This book is printed on acid-free paper.

Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at www.wiley.com/go/permissions.

Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.

Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

Karnopp, Dean.

System dynamics : modeling, simulation, and control of mechatronic systems / Dean C. Karnopp, Donald L. Margolis, Ronald C. Rosenberg. – 5th ed.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-88908-4 (cloth); ISBN 978-1-118-15281-2 (ebk); ISBN 978-1-118-15282-9 (ebk); ISBN 978-1-118-15283-6 (ebk); ISBN 978-1-118-15982-8 (ebk); ISBN 978-1-118-16007-7 (ebk); ISBN 978-1-118-16008-4 (ebk)

1. Systems engineering. 2. System analysis. 3. Bond graphs. 4. Mechatronics. I. Margolis, Donald L. II. Rosenberg, Ronald C. III. Title.

TA168.K362 2012

620.001′1–dc23

2011026141

Preface

This is the fifth edition of a textbook originally titled system Dynamics: A Unified Approach, which in subsequent editions acquired the title System Dynamics: Modeling and Simulation of Mechatronic Systems. As you can see, the subtitle has now expanded to be Modeling, Simulation, and Control of Mechatronic Systems.

The addition of the term control indicates the major change from previous editions. In older editions, the first six chapters of the book typically have been used as an undergraduate text and the last seven chapters have been used for more advanced courses. Now the latter part of Chapter Six can be used to introduce undergraduate students to a major use of mathematic models; namely, as a basis for the design control systems. In this case we are not trying to replace the many excellent books dealing with the design of automatic control systems. Rather we are trying to provide a contrasting approach to such books that often have a single chapter devoted to the construction of mathematical models from physical principles, while the rest of the book is devoted to discussing the dynamics of control systems given a model of the control system in the form of state equations, transfer functions, or frequency response functions. It is our contention that while the design of control systems is very important, the skills of modeling and computer simulation for a wide variety of physical systems are of fundamental importance even if an automatic control system is not involved.

Furthermore, we contend that the bond graph method is uniquely suited to the understanding of physical system dynamics. The basis of bond graphs lies in the study of energy storage and power flow in physical systems of almost any type. In this edition, we have tried to simplify the earlier chapters to focus on mechanical, electrical, and hydraulic systems that are relatively easy to model using bond graphs, leaving the more complex types of systems to the later chapters. It would be easy for an instructor to choose some topics of particular interest from these chapters to supplement the types of systems studied in the earlier part of the book if desired.

It has been gratifying to see that over the years, instructors and researchers world wide have learned that the bond graph technique is uniquely suited to the description of physical systems of engineering importance. It is our hope that this book will continue to provide useful information for engineers dealing with the analysis, simulation, and control of the devices of the future.

Chapter 1

Introduction

This book is concerned with the development of an understanding of the dynamic physical systems that engineers are called upon to design. The type of systems to be studied can be described by the term mechatronic, which implies that while the elements of the system are mechanical in a general sense, electronic control will also be involved. For the design of a computer-controlled system, it is crucial that the dynamics of systems that exchange power and energy in various forms be thoroughly understood. Methods for the mathematical modeling of real systems will be presented, ways of analyzing systems in order to shed light on system behavior will be shown, and techniques for using computers to simulate the dynamic response of systems to external stimuli will be developed. In addition, methods of using mathematical models of dynamic systems to design automatic control systems will be introduced. Before beginning the study of physical systems, it is worthwhile to reflect a moment on the nature of the discipline that is usually called system dynamics in engineering.

The word system is used so often and so loosely to describe a variety of concepts that it is hard to give a meaningful definition of the word or even to see the basic concept that unites its diverse meanings. When the word system is used in this book, two basic assumptions are being made:

1. A system is assumed to be an entity separable from the rest of the universe (the environment of the system) by means of a physical or conceptual boundary. An air conditioning system, for example, can be thought of as a system that reacts to its environment (the temperature of the outside air, for example) and that interchanges energy and information with its environment. In this case the boundary is physical or spatial. An air traffic control system, however, is a complex system, the environment of which is not only the physical surroundings but also the fluctuating demands for air traffic, which ultimately come from human decisions about travel and the shipping of goods. The unifying element in these two disparate systems is the conceptual boundary between what is considered to be part of the system and what represents an external disturbance or command originating from outside the system.

