Johannes Gerstmayr

Vibrations of the elasto-plastic pendulum

Abstract A numerical strategy for vibrations of elasto-plastic beams with rigid-body degrees-of-freedom is presented. Beams vibrating in the small-strain regime are considered. Special emphasis is laid upon the development of plastic zones. An elasto-plastic beam performing plane rotatory motions about a fixed hinged end is used as example problem. Emphasis is laid upon the coupling between the vibrations and the rigid body rotation of the pendulum. Plastic strains are treated as eigenstrains acting in the elastic background structure. The formulation leads to a non-linear system of differential algebraic equations which is solved by means of the Runge–Kutta midpoint rule. A low dimension of this system is obtained by splitting the flexural vibrations into a quasi-static and a dynamic part. Plastic strains are computed by means of an iterative procedure tailored for the Runge–Kutta midpoint rule. The numerical results demonstrate the decay of the vibration amplitude due to plasticity and the development of plastic zones. The pendulum approaches a state of plastic shake-down after sufficient time.

A continuum-mechanics interpretation of Reissner's non-linear shear-deformable beam theory

This article deals with the non-linear modelling of beams that are bent, sheared and stretched by external forces and moments. In the following, we restrict to plane-deformations and static conditions. Our task is to present a continuum mechanics-based interpretation of the celebrated large displacement finite deformation structural mechanics theory, which was presented by Eric Reissner [On one-dimensional finite-strain beam theory: the plane problem, J. Appl. Math. Phys. 23 (1972), pp. 795–804]. The latter formulation was restricted to the notions of structural mechanics and thus did not use the notions of stress and strain, which are fundamental for continuum mechanics. Thus, the common continuum mechanics-based constitutive modelling at the stress–strain level cannot be utilized in connection with Reissners original theory. Instead, Reissner suggested that constitutive relations between certain generalized strains (bending, shear and axial force strains) and generalized static entities (bending moments, shear and normal forces) should be evaluated from physical experiments. This means that the beam to be studied must be first built up, and the experiments must be performed for the real beam as a whole. Although such physical experiments are indeed to be performed in practice for safety reasons in sensible cases, for example, bridge decks or aircraft wings, it is nevertheless felt to be a drawback that the results of simple standardized stress–strain experiments concerning the constitutive behaviour of the materials, from which the beam is built up, cannot be used. Moreover, relying only on physical experiments on the whole beam means that computations (virtual experiments) can be made only after the beam has been built up. To overcome this problem, we subsequently present a continuum mechanics-based interpretation of Reissners structural mechanics modelling, by attaching a proper continuum mechanics-based meaning to both the generalized static entities and the generalized strains in Reissners theory [E. Reissner, On one-dimensional finite-strain beam theory: the plane problem, J. Appl. Math. Phys. 23 (1972), pp. 795–804]. Consequently, these generalized static entities can be related to the generalized strains on the basis of a constitutive modelling on the stress–strain level. We show this in some detail in this contribution for a hyperelastic material proposed by Simo and Hughes [Computational Inelasticity, Springer, New York, 1998]. An illustrative numerical example is given which shows the results of large bending and axial deformation behaviour for different constitutive relations. This article represents an extended version of a preliminary work published in [H. Irschik and J. Gerstmayr, A hyperelastic Reissner-type model for non-linear shear deformable beams, Proceedings of the Mathmod 09 Vienna, I. Troch and F. Breitenecker, eds., 2009, pp. 1–7].

Analysis of Stress and Strain in the Absolute Nodal Coordinate Formulation

Abstract Accurate values of stress and strain are required for the evaluation of comparative stresses in nonlinear material behavior. The absolute nodal coordinate formulation (ANCF) has been recently developed and focuses on the modeling of beams and plates under the presence of large deformation. The derivation of the equations of motion for an ANCF element is usually based on a solid finite element formulation and thus leads to finite elements that show locking behavior. While the problem of locking in the ANCF might be solved by means of standard techniques, the accuracy of stress and strain quantities within elements is still poor and needs to be improved in order to incorporate nonlinear material behavior. In the present paper, a higher order ANCF element is presented where locking is prevented by means of standard selective reduced integration techniques and the improved order and accuracy of stress and strain quantities is shown, in comparison with the original formulation. As an example of nonlinear material behavior, Prandl–Reuss plasticity is integrated in the absolute nodal coordinate formulation. Results of stress and strain components for the improved higher order element are compared to the solution of fully three-dimensional computations performed with the commercial software ABAQUS. Static and dynamic spatial examples are used to investigate the accuracy. Good agreement of the ANCF is found with the results of ABAQUS, as well as with examples of elasto-plastic multibody systems available from the literature.

Coupled optimization in MagOpt

Optimizing mechatronic components is of increasing importance, e.g. for minimizing energy consumption and the use of rare materials. MagOpt is a modular software tool for the simulation and optimization of mechatronic components. Parametric design optimization can be carried out with various different optimization strategies like gradient-based methods or multi-objective evolutionary or genetic algorithms. MagOpt features a flexible structure for the storage of complex data and an open and modular interface to existing third-party programs. One such third-party program which can be used by MagOpt for the optimization of mechanic components is the multi-body software HOTINT. This article describes MagOpt and how it was coupled with HOTINT to optimize a rotor geometry.

