Manfred Nader

Shape Control of Flexural Vibrations of Circular Plates by Shaped Piezoelectric Actuation

Vibrations of smart elastic plates with integrated piezoelectric actuators are considered. Piezoelastic layers are used to generate a distributed actuation of the plate. A spatial shape function of the piezoelastic actuators is sought such that flexural vibrations induced by external forces can be completely nullified. An analytic solution of this problem is worked out for the case of clamped circular plates with a spatially constant force loading. The Kirchhoff theory of thin plates is used to derive this analytic solution. Our result is successfully validated by means of coupled 3-dimensional finite-element computations.

Collocative PD Control of Circular Plates with Shaped Piezoelectric Actuators/Sensors

Abstract: In this paper, flexural vibrations of smart circular plates are considered. Distributed actuators and sensors are realized by means of spatially shaped piezoelastic layers. We use piezoelectric actuating layers shaped in order to annihilate deflections due to known external transverse forces. Such spatial shape functions correspond to the distribution of the static bending moment in the form of the so-called Marcus moment of the plate due to the external forces. When only the spatial distribution of the external forces is known, but their time evolution may be arbitrary, an automatic control system must be used in order to minimize the plate vibrations. To utilize the concept of collocated sensing, a shaped piezoelectric sensor is required that measures the so-called natural output. It is shown that the above shape function of the actuator can be used as the shape function of the sensor in order to achieve this goal. Hence, the shaped piezoelectric layer can be used as a self-sensing actuator without violating the requirements of collocated control. We develop the corresponding transfer function for the case of a clamped circular plate with a space-wise constant transverse force. This transfer function is used for the design of a self-sensing PD controller. It is proven that the energy of the closed-loop system becomes a positive definite function, its time derivative being negative semi-definite, such that the PD-controlled plate is stable. In a numerical study, output and input signals of the closed loop are discussed. This study successfully demonstrates the ability of the proposed method.

A Model Reduction Technique for High Speed Flexible Rotors

The present paper deals with a problem-oriented novel model reduction technique for elastic rotors with a high and/or rapidly changing axial speed. Numerical results are presented for various run-up simulations with different values of unbalance mass and internal damping ratio. An important result of this study is that the time-derivatives of the tilting angles and the flexible coordinates should not be considered as small when studying resonance and stability phenomena.

Mechatronics - The Innovation Request

Smart structures, which are equipped with piezoelectric actuators and sensors, and which involve automatic control, represent an important branch of Mechatronics. This paper gives a review over own research on smart structures, which has been performed during the last decade based on the principles of analogy and interdisciplinarity. The latter principles form a research strategy, which seems to be perfectly suited in order to answer the innovation request in Mechatronics, namely to decrease the time-lag between consecutive steps in the scientific development, and to keep fundamental and applied research in close co-operation. We start our report with a short excursion into the history of engineering sciences, in order to demonstrate this time-lag, where we use the history of elastic and piezoelastic plates as an example, and we discuss the notions of analogy and interdisciplinarity as means to systematically decrease the timelag. In our own work, we particularly have used an eigenstrain analogy as guideline. In the light of this analogy, various own works in the following fields are reviewed: Accurate electromechanically modeling; dynamic shape control by piezoelectric actuation and sensing; extension of dynamic shape control to closed loop control and active noise cancellation.

Compensation of Deformations in Elastic Solids and Structures in the Presence of Rigid-Body Motions

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.

Tracking of transient displacements of plates with support excitations

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.

Collocated actuator/sensor design for shape control of subregions of structures

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.

On a Momentum Based Version of Lagrange’s Equations

The present contribution intends to promote an alternative form of Lagrange ’ s Equations, which rests upon the notion of momentum. We first present a short derivation of the proposed momentum based version of Lagrange’s Equations. From this derivation it becomes apparent that the derivatives of the kinetic energy with respect to the generalized coordinates must cancel out in the original kinetic energy based version of Lagrange’s Equations, and thus need not to be computed. The presented momentum based formulation of Lagrange’s Equations is valid for deformable bodies, modeled in the framework of the Ritz approximation technique, where rigid-body degrees-of-freedom may be present. After having stated this momentum based version of Lagrange’s Equations, we restrict to plane motions of rigid bodies, and demonstrate our proposed formulation for the case of a rotational degree of freedom, where we present an additional connection to the notion of momentum of the rigid body, particularly to angular momentum. Finally, we present the exemplary application to systems consisting of two rigid bodies, namely the pendulum with a point mass and movable support, and the Sarazin pendulum consisting of a rigid rotating disc and an attached point mass.