Einbindung elastischer Körper in die Mehrkörpersimulationsbibliothek

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1 Einbindung elastischer Körper in die Mehrkörpersimulationsbibliothek von SimulationX Tobias Zaiczek, Sven Reitz, Olaf Enge-Rosenblatt, Peter Schneider Fraunhofer Institut für Integrierte Schaltungen, Institutsteil Entwurfsautomatisierung, Dresden Kurzfassung Dieses Manuskript umfasst ausgewählte Aspekte der Integration von Finite- Elemente-Modellen, die das linear-elastische Verhalten von Körpern beschreiben, in die bereits vorhandene Mehrkörperbibliothek von SimulationX. Hierbei spielen verschiedene Gesichtspunkte, wie die oftmals notwendige Modellordnungsreduktion, die geeignete Berechnung der Konnektorvariablen und die Berücksichtigung nichtlinearer Terme, die das dynamische Verhalten des Körpers bei großen Bewegungen beschreiben, ebenso eine Rolle wie spezielle Herausforderungen bei der Anpassung der mechanischen Modelle an die Mehrkörperbibliothek von SimulationX. Abstract This paper covers important aspects of the integration of Finite Element models, that describe the linear elastic behaviour of bodies, into the well-established multibody library of SimulationX. Here, some considerations about the often necessary model order reduction, the appropriate calculation of connector variables, and the significance of nonlinear terms describing the dynamical behaviour of the body s large motions play an essential role as well as specific challenges for the adaptation of these mechanical models to the multi-body library of SimulationX. Introduction The current multi-body library of SimulationX, MechanicsMBS, includes already a large variety of elements and supports in addition to the description of classical rigid body models also the integration of elastic beam elements. However, in order to represent the elastic behaviour of arbitrarily shaped solid bodies and beam elements within the multi-body simulation environment, the already existing library had to be extended. The task of including the dynamic behaviour of deformable bodies into classical multi-body libraries has already been investigated by several authors ([2], [5], [6]). Here, a different approach will be presented, as we try to derive the differential equations of motion directly from the parameters of the Finite Element model

2 (FEM), namely the mass, the damping, and the stiffness matrix. One essential advantage of this approach is the use of existing elaborate methods for the Finite Element discretization of the distributed parameter model derived from the linear theory of structural mechanics. In the following sections, different aspects of the import of Finite Element models into the multi-body library of SimulationX are presented. These sections cover the basic ideas of the chosen approach, of the required definition of appropriate interfaces to other subsystems of the simulation environment, of the model order reduction, of the inclusion of the nonlinear behaviour of large motions, and of the adaptation of the models to the specific needs of the algorithms used in the MechnicsMBS library. The second part of this paper is devoted to the developed FEM-Import-Tool for the import of flexible bodies. The general conception of this program as well as the work flow for the import of an deformable body are shown for an industrial relevant example. For the sake of shortness, we renounce the discussion of the relevant equations in more detail. More insights especially concerning the mathematical treatment of the following sections can be gained from the EOOLT paper [7], that is freely available online. Basics of FEM Import The starting point of the presented task is the output of a Finite Element simulation tool. Within the tool, the solid body or the beam element can be described concerning its geometry and its material properties. Using the Finite Element solver, a spatially discretized model can be generated that can be written down as (1) with the quantities as the vector of all (translational and maybe rotational) displacement variables, as the vector of all (translational and maybe rotational) forces, the mass matrix, the damping matrix, and the stiffness matrix. This linear system of differential equations gives a sufficiently good approximation of the elastic behaviour of the body under small deformations. Figure 1: Flexible body The basic idea for the import of such a model is that the motion of the body can be interpreted by a linear superposition of the possibly large motion of the undeformed body and the motion caused by its small deformations. The former can be represented by the motion of a reference frame with respect to (w. r. t.) an inertial frame while the latter may be described by the linear FEM (see Figure 1). This method is also called floating frame of reference method in the

