Abstract
In a dissipative gyroscopic system with four degrees of freedom and tensorial variables in contravariant (right upper index) and covariant (right lower index) forms, a Lagrangian-dissipative model, i.e., {L, D}-model, is obtained using second-order linear differential equations. The generalized elements are determined using the {L, D}-model of the system. When the prerequisite of a Legendre transform is fulfilled, the Hamiltonian is found. The Lyapunov function is obtained as a residual energy function (REF). The REF consists of the sum of Hamiltonian and losses or dissipative energies (which are negative), and can be used for stability by Lyapunov's second method. Stability conditions are mathematically proven.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ao P, 2004. Potential in stochastic differential equations: novel construction. J PhysA, 37(3):L25–L30. https://doi.org/10.1088/0305-4470/37/3/L01
Arnold VI, 1989. Mathematical methods of classical mechanics (2nd Ed.). Graduate Texts in Mathematics. Springer-Verlag, New York, USA. https://doi.org/10.1007/978-1-4757-2063-1
Barbashin EA, Krasovsky NN, 1952. On the stability of motion as a whole. Doklady Akademii Nauk SSSR, 86(3):453–546 (in Russian).
Chen J, Guo YX, Mei FX, 2018. New methods to find solutions and analyze stability of equilibrium of nonho-lonomic mechanical systems. Acta Mech Sin, 34(6):1136–1144. https://doi.org/10.1007/sl0409-018-0768-x
Chen LQ, Zu JW, Wu J, 2004. Principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string. Acta Mech Sin, 20(3):307–316. https://doi.org/10.1007/BF02486723
Civelek C, 2018. Stability analysis of engineering/physical dynamic systems using residual energy function. Arch Contr Sci, 28(2):201–222. https://doi.org/10.24425/123456
Civelek C, Diemar U, 2003. Stability Analysis Using Energy Functions. Internationales Wissenschaftliches Koloquium, Technische Universitat Ilmenau, Ilmenau (in German).
Hahn W, 1967. Stability of Motion. Springer-Verlag, Berlin, Heidelberg, Germany. https://doi.org/10.1007/978-3-642-50085-5
Heil M, Kitzka F, 1984. Grundkurs Theoretische Mechanik. Springer, Wiesbaden GmbH, Wiesbaden, Germany (in German), https://doi.org/10.1007/978-3-322-96697-1
Huang ZL, Zhu WQ, 2000. Lyapunov exponent and almost sure asymptotic stability of quasi-linear gyroscopic systems. Int J Nonl Mech, 35(4):645–655. https://doi.org/10.1016/S0020-7462(99)00047-5
Krasovskii NN, 1959. Problems of the Theory of Stability of Motion. Stanford University Press, California, USA.
Kwon C, Ao P, Thouless DJ, 2005. Structure of stochastic dynamics near fixed points. PNAS, 102(37):13029–13033. https://doi.org/10.1073/pnas.0506347102
Lasalle JP, 1960. Some extensions of Liapunov's second method. IRE Trans Circ Theory, 7(4):520–527. https://doi.org/10.1109/TCT.1960.1086720
Lyapunov AM, 1992. The general problem of the stability of motion. Int J Contr, 55(3):531–534. https://doi.org/10.1080/00207179208934253
Ma YA, Tan QJ, Yuan RS, et al., 2014. Potential function in a continuous dissipative chaotic system: decomposition scheme and role of strange attractor. Int J Bifurc Chaos, 24(2):1450015. https://doi.org/10.1142/S0218127414500151
Marino R, Nicosia S, 1983. Hamiltonian-type Lyapunov functions. IEEE Trans Autom Contr, 28(11):1055–1057. https://doi.org/10.1109/TAC.1983.1103168
Maschke BMJ, Ortega R, van der Schaft AJ, 2000. Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans Autom Contr, 45(8):1498–1502. https://doi.org/10.1109/9.871758
McLachlan RI, Quispel GRW, Robidoux N, 1998. Unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions or first integrals. Phys Rev Lett, 81(12):2399–2403. hhttps://doi.org/10.1103/PhysRevLett.81.2399
Rouche N, Habets P, Laloy M, 1977. Stability Theory by Liapunov's Direct Method. Springer-Verlag, New York, USA. https://doi.org/10.1007/978-1-4684-9362-7
Susse R, Civelek C, 2003. Analysis of engineering systems by means of Lagrange and Hamilton formalisms depending on contravariant, covariant tensorial variables. Forsch Ingen, 68(1):66–74. https://doi.org/10.1007/sl0010-003-0102-y
Siisse R, Civelek C, 2013. Analysis of coupled dissipative dynamic systems of engineering using extended Hamiltonian H for classical and nonconservative Hamiltonian H for higher order Lagrangian systems. Forsch Ingen, 77(1-2):1–11. https://doi.org/10.1007/sl0010-012-0158-7
Xu W, Yuan B, Ao P, 2011. Construction of Lyapunov function for dissipative gyroscopic system. Chin Phys Lett, 28(5):050201. https://doi.org/10.1088/0256-307X/28/5/050201
Yin L, Ao P, 2006. Existence and construction of dynamical potential in nonequilibrium processes without detailed balance. J Phys A, 39(27):8593–8601. https://doi.org/10.1088/0305-4470/39/27/003
Ying ZG, Zhu WQ, 2000. Exact stationary solutions of stochastically excited and dissipated gyroscopic systems. Int J Nonl Mech, 35(5):837–848. https://doi.org/10.1016/S0020-7462(99)00062-1
Yoshizawa T, 1966. Stability Theory by Liapunov's Second Method. Mathematical Society of Japan, Tokio, Japan.
Yuan RS, Wang XN, Ma YA, et al, 2013. Exploring a noisy van der Pol type oscillator with a stochastic approach. Phys Rev E, 87(6):062109. https://doi.org/10.1103/PhysRevE.87.062109
Yuan RS, Ma YA, Yuan B, et al., 2014. Lyapunov function as potential function: a dynamical equivalence. Chin Phys B, 23(1):010505. https://doi.org/10.1088/1674-1056/23/1/010505
Author information
Authors and Affiliations
Contributions
Cem CIVELEK designed the research and drafted the manuscript. Özge CIHANBEGENDi helped organize the manuscript. Cem CIVELEK revised and finalized the paper.
Corresponding author
Additional information
Compliance with ethics guidelines
Cem CİVELEK and Özge CİHANBEĞENDİ declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Civelek, C., Cihanbeğendi, Ö. Construction of a new Lyapunov function for a dissipative gyroscopic system using the residual energy function. Front Inform Technol Electron Eng 21, 629–634 (2020). https://doi.org/10.1631/FITEE.1900014
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1631/FITEE.1900014