fluid pfluid power & control (a series of textbooks & reference books)ower & control (a series of...

Marcel Dekker, Inc. New York BaselTM

INDUSTRIAL SERVO CONTROL SYSTEMS

Fundamentals andApplications

GEORGE W. YOUNKINIndustrial Controls Consulting, Inc.

Fond du Lac, Wisconsin, U.S.A.

Second Edition, Revised and Expanded

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved

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FLUID POWER AND CONTROL

A Series of Textbooks and Reference Books

FOUNDING EDITOR

Frank YeaplePresident

TEF EngineeringAllendale, New Jersey

1. Hydraulic Pumps and Motors: Selection and Application for Hydraulic PowerControl Systems, Raymond P. Lambeck

2. Designing Pneumatic Control Circuits: Efficient Techniques for PracticalApplication, Bruce E. McCord

3. Fluid Power Troubleshooting, Anton H. Hehn4. Hydraulic Valves and Controls: Selection and Application, John J. Pippenger5. Fluid Power Design Handbook, Frank Yeaple6. Industrial Pneumatic Control, Z. J. Lansky and Lawrence F. Schrader, Jr.7. Controlling Electrohydraulic Systems, Wayne Anderson8. Noise Control of Hydraulic Machinery, Stan Skaistis9. Interfacing Microprocessors in Hydraulic Systems, Alan Kleman

10. Fluid Power Design Handbook: Second Edition, Revised and Expanded, FrankYeaple

11. Fluid Power Troubleshooting: Second Edition, Revised and Expanded, Anton H.Hehn

12. Fluid Power Design Handbook: Third Edition, Revised and Expanded, FrankYeaple

13. Industrial Servo Control Systems: Fundamentals and Applications, George W.Younkin

14. Fluid Power Maintenance Basics and Troubleshooting, Richard J. Mitchell andJohn J. Pippenger

ADDITIONAL VOLUMES IN PREPARATION


Preface

In this second edition of Industrial Servo Control Systems, the chapters havebeen updated and expanded and a new chapter has been added on servocompensating techniques. The book continues to be dedicated to thepracticing engineer making the transition from academic theory to the real-world solution of engineering problems related to the application of servodrives to industrial machines.

Part I focuses on the evolution and classication of servos, withdescriptions of servo drive actuators, ampliers, feedback transducers,performance, and troubleshooting techniques. Part II discusses systemanalogs and vectors, with a review of differential equations. The concept oftransfer functions for the representation of differential equations isdiscussed in preparation for block diagram concepts. Discussion of themathematical equations for electric servo motors has been expanded toinclude both DC and brushless DC servo motors. The equations formechanical and electrical time constants are derived with additional analysison the effects of temperature on these time constants.

The representation of servo drive components by their transferfunction is followed by the combination of these servo drive buildingblocks into system block diagrams. Frequency response techniques areintroduced because of their usefulness in determining proper servocompensation. Also included are practical formulas for calculating inertiaand examples that show how machine components reect inertia to the


servo drive motor. Torque-to-inertia ratios for electric servomotors arediscussed along with calculations of the reected inertias for variousmachine congurations. These reected inertias are critical when sizing thedrive and motor and when determining time constants for a stabilityanalysis. Servo drive analysis and servo compensation techniques includethe use of lag and lead compensation for analog-type servo drive ampliers.This discussion is followed by a description of proportional, integral, anddifferential compensation, which is very popular with digital-type ampliers.

To classify practical servo performance criteria, the use of servoindexes of performance is discussed for both electrical and hydraulic servodrives. The chapter on servo plant compensation techniques addresses thecontinuing problems of structural dynamics and resonances in industrialmachinery. This type of compensation is different from feedback controlcompensation in that it uses special electrical devices and softwarealgorithms to correct for machine nonlinearities and mechanical resonanceproblems. Nonlinearities of dead zone and change-in-gain to overcome suchmachine problems as stiction and cogging of the servo drive at low velocitiesare covered. The topic of simple and practical notch networks used tocompensate for machine mechanical resonances has been expanded toinclude frequency selective feedback with suggestions for softwareimplementation. This edition also includes a discussion on the use of acontrol technique known as feedforward control to minimize the positionerrors of machine axes with widely varying dynamics. Chapters on servodrive stiffness and drive resolution are an integral part of the analysis ofindustrial servo drives. Coverage of speed and acceleration has beenbroadened to include the derivation and application of S typeacceleration. Machine considerations for ball screw resonances have beenexpanded to include ball screw critical speed, axial, and torsional resonancecalculations, and drive ratio considerations have been expanded to includeworm/wheel gear boxes. Friction considerations include the three types offriction and their relation to servo drives on industrial machines.

One important aspect of selecting a servo for a given application issizing the drive, because the servo must be large enough to meet the loadrequirements. Manual drive sizing forms for both electric and hydraulicservo drives are included in this book. The electric drive sizing forms havebeen expanded to include both DC and brushless DC drives.

Chapter 14 is entirely new and reviews the process of compensating anelectric servo drive. It is assumed that this industrial servo drive has abrushless DC motor with a current loop. An example is given for the case inwhich the motor and current loop are closed with a position loop, and alsowith a velocity and a position loop. The application of proportional plus


integral compensation is used throughout Chapter 14 and detailed in theinclusion of machine structural resonances in the position servo loop.

Once a servo drive has been properly sized for a given machine, it isnecessary to determine whether the machine and servo drive will be stable.There are many classical engineering methods that can be used, but it is alsopossible to use high-speed computer iterative techniques to simulate thecontrolservo drivemachine combination. Transient step responses in inputare used in this simulation since it is readily possible to duplicate them on anactual machine and compare the results. Machine simulation is used topredict performance before the machine is built. Machine simulationincludes the signicant parameters of the servo drive: the nonlinearities offeedback; coulomb, static, and viscous friction; machine structuralresonance; and machine mass. A number of examples are presented withcase histories for comparing simulated response with actual response.

My contemporaries encouraged me to pull together a lifetime of servodrive experience and write the rst edition of this book. The second editionupdates, expands, and increases the usefulness of this book to the practicingengineer.

I am sincerely grateful to Dr. Thomas Higgins for providing me, as astudent, with the academic tools needed in the eld of feedback control. Iam also grateful to Dr. John Bollinger and Dr. Robert Lorenz from theUniversity of Wisconsin, for their help over the last 45 years in researching,teaching, and applying principles that have been critical in improving theperformance of servo drives. Tom Rehm, my friend of 40 years and a fellowsoftware hacker, has been a great help in software technology. Inaddition, I would like to thank the Giddings and Lewis Machine ToolCompany for providing the atmosphere for study and research in the eld ofindustrial servo drives, and my family for their long years of patience as mycareer progressed.

George W. Younkin


Contents

Preface

I BASICS OF INDUSTRIAL SERVO DRIVES

1 The What and Why of a Machine ServoJust What Are Some of the Benets of a Servo System?

2 Types of Servos2.1 Evolution of Servo Drives2.2 Classication of Drives

3 Components of Servos3.1 Hydraulic/Electric Circuit Equations3.2 ActuatorsElectric3.3 ActuatorsHydraulic3.4 AmpliersElectric3.5 AmpliersHydraulic3.6 Transducers (Feedback)

4 Machine Servo Drives4.1 Types of Drives4.2 Feed Drive Performance


5 Troubleshooting Techniques5.1 Techniques by Drive5.2 Problems: Their Causes and Cures

6 Conclusion: Machine Feed DrivesAn Integral Part of aMachine Control System6.1 Advances in Technology6.2 Parameters for Making Application Choices

II ADVANCED APPLICATION OF INDUSTRIALSERVO DRIVES

7 Background7.1 Introduction7.2 Physical System Analogs, Quantities, and Vectors7.3 Differential Equations for Physical Systems7.4 Electric Servo Motor Transfer Functions and Time

Constants7.5 Transport Lag Transfer Function7.6 Servo Valve Transfer Function7.7 Hydraulic Servo Motor Characteristics7.8 General Transfer Characteristics

