Yury Vetyukov

On the Use of Piezoelectric Sensors in Structural Mechanics: Some Novel Strategies

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.

A Spatial Thin Beam Finite Element Based on the Absolute Nodal Coordinate Formulation Without Singularities

A three-dimensional nonlinear finite element for thin beams is proposed within the absolute nodal coordinate formulation (ANCF). The deformation of the element is described by means of displacement vector, axial slope and axial rotation parameter per node. The element is based on the Bernoulli-Euler theory and can undergo coupled axial extension, bending and torsion in the large deformation case. Singularities — which are typically caused by such parameterizations — are overcome by a director per element node. Once the directors are properly defined, a cross sectional frame is defined at any point of the beam axis. Since the director is updated during computation, no singularities occur. The proposed element is a three-dimensional ANCF Bernoulli-Euler beam element free of singularities and without transverse slope vectors. Detailed convergence analysis by means of various numerical examples and comparison to analytical solutions shows the performance and accuracy of the element.Copyright

Piezoelectric d 15 shear-response-based torsion actuation mechanism: An exact 3D Saint-Venant type solution

This paper is concerned with the detailed analysis of the behavior of a piezoceramic bi-morph torsion actuator using the d 15-effect. The bi-morph actuator is made of two oppositely polarized adjacent piezoceramic prismatic beams. The mathematical analysis is based on the Saint-Venant torsion theory; a formulation of the electromechanically coupled problem in terms of a stress function and of the electric potential is derived, which represents an exact solution of a specific three-dimensional problem; in particular, for the case when the axial stress and the axial component of the electric displacement vector are independent of the axial coordinate. The resulting boundary-value problem in the cross-section is solved using the method of finite differences. Solutions for the actuated rate of twist are presented and compared to three-dimensional electromechanically coupled finite element solutions using ABAQUS® for the case of a cantilevered bi-morph actuator. A very good agreement is found.

HOTINT: A Script Language Based Framework for the Simulation of Multibody Dynamics Systems

The multibody dynamics and finite element simulation code has been developed since 1997. In the past years, more than 10 researchers have contributed to certain parts of HOTINT, such as solver, graphical user interface, element library, joint library, finite element functionality and port blocks. Currently, a script-language based version of HOTINT is freely available for download, intended for research, education and industrial applications. The main features of the current available version include objects like point mass, rigid bodies, complex point-based joints, classical mechanical joints, flexible (nonlinear) beams, port-blocks for mechatronics applications and many other features such as loads, sensors and graphical objects. HOTINT includes a 3D graphical visualization showing the results immediately during simulation, which helps to reduce modelling errors. In the present paper, we show the current state and the structure of the code. Examples should demonstrate the easiness of use of HOTINT.Copyright

Optimal Continuous Strain-Type Sensors for Finite Deformations of Shell Structures

A continuously distributed strain-type sensor produces a signal proportional to the weighted integral of the local strains. In the geometrically linear setting this signal equals a desired kinematic entity, if the weight functions are chosen as a solution to a particular auxiliary problem of statics. In the present paper we study the design of such sensors for shell structures in a geometrically nonlinear regime using the modern variant of the theory of shells as material surfaces. In particular, we first suggest and numerically study the effect of the application of so-called nilpotent sensors, which produce a trivial signal in the geometrically linear range, to the determination of local buckling of shell structures. Using various sensors, one can estimate the location of the failure. Furthermore, we demonstrate the application of a recently suggested new general method to design strain-type sensors for measurements in the vicinity of a known pre-deformed state to shell structures.

On the combination of asymptotic and direct approaches to the modeling of plates with piezoelectric actuators and sensors

We suggest a method for the mathematical modeling of the response of a thin structure with integrated piezoelectric sensors and actuators under mechanical and electrical loads. The plate with attached piezoelectric patches is considered as a two-dimensional material surface with mechanical and electrical degrees of freedom of particles. The model is complemented with an asymptotic analysis of the three-dimensional problem. For the equations of a layered piezoelectric plate, we seek those terms in the solution, which dominate as the thickness tends to zero. The results serve as a basis for the analysis of geometrically nonlinear shell structures as well as for model-based monitoring and optimal sensor and actuator placement.

