Nonlinear Velocity and Shape Control of a Rotating Smart Flexible Beam-Like Structure


Rotating Flexible Structures
Nonlinear Systems
Piezoelectric Actuators
Smart Structures
Shape Control
Nonlinear Control
SDRE Control

How to Cite

Fenili, A. . (2022). Nonlinear Velocity and Shape Control of a Rotating Smart Flexible Beam-Like Structure . Annals of Applied Sciences, 1. Retrieved from


Smart materials and structures have a great appeal within the aerospace and automotive communities because they promise to enable better performance and functionality over existing structural and functional materials. The idea proposed in this work is part of this challenging and current scenario and can be applied to the study of morphing wings, helicopter blades, etc. Moreover, the mathematical modeling and velocity and shape control of a rotating flexible beam-like structure are investigated. The nonlinear partial differential governing equations of motion are derived using the extended Hamilton’s Principle and numerically integrated using a combination of finite difference and the fourth-order Runge–Kutta methods. In order to force the flexible structure to assume the desired shape and simultaneously control the velocity of the rotating axis, the optimal nonlinear control method named state-dependent Riccati equation (SDRE) is considered. This control technique is applied to piezoelectric actuators along the beam and an external torque coming from a DC motor and acting in the rotating axis. The numerical simulation results show that the proposed control technique is efficient when acting along the rotating beam to deform it into the desired shape while also acting on the motor axis to keep the rotation speed constant.



Weisshaar, T. A. 2006. “Morphing Aircraft Technology! New Shapes for Aircraft Design,” RTOMP-AVT-141, Neuilly-sur-Seine, France.

Weisshaar, T. A. 2013. “Morphing Aircraft Systems: Historical Perspectives and Future Challenges”, Journal of Aircraft, Vol. 50, No. 2, 2013, pp. 337-353.

Sofla, A. Y. N.; Meguid, S. A.; Tan, K. T.; Yeo, W. K. “Shape Morphing of Aircraft Wing: Status and Challenges”, Materials e Design, Vol. 31, No. 3, 2010, pp. 1284–1292.

Barbarino, S.; Saavedra Flores, E. I.; Ajaj, R. M.; Dayyani, I.; Friswell, M. I. A, “Review of Morphing Aircraft”, Journal of Intelligent Material Systems and Structures, Vol. 22, No. 9, 2011, pp. 823–877.

Barbarino, S.; Bilgen, O; Ajaj, R. M.; Friswell, M. I. Inman, D. J. “A Review on Shape Memory Alloys with Applications to Morphing Aircraft”, Smart Materials and Structures, Vol. 23, No. 6, 2014, pp. 063001.

Zhou, H., 2019. "Distributed Actuation and Control of Smart Structures". PhD thesis, Department of Mechanical Engineering, University of Bath. United Kingdon.

Yousefi-Koma, A., 1997. “Active Vibration Control of Smart Structures Using Piezoelements”. PhD thesis, Department of Mechanical and Aerospace Engineering, Carleton University. Ottawa/ Ontario. Canada.

Zhang, J.; He, L.; Wang, E. “Active Vibration Control of Piezoelectric Intelligent Structures”, Journal of Computers, Vol. 5, No. 3, 2010, pp. 401-409.

Çimen, T. “State-Dependent Riccati Equation (SDRE): A Survey”, Proceedings of the 17th World Congress, The International Federation of Automatic Control, Seoul, Korea, July 6-11, 2008, pp 3761-3775.

Shamma, J. S.; Cloutier, J. R. “Existence of SDRE Stabilizing Feedback”, IEEE Transactions on Automatic Control, 48, 2003, pp. 513-517.

Lu, E.; Li, W.; Yang, X.; Wang, Y.; Liu, Y. "Dynamic Modeling and Analysis of a Rotating Piezoelectric Smart Beam", International Journal of Structural Stability and Dynamics, Vol. 18, No. 1, 2018, 1850003 (19 pages). DOI: 10.1142/S0219455418500037.

Shoushtari N. D., 2013. "Optimal Active Control of Flexible Structures Applying Piezoelectric Actuators". PhD thesis, University of Waterloo/Ontario. Canada.

Agrawal, B. N; Treanor, K. E. "Shape control of a beam using piezoelectric actuators", Smart Materials and Structures, Vol. 8, 1999, pp. 729–740.

Yang, S.; Ngoi, B. "Shape Control of Beams by Piezoelectric Actuators", AIAA JOURNAL, Vol. 38, No. 12, 2000, pp. 2292-2298.

Agrawal, S. K.; Tong, D.; Nagaraja, K. "Control of shapes of elastic plates using embedded piezoelectric actuators", Proc. SPIE 2190, Smart Structures and Materials, 1994: Smart Structures and Intelligent Systems, Vol. 2190, 1994, pp. 463-470.,

Wang, K.; Alaluf, D.; Rodrigues, G.; Preumont, A. "Precision Shape Control of Ultra-thin Shells with Strain Actuators", Journal of Applied and Computational Mechanics, Vol. 7(SI), 2021, pp. 1130-1137. DOI: 10.22055/JACM.2020.31899.1987.

Lanczos, C. "The Variational Principles of Mechanics". Toronto: Dover publications, 1970.

L. Meirovitch, Analytical Methods in Vibrations, Macmillan Publishing Co., Inc., New York, 1967.

Fenili A., 2000. “Mathematical Modeling and Analysis of the Ideal and Nonideal Behavior of Slewing Flexible Structures”. PhD thesis, University of Campinas (UNICAMP), Faculty of Mechanical Engineering. Campinas/São Paulo. Brazil. In Portuguese.

Popov, E. P., "Introdução à Mecânica dos Sólidos", Editora Edgar Blücher Ltda, 1978.

Sah, J. J.; Mayne, R. W., "Modeling of a Slewing Motor-Beam System", Proceedings of the International Computers in Engineering Conference, Boston, 1990, pp 481-486.

Beache, K. V. A.; Fenili, A., "Active Vibration Control of a Smart Beam Under Rotation", Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering (CILAMCE 2016), Brasília, DF, Brazil, November 6-9, 2016.

LeVeque, R. J., “Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems” (Society for Industrial and Applied Mathematics), 2007. ISBN: 978-0-89871-629-0.

Kononenko, V. O., "Vibrating systems with a limited power supply", Iliffe Books Ltd., 1969.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Copyright (c) 2022 Fenili A.