8 Generalized Control Theory8.1 Servo Block Diagrams8.2 Frequency-Response Characteristics and Construction

of Approximate (Bode) Frequency Charts8.3 Nichols Charts8.4 Servo Analysis Techniques8.5 Servo Compensation

9 Indexes of Performance9.1 Denition of Indexes of Performance for Servo Drives9.2 Indexes of Performance for Electric and Hydraulic

DrivesSummary

10 Performance Criteria10.1 Percent Regulation10.2 Servo System Responses


11 Servo Plant Compensation Techniques11.1 Dead-Zone Nonlinearity11.2 Change-in-Gain Nonlinearity11.3 Structural Resonances11.4 Frequency Selective Feedback11.5 Feedforward Control

12 Machine Considerations12.1 Machine Feedback Drive Considerations12.2 Ball Screw Mechanical Resonances and Reected

Inertias for Machine Drives12.3 Drive Stiffness12.4 Drive Resolution12.5 Drive Acceleration12.6 Drive Speed Considerations12.7 Drive Ratio Considerations12.8 Drive Thrust/Torque and Friction Considerations12.9 Drive Duty Cycles

13 Drive Sizing Considerations13.1 Introduction13.2 Hydraulic Drive Sizing13.3 Electric Drive Sizing

14 Adjusting Servo Drive Compensation14.1 Motor and Current Loop14.2 Motor/Current Loop and Position Loop14.3 Motor/Current Loop with a Velocity Loop14.4 PI Compensation14.5 Position Servo Loop Compensation

15 Machine Simulation15.1 Introduction15.2 Simulation Fundamentals15.3 Machine Simulation Techniques to Predict

Performance15.4 Other Simulation Software

16 Conclusion

GlossaryKey Constants and Variables

Contents


Appendix: Hydraulic Drive and Electric Drive-Sizing FormsBibliography


IBASICS OF INDUSTRIALSERVO DRIVES


1The What and Why of aMachine Servo

The control user should be familiar with servos. The user will understandwhat a servo is and why it is required in so many applications. Thisdiscussion will answer these two basic questions: What is a servo? Why use aservo?

Any discussion of servos will have to employ the term feedback.Thousands of times every day we require information to be fed back to usso that we can perform normal activities. When controlling a car down ahighway, feedback is provided to our brain by the gift of sight. Howterrifying it would be if we were traveling at 70mph and we lost the ability tosee. Our brain, which is the center of our control system, would have littlefeedback to help it decide what corrective actions need to be taken tomaintain a proper path. The poorer feedback channels still available wouldbe the senses of hearing and touch, which would allow us to ride theshoulder. The result would be a lower speed, poorer control, a very irregularpath, and a greater chance for an accident. Inferior feedback on a machineblinds the operator or the control just as it does a driver. When usingnumerical control and servos, poor feedback can result in inferior parts,poor productivity, and high costs. Essentially, feedback is the retrieval ofinformation about the process being controlled. It veries that the machineis doing as commanded. There are two types of feedbacknegative and


positive. Positive feedback, used for instance in radios, is not discussed here.Negative feedback, required to make a servo work properly, subtracts fromcommands given to the servo so that a discrepancy or error between outputand input can be detected. This discrepancy initiates an action that willcause that discrepancy to approach zero. A perfect example of a negativefeedback system is a wall thermostat and furnace, as depicted in Figure 1. If,in a 658F room, we set the thermostat for 728F, then 728F can be consideredthe command. The 658F of the room feeds back, subtracts from the 728Fcommand, and results in a 78F discrepancy or error that instructs thefurnace to supply heat. The furnace supplies heat until the negative feedbackis sufcient to cancel the command so that no discrepancy exists and nofurther heat is required.

A servo or servomechanism is a system that works on the negativefeedback principle to induce an action to cause the output to be slaved to theinput. Our thermostat/furnace example was one of a servo which inducedthe generation of heat.

Any servo has two basic elements. These are a summing network andan amplier. The summing network, as shown in Figure 2, is simply a devicethat sums the negative feedback (F) with the command (C) to generate anerror discrepancy (E). Our drivers summing network (Figure 3) was hisbrain. The command would be to keep the car in the right-hand lane. If hewas straddling the center line, the feedback through his eyes to his mindwould so indicate. If he subtracts this feedback information from the

Fig. 1 Thermostat example.


Our driver (Figure 6) mentally computed an error (E) requiring thatthe car be moved 2 ft to the right. This instruction traveled from his brainthrough nerves to the muscles in his arms and hands, which turned thesteering wheel and with the muscle in his power steering caused the car tomove into the right-hand lane. In the machine feed axis example (Figure 7)the error of 1 in would be in the form of a small voltage, which wouldcause the servo motor to turn, and axis positioning motion would result. Ifwe connect the two elements together, as shown in Figure 8, a basic closed-loop servo system results. A new command will generate an error (E), whichwill activate the muscle until sufcient movement has caused the feedback(F) to be coincident with the command (C), at which point the error (E) iszero and motion is no longer instructed. The term closed loop suggeststhat after entering a command, signals traveled around the loop untilequilibrium is attained.

Fig. 4 Summing network for positioning.

Fig. 5 Amplier.


JUST WHAT ARE SOME OF THE BENEFITS OF ASERVO SYSTEM?

Here are six benets of a machine axis feed servo drive.

1. Shorter positioning time: The servo operates at maximumpositioning rate until the ideal time to decelerate, at which pointit slows down uniformly to the end point with no hesitation atintermediate feeds. Since it dynamically searches for zero error,variations in machine conditions are compensated for. Positioningtime is thus minimized.

2. Higher accuracy: A servo continually homes to the nal positionso that on January mornings it will continually strive to push theaxis toward the end point, whereas on the Fourth of July when themachine might have a tendency to overshoot, the servo error willreverse and force the machine back into position.

3. Better reliability: An outstanding feature of servos is the ability tocontrol acceleration and deceleration so that the mechanicalhardware will hold its specication tolerance much longer.

4. Improved repeatability: Repetitive moves to a particular com-manded point will show much better consistency. The result ismore consistency of parts that are intended for interchangeability.

5. Coordinated movements: Since all axes are closed-loop servos, theyare continually responding to the command at all feed rates.Coordinated movements thus require the generation of coordi-nated commands through employing interpolators with thecontrol.

6. Servo clamping: There is no longer a dependency on mechanicalclamps for servoed axes because of the continuous position-

Fig. 10 Closed-loop position servo.


holding capability of a servo. The stiffness of the servo must berelied on for any contouring movements requiring that both axesbe in motion. Properly designed, the servo can also hold the axisvery stify at a standstill.


10

Performance Criteria

10.1 PERCENT REGULATION

There are a number of ways to dene performance, such as servobandwidth, accuracy, regulation, etc. Percent regulation, for machine servodrives, can be dened as how much the controlled variable of a servo willreduce as load is applied to the servo. As such, percent regulation is usuallyapplied to type 0 velocity regulators to describe how well the output velocitywill be maintained as load is applied to the output of the drive. Percentregulation is often given as the equation

Percent regulation no load speed full load speedfull load speed

A general block diagram for a DC servo drive with a position loop is shownin Figure 1. Using block diagram algebra the servo drive block diagram canbe redrawn for speed as the controlled variable with the load torque as aninput, shown in Figure 2.

Equations (10.1-1) to (10.1-11) show the calculations for the regulationof a DC drive. Equations (10.1-12) to (10.1-14) rearrange Eq. (10.1-11) forvelocity-loop stiffness. Eq. (10.1-15) expresses the velocity stiffness.

The position-loop equations are shown in Eq. (10.1-16) to (10.1-27)for the position-loop stiffness.


