Vibration Suppression of Timoshenko Beams with Embedded Piezoelectrics Using POF
This paper deals with the design of a periodic output
feedback controller for a flexible beam structure modeled with
Timoshenko beam theory, Finite Element Method, State space
methods and embedded piezoelectrics concept. The first 3 modes are
considered in modeling the beam. The main objective of this work is
to control the vibrations of the beam when subjected to an external
force. Shear piezoelectric sensors and actuators are embedded into
the top and bottom layers of a flexible aluminum beam structure, thus
making it intelligent and self-adaptive. The composite beam is
divided into 5 finite elements and the control actuator is placed at
finite element position 1, whereas the sensor is varied from position 2
to 5, i.e., from the nearby fixed end to the free end. 4 state space
SISO models are thus developed. Periodic Output Feedback (POF)
Controllers are designed for the 4 SISO models of the same plant to
control the flexural vibrations. The effect of placing the sensor at
different locations on the beam is observed and the performance of
the controller is evaluated for vibration control. Conclusions are
finally drawn.
[1] 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.
[2] E. Crawley, and J. Luis, "Use Of Piezoelectric Actuators As Elements
Of Intelligent Structures," AIAA Journal, vol.25, no. 10, pp. 1373-
1385, 1987.
[3] S. Hanagud, M.W. Obal, and A.J. Callise, "Optimal Vibration Control
By The Use Of Piezoelectric Sensors And Actuators," Journal of
Guidance Control and Dynamics, vol. 15, no. 5, pp. 1199 - 1206,
1992.
[4] B. Bona, M. Indri, and A. Tornamble, "Flexible Piezoelectric
Structures-Approximate Motion Equations and Control Algorithms,"
IEEE Transactions on Automatic Control, vol. 42, no. 1, pp. 94- 101,
1997.
[5] J.E. Hubbard Jr., and T. Baily, "Distributed Piezoelectric Polymer
Active Vibration Control Of A Cantilever Beam," Journal Guidance
Dynamics and Control, vol. 8, no. 5, pp. 605 - 611, 1985.
[6] 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.
[7] 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.
[8] P. Donthireddy, and K. Chandrashekhara, "Modeling and Shape
Control of Composite Beam with Embedded Piezoelectric Actuators",
Composite Structures, vol. 35, no. 2, pp. 237- 244, 1996.
[9] C.T. Sun, and X.D. Zhang, "Use of Thickness-Shear Mode in
Adaptive Sandwich Structures", Smart Materials and Structures, vol.
4, no. 3, pp. 202 - 206, 1995.
[10] W. Hwang, and H.C. Park, "Finite Element Modeling of Piezoelectric
Sensors and Actuators", AIAA Journal, vol. 31, no. 5, pp. 930-937,
1993.
[11] P. Gahnet, A. Nemirovski, A. J. Laub, and M. Chilali, "LMI Tool box
for Matlab", The Math works Inc., Natick MA, 1995.
[12] 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.
[13] O.J. Aldraihem, R.C. Wetherhold, and T. Singh, "Distributed Control
of Laminated Beams: Timoshenko vs. Euler-Bernoulli Theory",
Journal of Intelligent Materials Systems and Structures, vol. 8, pp.
149-157, 1997.
[14] H. Abramovich, "Deflection Control of Laminated Composite Beam
with Piezoceramic Layers-Closed Form Solution", Composite
Structures, vol. 43, no. 3, pp. 217-231, 1998.
[15] X.D. Zhang, and C.T. Sun, "Formulation of an Adaptive Sandwich
Beam", Smart Materials and Structures, vol. 5, no. 6, pp. 814-823,
1996.
[16] S. Raja, G. Prathap, and P.K. Sinha, "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,
[17] A. Benjeddou, M.A. Trindade, and R. Ohayon, "New Shear Actuated
Smart Structure Beam Finite Element", AIAA Journal, vol. 37, pp.
378 - 383, 1998.
[18] O.J. Aldraihem, and A.A. Khdeir, "Smart Beams with Extension and
Thickness-Shear Piezoelectric Actuators", Smart Materials and
Structures, vol. 9, no. 1, pp. 1- 9, 2000.
[19] V.L. Syrmos, P. Abdallah, P. Dorato, and K. Grigoriadis, "Static
Output Feedback A Survey", Automatica, vol. 33, no. 2, pp. 125-137,
1997.
