Search

This paper is concerned with the dynamics of hyperelastic solids and structures. We seek for a smart control actuation that produces a desired (prescribed) displacement field in the presence of transient imposed forces. In the literature, this problem is denoted as displacement tracking, or also as shape morphing problem. One talks about shape control, when the displacements to be tracked do vanish. In the present paper, it is assumed that the control actuation is provided by imposed eigenstrains, e.g., by the electric field in piezoelectric actuators, or by thermal actuators, or via analogous physical effects, such as magneto-striction or pre-stress. Structures with a controlled eigenstrain-type actuation belong to the class of smart structures. The action of the eigenstrains can be conveniently characterized by actuation stresses. Our theoretical derivations are performed in the framework of the theory of small incremental dynamic deformations superimposed upon a statically pre-deformed configuration of a hyperelastic solid or structure. We particularly ask for a distribution of incremental actuation stresses, such that the incremental displacements follow exactly a prescribed trajectory field, despite the imposed incremental forces are present. An exact solution of this problem is presented under the assumption that the actuation stresses can be tailored freely and applied everywhere within the body. Extending a Neumann-type solution strategy, it is shown that the actuation stresses due to the distributed control eigenstrains must satisfy certain quasi-static equilibrium conditions, where auxiliary body-forces and auxiliary surface tractions are to be taken into account. The latter auxiliary loading can be directly computed from the imposed forces and from the desired displacement field to be tracked. Hence, despite the problem is a dynamic one, a straightforward computation of proper actuator distributions can be obtained in the framework of quasi-static equilibrium conditions. Necessary conditions for the functioning of this concept are presented. Particularly, it must be required that the intermediate configuration is infinitesimally superstable. Previous results of our group for the case of shape control and displacement tracking in linear elastic structures are included as special cases. The high potential of the solution is demonstrated via Finite Element computations for an irregularly shaped four-corner plate in a state of plain strain.

The contour motion of the belt drive, i.e., the motion with the constant trajectory, is addressed. The belt is considered as a closed Cosserat line whose particles have translational and rotational degrees of freedom. The problem is considered in the framework of geometrically nonlinear formulation with no restrictions on the smallness of displacements and rotations. The spatial (Eulerian) coordinate which is the arc coordinate in the actual configuration is introduced. The belt is divided into four segments: two contact segments on the pulleys and two free spans. The friction forces are assumed to obey the Coulomb law. The study is limited to the stationary case with the constant angular velocities of the pulleys and the equations in components are derived for both contact and free spans. In the contact segment two assumptions are employed to eliminate the unknown contact pressure and friction: (1) the full contact, i.e., coincidence between the pulley and the belt and (2) the stick condition, i.e., the belt velocity is related to the pulley angular velocity. A nondimensional coordinate is introduced in the segments to obtain the boundary value problem with fixed boundaries. The boundary coordinates of the contact zones are the integration constants of the derived problem along with the other constants. (VLID)4851961

In the present paper a concept of static and dynamic shape control for slender piezoelastic beams is addressed. The notion shape control means to find a weighting sequence for the piezoelectric control units, such that force-induced lateral deflections are completely annihilated along the beam axis. In our case this spatial control intensity for the piezoelectric transducers is realized with so-called resistive electrodes: by prescribing only the maximum and minimum voltage level, any desirable voltage distribution along the beam axis may be obtained. For the derivation of the shape control method an extended version of the Bernoulli–Euler beam theory is used, which unifies the kinematics of a slender piezoelectric beam and the Telegrapher’s equations for moderately conductive electrodes. The outcome is an actuator equation for the displacement, which is fourth order in space and second order in time, and a sensor equation, which is second order in space and first order in time (diffusion equation for the voltage). The developed shape control method is verified by several numerical examples in the static case as well as in the dynamic frequency regime for a clamped-free beam. We show that the presented method is perfectly suited for static loads, being able to cancel force-induced structural deformations along the beam axis, whereas the deflections in the higher frequency regime can be nearly canceled out. The method is efficient, as long as a non-dimensional constant, involving the resistance, the capacitance per unit length, the length of the electrodes and the excitation frequency, is small.

