Organizers: Jörg Fehr (Institute of Engineering and Computational Mechanics, University of Stuttgart)
Jürgen Pannek S 1 : Multi-body dynamics
Tuesday 14:00 - 16:00 Marienstr. 7, 1st floor, Room 105
Challenges in Modeling Flexible Bodies based on Experimental Data with Utilization in Elastic Multibody Simulation
C. Lein (TU Dresden, Fakultät Maschinenwesen), J. Woller (TU Dresden, Fakultät Maschinenwesen), M. Beitelschmidt (TU Dresden, Fakultät Maschinenwesen)
14:00
The Elastic Multi-Body Simulation (EMBS) progressively constitutes the established method when dealing with elastic deformations of components in complex mechani-cal systems. The Floating-Frame-of-Reference-Formulation (FFRF) is the state-of-the-art method, whereas the model of the elastic body is usually generated by the Finite-Element-Method (FEM). However, the FE-model consists of several uncertainties con-cerning geometry, mass distribution, local and directional stiffness as well as damping phenomena. For representing realistic components, in many cases an expensive model-updating based on measured data is necessary. A Model Order Reduction (MOR) is the next step in the conventional process condensing the elastic degrees of freedom, which embodies further approximation errors. Finally, the information of the elastic body model is passed to the EMBS using the Standard Input Data (SID) file format.
Due to the complex and error-prone conventional procedure, a novel approach is sug-gested, where the data of the elastic body model is directly gained from the results of an Experimental Modal Analysis (EMA) without using any FE-model or MOR. This novel approach yields five challenges that need to be dealt with: A proper measurement setup (free-free support and measurement points), the parameter identification (extraction of modal information), a correct representation of the mass distribution, the treatment of rotational coordinates and the creation of interface nodes for the EMBS-couplings.
The contribution presents solutions for the mentioned challenges and shows results of the novel approach using an U-section as validation example. The whole procedure is performed for the MBS-software SIMPACK version 9. The processing of the experimen-tal data and generation of the SID are realized by means of the software MORPACK
(Model Order Reduction PACKage), which is developed at the chair of dynamics and mechanism design.
Higher-Order Index-1 Co-Simulation Approach: Solver Coupling for Multi-body Systems
T. Meyer (TU Darmstadt), B. Schweizer (TU Darmstadt), P. Li (TU Darmstadt), D. Lu (TU Darmstadt)
14:20
This contribution attends to a co-simulation approach for solver coupling in time domain.
A general multibody system is divided into several subsystems, which are coupled by algebraic constraint equations. The coupling technique analyzed here is a linear-implicit predictor/corrector approach, i.e. coupling variables for the corrector step are calculated by one step of a Newton-iteration. At the communication-time points the coupling con-ditions together with its first and second derivatives are satisfied simultaneously, except for the error of the Newton-iteration. This index-1 approach uses cubic polynomials to approximate the coupling variables. The space of polynomials of degree≤ 3 is a four-dimensional vector space. One of the four degrees of freedom is used for a continuous approximation of the coupling variables at the communication-time points. The three remaining degrees of freedom are used in order to enforce the coupling equations on position, velocity, and acceleration level. Due to the higher order approximation, the numerical errors are very small and a good convergence behavior is achieved.
[1] Arnold, M.: Stability of sequential modular time integration methods for coupled multibody system models. J. Comput. Nonlinear Dyn. 5, 1–9 (2010)
[2] Schweizer, Bernhard, and Daixing Lu. ”Stabilized index-2 co-simulation approach for solver coupling with algebraic constraints.” Multibody System Dynamics 34.2 (2015):
129–161, doi:10.1007/s11044-014-9422-y.
[3] Schweizer, Bernhard, Pu Li, and Daixing Lu. ”Implicit co‐simulation methods: Sta-bility and convergence analysis for solver coupling approaches with algebraic con-straints.” ZAMM–Journal of Applied Mathematics and Mechanics (2015), DOI:
10.1002/zamm.201400087.