[20] H. Werner, and K. Furuta, "Simultaneous Stabilization Based on
Output Measurements", Kybernetika, vol. 31, no. 4, pp. 395 - 411,
1995
[21] H. Werner, H., "Robust multivariable control of a turbo-generator by
periodic output feedback," Proc. American Contr. Conf., New
Mexico, pp. 1979-1983, 1997.
[22] B. Culshaw, "Smart Structure A Concept or A Reality," Journal of
Systems and Control Engg., vol. 26, no. 206, pp. 1-8, 1992.
[23] T. C. Manjunath, and B. Bandyopadhyay, "Vibration control of a
smart flexible cantilever beam using periodic output feedback," Asian
Journal of Control, vol. 6, no. 1, pp. 74 - 87, Mar. 2004.
[24] T. C. Manjunath, and B. Bandyopadhyay, "Fault tolerant control of
flexible smart structures using robust decentralized periodic output
sampling feedback technique," International Journal of Smart
Materi. and Struct., vol. 14, no. 4, pp. 624-636, Aug. 2005.
[25] W. S. Levine, and M. Athans, "On the determination of the optimal
constant output feedback gains for linear multivariable systems,"
IEEE Trans. Auto. Contr., vol. AC-15, pp. 44-48, 1970.
[26] 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.
[27] L. E. Azulay, and H. Abramovich, "Piezoelectric actuation and
sensing mechanisms-Closed form solutions," Composite Structures J.,
vol. 64, pp. 443 - 453, 2004.
[28] 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.
[29] 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," in Computers and Structures,
82, 1349 - 1358, 2004.
[30] J.A. Zapfe, and G.A. Lesieutre, "A Discrete Layer Beam Finite
Element for The Dynamic Analysis of Composite Sandwiched Beams
With Integral Damping Layers," in Computers and Structures, vol.
70, pp. 647 - 666, 1999.
[31] J. Lee, "Free Vibration Analysis of Delaminated Composite Beams,"
in Computers and Structures, vol. 74, pp. 121 - 129, 2000.
[32] R.C. Louis, C.S. Edward, and M. Brian, "Induced Shear Piezoelectric
Actuators for Rotor Blade Trailing Edge Flaps," in Smart Materials
and Structures, vol. 11, pp. 24 - 35, 2002.
[33] H. Abramovich, and A. Lishvits, "Free Vibrations of Non-symmetric
Cross-ply Laminated Composite Beams," in Journal of Sound and
Vibration, vol. 176, no. 5, pp. 597 - 612, 1994.
[34] G.E. Blandford, T.R. Tauchert, and Y. Du, "Self-strained
piezothermoelastic composite beam analysis using first-order shear
deformation theory," Composites : Part B, vol. 30, pp. 51 - 63, 1999.
[35] P. Seshu, "Textbook of Finite Element Analysis," 1st Ed. Prentice
Hall of India, New Delhi, 2004.
[36] M. Umapathy, and B. Bandyopadhyay, "Control of flexible beam
through smart structure concept using periodic output feedback,"
System Science Journal, vol. 26, no. 1, pp. 23 - 46, 2000.
[37] T. C. Manjunath, and B. Bandyopadhyay, "Modeling and fast output
sampling feedback control of a smart Timoshenko cantilever beam",
Smart Structures and Systems, vol. 1, no. 3, pp. 283-308, Sept. 2005.
[38] T.C. Manjunath, and B. Bandyopadhyay, "Multivariable Control of
Smart Timoshenko Beam Structures using POF Technique,"
International Journal of Signal Processing, vol. 3, no. 2, pp. 74 - 90,
Apr. 2006.
[39] T.C. Manjunath, and B. Bandyopadhyay, "Modeling and FOS
Feedback Based Control of SISO Intelligent Structures With
Embedded Shear Sensors and Actuators," International Journal of
Intelligent Technology, vol. 1, no. 1, pp. 1 - 20, Jan. 2006.
[40] A. B. Chammas, and C. T. Leondes, "Pole placement by piecewise
constant output feedback," Int. J. Contr., vol. 29, pp. 31-38, 1979.
[41] A. B. Chammas, and C. T. Leondes, "On the design of LTI systems
by periodic output feedback, Part-I, Discrete Time pole assignment,"
Int. J. Ctrl., vol. 27, pp. 885-894, 1978.
[42] Chammas, A. B. and C. T. Leondes, "On the design of LTI systems
by periodic output feedback, Part-II, Output feedback controllability,"
Int. J. Ctrl., vol. 27, pp. 895-903, 1978.
[1] 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.
[2] E. Crawley, and J. Luis, "Use Of Piezoelectric Actuators As Elements
Of Intelligent Structures," AIAA Journal, vol.25, no. 10, pp. 1373-
1385, 1987.