SUMMARY Structural control has been comprehensively studied over the world as a multidisciplinary research field. The present work is motivated by an attempt to give a common frame to the recent research and applications of structural control technology in civil engineering across Europe. They include novel passive dampers, functional materials and semi-active dampers, active control systems, and their performance investigations. Design methods for the vibrations reduction of buildings, bridges, and wind turbines are discussed with reference to case studies. Control algorithms and dimension reduction techniques are also studied. Adaptation strategies and techniques based on the potential offered by piezoelectricity are reviewed. Copyright © 2014 John Wiley & Sons, Ltd. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with the discussion of a general methodology to design continuously distributed strain sensors; the latter are also known as spatial filters. The methodology is based on the application of the principle of virtual work. Hence, solutions for the design problem can be computed from static auxiliary problems; yet, they are valid for actual dynamic problems as well. Moreover, the use of the principle of virtual work enables a simple and straightforward translation of the results to a structural level. Different types of spatial filters, such as modal filters, displacement filters and compatibility filters are presented and their use for structural and health monitoring is discussed. Some illustrative examples for each type of spatial filter are presented, in which piezoelectric sensor networks are used to put the filters into practice; in each example experimental results validate the presented design methodology.

The setting of a looped drive belt on two equal pulleys is considered. The belt is modelled as a Cosserat rod, and a geometrically nonlinear model with account for tension and transverse shear is applied. The pulleys are considered as rigid bodies, and the belt–pulley contact is assumed to be frictionless. The problem has two axes of symmetry; therefore, the boundary value problem for the system of ordinary differential equations is formulated and solved for a quarter of the belt. The considered part consists of two segments, which are the free segment without the loading and the contact segment with the full frictionless contact. The introduction of a dimensionless material coordinate at both segments leads to a ninth-order system of ordinary differential equations. The boundary value problem for this system is solved numerically by the shooting method and finite difference method. As a result, the belt shape including the rotation angle, forces, moments, and the contact pressure are determined. The contact pressure increases near the end point of the contact area; however, no concentrated contact force occurs.

The drive belt set on two pulleys is considered as a nonlinear elastic rod deforming in plane. The modern equations of the nonlinear theory of rods are used. The static frictionless contact problem for the rod is derived. The nonlinear boundary value problems for the ordinary differential equations are solved by the finite differences method and by the shooting method by means of computer mathematics. The belt shape and the stresses are determined in the nonlinear formulation which delivers the contact reaction and the contact area. The developed method allows performing calculations for any set of geometrical and stiffness parameters. (VLID)4844522

The present paper is concerned with vibrations of an elastic bridge loaded by a moving elastic beam of a finite length, which is an extension of the authors\' previous study where the second beam was modeled as a semi-infinite beam. The second beam, which represents a train, moves with a constant speed along the bridge and is assumed to be connected to the bridge by the limiting case of a rigid interface such that the deflections of the bridge and the train are forced to be equal. The elastic stiffness and the mass of the train are taken into account. The differential equations are developed according to the Bernoulli-Euler theory and formulated in a non-dimensional form. A solution strategy is developed for the flexural vibrations, bending moments and shear forces in the bridge by means of symbolic computation. When the train travels across the bridge, concentrated forces and moments are found to take place at the front and back side of the train.

