Modelinq and Analisys of Dynamic Systems PDF

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MODELING AND ANALYSIS OF DYNAMIC SYSTEMS Third Edition EDITOR EDITORIAL ASSISTANT Steve Peterson MARKETING MANAGER Katherine Hnm11;cn SENIOR PRODUCTION EDITOR SENIOR DESIGNER Kevin Cover courtesy of NASA This book was set in Times Roman ancl bouncl Hamilton Press. This book was on acid-free paper. ...

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MODELING AND ANALYSIS OF DYNAMIC SYSTEMS Third Edition

EDITOR EDITORIAL ASSISTANT Steve Peterson MARKETING MANAGER Katherine Hnm11;cn SENIOR PRODUCTION EDITOR SENIOR DESIGNER Kevin

Cover

courtesy of NASA ancl bouncl

This book was set in Times Roman

Hamilton

Press. This book was

on acid-free paper. @

Copyright 2002 © John Wiley & Sons, Inc. All rights reserved. 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 Sections 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) 750-4470. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: [email protected]. To order books please call 1(800) 225-5945. Library of Congress Cataloging in Publication Data: Close, Charles M. Modeling and analysis of dynamic systems I Charles M. Close and Dean K. Frederick and Jonathan C. Newell-3rd ed. p. cm. Includes bibliographical references. ISBN 0-471-39442-4 (cloth: alk. paper) 1. System analysis. I. Frederick, Dean K., 1934- H. Newell, Jonathan C. HI. Title. QA402.C53 2001 003-dc21 2001033010

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

To my wife, Margo, and to our children and grandchildren CMC

To my mother, Elizabeth Dean Frederick, and to the memory of my father, Charles Elder Frederick DKF

To my wife, Sigrin, and my sons, Andrew and Raymond JCN

PREFACE The primary purpose of this edition remains the same as in the previous editions: to provide an introductory treatment of dynamic systems suitable for all engineering students regardless of discipline. We have, however, made significant changes as a result of experiences with our many students, comments from numerous professors around the country, and the increasing educational use of computer packages. We have maintained flexibility in the selection and ordering of material. The book can be adapted to several types of courses. One such use is for students who need a detailed treatment of modeling mechanical and electrical systems and of obtaining analytical and computer solutions before proceeding to more advanced levels. Such courses can serve as a foundation for subsequent courses in vehicular dynamics, vibrations, circuits and electronics, chemical process control, linear systems, feedback systems, nuclear reactor control, and biocontrol systems. The book also covers such general topics as transfer functions, state variables, the linearization of nonlinear models, block diagrams, and feedback systems. Hence it is suitable for a general dynamic systems course for students who have completed a disciplinary course such as machine dynamics, electrical circuits, or chemical process dynamics. This text can also be used for students with significant modeling and analysis experience who wish to emphasize computer techniques and feedback control systems. Topics include computer solutions for both linear and nonlinear models, as well as root-locus diagrams, Bode plots, block diagrams, and operational amplifiers. We explain some of the practical design criteria for control systems and illustrate the use of analytical and computer methods to meet those criteria. Finally, the book can provide a general introduction to dynamic systems for students in broad-based engineering programs or in programs such as biomedical and materials engineering who may have limited time for this subject. We assume that the reader has had differential and integral calculus and basic college physics, including mechanics and electrical phenomena. Many students will have had a course in differential equations at least concurrently. We have been careful to present the mathematical results precisely (although without the rigorous proofs required for a mathematics book), so that the concepts learned will remain valid in subsequent courses. For example, the impulse has been treated in a manner that is consistent with distribution theory but is no more difficult to grasp than the usual approach taken in introductory engineering books.