2. A system is composed of interacting parts. In an air conditioning system, we recognize devices with specific functions, such as compressors and fans, sensors that transmit information, and actuators that act on information, and so on. The air traffic control system is composed of people and machines with communication links between them. Clearly, the reticulation of a system into its component parts is something that requires skill and art, since most systems could be broken up into so many parts that any analysis would be swamped with largely irrelevant detail. The art and science of system modeling has to do with the construction of a model complex enough to represent the relevant aspects of the real system but not so complex as to be unwieldy.

These two aspects of systems can be recognized in everyday situations as well as in the more specific and technical applications that form the subject matter of most of this book. For example, when one hears a complaint that the transportation system in this country does not work well, one may see that there is some logic in using the word system. First of all, the transportation system is roughly identifiable as an entity. It consists of air, land, and sea vehicles and the human beings, machines, and decision rules by which they are operated. In addition, many parts of the system can be identified—cars, planes, ships, baggage handling equipment, computers, and the like. Each part of the transportation system could be further reticulated into parts (i.e., each component part is itself a system), but for obvious reasons we must exercise restraint in this division process.

The essence of what may be called the “systems viewpoint” is to concern oneself with the operation of a complete system rather than with just the operation of the component parts. Complaints about the transportation system are often real “system” complaints. It is possible to start a trip in a private car that functions just as its designers had hoped it would, transfer to an airplane that can fly at its design speed with no failures, and end in a taxi that does what a taxi is supposed to do, and yet have a terrible trip because of traffic jams, air traffic delays, and the like. Perfectly good components can be assembled into an unsatisfactory system.

In engineering, as indeed in virtually all other types of human endeavor, tasks associated with the design or operation of a system are broken up into parts that can be worked on in isolation to some extent. In a power plant, for example, the generator, turbine, boiler, and feed water pumps typically will be designed by separate groups. Furthermore, heat transfer, stress analysis, fluid dynamics, and electrical studies will be undertaken by subsets of these groups. In the same way, the bureaucracy of the federal government represents a splitting up of the various functions of government. All the separate groups working on an overall task must interact in some manner to make sure that not only will the parts of the system work, but also the system as a whole will perform its intended function. Many times, however, oversimplified assumptions about how a system will operate are made by those working on a small part of the system. When this happens, the results can be disappointing. The power plant may undergo damage during a full load rejection, or the economy of a country may collapse because of the unfavorable interaction of segments of government, each of which assiduously pursues seemingly reasonable policies.

In this book, the main emphasis will be on studying system aspects of behavior as distinct from component aspects. This requires knowledge of the component parts of the systems of interest and hence some knowledge in certain areas of engineering that are taught and sometimes even practiced in splendid isolation from other areas. In the engineering systems of primary interest in this book, topics from vibrations, strength of materials, dynamics, fluid mechanics, thermodynamics, automatic control, and electrical circuits will be used. It is possible, and perhaps even common, for an engineer to spend a major part of his or her professional career in just one of these disciplines, despite the fact that few significant engineering projects concern a single discipline. Systems engineers, however, must have a reasonable command of several of the engineering sciences as well as knowledge pertinent to the study of systems per se.

Although many systems may be successfully designed by careful attention to static or steady-state operation in which the system variables are assumed to remain constant in time, in this book the main concern will be with dynamic systems, that is, those systems whose behavior as a function of time is important. For a transport aircraft that will spend most of its flight time at a nearly steady speed, the fuel economy at constant speed is important. For the same plane, the stress in the wing spars during steady flight is probably less important than the time-varying stress during flight through turbulent air, during emergency maneuvers, or during hard landings. In studying the fuel economy of the aircraft, a static system analysis might suffice. For stress prediction, a dynamic system analysis would be required.