An extension of Mohr’s analogy to bending of shear-deformable beams with eigenstrain-type actuation, such as a piezoelectric actuation, is presented first. Various refined shear-deformable beam theories are included by means of a theory-dependent parameter. The one-dimensional version of Reissner’s sixth-order plate theory is exemplarily addressed. The Bernoulli–Euler theory of beams rigid in shear, as well as the shear-deformable theory of Timoshenko, are included as special cases. Afterwards, the extended Mohr analogy is applied to the bending of smart beams with piezoelectric patch actuators. The following special problem of static shape control is solved: Seek a placement of single patch actuators, such that the displacement and the cross-sectional rotation vanish at some pre-selected locations of the beam, despite the beam is loaded by external forces. Using Mohr’s analogy, it is shown that the auxiliary loading of the adjoint beam must form a self-equilibrated system of loading in order to achieve the latter goal. The high potential of the proposed actuator placement is demonstrated for the case of a cantilever beam with a single force acting at the tip. Placements of single actuators are represented such that the tip displacement and the tip cross-sectional rotation vanish. The outcomes of shear-deformable theories are compared to the Bernoulli–Euler theory and to a Finite Element computation using piezoelectrically coupled elements.

Adaptive and smart structures have become a major subject of contemporary research. Smart structures are structures that—like human beings—are capable of automatically reacting to disturbances exerted upon them by the environment they are operating in. Typically, smart structures are put into practice by embedding or attaching smart materials to a substrate structure. These materials have both actuating and sensing capabilities; a popular example would be piezoelectric materials that exhibit the direct and the converse piezoelectric effect. As many of the structures, which are considered as candidates for implementing smart materials, are continuous structures, there is an inherent need for properly distributing the actuation as well as the sensing. A method that has been successfully utilized for the design of distributed actuators is shape control. In general, shape control is concerned with finding a distributed actuation such that a structure assumes a desired shape. The latter shape may be the undist...

The present paper is concerned with the design of distributed sensors and actuators. Strain type sensors and actuators are considered with their intensity continuously distributed throughout a continuous structure. The sensors measure a weighted average of the strain tensor. As a starting point for their design we introduce the concept of collocated sensors and actuators as well as the so-called natural output. Then we utilize the principle of virtual work for an auxiliary quasi-static problem to assign a mechanical interpretation to the natural output of the sensors to be designed. Therefore, we take the virtual displacements in the principle of virtual work as that part of the displacement in the original problem, which characterizes the deviation from a desired one. We introduce different kinds of distributed sensors, each of them with a mechanical interpretation other than a weighted average of the strain tensor. Additionally, we assign a mechanical interpretation to the collocated actuators as well; for that purpose we use an extended body force analogy. The sensors and actuators are applied to solve the displacement tracking problem for continuous structures; i.e., the problem of enforcing a desired displacement field. We discuss feed forward and feed back control. In the case of feed back control we show that a PD controller can stabilize the continuous system. Finally, a numerical example is presented. A desired deflection of a clamped-clamped beam is tracked by means of feed forward control, feed back control and a combination of the two.

The present contribution is concerned with the active suppression of plane flexural vibrations of a slender, cantilever linear elastic beam. The vibrations of the beam are considered to be due to a prescribed large rigid-body motion of the beam support, as well as imposed forces. The rigid-body motion under consideration defines a floating reference configuration with respect to which the vibrations are studied. We assume these vibrations to take place in the moderately large strain regime. The beam is considered to be additionally equipped with distributed piezoelectric actuators, which are perfectly bonded to the beam. It is the scope of the present paper to derive a spatial shape of the latter actuators, such that the above vibrations can be completely suppressed by the piezoelectric actuation. This problem is also known as vibration compensation or shape control by piezoelectric actuation. In the present paper, an analytic solution for shape control is presented within the Bernoulli–Euler–von Karman theory of a slender cantilever beam, taking into account the so-called stress-stiffening effect. The presented solution for shape control makes the initial boundary value problem under consideration homogeneous, such that the vanishing of vibrations indeed represents a solution. Possible instabilities, such as parametrically excited vibrations, can be studied in the usual manner, this not being the content of the present paper. For the dynamically stable example of a rotating beam, the influence of the stress stiffening effect in the presence of a distributed piezoelectric actuation is studied in some detail. It turns out that the stress stiffening effect must indeed be taken into account for an adequate modelling. The corresponding analytic solution for shape control is validated by means of non-linear 3D finite element computations, showing excellent evidence of the appropriateness of the presented analytic beam-type solution for vibration compensation.

