A comprehensive text and reference for a first study of system dynamics and control, this volume emphasizes engineering concepts — modeling, dynamics feedback, and stability, for example — rather than mechanistic analysis procedures designed to yield routine answers to programmable problems. Its focus on physical modeling cultivates an appreciation for the breadth of dynamic systems without resorting to analogous electric-circuit formulation and analysis.
After a careful treatment of the modeling of physical systems in several media and the derivation of the differential equations of motion, the text determines the physical behavior those equations connote: the free and forced motions of elementary systems and compound "systems of systems." Dynamic stability and natural behavior receive comprehensive linear treatment, and concluding chapters examine response to continuing and abrupt forcing inputs and present a fundamental treatment of analysis and synthesis of feedback control systems. This text's breadth is further realized through a series of examples and problems that develop physical insight in the best traditions of modern engineering and lead students into richer technical ground.
As presented in this book, the concept of dynamics forms the basis for understanding not only physical devices, but also systems in such fields as management and transportation. Indeed, the fundamentals developed here constitute the common language of engineering, making this text applicable to a wide variety of undergraduate and graduate courses. 334 figures. 12 tables.
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DYNAMICS OF PHYSICAL SYSTEMS
By ROBERT H. CANNON JR.
Dover Publications, Inc.Copyright © 1967 Robert H. Cannon, Jr.
All rights reserved.
It is, at last, the day and the hour for launch. The forty-story gantry yawns slowly open and rolls back, leaving its sometime tenant to stand alone, tall, white, cloaked now only in a veil of liquid-oxygen vapor.
Minutes hence the huge vehicle will thunder skyward on a white-hot cone of fire, and in an hour its forebody will be pursuing an unerring path to a distant planet. But at this moment it stands silent in the spotlight of a hundred arc lamps, mixed with the early rays of dawn, while a thousand test signals are checked and rechecked by the men and machines who attend it.
The loudspeaker intones: "All systems are go." It is a commonplace phrase by now, but its connotation is almost unbelievable! Twenty months ago this vehicle—this collection of hundreds of intricate systems and subsystems—was no more than a concept, a bundle of specifications. Today every system will work, and in perfectly matched, intimate harmony with all the others.
When the start button is pushed, high-capacity pumps will pour fuel into the ravenous rocket chambers. The enormous fuel tanks will be emptied in one minute flat; but flow will be precisely uniform, and the last drop will be delivered exactly on schedule.
Quick, sure hydraulic pistons will swivel and aim the huge rocket engines, as the automatic pilot solves the problem of balancing a long, limber reed on one end.
Precise inertial instruments—each an intricate dynamic system in its own right—will sense the vehicle's path; and a miniature, high-capacity computer will solve the trajectory equations and generate continuous path-correction signals. Two dozen radio and television channels will handle instantly the voluminous communication between the vehicle and its ground base, and a world-wide radar tracking network will monitor its path through the sky.
In the vehicle's nose a myriad of delicate instrument systems are ready to measure, to observe, to photograph, to record and report their findings. Additional systems provide the essential environment for these instruments. A sophisticated air-conditioning system controls temperature and humidity. Carefully designed shock-mounting systems will isolate the delicate instruments from the tremendous vibration of the rocket engines. If there are human passengers, an elaborate complex of life-support systems will operate as well.
Few of the hundreds of systems in this vehicle existed until this decade. Some are based on technology that did not exist four years ago. Many will be making their first flight today. All have been developed specifically for this vehicle, and matched meticulously to one another.
How could so many interrelated systems be developed so rapidly at the same time? How can we be sure each will respond and perform as it must through the abruptly changing sequence of thunderous launch, searing air, and cold, empty space?
There are several underlying answers to these questions. The first is that we have come to understand some of nature's laws, and to know that they are perfectly dependable. This we call science.
Another answer is that we have learned how to use nature's laws to build systems of our own to perform tasks we wish done. This we call engineering.
