Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle

Kamilla Maratovna Fazlyeva; Timofei Sergeevich Shugailo; Mikhail Petrovich Yushkov

doi:doi:10.11648/j.ijmea.20170503.11

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Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle

Kamilla Maratovna Fazlyeva, Timofei Sergeevich Shugailo, Mikhail Petrovich Yushkov

Published in International Journal of Mechanical Engineering and Applications (Volume 5, Issue 3)

Received: 10 May 2017 Published: 10 May 2017

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Abstract

The paper studies the suppression of oscillation of a certain two-mass system when it is transferred from the initial state of rest to the given state of rest during a time interval prescribed. The problem is solved by the two methods: the Pontryagin maximum principle (first method) and the generalized Gauss principle (second method). Computational results are presented and the solutions are compared to each other. When the time of motion is short the both methods give practically the same results, but when the time of motion is long the results differ widely. If the time of motion is long then the second method is more preferable than the first one, since the control obtained by the second method sways the mechanical system less than the control obtained by the classical approach. This can be explained by the fact that the first method contains the control including harmonics with the natural frequency of the system, and this seeks to put the system into resonance. In contrast to this, in the second method the control is sought in the form of time polynomial that provides relatively smooth motion of the system. It is noted that the first method always finds the control with jumps at the beginning and at the end of motion. The second method also gives the same jumps when the time of motion is short, but when the time of motion is long the similar jumps vanish when one uses the generalized Gauss principle.

Published in	International Journal of Mechanical Engineering and Applications (Volume 5, Issue 3)
DOI	10.11648/j.ijmea.20170503.11
Page(s)	129-135
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Pontryagin Maximum Principle, Generalized Gauss Principle, Control Force, Suppression of Oscillation

References

[1]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko. The mathematical theory of optimal processes. Translated from Russian by K. N. Trirogoff; edited by L. W. Neustadt, Interscience Publishers John Wiley and Sons, Inc. NewYork–London, 1962. 360 p.
[2]	R. Bellman. Dynamic programming. M.: Izd-vo inostr. lit. 1960. 400 p. (in Russian).
[3]	N. N. Moiseev, Yu. P. Ivanilov, E. M. Stolyarova. Optimization methods. M.: Nauka. 1978. 352 p. (in Russian).
[4]	F. L. Chernous’ko, N. V. Banichuk. Variational problems in mechanics and control. Numerical methods. M.: Nauka. 1973. 238 p. (in Russian).
[5]	N. N. Polyakhov, S. A. Zegzhda, M. P. Yushkov. Generalization of the Gauss principle to the case of non-holonomic high-order systems // Doklady AN SSSR. 1983. V. 269. № 6. Pp. 1328-1330 (in Russian).
[6]	Sh. Kh. Soltakhanov, M. P. Yushkov, S. A. Zegzhda. Holonomic and non-holonomic constraint forces of any order // Proceedings of the Second International Conference on dynamics, vibration and control. August 23-26, 2006, Beijing, China. P.172.
[7]	Sh. Kh. Soltakhanov, M. P. Yushkov, S. A. Zegzhda. Application of the generalized Gaussian principle to the problem of suppression of mechanical systems oscillations // Proceedings of The Third International Conference on Dynamics, Vibration and Control (Hangzhou, China), 2010. CD: File 345.pdf.
[8]	S. A. Zegzhda, P. E. Tovstik, M. P. Yushkov. The Hamilton-Ostrogradski Generalized Principle and Its Application for Damping of Oscillations // ISSN 1028-3358, Doklady Physics, 2012, Vol. 57, No. 11, pp. 447-450. © Pleiades Publishing, Ltd., 2012.
[9]	S. A. Zegzhda, E. A. Shatrov, M. P. Yushkov. A new approach to finding the control that transports a system from one phase state to another // ISSN 1063-4541. Vestnik St. Petersburg University. Mathematics, 2016, Vol. 49, No. 2, pp. 183-190. © Allerton Press, Inc., 2016.
[10]	S. A. Zegzhda, E. A. Shatrov, M. P. Yushkov. Suppression of oscillation of a trolley with a double pendulum by means of control of its acceleration // Vestnik St. Petersburg University. Ser.1. 2016. Vyp.4. Pp.683-688 (in Russian).
[11]	S. A. Zegzhda, Sh. Kh. Soltakhanov. Application of a generalized Gauss principle to solving the problem of suppression of oscillation of mechanical systems // Izv. RAN. Theory and control systems. 2010. №2. Pp.20-25. (in Russian).
[12]	F. L. Chernousko, D. Akulenko, B. N. Sokolov. Control of oscillations. Nauka, Moscow, 1980. 384 р. (in Russian).
[13]	N. N. Polyakhov, S. A. Zegzhda, M. P. Yushkov. Theoretical Mechanics. M.: Vysshaya shkola, 2000. 592 p. (in Russian).
[14]	Sh. Kh. Soltakhanov, S. A. Zegzhda, M. P. Yushkov. Mechanics of non-holonomic systems. A New Class of control systems. Berlin Heidelberg: Springer-Verlag. 2009. 329 p.
[15]	B. A. Dubrovin, C. P. Novikov, A. T. Fomenko. Contemporary geometry. M.: Nauka.1979. 760 p. (in Russian).

