Computational Flexible Multibody Dynamics

A Differential-Algebraic Approach

verfasst von: Bernd Simeon

Verlag: Springer Berlin Heidelberg

Buchreihe : Differential-Algebraic Equations Forum

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This monograph, written from a numerical analysis perspective, aims to provide a comprehensive treatment of both the mathematical framework and the numerical methods for flexible multibody dynamics. Not only is this field permanently and rapidly growing, with various applications in aerospace engineering, biomechanics, robotics, and vehicle analysis, its foundations can also be built on reasonably established mathematical models. Regarding actual computations, great strides have been made over the last two decades, as sophisticated software packages are now capable of simulating highly complex structures with rigid and deformable components. The approach used in this book should benefit graduate students and scientists working in computational mechanics and related disciplines as well as those interested in time-dependent partial differential equations and heterogeneous problems with multiple time scales. Additionally, a number of open issues at the frontiers of research are addressed by taking a differential-algebraic approach and extending it to the notion of transient saddle point problems.

Inhaltsverzeichnis

Frontmatter

Mathematical Models

Frontmatter

Chapter 1. A Point of Departure

Abstract

What is a flexible multibody system? How can we derive an adequate mathematical model? And what are the major computational challenges that we are facing here? This introductory chapter gives some preliminary answers and, at the same time, illustrates the objectives pursued by this monograph.

Bernd Simeon

Chapter 2. Rigid Multibody Dynamics

Abstract

In this chapter, we give an overview on the mathematical models for the dynamics of systems of rigid bodies. Depending on the choice of coordinates for the position and orientation of each body, the governing equations form either a system of ordinary differential equations or, if constraints are present, a system of differential-algebraic equations. We analyze the structure of these equations, discuss practical aspects, and present several examples. Since differential-algebraic equations are a recurrent theme in this book, we furthermore summarize their most important theoretical properties, which includes the index concept and alternative formulations for the equations of constrained mechanical motion.

Bernd Simeon

Chapter 3. Elastic Motion

Abstract

In this chapter, we turn our attention to the more general class of flexible multibody systems that include rigid as well as elastic bodies. While an elastic body is described by a partial differential equation, the rigid body motion, on the other hand, satisfies an ordinary differential equation or, in the presence of Euler parameters or joints, a differential-algebraic equation. Thus, the overall mathematical model consists of a coupled system with a subtle structure. We apply a step-by-step procedure in order to derive the underlying models. This chapter is mainly devoted to the motion of a single elastic body under the assumption of linear elasticity. Large rotations and translations will be treated by the method of floating reference frames in the subsequent Chap. 4. Hamilton’s principle of stationary action is the starting point to generate the equations of motion. Constraints are appended by means of Lagrange multipliers and lead to the notion of a time-dependent saddle point formulation that can be viewed as the continuous analogue of the constrained system for rigid bodies.

Bernd Simeon

Chapter 4. Flexible Multibody Dynamics

Abstract

Flexible multibody systems contain both rigid and elastic components and aim at applications such as lightweight and high-precision mechanical systems where the elasticity of certain bodies needs to be taken into account. Since elastic bodies are governed by PDEs, as described in the previous chapter, and rigid bodies by ODEs or DAEs, the mathematical model of a flexible multibody system is heterogeneous by nature. This chapter introduces the method of floating reference frames as standard approach for flexible multibody dynamics. The treatment of constraints is extended to this problem class, and issues related to the modeling of joints are discussed. Moreover, we cover special bodies such as beams and also touch briefly upon additional nonlinearities, the so-called geometric stiffening terms. Several examples conclude this first part of the book.

Bernd Simeon

Numerical Methods

Frontmatter

Chapter 5. Spatial Discretization

Abstract

The model equations for elastic bodies developed in Chaps. 3 and 4 are now discretized with respect to the spatial variable. In case of the unconstrained equations of motion, this Galerkin projection is a widespread approach and usually applied with finite elements or eigenfunctions as spatial approximations. It leads in a natural way to a system of second order ODEs, which is then coupled with other components in the assembly to express the corresponding interactions. As argued in Part I, the most general model equations include the constraints already before discretization, which raises the question how to discretize the transient saddle point models from above. In this chapter, we generalize the Galerkin projection to such models and investigate the properties of the resulting differential-algebraic system. For this purpose, we employ some of the results that have become standard for the classes of mixed and hybrid finite element methods.

Bernd Simeon

Chapter 6. Stiff Mechanical System

Abstract

In the previous Chap. 5 we have seen how the spatial discretization of a flexible multibody system leads to a differential-algebraic equation in time. The partitioning into two types of state variables, namely, those for the gross motion, on the one hand, and those for the elastic deformations, on the other, quite often involves widely different time scales. This chapter is devoted to such stiff mechanical systems. In numerical analysis, the adjective “stiff” typically characterizes an ordinary differential equation whose eigenvalues have strongly negative real parts. However, numerical stiffness may also arise in case of second order differential equations with large eigenvalues on or close to the imaginary axis. If such high frequencies are viewed as a parasitic effect which perturbs a slowly varying smooth solution, implicit time integration methods with adequate numerical dissipation are an option and usually superior to explicit methods. For a mechanical system, this form of numerical stiffness is directly associated with large stiffness forces, and thus the notion of a stiff mechanical system has a twofold meaning.

Bernd Simeon

Chapter 7. Time Integration Methods

Abstract

In flexible multibody dynamics, the time integration of the semi-discretized equations of motion represents a challenging problem due to the simultaneous presence of constraints and different time scales. This combination leads, as analyzed in the foregoing chapter, to a stiff differential-algebraic system. We investigate here the behavior of numerical methods for such problems, with particular focus on the well-established implicit integrators that are either based on the BDF (Backward Differentiation Formulas) methods or on implicit Runge–Kutta methods of collocation type. The chapter starts, however, with an overview on time integration methods for constrained mechanical systems. Using a model equation with smooth and highly oscillatory solution parts, we then show that stiff methods suffer from order reductions which are directly related to the limiting DAE of index 3 to which the stiff mechanical system converges. At the end of this chapter, we also introduce extensions of the generalized-α method and of the implicit midpoint rule to the differential-algebraic case and investigate their potential for mechanical multibody systems.

Bernd Simeon

Chapter 8. Numerical Case Studies

Abstract

Flexible multibody dynamics has become a key methodology for various engineering fields, and state-of-the-art simulation software offers corresponding modules with powerful libraries for all kinds of applications. Since it would be beyond the scope of this monograph to cover such specific topics to a large extent, we focus instead in this final chapter on case studies that illustrate some of the numerical issues discussed before. As examples for challenging engineering tasks that can be modeled by means of a flexible multibody system, we mention, among many others, the rotorcraft dynamics in helicopter design, the landing of an aircraft, the treatment of dynamic contact and large deformation problems in a multibody framework, and the analysis of ride comfort and handling capabilities in vehicle dynamics.

Bernd Simeon

Backmatter

Titel: Computational Flexible Multibody Dynamics
verfasst von: Bernd Simeon
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-35158-7
Print ISBN: 978-3-642-35157-0
DOI: https://doi.org/10.1007/978-3-642-35158-7