Hans Irschik

A review on static and dynamic shape control of structures by piezoelectric actuation

A literature review is presented for shape control of structures, where special emphasis is laid upon smart structures with piezoelectric control actuation. In shape control one intends to specify the spatial distribution, or the shape, of an actuating control agency, such that the displacement field of a structure distorted from its original shape eventually vanishes, or such that the structure follows some desired field of trajectories. The disturbances that distort the shape of structures may be transient, or they may be slowly varying in time. While the problem of (quasi-) static shape control refers to the latter situation, one speaks about dynamic shape control in the former case. In the present review, some structural aspects are emphasized that contribute to the complexity of shape control. Particularly, we discuss the necessity of a distributed actuation, and we refer to the inverse character of the shape control problem. We shortly mention the requirements posed by the latter two aspects upon methods of automatic control for smart structures. Since an analogy exists between piezoelectric actuation and thermal actuation, some references are cited concerning the latter topic. In order to stimulate interdisciplinary work, a short account is further presented with respect to the similar strategies of shape control by shape memory alloys and shape control by distributed pre-stress.

On the influence of the electric field on free transverse vibrations of smart beams

Free transverse vibrations of smart beams are considered where distributed actuators and sensors are realized by means of piezoelastic layers. Utilizing a variational formulation, the direct piezoelectric effect is incorporated into beam theory via suitable approximations for the axial components of the electric field or the electric displacement, respectively. Influence of shear and rotatory inertia is taken into account in the manner suggested by Timoshenko. It is shown that the correction for electrical coupling leads to effective stiffness parameters. This advantageous behavior is utilized for studying its influence on natural frequencies of smart beams with various boundary conditions.

The equations of Lagrange written for a non-material volume

SummaryThe Lagrange equations are extended with respect to a non-material volume which instantaneously coincides with some material volume of a continuous body. The surface of the non-material volume is allowed to move at a velocity which is different from the velocity of the material surface. The non-material volume thus represents an arbitrarily moving control volume in the terminology of fluid mechanics. The extension of the Lagrange equations to a control volume is derived by using the method of fictitious particles. Within a continuum mechanics based framework, it is assumed that, the instantaneous positions of both, the original particles included in the material volume, and the fictitious particles included in the control volume, are given as function of their positions in the respective reference configurations, of a set of time-dependent generalized coordinates, and of time. The corresonding spatial formulations are also assumed to be available. Imagining that the fictitious particles do transport the density of kinetic energy of the original particles, the partial derivatives of the total kinetic energy included in the material volume with respect to generalized coordinates and velocities are related to the respective partial derivatives of the total kinetic energy contained in the control volume. Hence follow the Lagrange equations for a control volume by substituting the above relations into the classical formulations for a material volume. In the present paper, holonomic problems are considered. The correction terms in the newly derived version of the Lagrange equations contain the flux of kinetic energy appearing to be transported through the surface of the control volume. This flux comes into the play in the form of properly formulated partial derivatives. Our version of the Lagrange equations is tested using the rocket equation and a folded falling string as illustrative examples.

A Reissner-Mindlin-type plate theory including the direct piezoelectric and the pyroelectric effect

SummaryThis paper is concerned with flexural vibrations of composite plates, where piezoelastic layers are used to generate distributed actuation or to perform distributed sensing of strains in the plate. Special emphasis is given to the coupling between mechanical, electrical and thermal fields due to the direct piezoelectric effect and the pyroelectric effect. Moderately thick plates are considered, where the influence of shear and rotatory inertia is taken into account according to the kinematic approximations introduced by Mindlin. An equivalent single-layer theory is thus derived for the composite plates. It is shown that coupling can be taken into account by means of effective stiffness parameters and an effective thermal loading. Polygonal plates with simply supported edges are treated in some detail, where quasi-static thermal bending as well as free, forced and actuated vibrations are studied.

Shaping of Piezoelectric Sensors/Actuators for Vibrations of Slender Beams: Coupled Theory and Inappropriate Shape Functions

Flexural vibrations of smart slender beams with integrated piezoelectric actuators and sensors are considered. A spatial variation of the sensor/actuator activity is achieved by shaping the surface electrodes and/or varying the polarization profile of the piezoelectric layers, and this variation is characterized by shape functions. Seeking shape functions for a desired purpose is termed a shaping problem. Utilizing the classical lamination theory of slender composite beams, equations for shaped sensors and actuators are derived. The interaction of mechanical, electrical and thermal fields is taken into account in the form of effective stiffness parameters and effective thermal bending moments. Self-sensing actuators are included. From these sensor/actuator equations, shaping problems with a practical relevance are formulated and are cast in the form of integral equations of the first kind for the shape functions. As a practical interesting aspect of these inverse problems, shape functions which fail to measure or to induce certain structural deformations are investigated in the present paper. Such inappropriate shape functions are termed nilpotent solutions of the shaping problems. In order to derive an easy-to-obtain class of such nilpotent solutions, the homogeneous versions of the integral equations for the shaping problems are compared to orthogonality relations valid for redundant beams. Hence, by analogy, the presented nilpotent solutions are shown to correspond to solutions of the basic theory of thermoelastic structures, namely to thermally induced static bending moment distributions. This result beautifully reflects the close connection between the theory of thermally loaded structures and the theory of smart structures. A particular result for a nilpotent shape function previously investigated in the literature is explained in the context of the present theory, and examples of nilpotent shape functions for various structural systems are presented.