Generally, of course, no system can operate in a truly static or steady state, and both slow evolutionary changes in the system and shorter time transient effects associated, for example, with startup and shutdown are important. In this book, despite the importance of steady-state analysis in design studies, the emphasis will be on dynamic systems. Dynamic system analysis is more complex than static analysis but is extremely important, since decisions based on static analyses can be misleading. Systems may never actually achieve a possible steady state due to external disturbances or instabilities that appear when the system dynamics are taken into account.

Moreover, systems of all kinds can exhibit counterintuitive behavior when considered statically. A change in a system or a control policy may appear beneficial in the short run from static considerations but may have long-run repercussions opposite to the initial effect. The history of social systems abounds with sometimes tragic examples, and there is hope that dynamic system analysis can help avoid some of the errors in “static thinking” [1]. Even in engineering with rather simple systems, one must have some understanding of the dynamic response of a system before one can reasonably study the system on a static basis.

A simple example of a counterintuitive system in engineering is the case of a hydraulic power generating plant. To reduce power, wicket gates just before the turbine are moved toward the closed position. Temporarily, however, the power actually increases as the inertia of the water in the penstock forces the flow through the gates to remain almost constant, resulting in a higher velocity of flow through the smaller gate area. Ultimately, the water in the penstock slows down and power is reduced. Without an understanding of the dynamics of this system, one would be led to open the gates to reduce power. If this were done, the immediate result would be a gratifying decrease of power followed by a surprising and inevitable increase. Clearly, a good understanding of dynamic response is crucial to the design of a controller for mechatronic systems.

1.1 Models of Systems

A central idea involved in the study of the dynamics of real systems is the idea of a model of a system. Models of systems are simplified, abstracted constructs used to predict their behavior. Scaled physical models are well known in engineering. In this category fall the wind tunnel models of aircraft, ship hull models used in towing tanks, structural models used in civil engineering, plastic models of metal parts used in photoelastic stress analysis, and the “breadboard” models used in the design of electric circuits.

The characteristic feature of these models is that some, but not all, of the features of the real system are reflected in the model. In a wind tunnel aircraft model, for example, no attempt is made to reproduce the color or interior seating arrangement of the real aircraft. Aeronautical engineers assume that some aspects of a real craft are unimportant in determining the aerodynamic forces on it, and thus the model contains only those aspects of the real system that are supposed to be important to the characteristics under study.

In this book, another type of model, often called a mathematical model, is considered. Although this type of model may seem much more abstract than the physical model, there are strong similarities between physical and mathematical models. The mathematical model also is used to predict only certain aspects of the system response to inputs. For example, a mathematical model might be used to predict how a proposed aircraft would respond to pilot input command signals during test maneuvers. But such a model would not have the capability of predicting every aspect of the real aircraft response. The model might not contain any information on changes in aerodynamic heating during maneuvers or about high-frequency vibrations of the aircraft structure, for example.

Because a model must be a simplification of reality, there is a great deal of art in the construction of models. An unduly complex and detailed model may contain parameters virtually impossible to estimate, may be practically impossible to analyze, and may cloud important results in a welter of irrelevant detail if it can be analyzed. An oversimplified model will not be capable of exhibiting important effects. It is important, then, to realize that no system can be modeled exactly and that any competent system designer needs to have a procedure for constructing a variety of system models of varying complexity so as to find the simplest model capable of answering the questions about the system under study.

The rest of this book deals with models of systems and with the procedures for constructing models and for extracting system characteristics from models. The models will be mathematical models in the usual meaning of the term even though they may be represented by stylized graphs and computer printouts rather than the more conventional sets of differential equations.

System models will be constructed using a uniform notation for all types of physical systems. It is a remarkable fact that models based on apparently diverse branches of engineering science can all be expressed using the notation of bond graphs based on energy and information flow. This allows one to study the structure of the system model. The nature of the parts of the model and the manner in which the parts interact can be made evident in a graphical format. In this way, analogies between various types of systems are made evident, and experience in one field can be extended to other fields.