[3] S. Hanagud, M.W. Obal, and A.J. Callise, "Optimal Vibration Control
By The Use Of Piezoelectric Sensors And Actuators," Journal of
Guidance Control and Dynamics, vol. 15, no. 5, pp. 1199 - 1206,
1992.
[4] B. Bona, M. Indri, and A. Tornamble, "Flexible Piezoelectric
Structures-Approximate Motion Equations and Control Algorithms,"
IEEE Transactions on Automatic Control, vol. 42, no. 1, pp. 94- 101,
1997.
[5] J.E. Hubbard Jr., and T. Baily, "Distributed Piezoelectric Polymer
Active Vibration Control Of A Cantilever Beam," Journal Guidance
Dynamics and Control, vol. 8, no. 5, pp. 605 - 611, 1985.
[6] 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.
[7] 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.
[8] P. Donthireddy, and K. Chandrashekhara, "Modeling and Shape
Control of Composite Beam with Embedded Piezoelectric Actuators",
Composite Structures, vol. 35, no. 2, pp. 237- 244, 1996.
[9] C.T. Sun, and X.D. Zhang, "Use of Thickness-Shear Mode in
Adaptive Sandwich Structures", Smart Materials and Structures, vol.
4, no. 3, pp. 202 - 206, 1995.
[10] W. Hwang, and H.C. Park, "Finite Element Modeling of Piezoelectric
Sensors and Actuators", AIAA Journal, vol. 31, no. 5, pp. 930-937,
1993.
[11] P. Gahnet, A. Nemirovski, A. J. Laub, and M. Chilali, "LMI Tool box
for Matlab", The Math works Inc., Natick MA, 1995.
[12] 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.
[13] O.J. Aldraihem, R.C. Wetherhold, and T. Singh, "Distributed Control
of Laminated Beams: Timoshenko vs. Euler-Bernoulli Theory",
Journal of Intelligent Materials Systems and Structures, vol. 8, pp.
149-157, 1997.
[14] H. Abramovich, "Deflection Control of Laminated Composite Beam
with Piezoceramic Layers-Closed Form Solution", Composite
Structures, vol. 43, no. 3, pp. 217-231, 1998.
[15] X.D. Zhang, and C.T. Sun, "Formulation of an Adaptive Sandwich
Beam", Smart Materials and Structures, vol. 5, no. 6, pp. 814-823,
1996.
[16] S. Raja, G. Prathap, and P.K. Sinha, "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,
[17] A. Benjeddou, M.A. Trindade, and R. Ohayon, "New Shear Actuated
Smart Structure Beam Finite Element", AIAA Journal, vol. 37, pp.
378 - 383, 1998.
[18] O.J. Aldraihem, and A.A. Khdeir, "Smart Beams with Extension and
Thickness-Shear Piezoelectric Actuators", Smart Materials and
Structures, vol. 9, no. 1, pp. 1- 9, 2000.
[19] V.L. Syrmos, P. Abdallah, P. Dorato, and K. Grigoriadis, "Static
Output Feedback A Survey", Automatica, vol. 33, no. 2, pp. 125-137,
1997.
[20] H. Werner, and K. Furuta, "Simultaneous Stabilization Based on
Output Measurements", Kybernetika, vol. 31, no. 4, pp. 395 - 411,
1995
[21] H. Werner, H., "Robust multivariable control of a turbo-generator by
periodic output feedback," Proc. American Contr. Conf., New
Mexico, pp. 1979-1983, 1997.
[22] B. Culshaw, "Smart Structure A Concept or A Reality," Journal of
Systems and Control Engg., vol. 26, no. 206, pp. 1-8, 1992.
[23] T. C. Manjunath, and B. Bandyopadhyay, "Vibration control of a
smart flexible cantilever beam using periodic output feedback," Asian
Journal of Control, vol. 6, no. 1, pp. 74 - 87, Mar. 2004.
[24] T. C. Manjunath, and B. Bandyopadhyay, "Fault tolerant control of
flexible smart structures using robust decentralized periodic output
sampling feedback technique," International Journal of Smart
Materi. and Struct., vol. 14, no. 4, pp. 624-636, Aug. 2005.
[25] W. S. Levine, and M. Athans, "On the determination of the optimal
constant output feedback gains for linear multivariable systems,"
IEEE Trans. Auto. Contr., vol. AC-15, pp. 44-48, 1970.
[26] 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.
