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Thu, 11 Oct 2012 16:58:40 +0200Thu, 11 Oct 2012 16:58:40 +0200Integration of nonlinear models of flexible body deformation in Multibody System Dynamics
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3321
A simple transformation of the Equation of Motion (EoM) allows us to directly integrate nonlinear structural models into the recursive Multibody System (MBS) formalism of SIMPACK. This contribution describes how the integration is performed for a discrete Cosserat rod model which has been developed at the ITWM. As a practical example, the run-up of a simplified three-bladed wind turbine is studied where the dynamic deformations of the three blades are calculated by the Cosserat rod model.Martin Schulze; Stefan Dietz; Andrey Tuganov; Joachim Linn; Holger Langreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3321Thu, 11 Oct 2012 16:58:40 +0200Multibody dynamics simulation of geometrically exact Cosserat rods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2985
In this paper, we present a viscoelastic rod model that is suitable for fast and accurate dynamic simulations. It is based on Cosserat’s geometrically exact theory of rods and is able to represent extension, shearing (‘stiff’ dof), bending and torsion (‘soft’ dof). For inner dissipation, a consistent damping potential proposed by Antman is chosen. We parametrise the rotational dof by unit quaternions and directly use the quaternionic evolution differential equation for the discretisation of the Cosserat rod curvature. The discrete version of our rod model is obtained via a finite difference discretisation on a staggered grid. After an index reduction from three to zero, the right-hand side function f and the Jacobian \(\partial f/\partial(q, v, t)\) of the dynamical system \(\dot{q} = v, \dot{v} = f(q, v, t)\) is free of higher algebraic (e. g. root) or transcendental (e. g. trigonometric or exponential) functions and therefore cheap to evaluate. A comparison with Abaqus finite element results demonstrates the correct mechanical behavior of our discrete rod model. For the time integration of the system, we use well established stiff solvers like RADAU5 or DASPK. As our model yields computational times within milliseconds, it is suitable for interactive applications in ‘virtual reality’ as well as for multibody dynamics simulation.Holger Lang; Joachim Linn; Martin Arnoldreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2985Thu, 19 Apr 2012 07:33:08 +0200The difference of the solutions of the elastic and elastoplastic boundary value problem and an approach to multiaxial stress-strain correction
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1900
In the theoretical part of this thesis, the difference of the solutions of the elastic and the elastoplastic boundary value problem is analysed, both for linear kinematic and combined linear kinematic and isotropic hardening material. We consider both models in their quasistatic, rate-independent formulation with linearised geometry. The main result of the thesis is, that the differences of the physical obervables (the stresses, strains and displacements) can be expressed as composition of some linear operators and play operators with respect to the exterior forces. Explicit homotopies between both solutions are presented. The main analytical devices are Lipschitz estimates for the stop and the play operator. We present some generalisations of the standard estimates. They allow different input functions, different initial memories and different scalar products. Thereby, the underlying time involving function spaces are the Sobolov spaces of first order with arbitrary integrability exponent between one and infinity. The main results can easily be generalised for the class of continuous functions with bounded total variation. In the practical part of this work, a method to correct the elastic stress tensor over a long time interval at some chosen points of the body is presented and analysed. In contrast to widespread uniaxial corrections (Neuber or ESED), our method takes multiaxiality phenomena like cyclic hardening/softening, ratchetting and non-masing behaviour into account using Jiang's model of elastoplasticity. It can be easily adapted to other constitutive elastoplastic material laws. The theory for our correction model is developped for linear kinematic hardening material, for which error estimated are derived. Our numerical algorithm is very fast and designed for the case that the elastic stress is piecewise linear. The results for the stresses can be significantly improved with Seeger's empirical strain constraint. For the improved model, a simple predictor-correcor algorithm for smooth input loading is established.Holger Langdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1900Thu, 11 Oct 2007 11:20:18 +0200A piecewise analytical solution for Jiangs model of elastoplasticity
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1898
In this article, we present an analytic solution for Jiang's constitutive model of elastoplasticity. It is considered in its stress controlled form for proportional stress loading under the assumptions that the one-to-one coupling of the yield surface radius and the memory surface radius is switched off, that the transient hardening is neglected and that the ratchetting exponents are constant.Holger Lang; Klaus Dressler; Rene Pinnaureporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1898Tue, 02 Oct 2007 12:27:59 +0200A condition that a continuously deformed, simply connected body does not penetrate itself
