Asymptotic stabilization of a flexible beam with an attached mass

  • J. I. Kalosha Inst. Appl. Math. and Mech. Nat. Acad. Sci. Ukraine, Sloviansk
  • A. L. Zuyev Inst. Appl. Math. and Mech. Nat. Acad. Sci. Ukraine, Sloviansk; Otto von Guericke Univ. Magdeburg, Max Planck Inst. Dynamics of Complex Techn. Systems, Germany
Keywords: Euler--Bernoulli beam; stabilization; asymptotic stability; Lyapunov functional

Abstract

UDC 517.977

A mathematical model of a simply supported Euler – Bernoulli beam with attached spring-mass system is considered. The model is controlled by distributed piezo actuators and a lumped force. We address the issue of asymptotic behavior of solutions of this system driven by a linear feedback law. The precompactness of trajectories is established for the operator formulation of the closed-loop dynamics. Sufficient conditions for strong asymptotic stability of the trivial equilibrium are obtained.

References

J.-M. Coron, Control and nonlinearity, Amer. Math. Soc. (2007), https://doi.org/10.1090/surv/136 DOI: https://doi.org/10.1090/surv/136

R. Curtain, H. Zwart, Introduction to infinite-dimensional systems theory, Springer-Verlag, New York (2020), https://doi.org/10.1007/978-1-0716-0590-5 DOI: https://doi.org/10.1007/978-1-0716-0590-5_1

R. D´ager, E. Zuazua, Wave propagation, observation and control in 1-d flexible multi-structures, Springer-Verlag, Berlin, Heidelberg (2006), https://doi.org/10.1007/3-540-37726-3 DOI: https://doi.org/10.1007/3-540-37726-3

C. Dullinger, A. Schirrer, M. Kozek, Advanced control education: optimal & robust MIMO control of a flexible beam setup, IFAC Proc., vol., 47(3), 9019 – 9025 (2014). DOI: https://doi.org/10.3182/20140824-6-ZA-1003.02201

J. Kalosha, A. Zuyev, P. Benner, On the eigenvalue distribution for a beam with attached masses, Stabilization of Distributed Parameter Systems: Design Methods and Applications, Springer Intern. Publ., (2021), p. 43 – 56. DOI: https://doi.org/10.1007/978-3-030-61742-4_3

V. Komkov, Optimal control theory for thin plates, Springer, Berlin, Heidelberg (1972). DOI: https://doi.org/10.1007/BFb0058909

V. Komornik, P. Loreti, Fourier series in control theory, Springer-Verlag, New York (2005). DOI: https://doi.org/10.1007/b139040

W. Krabs, On moment theory and controllability of one-dimensional vibrating systems and heating processes, Springer-Verlag, Berlin, Heidelberg (1992), https://doi.org/10.1007/BFb0039513 DOI: https://doi.org/10.1007/BFb0039513

A. Lamei, M. Hayatdavoodi, On motion analysis and elastic response of floating offshore wind turbines, J. Ocean Engineering and Marine Energy, 6, № 1, 71 – 90 (2020), y (2020) 6:71–90

https://doi.org/10.1007/s40722-019-00159-2 DOI: https://doi.org/10.1007/s40722-019-00159-2

J. P. LaSalle, Stability theory and invariance principles, Dynamical systems, Acad. Press , p. 211 – 222 (1976)

Y. Le Gorrec, H. Zwart, H. Ramirez, Asymptotic stability of an Euler – Bernoulli beam coupled to nonlinear springdamper systems, IFAC-PapersOnLine, 50(1), 5580 – 5585 (2017), https://doi.org/10.1016/j.ifacol.2017.08.1102 DOI: https://doi.org/10.1016/j.ifacol.2017.08.1102

M. Liao, G. Wang, Z. Gao, Y. Zhao, R. Li, Mathematical modelling and dynamic analysis of an offshore drilling riser, Shock and Vibration, 2020 (2020), | https://doi.org/10.1155/2020/8834011 DOI: https://doi.org/10.1155/2020/8834011