An electromechanically coupled theory for piezoelastic beams taking into account the charge equation of electrostatics

SummaryThe present paper is devoted to the coupling between electrical and mechanical fields in piezoelastic structures. In the present contribution, an electromechanically coupled technical theory for flexural and extensional deformations of piezoelastic composite beams is developed. Such a technical theory should be of special interest for control applications, where a lower-order but sufficiently accurate modelling is required.In a first step, an equivalent single-layer theory of the Timoshenko-type for composite beams is utilized. The influence of shear, rotatory inertia as well as the influence of the electric field is taken into account in this technical beam theory. The electric field is unspecified so far in this formulation, but is coupled to the deformation by means of the charge equation of electrostatics. In order to incorporate this coupling, the electric potential is approximated by a power series in the thickness direction of the beam. Terms up to an order of two are considered in the approximation. The formulation then is adapted to the electric boundary conditions at the upper and lower sides of the electroded piezoelectric layers, namely that the electrodes have to be equipotential areas. Putting this distribution into an electrical variational principle, a weak one-dimensional formulation of the charge equation of electrostatics is obtained for the axial distribution of the electric potential. Prescribing the electric potential at the electrodes, and specifying the electrical boundary conditions at the vertical ends of the layer, this weak form completes the proposed electromechanically coupled technical theory for composite piezoelastic beams.In order to demonstrate the influence of the coupling between deformation and electric field, the quasi-static behavior and free flexural vibrations of a symmetrically laminated 3-layer beam are studied in detail. Results are compared to results of coupled finite element computations as well as to results obtained by a simplified theory, previously developed by the authors.

Eigenstrain Without Stress and Static Shape Control of Structures

The present contribution explores two fundamental aspects of eigenstrain analysis in three-dimensional bodies. At first, distributions of eigenstrain are derived that do not cause stresses, so-called stress-free or impotent eigenstrains. We consider bodies of finite extent with geometric surface constraints, such as imposed by immovable supports or rigidly clamped boundaries. Within the setting of anisotropic linear elastic bodies, it is verified that a field of eigenstrains that is equal to the field of strains produced by external forces is a stress-free one and that the deformations caused by these eigenstrains and the deformations caused by the forces are equal. Hence, the stress-free eigenstrain load represents an exact solution for the static shape control problem of bodies acted upon by forces. Additionally, nonuniqueness of this shape control problem is demonstrated, and three-dimensional eigenstrains responsible for that nonuniqueness are identified. This is performed by showing that incompatible distributions of eigenstrain and the strains generated by these fields, when applied as a compatible distribution of eigenstrain, result in identical deformations and stresses. Deformation-free fields then result by applying the difference between those fields of eigenstrain.

Modal analysis of elastic-plastic plate vibrations by integral equations

Abstract A direct boundary element method for the vibration problems of thne elastic-plastic plates is presented. Dynamic fundamental solutions of a suitably shaped finite domain are used in modal form. The series Greens functions are separated into a quasistatic and a dynamic part. Often the series of the quasistatic part can be written in a faster converging form than the equivalent modal series. Analytical integration in the vicinity of the singularity is performed on the closed form fundamental solutions of the infinite domain, and only the non-singular differences from the actual Greens functions are represented in series form. This paper gives a general formulation of this method for Kirchhoff plates on an arbitrary elastic foundation. After integration, the resulting algebraic equations are arranged in a form most convenient for a time-stepping analysis of inelastic response. This rearrangement has to be performed only once, if the time step is kept constant. Constitutive equations are integrated by an implicit backward Euler scheme for plane stress. Applications are shown for impacted circular plates on several different foundations.

Vibrations of the elasto-plastic pendulum

Abstract A numerical strategy for vibrations of elasto-plastic beams with rigid-body degrees-of-freedom is presented. Beams vibrating in the small-strain regime are considered. Special emphasis is laid upon the development of plastic zones. An elasto-plastic beam performing plane rotatory motions about a fixed hinged end is used as example problem. Emphasis is laid upon the coupling between the vibrations and the rigid body rotation of the pendulum. Plastic strains are treated as eigenstrains acting in the elastic background structure. The formulation leads to a non-linear system of differential algebraic equations which is solved by means of the Runge–Kutta midpoint rule. A low dimension of this system is obtained by splitting the flexural vibrations into a quasi-static and a dynamic part. Plastic strains are computed by means of an iterative procedure tailored for the Runge–Kutta midpoint rule. The numerical results demonstrate the decay of the vibration amplitude due to plasticity and the development of plastic zones. The pendulum approaches a state of plastic shake-down after sufficient time.

Maysel's formula generalized for piezoelectric vibrations - Application to thin shells of revolution

Either some or each of the layers of a composite shell made of piezoelectric materials behave as distributed actuators so that the shell becomes an intelligent or smart structure. To effectively suppress the vibrations of the shell, an electrical field with a proper control must be applied. The most efficient calculation of thermoelastic deformations is performed by Maysels formula, i.e., within a multiple-field analysis in the isothermal background. By generalizing Maysels formula, it becomes possible to include both the piezoelectric effects and inertia. A version is presented that allows the construction of the best auxiliary problem in the background, particularly preserving all kinds of symmetries present in the actual coupled problem. Applications of the three-dimensional formulation are illustrated for the special case of thin-layered shells of revolution, and, in addition, for a circular cylindrical shell with the piezoelectric influence function being presented. Because solutions of the auxiliary problem are more easily obtained in frequency space, the time convolution is replaced by a Fourier integral, which is eventually subjected to fast Fourier transform.