Fabien B. Analytical System Dynamics: Modeling and Simulation

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Analytical System Dynamics

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Brian C. Fabien

Analytical System Dynamics

Modeling and Simulation

ABC

Brian C. Fabien

University of Washington

Mechanical Engineering Department

Seattle, WA 98195

ISBN: 978-0-387-85604-9 e-ISBN: 978-0-387-85605-6

DOI: 10.1007/978-0-387-85605-6

Library of Congress Control Number: 2008935329

 Springer Science+Business Media, LLC 2009

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To Mary, Maurice and Zo¨e

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Preface

This book combines results from Analytical Mechanics and System Dynam-

ics to develop an approach to modeling constrained multidiscipline dynamic

systems. This combination yields a modeling technique based on the energy

method of Lagrange, which in turn, results in a set of diﬀerential-algebraic

equations that are suitable for numerical integration. Using the modeling ap-

proach presented in this book are able to model, and simulate, systems as

diverse as a six link closed-loop mechanism or a transistor power ampliﬁer.

The material in this text is used to teach dynamic systems modeling and

simulation to seniors and ﬁrst-year graduate students in engineering. In a

ten week course (4 hours per week) we cover most of the material in this

book. The basic prerequisites for the class are undergraduate level classes in;

(i) physics (speciﬁcally, Newtonian mechanics and basic circuit analysis); (ii)

ordinary diﬀerential equations; and (iii) linear algebra.

A brief summary of the chapters that make up the book are as follows.

Chapter 1 This chapter introduces Paynter’s uniﬁed system variables; eﬀort,

ﬂow, displacement and momentum. Here, we show how these fundamental

system variables are used to relate power and energy, within, and between, the

various engineering disciplines, i.e., mechanical, electrical, ﬂuid and thermal

systems.

Chapter 2 The modeling technique developed in this book requires that we

determine analytical expressions for the kinetic coenergy, potential energy

and the dissipation function. In the case of mechanical systems this requires

that we determine expressions the position and velocity of various points on

the system. In the case of electrical, ﬂuid and thermal networks this requires

that we determine the relationship between the ﬂow variables of the system.

This chapter develops an approach to the kinematic analysis of both planar

and spatial mechanical systems. In addition, we consider the structural prop-

erties of network systems that give rise to constraint relationships between

the ﬂow variables.

Chapter 3 Lagrange’s equation of motion is derived in this chapter by using

a diﬀerential/variational form of the ﬁrst law of thermodynamics. Here, via

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viii Preface

numerous examples, we also present a systematic approach to deriving the

equations of motion for multidiscipline dynamic systems.

Chapter 4 In this chapter Lagrange’s equation motion is extended to ac-

commodate systems with displacement, ﬂow, eﬀort and dynamic constraints.

This modeling approach leads to the Lagrangian diﬀerential-algebraic equa-

tions (LDAEs) of motion.

Chapter 5 This chapter presents numerical techniques for the solution of

diﬀerential equations, and diﬀerential-algebraic equations. Here, we give par-

ticular emphasis to the explicit and implicit Runge-Kutta methods. Also, we

consider some of the subtleties involved is solving diﬀerential-algebraic equa-

tions. For example, we address the problem of computing consistent initial

conditions, as well as the problem reducing the diﬀerentiation index of the

system in order to obtain accurate solutions.

Chapter 6 The concepts of equilibrium, and stability, in the sense of Lya-

punov, are introduced in this chapter. These ideas are used to analyze the

behavior of some simple dynamic systems. This chapter also presents sim-

ulation results for various the models developed in the book. Here, we use

two programs to demonstrate the eﬃcacy of the modeling technique devel-

oped here. The ﬁrst program, ldaetrans, translates an input ﬁle describing

the model into the LDAEs. This is accomplished via symbolic diﬀerentiation

of the system energy terms and constraints. The program ldaetrans also

generates MATLAB/Octave ﬁles that can be used to integrate the equations

of motion. A second program presented in this chapter is ride. This is a

MATLAB/Octave implementation of an implicit Runge-Kutta method that

is used to integrate the Lagrangian diﬀerential-algebraic equations. In ad-

dition, some of the example problems in this chapter are used to introduce

concepts from linear and nonlinear feedback control.

Acknowledgment A number of my graduate students have contributed to

the development of the material presented in this book. They include; Richard

A. Layton, Jonathan Alberts and Jae Suk. In addition, the research that has

culminated in this book was supported by the National Science Foundation

under Grant MSS-9350467. This support is gratefully acknowledged.

Brian C. Fabien

Seattle, Washington

August, 2008

Contents

1 A Uniﬁed System Representation . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Fundamental System Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Kinematic variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Kinetic variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Work, power and energy . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 System Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Ideal inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Ideal capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.3 Ideal resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.4 Constraint elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.5 Source elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.2.6 Paynter’s diagram and system models . . . . . . . . . . . . . . . 31

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1 Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1.1 A point moving in a ﬁxed frame . . . . . . . . . . . . . . . . . . . . 37

2.1.2 A point moving in a translating frame . . . . . . . . . . . . . . 41

2.1.3 A point ﬁxed in a rotating frame . . . . . . . . . . . . . . . . . . . 49

2.1.4 A point moving in a moving frame . . . . . . . . . . . . . . . . . . 54

2.1.5 Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.2 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.2.1 Mobility analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.2.2 Kinematic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.3 Network Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98