Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Acta Mechanica Sinica
Published in: Acta Mechanica Sinica 3/2020

05-06-2020 | Research Paper

One kind motion of controllable constrained Birkhoffian system: the absence of constraints

Authors: J. Chen, F. X. Mei, S. X. Liu, Y. X. Guo

Published in: Acta Mechanica Sinica | Issue 3/2020

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This paper is devoted to discuss the motion of controllable constrained Birkhoffian system along with its absence of constraints. The first step is to establish the autonomous and non-autonomous differential equations of motion of the system, based on Pfaff–Birkhoff principle. Secondly, the existence of constraint multipliers are exhaustively discussed. Thirdly, the definition of one kind motion of the system, called free motion, is given, which is described and analyzed by the absence of constraints that are determined by constraint multipliers. Lemma 2 illustrates that one system can realize its free motion by selecting proper control parameters. In particular, theorem 2 provides that one system can naturally realize free motion when we consider the integral of the unconstrained Birkhoffian system as the constraints of constrained Birkhoffian system. Finally, the results obtained are illustrated by several examples.
previous article Snap-through of an elastica under bilateral displacement control at a material point
next article 2D numerical model for studying frictional sliding

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
Literature
1.
Birkhoff, G.D.: Dynamical Systems. AMS College Publication, Providence (1927)MATH
2.
Santilli, R.M.: Foundations of Theoretical Mechanics II. Springer, New York (1983)CrossRef
3.
Aleksandrov, P.A., Elenin, G.G.: The possibility of constructing a conservative numerical method for the solution of the cauchy problem for Hamiltonian systems based on two-stage symmetric symplectic Runge–Kutta methods. Math. Model Comput. Simul. 7, 233–245 (2015)MathSciNetCrossRef
4.
Chartier, P., Darrigrand, E., Faou, E.: A regular fast multipole method for geometric numerical integrations of Hamiltonian systems. BIT Numer. Math. 50, 23–40 (2010)MathSciNetCrossRef
5.
D’Ambrosio, R.: Multi-value numerical methods for Hamiltonian systems. Numer. Mathe. Advan. Appl-ENUMATH. 185-193 (2013)
6.
Gkolias, I., Efthymiopoulos, C., Pucacco, G., et al.: Hamiltonian formulation of the spin-orbit model with time-varying non-conservative forces. Commun. Nonlinear. Sci. Numer. Simul. 51, 23–38 (2017)MathSciNetCrossRef
7.
Petrov, A.G., Uvarov, A.V.: A conservative numerical scheme for solving an autonomous Hamiltonian system. Dokl. Phys. 61, 471–475 (2016)CrossRef
8.
Kostoglotov, A.A., Lazarenko, S.V.: Synthesis of adaptive tracking systems based on the hypothesis of stationary of the Hamiltonian on the switching hypersurface. J. Commun. Technol. Electron. 62, 123–127 (2017)CrossRef
9.
Ciraolo, G., Chandre, C., Lima, R., et al.: Control of chaos in Hamiltonian systems. Celest. Mech. Dyn. Astr. 90, 3–12 (2004)MathSciNetCrossRef
10.
Chen, L.Q.: Chaos in perturbed planar non-Hamiltonian integrable systems with slowly-varying angle parameters. Appl. Math. Mech. 22, 1301–1305 (2011)MathSciNetCrossRef
11.
Mei, F.X., Shi, R.C., Zhang, Y.F., et al.: Dynamics of Birkhoffian System. Beijing Institute of Technology Press, Beijing (1996). (in Chinese)
12.
Santilli, R.M.: Foundations of Theoretical Mechanics I, 2nd edn. Springer, New York (1983)CrossRef
13.
Galiullin, A.S.: Analytical Dynamics. Nauka, Moscow (1989). (in Russian) MATH
14.
Galiullin, A.S., Gafarov, G.G., Malaishka, R.P., et al.: Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems. UFN, Moscow (1997). (in Russian)
15.
Guo, Y.X., Luo, S.K., Shang, M., et al.: Birkhoffian formulations of nonholonomic constrained systems. Rep. Math. Phys. 47, 313–322 (2001)MathSciNetCrossRef
16.
17.
Mei, F.X.: Stability of equilibrium for the autonomous Birkhoff system. Chin. Sci. Bull. 38, 816–819 (1993)MATH
18.
Wang, P.: Conformal invariance and conserved quantities of mechanical system with unilateral constraints. Commun. Nonlinear. Sci. Numer. Simul. 59, 463–471 (2018)MathSciNetCrossRef
