
Preface to the Third Edition The Third Edition is the consequence of publication modernization. Specifically, the text has been upgraded from the original DOS version of WordPerfect to a WinXP version. Additionally, all figures, equations, tables, etc have been fully crossreferenced so that subsequent updates will be facilitated. For example, if a new figure or equation has to be added, the work processing software will automatically renumber all of the downstream figures or equation. But with crossreferencing, all intext referrals to those figures and equations are automatically updated, too, making editing/correcting after revisions a much simplified chore. Anyone considering writing a technical book that needs occasional updating is urged to use crossreferencing. The reader will notice the small cross reference/target codes, usually two or three characters, eg, J%&. spread throughout the text. Sadly, WordPerfect does not allow the visible notations to be switched on and off, so they are all visible. This text attempts to present a more basic approach to hydraulic system analysis than is currently taught, emphasizing analytical methods, but without being too theoretical. Methods are presented which allow the student to advance beyond the mere content of the pages. And yet the approach is pragmatic, eschewing detailed theoretical developments in fluid mechanics, and instead jumping directly into the practical methods of circuit analysis. Adoption of Kirchoff's Laws are an example of the pragmatism as well as other analytical concepts that are so well documented and developed in electrical engineering. Mathematical level is that of about a sophomore or junior in mechanical engineering, or a sophomore in some of the more progressive twoyear Associate Degree curricula. Coverage of all the material requires two semesters. The material has evolved over the last 12 years, as electronic developments have advanced and made electronic control of hydraulic systems a practical reality. The first crude offerings were made at industrial seminars, both inplant and at the Milwaukee School of Engineering, and later, in both graduate and undergraduate classes at MSOE. Industrial events have demanded faster and more reliable machinery. This has required broad application of servo systems which in turn require the responsiveness of servo and proportional valves. In the development of industrial retraining programs, it became apparent that the present methods of teaching hydraulics were totally inadequate in preparing engineers for the design of servo systems. Certainly, feedback concepts are a stumbling block, but they are well documented at all technical levels. More basically, most fluid power curricula do not properly prepare the student for the hydraulic circuit analysis that is needed to understand and design feedback controlled servo systems and motion control. As for practicing engineers, there is an internal torment that abhors the inefficiency of high pressure drop valves, but loves the control they offer. Industry is filled with failed applications where the designer chose a servo or proportional valve for its continuous variability, and then oversized it, to "get rid of the pressure drop". Valve control works by virtue of high pressure drop and the consumption of excessive power. Designing that power loss out of the system works at cross purposes to the need for control. Many experienced fluid power system designers will find that they cannot make the transition from conventional design methods to servo system design. This book is aimed at helping to make the transition from conventional hydraulics to servo hydraulics as painless as possible, and should be of value in both academic and industrial settings. Prerequisites include a thorough knowledge of hydraulic componentry, and a course in basic electrical circuit analysis. Mathematics is always problematic, because it is not possible to truly understand the dynamic performance of systems without an understanding of differential equations, Laplace transforms and linear algebra. This book takes a different approach, because it focuses on the design of the closedloop, positional servomechanism as the final solution to the motion control problem. The generalized concepts of feedback control systems are not covered. This severely limits the breadth of mathematical theory needed to explain the system. Therefore, much of the math is introduced at the time it is needed. A background in basic algebra is assumed, however, Appendices A and B contain material on the basics of calculus, trigonometry, exponential functions, logarithms and dynamic response of second order systems. It can be covered as needed. I have found that in industrial seminars, coverage of the math is necessary for most graduates of twoyear programs, and serves as needed review for degreed engineers. The math, not