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 and 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), 46-53.

[2]

S. Avdonin and S. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, New York, Melbourne, 1995.

[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), 80-91. doi: 10.1109/7.135434.

[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, Math., 326 (1998), 453-458. doi: 10.1016/S0764-4442(97)89791-1.

[5]

M. L. Boas, Mathematical Methods in the Physical Sciences, Wiley, New York, 2006.

[6]

W. E. Boyce and G. H. Handelman, Vibrations of rotating beams with tip mass, Angew. Math. & Phys., 12 (1961), 369-392. doi: 10.1007/BF01600687.

[7]

F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass, SIAM J. Control. & Opt., 36 (1998), 1962-1986. doi: 10.1137/S0363012996302366.

[8]

M. Grobbelaar - Van Dalsen, Uniform stability for the Timoshenko beam with tip load, J. Math. Anal. & and Appl., 361 (2010), 392-400. doi: 10.1016/j.jmaa.2009.06.059.

[9]

B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM J. Control. & Opt., 39 (2001), 1736-1747. doi: 10.1137/S0363012999354880.

[10]

J. Humar and M. Ruban, Dynamics of Structures, CRC Press, Boca Raton, 2002.

[11]

A. E. Ingham, Some trigonometric inequalities in the theory of series, Math. Zeitschr., 41 (1936), 367-379. doi: 10.1007/BF01180426.

[12]

R. N. Jazar, Advanced Dynamics: Rigid Body, Multibody and Aerospace Applications, John Wiley & Sons, New York, 2011. doi: 10.1002/9780470950029.

[13]

J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, with Applications, Mathematics in Science and Engineering, Academic Press, New York and London, 1961.

[14]

W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura & Appl., 152 (1988), 281-330. doi: 10.1007/BF01766154.

[15]

W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Arch. Rat. Mech. & Anal., 103 (1988), 193-236. doi: 10.1007/BF00251758.

[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), 2140-2145. doi: 10.1109/9.328811.

[17]

B. P. Rao, Uniform stabilization of a hybrid system of elasticity, SIAM J. Control. & Opt., 33 (1995), 440-454. doi: 10.1137/S0363012992239879.

[18]

D. L. Russell, Nonharmonic Fourier Series in the Control Theory of Distributed Parameter Systems, J. Math. Anal. & Appl., 18 (1967), 542-560. doi: 10.1016/0022-247X(67)90045-5.

[19]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), 639-739. doi: 10.1137/1020095.

[20]

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

[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, With a foreword by William McCrea, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. doi: 10.1017/CBO9780511608797.

[22]

N. Yebari and M. D. Aouragh, Uniform stabilization of a hybrid system of elasticity with variable coefficients, Int. J. Tomogr. & Stat., 10 (2008), 125-140.

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), 46-53.

[2]

S. Avdonin and S. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, New York, Melbourne, 1995.

[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), 80-91. doi: 10.1109/7.135434.

[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, Math., 326 (1998), 453-458. doi: 10.1016/S0764-4442(97)89791-1.

[5]

M. L. Boas, Mathematical Methods in the Physical Sciences, Wiley, New York, 2006.

[6]

W. E. Boyce and G. H. Handelman, Vibrations of rotating beams with tip mass, Angew. Math. & Phys., 12 (1961), 369-392. doi: 10.1007/BF01600687.

[7]

F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass, SIAM J. Control. & Opt., 36 (1998), 1962-1986. doi: 10.1137/S0363012996302366.

[8]

M. Grobbelaar - Van Dalsen, Uniform stability for the Timoshenko beam with tip load, J. Math. Anal. & and Appl., 361 (2010), 392-400. doi: 10.1016/j.jmaa.2009.06.059.

[9]

B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM J. Control. & Opt., 39 (2001), 1736-1747. doi: 10.1137/S0363012999354880.

[10]

J. Humar and M. Ruban, Dynamics of Structures, CRC Press, Boca Raton, 2002.

[11]

A. E. Ingham, Some trigonometric inequalities in the theory of series, Math. Zeitschr., 41 (1936), 367-379. doi: 10.1007/BF01180426.

[12]

R. N. Jazar, Advanced Dynamics: Rigid Body, Multibody and Aerospace Applications, John Wiley & Sons, New York, 2011. doi: 10.1002/9780470950029.

[13]

J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, with Applications, Mathematics in Science and Engineering, Academic Press, New York and London, 1961.

[14]

W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura & Appl., 152 (1988), 281-330. doi: 10.1007/BF01766154.

[15]

W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Arch. Rat. Mech. & Anal., 103 (1988), 193-236. doi: 10.1007/BF00251758.

[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), 2140-2145. doi: 10.1109/9.328811.

[17]

B. P. Rao, Uniform stabilization of a hybrid system of elasticity, SIAM J. Control. & Opt., 33 (1995), 440-454. doi: 10.1137/S0363012992239879.

[18]

D. L. Russell, Nonharmonic Fourier Series in the Control Theory of Distributed Parameter Systems, J. Math. Anal. & Appl., 18 (1967), 542-560. doi: 10.1016/0022-247X(67)90045-5.

[19]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), 639-739. doi: 10.1137/1020095.

[20]

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

[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, With a foreword by William McCrea, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. doi: 10.1017/CBO9780511608797.

[22]

N. Yebari and M. D. Aouragh, Uniform stabilization of a hybrid system of elasticity with variable coefficients, Int. J. Tomogr. & Stat., 10 (2008), 125-140.

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