Vm

A KAK1T1s 1KeT2s 1TMTes2 TMs 1 KAK1T1s 1KTA

(10.1-4)

s ? 0Vm

A

s?0 KAK1Ke KAK1KTA (10.1-5)

Kv K2VmA

S?0 (10.1-6a) Kfb K2KAK1KfbKe KAK1KTA (10.1-6b)

Vm

TL 1Js6

1

1 KT KeK1KTAKAJsRa

LR s1

T1s1T2s1

(10.1-7)

Vm

TL 1Js KT

RaLR s1 6 Ke K1KTAKA

T1s1T2s1

h i (10.1-8)

s ?0Vm

TL 1

KTRa6Ke K1KTAKA

RaKT Ke K1KTAKAKT (10.1-9)

Velocity-Loop Regulation

For these equations, units are:

Ra, ohmKT , in.-lb/AKe, V/rad/secK1, V/VKTA, V/rad/secKA, V/V

Vm

TL RaKTKe K1KTAKAKT

rad=sec

in:-lb

(10.1-10)

Vm

Ti RaKTKe KVOKeKT

Vm

TL RaKTKe1 KVO (10.1-11)


Velocity-Loop Stiffness

TL

Vm KTKe K1KTAKAKT

Ra(10.1-12)

KVO K1KTAKAKe

open-loop gain (v=v) (10.1-13)KeKVO K1KTAKA (10.1-14)TL

Vm KT Ke KVOKe

Ra KTKe

Ra1 KVO in.-lb

rad/sec(10.1-15)

Position Loop

Vm

B 1Js KT

RaLs=R16 Ke K1KTAKAT1s1T2s1

h i (10.1-16)

Vm

B RaLs=R 1JsLs=R 1 KT Ke K1KTAKA T1s1T2s1

h i (10.1-17)

Vm

B RaLs=R 1T2s 1JsLs=R 1T2s 1 KTKeT2s 1 KT Ke K1KTAKAT1s 1

(10.1-18)

yTL

RaLs=R 1T2s 1s * 6

1

1 RaLs=R1T2s1s * 6

K1K2KAKTKfbT1s1Ls=r1T2s1

(10.1-19)

yTL

RaLs=R 1T2s 1s * RaLs=R 1T2s 1 K1K2KAKTKfbT1s1Ls=R1T2s1Ra

(10.1-20)

s?0 * denominator of10.1-18yTL

RaK1K2KAKTKfb

(10.1-21)

KV K1K2KAKTKfbC K1KTAKA (10.1-22)


KVO K1KTAKA6 KTLs=R 1RaJTs KeKT (10.1-23)

s?0

KVO K1KTAKAKe

(10.1-24)

KVO K1KTAKAKfbKe KVOKe (10.1-25)

yTL

RaK1K2KAKTKfbKT Ra

KvKTKe1 KVO (10.1-26)

Stiffness

Variables for this equation have the following units:

Ke, V/rad/secKT , in.-lb/AKv, 1/secKVO, v/vRa, ohms

TL

y KvKTKe1 KVO

Ra(10.1-27)

10.2 SERVO SYSTEM RESPONSES

In the analysis of different types of servos they are often modeled in theirsimplest form, which would include only the frequency characteristics belowthe servo bandwidth. This can be done for the type 0 and type 1 servos, asdiscussed in this section. The frequency characteristics (dynamics) above theservo bandwidth could introduce some errors in the analysis, but pastexperience demonstrates that the major contribution to system performanceoccurs in the frequency characteristics below the servo bandwidth. There-fore, assuming a type 1 positioning servo drive can be modeled as shown inFigure 3, the output response can be predicted depending on what type ofinput (velocity step, position step, velocity ramp, or position ramp) will be


applied to the model. For each input in Figure 3 the output response forvelocity, position, acceleration, and errors is shown. The models for the type1 servo of Figure 3 are often referred to as a naked servo, meaning there areno frequency characteristics below the bandwidth velocity constant Kv.

A type 0 velocity regulator servo drive can be modeled as shown inFigure 4. The rst model is simplied by not including any of the systemdynamics above the servo bandwidth. The second model is morecharacteristic of velocity servo drives using proportional and integral servocomposition. These response characteristics are more representative ofcommercially available velocity servo drives.

Fig. 4 (a) Velocity response for a velocity step input. (b) Velocity response for avelocity step input.


11

Servo Plant CompensationTechniques

Servo compensation usually implies that some type of lter network such aslead/lag circuits or proportional, integral, or differential (PID) algorithmswill be used to stabilize the servo drive. However, there are other types ofcompensation that can be used external to the servo drive to compensate forother things in the servo plant (machine) that can, for example, be structuralresonances or nonlinearities such as lost motion or stiction. These machinecompensation techniques are shown in Figure 1 and are valid for eitherhydraulic or electric servo drives.

11.1 DEAD-ZONE NONLINEARITY

Stiction, sometimes referred to as stick-slip, occurring inside a positioningservo, can result in a servo drive that will null hunt. The denition of a nullhunt is an unstable position loop that has a very low periodic frequency suchas 1Hz or less with a small (a few thousandths) peak-to-peak amplitude(limit cycle). The most successful way to avoid stiction problems is to useantifriction machine way (rollers or hydrostatics) or use a way linearmaterial that has minimal stiction properties. If stiction-free machine slideways cannot be provided, the use of a small dead-zone nonlinearity placedinside the position loop, preferably at the input to the velocity servo, has


in a turning machine application. At feed rates below 0.01 ipm, therequirement for a smooth surface may not be easily attainable because theservo drive may have a cogging problem at these low federates. Increasingthe forward loop gain to the velocity drive can overcome the lowfeedcogging problem but will result in an unstable servo drive. As a compromise,a change in gain nonlinear circuit can be used to improve the low-feed-ratesmoothness and still have a stable servo drive. The object is to have a highforward-loop gain in the velocity servo (which is inside a position loop). Fornormal operation, the high servo loop gain is reduced by the change-in-gaincircuit at a low velocity to its normal gain, thus maintaining a stable servodrive. This type of nonlinear circuit has been used successfully for smooth

Fig. 2 Dead-zone nonlinearity.


feed rates down to 0.001 ipm. The analog version of a change-in-gainnonlinearity is shown in Figure 3. With digital controls a digital algorithmcan be used.

11.3 STRUCTURAL RESONANCES

Structural resonances or machine dynamics, as it often referred to, iscertainly not a new subject. However, on the morning of November 7, 1940,the nation awoke to the destruction of the Tacoma Narrows Bridge. A42 mile-per-hour gale caused the bridge to oscillate thus exciting thestructural resonances of the bridge to a nal destruction frequency of about14Hz and a peak-to-peak amplitude of 28 ft. The destruction of the bridgewas a wake-up call to the importance of dynamic analysis in structural

Fig. 3 Change-in-gain nonlinearity.


design in addition to static analysis and design. Some sixty years later thetechnology of dynamic analysis is now well known.

To further investigate machine resonances, a typical linear industrialservo drive can be represented as in Figure 4. The mechanical componentsof this servo drive are referred to as the servo system plant. The servo plantmay have a multiplicity of resonant frequencies resulting from a number ofdegrees of freedom. In actual practice there will be some resonantfrequencies that are high in frequency and far enough above the servodrive bandwidth so that they can be ignored. In general there will be apredominant low resonant frequency that could possibly be close enough tothe servo drive bandwidth to cause a stability problem. Therefore a singledegree of freedom model as shown in Figure 5 can represent thepredominant low-resonant frequency, where:

BL viscous friction coefcient (lb-in.-min/rad)T driving torque, developed by the servo motor (lb-in.)Kmechanical stiffness of the spring mass model (lb-in./rad)JM inertia of the motor (lb-in.-sec2)JL inertia of the load (lb-in.-sec2)S laplace operator

From Newtons second law of motion, the classical equations for this servoplant (industrial machine system) can be written. In most industrialmachines it can be assumed that the damping BL is zero. Therefore the

Fig. 4 Block diagram of a machine feed drive.