G. Lumer, R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math., 11, № 2, 679 – 698 (1961). DOI: https://doi.org/10.2140/pjm.1961.11.679

Z.-H. Luo, B.-Z. Guo, O¨ . Morgu¨l, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag, London (1999), https://doi.org/10.1007/978-1-4471-0419-3 DOI: https://doi.org/10.1007/978-1-4471-0419-3_6

L. U. Odhner, A. M. Dollar, The smooth curvature model: an efficient representation of Euler – Bernoulli flexures as robot joints, IEEE Trans. Robotics, 28, № 4, 761 – 772 (2012). DOI: https://doi.org/10.1109/TRO.2012.2193232

J. Oostveen, Strongly stabilizable distributed parameter systems, SIAM (2000), https://doi.org/10.1137/1.9780898719864 DOI: https://doi.org/10.1137/1.9780898719864

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York (1983), https://doi.org/10.1007/978-1-4612-5561-1 DOI: https://doi.org/10.1007/978-1-4612-5561-1

D. L. Russell, Nonharmonic Fourier series in the control theory of distributed parameter systems, J. Math. Anal. and Appl., 18, № 3, 542 – 560 (1967), https://doi.org/10.1016/0022-247X(67)90045-5 DOI: https://doi.org/10.1016/0022-247X(67)90045-5

M. A. Shubov, L. P. Kindrat, Spectral analysis of the Euler – Bernoulli beam model with fully nonconservative feedback matrix, Mathematical Methods in the Applied Sciences, 41, № 12, 4691 – 4713 (2018), https://doi.org/10.1002/mma.4922 DOI: https://doi.org/10.1002/mma.4922

M. A. Shubov„ L. P. Kindrat, Asymptotics of the eigenmodes and stability of an elastic structure with general feedback matrix, IMA J. Appl. Math., 84, № 5, 873 – 911 (2019), https://doi.org/10.1093/imamat/hxz019 DOI: https://doi.org/10.1093/imamat/hxz019

M. Shubov, V. Shubov, Stability of a flexible structure with destabilizing boundary conditions, Proc. Roy. Soc. Math. Phis. and Eng Sci., 472 (2016), https://doi.org/10.1098/rspa.2016.0109 DOI: https://doi.org/10.1098/rspa.2016.0109

G. Sklyar, A. Zuyev, Stabilization of distributed parameter systems: design methods and applications, Springer Intern. Publ. (2021). DOI: https://doi.org/10.1007/978-3-030-61742-4

V. A. Trenogin, The functional analysis, Nauka, Moscow (1980).

A. Walsh, J. R. Forbes, Modeling and control of flexible telescoping manipulators, IEEE Trans. Robotics, 31, № 4, 936 – 947 (2015). DOI: https://doi.org/10.1109/TRO.2015.2441473

A. L. Zuev, Partial asymptotic stability of abstract differential equations, Ukr. Math. J., 58, № 5, 709 – 717 (2006), https://doi.org/10.1007/s11253-006-0096-3 DOI: https://doi.org/10.1007/s11253-006-0096-3

A. L. Zuyev, J. I. Kucher, Stabilization of a flexible beam model with distributed and lumped controls (in Russian),Dynamical Systems, 3(31), № 1-2, 25 – 35 (2013).

A. Zuyev, O. Sawodny, Stabilization of a flexible manipulator model with passive joints, IFAC Proc. Vol., 38(1), 784 – 789 (2005), https://doi.org/10.1155/2007/57238 DOI: https://doi.org/10.3182/20050703-6-CZ-1902.00531

A. Zuyev, O. Sawodny, Stabilization and observability of a rotating Timoshenko beam model, Math. Probl. Eng., 2007, 1 – 19 (2007), https://doi.org/10.1155/2007/57238 DOI: https://doi.org/10.1155/2007/57238

Published
11.10.2021
How to Cite
Kalosha, J. I., and A. L. Zuyev. “Asymptotic Stabilization of a Flexible Beam With an Attached Mass”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 10, Oct. 2021, pp. 1330-41, doi:10.37863/umzh.v73i10.6750.
Section
Research articles