Uwe Pichler

Mysel’s Formula for Small Vibrations Superimposed Upon Large Static Deformations of Piezoelastic Bodies

Maysel’s formula was originally developed for the linear static theory of thermoelasticity, [I]. It renders the thermoelastic displacement by a convenient volume integration, using isothermal influence functions as the kernels of the integrals. Maysel’s formula was brought to the knowledge of a wider audience through the book on thermoelasticity by W. Nowacki [2]. Later, Nowacki presented an important extension of this formula to the dynamic problem of piezo-thermoelasticity, [3]. Nowacki’s extension makes use of Green’s functions of the coupled piezo-thermoelastic problem. The proof of Maysel’s original formula presented by Parkus [4] demonstrates that the simple form of Maysel’s original formulation can be retained also in the case of coupling between temperature and elastic deformation, because the latter coupling needs not to be addressed in the proof The value of Maysel’s original formula thus lies in the fact that known solutions of an auxiliary isothermal force problem can be utilized for presenting a formal solution of the linear coupled thermoelastic problem. The coupling to the thermal field can be often neglected in practical applications, particularly in the case of quasi-static motions.

Annihilation of beam vibrations by shaped piezoelectric actuators: Coupled theory

Flexural vibrations of smart beams are studied in this paper. Layers made of piezoelectric material are used to perform a distributed actuation of the beam. Special emphasis is given to the following actuator shaping problem: A spatial shape function of the distributed actuator is sought such that vibrations induced by external forces are completely annihilated. The formulation is restricted to forces with a given spatial distribution and an arbitrary time evolution of their intensity. The scope is to derive a class of easy to obtain analytic solutions of this inverse problem. Actuator equations are used in the present contribution which take into account the interaction of mechanical and electrical fields. Extending the preliminary results, the above actuator shaping problem is solved in the context of these coupled equations. Beams with different boundary conditions are considered. Shape functions responsible for nonuniqueness of the shaping problem are also considered. These nilpotent solutions may be ...

A body-force analogy for dynamics of elastic bodies with eigenstrains

In the linear static theory of thermoelasticity, the body force analogy dates back to Duhamel. In its classical form, it reads: Consider the static deformation of an isotropic linear thermoelastic body under the action of a given temperature. Then the thermal stresses can be obtained by addition of an imaginary pressure to the isothermal stresses, which follow by solving the isothermal governing equations with certain imaginary body forces and surface tractions. Moreover, the thermal displacements due to the given temperature are identical to the isothermal displacements due to the imaginary body forces and surface tractions. In the present paper, a dynamic extension of this body force analogy is presented in the framework of the three-dimensional theory of linear anisotropic elastodynamics with eigenstrains. Our formulation thus includes not only effects such as thermal expansion strains or piezoelectric expansion strains, but also inelastic parts of strains in the framework of a geometrically linear theory. We treat two problems, namely a problem with a given distribution of eigenstrains and with assigned body and surface forces, and a second problem without eigenstrains, but with an auxiliary system of body and surface forces, which we determine such that the required body force analogy holds. It turns out that all what is needed for an extension of the classical static analogy to dynamics is the requirement of identical initial conditions and displacement boundary conditions in the two problems under consideration. We finally present the proper form of the jump conditions for balance of momentum that must be taken into account in the auxiliary problem in order that the body force analogy holds in the presence of a singular surface also.

Dynamic stress compensation by smart actuation

The actuating physical mechanisms utilized in smart materials can be described by eigenstrains. E.g., the converse piezoelectric effect in a piezoelastic body may be understood as an actuating eigenstrain. In the last decades, piezoelectricity has been extensively applied for the sake of actuation and sensing of structural vibrations. An important field of research in this respect has been devoted to the goal of compensating force-induced vibrations by means of eigenstrains. Considering the state-of-the-art in structural control and smart materials, almost no research has been performed on the problem of compensating stresses in force-loaded engineering structures by eigenstrains. It is well-known that stresses can influence the characteristics and the age of structures in various unpleasant ways. The present contribution is concerned with corresponding concepts for stress compensation which may have a highly beneficial influence upon the lifetime and structural integrity of the structure under consideration. We discuss the possibilities offered by displacement compensation to reduce the stresses to their quasi-static parts. As a numerical example, we consider the step response of an irregularly shaped cantilevered elastic plate under the action of an assigned traction at its boundary.

Dynamic shape control of flexural beam vibrations: an experimental setup

The topic of the present contribution in an experimental verification of the active control of flexural vibrations of smart beams. The spatial distribution of the piezoelectric actuator is determined in such a way that deformations induced by assigned forces with a given spacewise distribution and an arbitrary but known time-evolution are exactly eliminated by the piezoelectric actuation. In the present paper, the theoretical solution of this dynamic shape control problem is first derived from an electromechanically coupled theory in a three dimensional setting, where we make use of the theorem of work expended, and from Graffis theorem. This more general formulation is specialized to the case of beams, where the kinematic hypothesis of Bernoulli-Euler and a uni-axial stress state are assumed, and the direct piezoelectric effect is neglected. We thus re-derive some results for beams published by our group in earlier contributions. It has been found that if the piezoelectric actuator shape-function is chosen as the spanwise distribution of the quasi-static bending moment due to assigned transverse forces, and if additionally the time-evolution of the applied electrical potential difference is chosen to be identical to the negative time-evolution of the assigned forces, the beam deflections due to these forces are exactly eliminated by the piezoelectric actuation. In the present paper, the validity of this theoretical solution is studied in an experimental set-up. As a result of the performed experiments, the elimination of force-induced vibrations of smart beams by shaped piezoelectric actuators is demonstrated for various time-evolutions of exciting single forces. The obtained experimental results give evidence for the validity of the presented theoretical solution of the dynamic shape control problem.