Further solving for yM and yL by combining Eq. (11.3-4) and (11.3-6):

yM TM KKyM=JLs2 K

JMs2 K (11.3-7)

yM TMJLs2 K K2yM

JMs2 KJLs2 K (11.3-8)

yM TMJLs2 K

JMs2 KJLs2 K K2yM

JMs2 KJLs2 K (11.3-9)

yM K2yM

JMs2 KJLs2 K TMJLs2 K

JMs2 KJLs2 K (11.3-10)

JMs2 KJLs2 K K2yM TMJLs2 K (11.3-11)

yM TMJLs2 K

JMs2 KJLs2 K K2 (11.3-12)

yM TMJLs2 K

JMJLs4 KJM JLs2 (11.3-13)

yM TMJLs2 K

s2JMJLs2 JM JLK (11.3-14)

yM TMJLs2 K

s2JM JLJMJL=JM JLs2 K (11.3-15)

Let:

J JM JLJP JMJL=JM JL

yM TMJLs2 K

s2Js2JP 1 (11.3-16)

Also:

yL KyMJLs2 K (11.3-17)


yL KJLs2 K6TMJLs2 Ks2Js2JP K (11.3-18)

yL TMKs2Js2Jp K (11.3-19)

yLTM

1J=Ks2s2Jp=K 1 (11.3-20)yLTM

1J=Ks2s2=o2r 1(11.3-21)

or load resonant frequency

K

Jp

s

KJM JLJMJL

s

(11.3-22)

From a practical point of view industrial machines and their servo drives(hydraulic and electric) are to this day still subject to resonant frequencystability problems. Most industrial servo drives use an inner velocity servoinside a position servo loop. Hydraulic servo drives have the added variableof hydraulic uid resonance, which can be a limiting factor of stability. Thehydraulic resonance or can be observed as a typical second order responsein the Bode frequency response of Figure 6. For hydraulic drives having alow damping factor dh, the resonant peak may be higher than 0 dB gain,which will result in a resonant oscillation. There are a number of methods tocompensate for this resonant oscillation. First, a small cross-port dampinghole of about 0.002 in. can be used across the motor ports. Secondly, thevelocity loop differential compensation can be varied, which quite ofteneliminates the oscillation. Lastly, the velocity loop gain could be lowered,which can also lower the velocity servo bandwidth. As an index ofperformance (I.P.) the hydraulic resonance should by proper sizing be above200Hz, and the separation between the velocity servo loop bandwidth ocand the hydraulic resonance oh should be three to one or greater. BrushlessDC electric drives do not usually have velocity loop resonance problemsunless a more compliant coupling is used internally in the motor to couple aposition transducer to the motor shaft.

Both hydraulic and brushless DC electric drives can have resonance(stability) problems if the machine is included in the position servo loop.This is an ongoing problem with industrial machines, in spite of all theavailable technology to minimize stability problems. A typical positionservo Bode frequency response is shown in Figure 7. As a gure of merit theseparation between the velocity loop bandwidth oc and the position-loopvelocity constant Kv (gain) should be two to one or greater. The machine


show the change in the depth of the notch lter as the potentiometer isvaried.

A few case histories are of interest. In a hydraulic servo valve feeddrive, pump pulsations of 500Hz traveled through the machine piping to theservo valve, the hydraulic motor, and nally the feedback tachometer of thevelocity loop. The high sensitivity of the tachometer (100V/1000 rpm)sensed the 500-Hz vibration and generated this voltage into the servo driveelectronics, where it was amplied through the entire drive, causing anundesirable vibration. A 500-Hz notch lter at the tachometer output feedto the servo amplier eliminated the vibration problem.

In another case, the switching frequency of a numerical control(125Hz) beat with the single-phase, full-wave DC SCR drive frequency(120Hz) producing a 5-Hz signal that appeared in the machine servo drive,causing what appeared to be a 5-Hz instability. Using a notch lterfrequency of the control switching frequency, the 5-Hz beat frequency signalwas eliminated.

In another case history a 45-Hz resonance existed in an air bearing of arotary position feedback transducer. Once this resonance was excited it wasamplied through the electronics drive and the machine slide vibrated at45Hz.

A 45-Hz notch lter applied at the input to the velocity driveeliminated the problem. There are numerous possibilities where unwanted

Fig. 8 Notch lter circuit. (Reprinted with permission from Penton Publishing.)


Fig. 9 Notch lter resistance nomogram.


Fig. 10 Notch lter characteristics.

Fig. 11 Depth of notch lter characteristics.


signal frequencies appear in industrial servo drives. This simple notch ltercan readily be used to eliminate all kinds of undesirable signals.

11.4 FREQUENCY SELECTIVE FEEDBACK

Another technique that has been very successful with industrial machineshaving low-frequency machine resonances, is known as frequency selectivefeedback. In abbreviated form it requires that the position feedback belocated at the servo motor eliminating the mechanical resonance from theposition servo loop, resulting in a stable servo drive but with signicantposition errors. These position errors are compensated for by measuring theslide position through a low-pass lter; taking the position differencebetween the servo motor position and the machine slide position; andmaking a correction to the position loop, which is primarily closed at theservo motor.

Compensator Operation

The complete control circuit has more complexity than comparing twoposition transducer outputs. Figure 12 can be used to discuss the actualoperation of the compensating system. This particular compensating schemeused an instrument servo to perform the compensating function. A softwarebased frequency selective feedback system will also be discussed. For thiscompensating system, the machine-feed servo drive uses a positionmeasuring feedback resolver (1) connected electrically in series with acorrection resolver (5). Any correction required during positioning isintroduced into the numerical control feedback circuit with the correctionresolver (5).

The compensator circuit includes the positioning servo-motor positionmeasuring resolver called a compensator feedback resolver (2), a machineslide linear position measuring transducer (3), and an instrument typecorrection servo drive. The difference between the feed servo-drive motorposition and the machine slide position is measured with the compensatorfeedback resolver (2) and the linear resolver (3). However, an additionalcorrection resolver (4) is included in the circuit. Therefore, the instrumentcorrection servo error is a function of three resolver positions. This is shownin the block diagram of Figure 13. The resolver positions are shown as angley. The total correction error is shown as a function of the three resolverpositions

yc ym yc ys: (11.4-1)


As a position error is developed between the feed servo-drive motorposition ym and machine slide position ys, an error is developed at theinstrument servo drive (Figure 13). The error is also a function of yc.

ye ym yc ys ym ys yc (11.4-2)

It is signicant to note that the bandwidth of the instrument correctionservo must be a low frequency such as 1.5Hz. This is required to eliminatethe machine structural dynamics from the machine slide feedback positionys and is the key to the frequency selective feedback control function.

A correction yc is developed at the output of the instrument servodrive, which is a function of:

yc G1ym ys yc (11.4-3)

The correction yc is added to the main-feed servo drive by means of resolver(5). The correction yc causes the feed servo drive to move by the amount ofthe correction. Therefore both ym (at the motor) and ys (at the machine slide)move by the amount of the correction yc. The correction can be shownmathematically to be approximately a function of ym ys as follows:

Since: yc Glym ys yc from Eq: (11.4-3)yc ycGl ym ysGl (11.4-4)

yc ym ysG11 G1 (11.4-5)

yc ym yc 11=G1 1 (11.4-6)

1=G15much smaller than 1

Therefore: yc%ym ys (11.4-7)

The correction can also be shown graphically in Figure 14. Forillustration the relation of the feed servo-drive position ym to the machineslide position ys will remain constant during the correction. Figure 14 showsthat the machine slide position ys moves by the amount of the correction.The feed servo-drive motor position moves by the same amount.

From another point of view the instrument servo-drive error must bereduced to zero after the correction is made. From Eq. (11.4-2) the error is:

ye ym ys yc (11.4-2)

Assuming the initial condition at the start of the correction is such that the


Meanwhile the correction yc (which was equivalent to 0.01 in.) had tobe included in the error of the instrument servo. Since the relation of ym ys remains constant, a correction must be made in the instrument servoloop to reduce the error to zero:

yc 0 ym ys yc (11.4-8)The compensator correction resolver (4) serves the purpose to reduce theinstrument servo error to zero.

As the correction process becomes a continual process, as in normalmachine operation, the resolver (4) will continually be in motion to reducethe error to zero in the correction loop of the instrument servo. At machinetraverse feeds the correction will not be effective, but this is not importantsince there will not be any machining operations at the traverse feeds.