3 literature. From this idea, there result some questions that are discussed in the subsequent subsections. Inclusion of nonlinear terms For the import of the mechanical structures, we have to take the nonlinear dynamic forces into account resulting from the large motions of the body in the three-dimensional space. As proposed in the section above the motion can be splitted up into two parts, the motion of the undeformed body including all related nonlinear terms and the motion resulting from the deformation of this body. As it turned out, these motions are not independent of each other. For the inclusion we consider the original equations (1) in a moving frame. So, we replace all time derivatives w. r. t. the inertial frame by time derivatives w. r. t. the moving reference frame and express the acceleration as a linear superposition of the acceleration w. r. t. the moving reference frame and the acceleration of the moving reference frame w. r. t. to the inertial frame. For pure solid bodies with solely translational degrees of freedom, we can write the equations of motion as with consisting of For beam elements, the equations get some more involved. Connector definition The import of Finite Element models requires the definition of an appropriate interface to other subsystems of the simulation environment. For the sake of exchangeability, this interface, called connector, must be compatible to the connectors of the other library elements. By introducing appropriate connector variables for every connector, the model is extended by some new degrees of freedom (DoFs). There might be different possible methods to add a relation between the new DoFs and the DoFs of the original FEM. However, in the current version of the FEM-Import-Tool, a certain method has been chosen for the generation of these mechanical connector elements: For every connector a special region of the body has to be chosen together with a location of a virtual master node. All nodes lying within the region are called slave nodes and are rigidly connected to the master node. The resulting rigid part of the body is called flange and allows the interconnection to other rigid bodies. Even though this approach may change the model properties, this method seems very reasonable and has an essential advantage, as it ensures under certain conditions that the model remains independent of the signal flow direction within the simulation. For that, we need to fulfil the requirement that the knowledge about the position (and possibly the orientation) of the rigidly linked slave nodes uniquely determine the position and orientation of the flange and vise versa. Therefore, for pure solid bodies (with no rotational DoFs), there exists the.

4 restriction that at least three nodes not lying on one line have to be used as slave nodes for one flange. Model order reduction The linear system of differential equations (1) is typically large in scale to achieve a good approximation of the constraint partial differential equation for all nodes over a wide range of the frequency domain. Anyway, SimulationX as an equationbased simulation tool applies computer algebra to derive a solvable system of differential algebraic equations. The computational efforts and the memory consumption for these operations increase dramatically for a growing number of equations and variables. Thus, SimulationX is generally not able to handle largescale dynamic systems directly. However, for many applications it is already sufficient to approximate the behaviour between the connector variables in a relevant range of the frequency domain. Hence, it is preferable and often necessary to reduce the size of the system drastically by an appropriate reduction method. The reduction method should of course take into account the interesting range of the frequency domain. There are many different methods ([1], [3]) for the linear model order reduction and they all produce a reduction matrix defining a linear mapping between the vector of reduced variables and the vector of the original variables. For our model we used a sophisticated model order reduction algorithm that has been implemented at the Fraunhofer Institute for Integrated Circuits, Design Automation Division in Dresden ([4]). However, for a proper inclusion of the large motion behaviour, it is necessary that the reduction matrix includes the six rigid body modes of the compound, i. e. that this matrix includes six columns with the displacements of all nodes when moving the undeformed body a little according to its six DoFs. Definition of the Reference Frame and the Recursive Algorithm As there is still no relation between the motion of the reference frame and the motion of the body, we still have to make some assumptions on the model. For the sake of simplicity, in the current version of the FEM-Import-Tool only one method has been implemented, that fixes the reference frame to any of the used flanges. The multi-body library of SimulationX uses a recursive formalism for the generation of the equations of motion. In order to ease the application of the desired formalism, the models have to be transformed to a special structure. Technical Issues of the FEM Import In this part of the paper, the work flow for the import of a mechanical FEM into the library MechanicsMBS is sketched introducing the new FEM-Import-Tool and its auxiliary programs.