Approach The book reflects the approach we have used for many years in teaching basic courses in dynamic systems. Whether for a particular discipline or for a general engineering

Vil

viii flJ> Preface

course, we have found it valuable to include systems from at least two disciplines in some depth. This illustrates the commonality of the modeling and analysis techniques, encourages students to avoid compartmentalizing their knowledge, and prepares them to work on projects as part of an interdisciplinary team. Mechanical systems are examined first because they are easily visualized and because most students have had previous experience with them. The basic procedures for obtaining models for analytical and computer solutions are developed in terms of translational systems. Models include those in state-variable, input-output, and matrix form, as well as block diagrams. The techniques are quickly extended to rotational systems, and there are also chapters on electrical, electromechanical, thermal, and fluid systems. Each type of system is modeled in terms of its own fundamental laws and nomenclature. After introducing block diagrams in Chapter 4, we show how to use Simulink and MATLAB to obtain responses of simple systems. We introduce Laplace transforms fairly early and use them as the primary means of finding analytical solutions. We emphasize the transfer function as a unifying theme. We treat both linear and nonlinear models, although the book allows nonlinear systems to be deemphasized if desired. Students should realize that inherent nonlinearities generally cannot be ignored in the formulation of an accurate model. Techniques are introduced for approximating nonlinear systems by linear models, as well as for obtaining computer solutions for nonlinear models. We believe that obtaining and interpreting computer solutions of both linear and nonlinear models constitute an important part of any course on dynamic systems. We introduce, early on, MATLAB and Simulink, two computer packages that are widely used in both educational and industrial settings. Although it is possible to use the book without detailed computer work, the inclusion of such methods enhances the understanding of important concepts, permits more interesting examples, allows the early use of computer projects, and prepares the students for real-life work. An important feature is providing motivation and guidance for the reader. Each chapter except Chapter 1 has an introduction and summary. There are approximately 200 examples to reinforce new concepts as soon as they are introduced. Before the examples, there are explicit statements about the points to be illustrated. Where appropriate, comments about the significance of the results follow the examples. There are over 400 end-of-chapter problems. The answers to selected problems are contained in Appendix G.

Organization The majority of the material can be covered in a one semester course, but the book can also be used as the basis for a two quarter or year long course. A number of chapters (including Chapters 10, 11, 12, 14, and 15) and a number of individual sections (including Sections 3.3, 6.7, 8.6, and 9.4) can be omitted or abbreviated without any loss of continuity. The chapters can be grouped into the following four blocks:

1. Modeling of mechanical and electrical systems: Chapters 1 through 6. The sections of Chapter 4 on Simulink and MATLAB can be deferred. However, we believe it is beneficial to encourage students to prepare simulations and to note typical features of the responses even before complete analytical explanations are presented.

Preface

2.

'"''"'"'''"'' ..":ir

ix

no:ndinear models: tool for

3,

4. A core sequence for students without a nrc'""'"Q include 1 9

to oilier we suggest including one or more of 11, and 12. 13 15 can be used as an introduction to the""'~~'""'"' and design of feedback control In '-'"'"'-'"~' of MATLAB's Control '"Q""'rn Rather than provide lengthy ae~;cnpuons we them in an iterative provH>IJ"'""·"'"'-' interpreting crnmpute:rcess to meet some design criteria. in material from The following table of prerequisites should be the later chapters:

matelial Linearization

1

9

Chapter 5 1

Electromechanical systems

10

Chapters 5-8

Thermal systems

11

Chapters 8, 9

Fluid systems

u

8,9

Block

13

8

Feedback systems

14

13

Section 9.2 also requires Chapter 6.

to l,

methods are introduced in a vV,cHj.'''"'''-''.J the rest of the book. the many the instructor's to add features to an eadier the simulation in 4 of ~U l s. a. b. c. d.

with is zero for

an t ~ 1 s and is

"'~HY"'·""'·"'·"''" x for 0 ~ t ~ 8 when M = 4

whenM=20 How does the value of M affect the trial and error, determine the range of values of M for which the x never overshoots the steady-state value.