The Model of a Deformable String with Discontinuities at Spatial Description in the Dynamics of a Belt Drive

We study dynamics of a belt drive with a nonlinear model of an extensible string at contour motion, in which the trajectories of particles of the belt are pre-determined. Writing the equations of string dynamics in a spatial frame and assuming the absence of slip of the belt on the surface of the pulleys we arrive at a new model with a discontinuous velocity field and concentrated contact forces. It is demonstrated how the model may be applied for analytical study of stationary regimes, and differences in the behavior of synchronous and friction belt drives are pointed out. We also present a novel approach for modeling transient behavior of the system, e.g. for the analysis of start up of the belt drive or its frequency response. Both numerical and analytical solutions of the resulting problem are possible.

Nonlinear electro-elastic modeling of thin dielectric elastomer plate actuators

Electro-active polymers undergo large deformations while being typically very thin; this encourages us to study the geometric nonlinear set up within the structural mechanics framework of thin plates and shells as a material surface. In this paper, the full set of three dimensional, geometric nonlinear field equations are incorporated to develop constitutive relations by introducing a generalized free energy function, which takes parts from a pure mechanical strain energy (e.g. neo-Hookean) and a mixed electro-mechanical free energy. The key feature is the multiplicative decomposition of the deformation gradient tensor, which allows for separate constitutive models for any electro-mechanic coupling phenomenon. We apply this model exemplary to the case of electrostriction and use the Gauss law of electrostatics in order to incorporate charge controlled actuation, which has been reported to omit pull-in instability. In order to translate the resulting equations to their two-dimensional geometrically nonlinear counterparts for thin plates, a plane stress condition is imposed on the total stress tensor and the effect of the electrostrictive coupling is investigated on voltage controlled as well as on charge controlled actuation, employing non-linear Finite Elements. Finally, results are compared to numerical as well as experimental results on electrostrictive coupling and charge controlled actuation.

Plane Bending of a Curved Rod

We present a combination of the asymptotic, direct, and numerical methods on the example plane problem of finite deformations of a thin curved strip. The study includes both static and dynamic analyses; the inhomogeneity of the strip is taken into account. The method of asymptotic splitting allows for a consistent dimensional reduction of the equations of the original two-dimensional continuous problem into a one-dimensional formulation of the reduced theory and a problem in the cross section. It also provides a consistent way to recover the distributions of stresses, strains, and displacements in two dimensions. The direct approach to a material line extends the results to the geometrically nonlinear range. Several analytical solutions and the implementation of a finite element scheme are demonstrated with the Mathematica computer system. We investigate the convergence of solutions of various problems in the original (two-dimensional) and reduced (one-dimensional) models with respect to the thickness. This justifies the analytical conclusion that the classical Kirchhoff theory of rods remains asymptotically accurate irrespective from both the curvature of the strip and the variation of the material properties over the thickness. This chapter quotes extensively from Vetyukov (Acta Mech., 223(2):371–385, 2012) with kind permission from Springer Science and Business Media.

Mechanics of Thin-Walled Rods of Open Profile

We consider rods, whose cross section is a thin open-ended strip. The effect of warping, when the points of the rod move axially at torsion, needs to be included in the analysis. Beginning with the discussion of traditional approaches, based on certain hypotheses and approximations, we proceed with the asymptotic study of the corresponding model of a shell. The procedure of asymptotic splitting justifies known equations with an additional force factor (bi-moment) and warping, related to the rate of twist. Moreover, it allows for accurate recovery of the stressed state in a cross section first in terms of the shell theory, and then in the three-dimensional body. Further, we extend the results to the geometrically nonlinear range with the direct approach to a material line with additional degrees of freedom of particles due to the warping. The incremental analysis results in linear equations in the vicinity of a pre-stressed state. Solutions of particular problems show the importance of cubic terms in the strain energy of the rod, which is deduced from the equations of the nonlinear shell theory. Considered examples include linear problems, several types of buckling behavior as well as the analysis of higher order effects in comparison to numerical solutions with the shell model. The presented form of the direct approach was originally suggested by Vetyukov and Eliseev (Sci. Tech. Bull. St. Petersbg. State Polytechn. Univ., 1:49–53, 2007). This chapter quotes extensively from Vetyukov (Acta Mech., 200(3–4):167–176, 2008; J. Elast., 98(2):141–158, 2010) with kind permission from Springer Science and Business Media.