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

A Detailed Comparison of the Absolute Nodal Coordinate and the Floating Frame of Reference Formulation in Deformable Multibody Systems

In extension to a former work, a detailed comparison of the absolute nodal coordinate formulation (ANCF) and the floating frame of reference formulation (FFRF) is performed for standard static and dynamic problems, both in the small and large deformation regimes. Special emphasis is laid on converged solutions and on a comparison to analytical and numerical solutions from the literature. In addition to the work of previous authors, the computational performance of both formulations is studied for the dynamic case, where detailed information is provided, concerning the different effects influencing the single parts of the computation time. In case of the ANCF finite element, a planar formulation based on the Bernoulli-Euler theory is utilized, consisting of two position and two slope coordinates in each node only. In the FFRF beam finite element, the displacements are described by the rigid body motion and a small superimposed transverse deflection. The latter is described by means of two static modes for the rotation at the boundary and a user-defined number of eigenmodes of the clamped-clamped beam. In numerical studies, the accuracy and computational costs of the two formulations are compared for a cantilever beam, a pendulum, and a slider-crank mechanism. It turns out that both formulations have comparable performance and that the choice of the optimal formulation depends on the problem configuration. Recent claims in literature that the ANCF would have deficiencies compared with the FFRF thus can be refuted.

A Large Deformation Planar Finite Element for Pipes Conveying Fluid Based on the Absolute Nodal Coordinate Formulation

A novel pipe finite element conveying fluid, suitable for modeling large deformations in the framework of Bernoulli Euler beam theory, is presented. The element is based on a third order planar beam finite element, introduced by Berzeri and Shabana, on basis of the absolute nodal coordinate formulation. The equations of motion for the pipe-element are derived using an extended version of Lagrange’s equations of the second kind for taking into account the flow of fluids, in contrast to the literature, where most derivations are based on Hamilton’s Principle or Newtonian approaches. The advantage of this element in comparison to classical large deformation beam elements, which are based on rotations, is the direct interpolation of position and directional derivatives, which simplifies the equations of motion considerably. As an advantage Lagrange’s equations of the second kind offer a convenient connection for introducing fluids into multibody dynamic systems. Standard numerical examples show the convergence of the deformation for increasing number of elements. For a cantilever pipe, the critical flow velocities for increasing number of pipe elements are compared to existing works, based on Euler elastica beams and moving discrete masses. The results show good agreements with the reference solutions applying only a small number of pipe finite elements.Copyright

A 3D Shear Deformable Finite Element Based on the Absolute Nodal Coordinate Formulation

The absolute nodal coordinate formulation (ANCF) has been developed for the modeling of large deformation beams in two or three dimensions. The absence of rotational degrees of freedom is the main conceptual difference between the ANCF and classical nonlinear beam finite elements that can be found in literature. In the present approach, an ANCF beam finite element is presented, in which the orientation of the cross section is parameterized by means of slope vectors. Based on these slope vectors, a thickness as well as a shear deformation of the cross section is included. The proposed finite beam element is investigated by an eigenfrequency analysis of a simply supported beam. The high frequencies of thickness modes are of the same magnitude as the shear mode frequencies. Therefore, the thickness modes do not significantly influence the performance of the finite element in dynamical simulations. The lateral buckling of a cantilevered right-angle frame under an end load is investigated in order to show a large deformation example in statics, as well as a dynamic application. A comparison to results provided in the literature reveals that the present element shows accuracy and high order convergence.

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

On the Accuracy and Computational Costs of the Absolute Nodal Coordinate and the Floating Frame of Reference Formulation in Deformable Multibody Systems

In the present paper, a comparison of the absolute nodal coordinate formulation (ANCF) and the floating frame of reference formulation (FFRF) is performed for standard static and dynamic problems, both in the small and large deformation regime. Special emphasis is laid on the converged solutions and a comparison to analytical and numerical solutions from the literature. In addition to the work of previous authors, the computational performance of both formulations is studied for the dynamic case, where detailed information is provided concerning the different effects influencing the single parts of the computation time. In case of the ANCF finite element, a planar formulation based on the Bernoulli-Euler theory is utilized, consisting of two position and two slope coordinates in each node only. In the FFRF beam finite element, the displacements are described by the rigid body motion and a small superimposed transverse deflection. The latter is described by means of two static modes for the rotation at the boundary and a user-defined number of eigenmodes of the clamped-clamped beam. In numerical studies, the accuracy and computational costs of the two formulations are compared for a cantilever beam, a pendulum and a slider-crank mechanism. It turns out that both formulations have comparable performance and that the choice of the optimal formulation depends on the problem configuration. Recent claims in the literature that the ANCF would have deficiencies compared to the FFRF thus can be refuted.Copyright