The present paper is concerned with vibrations of an elastic bridge, loaded by a second elastic beam moving with a constant speed. Both, the elastic stiffness and the mass of this “train” are taken into account in extension of the usual model of distributed forces. The train is assumed to be connected to the bridge by means of a rigid interface. We develop a solution strategy for the flexural vibrations, bending moments and shear forces in the bridge by means of symbolic computation. Concentrated force and moment are found to take place at the front of the train.

This paper is concerned with the force-induced vibrations of linear elastic solids and structures. We seek a transient distribution of actuating stresses produced by additional eigenstrain, such that the vibrations produced by a given set of imposed forces are exactly compensated. This problem, known as dynamic shape control problem in structural engineering, or as dynamic displacement compensation problem in automatic control, is inverse to the usual direct problem of determining displacements due to imposed forces and actuation stresses. In the present paper, we extend a method, which was introduced by F.E. Neumann for demonstrating the uniqueness of direct elastodynamic problems. We use this extended Neumann method in order to show that the distribution of the actuating stresses for shape control must be equal to any statically admissible stress distribution that is in temporal equilibrium with the imposed forces. We furthermore discuss the role of stresses corresponding to this class of solutions in some detail, emphasizing the non-unique nature of a statically admissible stress. As an analytical justification of our formulations, we show that our method reveals some static results by J.M.C. Duhamel and by W. Voigt and D.E. Carlson. Particularly, our method can be interpreted as a dynamic extension of the Duhamel body-force analogy. We moreover present numerical results for a dynamically loaded, irregularly shaped domain in a state of plane strain. These finite element computations give excellent evidence for the validity of the presented method of shape control for both, the case of a step-input and the case of a harmonic excitation.

Statics of elasto-plastic media is stated in terms of eigenstrains acting upon a background elastic problem with fixed (initial) stiffness. In order to minimize the number of unknowns and to provide computationally cheap algorithms plastic multipliers are used as main driving variables. Fixed-point type iterations are suggested for computing plastic multipliers within the load increment. Numerical experiments results are shown to be in good correlation with other software results. A range of problems is outlined where the usage of the proposed algorithms can be advantageous.

The present paper is concerned with the motion of an elastically supported beam that carries an elastic beam moving at constant speed. This problem provides a limiting case to the assumptions usually considered in the study of trains moving on rail tracks. In the literature, the train is commonly treated as a moving line-load with space-wise constant intensity, or as a system of moving rigid bodies supported by single springs and dampers. In extension, we study an elastically supported infinite beam, which is mounted by an elastic beam moving at a constant speed. Both beams are considered to have distributed stiffness and mass. The moving beam represents the train, while the elastically supported infinite beam models the railway track. The two beams are connected by an interface modeled as an additional continuous elastic foundation. Here, we follow a strategy by Stephen P. Timoshenko, who showed that a beam on discrete elastic supports could be modeled as a beam on a continuous elastic Winkler (one-parameter) foundation without suffering a substantial loss in accuracy. The celebrated Timoshenko theory of shear deformable beams with rotatory inertia is used to formulate the equations of motion of the two beams under consideration. The resulting system of ordinary differential equations and boundary conditions is solved by means of the powerful methods of symbolic computation. We present a nondimensional study on the influence of the train stiffness and the interface stiffness upon the pressure distribution between train and railway track. Considerable pressure concentrations are found to take place at the ends of the moving train.

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.

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.

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.

