Sensor and actuator design for displacement control of continuous systems

Krommer, Michael,Irschik, Hans Techno-Press 2007 Smart Structures and Systems, An International Jou Vol.3 No.2

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.

Sensor and actuator design for displacement control of continuous systems

Michael Krommer,Hans Irschik 국제구조공학회 2007 Smart Structures and Systems, An International Jou Vol.3 No.2

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.

Modeling and numerical simulation of electrostrictive materials and structures

Michael Krommer,Astrid Pechstein,Alexander Humer 국제구조공학회 2022 Smart Structures and Systems, An International Jou Vol.30 No.3

This paper is concerned with nonlinear modeling and efficient numerical simulation of electrostrictive materials and structures. Two types of such materials are considered: relaxor ferroelectric ceramics and electrostrictive polymers. For ceramics, a geometrically linear formulation is developed, whereas polymers are studied in a geometrically nonlinear regime. In the paper, we focus on constitutive modeling first. For the reversible constitutive response under consideration, we introduce the augmented Helmholtz free energy, which is composed of a purely elastic part, a dielectric part and an augmentation term. For the elastic part, we involve an additive decomposition of the strain tensor into an elastic strain and an electrostrictive eigenstrain, which depends on the polarization of the material. In the geometrically nonlinear case, a corresponding multiplicative decomposition of the deformation gradient tensor replaces the additive strain decomposition used in the geometrically linear formulation. For the dielectric part, we first introduce the internal energy, to which a Legendre transformation is applied to compute the free energy. The augmentation term accounts for the contribution from vacuum to the energy. In our formulation, the augmented free energy depends not only on the strain and the electric field, but also on the polarization and an internal polarization; the latter two are internal variables. With the constitutive framework established, a Finite Element implementation is briefly discussed. We use high-order elements for the discretization of the independent variables, which include also the internal variables and, in case the material is assumed incompressible, the hydrostatic pressure, which is introduced as a Lagrange multiplier. The elements are implemented in the open source code Netgen/NGSolve. Finally, example problems are solved for both, relaxor ferroelectric ceramics and electrostrictive polymers. We focus on thin plate-type structures to show the efficiency of the numerical scheme and its applicability to thin electrostrictive structures.

Nonlinear modelling and analysis of thin piezoelectric plates: Buckling and post-buckling behaviour

Michael Krommer,Yury Vetyukov,Elisabeth Staudigl 국제구조공학회 2016 Smart Structures and Systems, An International Jou Vol.18 No.1

In the present paper we discuss the stability and the post-buckling behaviour of thin piezoelastic plates. The first part of the paper is concerned with the modelling of such plates. We discuss the constitutive modelling, starting with the three-dimensional constitutive relations within Voigt\'s linearized theory of piezoelasticity. Assuming a plane state of stress and a linear distribution of the strains with respect to the thickness of the thin plate, two-dimensional constitutive relations are obtained. The specific form of the linear thickness distribution of the strain is first derived within a fully geometrically nonlinear formulation, for which a Finite Element implementation is introduced. Then, a simplified theory based on the von Karman and Tsien kinematic assumption and the Berger approximation is introduced for simply supported plates with polygonal planform. The governing equations of this theory are solved using a Galerkin procedure and cast into a non-dimensional formulation. In the second part of the paper we discuss the stability and the post-buckling behaviour for single term and multi term solutions of the non-dimensional equations. Finally, numerical results are presented using the Finite Element implementation for the fully geometrically nonlinear theory. The results from the simplified von Karman and Tsien theory are then verified by a comparison with the numerical solutions.

Dynamic displacement tracking of a one-storey frame structure using patch actuator networks: Analytical plate solution and FE validation

Huber, Daniel,Krommer, Michael,Irschik, Hans Techno-Press 2009 Smart Structures and Systems, An International Jou Vol.5 No.6

The present paper is concerned with the design of a proper patch actuator network in order to track a desired displacement of the sidewalls of a one-storey frame structure; both, for the static and the dynamic case. Weights for each patch of the actuator network found in our previous work were based on beam theory; in the present paper a refinement of these weights by modeling the sidewalls of the frame structure as thin 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 solutions based on the analytical plate model are compared with three-dimensional Finite Element results computed in the commercially available code ANSYS. The patch actuator network is put into practice by means of four piezoelectric patches attached to each of the two sidewalls of the frame structures, to which electric voltages proportional to the analytically refined patch weights are applied. Analytical and numerical results coincide very well over a broad frequency range.

