June  2014, 3(2): 305-329. doi: 10.3934/eect.2014.3.305

Modeling and control of hybrid beam systems with rotating tip component

1. 

Department of Mathematics, Virginia Tech, 460 McBryde Hall, MS 0123, Blacksburg, VA 24061-0123, United States

Received  December 2013 Revised  March 2014 Published  May 2014

We study control and stability for two types of hybrid elastic structures consisting of distributed parameter, beam and rod type, elements coupled at one end to a rotating lumped mass. Applications to control of structural vibrations in wind energy units are indicated but not treated explicitly.
Citation: David L. Russell. Modeling and control of hybrid beam systems with rotating tip component. Evolution Equations & Control Theory, 2014, 3 (2) : 305-329. doi: 10.3934/eect.2014.3.305
References:
[1]

M. D. Aouragh and N. Yebari, Riesz basis approach and exponential stabilization of a nonhomogeneous flexible beam with a tip mass,, Int. J. Math. & Stat., 7 (2010), 46. Google Scholar

[2]

S. Avdonin and S. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems,, Cambridge University Press, (1995). Google Scholar

[3]

M. Azam, S. N. Singh, A. Iyer and Y. P. Kakad, Detumbling and reorientation maneuvers and stabilization of NASA SCOLE system,, IEEE Trans. Aerosp. & Electr. Syst., 28 (1992), 80. doi: 10.1109/7.135434. Google Scholar

[4]

C. Baiocchi, V. Komornik and P. Loreti, Théorèmes du type Ingham et application à la théorie du contrôle,, C. R. Acad. Sci. Paris Sér. I, 326 (1998), 453. doi: 10.1016/S0764-4442(97)89791-1. Google Scholar

[5]

M. L. Boas, Mathematical Methods in the Physical Sciences,, Wiley, (2006). Google Scholar

[6]

W. E. Boyce and G. H. Handelman, Vibrations of rotating beams with tip mass,, Angew. Math. & Phys., 12 (1961), 369. doi: 10.1007/BF01600687. Google Scholar

[7]

F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass,, SIAM J. Control. & Opt., 36 (1998), 1962. doi: 10.1137/S0363012996302366. Google Scholar

[8]

M. Grobbelaar - Van Dalsen, Uniform stability for the Timoshenko beam with tip load,, J. Math. Anal. & and Appl., 361 (2010), 392. doi: 10.1016/j.jmaa.2009.06.059. Google Scholar

[9]

B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass,, SIAM J. Control. & Opt., 39 (2001), 1736. doi: 10.1137/S0363012999354880. Google Scholar

[10]

J. Humar and M. Ruban, Dynamics of Structures,, CRC Press, (2002). Google Scholar

[11]

A. E. Ingham, Some trigonometric inequalities in the theory of series,, Math. Zeitschr., 41 (1936), 367. doi: 10.1007/BF01180426. Google Scholar

[12]

R. N. Jazar, Advanced Dynamics: Rigid Body, Multibody and Aerospace Applications,, John Wiley & Sons, (2011). doi: 10.1002/9780470950029. Google Scholar

[13]

J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, with Applications,, Mathematics in Science and Engineering, (1961). Google Scholar

[14]

W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping,, Ann. Mat. Pura & Appl., 152 (1988), 281. doi: 10.1007/BF01766154. Google Scholar

[15]

W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity,, Arch. Rat. Mech. & Anal., 103 (1988), 193. doi: 10.1007/BF00251758. Google Scholar

[16]

Ö. Morgül, B. P. Rao and F. Conrad, On the stabilization of a cable with a tip mass,, IEEE Trans. Automat. Control, 39 (1994), 2140. doi: 10.1109/9.328811. Google Scholar

[17]

B. P. Rao, Uniform stabilization of a hybrid system of elasticity,, SIAM J. Control. & Opt., 33 (1995), 440. doi: 10.1137/S0363012992239879. Google Scholar

[18]

D. L. Russell, Nonharmonic Fourier Series in the Control Theory of Distributed Parameter Systems,, J. Math. Anal. & Appl., 18 (1967), 542. doi: 10.1016/0022-247X(67)90045-5. Google Scholar

[19]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Review, 20 (1978), 639. doi: 10.1137/1020095. Google Scholar

[20]

D. L. Russell, Control via decoupling of a class of second order linear hybrid systems,, to appear in Disc. & Cont. Dyn. Syst., (2014). Google Scholar

