Influence of model reduction techniques on the impact force calculation of two flexible bodies

One important issue for the simulation of flexible multibody systems is the reduction of the flexible body's degrees of freedom. For the reduction process finite element data and user inputs are necessary. The model reduction program for elastic multibody systems MOREMBS, which is developed at the ITM, has an easy-to-use interface and the data can be gained from the programs ABAQUS or ANSYS. In this work, the simulation of a fuel injection process is investigated with MOREMBS. We focus on the interaction between valve and armature. These two bodies impact in every injection circle. The impacting bodies are modeled as flexible and the contact force is calculated by a penalty approach. One essential part of this work is the investigation of the influence of different model reduction techniques on the impact force calculation of the flexible multibody system. The main reduction techniques modal reduction, Krylov-subspace based and Gramian matrix based techniques are compared. The results achieved with modal reduction, the state of the art reduction method, are not acceptable here. Krylov-subspace based techniques are especially well-suited for large sparse systems but are not error controlled. However, by choosing appropriate moment-matching properties the impact force calculation is nearly as good as with a full finite element model. The Gramian matrix based reduction techniques can be fully automated and are error controlled but require high computational effort. Hence, appropriate approximation schemes have to be used for them. With Gramian matrix based methods we can even further reduce the size of the subsystems compared to Krylov-subspace based methods and still have an impact force calculation nearly as good as with finite element results, but we gain a simulation speedup by the factor 4000. In addition, a parameter study of the parameters involved in the model reduction process is presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Contact Modeling in Multibody Systems with Elastic Bodies in High-Frequency Applications

This paper discusses the refinement of multibody models by integration of flexible bodies and by considering nonlinearities from contacts. It presents common approaches for contact modeling in multibody simulations and strategies to include flexible bodies. A contact model is implemented in the elastic multibody model. Experimental results show that significant effects of system dynamics can be modeled by use of a multibody model including elastic bodies and contacts. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rotationless formulation for large deformations in flexible multibody dynamics

Especially for specific applications, such as contact problems, computer methods for flexible multibody dynamics that are able to treat large deformation phenomena are important. Classical formalisms for multibody dynamics are based on rigid bodies. Their extension to flexible multibody systems is typically restricted to linear elastic material behavior whereas large deformation phenomena are formulated in the framework of the nonlinear finite element method. In the talk we address computer methods that can handle large deformations in the context of multibody systems. In particular, the link between nonlinear continuum mechanics and multibody systems is facilitated by a specific formulation of rigid body dynamics [1]. It makes possible the incorporation of state-of-the-art computer methods for large deformation problems. In the talk we focus on the treatment of large deformation contact whithin flexible multibody dynamics [2]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient integration of flexible multibody dynamics

The modelling of flexible multibody dynamics as finite dimensional Hamiltonian system subject to holonomic constraints constitutes a general framework for a unified treatment of rigid and elastic components. Internal constraints, which are associated with the kinematic assumptions of the underlying continuous theory, as well as external constraints, representing the interconnection of different bodies by joints, can be accounted for in a likewise systematic way. The discrete null space method developed in [0] provides an energy-momentum conserving integration scheme for the DAEs of motion of constrained mechanical systems. It relies on the elimination of the constraint forces from the discrete system along with a reparametrisation of the nodal unknowns. The resulting reduced scheme performs advantageously concerning different aspects: the constraints are fulfilled exactly, the condition number of the iteration matrix is independent of the time step and the dimension of the system is reduced to the minimal possible number saving computational costs. A six-body-linkage possessing a single degree of freedom is analysed as an example of a closed loop structure. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the influence of the lamella's elasticity on self-excited vibrations in gearboxes

Eek noise in a gearbox of a vehicle drivetrain is a phenomenon, which can arise while shifting between gears and which is not accepted by customers. Beneath audible squeaking, it can cause damage of mechanical components. There is a wide range of possible reasons for the occurrence of this effect, which strongly depends on properties of the considered gearbox (physical parameters, geometry, operation, …). From the mathematical point of view, the occurrence can be predicted using linear stability analysis of the stationary behaviour of a physically motivated gearbox model. The components of a gearbox are clutch discs being in contact, gears and elastically supported shafts. In this contribution, a rigid multibody model of the device [4] is extended by the elastic modelling of the motor's side disc (rotating Kirchhoff plate). The aim of the overall system is to analyze the shifting process. The analysis reveals that beneath instability mechanisms which are known from systems with rigid bodies, new instabilities occur incorporating of out-of-plane vibrations of the plate. In a reasonable parameter region, the first two unsymmetrical modes of the lamella have the main contribution to the instability. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Various Feedforward Schemes and a Passivity Based Backstepping Feedback Control of an Elastic Robot