4.19. Change the value of to 2 and repeat Problem 4.12. Give an explanation in physical terms for the change in the response from the response of Problem 4.12. 4.20. Problem 4.11 wi.th the same values, = 20 and = 20 Give an explanation in nnvQ'"'" in the response compared to that of Problem 4.1 L

for

= 30 N-s/m,

5 ROTATIONAL MECHANICAL SYSTEMS In Chapter 2 we presented the laws governing translational systems and introduced the use of free-body diagrams as an aid in writing equations describing the motion. In Chapter 3 we showed how to rearrange the equations and develop state-variable and input-output models. Extending these procedures to rotational systems requires little in the way of new concepts. We first introduce the three rotational elements that are analogs of mass, friction, and stiffness in translational systems. Two other elements, levers and gears, are characterized in a somewhat different way. The use of interconnection laws and free-body diagrams for rotational systems is very similar to their use for translational systems. In the examples, we seek models consisting of sets of state-variable and output equations, or input-output equations that contain only a single unknown variable. We include examples of combined translational and rotational systems, followed by a computer simulation. ~

5.1 VARIABLES For rotational mechanical systems, the symbols used for the variables are (), angular displacement in radians (rad)

w, angular velocity in radians per second (rad/s) o:, angular acceleration in radians per second per second (rad/s2 )

r, torque in newton-meters (N·m) all of which are functions of time. Angular displacements are measured with respect to some specified reference angle, often the equilibrium orientation of the body or point in question. We shall always choose the reference arrows for the angular displacement, velocity, and acceleration of a body to be in the same direction so that the relationships

w = ()

o:=w=B hold. The conventions used are illustrated in Figure 5 .1, where r denotes an external torque applied to the rotating body by means of some unspecified mechanism, such as by a gear on the supporting shaft. Because of the convention that the assumed positive directions for (), w, and o: are the same, it is not necessary to show all three reference arrows explicitly. The power supplied to the rotating body in Figure 5 .1 is

p=rw

(1)

95

96

Rotational Mechanical

The power is the derivative of the energy w, and the energy t is

to the

up to time

+

devices in rotational are moment and gears. We shall restrict our consideration to elements that rotate about fixed axes in an inertial reference frame.

When Newton's second law is to the differential mass element dm in Figure 5.1 and the result is over the entire body, we obtain d

dt(Jw)=r

(2)

where J w is the angular momentum of the body and where T denotes the net torque applied about the fixed axis of rotation. The J denotes the moment of inertia in kilogrammeters2 We can obtain it out the of r2 dm over the entire body. whose mass M can be considered to concentrated The moment of inertia for a is , where L is the distance from the point to the axis of rotation. Figure 5 .2 at a shows a slender bar and a solid each of which has a total mass M that is uniformly distributed the and of the for J are the center of mass. The results for the case where the axis of rotation passes for other common can be found in basic the center of mass, we can use the theorem. axis that does not pass Let denote the moment of inertia about the axis that passes the center of mass, and let a be the distance between the two axes. Then the desired moment of inertia is l= +

2

l=

+M 2

(4)

502 Element Laws

97

l= (a)

(b)

(c)

(cl)

5.2 Moments of inertia" (a) Slender bar" (b) Disk Slender bar where axis of rotation does not pass through the center of mas so

For two or more components rotating about the same we can find the total moment of inertia by the individual contributions" ff the uniform bar shown in Figure then the masses for the sections of and will be has a total mass (d1 + and + Using (4), we see that the total moment of inertia is J= l M d2 - M(df +dD + 3 2 2 - 3(d1 +dz) We consider nonrelativistic so and constant moments of reduces to fW=T

where wis the angular acceleration" As is the case for a mass translational """"''"' a rotating body can store energy in both kinetic and forms" The kinetic energy is

Wp

=

where M is the mass, g the gravitational constant, and h the height of the center of mass above its reference If the fixed axis of rotation is vertical or passes the center of mass, there is no change in the energy as the rotates, and (8) is not needed" To find the complete response of a containing a we must know its initial angular velocity If its potential energy can vary or if we want to find e(t), then we must also know

Friction A irotatfonal frktiion element is one for which there is an the tmque and the relative friction arises when two two concentric 'v"'""'"' of the T

= B!iw

98

Rotational Mechanical Oil film, B

B

B

(a)

(c)

(b)

5.3 Rotational devices characterized

viscous friction.

