Mathematical Modeling of SISO based Timoshenko Structures – A Case Study

Authors: T.C. Manjunath, Student Member, B. Bandyopadhyay

Abstract:

This paper features the mathematical modeling of a single input single output based Timoshenko smart beam. Further, this mathematical model is used to design a multirate output feedback based discrete sliding mode controller using Bartoszewicz law to suppress the flexural vibrations. The first 2 dominant vibratory modes is retained. Here, an application of the discrete sliding mode control in smart systems is presented. The algorithm uses a fast output sampling based sliding mode control strategy that would avoid the use of switching in the control input and hence avoids chattering. This method does not need the measurement of the system states for feedback as it makes use of only the output samples for designing the controller. Thus, this methodology is more practical and easy to implement.

Keywords: Smart structure, Timoshenko beam theory, Discretesliding mode control, Bartoszewicz law, Finite Element Method, State space model, Vibration control, Mathematical model, SISO.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1333979

References:

[1] H. Abramovich, and A. Lishvits, ''Free vibrations of non-symmetric cross-ply laminated composite beams,'' Journal of Sound and Vibration,vol. 176, no. 5, pp. 597 - 612, 1994.
[2] O. J. Aldraihem, R. C. Wetherhold, and T. Singh, ''Distributed control of laminated beams : Timoshenko Vs. Euler-Bernoulli Theory,'' J. of Intelligent Materials Systems and Structures, vol. 8, pp. 149-157, 1997.
[3] H. Abramovich, ''Deflection control of laminated composite beam with piezoceramic layers-Closed form solutions,'' Composite Structures, vol. 43, no. 3, pp. 217-131, 1998.
[4] O. J. Aldraihem, and K. A. Ahmed, ''Smart beams with extension and thickness-shear piezoelectric actuators,'' Smart Materials and Structures,vol. 9, no. 1, pp. 1-9, 2000.
[5] A. K. Ahmed, and J. A. Osama, ''Deflection analysis of beams with extension and shear piezoelectric patches using discontinuity functions'' Smart Materials and Structures, vol. 10, no. 1, pp. 212-220, 2001.
[6] L. E. Azulay, and H. Abramovich, ''Piezoelectric actuation and sensing mechanisms-Closed form solutions,'' Composite Structures J., vol. 64, pp. 443-453, 2004.
[7] T. Baily, and J. E. Hubbard Jr., ''Distributed piezoelectric polymer active vibration control of a cantilever beam,'' J. of Guidance, Control and Dynamics, vol. 8, no.5, pp. 605-611, 1985.
[8] M. J. Balas, ''Feedback control of flexible structures,'' IEEE Trans. Automat. Contr., vol. AC-23, no. 4, pp. 673-679, 1978.
[9] J. S. Burdess, and J. N. Fawcett, ''Experimental evaluation of piezoelectric actuator for the control of vibrations in a cantilever beam,'' J. Syst. Control. Engg., vol. 206, no. 12, pp. 99-106, 1992.
[10] M. J. Brennan, J. G. Bonito, S. J. Elliot, A. David, and R.J. Pinnington, ''Experimental investigation of different actuator technologies for active vibration control,'' Journal of Smart Materials and Structures, vol. 8, pp. 145-153, 1999.
[11] B. M. Bona, M. Indri, and A. Tornamble, ''Flexible piezoelectric structures-approximate motion equations and control algorithms,'' IEEETrans. on Auto. Contr., vol. 42, no. 1, pp. 94-101, 1997.
[12] A. Benjeddou, M. A. Trindade, and R. Ohayon, ''New shear actuated smart structure beam finite element,'' AIAA J., vol. 37, pp. 378-383, 1999.
[13] E. F. Crawley, and J. De Luis, ''Use of piezoelectric actuators as elements of intelligent structures,'' AIAA J., vol. 25, pp. 1373-1385, 1987.
[14] K. Chandrashekhara, and S. Varadarajan, ''Adaptive shape control of composite beams with piezoelectric actuators,'' J. of Intelligent Materials Systems and Structures, vol. 8, pp. 112-124, 1997.
[15] B. Culshaw, ''Smart Structures : A concept or a reality,'' J. of Systems and Control Engg., vol. 26, no. 206, pp. 1-8, 1992.
[16] C. R. Cooper, ''Shear coefficient in Timoshenko beam theory,'' ASME J. of Applied Mechanics, vol. 33, pp. 335-340, 1966.
[17] S. B. Choi, C. Cheong, and S. Kini, ''Control of flexible structures by distributed piezo-film actuators and sensors,'' J. of Intelligent Materials and Structures, vol. 16, pp. 430-435, 1995.