surprisingly, is the tool for proof of concepts and derivation of the application equations, but is not needed in the basic design of the servo system. For example, Laplace transforms are required to explain why the open loop gain defines the closed loop bandwidth, however, only basic arithmetic and technical insight are needed to calculate the bandwidth, once the proof is accepted. Therefore, in Chapter 17, the integrator is replaced with 1/s without proof, so that bandwidth can be derived, producing a valuable design formula. Chapter 1 provides an overall summary of the entire book. It introduces the closed loop positional servomechanism as the ultimate set of hardware, and the command profile as the motion control method. It is important that both concepts be well understood, because they are called upon as the focal points of the details in subsequent chapters. This understanding does not have to be quantitative, it needs to be only qualitative. The homework questions provide a guide as to the important ideas. Chapter 2 introduces basic nonlinear, hydraulic circuit analysis. The problems and circuits, are of limited practical applicability, but serve to solidify basic circuit analysis methods. The material is essential to all subsequent chapters, and represents that which is missing in traditional hydraulic curricula. Chapter 3 is included because it shows how the orifice equation and basic mechanics combine to develop valve design methods which are based upon required valve performance. It has been included because there is no better way to understand the functions of valves than to engage in their design. But, it is not necessary to know how to design, for example, a pressure compensated flow control valve, in order to design a motion control system. Therefore, chapter 3 can be omitted if time is at a premium. Chapter 4 is a semiquantitative overview of electrohydraulic servo and proportional valves and is basic to the material that follows. Chapters 5 and 6 deal with the mathematical modeling of the fourway directional control valve. The nonlinear model of Chapter 5 forms a background for the use of the orifice equation and is essential for anyone wanting to computerize the valve model, such as in a simulation. The linear model of Chapter 6 is valuable to anyone who is interested in going beyond the content of this book in linear control system design. It is included only in the interest of making the book complete, but it is not vital to subsequent chapters. Interpretation of valve catalog data is covered in Chapter 7. It can be taken in the sequential position it occupies, or can be a reference chapter, being accessed as needed to solve problems in later chapters. Chapter 8 fills a void in the teaching technology, because it takes a generalized approach to analysis of cylinder circuits, rather than the more traditional, positive displacement methods. In those methods, the assumption is made that if the pump flow is known, the cylinder speed is known. This is an unfortunate and pervasive misconception of hydraulic systems. This chapter is vital to dispelling the myths of cylinder circuits because it forms the basis for valve control of cylinder motion. Chapter 9 is a key design chapter, because it presents the basic methods for sizing the hydraulic system needed to achieve the required performance. It is a focal point and vital to the motion control design process. Stopping the cylinder with a valve, the subject of Chapter 10, is basic to understanding the imperfections in the positional servo mechanism. It is true that the servo system can be designed around its imperfections. That is the design methodology of the book. One cannot understand the imperfections of the servo loop without understanding how the valve's pressure metering teams up with the cylinder and load to effect cylinder stoppage. Closure of the positional loop guarantees only that the loop will automatically settle on those conditions needed to stop the cylinder. Whether or not it stops and/or maintains the desired position is completely dependent upon how well the servo has been designed. This chapter is crucial. The fluid compressibility phenomenon is covered in Chapter 11. The approach has been to deal with compressibility as a hydraulic capacitance. The equations that ensue are identical to those of electrical circuits, so I have borrowed heavily from those techniques. Too often, the hydraulics industry has chosen to treat fluid as being incompressible. This is simply not the case. Hydraulic and electrical drives can be compared on the basis of power level and bandwidths. When doing so, we find, perhaps to the dismay of some, that wattforwatt, the frequencies in hydraulic systems are no higher than those of electrical systems. It is necessary to include compressibility effects in order to draw such conclusions. This chapter attempts to fill