Simulation of the dynamics of flexible mounted machines with nonlinear spring elements
E. Gerlach (TU Ilmenau), B. Fiedler (TU Ilmenau) 14:40
This contribution presents ideas, how influence undesired interactions between ground and machines, e.g., vibrations. While designing machines hook up, it is often important to minimize the dynamic load on the ground. One possibility is the optimal design of the stiffness and damping properties of the elastic mountings to reduce the emission of vibrations [1]. For this purpose, rubber elements, metal cushions, hydraulic bearings or form springs are often used. But to design these components well, an accurate knowledge of the excitation is required. If there are no or only few information about the excitation,
numeric simulations could be performed to analyze different scenarios. The multi-body simulation software ALASKA is used to analyze a simplified model of the system (e.g.
centrifuge). The model consists of rigid bodies and the elastic support. The elastic support is represented by a combination of commonly used elements incorporated in ALASKA. They form a series circuit. Element one is a spring/damper element with nonlinear properties, it is modelling within the limits of the Kevin-Voigt model. Model parameter are measured by a real rubber element. The second element is characterized by the behavior of a nonlinear form spring.
The dynamic behavior of the mechanical system could be described by a system of differential equations of the form:
M ¨⃗q + K ˙⃗q + C⃗q = ⃗Q
The elements of the matrices M , K and C are constant or variable in time. Matrices K and C depend on the elastic support. Changing the coefficients and the loading different simulations are done to study the system behavior.
[1] H.Dresig, F.Holzweißig: Maschinendynamik, 12. Auflage, Springer, Berlin, Heidel-berg, 2016
Effective preconditioning techniques for domain decomposition methods for nonlinear dynamic systems arising in multibody dynamics
E. Dewes (Lehrstuhl für Angewandte Mechanik, TU München), D. Rixen (Lehrstuhl für Angewandte Mechanik, TU München)
15:00
Domain decomposition methods, in particular the finite element tearing and intercon-necting (FETI) method, have proven to be highly efficient in solving large-scale finite element problems arising in structural dynamics. Their good numerical and parallel scalability is also promising for application to flexible multibody system dynamics.
In many engineering applications, one can assume that the deformations within each flexible body are small, but the overall motion is non-linear and described typically by a co-rotated floating frame. The challenge is then to solve the overall non-linear system efficiently with domain decomposition techniques, which seems a natural approach since each body can be considered as a domain. Especially non-constant jacobians originating from nonlinear constraint equations require good preconditioning techniques to keep computational costs low and preserve scalability.
In this contribution, in order to simulate the dynamic behaviour of constrained multi-body systems, we present a time integration scheme, which is based on a generalized-α method with an embedded FETI method to solve the linear systems arising in ev-ery time step within the Newton iterations. We investigate different types of FETI-preconditioners in the context of multibody system dynamics concerning perfomance, numerical stability and scalability. We validate our results numerically based on an academical test model.
[1] Arnold, Martin ; Brüls, Olivier: Convergence of the generalized-α scheme for con-strained mechanical systems. In: Multibody System Dynamics 18 (2007), Nr. 2, 185–
202. http://dx.doi.org/10.1007/s11044-007-9084-0. – DOI 10.1007/s11044–
007–9084–0. – ISSN 1573–272X
[2] Farhat, Charbel ; Roux, Francois-Xavier: A method of Finite Element Tearing and Interconnecting and its parallel solution algorithm. In: International Journal for Numerical Methods in Engineering 32 (1991), S. 1205–1227
[3] Rixen, Daniel J.: Extended preconditioners for the FETI method applied to constrained problems. In: International Journal for Numerical Methods in Engi-neering 54 (2002), Nr. 1, 1–26. http://dx.doi.org/10.1002/nme.412. – DOI 10.1002/nme.412. – ISSN 1097–0207
Interface-Reduction for Substructured Mechanical Systems with Constraints Using General Singular Value Decomposition
N. Walker (University of Stuttgart), P. Eberhard (University of Stuttgart) 15:20
During the development of new products, different components are usually designed separately. Therefore, a modular setup of simulation models, as provided in substruc-tured finite-element models or in flexible multibody systems, is advantageous. If the finite-element models of the components are densely meshed, the simulation might get expensive. Therefore, a model order reduction is performed. In substructured settings, model order reduction including moment matching has shown to be advantageous. One major drawback of these methods is that the minimal order of the reduced model depends directly on the number of interactions. Especially when structures are coupled via sur-faces, this might be a decisive restriction. This motivates the use of interface-reduction methods on such surfaces. The coupling between the bodies can be modeled via force el-ements or via constraints. For coupled systems with bushings, interface-reduction using General Singular Value Decomposition (GSVD) has shown good approximation qual-ity. However, the use of the method for bodies coupled via constraints is not straight forward. In this contribution an approach for interface reduction using GSVD for sub-structured mechanical systems with constraints is presented and compared to other interface-reduction methods.