Design of piezoelectric transducer arrays for passive and active modal control of thin plates

Georg Zenz,Michael Krommer,Wolfgang Berger,Johannes Gerstmayr,Manfred Nader 국제구조공학회 2013 Smart Structures and Systems, An International Jou Vol.12 No.5

To suppress vibration and noise of mechanical structures piezoelectric ceramics play an increasing role as effective, simple and light-weighted damping devices as they are suitable for sensing and actuating. Out of the various piezoelectric damping methods this paper compares mode based active control strategies to passive shunt damping for thin plates. Therefore, a new approach for the optimal placement of the piezoelectric sensors/actuators, or more general transducers, is proposed after intense theoretical investigations based on the Kirchhoff kinematical hypotheses of plates; in particular, modal and nilpotent transducers are discussed in detail. Based on the proposed distribution a discrete design for modal transducers is implemented, tested and verified on an experimental setup. For active control the modal sensors clearly identify the eigenmodes, whereas the modal actuators impose distributed eigenstrains in order to reduce the transverse plate vibrations. In contrast to the modal control, passive shunt damping works without requiring additional actuators or auxiliary power and can therefore act as an autonomous system, but it is less effective compensating the flexible vibrations. Exemplarily, an acryl glass plate disturbed by an arbitrary force initialized by a loudspeaker is investigated. Comparing the different methods their specific advantages are highlighted and a significant broadband reduction of the vibrations of up to -20dB is obtained.

Experimental assessment of the piezoelectric transverse d₁₅ shear sensing mechanism

Berik, Pelin,Benjeddou, Ayech,Krommer, Michael Techno-Press 2014 Smart Structures and Systems, An International Jou Vol.13 No.4

The piezoelectric transverse $d_{15}$ shear sensing mechanism is firstly assessed experimentally for a cantilever smart sandwich plate made of a piezoceramic axially poled patched core and glass fiber reinforced polymer composite faces. Different electrical connections are tested for the assessment of the sensor performance under a varying amplitude harmonic (at 24 Hz) force. Also, the dynamic response of the smart sandwich composite structure is monitored using different acquisition devices. The obtained experimentally sensed voltages are compared to those resulting from the benchmark three-dimensional piezoelectric coupled finite element simulations using a commercial code where realistic features, like equipotential conditions on the patches' electrodes and mechanical updating of the clamp, are considered. Numerically, it is found that the stiffness of the clamp, which is much softer than the ideal one, has an enormous influence on the sensed voltage of its adjacent patch; therefore, sensing with the patch on the free side would be more advantageous for a cantilever configuration. Apart from confirming the latter result, the plate benchmark experimental assessment showed that the parallel connection of its two oppositely poled patches has a moderate performance but better than the clamp side patch acting as an individual sensor.

Displacement tracking of pre-deformed smart structures

Hans Irschik,Michael Krommer,Christian Zehetner 국제구조공학회 2016 Smart Structures and Systems, An International Jou Vol.18 No.1

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.

Piezoceramic d₁₅ shear-induced direct torsion actuation mechanism: a new representative experimental benchmark

Berik, Pelin,Benjeddou, Ayech,Krommer, Michael Techno-Press 2013 Smart Structures and Systems, An International Jou Vol.12 No.5

A new piezoceramic $d_{15}$ shear-induced torsion actuation mechanism representative benchmark is proposed and its experimentations and corresponding 3D finite element (FE) simulations are conducted. For this purpose, a long and thin smart sandwich cantilever beam is dimensioned and built so that it can be used later for either validating analytical Saint Venant-type solutions or for analyzing arm or blade-based smart structures and systems applications. The sandwich beam core is formed by two adjacent rows of 8 oppositely axially polarized d15 shear piezoceramic patches, and its faces are dimensionally identical and made of the same glass fiber reinforced polymer composite material. Quasi-static and static experimentations were made using a point laser sensor and a scanning laser vibrometer, while the 3D FE simulations were conducted using the commercial software $ABAQUS^{(R)}$. The measured transverse deflection by both sensors showed strong nonlinear and hysteretic (static only) variation with the actuation voltage, which cannot be caught by the linear 3D FE simulations.

Experimental assessment of the piezoelectric transverse d15 shear sensing mechanism

Pelin Berik,Ayech Benjeddou,Michael Krommer 국제구조공학회 2014 Smart Structures and Systems, An International Jou Vol.13 No.4

The piezoelectric transverse d15 shear sensing mechanism is firstly assessed experimentally for a cantilever smart sandwich plate made of a piezoceramic axially poled patched core and glass fiber reinforced polymer composite faces. Different electrical connections are tested for the assessment of the sensor performance under a varying amplitude harmonic (at 24 Hz) force. Also, the dynamic response of the smart sandwich composite structure is monitored using different acquisition devices. The obtained experimentally sensed voltages are compared to those resulting from the benchmark three-dimensional piezoelectric coupled finite element simulations using a commercial code where realistic features, like equipotential conditions on the patches\' electrodes and mechanical updating of the clamp, are considered. Numerically, it is found that the stiffness of the clamp, which is much softer than the ideal one, has an enormous influence on the sensed voltage of its adjacent patch; therefore, sensing with the patch on the free side would be more advantageous for a cantilever configuration. Apart from confirming the latter result, the plate benchmark experimental assessment showed that the parallel connection of its two oppositely poled patches has a moderate performance but better than the clamp side patch acting as an individual sensor. Key