[27] L. E. Azulay, and H. Abramovich, "Piezoelectric actuation and
sensing mechanisms-Closed form solutions," Composite Structures J.,
vol. 64, pp. 443 - 453, 2004.
[28] 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.
[29] 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," in Computers and Structures,
82, 1349 - 1358, 2004.
[30] J.A. Zapfe, and G.A. Lesieutre, "A Discrete Layer Beam Finite
Element for The Dynamic Analysis of Composite Sandwiched Beams
With Integral Damping Layers," in Computers and Structures, vol.
70, pp. 647 - 666, 1999.
[31] J. Lee, "Free Vibration Analysis of Delaminated Composite Beams,"
in Computers and Structures, vol. 74, pp. 121 - 129, 2000.
[32] R.C. Louis, C.S. Edward, and M. Brian, "Induced Shear Piezoelectric
Actuators for Rotor Blade Trailing Edge Flaps," in Smart Materials
and Structures, vol. 11, pp. 24 - 35, 2002.
[33] H. Abramovich, and A. Lishvits, "Free Vibrations of Non-symmetric
Cross-ply Laminated Composite Beams," in Journal of Sound and
Vibration, vol. 176, no. 5, pp. 597 - 612, 1994.
[34] G.E. Blandford, T.R. Tauchert, and Y. Du, "Self-strained
piezothermoelastic composite beam analysis using first-order shear
deformation theory," Composites : Part B, vol. 30, pp. 51 - 63, 1999.
[35] P. Seshu, "Textbook of Finite Element Analysis," 1st Ed. Prentice
Hall of India, New Delhi, 2004.
[36] M. Umapathy, and B. Bandyopadhyay, "Control of flexible beam
through smart structure concept using periodic output feedback,"
System Science Journal, vol. 26, no. 1, pp. 23 - 46, 2000.
[37] T. C. Manjunath, and B. Bandyopadhyay, "Modeling and fast output
sampling feedback control of a smart Timoshenko cantilever beam",
Smart Structures and Systems, vol. 1, no. 3, pp. 283-308, Sept. 2005.
[38] T.C. Manjunath, and B. Bandyopadhyay, "Multivariable Control of
Smart Timoshenko Beam Structures using POF Technique,"
International Journal of Signal Processing, vol. 3, no. 2, pp. 74 - 90,
Apr. 2006.
[39] T.C. Manjunath, and B. Bandyopadhyay, "Modeling and FOS
Feedback Based Control of SISO Intelligent Structures With
Embedded Shear Sensors and Actuators," International Journal of
Intelligent Technology, vol. 1, no. 1, pp. 1 - 20, Jan. 2006.
[40] A. B. Chammas, and C. T. Leondes, "Pole placement by piecewise
constant output feedback," Int. J. Contr., vol. 29, pp. 31-38, 1979.
[41] A. B. Chammas, and C. T. Leondes, "On the design of LTI systems
by periodic output feedback, Part-I, Discrete Time pole assignment,"
Int. J. Ctrl., vol. 27, pp. 885-894, 1978.
[42] Chammas, A. B. and C. T. Leondes, "On the design of LTI systems
by periodic output feedback, Part-II, Output feedback controllability,"
Int. J. Ctrl., vol. 27, pp. 895-903, 1978.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:63858", author = "T. C. Manjunath and B. Bandyopadhyay", title = "Vibration Suppression of Timoshenko Beams with Embedded Piezoelectrics Using POF", abstract = "This paper deals with the design of a periodic output
feedback controller for a flexible beam structure modeled with
Timoshenko beam theory, Finite Element Method, State space
methods and embedded piezoelectrics concept. The first 3 modes are
considered in modeling the beam. The main objective of this work is
to control the vibrations of the beam when subjected to an external
force. Shear piezoelectric sensors and actuators are embedded into
the top and bottom layers of a flexible aluminum beam structure, thus
making it intelligent and self-adaptive. The composite beam is
divided into 5 finite elements and the control actuator is placed at
finite element position 1, whereas the sensor is varied from position 2
to 5, i.e., from the nearby fixed end to the free end. 4 state space
SISO models are thus developed. Periodic Output Feedback (POF)
Controllers are designed for the 4 SISO models of the same plant to
control the flexural vibrations. The effect of placing the sensor at
different locations on the beam is observed and the performance of
the controller is evaluated for vibration control. Conclusions are
finally drawn.", keywords = "Smart structure, Timoshenko beam theory, Periodic
output feedback control, Finite Element Method, State space model,
SISO, Embedded sensors and actuators, Vibration control.", volume = "2", number = "3", pages = "359-10", }