Software version of frequency selective feedback

The software version for the correction is shown in Figure 15. This versioncan be added to a machine axis that exhibits a structural resonance problem.The drive is assumed to be a typical commercial electric servo drive with acurrent loop and a velocity loop inside a position servo loop.

The difference between the motor position and the machine slideposition is used as an input to a low-pass digital lter. This lter has a verylow bandwidth of about 1.5Hz. The reason for this is to remove structuraldynamic frequencies from the correction process.

Fig. 15 Software version of frequency selective feedback.


The machine axis prior to adding frequency selective feedback, isassumed to have a position transducer on the machine slide. To addfrequency selective feedback, an instrument gearbox must be added to therear of the servo motor. A position transducer such as a resolver will begeared to the motor shaft with a ratio determined by the resolution of themachine slide feedback transducer.

11.5 FEEDFORWARD CONTROL

The problem of developing an industrial servo drive with high-performancecapabilities for accurate positioning is a subject of much importance. Onmultiaxis industrial machine servos using classical type 1 servo control, it isa requirement that each machine axis have matched position-loop gains tomaintain accuracy in positioning. Quite often this means that all machineaxis servo drives must have their position-loop gains Kv adjusted to thepoorest performing axis. Consider the basic approach to the design of apoisoning servo drive illustrated in Figure 16.

This is the classical type 1 servo, which exhibits characteristic errors ein position that are well known for various inputs yi.

Consider a simplied block diagram of the system where Gs is theservo drive and inner-loop transfer function typically of the following form:

Gs KvsFs

(11.5-1)

F s is a polynomial that represents the dynamics of the servo drive andservo plant. What is really desired of the servo of Figure 17 is that yi y0under all conditions. That is, for any yi; e 0. Clearly, this is not possiblefor a type 1 servo described by Figure 17.

Fig. 16 Type 1 servo block diagram.


Fig. 20 Zero-error feedforward block diagram.

Fig. 18 Compensator with position servo.

Fig. 19 Block diagram with feedforward.


Because the inverse of a type 1 system differentiates rather thanintegrates the compensating path is simply a velocity signal rather thana position signal. Consider the nal design of the system described inFigure 22.

From Figure 22

yos yis yos oisKv

Kv

sFs(11.5-7)

yos 1 KvsFs

yis oisKv

Kv

sFs(11.5-8)

yos Kvyis oissFs Kv

(11.5-9)

Fig. 21 Complete feedforward block diagram.

Fig. 22 Zero-error position servo.


Equation (11.5-9) may be used to illustrate how in the ideal case (whereF s 1) the system is zero error in position regardless of the input.

yi Ct therefore yis Cs2

(11.5-10)

oi C ois Cs

(11.5-11)

Therefore

yos KvC=s2 C=s

sFs Kv s KvCs2sFs Kv

(11.5-12)

Or for F s&1, the lag-lead cancel and

yos Cs2

yis therefore zero error: (11.5-13)

Consider next the steady-state error for the constant velocity case:

es yis yos Cs2Cs2

s KvsFs Kv

(11.5-14)

es Cs2

1 s KvsFs Kv

(11.5-15)

Applying the nal value theorem, the steady-state position error is:

e? lim ses lims?0

Cs

1 s KvsFs Kv

(11.5-16)

limCs?0

1 s KvsFs Kv

1

s(11.5-17)

limCs?0

sFs Kv s KvsFs Kv

1

s(11.5-18)

limCs?0

Fs 1sFs Kv

(11.5-19)

However, for all possible forms of F s

lim

s?0F s 1

Thus: e? 0


For constant acceleration input:

yit 12at2 yis a

s3(11.5-20)

oit at ois as2

(11.5-21)

From Eq. 11.5-9

yis Kva=s3 a=s2sFs Kv

as3

s KvsFs Kv

(11.5-22)

And e? limsess?0

as2

sFs Kv s KvsFs Kv

(11.5-23)

e? lims?0

as

Fs 1sFs Kv

(11.5-24)

e? alims?0

Fs 1s2Fs Kvs

alims?0

dFs=dsds2Fs=ds Kv

(11.5-25)

Thenlim

s?0dFsds

?finite value (11.5-26)

Whilelimit

s?0ds2Fsds

?0 (11.5-27)

Thus e? 1Kv

(11.5-28)

The magnitude of the nite position error will depend in this case on thecoefcient of the s term of F(s). If b is this coefcient, then

e? abKv

(11.5-29)

The compensator technique described holds excellent potential for provid-ing an outstanding servo drive for the industrial machine. When properlyexecuted in a well-designed position servo it should virtually eliminatevelocity lag errors and reduce acceleration lag errors to low levels. Thevelocity feedforward approach will eliminate position errors for constantvelocity moves on a machine axis if the machine dynamics represented byF s in the feedforward term F s=Kv exactly respresents the G(s) term of theforward position loop. Matching of position-loop gains Kv on all axes willnot be required. For the case of acceleration and deceleration the velocityfeedforward approach will not be effective. For these situations commercialservo-drive manufacturers will use an additional technique of accelerationfeedforward.


12

Machine Considerations

12.1 MACHINE FEEDBACK DRIVE CONSIDERATIONS

Machine Feed Drive Considerations for ResolverFeedback

The usual conguration for resolver position feedback is to have the resolvergeared to the servo drive motor. Computer-aided design programs forhydraulic drive sizing are based on the fact that the hydraulic resonance isthe predominant resonance in the servo loop. With electric drives themechanical time constant is not of prime consideration since it can becompensated for with the drive compensation. These are two importantdynamic considerations with electric and hydraulic servo drives usingresolver feedback at the servo motor. Since the position feedback resolver isclose-coupled to the drive servo motor, all other mechanical resonances andnonlinearities are excluded from the position servo loop. Recommendationsfor the required indexes of performance (I.P.) are identical with those givenin Section 9.2.


Machine Feed Drive Considerations for Direct MachineSlide Position Feedback

The main reason that direct feedback could be used was the universal use ofthe soft servo with its low position-loop gain. By denition a soft servohas a low position-loop gain of about 1 ipm/mil with no frequency breaksbelow the servo bandwidth. The range of gains for the soft servo rangefrom 0.5 ipm/mil to 2 ipm/mil. As long as no resonances are within six timesthe high end of this range (200 rad/sec or 32Hz, Eq. [9.2-27]), it should bepossible to close the position loop and be stable.

As previously stated, if the feedback position transducer (resolver) islocated at the drive motor, any mechanical resonance in the machine will beoutside the servo loop. The effects of these mechanical resonances on theclosed-position loop will be reected load disturbances.

However, if direct (e.g., Inductosyn-linear scales) position feedback isto be used, the mechanical resonance in the mechanical drive componentswill be inside the position loop. For this situation, some guidelines,recommendations, etc. should be put forth to prevent a situation wheredirect feedback is used on a feed drive that will not be stable.

12.2 BALL SCREWMECHANICAL RESONANCES ANDREFLECTED INERTIAS FOR MACHINE DRIVES

As a constraint the lower limit on any resonance (hydraulic or mechanical)inside the velocity loop should not be less than 200 rad/sec (Eq. [9.2-27]).Likewise, the bandwith oc of the closed velocity loop should not begreater than one-third the lowest resonance in the servo loop, which isusually the hydraulic resonance, oh, in hydraulic drives (Eq. [9.2-23]).

With the position loop, there may be a number of different mechanicalresonances. These resonances are outside the velocity loop but inside theposition loop. Considering the mechanical feed drive components, the oneof greatest concern is the drive screw axial resonance. This resonance willprobably be the predominant low mechanical resonance in a ball-screw feeddrive.

Typical values of ball screw stiffness are shown in the graph of Figure 1for the variables of screw diameter and length. These values are based onconstant end bearing and ball nut stiffnesses. In actual practice, these wouldbe varied according to the screw diameter. In Figures 2 to 5, the ball screwresonances are plotted for various screw diameters, lengths, and appliedload weights.

Resonances occurring outside the velocity loop and inside the positionloop will contribute phase shift to the open-position loop frequency-


Fig. 3 Ball screw resonance.

