Table of Contents
 

1 Multibody Systems
1.1 What is a Multibody System?
1.2 Basic Mathematical Tasks in Multibody Dynamics
1.3 Equations of Motion of Multibody Systems
1.3.1 Unconstrained Planar Multibody Systems
1.3.2 Constrained Planar Multibody Systems
1.3.3 Nonholonomic Constraints
1.3.4 System with Dynamical Force Elements
1.3.5 General Systems
1.4 Relative and Absolute Coordinates
1.4.1 Mixed Coordinate Formulation
1.5 Linearization of Equations of Motion
1.6 Nominal Interaction Forces
2 Linear Systems
2.1 State Space Form of Linear Constrained Systems
2.2 Numerical Reduction to State Space Form
2.3 Constrained Least Squares Problems
2.3.1 Pseudo-Inverses
2.3.2 Numerical Computation
2.3.3 Underdetermined Linear Systems
2.4 The Transition Matrix
2.5 The Frequency Response
2.6 Linear Constant Coefficient DAEs
2.6.1 The Matrix Pencil
2.6.2 The Matrix Pencil and the Solution of the DAE
2.6.3 Construction of the Drazin Inverse
2.7 DAEs and the Generalized Eigenvalue Problem
2.7.1 State Space Form and the Drazin ODE
3 Nonlinear Equations
3.1 Static Equilibrium Position
3.2 Solvability of Nonlinear Equations
3.3 Fixed Point Iteration
3.4 Newton's Method
3.5 Numerical Computation of Jacobians
3.6 Reevaluation of the Jacobian
3.7 Limitations and Extensions
3.8 Continuation Methods in Equilibrium Computation
3.8.1 Globalizing the convergence of Newton's Method
3.8.2 Theoretical Background
3.8.3 Basic Concepts of Continuation Methods
4 Explicit Ordinary Differential Equations
4.1 Linear Multistep Methods
4.1.1 Adams Methods
4.1.2 Backward Differentiation Formulas (BDF)
4.1.3 General Form of Multistep Methods
4.1.4 Accuracy of a Multistep Method
4.1.5 Order and Step Size Selection
4.1.6 Solving the Corrector Equations
4.2 Explicit Runge-Kutta Methods
4.2.1 The Order of a Runge-Kutta Method
4.2.2 Embedded Methods for Error Estimation
4.2.3 Stability of Runge-Kutta Methods
4.3 Implicit Runge-Kutta Methods
4.3.1 Collocation Methods
4.3.2 Corrector Equations in Implicit Runge-Kutta Methods
4.3.3 Accuracy of Implicit Runge-Kutta Methods
4.3.4 Stability of Implicit Runge-Kutta Methods
4.4 Stiff Systems
4.5 Dense Output
5 Implicit Ordinary Differential Equations
5.1 Implicit ODEs
5.1.1 Types of Implicit ODEs
5.1.2 Existence and Uniqueness of Solutions
5.1.3 Sensitivity under Perturbations
5.1.4 ODEs with Invariants
5.2 Linear Multistep Methods for DAEs
5.2.1 Euler's Method and Adams-Moulton for DAEs
5.2.2 Multistep Discretization Schemes for DAEs
5.2.3 Convergence of Multistep Methods
5.2.4 Solving the Corrector Equations in Discretized DAEs
5.3 Stabilization and Projection Methods
5.3.1 Coordinate Projection Method
5.3.2 Implicit State Space Form
5.3.3 Projection Methods for ODEs with Invariants
5.4 One Step Methods
5.4.1 Runge-Kutta Methods
5.4.2 Half-Explicit Methods
5.5 Contact Problems as DAEs
5.5.1 Single Point Contact of Planar Bodies
5.5.2 Problems
6 ODEs with Discontinuities
6.1 Difficulties with Discontinuities
6.2 Differential Equations with Switching Conditions
6.3 Switching Algorithm
6.3.1 Test for Occurrence of a Discontinuity
6.3.2 Localization of Switching Points
6.3.3 Change to the New Right Hand Side and Restarting
6.3.4 Adaptation of the Discretization
6.3.5 A Model Problem for Setting up a Switching Logic
6.3.6 Aspects of Realization
6.3.7 Other Methods for Discontinuous ODEs
6.3.8 DAEs with Discontinuities
6.4 Coulomb Friction: Difficulties with Switching
6.4.1 Introductory Example
6.4.2 Coulomb Friction: Mathematical Background
6.5 Other Special Classes of Discontinuities
6.5.1 Time Events
6.5.2 Structure Varying Contact Problems with Friction
6.5.3 Computer-controlled Systems
6.5.4 Hysteresis
6.5.5 Approximations by Piecewise Smooth Functions
6.5.6 Implementation of Switching Algorithms
7 Parameter Identification Problems
7.1 Problem Formulation
7.2 Numerical Solution
7.2.1 Elimination of the ODE: Integration
7.2.2 Gauss Newton Methods
7.2.3 Evaluation of Functions and Jacobians
7.2.4 Summary of the Algorithm
7.2.5 The Boundary Value Approach
7.3 Extensions
7.3.1 Differential Algebraic Equations
7.3.2 Discontinuous Systems
A The Truck Model
A.1 Data of the Truck Model
A.2 Model of a nonlinear pneumatic spring
A.3 Matlab m-files for the Unconstrained Truck