Summary Two different methods, local and modal, are suggested to control systems with distributed parameters, each of them having its own advantages and drawbacks. The aim of the present research is to carry out experiments aiming in comparison of these 2 methods for the problem of suppression of forced bending vibrations of a thin metal beam. Two pairs of piezoelectric sensors and actuators are attached to the beam in each control system considered. Their locations are chosen so as to provide the most efficient measurement and excitation of the first and second vibration modes of the beam. Frequency methods of the automatic control theory are employed to design stable control systems that can efficiently suppress bending vibrations of the beam with the first and second resonance frequencies. As a result, control systems with the desired performance are created on the basis of both local and modal methods. The obtained modal system works efficiently for both resonances, whereas the local systems demonstrate appropriate performance either at the first or at the second resonance frequency only. This difference is due to the fact that in the case of modal control, each control loop corresponds to a particular vibration mode and can be designed to provide optimal performance at the required frequency. The performed benchmark study demonstrates the advantages of the modal method over the local one for the cases where it is necessary to suppress vibrations in the frequency range that contains more than one resonance frequency of the control object.

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.

A numerical algorithm for studying the development of plastic and damaged zones in a vibrating structural element with a large, guided rigid-body motion is presented. Beam-type elements vibrating in the small-strain regime are considered. A machine element performing rotatory motions, similar to an element of a slider-crank mechanism, is treated as a benchmark problem. Microstructural changes in the deforming material are described by the mesolevel variables of plastic strain and damage, which are consistently included into a macroscopic analysis of the overall beam motion. The method is based on an eigenstrain formulation, considering plastic strain and damage to contribute to an eigenstrain loading of a linear elastic background structure. Rigid-body coordinates are incorporated into this beam-type structural formulation, and an implicit numerical scheme is presented for iterative computation of the eigenstrains from the mesolevel constitutive behavior. Owing to the eigenstrain formulation, any of the existing constitutive models with internal variables could in principle be implemented. Linear elastic/perfectly plastic behavior is exemplarily treated in a numerical study, where plastic strain is connected to the Kachanov damage parameter by a simple damage law. Inelastic effects like plastic shakedown and damage-induced low-cycle rupture are shown to occur in the examplary problems.

In the present paper, spatially distributed self-sensing layers with an axially varying intensity of piezoelectric activity are studied within the context of an electromechanically coupled theory for plane flexural deformations of slender smart beams. The main purpose of our derivations is to find shape functions for a piezoelectric self-sensing layer such that quasi-static deflections due to known external forces can be exactly annihilated by the piezoelectric actuation. It is shown that shape functions corresponding to the bending moment distributions due to these external forces do represent solutions of this static shape control problem. In the context of self-sensing layers, the shaped actuator can be also used as a collocated sensor, which is discussed in detail in the present paper. If the span-wise distribution of the external forces is not known, it may be useful to design self-sensing layers with the purpose of controlling the deflection at specific locations of the beam axis by means of shape functions with a particularly simple form. Such a concept is also presented. Furthermore, we make reference to so-called nilpotent shape functions which are neither able to produce actuating effects nor to measure deflections of the beam. The presented theoretical findings are demonstrated for one-span beams with various boundary conditions.

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.

An infinite-dimensional approach for the active vibration control of a multilayered straight composite piezoelectric beam is presented. In order to control the excited beam vibrations, distributed piezoelectric actuator and sensor layers are spatially shaped to achieve a sensor/actuator collocation which fits the control problem. In the sense of von Karman a nonlinear formulation for the axial strain is used and a nonlinear initial boundary-value problem for the deflection is derived by means of the Hamilton formalism. Three different control strategies are proposed. The first one is an extension of the nonlinear H∞-design to the infinite-dimensional case. It will be shown that an exact solution of the corresponding Hamilton–Jacobi–Isaacs equation can be found for the beam under investigation and this leads to a control law with optimal damping properties. The second approach is a PD-controller for infinite-dimensional systems and the third strategy makes use of the disturbance compensation idea. Under certain observability assumptions of the free system, the closed loop is asymptotically stable in the sense of Lyapunov. In this way, flexural vibrations which are excited by an axial support motion or by different time varying lateral loadings, can be suppressed in an optimal manner. A numerical example serves both to illustrate the design process and to demonstrate the feasibility of the proposed methods.