a void existing in present curricula. Chapters 12, 13, 14 and 15, are a stepbystep introduction of the effects of leakage, friction, mass and capacitance, with Chapter 15 as the culmination, combining all effects at once. Proofs and derivations require a knowledge of differential equations and their solutions, however, the focal point of Chapter 15 is the equation for resonant frequency and damping ratio. Homework problems are of both the derivation and formulause types, so it is possible to tailor the course to the math backgrounds of the students at hand. Frequency response is covered in a semiquantitative manner in Chapter 16. The minimum outcome is that the student should understand that frequency response testing is one method of describing how fast a component (valve, eg) will respond, and that competing valves can be compared with their frequency responses. The maximum outcome is to understand and know how to combine the frequency response curves of individual components in order to create the frequency response of the system. It is hoped that the student will investigate one or more of the thousands of very good books that cover the subject in quantitative detail. Chapter 17 is the focal point of the book. All of the concepts of the other chapters are brought to bear on the system design problem. Chapter 18 deals with the physics of acceleration, speed and position in the form of motion control profiles. It appears in a position which is out of order, because it must be covered before Chapter 17, and yet, there is no logical place except before Chapter 17. The instructor is urged to stick it in wherever it seems to fit best. Chapter 19 has been included because of the popularity of the Programmable Logic Controller in industry. But instead of concentrating on its strengths, which I believe to be wellknown, it concentrates on its limitations. This may seem unfair to a perfectly wonderful machine, however, my experience has been that PLC's are too simplistically understood, and that simplicity leads to their misapplication. The PLC is covered in the context of open loop motion control and exposes many of the problems. The closed loop methods as covered in the other chapters, it will be seen, offer the very best method of motion control in the most critical and demanding applications. The material in this book has been evolving over several years. A number of people have been extremely helpful in completing this book who deserve recognition for their contributions. First, my wife Louise, and daughter, Barbara, set the type for the entire First Edition in WordPerfect. They did the text, all the equations, page lay out, book lay out and organized the work in a most expedient and efficient manner. Rick Ottman, Associate Director of the Applied Technology Center at the Milwaukee School of Engineering and his assistant Jeff Lawrence, drew all the figures in Fastcad and Quattro Pro, and figured out how to import them into Wordperfect. Jeff Cullman, Manager of Hydraulic Application Engineering and Larry Schrader, Director of Training, Parker Hannifin created the environment and the cause for this book. For all of them and all their efforts, I extend a very sincere and warm personal thank you. For this, the Third Edition, I give my most sincere thank you to my granddaughter, Rebecca Johnson, who converted the original WP5.1 manuscript to WP12.0. Unlikely that this may seem, since both products are from Corel, the conversion of such a book, especially with its many figures and equations, was indeed formidable. The learning curve has been steep and she has climbed it admirably. Jack L. Johnson, PE East Troy, Wisconsin December, 2006
TABLE OF CONTENTS Chapter 1  Background and Overview 11 Industrial Needs and Problems 11 Motion Control 13 Two Methods of Control in Hydraulic Systems 15 Motion Control Profiles 19 System Specifications and Performance 111 Factors Which Affect overshoot 111 Factors Which Increase Lag 112 Solutions to the Motion Control Problem 113 The Closed Loop Positional Servo and Profiling as the Motion Control Solution 115 The Design methodology 117 Conclusions 121 Problems Chapter 2  Basic Hydraulic Circuit Analysis 224 Resistance in Hydraulic Circuits 224 Four Profound Differences Between Hydraulic and Electrical Circuits 225 Basic Hydraulic Circuit Laws 225 Circuit Concepts 229 Power in Hydraulic Circuits 230 Hydraulic Sources 230 PressureFlow Relationships in Hydraulic Restrictions 233 Hydraulic Circuits With Sources and Restrictions 236 Circuits Which Contain Both Linear and Square Law Orifices 241 Problems 246 Chapter 3  Math Modeling for Designing Hydraulic Valves Introduction 349 Flow Forces in Spool Valves 350 Relief Valves 352 Math Modeling of a Direct Acting Relief Valve 354 Pressure Compensated Flow Control Valve  Pressure Reducing Type 364 Problems 368 Chapter 4  Hydraulic Directional Control