Fig. 4 Ball screw resonance.


enough (at least 200 rad/sec) such that the resonance will not have adetrimental effect on the drive.

For large machines where the velocity constant Kv usually cannot belarger than 0.5 ipm/mil for reasons of stability, the lowest allowableresonance (from Figure 6) could be as low as 50 rad/sec. For small machinesusing the soft servo with velocity constants of 2 ipm/mil, the lowestresonance in the position servo loop should not be less than 200 rad/sec.Therefore, it is difcult to make an across-the-board recommendation withchanging velocity constants Kv from machine to machine.

One possible way to arrive at a recommendation is to assume that allmachines being built use position loop gains up to 2 ipm/mil, and therefore,machines with ball screw resonances under 200 rad/sec cannot use directfeedback. Another possibility, which is more practical, is to list the machineswith the recommended maximum allowable position servo loop gains Kvthat can be used with direct feedback. Since all axes on a machine must havethe same position loop gain, the machine axis with the lowest resonance willbe the determining factor in how low the position loop gain or velocityconstant Kv must be set.

Other Resonance Considerations

Additional mechanical resonances inside the position loop that may causestability problems should also be considered. Reected structural resonances

Fig. 6 Lowest recommended resonance inside the position loop.


due to large and sometimes limber machine structures may reect theirresonances into the machine slide as the result of an antinode vibration.Likewise, very large and heavy workpieces placed on an otherwise stablemachine slide can cause stability problems when direct position feedback isused.

Additionally, mechanical power transmission devices, such as woundup gear trains, can be a source of stability problems when they are includedinside the position loop with direct slide position feedback. The gearbox canhave two types of problems. First, the amount of windup can vary causing achange in spring rate that can cause instability. Second, any nonlinearitysuch as backlash that occurs as a result of lost windup will result in anunstable servo drive.

Reected Inertias for Machine Drives

An important parameter to consider in sizing a machine servo drive is thetotal inertia at the servo motor. The denition of inertia is the property of abody, by virtue of which it offers resistance to any change in its motion.

For an industrial machine slide the total reected inertia to the servomotor is made up of the reected belt and pulley inertia (or gearboxes), thereected drive screw, and the reected machine slide with its load weight.Knowing the total inertia at the servo drive motor is a requirement for anyform of servo analysis or drive sizing.

The reected inertia at the drive motor can be calculated as

Jreflected JloadD2=D12(lb-in.-sec2 (12.2-1)

Where D1 diameter of the motor pulley or gear (in.)D2 diameter of the load pulley or gear (in.)

A very important consideration in applying a drive to a machine axis is thetorque to inertia ratio. A mechanical equation for acceleration torque is:

TT J a (lb-in.) (12.2-2)Where JT total inertia at the motor shaft (lb-in.)

a acceleration at motor shaft (rad=sec2)


Equation (12.2-1) can be rearranged as

a TJ

(rad=sec2) (12.2-3)

Equation (12.2-3) is known as the torque to inertia ratio, and has importantsignicance. In a given drive application the machine will have a total inertia(motor inertia plus reected inertia) that exists by design. For productivityrequirements there will be a required acceleration. The question must nowbe asked: Is there enough acceleration torque to meet these requirements?Quite often this becomes a critical drive-sizing problem providing enoughavailable current from the servo amplier to accelerate the load required tomeet a productivity requirement?

For industrial servos using brushless DC drives there are two types ofmotors to be considered. The brushless DC motor in general industrialapplications uses ceramic magnets. For high-performance applications, lowinertia motors are used. These motors use Neodymium-iron-boron (NdFeB)magnet material or Samarium cobalt (SmCo) magnet material. Acomparison of torque to inertia ratios T/J with various total inertia loadsis compared in Figure 7.

At small inertia loads the torque to inertia ratios for low inertiamotors are signicantly higher than standard ceramic motors, which is anindication that the low inertia motors are capable of higher performance(and acceleration) with small total inertia loads. These curves also indicatethat the low inertia motors have a signicant reduction in performance astotal load inertia increases. The standard ceramic magnet-type motors haveless reduction in performance with added total load inertia. The low inertiadrive is usually more expensive than the standard ceramic-magnet brushlessDC motor. For larger total inertia loads it is not economically justiable touse the low inertia type motor since performance will be compromised. Ifhigh performance is a requirement, the reected inertia load should bereduced with a ratio if possible.

To further study the performance of brushless DC drives undervarious inertial loads, a transient step response test will be made for avelocity servo using a lag/lead compensation in the servo amplier with again of 1000 volts/volt. A more detailed analysis is presented in Chapter 14.A 3-volt/1000 rpm tachometer feedback will be used. The block diagram forthis servo drive is shown in Figure 8. A simple velocity servo was chosen forthis analysis to minimize complexity.

A standard ceramic brushless DC motor and a low inertia brushlessDC motor will be used for this analysis. The motor parameters are identiedas the following:


MOTOR STANDARD MOTOR LOW INERTIA MOTOR

(Kollmorgan M607B) (Kollmorgan EB606B)

KT 9.9 lb-in./A 9.9 lb-in./AKe 0.3734 v-sec/rad 0.3734 v-sec/radRmotll 0.14 ohm 0.14 ohmSRmotll 0.189 ohm 0.189 ohmSRmotphase 0.0945 ohm 0.0945 ohmLll 0.0038 henry 0.0038 henryJmot 0.1872 lb-in.-sec2 0.02688 lb-in.-sec2

te Lll

SRmotll 0:0038

0:189 0:02 sec 0:02 sec (12.2-4)

tm SRmotphaseJtotKephaseKT

(12:2-5)

Jtotal Jmot Jreflected (12.2-6)

Transient responses for a multiplicity of load inertias are shown in Figures 9to 16. In this analysis it is assumed that each drive will be analyzed withidentical inertia loads as if a machine axis was being tested with a standardbrushless DC motor and then tested again with a low inertia motor.

Load inertia Standard motor Low inertia motor

(lb-in.-sec2) transient responses transient responses

1XMOTOR Fig. 9 Fig. 10

0.2 Fig. 11 Fig. 12

0.8 Fig. 13 Fig. 14

2.0 Fig. 15 Fig. 16

A ceramic-magnet brushless DC servo motor will have some reduction inservo bandwidth with added load inertia (Figures 9, 11, 13, and 15), butthese drives are tolerant of added load inertia without having to berecompensated. Rare earth brushless DC drives are load inertia sensitive(Figures 10, 12, 14, and 16). For brushless DC servo drives with ceramic-magnet armatures, a ratio of approximately four times reected load inertiato the motor armature can be tolerated. For low-inertia brushless DCmotors it is recommended that the reected load inertia match the motorinertia. Since these motors are used for high-performance applications, it isadvisable not to reduce the servo bandwidth with inertia mismatch ratioslarger than one. Reected load inertias can be computed as follows:


where:

N NOUTNIN

NIN < NOUT

L lead of belt inrev

JREF WT lb386in:=sec26

L

N(in./rev)6(rev=2p rad)

2

JREF WTL=N20:0000656 (lb-in.-sec2)

where:

L lead of rack and pinion inrev

JREF WTL=N20:0000656 (lb-in.-sec2)

Fig. 18 Reected inertia for a rack drive.

Fig. 17 Reected inertia for a belt drive.


where:

N NOUTNIN

NIN < NOUT

JREF TO MOTOR WT6RADIUS2

263866RATIO2(lb-in.-sec2)

where:

N NOUTNIN

NIN < NOUT

Ratio of table ring gear/pinion lead revolutions of table/revolu-tions of pinion.

JREF TO MOTOR WT6RADIUS2

263866RATIO6LEAD2 (lb-in.-sec2)

WT lbsLead ln=revN ratioRadius in:

Fig. 19 Reected inertia for a rotary drive.

Fig. 20 Reected inertia for a rotary drive using a ring gear.


12.3 DRIVE STIFFNESS

Machine feed drives should have sufcient static stiffness to be insensitive toload disturbances. In addition, a feed drive in a contouring numericalcontrol system must remain stationary or clamped when not in motion. Thisis called servo clamping. Also during the standstill period, the axis at restmust resist the load disturbances caused by the cutter reaction forces.