Using the language of bond graphs, one may construct models of electrical, magnetic, mechanical, hydraulic, pneumatic, thermal, and other systems using only a rather small set of ideal elements. Standard techniques allow the models to be translated into differential equations or computer simulation schemes. Historically, diagrams for representing dynamic system models developed separately for each type of system. For example, parts a, b, and c of Figure 1.1 each represent a diagram of a typical model. Note that in each case the elements in the diagram seem to have evolved from sketches of devices, but in fact a photograph of the real system would not resemble the diagram at all. Figure 1.1a might well represent the dynamics of the heave motion of an automobile, but the masses, springs, and dampers of the model are not directly related to the parts of an automobile visible in a photograph. Similarly, symbols for resistors and inductors in diagrams such as Figure 1.1b may not correspond to separate physical elements called resistors and chokes but instead may correspond to the resistance and inductance effects present in a single physical device. Thus, even semipictorial diagrams are often a good deal more abstract than they might at first appear.

Figure 1.1. (a) Typical schematic diagram; (b) typical electric circuit diagram; (c) typical hydraulic diagram; (d) schematic diagram of system containing mechanical, electrical, and hydraulic components.

When mixed systems such as that shown in Figure 1.1d are to be studied, the conventional means of displaying the system model are less well developed. Indeed, few such diagrams are very explicit about just what effects are to be included in the model. The basic structure of the model may not be evident from the diagram. A bond graph is more abstract than the type of diagrams shown in Figure 1.1, but it is explicit and has the great advantage that all the models shown in Figure 1.1 would be represented using exactly the same set of symbols. For mixed systems such as that shown in Figure 1.1d, a universal language such as bond graphs provide is required in order to display the essential structure of the system model.

1.2 Systems, Subsystems, and Components

To model a system, it is usually necessary first to break it up into smaller parts that can be modeled and perhaps studied experimentally and then to assemble the system model from the parts. Often, the breaking up of the system is conveniently accomplished in several stages. In this book major parts of a system will be called subsystems and primitive parts of subsystems will be called components or, at the most primitive level, elements. Of course, the hierarchy of components, subsystems, and systems can never be absolute, since even the most primitive part of a system could be modeled in such detail that it would be a complex subsystem. Yet in many engineering applications, the subsystem and component categories are fairly obvious.

Basically, a subsystem is a part of a system that will be modeled as a system itself; that is, the subsystem will be broken into interacting component parts. A component, however, is modeled as a unit and is not thought of as composed of simpler parts. One needs to know how the component interacts with other components and one must have a characterization of the component, but otherwise a component is treated as a “black box” without any need to know in detail what causes it to act as it does.

To illustrate these ideas, consider the vibration test system shown in Figure 1.2. The system is intended to subject a test structure to a vibration environment specified by a signal generator. For example, if the signal generator delivers a random-noise signal, it may be desired that the acceleration of the shaker table be a faithful reproduction of the electrical noise signal waveform. In a system that is assembled from physically separate pieces, it is natural to consider the parts that are assembled by connecting wires and hydraulic lines or by mechanical fasteners as subsystems. Certainly, the electronic boxes labeled signal generator, controller, and electrical amplifier are subsystems, as are the electrohydraulic valve, the hydraulic shaker, and the test structure. It may be possible to treat some of these subsystems as components if their interactions with the rest of the system can be specified without knowledge of the internal construction of the subsystem. The electrical amplifier is obviously composed of many components, such as resistors, capacitors, transistors, and the like, but if the amplifier is sized correctly so that it is not overloaded, then it may be possible to treat the amplifier as a component specified by the manufacturer's input–output data. Other subsystems may require a subsystem analysis in order to achieve a dynamic description suitable for the overall system study.

Figure 1.2. Vibration test system.

Consider, for example, the electrohydraulic valve. A typical servo valve is shown in Figure 1.3. Clearly, the valve is composed of a variety of electrical, mechanical, and hydraulic parts that work together to produce the dynamic response of the valve. For this subsystem the components might be the torque motor, the hydraulic amplifier, mechanical springs, hydraulic passages, and the spool valve. A subsystem dynamic analysis can reveal weaknesses in the subsystem design that may necessitate the substitution of another subsystem or a reconfiguration of the overall system. Yet such an analysis may indicate that, from the point of view of the overall system, the subsystem may be adequately characterized as a simple component. A skilled and experienced system designer often makes a judgment on the appropriate level of detail for the modeling of a subsystem on an intuitive basis. A major purpose of the methods presented in this book is to show how system models can be assembled conveniently from component models. It is then possible to experiment with subsystem models of varying degrees of sophistication in order to verify or disprove initial modeling decisions.