Maysel's formula of thermoelasticity extended to anisotropic materials at finite strain

Abstract The present paper is devoted to an extension of Maysels formula from the linear theory of thermoelasticity to the geometrically non-linear theory of anisotropic solids and structures. The material description of continuum mechanics is used, and the constitutive equations of St. Venant and Kirchhoff are considered under two circumstances. First, we model the thermally induced deformation of anisotropic solids from an undistorted reference state. This leads to a non-linear integral equation for the thermal deformation, extending Maysels formula of the infinitesimal theory to the regime of moderately large strains. Secondly, the linearized form of the constitutive relation is used to describe an infinitesimal strain superimposed upon a given, intermediate state of stress in a hyperelastic material of arbitrary type. The intermediate strain needs not be small. It is shown that, when properly interpreted, Maysels formula may be applied directly to this second case, without any additional terms. The presented results should be of a rather general interest, since they lead to an efficient and powerful representation of the thermal response of anisotropic materials. As a structural application, Maysels formula is subsequently derived for shear-deformable beams made of a St. Venant–Kirchhoff material. Geometric non-linearity is taken into account according to the assumptions of v. Karman, and the influence of shear is considered in the sense of Timoshenko. A semi-analytic solution procedure is derived for the case of simply supported beams with fixed ends. The thermally induced deflection is derived in closed form, and a non-linear equation is presented for the corresponding normal force. In the post-buckling regime, three branches of the solution are found in the considered range of thermal loading. Thermally loaded beams made of pyrolitic–graphite type material are studied, and the strong influence of the characteristic parameter of anisotropy is demonstrated. Stability of the solution is discussed, and results are compared to finite element computations in the isotropic case. An excellent agreement is found, and a paradoxical behavior of the finite element code abaqus is clarified.

Design of sensors/actuators for structural control of continuous CMA systems

Smart structure technology has become a key technology in the design of modern, so-called intelligent, civil, mechanical and aerospace (CMA) systems. One key aspect for a successful design is the communication between structure and controller, for which sensors and actuators are responsible. In continuous CMA systems a crucial point is the distribution of sensors to obtain proper information and the distribution of actuators to influence the behavior of the structure properly. Finding these distributions is the topic of this paper. A common strategy for the modeling of continuous CMA systems is based on the linearized theory of elasticity; within this paper we consider a three-dimensional linear elastic background body with sources of self-stress. These self-stresses can be produced by smart materials, which exhibit the well known strain induced actuation mechanism; as many of the modern smart materials have both, actuation and sensing properties, we assume the sensing be based on the same mechanism. We show that a suitable distribution of sensors results into a sensor signal proportional to kinematical entities (e.g. displacement), whereas a suitable distribution of the actuation results in actuators that act like dynamical entities (e.g. force). Our design strategy automatically results into collocated sensor/actuator pairs; this design is highly suitable from a control point of view, because it allows the application of common control strategies in a straightforward manner; e.g. a simple PD-controller ensures stability of the closed loop system.

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.

STATIC SHAPE CONTROL OF FORCE-INDUCED INFINITESIMAL DEFORMATIONS SUPERIMPOSED ON LARGE DEFORMATIONS OF THERMOELASTIC BODIES

The present paper is concerned with the geometrically nonlinear static theory of anisotropic thermoelastic solids and structures. We consider infinitesimal incremental deformations superimposed on a given state of possibly large strain, the latter being called the intermediate state. Our goal is to derive a distribution of incremental thermal actuation stresses, which, when applied to the intermediate state together with a given set of incremental body forces and surface tractions, give zero incremental displacements everywhere in the body under consideration. This problem belongs to the field of static shape control, a notion originally introduced by Haftka and Adelman, who developed a procedure for determining temperatures in control elements to minimize the infinitesimal distortion of a large space antenna from its original shape. The present paper is concerned with the extension of shape control to infinitesimal force-induced static distortions from a large pre-deformation. Referring to the intermediate configuration as the reference configuration, we show that, in order to compensate the incremental force-induced deformations everywhere within the body, the incremental thermal actuation stress tensor must be equal to any statically admissible incremental first-order Piola–Kirchhoff stress tensor, a relation that is derived under the assumption that the intermediate state is infinitesimally superstable. We also discuss under which conditions it is possible to work with an isotropic thermal actuation stress. Finally, we present a formulation for shape control of infinitesimal deflections superimposed on a state of possibly large deflections of a slender beam.