The static stiffness of a feed servo drive is the equivalent springstiffness of the drive measured at the output. The stiffness can be measuredat several different places, such as the drive motor shaft, at the drive input tothe ball screw, or at the machine slide. When the stiffness is measured at therotary components the units of measurement are lb-in./rad. When thestiffness is measured at the machine slide the unit of measurement is lb/in.Drive stiffness, measured as the machine slide displacement in inches causedby a displacement force in pounds, increases as the square of the ratio andwith the neness of the lead of the screw. Therefore, in sizing a servo drive, itis important to consider the benets of using a drive ratio and small lead(in./rev).

Fig. 21 Rotary inertia conversion table.


Fig. 22 Hydraulic servo-drive stiffness.


Fig. 23 Electric servo-drive stiffness.


It is possible to have a stiffness measurement for each type of feedbackservomechanism. For example, a velocity servo drive with tachometerfeedback would have a velocity stiffness with the units of lb-in./rad/sec. Forthe purpose of this discussion the drive stiffness is derived for the case of apositioning servo drive with an internal tachometer loop.

To illustrate the engineering analysis of the static drive stiffness, twocases of direct slide positioning are considered: a hydraulic servo drive andan electric servo drive. Servo-drive stiffness equations for both hydraulicand electric drives are summarized in Figures 22 and 23. The derivation forthe stiffness of the hydraulic servo drive is followed by the electric servo-drive stiffness derivation.

Hydraulic Servo-Drive Stiffness with Direct Feedback

A mechanical model of a machine feed slide is shown in Figure 24. It isassumed that the drive screw spring rate is represented by GT . Themachines slide inertia reected to the drive screw plus the inertia of thedrive screw is represented by JL. The load forces acting on the machine slideare reected to the drive screw as TL. The friction forces of the machine slideare reected to the drive screw and represented by BL. The mechanical driveratio is represented by N.

Fig. 24 Machine slide free-body diagram.


The equations for the mechanical model of Figure 24 are written in thefollowing steps:

y1 ymN

T1 NTm T1 GTy1 yo

For dynamic equilibrium we have:

JLs2yo BLsyo TL GTy1 yo (12.3-1)

JLs2 BLs GTyo TL GTy1 (12.3-2)JLs2 BLs GTyo TL GT

Nym (12.3-3)

From the basic torque relationship,

Tm T1N

GTN

y1 yo (12.3-4)

Tm GTN

ymN

yo

(12.3-5)

Tm GTN2

ym GTN

yo (12.3-6)

From Eq. (12.3-3) the output position of the drive screw is

yos TL GTN ym

JLs2 BLs GT (12.3-7)

An additional set of equations can be written for the hydraulic drive. Thebasic ow equations are developed as a rst step from Figure 25, using theterms

Q1;Q2 hydraulic oil flow in and out (in:3=sec)P1;P2 hydraulic pressure (psi)

KC hydraulic valve pressure coefficient (in:3=sec psi)Kv hydraulic valve gain (in:3=sec-in:)Xv hydraulic valve displacement (in.)

KC qQqPL DQDPL

(12.3-8)

Therefore

PL DP1 DP2 (12.3-9)


Motor ow equations are developed as follows for each chamber:

Flow in flow out theoretical flow compressibilityQ1 KLcpP1 P2 KLexP1 sV1 V1b sP1 (12.3-17)

Q2 KLcpP1 P2 KLexP2 sV2 V2b sP2 (12.3-18)

where:

V1 Vc fvym (12.3-19)V2 Vc fvym (12.3-20)sV1 Dmsym sV2 (12.3-21)KLcp cross port leakage (in:3=sec-psi)KLex hydraulic external leakage (in:3=sec-psi)

b bulk modulus (16105 lb=in:2)V1;V2 hydraulic volume in and out (in:3)fvym variation in each chamber volume (in:3)

Dm hydraulic motor displacement (in:3=rad)Vc total trapped volume on one side of motor

(that oil under compression); valve; and manifold (in:3)

s Laplace operator

Combining Eq. (12.3-17) to (12.3-21) results in

Q1 KLcpP1 KLcpP2 KLexP1 Dmsym Vcb

fvymb

sP1 (12.3-22)

Q2 KLcpP1 KLcpP2 KLexP2 Dmsym Vcb

fvymb

sP2 (12.3-23)

Combining Eq. (12.3-22) and (12.3-23) yields the total motor ow equation:

Q1 Q2 2KLcp KLexP2 P1 2Dmsym Vcb sP1 P2

fvymb

sP1 P2 (12.3-24)


Since

Q1 Q22

QL (12.3-25)PL P1 P2 (12.3-26)

and

sP1 sP2 0 (12.3-27)

since the time rate of change of P1 is equal and opposite to that of P2.Combining Eq. (12.3-24) to (12.3-27) results in the total motor ow

equation

QL KLcp KLex2

PL Dmsym Vc2b

sPL (12.3-28)

Rearranging, the total motor ow equation becomes

Dmsym KLcp KLex2

PL Vc2b

sPL QL (12.3-29)

motor displacement leakage compressability

Simplifying the leakage by

KLcp KLex2

KL (12.3-30)

results in a simplied motor ow equation:

Dmsym KLPL Vc2b

sPL QL (12.3-31)

Applying the torque equation for a hydraulic motor gives

Torque generated DmPL Jms2ym Tm (12.3-32)

Rearranging terms yields

PL Jms2

Dmym Tm

Dm(12.3-33)


Differentiating yields

sPL Jms3

Dmym Tm

Dms (12.3-34)

Combining the motor ow Eq. (12.3-31) with Eq. (12.3-33) and (12.3-34)equals the total ow equation with load torque:

Dmsym KL Jms2

Dmym TmKL

Dm Vc2b

Jms3

Dmym

TmVcsDm2b

QL (total flow, in:3=sec) (12.3-35)

Letting QL Q and factoring ym yields

sJmVc

2bDms2 KLJm

DmsDm

ym Q TmKLDm

TmVc2bDm

(12.3-36)

ym Q TmDm KL

Vc2b s

s JmVc2bDm

s2 KLJmDm

sDm (12.3-37)

ym QDm Tm KL Vc2b s

s JmVc2b s

2 KLJmsD2m (12.3-38)

Combining with Eq. (12.3-5) yields

ym QDm KL Vc2b s

GTN2ym KL Vc2b s

GTNyo

h i

s JmVc2b s

2 KLJmsD2m (12.3-39)

It is desired to combine the mechanical torque equations and the hydraulicow equations to arrive at one system equation. By rearranging Eq. (12.3-39) in several steps, Eq. (12.3-43) results.

A second-order equation can be written as s2=o2h 2d s=oh 1.Therefore in Eq. (12.3-39) oh

2bD2m=JmVcp

is dened as the hydraulicresonance.


sJmVc

2bs2 KLJmsD2m

ym

QDm KL Vc2b

s

GT

N2ym KL Vc

2bs

GT

Nyo (12.3-40)

s3JmVc

2b KLJms2 D2ms KL

Vc

2bs

GT

N2

ym

QDm KL Vc2b

s

GT

Nyo (12.3-41)

s3JmVc

2b s2KLJm D2m

VcGT

2bN2

s KLGTN2

ym

QDm KL Vc2b

s

GT

Nyo (12.3-42)

ym QDm KL Vc2b s

GTNyo

JmVc2b s

3 KLJms2 D2m VcGT2bN2

s KLGTN2

h i (12.3-43)

Drive system Eq. (12.3-7) and (12.3-43) can now be included in apositioning servo drive with an inner tachometer loop as shown in the blockdiagram of Figure 26. To simplify the solution of the drive system equations,substitutions for some of the blocks are introduced in Figure 26.Rearranging the block diagram of Figure 26 yields the block diagram ofFigure 27 for a slide displacement output on a load force input. This blockdiagram can then be used to solve for the drive system stiffness FL=Xo.