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APA Style

Kamilla Maratovna Fazlyeva, Timofei Sergeevich Shugailo, Mikhail Petrovich Yushkov. (2017). Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle. International Journal of Mechanical Engineering and Applications, 5(3), 129-135. https://doi.org/10.11648/j.ijmea.20170503.11

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ACS Style

Kamilla Maratovna Fazlyeva; Timofei Sergeevich Shugailo; Mikhail Petrovich Yushkov. Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle. Int. J. Mech. Eng. Appl. 2017, 5(3), 129-135. doi: 10.11648/j.ijmea.20170503.11

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AMA Style

Kamilla Maratovna Fazlyeva, Timofei Sergeevich Shugailo, Mikhail Petrovich Yushkov. Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle. Int J Mech Eng Appl. 2017;5(3):129-135. doi: 10.11648/j.ijmea.20170503.11

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@article{10.11648/j.ijmea.20170503.11,
  author = {Kamilla Maratovna Fazlyeva and Timofei Sergeevich Shugailo and Mikhail Petrovich Yushkov},
  title = {Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle},
  journal = {International Journal of Mechanical Engineering and Applications},
  volume = {5},
  number = {3},
  pages = {129-135},
  doi = {10.11648/j.ijmea.20170503.11},
  url = {https://doi.org/10.11648/j.ijmea.20170503.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmea.20170503.11},
  abstract = {The paper studies the suppression of oscillation of a certain two-mass system when it is transferred from the initial state of rest to the given state of rest during a time interval prescribed. The problem is solved by the two methods: the Pontryagin maximum principle (first method) and the generalized Gauss principle (second method). Computational results are presented and the solutions are compared to each other. When the time of motion is short the both methods give practically the same results, but when the time of motion is long the results differ widely. If the time of motion is long then the second method is more preferable than the first one, since the control obtained by the second method sways the mechanical system less than the control obtained by the classical approach. This can be explained by the fact that the first method contains the control including harmonics with the natural frequency of the system, and this seeks to put the system into resonance. In contrast to this, in the second method the control is sought in the form of time polynomial that provides relatively smooth motion of the system. It is noted that the first method always finds the control with jumps at the beginning and at the end of motion. The second method also gives the same jumps when the time of motion is short, but when the time of motion is long the similar jumps vanish when one uses the generalized Gauss principle.},
 year = {2017}
}

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TY  - JOUR
T1  - Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle
AU  - Kamilla Maratovna Fazlyeva
AU  - Timofei Sergeevich Shugailo
AU  - Mikhail Petrovich Yushkov
Y1  - 2017/05/10
PY  - 2017
N1  - https://doi.org/10.11648/j.ijmea.20170503.11
DO  - 10.11648/j.ijmea.20170503.11
T2  - International Journal of Mechanical Engineering and Applications
JF  - International Journal of Mechanical Engineering and Applications
JO  - International Journal of Mechanical Engineering and Applications
SP  - 129
EP  - 135
PB  - Science Publishing Group
SN  - 2330-0248
UR  - https://doi.org/10.11648/j.ijmea.20170503.11
AB  - The paper studies the suppression of oscillation of a certain two-mass system when it is transferred from the initial state of rest to the given state of rest during a time interval prescribed. The problem is solved by the two methods: the Pontryagin maximum principle (first method) and the generalized Gauss principle (second method). Computational results are presented and the solutions are compared to each other. When the time of motion is short the both methods give practically the same results, but when the time of motion is long the results differ widely. If the time of motion is long then the second method is more preferable than the first one, since the control obtained by the second method sways the mechanical system less than the control obtained by the classical approach. This can be explained by the fact that the first method contains the control including harmonics with the natural frequency of the system, and this seeks to put the system into resonance. In contrast to this, in the second method the control is sought in the form of time polynomial that provides relatively smooth motion of the system. It is noted that the first method always finds the control with jumps at the beginning and at the end of motion. The second method also gives the same jumps when the time of motion is short, but when the time of motion is long the similar jumps vanish when one uses the generalized Gauss principle.
VL  - 5
IS  - 3
ER  -