The Boundary Layer Machine
G. Ostermeyer (TU Braunschweig), T. Vietor (TU Braunschweig), M.
Müller (TU Braunschweig), D. Inkermann (TU Braunschweig), J. Otto (TU Braunschweig), H. Lembeck (TU Braunschweig)
15:40
In multi-body systems, friction often occurs as a result of the contact between two bodies. These forces may be desirable or undesirable, depending on the intended function of a technical system. Although many conceivable causes of these forces have been described, the scientific understanding of friction is still incomplete.
A new, holistic approach is to be taken, in order to recognize and describe the general design principles of friction contacts. These principles may also be employed towards the formulation of design guidelines. When two bodies are in contact, many additional effects can be observed along with friction. From a holistic viewpoint, the design of
”two bodies in contact” can be understood as a highly complex machine. The capabilities of this machine include, but are not limited to:
- generation of forces, - generation of heat,
- generation of elastic and acoustic waves including ultrasonic waves, - generation of wear particles,
- generation of material transport with various forms of self-organization, - generation of surface changes on highly differing scale sizes, and
- generation of chemical reactions.
These effects are all dynamically coupled with one another. The boundary layer machine represents the simplest possible ”machine” in a multi-body system.
S 1 : Multi-body dynamics
Tuesday 16:30 - 18:30 Marienstr. 7, 1st floor, Room 105
The long History of Impact Mechanics, Rolling Contact and Multibody Sys-tem Dynamics
W. Schiehlen (University of Stuttgart) 16:30
Contact mechanics is a branch of solid mechanics where two or more bodies are involved.
The contacting points are characterized by a common tangential plane and a common normal vector given by the surface curvature of the bodies. Additional parameters of contact problems are the material of the bodies, the forces and torques acting on the bodies as well as their time history, location of the centers of mass and the roughness of the surface around the contact point. The classical contact problem solved by Hertz (1882) is based on spherical bodies surfaces, linear-elastic material, constant static load-ing in normal direction and smooth surfaces. The rigorous results are often used as benchmarks. Details can be found in the monographs of Johnson (2003) and Popov (2015). The bodies of mechanical systems in engineering applications get in contact to each other mainly by impacts, rolling or sliding, or by joints.
Impacts of bodies by collisions result in strongly augmented forces what is known to humans since ancient times. The first human tools have been originally hand-axes made of rock which were replaced by hammers about 10 thousand years ago. Today, hammers are used by everybody. They are now properly designed with shaft lengths considering the center of impact to avoid reaction forces in the worker’s hand. In antiquity already the philosopher Aristotle stated the question on the augmentation of impact forces, he
imagined a lever action. In the Middle Ages many scientists started working on impact problems as reviewed by Szabo (1977). The recent state of the art was presented by Sändig, Schiehlen and Wendland (2000).
Rolling contact plays a unique, central and dominant role for human mobility. The two contacting bodies involved are the wheel and the guideway. And these two parts are not available in our natural environment, they were invented by humans. Most of the other kinds of transportation like flying, swimming or diving have been inspired by birds and fishes. This is the reason why the wheel was invented very recently, namely 3.500 years before Christ. For comparison, the life on our planet started 4 billion years ago.
