Damping of vibrations is often required to improve both the performance and the integrity of engineering structures, for example, gas turbine blades. In this paper, we explore the possibility of using piezoelectric plates to control the multimode vibrations of a cantilever beam. To develop an effective control strategy and optimize the placement of the active piezoelectric elements in terms of vibrations amplitude reduction, a procedure has been developed and a new analytical solution has been proposed. The results obtained have been corroborated by comparison with the results from a multiphysics finite elements package (COMSOL), results available in the literature, and experimental investigations carried out by the authors. 1. Introduction The vibration control is a problem of great interest in many engineering fields since it allows avoiding problems connected with the vibrations. Blade vibrations in aircraft engines, for example, are often induced by interactions between blades and fluid and the associated fatigue phenomena can give rise to catastrophic failures [1–3]. Typically, passive damping systems, such as friction damping, are used to increase the blade life. These systems are very effective, but, in contrast to active damping elements, they are not able to change their characteristics depending on the system response. In the last two decades, the adoption of piezoelectric elements has received considerable attention by many researchers for its potential applicability to different areas of mechanical, aerospace, aeronautical, and civil engineering. These elements have an interesting coupling between electrical and mechanical properties: a deformation appears when an electric field is applied and vice versa [4, 5]. Their effectiveness to damp a particular excited mode or a multimode combination strongly depends on their position; in fact the study of their optimal position has received increasing attention. Typically, the aim of these studies is to find the position that minimizes an objective function or maximizes the degree of modal controllability (see [6, 7] for a review). The first study concerned with the optimal position to damp a specified mode has been that of Crawley and de Luis [4]. They found that the actuators should be in regions of higher average strain. Analogous results have been found by other researchers [8, 9]. For a cantilever beam Sunar and Rao [10], Demetriou [11], and Bruant et al. [12] found that the closer the piezoelectric actuators are to the fixed end, the more efficient they are. For a simply supported beam Yang et
References
[1]
E. Poursaeidi and M. Salavatian, “Fatigue crack growth simulation in a generator fan blade,” Engineering Failure Analysis, vol. 16, no. 3, pp. 888–898, 2009.
[2]
L. Witek, “Experimental crack propagation and failure analysis of the first stage compressor blade subjected to vibration,” Engineering Failure Analysis, vol. 16, no. 7, pp. 2163–2170, 2009.
[3]
J. Kubiak Sz, G. Urquiza B, J. García C, and F. Sierra E, “Failure analysis of steam turbine last stage blade tenon and shroud,” Engineering Failure Analysis, vol. 14, no. 8, pp. 1476–1487, 2007.
[4]
E. F. Crawley and J. de Luis, “Use of piezoelectric actuators as elements of intelligent structures,” AIAA Journal, vol. 25, no. 10, pp. 1373–1385, 1987.
[5]
F. Botta and G. Cerri, “Wave propagation in Reissner-Mindlin piezoelectric coupled cylinder with non-constant electric field through the thickness,” International Journal of Solids and Structures, vol. 44, no. 18-19, pp. 6201–6219, 2007.
[6]
M. I. Frecker, “Recent advances in optimization of smart structures and actuators,” Journal of Intelligent Material Systems and Structures, vol. 14, no. 4-5, pp. 207–216, 2003.
[7]
V. Gupta, M. Sharma, and N. Thakur, “Optimization criteria for optimal placement of piezoelectric sensors and actuators on a smart structure: a technical review,” Journal of Intelligent Material Systems and Structures, vol. 21, no. 12, pp. 1227–1243, 2010.
[8]
K. D. Dhuri and P. Seshu, “Piezo actuator placement and sizing for good control effectiveness and minimal change in original system dynamics,” Smart Materials and Structures, vol. 15, no. 6, pp. 1661–1672, 2006.
[9]
K. R. Kumar and S. Narayanan, “Active vibration control of beams with optimal placement of piezoelectric sensor/actuator pairs,” Smart Materials and Structures, vol. 17, no. 5, Article ID 055008, 2008.
[10]
M. Sunar and S. S. Rao, “Distributed modeling and actuator location for piezoelectric control systems,” AIAA Journal, vol. 34, no. 10, pp. 2209–2211, 1996.
[11]
M. A. Demetriou, “Numerical algorithm for the optimal placement of actuators and sensors for flexible structures,” in Proceedings of the American Control Conference, pp. 2290–2294, Chicago, Ill, USA, June 2000.
[12]
I. Bruant, G. Coffignal, F. Lene, and M. Verge, “A methodology for determination of piezoelectric actuator and sensor location on beam structures,” Journal of Sound and Vibration, vol. 243, no. 5, pp. 861–882, 2001.