A third answer is part of the second: We have learned to predict the dynamic behavior of systems not yet built. This is called dynamic analysis. When used in careful support of design and testing programs, dynamic analysis is the key to telescoping the development time of new systems, the key to confidence that they will work properly together in a strange new environment.
A fourth answer is in turn part of the third: Nature is orderly and systematic, and the dynamic behavior of large, intricate compound systems is found to be made up of elementary behavior patterns which can be discerned and studied one by one. The process of discernment is sometimes an involved one; but it is straightforward and it can be accomplished by repeated (and astute) application of relatively elementary analytical techniques ; and thereon hangs our ability to contemplate very involved systems of systems, and to predict their behavior with confidence.
Thus, in the development of a space-vehicle system—or of a television network, a power complex, or any other compound dynamic system—a program of dynamic investigation is the central cord that threads together the myriad of physical systems at their inception. It writes the script from which specifications are established, preliminary designs are evolved, "breadboard" models are constructed, early tests are conceived, performed, and analyzed, and final design decisions are made. This is the script against which the final performance of each system will be measured, first alone and then in concert with its teammates. This is the script by which several hundred complex systems have been developed in months, have been tested and integrated together and tested again and again. And now this system of systems is ready to be launched.
This process of dynamic investigation—its fundamental concepts, its basic building blocks, and its applications—is the subject of this book.
1.1 THE SCOPE OF DYNAMIC INVESTIGATION
Dynamics is the study of how things change with time, and of the forces that cause them to do so. It is an intriguing discipline. Moreover, in this transilient era of space travel, instant communication, and pervasive automation, it is a pivotal discipline: Analysis of the dynamic behavior of physical systems has become a keystone to modern technology.
The motion of a space vehicle, for example, must be thoroughly understood, and its precision control correctly provided for early in the design of the vehicle, many months before it is actually launched. The design of an atomic power plant is predicated in part on the predicted dynamic response of the plant to sudden changes in load. The circuit design and component selection for a high-fidelity radio receiver system are based on calculations of how the electronic section will combine with the speakers, in their enclosures, to produce dynamic response that will match to the desired degree the original sound at the broadcasting station.
In these engineering problems and countless others the first requirement is to predict, before construction, the dynamic behavior a physical system will have—its natural motions when disturbed, and its response to commands and stimuli. More, perhaps, than any other field, the study of dynamic behavior links the engineering disciplines.
In a larger sense, the field of dynamics extends well beyond the realm of physical phenomena. In the field of biology, the response of the eye pupil to a sudden change in light intensity, or of the hand to motor commands from the brain, and the transient adjustment of the body's energy balance to the trauma of major surgery are exciting subjects for dynamic analysis. In economics, the response of a banking system to fluctuations in market activity, of an industry to variations in consumer demand, and, more broadly, the dynamic behavior of the entire economy, have become the subject of dynamic analysis of increasing penetration and importance. Even the phenomena of "group dynamics"—the collective dynamic behavior of teams of individuals having specific tasks to perform—are being studied quantitatively with useful results.
The twofold objective of this book is to develop familiarity with the elementary concepts of dynamic behavior, and to develop proficiency with the techniques of linear dynamic analysis.
Consideration is confined to physical systems in the interest of efficient exposition, because physical phenomena are more familiar and because their behavior is more easily analyzed. Once a degree of proficiency and insight has been attained for one medium, the extension to others will be found to follow by analogy (and to be most intriguing).
Moreover, the similarity in dynamic behavior of different physical systems is accompanied by a striking consistency in the pattern of analytical investigation by which that behavior can most effectively be studied. There are certain broad stages through which an investigation nearly always proceeds; at each stage a small number of basic concepts and analytical methods will be found to be the keys to efficient, successful prediction of dynamic behavior, whatever the physical medium or the particular arrangement.
