The present paper is concerned with dynamic shape control of linear elastic plates under the action of transient forces,
with prescribed time-dependent boundary conditions, and with given initial conditions. We consider anisotropic linear
elastic plates. The following displacement tracking problem is treated: We ask for an additional distribution of actuation
stresses such that the resulting displacements of the plates under consideration follow exactly some desired trajectories in
every point and at every time instant. We present relations that must be satisfied for the actuation stresses in order that
this goal of transient displacement tracking is reached. The actuation stresses we have in mind for enforcing tracking of
transient displacements are induced by eigenstrains, such as thermal expansion strains or, more technologically
important, piezoelectric parts of strain. Transient vibrations of circular plates in axi-symmetric bending are studied as an
exemplary case. The vibrations are excited by support excitations. Actuation stresses are superimposed, which enforce
the plate to track prescribed transient deflections. We present analytical solutions for the tracking of prescribed plate
deflections with time-dependent support excitation. Coupling between electric and mechanical field is taken into account
already at the level of plate theory. The analytical plate solutions are validated by Finite Element computations.
Electromechanically coupled three-dimensional piezoelectric elements are used in these numerical calculations.
Excellent coincidence between the analytical and the Finite Element computations is observed.
The control of the shape of a sub - region of a structure has many important practical applications; for instance the control of the shape of a conformal antenna that is mounted to the surface of a sub - region of a structure. Given the desired shape of the sub - region one can use self - stress actuators, which only act in the sub - region itself, to implement the required control. However, if the structure is disturbed by external excitations, the actuators have to compensate the additional vibrations too; therefore, also sensors have to be designed. A proper sensor design requires the sensor to be an integrated part of the structure and to be located in the sub - region only. Using self - stress sensors is near at hand; moreover, one may even use the actuators as sensors, resulting in so - called self - sensing actuator / sensor pairs. Clearly, this procedure provides collocation between actuator and sensor automatically, which, from a control point of view, is highly desirable.
In the present paper we summarize the design of actuators for the sub - region control of a structure. Then we discuss the design of collocated sensors, their output signal and the application of a PD - control law. We show that the output signal is the natural output of the system and that the closed loop system is stable. Finally, we present numerical results for a beam type structure.
The present contribution is concerned with a thin shell, which is
excited to elastic vibrations by an imposed motion of a supporting
boundary. Piezoelectric actuation is used to generate an
additional actuation of the shell. As a practical application of
dynamic shape control, we consider the suppression of flexural and
extensional elastic vibrations, such that the shell performs a
rigid body motion only. The idea of this procedure is to eliminate
the disturbing acoustic noise caused by elastic deformations. We
first point out that a suppression of the elastic vibrations can
be achieved provided the distributed piezoelectric actuation
coincides with a statically admissible (quasi-static) membrane
force and bending moment distribution due to a fictitious inertial
body force loading of the shell. For practical applications we
assume the exciting support motion to be either translational or
rotational, and to be given in advance. As an example of practical
relevance, noise radiation of the support-excited shell-type
funnel of a magnetic resonance tomograph is considered. Due to the
complex geometry of this thin shell made of plastics, numerical
methods are used in order to treat the shape control problem. The
statically admissible membrane forces and bending moments due to
the fictitious body force loading are computed by means of the
Finite-Element-Code ANSYS. A distributed piezoelectric actuation
coinciding with these forces and moments is derived and is
approximated by a sparse distribution of piezoelectric patches. It
is numerically demonstrated that this sparse distribution is able
to suppress the elastic vibrations caused by the support
excitation of the funnel.