5 Export Routines Here, it is very important to mention, that for the consistancy of all subsequent calculations, the body has to be modeled as a free body within the Finite Element simulation tool, i. e. without any constraints on the undeformed motion of the body. According to the explanations above we have to export the matrices,, and for our further calculations. In addition, we need to export the position of all nodes of the undeformed body. Furthermore, a file must be generated to determine all possible flanges. Therein, for every flange, a name, the location of the virtual master node w. r. t. the reference frame and all node numbers must be specified. For a possible visualisation of the body, one could also export some mesh or element information. For two Finite Element programs, ANSYS and Comsol Multiphysics, we implemented export routines that enables the user to export all necessary data in a quite comfortable way. The interested reader can find more information within the documentation of the FEM-Import-Tool. The FEM-Import-Tool The FEM-Import-Tool is a program that imports the data generated by the export routines of the Finite Element simulation tool, that does all the necessary preprocessing described in the last section, and that exports the desired models into some Modelica source code files. Figure 2 shows the program structure and the data flow for the import of a flexible model in SimulationX. After the export of all necessary data from the Finite Element simulation tool, the user can start the FEM-Import-Tool using the graphical user interface of SimulationX. Figure 2: Work flow of FEM import into SimulationX Import of Models into SimulationX As soon as the model has been generated by the FEM-Import-Tool, the model can be included into SimulationX as an external model. For more detailed information refer to the user manual of SimulationX.

6 Example As an example, the control of a double pendulum will be investigated. Figure 3 shows a sketch of the experiment setup. The controller consists of two parts: a feedforward branch using the method of computed torques and a feedback PID controller. In order to investigate the influence of the flexibility of the second arm of the robot on the controller performance, the rigid beam has been substituted by a flexible body of the same shape and mass. Figure 3: Sketch of double pendulum The schematic of the model in SimulationX is depicted in Figure 4. In absence of disturbances, the method of computed torques is supposed to ideally control the system outputs and of the rigid body model along the reference trajectories without any control action of the feedback loop. Figure 4: SimulationX model of double pendulum However, when using the model of the flexible body for the second arm, the task of the feedback controller is to compensate the different behaviour of the flexible robot. As shown in Figure 5, the controller is able to stabilize the system along the reference trajectory. The right-hand side of Figure 5 shows the error between the reference trajectory and the actual trajectory passed by the revolute joints. This example shows, how the integration of elastic bodies can be utilized to analyse the system behavior, in terms of stability, eigenfrequencies or controller performance.

7 Figure 5: Simulation results: reference and actual values (left) and control errors (right) for both revolute joint angles and Summary The paper presented an approach for the integration of Finite Element models, that describe the linear elastic behaviour of bodies, into the multi-body library of SimulationX. Special issues as the model order reduction, the appropriate calculation of connector variables, and the significance of nonlinear terms describing the dynamical behaviour of the body s large motions have been pointed out. In addition, specific challenges for the adaptation of these mechanical models to the multi-body library of SimulationX have been mentioned. Finally, some simulation results have been presented for the analysis of a simple control example. [1] A. C. Antoulas: Approximation of large-scale dynamical systems. Society for Industrial & Applied Mathematics, [2] H. Bremer and F. Pfeiffer: Elastische Mehrkörpersysteme. Teubner, [3] R. W. Freund: Model reduction methods based on Krylov subspaces. Acta Numerica, 12: , [4] A. Köhler: Modellreduktion von linearen Deskriptorsystemen erster und zweiter Ordnung mit Hilfe von Block-Krylov-Unterraumverfahren. Diplomarbeit. TU Dresden, Germany, [5] A. A. Shabana: Dynamics of Multibody Systems. Cambridge University Press, 2nd Edition, [6] R. Schwertassek and O. Wallrapp: Dynamik flexibler Mehrkörpersysteme. Vieweg-Verlag, [7] T. Zaiczek and O. Enge-Rosenblatt: Import of distributed parameter models into lumped parameter model libraries for the example of linearly deformable solid bodies. 3rd International Workshop on Equation-Based Object-Oriented Languages and Tools. Oslo, Norway, October 3, 2010, Proc., pages