Claus Führer

Numerical methods in vehicle system dynamics: state of the art and current developments

Robust and efficient numerical methods are an essential prerequisite for the computer-based dynamical analysis of engineering systems. In vehicle system dynamics, the methods and software tools from multibody system dynamics provide the integration platform for the analysis, simulation and optimisation of the complex dynamical behaviour of vehicles and vehicle components and their interaction with hydraulic components, electronical devices and control structures. Based on the principles of classical mechanics, the modelling of vehicles and their components results in nonlinear systems of ordinary differential equations (ODEs) or differential-algebraic equations (DAEs) of moderate dimension that describe the dynamical behaviour in the frequency range required and with a level of detail being characteristic of vehicle system dynamics. Most practical problems in this field may be transformed to generic problems of numerical mathematics like systems of nonlinear equations in the (quasi-)static analysis and explicit ODEs or DAEs with a typical semi-explicit structure in the dynamical analysis. This transformation to mathematical standard problems allows to use sophisticated, freely available numerical software that is based on well approved numerical methods like the Newton–Raphson iteration for nonlinear equations or Runge–Kutta and linear multistep methods for ODE/DAE time integration. Substantial speed-ups of these numerical standard methods may be achieved exploiting some specific structure of the mathematical models in vehicle system dynamics. In the present paper, we follow this framework and start with some modelling aspects being relevant from the numerical viewpoint. The focus of the paper is on numerical methods for static and dynamic problems, including software issues and a discussion which method fits best for which class of problems. Adaptive components in state-of-the-art numerical software like stepsize and order control in time integration are introduced and illustrated by a well-known benchmark problem from rail vehicle simulation. Over the last few decades, the complexity of high-end applications in vehicle system dynamics has frequently given a fresh impetus for substantial improvements of numerical methods and for the development of novel methods for new problem classes. In the present paper, we address three of these challenging problems of current interest that are today still beyond the mainstream of numerical mathematics: (i) modelling and simulation of contact problems in multibody dynamics, (ii) real-time capable numerical simulation techniques in vehicle system dynamics and (iii) modelling and time integration of multidisciplinary problems in system dynamics including co-simulation techniques.

A nonlinear truck model and its treatment as a multibody system

Abstract A planar vertical truck model with nonlinear suspension and its multibody system formulation are presented. The equations of motion of the model form a system of differential-algebraic equations (DAEs). All equations are given explicitly, including a complete set of parameter values, consistent initial values, and a sample road excitation. Thus the truck model allows various investigations of the specific DAE effects and represents a test problem for algorithms in control theory, mechanics of multibody systems, and numerical analysis. Several numerical tests show the properties of the model.

A computer-oriented method for reducing linearized multibody system equations by incorporating constraints

Consider a spatial multibody system with rigid and elastic bodies. The bodies are linked by rigid interconnections (e.g. revolute joints) causing constraints, as well as by flexible interconnections (e.g. springs) causing applied forces. Small motions of the system with respect to a given nominal configuration can be described by linearized dynamic equations and kinematic constraint equations. We present a computer-oriented procedure which allows to develop a minimum number of these equations. There are three problems. First: algorithmic selection of position coordinates; second: condensation of the dynamic equations; third: evaluation of the constraint forces. To demonstrate the procedure, a closed loop multibody system is used as an example.

A new class of generalized inverses for the solution of discretized Euler - Lagrange equations

For a wide class of mechanical systems, so-called multibody systems, there are highly developed methods for the automatic generation of the corresponding equations of motion. As these equations can be stated in various, equivalent forms the question which of these forms are best suited for numerical treatment has become an important topic of research in computational mechanics.

Assimulo: A unified framework for ODE solvers

During the last three decades, a vast variety of methods to numerically solve ordinary differential equations and differential-algebraic equations has been developed and investigated. The methods are mostly freely available in different programming languages and with different interfaces. Accessing them using a unified interface is a need not only of the research community and for education purposes but also to make them available in industrial contexts.An industrial model of a dynamic system is usually not just a set of differential equations. The models today may contain discrete controllers, impacts or friction resulting in discontinuities that need to be handled by a modern solver in a correct and efficient way. Additionally, the models may produce an enormous amount of data that puts strain on the simulation software.In this paper, Assimulo is presented. It is a unified high-level interface to solvers of ordinary differential equations and is designed to satisfy the needs in research and education together with the requirements for solving industrial models with discontinuities and data handling. It combines original classical and modern solvers independent of their programming language with a well-structured Python/Cython interface. This allows to easily control parameter setting and discontinuity handling for a wide range of problem classes.

A collocation formulation of multistep methods for variable step-size extensions

Multistep methods are classically constructed by specially designed difference operators on an equidistant time grid. To make them practically useful, they have to be implemented by varying the step-size according to some error-control algorithm. It is well known how to extend Adams and BDF formulas to a variable step-size formulation. In this paper we present a collocation approach to construct variable step-size formulas. We make use of piecewise polynomials to show that every k-step method of order k + 1 has a variable step-size polynomial collocation formulation.

Stabilized multistep methods for index 2 Euler-Lagrange DAEs

We consider multistep discretizations, stabilized by β-blocking, for Euler-Lagrange DAEs of index 2. Thus we may use “nonstiff” multistep methods with an appropriate stabilizing difference correction applied to the Lagrangian multiplier term. We show that orderp =k + 1 can be achieved for the differential variables with orderp =k for the Lagrangian multiplier fork-step difference corrected BDF methods as well as for low orderk-step Adams-Moulton methods. This approach is related to the recently proposed “half-explicit” Runge-Kutta methods.

The befenits of parallel multibody simulation and its application to vehicle dynamics

In summer 1987 most of the multibody dynamics community met at the JPL, Pasadena, to discuss the needs and the open problems in multibody system simulation, especially for space applications. P. W. Likins stated in his survey [16]: “Computational questions focused initially on the selection of subroutines for numerical integration, matrix inversion, or eigensystem analysis,and lately have shifted to preprocessors and postprocessors for user convenience. More fundamental issues are raised by the potential of symbolic manipulation and parallel processing, both of which present the possibility of revolutionizing the field.” Concepts for symbolic implementation have been pursued at various places, e.g. [14, 21]. This paper presents results of our efforts to exploit the potential of parallel computer architectures for multibody simulation. It has its roots in an analysis of the status of knowledge at the time, the above statement was made.

MEDYNA — An Interactive Analysis and Design Program for Geometrically Linear and Flexible Multibody Systems

Simulation and computer-aided analysis of complex mechanical systems has become a task of increasing importance in computational mechanics. It requires software tools combining modeling support, efficient generation of the equations of motion as well as modern numerical solution and system analysis techniques.

Collocation Methods for the Investigation of Periodic Motions of Constrained Multibody Systems

The investigation of periodic motions of constrained multibodysystems requires the numerical solution of differential-algebraicboundary value problems. After briefly surveying the basics of periodicmotion analysis the paper presents an extension of projected collocationmethods [6] to a special class of boundary value problems for multibodysystem equations with position and velocity constraints. These methodscan be applied for computing stable as well as unstable periodicmotions. Furthermore they provide stability information, which can beused to detect bifurcations on periodic branches. The special class ofequations stemming from contact problems like in railroad systems [22]can be handled as well. Numerical experiments with a wheelset modeldemonstrate the performance of the algorithms.