Figure 1.3. Electrohydraulic valve.

1.3 State-Determined Systems

The goal of this book is to describe means for setting up mathematical models for systems. The type of model that will be found is often described as a state-determined system. In mathematical notation, such a system model is often described by a set of ordinary differential equations in terms of so-called state variables and a set of algebraic equations that relate other system variables of interest to the state variables. In succeeding chapters an orderly procedure, beginning with physical effects to be modeled and ending with state differential equations, will be demonstrated. Even though some techniques of analysis and computer simulation do not require that the state equations be written explictly, from a mathematical point of view all the system models that will be discussed are state-determined systems.

The future of all the variables associated with a state-determined system can be predicted if [1] the state variables are known at some initial time and [2] the future time history of the input quantities from the external environment is known.

Such models, which are virtually the only types used in engineering, have some built-in philosophical implications. For example, events in the future do not affect the present state of the system. This implication is correlated with the assumption that time runs only in one direction—from past to future. That models should have these properties probably seems plausible, if not obvious, yet it is remarkably difficult to conceive of a demonstration that real systems always have these properties.

Clearly, past history can have an effect on a system; yet the influence of the past is exhibited in a special way in state-determined systems. All the past history of a state-determined system is summed up in the present values of its state variables. This means that many past histories could have resulted in the same present value of state variables and hence the same future behavior of the system. It also means that if one can condition the system to bring the state variables to some particular values, then the future system response is determined by the future inputs and nothing is important about the past except that the state variables were brought to those values.

Scientific experiments are run as if the systems to be studied were state determined. The system is always started from controlled conditions that are expressed in terms of carefully monitored variables. If the experiment is repeatable, then the assumption is that the state variables are properly initialized by the operations used to set up the experiment. If the experiment is not repeatable, then the assumption is that some important influence has not been controlled. This influence can be either a state variable that was not monitored and initialized properly or an unrecognized input quantity through which the environment influences the system.

State-determined system models have proved useful over centuries of scientific and technical work. For the usual macroscopic systems encountered in engineering, state-determined system models are nearly universal, and there is continuing interest in developing such models for social and economic systems. This book can be regarded as a textbook devoted to the establishment and study of state-determined system models using well-defined physical systems of interest to engineers as examples.

1.4 Uses of Dynamic Models

In Figure 1.4 a general dynamic system model is shown schematically. The system S is characterized by a set of state variables, indicated by X, that are influenced by a set of input variables U that represent the action of the system's environment on the system or variables that could be manipulated by a control system. The set of output variables Y are either back effects from the system onto the environment or variables that can be sensed and used by a control system. This type of dynamic system model may be used in four quite distinct ways:

1. Analysis. Given U for the future, X at the present, and the model S, predict the future of Y. Assuming that the system model is an accurate representation of the real system, analysis techniques allow one to predict system behavior. In some cases, this analysis will indicate the need of a control system in order for the system to respond in a desirable way.

2. Identification. Given time histories of U and Y, usually by experimentation on real systems, find a model S and state variables X that are consistent with U and Y. This is the essence of scientific experimentation. Clearly, a “good” model is one that is consistent with a great variety of sets U and Y.

3. Synthesis. Given U and some desired Y, find S such that U acting on S will produce Y. Most of engineering deals with synthesis, but only in limited contexts are there direct synthesis methods. Often we must be content to accomplish synthesis of systems via a trial-and-error process of repetitive analysis of a series of candidate systems. In this regard, dynamic models pay a vital role, since progress would be slow indeed if one had to construct each candidate system “in the metal” in order to discover its properties.

4. Control. Given the system model S, design a control system that uses those variables in the output Y available from sensors to produce the some of the inputs in U in order to make the system respond properly.

Figure 1.4. General dynamic system model.