19.
Luo, S.K.: First integrals and integral invariants of relativistic Birkhoffian systems. Commun. Theor. Phys. 40, 133–136 (2003)MathSciNetCrossRef
20.
Zhang, Y.: Variational problem of Herglotz type for Birkhoffian system and its Noether’s theorems. Acta Mech. 228, 1481–1492 (2017)MathSciNetCrossRef
21.
Waldyr, M.O., Gláucio, T.: Birkhoffian systems in infinite dimensional manifolds. J. Dyn. Diff. Eq. 22, 193–201 (2010)MathSciNetCrossRef
22.
He, L., Wu, H.B., Mei, F.X.: Variational integrators for fractional Birkhoffian systems. Nonlinear Dyn. 87, 2325–2334 (2017)CrossRef
23.
Mei, F.X.: Stability of motion for a constrained Birkhoff’s system in terms of independent variables. Appl. Math. Mech. 18, 55–60 (1997)MathSciNetCrossRef
24.
Chen, X.W.: Closed orbits and limit cycles of second-order autonomous Birkhoff system. Chin. Phys. 12, 586–589 (2003)CrossRef
25.
Kong, X.L., Wu, H.B., Mei, F.X.: Variational discretization for the planar Lotka-Volterra equations in the Birkhoffian sense. Nonlinear Dyn. 84, 733–742 (2016)MathSciNetCrossRef
26.
Zhang, H.B., Chen, H.B.: Generalized variational problems and Birkhoff equations. Nonlinear Dyn. 83, 347–354 (2016)MathSciNetCrossRef
27.
28.
29.
Kirgetov, V.I.: On the equations of motion of controllable mechanical systems. J. Appl. Math. Mech. 28, 232–241 (1964)MathSciNetCrossRef
30.
Rumyantsev, V.V.: On the motion of controllable mechanical systems. J. Appl. Math. Mech. 40, 771–781 (1976)MathSciNet
31.
Mei, F.X.: Jourdain’s principle and Nielsen equations of controllable mechanical systems. J. BIT 2, 1–14 (1983). (in Chinese)
32.
Chen, L.Q.: The generalized Nielsen’s equations for controllable variable mass systems. Acta. Mech. Sol. Sin. 2, 483–490 (1989)
33.
Mei, F.X.: A filed method for integrating the equations of motion of nonholonomic controllable systems. Appl. Math. Mech. 13, 181–187 (1992)CrossRef
34.
Luo, S.K.: Relativistic variation principle and equations of motion for variable mass controllable mechanical systems. Appl. Math. Mech. 17, 683–692 (1996)MathSciNetCrossRef
35.
Xia, L.L., Li, Y.C., Wang, J., et al.: Symmetries and Mei conserved quantities of nonholonomic controllable mechanical systems. Commun. Theor. Phys. 46, 415–418 (2006)MathSciNetCrossRef
36.
Mei, F.X.: The free motion of nonholonomic system and disappearance of the nonholonomic property. Acta. Mech. Sin. 26, 470–476 (1994)
37.
Chen, J., Wu, H.B., Mei, F.X.: The free motion of holonomic system with redundant coordinates. Chin. J. Theor. Appl. Mech. 48, 971–975 (2016)
38.
Zegzhda, S.A., Soltakhanov, Sh.Kh., Yushkov, M.P.: Equations of Motion of Nonholonomic Systems and Variational Principles of Mechanics. A New Class of Control Systems. Moscow: FIZMATLIT (2005). (in Russian)
39.
Mušicki, D., Zeković, D.: Energy integrals for the systems with nonholonomic constraints of arbitrary form and origin. Acta Mech. 227, 467–493 (2016)MathSciNetCrossRef
40.
Block, A.M.: Nonholonomic Mechanics and Control, 2nd edn. Springer, New York (2015)CrossRef
Metadata
Title
One kind motion of controllable constrained Birkhoffian system: the absence of constraints
Authors
J. Chen
F. X. Mei
S. X. Liu
Y. X. Guo
Publication date
05-06-2020
Publisher
The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences
Published in
Acta Mechanica Sinica / Issue 3/2020
Print ISSN: 0567-7718
Electronic ISSN: 1614-3116
DOI
https://doi.org/10.1007/s10409-020-00961-4

Other articles of this Issue 3/2020

Acta Mechanica Sinica 3/2020 Go to the issue

Review Paper

Review on fatigue life prediction models of welded joint

Research Paper

Parameter identification of nonlinear system via a dynamic frequency approach and its energy harvester application

Research Paper

Nonlinear characterization and performance optimization for broadband bistable energy harvester

Research Paper

Improving the off-resonance energy harvesting performance using dynamic magnetic preloading

Research Paper

Survey of the mechanisms of power take-off (PTO) devices of wave energy converters

Research Paper

Snap-through of an elastica under bilateral displacement control at a material point

Premium Partners

    Image Credits