Abstract
The present Lecture is concerned with vibrations of linear elastic solids and structures. Some part of the boundary of the structure is suffering a prescribed large rigid-body motion, while an imposed external traction is acting at the remaining part of the boundary, together with given body forces in the interior. Due to this combined loading, vibrations take place. The latter are assumed to remain small, such that the linear theory of elasticity can be applied. As an illustrative example for the type of problems in hand, we mention the flexible wing of an aircraft in flight. In this example, the rigid-body motion is defined through the motion of the comparatively stiff fuselage to which a part of the boundary of the wing is attached. The goal of the present paper is to derive a time-dependent distribution of actuating stresses produced by additional eigenstrains, such that the deformations produced by the imposed forces and the rigid-body motion are exactly compensated. This is called a shape control problem, or a deformation compensation problem. We show that the distribution of the actuating stresses for shape control must be equal to a quasi-static stress distribution that is in temporal equilibrium with the imposed forces and the inertia forces due to the rigid-body motion. Our solution thus explicitly reflects the non-uniqueness of the inverse problem under consideration. The present Lecture extends previous results by Irschik and Pichler (2001, 2004) for problems without rigid-body degrees of freedom. As a computational example, we present results for a rectangular domain in a state of plane strain under the action of a translatory support motion.
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Belyaev, A.K. (2004). Basics of Continuum Mechanics. In: Irschik, H. and Schlacher, K., eds., Advanced Dynamics and Control of Structures and Machines. CISM Courses and Lectures No. 444, Wien—New York: Springer-Verlag.
Carlson, D. E. (1972). Linear Thermoelasticity. In Flügge, S., ed., Handbuch der Physik, Vol. VIa/2, Berlin: Springer-Verlag.
Chandrasekharaiah, D. S. and Debnath, L. (1994). Continuum Mechanics. Boston: Ac. Press.
Gurtin, M. E. (1972). The linear theory of elasticity. In Flügge, S., ed., Handbuch der Physik, Vol. VIa/2, Berlin: Springer-Verlag.
Haftka, R.T., and Adelman, H.M. (1985). An analytical investigation of shape control of large space structures by applied temperatures. AIAA Journal 23: 450–457.
Haupt, P. (2002). Continuum Mechanics and Theory of Materials. Berlin: Springer-Verlag.
Irschik, H. (2002). A Review on Static and Dynamic Shape Control of Structures by Piezoelectric Actuation. Engineering Structures 24: 5–11.
Irschik, H. (2004). A Treatise on the Equations of Balance and on the Jump Relations in Continuum Mechanics. In: Irschik, H. and Schlacher, K., eds., Advanced Dynamics and Control of Structures and Machines. CISM Courses and Lectures No. 444, Wien—New York: Springer-Verlag.
Irschik, H., Holl, H. J. and Hammelmüller, F. (2004). The Rayleigh-Ritz Technique and the Lagrange Equations in Continuum Mechanics: Formulations for Material and Non-Material Volumes. In: Irschik, H. and Schlacher, K., eds., Advanced Dynamics and Control of Structures and Machines. CISM Courses and Lectures No. 444, Wien—New York: Springer-Verlag.
Irschik, H., and Pichler, U. (2001). Dynamic shape control of solids and structures by thermal expansion strains. Journal of Thermal Stresses 24: 565–576.
Irschik, H., and Pichler, U. (to appear 2004 ). An Extension of Neumann’s Method for Shape Control of Force-Induced Elastic Vibrations by Eigenstrains. International Journal of Solids and Structures.
Irschik, H., Krommer, M., and Pichler, U. (2001). Collocative Control of Beam Vibrations with Piezoelectric Self-Sensing Layers. In Gabbert, U., and Tzou, H.S.., eds., Proceedings of the IUTAM-Symposium on Smart Structures and Structronic Systems, Dordrecht 2001. Dordrecht: Kluwer, 315–322.
Irschik, H., Krommer, M., and Pichler, U. (2003). Dynamic shape control of beam-type structures by piezoelectric actuation and sensing. Int. Journal Applied Electromagnetics and Mechanics 17 (1–3): 251–258.
Irschik H., Nader M., Zehetner C. (2003). Exact Cancellation of Vibrations in Elastic Structures Performing Large Rigid Body Motions (Sectional Key Note Lecture). Proc. of 10th International Congress on Sound and Vibration, Stockholm, Sweden, Vol. 7: 3487–3498.
Irschik, H., and Ziegler, F., (2001). Eigenstrain without Stress and Static Shape Control of Structures. AIAA-Journal 39: 1985–1999.
Nader, M., Gattringer, H., Krommer, M., and Irschik, H., (2003). Control of Flexural Vibrations of Circular Plates by Shaped Piezoelectric Actuation. ASME Journal of Vibration and Acoustics 125: 88–94.
Nader M., Irschik H. (2003). Suppression of Force–Induced Circular Plate Vibrations by Thermal Expansion Strains. Proc. of 5th International Congress on Thermal Stresses and Related Topics, Blacksburg, Virgina, USA, MA–3–4–1 – MA–3–4–4.
Truesdell, C., Toupin, R. (1960). The Classical Field Theories. In: Handbuch der Physik, Vol. III/1, Springer-Verlag.
Ziegler, F., (1998). Mechanics of Solids and Fluids (2nd edition, 2nd corr. print). New York: Springer-Verlag.
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Irschik, H., Pichler, U., Nader, M., Zehetner, C. (2004). Compensation of Deformations in Elastic Solids and Structures in the Presence of Rigid-Body Motions. In: Irschik, H., Schlacher, K. (eds) Advanced Dynamics and Control of Structures and Machines. International Centre for Mechanical Sciences, vol 444. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2774-2_5
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DOI: https://doi.org/10.1007/978-3-7091-2774-2_5
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