In this book we will concentrate on setting up system models and predicting the behavior of the systems using analytical or computational techniques. Thus, we will concentrate on analysis, but it is important to remember that the techniques are useful for identification problems and that the major challenge to a systems engineer is to synthesize desirable systems. It may not be too much to say that analysis, except in the service of synthesis, is a rather sterile pursuit for an engineer. Also, a brief introduction to automatic control techniques will be given.

1.5 Linear and Nonlinear Systems

An overall system model, consisting of subsystems and their components, requires modeling decisions as to what dynamic effects must be included in order to use the model for its intended purpose. The result of these modeling decisions is typically a system schematic that indicates the important dynamic effects. Figures 1.1 and 1.2 are examples of system schematics where modeling decisions have been made. In Figure 1.1d, the important dynamic effects at the component level are indicated by labeling inertial, compliance, and resistance effects. In Figure 1.2, modeling decisions are indicated at the subsystem level, while the detail modeling of each subsystem remains to be done. A very important aspect of the modeling process is whether components of subsystems behave linearly or nonlinearly. As we progress through the chapters, it will become very clear as to what is meant by linear or nonlinear behavior. For now it is simply stated that linear systems are represented by sets of linear, first-order differential state equations, and nonlinear systems, while still state determined, are represented by sets of nonlinear, first-order differential equations.

If it is justified to assume that an overall system can be represented as linear, then there exist an abundance of analytical tools for obtaining exact analytical solutions to the linear equations and for extracting detailed information about the response of the system. Some of the analytical information that is covered in later chapters includes eigenvalues, transfer functions, and frequency response. If the systems have large numbers of state variables, then pencil and paper analysis may be virtually impossible, and one must resort to numerical computation to obtain the properties of the system.

If a single component in a system model is represented as a nonlinear element, then the system is nonlinear, and linear analysis tools will not work. Sliding friction is an example of a common nonlinear phenomenon. If friction is important in a system, eigenvalues, transfer functions, and frequency response concepts do not apply as they would to a strictly linear system. To extract information about the response of nonlinear systems, one must resort to time step simulation. Fortunately, there is an abundance of commercial programs to simulate nonlinear systems so that the use of nonlinear models is not the stumbling block it was before the advent of computers.

The fact is that there are no real physical systems that are truly linear. However, in order to introduce the concepts of constructing overall system models of interacting electrical, mechanical, hydraulic, and thermal components, it is easier to start with linear systems and then extend the procedures to nonlinear systems. In addition, it is hard to understand the dynamics of system without first concentrating of the special case of linear systems. In the following five chapters, the emphasis is on linear system models, but whenever possible, the reader is reminded that real physical systems are nonlinear and simulation tools must be used to obtain system responses. The more advanced topics dealt with in Chapters 7–13 are largely concerned with nonlinear phenomena.

1.6 Automated Simulation

Mathematical models of dynamic physical systems have been made ever since the invention of differential equations. But until the development of powerful computers, there were severe limitations on the analysis of these models. Practically speaking, dynamic system behavior could generally be predicted only for low-order linear models that often were not very accurate representatives of real systems.

There is a lot to be said for the study of low-order linear models in order to gain an appreciation of system dynamics, and the first six chapters of this book deal primarily with just such system models. However, computer simulation can now be used to gain experience with system dynamics even when the system models become large and when they contain nonlinear elements. Chapter 13 discusses some of the issues that arise when dealing with complex but realistic models.

The next Chapters, 2, 3, and 4, present techniques for representing elements of mechanical, electrical, and fluid systems (and combination systems) in the abstract form of bond graphs instead of the schematic diagrams usually used to show vibratory systems, electric circuits, or hydraulic systems. For some, this may seem to be an unnecessary step away from physical reality, but it has useful consequences.

First of all, a bond graph is a precise way to represent a mathematical model. Often schematic diagrams are not entirely clear about whether certain effects are to be included or neglected in the model. Second, for many systems involving two or more forms of energy, such as mechanical, electrical, and hydraulic, there are no standard schematic diagrams that clearly indicate assumptions made in the modeling process. Finally, it is much easier to communicate a bond graph model unambiguously to a computer than a schematic diagram.

The bond graph uses only a few standard symbols, whereas typical schematic diagrams for the same system model drawn by different people are almost never identical. Just as computers more easily read bar codes than handwriting, they more easily interpret bond graphs than schematic diagrams.