The case where one structural member of a larger complex structure experiences a dynamic instability is addressed. An averaged integral representation of the overall structure is chosen, except for the structural member of interest which is modelled precisely by means of the conventional methods of vibration theory. Material damping and the energy dissipation between components are taken into account by means of a distributed elastic-plastic rheological model. Particular attention is paid to keeping the intrinsic nonlinear properties of the energy dissipation. Structural members prove to act as dynamic absorbers with respect to the primary structure. The phenomenon of vibration saturation in complex structures is revealed and discussed. Full-scale vibration testing of a complex structure driven by an electromagnetic shaker is considered as an application of the concept. The moving coil of an electromagnetic shaker is shown to be dynamically unstable if an experiment is not properly designed. The influence of the large vibrating structure on the stability chart of the shaker turns out to be considerable and it changes the shaker’s stability chart drastically.

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 Maysel's formula, i.e., within a multiple-field analysis in the isothermal background. By generalizing Maysel's 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.

In the present paper, a review on piezoelectric sensing of mechanical deformations and vibrations of so-called smart or intelligent structures is given. After a short introduction into piezoelectric sensing and actuation of such controlled structures, we pay special emphasis on the description of some own work, which has been performed at the Institute of Technical Mechanics of the Johannes Kepler University of Linz (JKU) in the last years. Among other aspects, this work has been motivated by the fact that collocated control of smart structures requires a sensor output that is work-conjugated to the input by the actuator. This fact in turn brings into the play the more general question of how to measure mechanically meaningful structural quantities, such as displacements, slopes, or other quantities, which form the work-conjugated quantities of the actuation, by means piezoelectric sensors. At least in the range of small strains, there is confidence that distributed piezoelectric sensors or sensor patches in smart structures do measure weighted integrals over their domain. Therefore, there is a need of distributing or shaping the sensor activity in order to be able to re-interpret the sensor signals in the desired mechanical sense. We sketch a general strategy that is based on a special application of work principles, more generally on displacement virials. We also review our work in the past on bringing this concept to application in smart structures, such as beams, rods and plates.

T. Adachi: Energy Absorption of Axially-Impacted Column Controlled by Transverse Impact - C. Adam, T. Furtmuller: Seismic Performance of Tuned Mass Dampers - H. Bremer: Problems in Fast Moving Non-Holonomic Elastic Systems - F. Casciati, Z. Chen: Using GPS Sensors in Structural Mechanics - L. Faravelli, C. Fuggini, F. Ubertini: Hybrid Control Procedures in Mitigating Cable Vibrations - T. Furukawa: Thermal Stress Analysis in a Functionally Graded Material Considering Finite Thermal Wave Speed - U. Gabbert, S. Kari, N. Bohn, H. Berger: Numerical Homogenization and Optimization of Smart Composite Materials - R. Heuer, S. Mehdi Yousefi: Hybride Bell Tower Like Structures in Earthquake Environment - H. Irschik, M. Krommer, M. Gusenbauer: Tracking of Stresses: A Further Step Towards Ageless Structures - M. Ishihara, Y. Watanabe, N. Noda: Non-Linear Dynamic Deformation of a Piezothermoelastic Laminate - B. Jakoby, E. K. Reichel, F. Lucklum, B. Weiss, C. Riesch, F. Keplinger, R. Beigelbeck, W. Hilber: Determining Liquid Properties Using Mechanically Vibrating Sensors - R. Kawamura, H. Fujita, K. Heguri, Y. Tanigawa: Mathematical Analysis of Flexural Vibration for a Functionally Graded Material Plate and Vibration Suppression by Flexural Wave Control - M. Krommer, M. Zellhofer: Monitoring and Control of Multi-Storey Frame Structures by Strain-Type Actuators and Sensors - X. Ootao: Transient Piezothermoelastic Problem of a Functionally Graded Thermopiezoelectric Cylindrical Panel - M. Reichhartinger, M. Horn, A. Hofer: Control of an Electronic Throttle Valve for Drive-by-Wire Applications - T. Rittenschober, K. Schlacher: Output Regulation of Smart Structures, Theory and Practice - A. Saimoto: Analysis of Weld Induced Plasticity by BFM - Y. Shibuya: Evaluation of Internal Friction of Viscoelastic Composites with Meso-Scale Structures for Vibration Damping of Mechanical Structures - T. Tsuji, T. Kurimoto, T. Shibuya: An Identification Method of the Time Dependence of theImpact Force by Using Acoustic Response and FEM Analysis - S. Ueda: Infinite Row of Parallel Cracks in a Piezoelectric Material Strip Under Mechanical and Transient Thermal Loadings - K. Watanabe, N. Sumi: Elastodynamic Doppler Effects by a Moving Interface - C. Zehetner, J. Gerstmayr: Compensation of Flexible Vibrations in a Two-Link Robot by Piezoelectric Actuation - F. Ziegler: The Basis of Optimal Active (Static and Dynamic) Shape- and Stress-Controlby Means of Smart Materials