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Author Information

Kamilla Maratovna Fazlyeva

Department of Theoretical and Applied Mechanics, St. Petersburg University, St. Petersburg, Russia
Timofei Sergeevich Shugailo

Department of Theoretical and Applied Mechanics, St. Petersburg University, St. Petersburg, Russia
Mikhail Petrovich Yushkov

Department of Theoretical and Applied Mechanics, St. Petersburg University, St. Petersburg, Russia

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APA Style

Kamilla Maratovna Fazlyeva, Timofei Sergeevich Shugailo, Mikhail Petrovich Yushkov. (2017). Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle. International Journal of Mechanical Engineering and Applications, 5(3), 129-135. https://doi.org/10.11648/j.ijmea.20170503.11

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ACS Style

Kamilla Maratovna Fazlyeva; Timofei Sergeevich Shugailo; Mikhail Petrovich Yushkov. Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle. Int. J. Mech. Eng. Appl. 2017, 5(3), 129-135. doi: 10.11648/j.ijmea.20170503.11

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AMA Style

Kamilla Maratovna Fazlyeva, Timofei Sergeevich Shugailo, Mikhail Petrovich Yushkov. Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle. Int J Mech Eng Appl. 2017;5(3):129-135. doi: 10.11648/j.ijmea.20170503.11

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@article{10.11648/j.ijmea.20170503.11,
  author = {Kamilla Maratovna Fazlyeva and Timofei Sergeevich Shugailo and Mikhail Petrovich Yushkov},
  title = {Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle},
  journal = {International Journal of Mechanical Engineering and Applications},
  volume = {5},
  number = {3},
  pages = {129-135},
  doi = {10.11648/j.ijmea.20170503.11},
  url = {https://doi.org/10.11648/j.ijmea.20170503.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmea.20170503.11},
  abstract = {The paper studies the suppression of oscillation of a certain two-mass system when it is transferred from the initial state of rest to the given state of rest during a time interval prescribed. The problem is solved by the two methods: the Pontryagin maximum principle (first method) and the generalized Gauss principle (second method). Computational results are presented and the solutions are compared to each other. When the time of motion is short the both methods give practically the same results, but when the time of motion is long the results differ widely. If the time of motion is long then the second method is more preferable than the first one, since the control obtained by the second method sways the mechanical system less than the control obtained by the classical approach. This can be explained by the fact that the first method contains the control including harmonics with the natural frequency of the system, and this seeks to put the system into resonance. In contrast to this, in the second method the control is sought in the form of time polynomial that provides relatively smooth motion of the system. It is noted that the first method always finds the control with jumps at the beginning and at the end of motion. The second method also gives the same jumps when the time of motion is short, but when the time of motion is long the similar jumps vanish when one uses the generalized Gauss principle.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle
AU  - Kamilla Maratovna Fazlyeva
AU  - Timofei Sergeevich Shugailo
AU  - Mikhail Petrovich Yushkov
Y1  - 2017/05/10
PY  - 2017
N1  - https://doi.org/10.11648/j.ijmea.20170503.11
DO  - 10.11648/j.ijmea.20170503.11
T2  - International Journal of Mechanical Engineering and Applications
JF  - International Journal of Mechanical Engineering and Applications
JO  - International Journal of Mechanical Engineering and Applications
SP  - 129
EP  - 135
PB  - Science Publishing Group
SN  - 2330-0248
UR  - https://doi.org/10.11648/j.ijmea.20170503.11
AB  - The paper studies the suppression of oscillation of a certain two-mass system when it is transferred from the initial state of rest to the given state of rest during a time interval prescribed. The problem is solved by the two methods: the Pontryagin maximum principle (first method) and the generalized Gauss principle (second method). Computational results are presented and the solutions are compared to each other. When the time of motion is short the both methods give practically the same results, but when the time of motion is long the results differ widely. If the time of motion is long then the second method is more preferable than the first one, since the control obtained by the second method sways the mechanical system less than the control obtained by the classical approach. This can be explained by the fact that the first method contains the control including harmonics with the natural frequency of the system, and this seeks to put the system into resonance. In contrast to this, in the second method the control is sought in the form of time polynomial that provides relatively smooth motion of the system. It is noted that the first method always finds the control with jumps at the beginning and at the end of motion. The second method also gives the same jumps when the time of motion is short, but when the time of motion is long the similar jumps vanish when one uses the generalized Gauss principle.
VL  - 5
IS  - 3
ER  -

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