Displacement tracking of pre-deformed smart structures

- Journal title : Smart Structures and Systems
- Volume 18, Issue 1, 2016, pp.139-154
- Publisher : Techno-Press
- DOI : 10.12989/sss.2016.18.1.139

Title & Authors

Displacement tracking of pre-deformed smart structures

Irschik, Hans; Krommer, Michael; Zehetner, Christian;

Irschik, Hans; Krommer, Michael; Zehetner, Christian;

Abstract

This paper is concerned with the dynamics of hyperelastic solids and structures. We seek for a smart control actuation that produces a desired (prescribed) displacement field in the presence of transient imposed forces. In the literature, this problem is denoted as displacement tracking, or also as shape morphing problem. One talks about shape control, when the displacements to be tracked do vanish. In the present paper, it is assumed that the control actuation is provided by imposed eigenstrains, e.g., by the electric field in piezoelectric actuators, or by thermal actuators, or via analogous physical effects, such as magneto-striction or pre-stress. Structures with a controlled eigenstrain-type actuation belong to the class of smart structures. The action of the eigenstrains can be conveniently characterized by actuation stresses. Our theoretical derivations are performed in the framework of the theory of small incremental dynamic deformations superimposed upon a statically pre-deformed configuration of a hyperelastic solid or structure. We particularly ask for a distribution of incremental actuation stresses, such that the incremental displacements follow exactly a prescribed trajectory field, despite the imposed incremental forces are present. An exact solution of this problem is presented under the assumption that the actuation stresses can be tailored freely and applied everywhere within the body. Extending a Neumann-type solution strategy, it is shown that the actuation stresses due to the distributed control eigenstrains must satisfy certain quasi-static equilibrium conditions, where auxiliary body-forces and auxiliary surface tractions are to be taken into account. The latter auxiliary loading can be directly computed from the imposed forces and from the desired displacement field to be tracked. Hence, despite the problem is a dynamic one, a straightforward computation of proper actuator distributions can be obtained in the framework of quasi-static equilibrium conditions. Necessary conditions for the functioning of this concept are presented. Particularly, it must be required that the intermediate configuration is infinitesimally superstable. Previous results of our group for the case of shape control and displacement tracking in linear elastic structures are included as special cases. The high potential of the solution is demonstrated via Finite Element computations for an irregularly shaped four-corner plate in a state of plain strain.

Keywords

displacement tracking;shape control;smart structures;piezoelectric actuation;pre-deformed configuration;small superimposed displacements;

Language

English

Cited by

References

1.

Chandrasekharaiah, D.S. and Debnath, L. (1994), Continuum Mechanics, Academic Press, Boston.

2.

Eringen A.C. and Maugin G.A. (1989), Electrodynamics of Continua I. Foundations and Solid Media. Springer, New York.

3.

Gurtin, M.E. (1972), The Linear Theory of Elasticity, (Ed., S. Flugge), Handbuch der Physik, Vol. VIa/2, (Ed., C. Truesdell ), Festkorpermechanik II, 1-296, Springer-Verlag, Berlin.

4.

Irschik, H. (2002), "A review on static and dynamic shape control of structures by piezoelectric actuation", Eng. Struct., 24(1), 5-11.

5.

Irschik, H., Krommer, M., Nader, M., Schoftner, J. and Zehetner, Ch. (2012), "Active and passive shape control of structures", invited Plenary Lecture, CD-Rom Proceedings of the 5th European Conference on Structural Control (EACS 2012), (Eds., E. Del Grosso and P. Basso), Genua, Italy 2012, Paper No. # 146.

6.

Irschik, H. and Krommer, M. (2013), "A review on static and dynamic shape control of structures: The period 2002-2012", invited Plenary Lecture, in: USB Proceedings Vienna Congress on Recent Advances in Earthquake Engineering and Structural Dynamics 2013 (VEESD 2013), (Eds., C. Adam, R. Heuer, W. Lenhardt and C. Schranz ), Vienna, Austria 2013, Paper No. 581.

7.

Irschik, H. and U. Pichler (2004), "An extension of Neumann's method for shape control of force-induced elastic vibrations by eigenstrains", Int. J. Solids Struct., 41(3-4), 871-88.

8.

Irschik, H. and M. Krommer (2005), "Dynamic displacement tracking of force-loaded linear elastic or viscoelastic bodies by eigenstrain-induced actuation stresses", IDETC/CIE 2005 ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference 2005, Long Beach, California, USA, ASME Paper No. DETC2005-84835.

9.

Irschik, H., Krommer, M., Nader, M., Zehetner, Ch. And Zellhofer, M. (2007), "Structural displacement tracking by eigenstrain actuation: Computational and experimental validations", CD-Rom Proc. of the Eleventh International Conference on Civil, Structural and Environmental Engineering Computing, St.

10.

Julians, Malta (2007), (Eds. B.H.V. Topping), Civil-Comp. Press, Stirlingshire, Scotland.

11.

Kamlah, M. (2001), "Ferroelectric and ferroelastic piezoceramics - modeling of electromechanical hysteresis phenomena", Continuum Mech. Therm., 13(4), 219-268

12.

Knops. R.J. and Wilkes. E.W. (1973), Theory of Elastic Stability, (Ed., S. Flugge), Handbuch der Physik, Vol. VIa, (Ed., C. Truesdell), Festkorpermechanik II, 125-302, Springer-Verlag, Berlin.

13.

Krommer, M. and Irschik, H. (2007), "Sensor and actuator design for displacement control of continuous systems", Smart Struct. Syst., 3(2), 147-172.

14.

Parkus, H. (1976), Thermoelasticity, 2nd Ed., Springer, New York.

15.

Rao, S.S. and Sunar, M. (1994), "Piezoelectricity and its use in disturbance sensing and control of flexible structures: a survey", Appl. Mech. Rev., 47(4), 113-123.

16.

Rao, S.S. and Sunar, M. (1999), "Recent advances in sensing and control of flexible structures via piezoelectric material technology", Appl. Mech. Rev., 52(1), 1-16.

17.

Schoeftner, J. and Irschik, H. (2011), "Passive shape control of force-induced harmonic lateral vibrations for laminated piezoelastic Bernoulli-Eulerbeams-theory and practical relevance", Smart Struct. Syst., 7(5), 417-432.

18.

Schoeftner, J. and Buchberger, G. (2013), "Active shape control of a cantilever by resistively interconnected piezoelectric patches", Smart Struct. Syst., 12(5), 501-521.

19.

Tiersten, H.F. (1969), Linear Piezoelectric Plate Vibrations, Plenum Press, New York.

20.

Tiersten, H.F. (1978), "Perturbation theory for linear electroelastic equations for small fields superposed on a bias", J. Acoust. Am., 64, 832-837.

21.

Wang, C.C. and Truesdell, C. (1973), Introduction to Rational Elasticity, Springer, New York.

22.

Yang, J. (2006), An Introduction to the Theory of Piezoelectricity, Springer, Boston.