This paper is concerned with the design of a proper piezoelectric patch actuator network in order to track the displacement of the sidewalls of a one-story frame structure; both, for the static and the dynamic case. Weights for each network member found in our previous work were based on beam theory; in the present paper a refinement of the weights by modeling the sidewalls of the frame structure as Kirchhoff plates is presented. For the sake of calculating the refined weights approximate solutions of the plate equations are calculated by an extended Galerkin method.

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.

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.

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.

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.

Following the guidelines of the European Union (EU), Austria supports scientific research and technological development by publicly funding of research centers. Such centers are positioned between scientific and industrial stakeholders and have to simultaneously fulfill contradictory demands. This paper deeply analyzes the Austrian Center of Competence in Mechatronics (ACCM) and exposes why this publicly funded research center is able to effectively manage these conflicting demands. Using the theory of ambidexterity, the study highlights that appropriate structures and strategies are preliminarily needed. In particular we found that the fundamental abilities for managing contradictory demands are located on an individual level and argue that especially the autonomous, well-educated people and their competences of self-organization enable the research center to be ambidextrous.

An elasto-plastic pendulum performing large rigid-body rotations and vibrating in the small-strain regime is studied. Spatial distribution of plastic zones within the beam-type pendulum is taken into account. The problem is described by a differential algebraic system of equations for the flexural coordinates and the rotation of the pendulum. This system of equations can be interpreted as a model for an elastic background pendulum under the action of additional sources, formed by the plastic parts of strain. Since the elastic pendulum is a Hamiltonian system, it is possible to control the motion of the elastic pendulum by means of a collocated PD-controller. Especially, we consider the problem of bringing the tip of the elastic pendulum into its upward (inverted) position by means of a control moment acting at the fixed end. We then apply the controller designed for the elastic pendulum to the elasto-plastic model, assuming that the yield level of the material was lowered considerably by some environmental influences. Since the effect of plasticity is dissipative, the controlled elasto-plastic pendulum turns out to reach an equilibrium position, which however does not exactly coincide with the upward target position. This deviation is due to the permanent deformations induced in the pendulum by plasticity, and it is demonstrated in a numerical study.

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 Graffi's 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.