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1874
In this article we give a sufficient condition that a simply connected flexible body does not penetrate itself, if it is subjected to a continuous deformation. It is shown that the deformation map is automatically injective, if it is just locally injective and injective on the boundary of the body. Thereby, it is very remarkable that no higher regularity assumption than continuity for the deformation map is required. The proof exclusively relies on homotopy methods and the Jordan-Brouwer separation theorem.Holger Langreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1874Sat, 23 Jun 2007 13:31:37 +0200Notes at the embeddedness of the minimal surface of Costa, Hoffman and Meeks
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1777
The existence of a complete, embedded minimal surface of genus one, with three ends and whose total Gaussian curvature satisfies equality in the estimate of Jorge and Meeks, was a sensation in the middle eighties. From this moment on, the surface of Costa, Hoffman and Meeks has become famous all around the world, not only in the community of mathematicians. With this article, we want to fill a gap in the injectivity proof of Hoffman and Meeks, where there is a lack of a strict mathematical justification. We exclusively argue topologically and do not use additional properties like differentiability or even holomorphy.Holger Langreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1777Sat, 23 Sep 2006 13:08:22 +0200A homotopy between the solutions of the elastic and elastoplastic boundary value problem
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1776
In this article, we give an explicit homotopy between the solutions (i.e. stress, strain, displacement) of the quasistatic linear elastic and nonlinear elastoplastic boundary value problem, where we assume a linear kinematic hardening material law. We give error estimates with respect to the homotopy parameter.Holger Lang; Klaus Dressler; Rene Pinnau; Gerd Bitschreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1776Fri, 22 Sep 2006 16:05:14 +0200Error estimates for quasistatic global elastic correction and linear kinematic hardening material
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1775
We consider in this paper the quasistatic boundary value problems of linear elasticity and nonlinear elastoplasticity with linear kinematic hardening material. We derive expressions and estimates for the difference of solutions (i.e. stress, strain and displacement) of both models. Further, we study the error between the elastoplastic solution and the solution of a postprocessing method, that corrects the solution of the linear elastic problem in order to approximate the elastoplastic model.Holger Lang; Klaus Dressler; Rene Pinnau; Michael Speckertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1775Thu, 21 Sep 2006 20:54:30 +0200Lipschitz estimates for the stop and the play operator
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1773
In this article, we give some generalisations of existing Lipschitz estimates for the stop and the play operator with respect to an arbitrary convex and closed characteristic a separable Hilbert space. We are especially concerned with the dependency of their outputs with respect to different scalar products.Holger Lang; Klaus Dressler; Rene Pinnaureporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1773Thu, 21 Sep 2006 12:05:33 +0200Parameter optimization for a stress-strain correction scheme
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1702
A gradient based algorithm for parameter identification (least-squares) is applied to a multiaxial correction method for elastic stresses and strains at notches. The correction scheme, which is numerically cheap, is based on Jiang's model of elastoplasticity. Both mathematical stress-strain computations (nonlinear PDE with Jiang's constitutive material law) and physical strain measurements have been approximized. The gradient evaluation with respect to the parameters, which is large-scale, is realized by the automatic forward differentiation technique.Holger Lang; Rene Pinnau; Klaus Dreßlerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1702Sat, 21 Jan 2006 15:51:35 +0100A multiaxial stress-strain correction scheme
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1701
A method to correct the elastic stress tensor at a fixed point of an elastoplastic body, which is subject to exterior loads, is presented and analysed. In contrast to uniaxial corrections (Neuber or ESED), our method takes multiaxial phenomena like ratchetting or cyclic hardening/softening into account by use of Jiang's model. Our numerical algorithm is designed for the case that the scalar load functions are piecewise linear and can be used in connection with critical plane/multiaxial rainflow methods in high cycle fatigue analysis. In addition, a local existence and uniqueness result of Jiang's equations is given.Holger Lang; Rene Pinnau; Klaus Dreßlerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1701Sat, 21 Jan 2006 15:51:16 +0100