials, the construction of the basis systems forming the moment problem is a special challenge, since the hyperbolic equation contains a mixed derivative (Coriolis acceleration). Therefore, the classical method of separating variables is not applicable in this case. Instead, a new method is used to find self-similar solutions of non-stationary equations, which makes it possible to find the basis systems explicitly.
Results. In the case of paper web, it is necessary to find a minimal in the whole class of admissible perturbations time interval, within which the trigonometric system forming the problem of moments is the Riesz basis, that make it possible through using the system conjugate with it to find the optimal control way in the form of a series and, therefore, to build a so-called optimal damper.
Conclusions. As a result of the study, a generalized solution of the problem of transverse oscillations is constructed. For the problem of damping oscillations, the exact damping time is obtained, namely, a time T₀ at which the total energy of the system is zero. Optimum control is found in the form of a Fourier series.

Keywords: damping oscillations, hyperbolic equation, Coriolis acceleration, trigonometric moment problem, Riesz base

For citation: Muravey L. A., Petrov V. M., Romanenkov A. M. The Problem of Damping the Transverse Oscillations on a Longitudinally Moving String. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2018; 28(4):472–485. DOI: https://doi.org/10.15507/0236-2910.028.201804.472-485

Acknowledgements: The work was supported by grant No. 16-01-00425 A from the Russian Foundation for Basic Research.

Contribution of the authors: L. A. Muravey – formulation of the problem, method development, analysis of the methodological base the research, literature review, proof of Riesz basicity for a special type of trigonometric system; V. M. Petrov – obtaining the speed limit in a moving material, posing an extremal problem, and deriving the representation of the minimized functional; A. M. Romanenkov – obtaining an estimate for optimal control.

All authors have read and approved the final version of the paper.

Received 19.06.2018; revised 15.08.2018; published online 28.12.2018

REFERENCES

1. Lagness J. Control of wave process with distributed controls supported on a subregion. SIAM Journal of Control and Optimization. 1983; 21(1):68–85.

2. Archibald F. R., Emslie A. G. The vibration of a string having a uniform motion along its length. ASME Journal of Applied Mechanics. 1958; 25(1):347–348.

3. Lions J. L. Exact Controllability, Stabilization and Perturbations for Distributed Systems. SIAM Review. 1983; 30(1):1–68.

4. Levinson N. Gap and density theorem. Colloquium Publications. 1940; 26. 246 p.

5. Mahalingam S. Transverse vibrations of power transmissions chains. British Journal of Applied Physics. 1957; 8(4):145–148.

6. Sack R. A. Transverse oscillations in traveling strings. British Journal of Applied Physics. 1954; 5(6):224–226.

7. Banichuk N., Jeronen J., Neittaanmäki P., Saksa T., Tuovinen T.Mechanics of Moving Materials. Switzerland: Springer; 2014. 207 p.

8. Bilalov B. T. On the basis property of the system {e^{iα nx}sin(nx)} of exponentials with a shift. Doklady Akademii nauk = Reports of the Academy of Sciences. 1995; 345(2):644–647. (In Russ.)

9. Russel D. L. Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Review. 1978; 20(4):639–739.

10. Butkovsky A. G. [Methods of Control of Systems with Distributed Parameters]. Moscow: Nauka Publ.; 1975. 568 p. (In Russ.)

11. Aslanov S. Zh., Mikhailov I. E., Muravey L. A. [Analytical and numerical methods in the problem of string oscillation damping by a point damper]. Mekhatronika, avtomatizatsiya, upravlenie = Mechatronics, Automation, Control. 2006; 7:28–35. (In Russ.)

12. Aslanov S. Zh., Mikhailov I. E., Muravey L. A. [On damping oscillations of a circular membrane using a ring damper]. Trudy ISA RAN = Proceeding of the Institute for Systems Analysis of the Russian Academy of Science. 2007; 29(1)11:54–59 (In Russ.)

13. Mikhailov V. P. [Partial Differential Equations]. Moscow: Nauka Publ.; 1983. 424 p. (In Russ.)

14. Atamuratov A. Zh., Mikhailov I. Ye., Muravey L. A. The problem of moments in problems of controlling elastic dynamical systems. Mekhatronika, avtomatizatsiya, upravlenie = Mechatronics, Automation, Control. 2016; 9(17):587–598. (In Russ.)

15. Muravey L. A., Romanenkov A. M., Petrov V. M. Optimal Control of Nonlinear Processes in Problems of Mathematical Physics. Moscow: MAI Publ.; 2018. 159 p. (In Russ.).

16. Il’in V. A., Moiseev E. I. Optimization of the boundary control by shift or elastic force at one end of string in a sufficiently long arbitrary time. Automation and Remote Control. 2008; 69(3):354–362.

17. Il’in V. A., Moiseev E. I. Boundary control of string vibrations that minimizes the integral of power p ≥ 1 of the module of control or its derivative. Automation and Remote Control. 2007; 68(2):313–319.

18. Il’in V. A., Moiseev E. I. Optimization of boundary controls of string vibrations. Russian Mathematical Surveys. 2005; 60(6):1093–1119.

19. Malookani R., Van Horssen W. T. On the vibrations of the moving string with a time-dependent velocity. ASME 2015 International Mechanical Engineering Congress and Exposition. 2015; V04BT04A061.

20. Muravey L. A. The problem boundary control for elliptic equation. Vesntik Moskovskogo universiteta. Ser. 15: Vychislitelnaya matematika i kibernetika = MSU Bulletin. Series 15: Computational Mathematics and Cybernetics. 1998; 3:7–13. (In Russ.)

21. Gurchenkov A. A., Muravey L. A., Romanenkov A. M. Modeling and optimization of the technological process of ion-beam etching. Inzhenernyy zhurnal: nauka i innovatsii = Engineering Journal: Science and Innovation. 2014; 2(26). (In Russ.)

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