[13]
Y. Yang, Z. Jin, and C. K. Soh, “Integrated optimal design of vibration control system for smart beams using genetic algorithms,” Journal of Sound and Vibration, vol. 282, no. 3–5, pp. 1293–1307, 2005.
[14]
R. Barboni, A. Mannini, E. Fantini, and P. Gaudenzi, “Optimal placement of PZT actuators for the control of beam dynamics,” Smart Materials and Structures, vol. 9, no. 1, pp. 110–120, 2000.
[15]
O. J. Aldraihem, T. Singh, and R. C. Wetherhold, “Optimal size and location of piezoelectric actuator/sensors: practical considerations,” Journal of Guidance, Control, and Dynamics, vol. 23, no. 3, pp. 509–515, 2000.
[16]
A. Baz and S. Poh, “Performance of an active control system with piezoelectric actuators,” Journal of Sound and Vibration, vol. 126, no. 2, pp. 327–343, 1988.
[17]
S. M. Yang and Y. J. Lee, “Optimization of non-collocated sensor/actuator location and feedback gain in control systems,” Smart Materials and Structures, vol. 2, no. 2, pp. 96–102, 1993.
[18]
S. M. Yang and Y. J. Lee, “Vibration suppression with optimal sensor/actuator location and feedback gain,” Smart Materials and Structures, vol. 2, no. 4, pp. 232–239, 1993.
[19]
Q. Wang and C. M. Wang, “Optimal placement and size of piezoelectric patches on beams from the controllability perspective,” Smart Materials and Structures, vol. 9, no. 4, pp. 558–567, 2000.
[20]
A. Hohl, M. Neubauer, S. M. Schwarzendahl, L. Panning, and J. Wallaschek, “Active and semiactive vibration damping of turbine blades with piezoceramics,” in Active and Passive Smart Structures and Integrated Systems, vol. 7288 of Proceedings of SPIE, San Diego, Calif, USA, March 2009.
[21]
I. Goltz, H. Bhmer, R. Nollau, J. Belz, B. Grueber, and J. R. Seume, “Piezo-electric actuation of rotor blades in an axial compressor with piezoceramics,” in Proceedings of 8th European Conference on Turbomachinery (ETC '09), Graz, Austria, March 2009.
[22]
A. J. Provenza and C. R. Morrison, “Control of fan blade vibrations using piezo-electric and bi-directional telemetry,” in Proceedings of the ASME Turbo Expo, Vancouver, Canada, June 2011.
[23]
S. M. Lin, I. C. Mao, and J. H. Lin, “Vibration of a rotating smart beam,” AIAA Journal, vol. 45, no. 2, pp. 382–389, 2007.
[24]
Comsol Multiphysics User’s Guide Version 4.3.
[25]
F. Botta, N. Marx, S. Gentili et al., “Optimal placement of piezoelectric plates for active vibration control of gas turbine blades: experimental results,” in Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems, vol. 8345 of Proceedings of SPIE, San Diego, Calif, USA, March 2012.
[26]
F. Botta, N. Marx, D. Dini, R. de Lieto Vollaro, and G. Battista, “Experimental results for optimal placement of piezoelectric plates for active vibration control of a cantilever beam,” International Journal of Engineering and Technology, vol. 5, no. 5, 2013.
[27]
N. Marx, Counter vibration on gas turbine compressor blades via piezoelectric plates [M.S. thesis], Imperial College, London, UK, 2012.
[28]
F. Botta, N. Marx, C. W. Schwingshackl, G. Cerri, and D. Dini, “A wireless vibration control technique for gas turbine blades using piezoelectric plates and contactless energy transfer,” in Proceedings of the ASME Turbo Expo, San Antonio, Tex, USA, June 2013.
[29]
F. Botta, D. Dini, and R. de Lieto Vollaro, “A new function for the optimal placement of piezoelectric plates to control multimode vibrations of a rotating beam,” International Journal of Engineering and Technology, vol. 5, no. 5, 2013.
[30]
G. Cerri, M. Gazzino, F. Botta, and C. Salvini, “Production planning with hot section life prediction for optimum gas turbine management,” International Journal of Gas Turbine, Propulsion and Power Systems, vol. 2, no. 1, pp. 9–16, 2008.
[31]
F. Botta and G. Cerri, “Shock response spectrum in plates under impulse loads,” Journal of Sound and Vibration, vol. 308, no. 3–5, pp. 563–578, 2007.