[21]

E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. With an Introduction to the Problem of Three Bodies,, Reprint of the 1937 edition, (1937). doi: 10.1017/CBO9780511608797. Google Scholar

[22]

N. Yebari and M. D. Aouragh, Uniform stabilization of a hybrid system of elasticity with variable coefficients,, Int. J. Tomogr. & Stat., 10 (2008), 125. Google Scholar

show all references

References:
[1]

M. D. Aouragh and N. Yebari, Riesz basis approach and exponential stabilization of a nonhomogeneous flexible beam with a tip mass,, Int. J. Math. & Stat., 7 (2010), 46. Google Scholar

[2]

S. Avdonin and S. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems,, Cambridge University Press, (1995). Google Scholar

[3]

M. Azam, S. N. Singh, A. Iyer and Y. P. Kakad, Detumbling and reorientation maneuvers and stabilization of NASA SCOLE system,, IEEE Trans. Aerosp. & Electr. Syst., 28 (1992), 80. doi: 10.1109/7.135434. Google Scholar

[4]

C. Baiocchi, V. Komornik and P. Loreti, Théorèmes du type Ingham et application à la théorie du contrôle,, C. R. Acad. Sci. Paris Sér. I, 326 (1998), 453. doi: 10.1016/S0764-4442(97)89791-1. Google Scholar

[5]

M. L. Boas, Mathematical Methods in the Physical Sciences,, Wiley, (2006). Google Scholar

[6]

W. E. Boyce and G. H. Handelman, Vibrations of rotating beams with tip mass,, Angew. Math. & Phys., 12 (1961), 369. doi: 10.1007/BF01600687. Google Scholar

[7]

F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass,, SIAM J. Control. & Opt., 36 (1998), 1962. doi: 10.1137/S0363012996302366. Google Scholar

[8]

M. Grobbelaar - Van Dalsen, Uniform stability for the Timoshenko beam with tip load,, J. Math. Anal. & and Appl., 361 (2010), 392. doi: 10.1016/j.jmaa.2009.06.059. Google Scholar

[9]

B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass,, SIAM J. Control. & Opt., 39 (2001), 1736. doi: 10.1137/S0363012999354880. Google Scholar

[10]

J. Humar and M. Ruban, Dynamics of Structures,, CRC Press, (2002). Google Scholar

[11]

A. E. Ingham, Some trigonometric inequalities in the theory of series,, Math. Zeitschr., 41 (1936), 367. doi: 10.1007/BF01180426. Google Scholar

[12]

R. N. Jazar, Advanced Dynamics: Rigid Body, Multibody and Aerospace Applications,, John Wiley & Sons, (2011). doi: 10.1002/9780470950029. Google Scholar

[13]

J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, with Applications,, Mathematics in Science and Engineering, (1961). Google Scholar

[14]

W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping,, Ann. Mat. Pura & Appl., 152 (1988), 281. doi: 10.1007/BF01766154. Google Scholar

[15]

W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity,, Arch. Rat. Mech. & Anal., 103 (1988), 193. doi: 10.1007/BF00251758. Google Scholar

[16]

Ö. Morgül, B. P. Rao and F. Conrad, On the stabilization of a cable with a tip mass,, IEEE Trans. Automat. Control, 39 (1994), 2140. doi: 10.1109/9.328811. Google Scholar

[17]

B. P. Rao, Uniform stabilization of a hybrid system of elasticity,, SIAM J. Control. & Opt., 33 (1995), 440. doi: 10.1137/S0363012992239879. Google Scholar

[18]

D. L. Russell, Nonharmonic Fourier Series in the Control Theory of Distributed Parameter Systems,, J. Math. Anal. & Appl., 18 (1967), 542. doi: 10.1016/0022-247X(67)90045-5. Google Scholar

[19]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Review, 20 (1978), 639. doi: 10.1137/1020095. Google Scholar

[20]

D. L. Russell, Control via decoupling of a class of second order linear hybrid systems,, to appear in Disc. & Cont. Dyn. Syst., (2014). Google Scholar

[21]

E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. With an Introduction to the Problem of Three Bodies,, Reprint of the 1937 edition, (1937). doi: 10.1017/CBO9780511608797. Google Scholar

[22]

N. Yebari and M. D. Aouragh, Uniform stabilization of a hybrid system of elasticity with variable coefficients,, Int. J. Tomogr. & Stat., 10 (2008), 125. Google Scholar

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