Computer programs have been developed that recognize bond graphs and can process them in the same manner that a human would in order to extract differential equations for analysis or simulation. In the process, useful facts about the mathematics of the model are discovered even before any numerical parameters or laws have been supplied. Furthermore, when the parameters of the elements and the forcing functions for the system have been specified, the programs can then simulate the response of the system. In this process, only a minimum of human intervention is required.

Although it is important for a system engineer to understand the entire process of modeling and simulation, the use of bond graphs and bond graph simulation programs allows a beginner to start developing the skills associated with computer simulation even before all the bond graph modeling techniques have been learned. This has proved to be very effective in teaching. From the first day, a student given a bond graph model and a bond graph simulation program can see how the model reacts to various input forcing functions and to variations in system parameters. Simple design studies on dynamic systems can be assigned without waiting until the student has learned to make models, derive equations, and use an equation solver. This provides motivation to learn about bond graph dynamic system modeling and numerical simulation techniques.

The fact that the simulation programs are effective for nonlinear as well as linear models, and for large models as well as small ones, may not be fully appreciated by a beginning student. However, in the course of time the significance of this fact should become apparent.

References [2-4] give the names of some of the more well-known commercial bond graph processors, some of which are used in conjunction with simulation programs to solve the differential equations. A web search will reveal that there are a number of other bond graph processor programs.

Another category of program is based on a stored library of predetermined bond graph submodels, but in use, replaces bond graph submodels with icons. See Reference [5], for example. Such programs are useful for studying large engineering models but they are less useful for learning about bond graph modeling.

References

[1] J. W. Forrester, Urban Dynamics, Cambridge, MA: MIT Press, 1969.

[2] CAMP-G, Cadsim Engineering, P.O. Box 4083, Davis, CA 95616.

[3] 20-SIM, Controllab Products B.V., Drienerlolaan 5, EL-CE, 7255 NB Enschede, Netherlands.

[4] SYMBOLS 2000, Hightech Consultants, STEP IIT Ktragpur-721302, WB, India.

[5] The LMS Imagine. Lab AMESim Introduction, http://www.imsintl.com/imagine-amesim-intro.

Problems

1-1. Suppose you were a heating engineer and you wished to consider a house as a dynamic system. Without a heater, the average temperature in the house would clearly vary over a 24-h period. What might you consider for inputs, outputs, and state variables for a simple dynamic model? How would you expand your model so that it would predict temperatures in several rooms of the house? How does the installation of a thermostatically controlled heater change your model?

1-2. For a particular car operated on a level road at steady conditions there is a relation between throttle position and speed. Sketch the general shape you would expect for this curve. If recordings were made of instantaneous speed and throttle position while the car was driven normally for several miles on ordinary roads, do you think that the instantaneous values of speed and throttle position would fall on the steady-state curve? What inputs, outputs, and state variables might prove useful in trying to find a dynamic model useful in predicting dynamic speed variations?

1-3. A car is driven over a curb twice—once very slowly and once quite rapidly. What would you need to know about the car in the second case that you did not need to know in the first case if you were required to find the tire force that resulted from going over the curb?

1-4. In the steady state a good weather vane points into the wind, but when the wind shifts, the vane cannot always be trusted to be pointing into the wind. Identify inputs, outputs, and the parameters of the weather vane system that affect its response to the wind. Sketch your idea of how the position of the vane would change in time if the wind suddenly shifted 10°.

1-5. The height of water in a reservoir fluctuates over time. If you had to construct a dynamic system model to help water resource planners predict variations in the height, what input quantities would you consider? How many state variables do you think you would need for your model?

1-6. A mass, M, and spring, k, are at rest in a gravity field, about to be struck by a mass, m, falling from a height, h. The mass, m, sticks to M, and the two move downward. The variable, x, keeps track of the displacement after impact. Sketch the general motion of x for some period of time after impact. How many equations do you think are needed to describe this system mathematically? If the system ever came to rest, what would be the deflection of the spring?