[18] A. B. Chammas, and C. T. Leondes, ''Pole placement by piecewise constant output feedback,'' Int. J. Contr., vol. 29, pp. 31-38, 1979.
[19] C. Doschner, and M. Enzmann, ''On model based controller design for smart structure,'' Smart Mechanical Systems Adaptronics SAE International USA, pp. 157-166, 1998.
[20] P. Donthireddy, and K. Chandrashekhara, ''Modeling and shape control of composite beam with embedded piezoelectric actuators,'' Comp.Structures, vol. 35, no. 2, pp. 237-244, 1996.
[21] J. L. Fanson, and T. K. Caughey, ''Positive position feedback control for structures,'' AIAA J., vol. 18, no. 4, pp. 717-723, 1990.
[22] P. Forouza, ''Distributed controllers for flexible structures using piezo-electric actuators / sensors,'' Proc. the 32nd Conference on Decision and Control, Texas, pp. 1367-1369, Dec. 1993.
[23] J. C. Geromel, C. C. De Souza, and R. E. Skeleton, ''LMI Numerical solution for output feedback stabilization,'' Proc. American Contr. Conf.,pp. 40-44, 1994.
[24] S. V. Gosavi, and A. V. Kelkar, ''Modeling, identification, and passivity-based robust control of piezo-actuated flexible beam,'' Journal of Vibration and Acoustics, vol. 129, pp. 260-271, Apr. 2004.
[25] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, ''LMI Tool box for Matlab'', The Math works Inc., Natick MA, 1995.
[26] S. Hanagud, M. W. Obal, and A. J. Callise, ''Optimal vibration control by the use of piezoceramic sensors and actuators,'' J. of Guidance, Control and Dyn., vol. 15, no. 5, pp. 1199-1206, 1992.
[27] W. Hwang, and H. C. Park, ''Finite element modeling of piezoelectric sensors and actuators'', AIAA J., vol. 31, no. 5, pp. 930-937, 1993.
[28] J. B. Kosmataka, and Z. Friedman, ''An improved two-node Timoshenko beam finite element'', Computers and Struct., vol. 47, no. 3, pp. 473-481, 1993.
[29] W. S. Levine, and M. Athans, ''On the determination of the optimal constant output feedback gains for linear multivariable systems,'' IEEETrans. Auto. Contr., vol. AC-15, pp. 44-48, 1970.
[30] T. C. Manjunath and B. Bandyopadhyay, ''Modeling and fast output sampling feedback control of a smart Timoshenko cantilever beam,'' International Journal of Smart Structures and Systems, Vol. 1, no. 3, pp. 283-308, Sep. 2005.
[31] G. Murali, G.A. Pajunen, ''Model reference control of vibrations in flexible smart structures,'' Proc. 34th Conference on Decision and Control, New Orleans, USA, pp. 3551-3556, Dec. 1995.
[32] J. S. M. Moita, I. F. P. Coreia, C. M. M. Soares, and C. A. M. Soares, ''Active control of adaptive laminated structures with bonded piezoelectric sensors and actuators,'' Computers and Structures, vol. 82, pp. 1349 - 1358, 2004.
[33] Robin Scott, Michael Brown and Martin Levesley, ''Robust multivariable control of a double beam cantilever smart structure,'' J. of Smart Materials and Structures, vol. 13, pp. 731-743, 2003.
[34] S. Raja, G. Prathap, and P. K. Sinha, P.K., ''Active vibration control of composite sandwich beams with piezoelectric extension-bending and shear actuators,'' Smart Materials and Structures, vol. 11, no. 1, pp. 63?71, 2002.
[35] S. Rao, and M. Sunar, ''Piezoelectricity and its uses in disturbance sensing and control of flexible structures : A Survey,'' Applied Mechanics Rev., vol. 47, no. 2, pp. 113-119, 1994.
[36] C. T. Sun, and X. D. Zhang, ''Use of thickness-shear mode in adaptive sandwich structures,'' Smart Materials and Structures J., vol. 3, no. 4, pp. 202-206, 1995.
[37] V. L. Syrmos, P. Abdallah, P. Dorato, and K. Grigoriadis, ''Static output feedback : A survey,'' Automatica, vol. 33, no. 2, pp. 125-137, 1997.
[38] W. Schiehlen, and H. Schonerstedt, ''Controller design for the active vibration damping of beam structure,'' Smart Mechanical Systems Adaptronics SAE International, USA, pp. 137-146, 1998.
[39] P. Seshu, ''Textbook of Finite Element Analysis,'' 1st Ed. Prentice Hall of India, New Delhi, 2004.
[40] W. Shiang Lee, ''System identification and control of smart structures using neural networks,'' Automatica, vol. 38, no. 4-8, pp. 269-276, 1996.
[41] J. Thomas, and B. A. H. Abbas, ''Finite Element Methods for dynamic analysis of Timoshenko beam,'' J. of Sound and Vibration, vol. 41, pp. 291-299, 1975.
[42] M. O. Tokhi, ''Self tuning active vibration control in flexible beam structures,'' Journal of Systems and Control Engineering - Proceedings of the Institute of Mechanical Engineers, vol. 208, pp. 263-277, 1994.
