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.
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.
Control of continuous structures requires suitably distributed actuation and sensing. In the present contribution, we consider thin composite plates with piezoelastic layers under the action of a given set of imposed forces. Our goal is to suppress the force induced plate vibrations by means of distributed (shaped) piezoelectric actuation. This problem is referred to as the "Shape Control Problem". The present contribution investigates the dynamic shape control problem in the special but practically important case of small vibrations superposed on large deformations of a quasi-static intermediate state. Moderately large deformations are taken into account by means of the kinematic approximation of von Karman. Linearization of the non-linear electromechanical field equations, with respect to the static intermediate state, results in a set of linear partial differential equations for the superposed vibrations. These equations are cast into convolution integral formulations for both the transient piezoelectric actuation and the transient external forces. Comparing the kernels of the convolution integrals, a distributed piezoelectric actuation is found, which exactly eliminates the forced vibrations. The distribution (shape) of the actuation coincides with the distribution of the statically admissible stress due to the transient external forces.
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.
This paper is concerned with flexural vibrations of smart beams. Layers made of piezoelectric material are used to perform a distributed sensing of strains. In the present contribution, special emphasis is given to the sensor shaping problem, which can be stated as follows: Seek a shape function of the distributed sensor such that a mechanical interpretation of the sensor output is possible, e.g. to interpret the output as deflection, or as a slope. The scope of the present contribution is to find a class of easy to obtain analytic solutions of this inverse problem, and to present an experimental verification. Within the context of the Bernoulli-Euler beam theory, sensor equations are derived taking into account the coupling of mechanical and electrical fields. The principle of virtual work is then applied to derive integral equations for the structural deformations e.g. deflection, slope, curvature. Comparing these integral equations, the above sensor shaping problem is solved. Beams with different boundary conditions are considered. Furthermore, shape functions responsible for non-uniqueness of the shaping problem are derived. These nilpotent solutions may be added to the above derived solution of the sensor shaping problem without influencing the measured sensor signal. The analytical results of the sensor shaping problem are realized in a series of experiments for a cantilever beam, without and including a redundant support. Deflections measured by the new type of distributed piezoelectric sensor are compared to laser based distance measurements. Excellent coincidence between these measurements is found.
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