Valve Fundamentals Introduction 471 Hydraulic Valve Symbology Fundamentals 474 Valves as Bridges 474 An Overview of Proportional Electrohydraulic Interface Devices 475 Details of the Parker/Nichols Proportional Valve 483 Positioning the Main Spool 484 Table of Contents Valve Testing and Valve Characteristics 485 Problems 496 Chapter 5  NonLinear Model of a Directional Valve Operating out of the Null Zone Introduction 599 Model Development 599 Split Metering Servo Valve Model 5101 Summary 5102 Problems 5104 Chapter 6  Linear Valve Models Introduction 6105 Simplified Servo Valve Model 6105 Derivation of a TwoSource, Assymmetrical Linear Valve Model 6106 Chapter 7  Interpreting Valve Catalog Data Introduction 7108 Constant Pressure, NoLoad Flow Metering Characteristics 7109 Linearity 7112 Hysteresis 7113 Valve Threshold 7113 Pressure Gain, Port 7114 Pressure Gain, Differential 7114 Pressure Gain in Large Overlap Valves 7116 Rated Current 7118 Servo Valve Null Characteristics 7119 Valve Coefficient 7121 Chapter 8  Analysis Methods for Basic Cylinder Circuits Introduction 8123 Cylinders 8123 Flow in Hydraulic Cylinders 8125 SteadyState Cylinder Operation with Throttling Orifices 8127 Power in Cylinder Circuits 8130 Metering Methods in Hydraulic Systems 8132 ThreeWay Valve Circuits 8136 Problems 8140 Chapter 9  Valve Control of Cylinder Motion (VCCM) Derivation of Generalized VCCM Equation 9145 VCCM Equation and the ForceVelocity Operating Envelope 9148 Derivations for Design and Analysis of VCCM Systems 9152 Optimal Control of Cylinder Motion 9158 OverRunning Load and Cavitation 9161 Maximum Deceleration Without Cavitation 9162 Table of Contents Design Strategies 9167 Changing Design Scenarios 9170 Sizing the Pump 9170 Problems 9172 Chapter 10  Stopping the Cylinder with the Valve Conditions for Stopping the Cylinder 10177 Effects of Load Force Change Upon Stopping Conditions 10181 Effects of Tank Pressure Variation upon Stopping Conditions 10182 Problems 10187 Chapter 11  Fluid Compressibility, Hydraulic Capacitance and Cavitation A Fluid Compressibility Experiment 11189 Capacitance Calculations 11191 Capacitances in Parallel 11192 From Circuit Schematic to Analytical Schematic 11193 A Series Implementable Capacitance 11194 Capacitances in Series 11195 Effective Capacitance of a Hydraulic Cylinder 11195 Effects of Envelope Expansion and Apparent Bulk Modulus 11205 Line Length Limitaitons 11205 Problems 11207 Chapter 12  Circuits with Capacitance and Leakage Circuits with Capacitance and Leakage 12210 When Pressure Waveshape is Known and the Flow is Unknown 12210 When Flow is Known and Pressure is Unknown 12212 A Summary of the RC Process 12214 Application of the RC Circuit 12215 Problems 12216 Chapter 13  Friction Friction 13217 Problems 13220 Chapter 14  Cylinder Circuits with Mass and Friction Loads Cylinders with Mass and Friction Loads 14222 Problems 14225 Chapter 15  Cylinders with Capacitance and MassFriction Loading Cylinders with Mass, Leakage and Capacitance 15226 Analytical Approach 15229 Derivation of CylinderMassCapacitance Response With Linear, Viscous Friction 15234 Table of Contents Response of a Cylinder System Which Has Leakage, Capacitance, Mass Load and Viscous Friction 15236 Summary of CylinderMass Systems 15238 Estimating Resonant Frequency and Damping Ratio 15239 Problems 15242 Chapter 16  Frequency Response Frequency Response and Transfer Functions 16244 Phase and Gain Margins 16247 Stability Criterion  Frequency Domain 16249 Combining Frequency Response Curves 16251 Frequency Response of and Integrator 16257 Universal, Second Order Frequency Response Graphs 16256 Graphical Construction of the Closed Loop Frequency Response from the Open Loop Nyquist Plot 16259 Frequency Response Testing 16259 Example Frequency Response 16260 Problems 16262 Chapter 17  Closure of the Position Loop Canonical Forms of Control Systems 17266 Errors in Positional Servomechanisms 17276 Electrohydraulic Positional Servomechanism 17280 Disturbances and Errors in the Electrohydraulic Positional Servomechanism 17284 Calculation of ÄI Due to Error Contributors 17292 Bandwidth Criteria 17296 Requirements of the Profile 17297 Myths and Realities of Electronic Deadband Corrections 17299 Phasing the Positional Servomechanism 17300 Problems 17301 Chapter 18  Basics of Motion Control Profiles The Motion Control System 18303 Motion Control Defined 18304 The Mathematical Approach 18304 Acceleration, Velocity and Position are Not Independent of one Another 18306 The Geometric Approach 18306 Problems 18312 Chapter 19  The PLC and Open Loop Motion Control The PLC As a Control Device 19314 The PLC and the Motion Profile 19314 Slow Down and Creep to Brake Point 19316 Other Factors Affecting Open Loop Positioning Accuracy 19318 Problems 19321 APPENDIX A  Math Background A322 Table of Contents APPENDIX B  Second Order Responses B345 INDEX I356 