This "universality" of the concepts and methods of dynamic analysis is a remarkable thing; and it is, of course, a very fortunate thing for people who enjoy moving to new fields of study, and for people responsible for the development of large systems of systems in which numerous physical media interact dynamically. In particular, it will add substantial breadth to our study, in this book, of the process of dynamic investigation.
We begin our study with an overview of the process.
1.2 THE STAGES OF A DYNAMIC INVESTIGATION
The objective of a dynamic investigation is to understand and predict the dynamic behavior of a given system and sometimes to improve upon it.
The detailed tasks in a program of dynamic investigation depend, of course, on the physical media involved, the size and complexity of the system, the stringency with which it must perform, and so on. But whatever the particular physical system under study, the procedure for analytical investigation usually incorporates each of the following four stages:
I. Specify the system to be studied and imagine a simple PHYSICAL MODEL whose behavior will match sufficiently closely the behavior of the actual system.
II. Derive a mathematical model to represent the The stages physical model; that is, write the differential EQUATIONS OF MOTION of the physical model.
III. Study the DYNAMIC BEHAVIOR of the mathematical model, by solving the differential equations of motion.
IV. Make DESIGN DECISIONS; i.e., choose the physical parameters of the system, and/or augment the system, so that it will behave as desired.
The stages of a dynamic investigation
In Stages I, II, and III the emphasis is on analysis, while in Stage IV it is on synthesis.
To illustrate the stages let us single out one of the hundreds of dynamic systems that make up a rocket vehicle, and focus attention on the dynamic investigation that has accompanied its development during the months preceding launch day. Consider, for example, the rather straightforward but vital problem of designing a suitable mounting for a "payload" of delicate scientific instruments. Figure 1.1 shows a panel containing such a payload.
The vibration of the vehicle is so severe during rocket firing that if the panel were mounted rigidly to the vehicle structure the instruments would fail to function properly. A simple cushioning arrangement to protect the instruments from this vibration must be designed. Rubber cups, such as those depicted in Fig. 1.1a, are considered by the dynamic analyst as candidates to furnish the necessary cushioning effect. His next task is to specify the size and stiffness of the cups to achieve the required degree of isolation in the face of expected vehicle vibration.
In Stage I the analyst contrives an imaginary model of the system he wishes to study—a model essentially like the real system, but much easier to analyze mathematically. For the real system of Fig. 1.1a he imagines (and sketches) a simplified model such as that shown in Fig. 1.1b, where the vehicle structure has been replaced by a completely rigid member, the instrument panel by another, and the rubber cups by linear springs plus viscous dashpots. In this physical model, distortions of the vehicle frame and motions of the instrument parts have been ignored, and the rather nonlinear rubber cups have been replaced by the simpler linear spring-and-dashpot model. In his studies the analyst decides to consider only motions in one direction at a time (e.g., vertical). The symmetry of the mount indicates that this is reasonable, and one-dimensional motion is much easier to study.
To complete the model, the analyst inspects the boundaries of his chosen system, Fig. 1.1a and judges that the motions of the vehicle structure segments will be affected negligibly by motions of the instrument panel. Accordingly in the model, Fig. 1.1b, he considers [x.sub.1] a prescribed input to the system.
Next, in Stage II, the analyst applies the appropriate physical laws—in this case Newton's laws of motion—to obtain a set of "dynamic equations of motion" which, for this model, will be a set of ordinary differential equations in the variables x1, and x2.
In Stage III, the equations of motion are solved to obtain explicit expressions for the time variation of x2 as a function of input motion x1. Further, expressions are obtained for the actual excursions and accelerations which would be produced by the "worst inputs," x1(t), that the rocket vehicle is expected to impose on the panel. A major task is to estimate accurately the maximum magnitudes of inputs of various types—sudden shock, steady vibration, and so on.
Finally, in Stage IV, the calculated behavior from Stage III is studied to determine what combinations of spring and dashpot characteristics will do the job—will insure that the resulting panel motions are small enough not to impair instrument performance.