The first rail tracks were laid 1767 in UK for horse drawn vehicles and since 1825 for railways. The first asphalt roads were built around 1850 in France, and in 1885 the first car powered by a gasoline engine was successfully completed by Benz in Germany. This means that by the end of the 19th century the basic components for the mechanization of transport were invented. The rolling contact, Carter (1926) and Kalker (1967) or Pacejka (1975), respectively, considered linear or nonlinear material, cylindrical wheel disks and planar guideways, rough surface with friction, static loading and steady-state rotating wheels. For details see Popp and Schiehlen (2010).
Multibody System Dynamics provides engineering software tools for design, simulation and testing of complex dynamical systems, too, in particular for vehicle systems. The fundamental equations are based on classical mechanics. Newton published his three laws in 1686 while Euler added his equations in 1775. During the late Middle Ages many dynamics principles were published, see Szabo (1977). A major progress was achieved by the availability of digital computers half a century ago. Two more recent review papers by Schiehlen (1997) and Shabana (1997) deal with multibody dynamics. Thus, not only the analytical beauty but also the computational efficiency of the algorithms is now important for modelling and visualization of vehicle systems. The contact between the bodies by joints is in multibody system codes as a standard implemented but the wheel-guideway forces of the rolling contact have to be added to complete the vehicle system as discussed in this session on multibody dynamics.
[1] Carter, F.W.: On the action of a locomotive driving wheel. Proc. Roy. Soc. A;
112-151 (1926)
[2] Hertz, H.: Über die Berührung fester elastischer Körper (in German). Journal reine angewandte Mathe. 92, 156 - 171 (1882)
[3] Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (2003) [4] Kalker, J.J.: On the Rolling Contact of Two Elastic Bodies in the Presence of Dry
Friction. PhD Thesis, TH Delft (1967)
[5] Pacejka, H.B.: Principles of Plane Motions of Automobiles. In: Pacejka, H.B. (ed.) Proc. IUTAM Symp. Dynamics Vehicles Roads Railway Tracks, pp. 33-59. Swets and Zeitlinger, Amsterdam (1975)
[6] Popov, V.: Kontaktmechanik und Reibung (in German). Springer (2015)
[7] Popp, K. and Schiehlen, W.: Ground Vehicle Dynamics. Springer (2010)
[8] Sändig, A.M.; Schiehlen, W. and Wendland, W.L.: Multifield Problems – State of the Art. Springer, Berlin 2000.
[9] Schiehlen, W.: Multibody System Dynamics: Roots and Perspectives. Multibody System Dynamics 1, 149 – 188 (1997)
[10] Shabana, A.A.: Flexible Multibody Dynamics: Review of Past and Recent Devel-opments. Multibody System Dynamics 1, 189 – 222 (1997)
[11] Szabo, I.: Geschichte der mechanischen Prinzipien (in German). Birkhaeuser, Basel (1977)
Vehicle Modeling by non-perfect Multi-body Systems
G. Rill (OTH Regensburg), T. Schaeffer 16:50
Virtual prototyping has become a common tool in the automotive industry. Commercial software packages make it possible to set up sophisticated and complex three-dimensional vehicle models. Some providers even offer complete vehicle models with a comfortable graphical user interface. However, sophisticated vehicle models are not only required in the automotive industry and their suppliers but also in the academic research com-munity that develops new control strategies in order to improve the comfort and ride safety of vehicles. To avoid costly licence fees, rather simple models, generated by the specific researcher, are commonly used for that purpose. However, a modeling technique tailored to the specific properties of vehicles makes it possible to set up fully non-linear and three-dimensional vehicle models with reasonable effort. This models require a minimum number of floating point operations and will run in real-time even on small computers. The paper shows, that Jourdain’s principle combined with non-trivial gener-alized velocities results in a straight forward and rather simple procedure for generating the equations of motion for road vehicles. In addition, some acceleration terms that are very cumbersome to calculate, may be neglected due to the specific properties of vehicle suspension systems. Simulation results illustrate that the resulting non-perfect multi-body vehicle system is valid in all operating conditions.
