Michael Krommer

Transient modelling of flexible belt drive dynamics using the equations of a deformable string with discontinuities

ABSTRACT Dynamics of a belt drive is analysed using a non-linear model of an extensible string at contour motion, in which the trajectories of particles of the belt are predetermined. The equations of string dynamics at the tight and slack spans are considered in a fixed domain by transforming into a spatial frame. 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. Finite difference discretization allows numerical analysis of the resulting system of partial differential equations with delays. Example solution for the acceleration of a belt drive and an investigation of its frequency response depending on the velocity are presented and discussed.

Post-Buckling of Piezoelectric Thin Plates

In the present paper, the geometrically nonlinear behavior of piezoelastic thin plates is studied. First, the governing equations for the electromechanically coupled problem are derived based on the von Karman–Tsien kinematic assumption. Here, the Berger approximation is extended to the coupled piezoelastic problem. The general equations are then reduced to a single nonlinear partial differential equation for the special case of simply supported polygonal edges. The nonlinear equations are approximated by using a problem-oriented Ritz Ansatz in combination with a Galerkin procedure. Based on the resulting equations the buckling and post-buckling behavior of a polygonal simply supported plate is studied in a nondimensional form, where the special geometry of the polygonal plate enters via the eigenvalues of a Helmholtz problem with Dirichlet boundary conditions. Single term as well as multi-term solutions are discussed including the effects of piezoelectric actuation and transverse force loadings upon the solution. Novel results concerning the buckling, snap through and snap buckling behavior are presented.

Finite deformations of thin plates made of dielectric elastomers: Modeling, numerics, and stability:

In this article, we present a nonlinear theory for thin plates, which are made of incompressible electroded dielectric elastomer layers. The layers are assumed to exhibit a neo-Hookean elastic behavior, and the effect of the electrostatic forces is taken into account by means of the electrostatic stress tensor. A plane state of stress is imposed on the total stress tensor, based on which two-dimensional constitutive relations for the plate are derived. A geometrically nonlinear formulation for the plate as a material surface is devloped, and solutions are computed using nonlinear finite elements. The numerical results are compared to available results from the literature verifying our approach, and an additional nonsymmetric example problem is studied with respect to stability.

Modeling of Dielectric Elastomers Accounting for Electrostriction by Means of a Multiplicative Decomposition of the Deformation Gradient Tensor

Nonlinear modeling of inelastic material behavior by a multiplicative decomposition of the deformation gradient tensor is quite common for finite strains. The concept has proven applicable in thermoelasticity, elastoplacticity, as well as for the description of residual stresses arising in growth processes of biological tissues. In the context of advanced materials, the multiplicative decomposition of the deformation gradient tenser has been introduced within the fields of electroelastic elastomers, shape-memory alloys as well as piezoelastic materials. In the present paper we apply this multiplicative approach to the special case of dielectric elastomers in order to account for the electrostrictive effect. Therefore, we seek to include the two main sources of electro-mechanical coupling in dielectric elastomers. These are elastostatic forces acting between the electric charges and electrostriction due to intramolecular forces of the material. In particular we intend to study the significance of electrostriction for the particular case of dielectric elastomers, in the form of a thin layer with two compliant electrodes.

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.