[43] H. Werner, ''Robust multivariable control of a turbo-generator by periodic output feedback,'' vol. 31, no. 4, pp. 395 - 411, 1997.
[44] Ulrich Gabbert1, Tamara Nestorovie Trajkov1, and Heinz Köppel, ''Modeling, control and simulation of piezoelectric smart structures using finite element method and optimal LQ control,'' Facta Universitatis Series: Mechanics, Automatic Control and Robotics, Vol. 3, no. 12, pp. 417 - 430, 2002.
[45] Vukovich, G., and A.Y. Koma. Vibration suppression of flexible beams with bonded piezo-transducers using wave-absorbing controllers in J.Guidance, Control and Dynamics, pp. 347-354, Mar-Apr. 2000.
[46] H. Werner, and K. Furuta, ''Simultaneous stabilization based on output measurements,'' Kybernetika, vol. 31, no. 4, pp. 395-411, 1995.
[47] H. Werner, H., ''Multimodal robust control by fast output sampling-An LMI approach,'' Automatica, vol. 34, no. 12, pp. 1625-1630, 1998.
[48] Young-Hun Lim, V. Senthil Gopinathan, Vasundara V Varadhan, and K. Vijay K Varadan, ''Finite element simulation of smart structures using an optimal output feedback controller for vibration and noise control,'' Journal of Smart Materials and Structures, vol. 8, pp. 324-337, 1999.
[49] S. M. Yang, and Y. J. Lee, ''Optimization of non-collocated sensor / actuator location and feedback gain in control systems,'' Journal of Smart Materials Structures, vol. 8, pp. 96-102, 1993.
[50] Y. C. Yan, J. Lam, and Y. X. Sun, ''Static output feedback stabilization: An LMI approach,'' Automatica, vol. 34, no. 12, pp. 1641-1645, 1998.
[51] X. D. Zhang, and C. T. Sun, ''Formulation of an adaptive sandwich beam,'' Smart Mater. and Struct., vol. 5, no.6, pp. 814-823, 1996.
[52] V.I. Utkin, ''Variable structure systems with sliding modes'', IEEETrans. Automat. Contr., vol. AC-22, pp. 212-222, 1977.
[53] J.Y. Hung., W. Gao, and J.C. Hung, ''Variable structure control : A survey'', IEEE Trans. Ind. Electron., Vol. 40, no. 1, pp. 2-21, 1993.
[54] K.D. Young, V.I. Utkin, U. Ozguner, ''A control engineer's guide to sliding mode control'', IEEE Trans. Contr. Syst., vol. 7, no. 3, pp. 328-342, 1999.
[55] S. Sarpturk, Y. Istefanopulos, and O. Kaynak, ''On the stability of discrete-time sliding mode systems '', IEEE Trans. Automat. Contr., vol. AC-32, pp. 930-932, 1987.
[56] K. Furuta, ''Sliding mode control of a discrete system'', Syst. Control Lett., vol. 14, pp. 145-152., 1990.
[57] W. Gao, Y. Wang, and A. Homaifa, ''Discrete-time variable structure control systems,'' IEEE Trans. Ind. Electron., vol. 42, pp. 117-122, 1995.
[58] A. Bartoszewicz, ''Discrete-time quasi-sliding-mode control strategies,'' IEEE Trans. Ind. Electron., vol. 45, no. 1, pp. 633-637, 1998.
[59] Werner, H., Robust Control of a Laboratory Flight Simulator by Nondynamic Multirate Output Feedback, Proc. IEEE Conf. Decision and Control, pp. 1575-1580, 1996.
[60] S. Janardhanan, B. Bandyopadhyay and T. C. Manjunath, Fast Output Sampling Based Output Feedback Sliding Mode Control Law For Uncertain Systems'', Proc. Third Intl. Conf. on System Identification and Control Problems, SICPRO - 2004, Moscow, Russia, no. 23010, pp. 1300 - 1312, Jan. 2004.
[61] T.C. Manjunath, B. Bandyopadhyay and S. Janardhanan, ''Multirate Output Feedback based Sliding Mode Controller Design for Active Vibration Control of SISO Smart Structures,'' Proc. 29th National Systems Conference (NSC-2005), IIT Bombay, Mumbai, India, Paper no. 54, Dec. 2005.
[62] T.C. Manjunath, B. Bandyopadhyay and S. Janardhanan, ''Vibration Control of Smart Structures Using Multirate Output Feedback Sliding Mode Control Law,'' Proc. 1st National Conference on Control and Dynamical Systems (NCCDS 05), IIT Bombay, Mumbai, India, Paper no. 61, Jan. 2005.
[63] T.C. Manjunath and B. Bandyopadhyay, ''Smart Control of Cantilever Structures Using Output Feedback,'' International Journal of Simulation, System Science and Technology, A Special issue on Modelling, Simulation and Decision Support, vol. 7, nos. 4-5, pp. 51-68, Jul. 2006.
[64] B. Bandyopadhyay and S. Janardhanan, ''Discrete time sliding mode control - A multirate output feedback approach,'' Springer Publications.,Lecture Notes in Control and Information Sciences, LNCIS No. 323.