The present paper studies the influence of lateral eigenstrains on the transverse deflection of wide beams. We show that in this case a laterally nonuniform transverse displacement becomes notable; moreover, it turns out that the axial variation of the transverse displacement is significantly altered in comparison to the results obtained from beam theory. In order to derive a corrected analytical solution for the transverse displacement of wide beams, the latter are modeled as thin plates with induced eigenstrains in both in–plane directions. A Galerkin method is utilized to solve the plate equations, in which solutions for the transverse displacement resulting from beam theory are used as shape functions for the plate deflection. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Flexural 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 and the pyroelectric effect are incorporated into beam theory via suitable approximations for the axial components of electric field or electric displacement, respectively. Influence of shear and rotary inertia is taken into account in the manner suggested by Timoshenko. It is shown that the effects of the direct piezoelectric and the pyroelectric effect lead to effective stiffness parameters. The present coupled Timoshenko-type theory thus is equal to computational complexity when compared to the commonly used decoupled theory. This advantageous behavior is utilized for studying thermally induced deflections and calculating natural frequencies.

Institute of Technical Mechanics, Johannes Kepler University of Linz, Altenbergerstrasse 69, A-4040 Linz, AustriaInthepresentcontribution,thecompletesuppressionofelasticvibrationsofstructuresbymeansofanadditionalpiezoelectriceigenstrain distribution is considered. A vibrating cantilever is used as a descriptive example. First, according to the theory ofdynamicshapecontrol,acontinuousdistributionfortheactuationisderived. Then,positionsfordiscretepiezoelectricpatchesare suggested. The introduced theoretical strategy is validated by finite element computations and an appropriate laboratoryexperiment. Based on frequency response measurements, a feedback controller is designed in the frequency domain.

The present contribution is concerned with active suppression of plane flexural vibrations of a cantilevered panel. The vibrations of the panel are produced by imposed forces and by a prescribed large rigid-body motion of the cantilevered end, such that the panel appears to fly with respect to an inertial frame. The rigid body motion defines a floating reference configuration of the panel. With respect to the latter, the vibrations are assumed to remain in the elastic regime and to be moderately large. The cantilever is considered to be equipped with integrated piezoelectric actuators. It is the scope of the present paper to derive a spatial shape of these actuators such that the vibrations can be completely suppressed by the piezoelectric actuation. This inverse problem is exactly solved in the context of an equivalent single-layer Kirchhoff-theory for thin panels rigid in shear. Coupling to the electric field is taken into account. The presented solution of the shape-control problem is intended to serve as a benchmark test for the field of intelligent wings. For comparison sake, 3-dimensional Finite Element computations are performed.

Storage tanks of standard design have natural periods well above one second and, thus, are to be considered likewise to structures with a soft first storey. In case of a site with high seismic risk and soft soil-layers resonance loading at low frequencies will occur. Contrary to ordinary structures the stiffness of the tank cannot be increased considerably. A deep-pile foundation traversing an intermediate layer of low mechanical impedance is analysed as an alternative to the cheaper shallow sand or concrete foundation used commonly under sufficiently good soil conditions. Such dynamic interaction problems of relatively stiff structures on pile foundations are treated by Wolf [5.2–1] and nonlinear constitutive relations of the viscous soil are considered by Penzien [5.2–2]. The behavior of the piles in layered media is discussed by Novak and Aboul-Ella [5.2–3] and, more, recently, by Nogami [5.2–4]. The analysis is kept linear following the latter references and, by considering also the fact that not sufficient data are commonly available to model the soil-layers more sophistically than linear elastic and hysteretically damped. Contrary to the vibrations of a stiff structure on a deep-pile foundation the low-pass filter properties dominate the overall resonance behavior. For a dense population of piles the model of reinforced soil reflects the dynamic properties well in the low frequency range and the foundation comes close to a Thimoshenko beam model. Individual piles contribute to thickness vibrations only at much higher frequencies. The inhomogeneous cross-section of the Timoshenko beam is assumed to exhibit circular symmetry, and its effective rigidities with respect to bending and shear as well as its mass per unit of length vary from layer to layer according to the soil properties. The effective modulus theory is applied in a standard fashion.

A method of analyzing inelastic structures is presented, in which inelastic strains are interpreted as sources of eigenstresses in the linear elastic structure. Stresses due to inelastic strains are calculated by means of influence functions. The method is applied to viscoplastic sandwich beams.