Configuration spaces with Lie group structure: A novel approach and its application to a Cosserat beam model
S. Hante (Martin-Luther-Universität Halle-Wittenberg), M. Arnold (Martin-Luther-Universität Halle-Wittenberg)
17:10
In multibody dynamics, the parametrization of the bodies’ configuration plays a crucial role. Large rotations often appear and thus have to be described in a way that is free from singularities. Popular parametrizations are SO(3) and unit quaternions S3. We present the semi-direct productS3⋉ R3 as a novel approach to configuration spaces of rigid bodies and the relation to and advantages over established configuration spaces such as SE(3) or dual quaternions.
Subsequently, we present a Cosserat beam model, where the beam is parametrized by a time-dependent curve in S3 ⋉ R3. We show the advantages over parametrizations involving the direct productS3×R3. While the equations of motion may be obtained by a two-dimensional variational principle (Lang, Linn 2009), we use a discretized variational principle in order to obtain semidiscrete equations of motion, thus applying a method of lines. The resulting equations can be interpreted as the equations of motion of n rigid bodies and therefore a differential equation of second order on the Lie group(
S3⋉ R3)n
. We show results, where a generalized-α Lie group time integrator was applied for time integration.
S 1 : Multi-body dynamics
Wednesday 14:00 - 16:00 Marienstr. 7, 1st floor, Room 105
Dynamical Analysis of a Tip Balancing Cube
H. Gattringer (Johannes Kepler University Linz), A. Reiter, C. Stöger (Johannes Kepler University Linz), M. Jörgl, A. Müller (Johannes Kepler University Linz)
14:00
The paper discusses some dynamical aspects of a self-balancing Inertia Wheel Cube (IWC). The acceleration and deceleration of actuated flywheels is used to stabilize the IWC balancing on one of its corners, which is an unstable equilibrium position. The orientations as well as the rotational velocity of the device are measured by an iner-tial measurement unit (IMU). The equations of motions are derived with the Projection Equation. Using the rotational velocity of the IWC as part of the generalized velocities leads to well-structured and interpretable dynamical equations. These equations are the basis for simulation, parameter identification, control synthesis, and stability anal-ysis. The stability of the unstable equilibrium is investigated by means of the linearized motion equations. It turns out that there are always uncontrollable states due to the conservation of momentum. A further analysis considers offsets in the orientation mea-surement with the IMU and the imprecisely knowledge of the center of gravity. These offsets lead to constant flywheel speeds for the PD controlled IWC. In order to compen-sate this negative effect an observer for the center of gravity is introduced. Simulations as well as experimental results are presented.
A robot inspired by a non-smooth point mass model of a worm
T. Winandy (University of Stuttgart), S. Eugster (University of Stuttgart) 14:20
An earthworm travels using waves of muscular contractions which alternately shorten and lengthen the body. The skin of the earthworm carries claw-like bristles, which anchor the shortened part of the body to the soil.
We present a planar model of a chain of five point masses that mimics the worm-like locomotion. The point masses are aligned horizontally on the ground. The muscles of
the worm are modelled by including a force law between each pair of neighbouring point masses. The four force laws allow the actuation of the chain. The choice of the force laws allows to consider different types of actuators and to compare various actuation patterns. The contacts of the elements with the ground are idealized as being unilateral constraints subjected to friction. It is the inclusion of friction, which enables the chain to
the worm are modelled by including a force law between each pair of neighbouring point masses. The four force laws allow the actuation of the chain. The choice of the force laws allows to consider different types of actuators and to compare various actuation patterns. The contacts of the elements with the ground are idealized as being unilateral constraints subjected to friction. It is the inclusion of friction, which enables the chain to