Lightweight constructions in industry plants lead to elastic deflections causing vibrations and a loss in tracking precision. In order to keep the tracking error for these elastic multibody systems low, the proposed control strategies are combinations of feedforward and feedback schemes. In this work various implemented strategies for computing a feedforward control are proposed and compared, which can be calculated with some simplifications from the mathematical model of the elastic multibody system. Some of these are considering the elastic deflections. The stabilization of the error dynamics is achieved by a simple PD-joint control or passivity based backstepping. In this algorithm the system is split into subsystems and for these subsystems simple control concepts can be applied. The feedback control law of the total system is obtained by means of backstepping theory, considering the internal energy flows in the system. Experimental results are presented to verify the control strategies. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Approach to Feed-Forward Controller Design for Underactuated Multibody Systems in the Presence of Uncertainty

The performance of a model-based tracking controller depends on the quality of the underlying model. Especially for flexible multibody systems, the derivation of a suitable model and the subsequent controller design are challenging tasks. In the paper, it is shown how in a straightforward approach a feed-forward controller for a flexible multibody system is designed based on a simplified model which approximates an elastic beam by a combination of rigid beams and force elements. Furthermore, the modelling error due to this harsh simplification is included as uncertainty in the simplified model and considered in the model-based feed-forward controller design using fuzzy arithmetic. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

DAEs and PDEs in elastic multibody systems

Elastic multibody systems arise in the simulation of vehicles, robots, air- and spacecrafts. They feature a mixed structure with differential-algebraic equations (DAEs) governing the gross motion and partial differential equations (PDEs) describing the elastic deformation of particular bodies. We introduce a general modelling framework for this new application field and discuss numerical simulation techniques from several points of view. Due to different time scales, singular perturbation theory and model reduction play an important role. A slider crank mechanism with a 2D FE grid for the elastic connecting rod illustrates the techniques. This revised version was published online in June 2006 with corrections to the Cover Date.     

Effect of wing deformation on highlift systems of transport aircraft

An elastic multibody model is developed that combines highlift system models consisting of flap and slat mechanisms with an elastic wing model. It describes the motions and deformations of the highlift systems with airloads applied and, in particular, the effect of wing deformations on the highlift systems. In addition the model contains a flexible wing with the engine and the middle section of the fuselage that is fixed at the symmetry plane of the aircraft. While previously wing deformation was taken into account in static FE simulations only the developed multibody model provides the investigation of the motions and inner loads of the highlift mechanisms with consideration of the wing deformations with dynamical effects. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equations of Motion for Constrained Multibody Systems and their Control

This paper presents some recent advances in the dynamics and control of constrained multibody systems. The constraints considered need not satisfy the D’Alembert principle and therefore the results are of general applicability. They show that, in the presence of constraints, the constraint force acting on the multibody system can always be viewed as made up of the sum of two components whose explicit form is provided. The first of these components consists of the constraint force that would have existed were all the constraints ideal; the second is caused by the nonideal nature of the constraints, and though it needs specification by the mechanician who is modeling the specific system at hand, it has a specific form. The general equations of motion obtained herein provide new insights into the simplicity with which Nature seems to operate. They point toward the development of new and novel approaches for the exact control of complex multibody nonlinear systems. In honor of Bob kalaba, friend, colleague, and mentor.     

Non-linearity of multibody dynamic equations with respect to Lagrange multipliers: application to railway dynamics

In this paper, the dynamics of multibody systems with closed kinematical chains of bodies is considered. The main focus is set on non-linearity of the multibody equations with respect to the Lagrange multipliers. When closed chains are considered, loop cutting procedure is a solution to express the constraint equations associated with the loops. Dynamic equations of the multibody tree-like structure are thus completed with the constraint forces via the Lagrange multipliers. In the considered case of railway vehicles, constraints arise from the contact between the rigid wheels and the rails. Corresponding contact forces applied to the wheels appears via the Lagrange multipliers λ and the tangent creep forces as well. Resulting differential-algebraic equations can be transformed into an ODE system and then time-integrated using the coordinate partitioning method [3], when the system is linear with respect to λ. This paper presents an algorithm allowing us to solve this system in case of nonlinearities with respect to λ, which is typical of wheel/rail contact force models. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Real-Time Simulation of Elastic Multibody Systems with Application in Vehicle Dynamics

Real-time simulation models are widely used for vehicle development, usually built up as rigid multibody systems. However, since lightweight structures are commonly used, body deformation is no longer negligible and rigid multibody simulations may be inaccurate. This work presents a real-time capable full vehicle model with a flexible car body, derived from a finite element model, whose performance has been improved by model order reduction. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonlinear finite element framework for flexible multibody dynamics

We present a uniform treatment of rigid body dynamics and nonlinear structural dynamics. The advocated approach is based on a rotationless formulation of rigid bodies, nonlinear beams and shells. In this connection, the specific kinematic assumptions are taken into account by the explicit incorporation of holonomic constraints. This approach facilitates the straightforward extension to flexible multibody dynamics by including additional constraints due to the interconnection of rigid and flexible bodies. We further address the design of energy-momentum schemes for the stable numerical integration of the underlying finite-dimensional Hamiltonian systems. To demonstrate the superior numerical performance of the proposed methodology, the numerical examples deals with a multibody system containing both rigid and flexible bodies undergoing large deformations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)