MechanicalSystems

Kinematic and Dynamic Analysis
in Mathematica

Version 2.0

Robert Beretta
Dynamic Modeling

Copyright

September 2005
Third edition
Intended for use with Mathematica 5 or later

Software and manual written by Robert Beretta

Partnerships manager: Joy Costa
Developer support specialist: Nirmal Malapaka
Project manager: Jennifer Peterson
Software quality assurance: Rachelle Bergmann, Jay Hawkins, Cindie Strater
Document quality assurance: Jan Progen, Marcia Krause, Angela Latham, Wendy Leung, Carol Ordal
Document production: Richard Miske
Graphics design: Megan Gillette, Jody Jasinski, Kimberly Michael
Marketing product manager: Yezabel Dooley

Special thanks to Karen Beretta, Joy Costa, Monica Davis, Ed Garner, Joe Grohens, Jay Hawkins, Gary Miller, Leszek Sczaniecki, Kristina Zack

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Contents

Preface

1. Getting Started

1.1    Starting MechanicalSystems
1.2    Basic 2D Model
1.3    Basic 3D Model

2. Kinematic Modeling

2.1    Body Objects
2.2    Points, Vectors, and Axes
2.3    Constraint Objects
2.4    Building the Model
2.5    Running the Model
2.6    Variable Management

3. Output Functions

3.1    2D Output Functions
3.2    3D Output Functions
3.3    Plots and Graphs
3.4    2D Mechanism Images
3.5    3D Mechanism Images

4. Velocity and Acceleration

4.1    Velocity and Acceleration Solutions
4.2    Velocity and Acceleration Output

5. Multistage Mechanisms

5.1    Stage Switching Constraints
5.2    Time Switching Constraints

6. Cams and Gears

6.1    Cam Constraints
6.2    Gear Constraints

7. Loads and Reactions

7.1    Load Objects
7.2    Static Solutions
7.3    Resultant Forces
7.4    Multistage Loads

8. Inverse Dynamics

8.1    Inertia Properties
8.2    Kinematic and Dynamic Solutions
8.3    Friction and Damping

9. Parametric Design

9.1    Coupled Models
9.2    Multiple Configuration Couples
9.3    Velocity and Acceleration Couples
9.4    Load-Dependent Couples
9.5    Sensitivity

10. Underconstrained Systems

10.1    Equilibrium Configuration
10.2    Equilibrium Velocity
10.3    Free Acceleration
10.4    Dynamic Motion Synthesis
10.5    Nonholonomic Systems

11. Generalized Coordinates

11.1    Generalized Coordinate Constraints
11.2    Loads and Reactions
11.3    Dynamic Modeling

12. Performance Improvement

12.1    Updating Solutions
12.2    Simplifying Expressions
12.3    Compiling Expressions
12.4    Coordinate Systems
12.5    Locking Variables

13. Debugging Models

13.1    Model Building Errors
13.2    Failure to Converge
13.3    Mathematical Anomalies
13.4    Equations of Motion

Appendix. Reference Guide

A.    Modified Mathematica Built-in Symbols
B.    MechanicalSystems Function Listing
Index

Preface

MechanicalSystems is a set of Mathematica packages designed to assist in the analysis and design of planar and spatial rigid body mechanisms. The package is written entirely in the Mathematica programming language and is completely portable and platform independent. The source code is written in the preferred package format as generally used by Wolfram Research, Inc. for the standard packages that are included with Mathematica. MechanicalSystems is intended to run under Mathematica 5.0 or later.