1-7. The system is an electric circuit consisting of an input voltage, e(t), and a capacitor, resistor, and inductor, C, R, L. As will be seen in later chapters, if a voltage is applied to a capacitor, current flows easily at first and then slows as the capacitor becomes charged. Inductors behave just the opposite, in that they reluctantly pass current when a voltage is first applied, and then the current passes easily as time passes. If the input voltage is suddenly raised from zero to some constant value, sketch the current in the capacitor, iC, and the inductor, iL, as a function of time. What is the steady-state current in the capacitor and inductor?

1-8. A hydraulic system consists of a supply pressure, Ps, and a long fluid-filled tube. Branching from the tube is an accumulator, Ca, consisting of a compliant gas separated from the fluid by a diaphragm. The long tube has inertia and resistance, If and Rf. It may be hard to believe at this point, but this hydraulic system exhibits identical behavior to that of the electric circuit of Problem 1-7. Armed with this information, sketch the volume flow rate of the fluid into the accumulator, Qa, and in the tube, QI, as a function of time.

1-9. Shown here is the hydraulic system in Problem 1-8 connected to a hydraulic cylinder of piston area Ap. The piston is connected to a mass, m, attached to the ground through the spring and damper, k and b. This is a “system” consisting of interacting hydraulic and mechanical components. How many variables do you think are needed to fully describe the motion of the system? Sketch how you think the volume flow rate in the accumulator and in the tube will respond to a sudden elevation of the supply pressure. Sketch the motion, x, of the mass.

Chapter 2

Multiport Systems and Bond Graphs

In this chapter the first steps are taken toward the development of system-modeling techniques for engineering systems involving power interactions. First, major subsystems are identified, and the means by which the subsystems are interconnected are studied. The fact that interacting physical systems must transmit power then is used to unify the description of interconnected subsystems. A uniform classification of the variables associated with power and energy is established, and bond graphs showing the interconnection of subsystems are introduced. Finally, the notions of inputs, outputs, and pure signal flows are discussed.

2.1 Engineering Multiports

In Figure 2.1 a representative collection of subsystems or components of engineering systems is shown. Although the subsystems sketched are quite elementary, they will serve to introduce the concept of an engineering multiport. The term “engineering” is used to imply that the devices are physical subsystems used to build up systems such as automobiles, television sets, machine tools, or electric power plants that are designed to accomplish some specific objectives. The term “multiport” refers to a point of view taken in the description of the subsystems.

Figure 2.1. A collection of engineering multiports. (a) Electric motor: torque , angular speed , voltage e, current i; (b) hydraulic pump: torque , angular speed , pressure P, volume flow rate Q; (c) drive shaft: torque , angular speeds 1 and 2; (d) spring shock absorber unit: force F, velocities V1 and V2; (e) transistor: voltages e1 and e2, currents i1 and i2; (f) loudspeaker: voltage e, current i; (g) crank and slider mechanism: torque , angular speed , force F, velocity V; (h) wheel: force F, velocity V, torque , angular speed ; (i) separately excited direct current (dc) motor: torque , angular speed , voltages ea and ef, currents ia and if.

Inspection of Figure 2.1 reveals that a number of variables have been labeled on the subsystems. These variables are torques, angular speeds, forces, velocities, voltages, currents, pressures, and volume flow rates. The variables occur in pairs associated with points at which the subsystems could be connected with other subsystems to form a system. It would be possible, for example, to couple the shaft of the electric motor () to one end of the drive shaft () and the hydraulic motor shaft () to the other end of the drive shaft. After the coupling, the motor torque and speed would be identical to the torque and speed of one end of the drive shaft. Similarly, the torque and speed of the other end of the drive shaft would be identical to the torque and speed of the hydraulic pump. (If the drive shaft were not rigid, then the two angular speeds, and , on the drive shaft ends would not necessarily be equal at all times.) Similarly, one could connect the two terminals of the transistor () associated with and to the terminals of the loudspeaker (). After the connection, the voltage and current associated with one terminal pair of the transistor would be identical to the voltage and current associated with the loudspeaker terminals. Generally, when two subsystems or components are joined together physically, two complementary variables are simultaneously constrained to be equal for the two subsystems.

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!