win be exerted on each cylinder, in directions that tend to reduce the relative angular velocity Aw. Hence the sense of the frictional must be counterclockwise on the The friction coefficient B has units of inner cylinder and clockwise on the outer newton-meter-seconds. Note that the same is used for translational viscous friction, where it has units of newton-seconds per meter. Equation also to a rotational ua.>UIJU• drive system, such as that shown in Figure or 5 .3 (c). The inertia of the parts is assumed to be negligible or else is accounted for in the mathematical model by separate moments of inertia. If the rotational friction element is assumed to have no inertia, then when a T is applied to one side, a torque of equal magnitude but opposite direction the or some other component), as shown must be exerted on the other side (by a in Figure 5.4(a), where T = B(w2 . Thus in Figure the torquer passes through the first friction element and is exerted directly on the moment of inertia J. Other types of friction, such as the damping vanes shown in Figure 5.5(a), may exert a torque that is not directly proportional to the angular velocity but that may be as shown in Figure 5.5(b). For a linear element, the described by a curve of r versus curve must be a straight line passing through the The power to the friction "'""""""' r is immediately lost from the mechanical in the form of peat.

Rotational stiffness is associated with a torsional such as the ""~U.JI-'• of a or with a relatively thin shaft. It is an element for which there is an

B

B

(b)

(a)

5.41 (a) Rotational

with u'-l';ui;.w'" inertia. element

Torque transmitted through a friction

5.2 Element Laws

e1' so the must be labeled By the law of reaction torques, the effect of the connecting shaft on disk 1 is a with its sense in the clockwise direction. We can reach the same conclusion first selecting a clockwise sense for the arrow in Figure that on disk 1, and then noting that disk 2 wm tend to drive disk 1 in the ~~''""'a if > 81. Thus the correct ex1ore:ss1on is Of course, if we had selected a counterclockwise arrow in we would have labeled the arrow either - 81) or

5.4 Obtaining the System Model

~

107

For each of the free-body diagrams, the algebraic sum of the torques may be set equal to zero by D' Alembert's law, giving the pair of equations fiw1 +B1w1+K181 - K2(82 -81)= 0 fzw2 +B2w2 +K2(82 - 81) - Ta (t)= 0

(~?)

Two of the state-variable equations are B1 = w1 and 02 = wz, and we can find the other two by solving the two equations in (26) for w1 and w2, respectively. Thus

01= w1 1

w1 = -[-(K1 +K2)81 - B1w1+K282] 11

(27)

B2= w2 W2 = ; 2 [K281 - K282 -B2w2 + Ta(t )] The output equations are

Kz(82 -fh) mr = fi w1+ fzw2

TK2

=

where mr is the total angular momentum. To obtain an input-output equation, we rewrite (26) in te1ms of the angular displacements 81 and 82. A slight rearrangement of tenns yields

liB1 + B1B1 + (K1 + K2)B1 - K2B2 = 0

(28a) (28b)

Neither of the equations in (28) can be solved separately, but we want to combine them into a single differential equation that does not contain B1. Because Bi appears in (28b) but none of its derivatives do, we rearrange that equation to solve for 81 as .. . 1 81 = - [JzB2 +B2B2 + K2B2 -Ta(t )J

Kz

Substituting this result into (28a) gives

lihBiiv) + (liB2 + JzB1)eiiii} + (liK2 + hK1 + JzK2 + B1B2)B2 + (B1K2 + B2K1 +B2K2 )f)z +K1K202

= l 1fa +Bda + (K1 + K2) Ta(t)

(29)

which is the desired result. Equation (29) is a fourth-order differential equation relating 02 and Ta(t ), in agreement with the fact that four state variables appear in (27).

In the last example, note the signs when like terms are gathered together in the torque equations corresponding to the free-body diagrams. In (28a) for Ii , all the terms involving 81 and its derivatives have the same sign. Similarly in (28b) for ]z, the signs of all the terms with B2 and its derivatives are the same. This is consistent with the comments made after Example 2.2, and can be used as a check on the work. Some insight into the reason for this can be obtained from the discussion of stability in Section 8.2. ~· EXAMPLE 5.3

Find state-variable and input-output models for the system shown in Figure 5.14(a) and studied in Example 5.2, but with the...