When the final stage has been completed, and a preliminary design established, the entire analysis will be repeated with some of the simplifications omitted, to get a more precise estimate of the performance to be expected.
The analyst will also arrange to have a hardware model of the panel constructed, using actual rubber shock mounts, to give him added experience by allowing him to check his imagined model against physical reality. Eventually an actual instrument panel will be vibration-tested to make sure the design is acceptable.
Clearly, without an astute initial analysis, costly experimental time could be wasted in a trial-and-error approach to solving even this rather elementary problem in dynamic design.
The four stages of dynamic investigation are depicted in a more general way in Fig. 1.2. Note particularly the "feedback" feature in the procedure: that errors in the analysis may be detected by comparison with actual test behavior, and corrections then made ("fed back") to the original physical model to make it more realistic. The application of design specifications to the actual system is also in the nature of a "feedback" process.
In the five parts of this book we shall describe in greater detail, and with numerous examples, the analytical procedures involved in each of the above stages. In Part A we shall study the process of deriving the equations of motion of physical systems, first imagining a physical model, and then applying physical laws to the model to obtain the governing equations (Stages I and II). We shall study simple mechanical, electrical, electromechanical, fluid, and thermal systems, and will note the detailed analogy between their mathematical descriptions.
In Parts B, C, and D of this book we shall be concerned with determining the dynamic behavior of systems (Stage III), once their equations of motion have been derived.
Finally, in Part E we shall introduce the synthesis of systems (Stage IV) and, in particular, of automatic-control systems.
1.3 THE BLOCK DIAGRAM: A CONCEPTUAL TOOL
Block diagrams, for portraying a sequence of events or interrelationships, have important technical, as well as organizational, applications. In working with the physical model (Stage I) or the mathematical model (Stage II) of a complicated system, it is sometimes convenient to indicate the interrelation between subsystems in a block diagram. This is especially easy when the interaction is in one direction only.
For example, the shock-mount in Fig. 1.1 was considered to have no effect on the vehicle. The whole shock-mount "subsystem" might be represented by a single block in a block diagram, Fig. 1.3, that indicates the propagation of vibration from the rocket engine through the vehicle frame to the instrument mount, and thence to the separate instruments where it contaminates their output signals. Such a diagram dramatizes the sequence of cause and effect.
Excerpted from DYNAMICS OF PHYSICAL SYSTEMS by ROBERT H. CANNON JR.. Copyright © 1967 Robert H. Cannon, Jr.. Excerpted by permission of Dover Publications, Inc..
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Table of ContentsContents:
Preface to the Dover Edition.
1. Dynamic Investigation.
A. Equations of Motion for Physical Systems.
2. Equations of Motion for Simple Physical Systems: Mechanical, Electrical, and Electromechanical.
3. Equations of Motion for Simple Heat-Conduction and Fluid Systems.
5. Equations of Motion for Mechanical Systems in Two and Three Dimensions.
B. Dynamic Responses of Elementary Systems.
6. First-Order Systems.
7 Undampened Second-Order Systems: Free Vibrations.
8. Damped Second-Order Systems.
9. Forced Oscillations of Elementary Systems.
10. Natural Motions of Nonlinear Systems and Time-Varying Systems.
C. Natural Behavior of Compound Systems.
11. Dynamic Stability.
12. Coupled Modes of Natural Motion: Two Degrees of Freedom.
13 Coupled Modes of Natural Motion: Many Degrees of Freedom.
D. Total Response of Compound Systems.
14. est and Transfer Functions.
15. Forced Oscillations of Compound Systems.
16. Response to Periodic Functions: Fourier Analysis.
17. The Laplace Transform Method.
18. From LaPlace Transform to Time Response by Partial Fraction Expansion.
19. Complete System Analysis: Some Case Studies.
E. Fundamentals of Control-System Analysis.
20. Feedback Control.
21 Evans' Root-Locus Method.
22. Some Case Studies in Automatic Control.
Answers to Odd-Numbered Problems.