Optimal motions of an elastic rod controlled by a piezoelectric actuator

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Abstract

The longitudinal vibrations of an elastic rod controlled by normal forces in the cross section, which are uniformly distributed along the length over a selected interval, are studied. Such a system can be implemented using an actuator consisting of piezoelectric elements located along the axis of the rod. Criteria for the uncontrollability of individual vibration modes are given. A generalized solution to the initial-boundary value problem is found applying d’Alembert traveling waves, which are determined on the space-time mesh formed by characteristics. Linear combinations of the traveling wave and control functions define the sought displacements and dynamic potential in the energy space. The latter in a certain way relates the momentum density and the force in the cross section. The problem is to transfer the rod to a prescribed state in a fixed time while minimizing the norm of the control force. The optimal motion and the corresponding feedforward control law are found by reducing the original problem to a one-dimensional variational one. The example shows the control of vibrations for certain geometric parameters of the piezoelectric actuator.

About the authors

G. V. Kostin

Ishlinsky Institute for Problems in Mechanics RAS

Author for correspondence.
Email: kostin@ipmnet.ru
Russian Federation, Moscow

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