Multi-layered Shell Actuators

  • Hornsen (HS) TzouEmail author
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 247)


In this chapter, a theoretical development of a multi-layered thin shell distributed actuator is presented. The distributed actuator layers can be made of electromechanical sensitive materials which respond to externally supplied voltages and generate local control forces for active distributed vibration controls. Based on the assumptions, dynamic equations for the generic multi-layered thin shell actuator (with distributed control layers) were developed using Kirchhoff-Love’s theory and Hamilton’s principle. The system equations are generic and can be simplified to apply to many other common geometries and structures, such as plates (e.g., circular or rectangular), other conventional shells (e.g., cylindrical shell, spheres), beams, etc. The common geometries can be defined by the fundamental form, Lamé parameters, radii of curvatures, etc. It should be noted that the deformations resulting from transverse shears and rotatory inertias were neglected in the derivations.


  1. Baily, T. and Hubbard, J.E., 1987, “Distributed Piezoelectric Polymer Active Vibration Control of a Cantilever Beam,” J. of Guidance, Control, and Dynamics, Vol.8 No.5, pp.605-611.Google Scholar
  2. Chai, W.K., DeHaven, J.G. and Tzou, H.S., 2005, “Distributed Micro-control Characteristics of Shallow Conical Shell Sections,” Journal of Vibration and Control, Vol.11, No.11, pp.1397–1411.Google Scholar
  3. Chai, W.K., Tzou, H.S. and Higuchi, K., 2006, “Micro-actuation Characteristics of Conical Shell Sections,” Journal of Sound & Vibration. Vol.293, No.1–2, pp.286–298.Google Scholar
  4. Crawley, E.F. and de Luis, J., 1987, “Use of Piezoelectric Actuators as Elements of Intelligent Structures,” AIAA Journal, Vol.25, No.10, pp.1373-1385.CrossRefGoogle Scholar
  5. DeHaven, J.G., Han, Y. and Tzou, 2007, “Transition of Membrane/bending Neural Signals on Transforming Adaptive Shells,” Journal of Vibration and Control, Vol.13, No.7, pp.1007–1029.CrossRefGoogle Scholar
  6. Hanagud, S. and Obal, M.W., 1988, “Identification of Dynamic Coupling Coefficients in a Structure with Piezoelectric Sensors and Actuators,” Proc. of AIAA/ASME/AHS 29th Structures, Structural Dynamics, and Materials Conference, (Paper No.88-2418), Part-3, pp.1611-1620.Google Scholar
  7. Howard, R.V., Chai, W.K. and Tzou, H.S., 2001, “Modal Voltages of Linear and Non-Linear Structures Using Distributed Artificial Neurons,” Mechanical Systems and Signal Processing 15, 629–640.CrossRefGoogle Scholar
  8. Hu, S.D., Li, H. and Tzou, H.S., 2013a, “Flexoelectric responses of circular rings,” Journal of Vibration and Acoustics -Transactions of the ASME, 2013, 135(2): 021003.CrossRefGoogle Scholar
  9. Hu, S.D., Li, H. and Tzou, H.S., 2013b, “Distributed Neural Signals on Parabolic Cylindrical Shells,” Journal of Sound and Vibration 332, 2984–3001.CrossRefGoogle Scholar
  10. Li, H.Y., Li, H. and Tzou, H.S., 2015, “Frequency Control of Beams and Cylinrical Shells with Light-Activated Shape Memory Polymers,” ASME Journal of Vibration and Acoustics, 37, p.011006.Google Scholar
  11. Li, H., Hu, S. D., Tzou, H. S., and Chen, Z. B., 2012, “Optimal Vibration Control of Conical Shells with Collocated Helical Sensor/Actuator Pairs,” Journal of Theoretical and Applied Mechanics, 50(3), pp.769–784.Google Scholar
  12. Love, A.E.N., 1888, “On the Small Free Vibrations and Deformations of Thin Elastic Shells,” Phil. Trans. Royal Society (London), Vol.179A, pp.491-546.Google Scholar
  13. Smithmaitrie, P. and Tzou, H.S., 2004, Micro-control actions of actuator patches laminated on hemispherical shells, Journal of Sound and Vibration, 277, 691–710.CrossRefGoogle Scholar
  14. Smithmaitrie, P. and Tzou, H.S., 2005, “Electro-dynamics, Micro-actuation and Design of Ultrasonic Curvilinear Arc Stators,” Journal of Sound & Vibration, Vol.284, pp.635–650.Google Scholar
  15. Soedel, W., 1981, Vibrations of Shells and Plates, Marcel Dekker Inc., New York.Google Scholar
  16. Tzou, H.S., 1987, “Active Vibration Controls of Flexible Structures Via Converse Piezoelectricity,” Developments in Mechanics, Vol.14(c), pp.1201-1206. The 20th Midwest Mechanics Conference, West Lafayette, IN, August 1987.Google Scholar
  17. Tzou, H.S., 1988, “Integrated Sensing and Adaptive Vibration Suppression of Distributed Systems,” Recent Advances in Control of Nonlinear and Distributed Parameter Systems, ASME-DSC-Vol.(10), pp.51-58, 1988 ASME WAM, Chicago, Illinois, Nov.27-Dec.2, 1988.Google Scholar
  18. Tzou, H.S. and Anderson, G.L. (Editors), Intelligent Structural Systems, ISBN No.0-7923-1920-6, 488 pages, Book, Kluwer Academic Publishers, August 1992.Google Scholar
  19. Tzou, H.S., Bao, Y., Venkayya, V.B., 1996a, “Study of Segmented Transducers Laminated on Cylindrical Shells, Part 1: Sensor Patches,” Journal of Sound & Vibration, Vol.197, No.2, pp.207–224.Google Scholar
  20. Tzou, H.S., Bao, Y., Venkayya, V.B., 1996b, “Study of Segmented Transducers Laminated on Cylindrical Shells, Part 2: Actuator Patches,” Journal of Sound & Vibration, Vol.197, No.2, pp.225–249, October 1996.Google Scholar
  21. Tzou, H. S., Ding, J. H., and Hagiwara, I., 2002, “Micro-Control Actions of Segmented Actuator Patches Laminated on Deep Paraboloidal Shells,” JSME International Journal Series C-Mechanical Systems Machine Elements and Manufacturing, 45(1), pp.8–15.CrossRefGoogle Scholar
  22. Tzou, H.S., Chai, W.K. and Wang, D.W., 2003, “Modal Voltages and Micro-Signal Analysis of Conical Shells of Revolution,” Journal of Sound and Vibration 260, 589–609.CrossRefGoogle Scholar
  23. Tzou, H. S., Chai, W. K., and Wang, D. W., 2004, “Micro-Control Actions and Location Sensitivity of Actuator Patches Laminated on Toroidal Shells,” Journal of Vibration and Acoustics-Transactions of the ASME, 126(2), pp.284–297.CrossRefGoogle Scholar
  24. Tzou, H. S. and Ding, J. H., 2004, “Optimal Control of Precision Paraboloidal Shell Structronic Systems,” Journal of Sound and Vibration, 276(1–2), pp.273–291.MathSciNetCrossRefGoogle Scholar
  25. Tzou, H.S. Lee, H.-J. and Arnold, S.M., “Smart Materials, Precision Sensors/Actuators, Smart Structures and Structronic Systems,” Mechanics of Advanced Materials and Structures, Vol.11, pp.367–393, 2004.CrossRefGoogle Scholar
  26. Tzou, H. S., Smithmaitrie, P. and Ding, J. H., 2002, “Sensor Electromechanics and Distributed Signal Analysis of Piezo(Electric)-Elastic Spherical Shells,” Mechanical Systems and Signal Processing (Journal of), Vol.16(2–3), pp.185–199.CrossRefGoogle Scholar
  27. Tzou, H.S. and Tseng, C.I., 1988a, “Sensing and Adaptive Vibration Control of Flexible Distributed Mechanical Systems,” Machine Dynamics and Engr. Applications, Xian Jiaotong University Press, China, Vol.1, pp.G1-G6, August 1988.Google Scholar
  28. Tzou, H.S. and Tseng, C.I., 1988b, “Active Vibration Controls of Distributed Parameter Systems by Finite Element Method,” ASME Computers in Engineering 1988, Vol.3, pp.599-604.Google Scholar
  29. Tzou, H.S. and Tseng, C.I., 1988c, “Development of a Thin Piezoelectric Finite Element Applied to Distributed Sensing and Vibration Controls,” ASME Paper No. 88-WA/CIE-2, 1988 WAM, Chicago, Illinois, Nov.27-Dec.2, 1988.Google Scholar
  30. Tzou, H.S. and Wang, D.W., 2002, “Micro-Sensing Characteristics and Modal Voltages Of Linear/Non-Linear Toroidal Shells,” Journal of Sound and Vibration, 2002, 254(2): 203–218.CrossRefGoogle Scholar
  31. Tzou, H.S. and Wang, D.W., 2003, “Vibration Control of Toroidal Shells with Parallel Aand Diagonal Piezoelectric Actuators,” Journal of Pressure Vessel Technology -Transactions of the ASME, 2003, 125(2): 171–176.CrossRefGoogle Scholar
  32. Tzou, H.S. and Zhong, J., 1996, “Spatially Filtered Vibration Control of Cylindrical Shells,” Shock and Vibration Journal, Vol.3, No.4, pp.269–278.CrossRefGoogle Scholar
  33. Tzou, H.S., and Zhou, Y., 1997, “Nonlinear Piezothermoelasticity and Multi field Actuations, Part 2: Control of Nonlinear Buckling and Dynamics,” ASME Transactions, Journal of Vibration & Acoustics, Vol.119, pp.382–389.Google Scholar
  34. Tzou, H.S., Zhong, J.P., and Natori, M.C., 1993, “Sensor Mechanics of Distributed Shell Convolving Sensors Applied to Flexible Rings,” ASME Transactions, Journal of Vibration & Acoustics, Vol.115, No.1, pp.40–46.CrossRefGoogle Scholar
  35. Tzou, H.S., Zhu, Y.T. and Hagiwara, I, 2002, “Distributed Precision Control of Structronic Shells and Common Shapes– A New Approach,” Recent Research Developments in Sound & Vibration, ISBN: 81-7895-031-6, Vol.1, Part-2, pp.613–646, TransWorld Research, 2002.Google Scholar
  36. Wada, B.K, Fanson, J.L. & Crawley, E.F., 1989, “Adaptive Structures,” Adaptive Struatures, ASME AD-Vol.15, pp.1-8, 1989 ASME Winter Annual Meeting, San Francisco, CA, December. (5/MultiShlAct.BkPzSm2).Google Scholar
  37. Wang, D.W., Tzou, H.S., Arnold, S.M. and Lee, H.J., 2006, “Control of Static Shape, Dynamic Oscillation and Thermally Induced Vibration of Nozzles,” ASME Transactions, Journal of Pressure Vessel Technology, Vol.128, pp.357–363.CrossRefGoogle Scholar
  38. Yue, H. H., Deng, Z. Q., and Tzou, H.S., 2008, “Optimal Actuator Locations and Precision Micro-Control Actions on Free Paraboloidal Membrane Shells,” Communications in Nonlinear Science and Numerical Simulation, 13(10), pp.2298–2307.CrossRefGoogle Scholar
  39. Zhou, Y.H., and Tzou, H.S., 2000, “Active Control of Nonlinear Piezoelectric Circular Shallow Spherical Shells,” International Journal of Solids and Structures, 37(12), pp.1663–1677.CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Mechanics and Control of Mechanical Structures; Interdisciplinary Research Institute, College of Aerospace EngineeringNanjing University of Aeronautics and AstronauticsNanjingChina

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