Using the substitutions for the block in Figure 26, drive systemequations can be written in Eq. (12.3-44) to (12.3-52).

From Figure 26

Q XoKfbKDGv ymKTAsGv Xi (12.3-44)

But Xi 0, so

ym QDmGm GmGF 2pLXo (12.3-45)

where:

2pXoL

yo2pLXo GL GT

Nym FLGL L

2p(12.3-46)


Combining Eq. (12.3-44) to (12.3-46) yields

ym DmGmKfbKDGvXo KTAsGvym GmGF 2pLXo (12.3-47)

ym DmGmKfbKDGvXo DmGmKTAGvsym GmGFL

Xo2p (12.3-48)

Rearranging Eq. (12.3-48) yields

ym1DmGmKTAsGv DmGmKfbKDGv 2pGmGFL

Xo (12.3-49)

ym DmGmKfbKDGv 2pL GmGF

1DmGmKTAsGv Xo (12.3-50)

Using Figure 26, Eq. (12.3-50) can be modied to yield

2pLXo GL GT

N

DmGmKfbKDGv 2pL GmGF

1DmGmKTAsGv Xo

L2p

GLFL (12.3-51)

Rearranging yields

2pL

GL GTN


1DmGmKTAsGv

Xo L2p

GLFL (12.3-52)

FL

Xo 2p

L GL GT

N6


1DmGmKTAsGv

2pLGL

(12.3-53)

FL

Xo 2p

L

21

GL 2pGT DmGmKfbKDGv

2pLGmGF

LN1DmGmKTAsGv (12.3-54)

for o 0 at steady state.From Figure 26 the following simplications are possible.

Gm N2

KLGT(12.3-55)

GF GTKLN

(12.3-56)

GL 1GT

(12.3-57)

Gv K1Kv (12.3-58)

Equations (12.3-55) to (12.3-58) can then be substituted into Eq. (12.3-54),


yielding

F

Xo

o0 2p

L

2

GT 2pGTLN

6 Dm N2

KLGTKfbKDK1Kv 2pN

2GTKL

LKLGTN

(12.3-59)


F

Xo 2p

L

2

GT 2pDmNKfbKDK1KvLKL

2pL

2

GT (12.3-60)

FL

X 2p

L

DmNKfbKDK1Kv

KLstiffness for direct feedback (12.3-61)

It can be shown that the velocity closed-loop equation from Eq. (12.3-35) is

ymvr

K1KvDm 1 K1KvDm KTA

(also see Figure 26) (12.3-62)

The velocity open-loop gain is

Kvo K1KvKTADm

(V/V) (12.3-63)

Substituting into Eq. (12.3-62) yields

ymvr

K1KvDm1 Kvo (12.3-64)

From Figure 26 the position-loop velocity constant is

Kv KDKfbL2pN

6K1Kv

Dm1 Kvo (12.3-65)

Rewriting Eq. (12.3-65) yields

K1KvKDKfb Kv2pNDm1 KvoL

(12.3-66)

Rewriting Eq. (12.3-61) yields

FL

Xo 2p

L6

DmN

KL6KfbKD6K1Kv (12.3-67)


Substituting Eq. (12.3-66) into (12.3-67) yields

FL

Xo 2p

L

DmN

KL

Kv2pNL

6Dm1 Kvo (12.3-68)

The drive stiffness equation for direct position feedback and measured at themachine slide is

FL

Xo 2p

L

2D2mN

2KvKvo 1KL

(lb/in.) (12.3-69)

As a dimensional check,

2rad2

rev26

rev2

in:26

in:6

rad66

1

sec6

V

V6

sec lb

in:3 in:2

lbin

Linear Electric Servo-Drive Stiffness with ResolverFeedback at the Drive Motor

The rst step in deriving the drive system equations is to derive the DCelectric motor equations, which can be done with the equivalent electricalcircuit shown in Figure 28.

The equation for the equivalent circuit of Figure 28 is

es LaRa

s 1

iaRa KeVms (12.3-70)es KeVms

LaRas 1

Raia (12.3-71)

Fig. 28 Electric motor diagram.


The torque equation for Figure 28 is

Torque KTia JTsVms TL (12.3-72)ia JTs

KTVms TL

KT(12.3-73)

Solving Eq. (12.3-73) for the motor velocity Vms yields

ia TLKT

KT

JTs Vms (12.3-74)

Equation (12.3-74) can be reduced as follows:

iaKT

JTs TLJTs

Vms iaKT TLJTs

Vms (12.3-75)

Equations (12.3-71) and (12.3-75) can be represented by the block diagramof Figure 29:

The motor block diagram can now be included in an electricpositioning servo drive with an internal tachometer loop as shown inFigure 30. This drive system block diagram can be rearranged as shown inthe block diagram of Figure 31 for an output displacement on a torque loadinput while holding the reference input position yi at zero. This solution willbe a linear solution considering all components as linear devices.

Fig. 29 Electric motor block diagram.


The output/input of the block diagram of Figure 31 can then be solved for and isshown in Eq. (12.3-77).

yosTLs

RfRaLaRas 1

t2s 1s JTsRa

LaRas 1

Rf t1s 1 KTKeRf t2s 1 K1KTAR2t1s 1h i

Rf t1s 1K1KDKT R2Rin

Kfb

(12.3-77)

Letting s jo ! 0 for the steady-state case,

yoTL

RfRaRfK1KDKTR2=RinKfb (12.3-78)

TLyo

K1KDKTR2KfbRaRin

(12.3-79)

TLyo

KDRin6

R2K1KTKfb

Ra(12.3-80)

TLyo

KTRa6

K1R2KDKfb

Rin(12.3-81)

It is necessary to present the drive stiffness in terms of readily availableparameters. Therefore, the following solutions are presented starting withthe motor closed loop followed by the velocity closed loop and nally theposition-loop velocity constant.

Motor closed-loop equations can be written from Figure 30:

Vmses

KT

JTsRaLaRas 1

61

1 KTKeJT sRa

LaRas1

KTJTsRa

LaRas 1

KTKe(12.3-82)

Vmses

1=KeJTRaKTKe

LaRas2 JTRa

KTKes 1

1=Ketmetes2 tmes 1 (12.3-83)


where:

tme JTRaKTKe

te LaRa

Velocity closed-loop equations can be written from Figure 30:

VmsEis

R2t1s 1K1Ket2s 1tmetes2 tmes 16

1

1 R2t1s1K1KTAKet2s1tmetes2tmes1Rf

(12.3-84)

Equation (12.5-84) reduces to

VmsEis

R2t1s 1K1Ket2s 1tmetes2 tmes 1 R2t1s1K1KTARf

(12.3-85)

The open position-loop velocity constant from Figure 30 is

Kv KDRin6

VmsEis 6Kfb with s 0 (12.3-86)

The position-loop velocity constant including Eq. (12.3-85) yields

Kv KDKfbRin

6R2t1s 1K1

Ket2s 1tmetes2 tmes 1 R2t1s1K1KTARf(12.3-87)

Kv KDKfbR2K1RinKe R2K1KTARinRf

(12.3-88)


Kv KDKfbR2K1RfRinKeRf R2K1KTARin (12.3-89)

Rearranging,

KvRinKe KvR2K1KTARinRf

KDK1R2Kfb (12.3-90)


Combining Eq. (12.3-81) and (12.3-90) yields

TL

yo KTRaRin

6 KvRinKe KvR2K1KTARinRf

(12.3-91)

The velocity open-loop gain from Figure 30 is

Kvo R2t1s 1K1KTAKet2s 1Rf tmetes2 tmes 1 (12.3-92)

For the steady-state condition s ! 0, the velocity open-loop gain is

Kvo R2K1KTAKeRf

(12.3-93)

Substituting Eq. (12.3-93) into Eq. (12.3-91) yields

TL

yo KTRaRin

KvRinKe KvoKe (12.3-94)

Therefore, the static stiffness equation for a DC electric position loop withan internal tachometer loop is

TL

yo KTKe

RaKv1 Kvo in.-lb

rad

(12.3-95)

Drive Stiffness for Three-Phase, Half-Wave SCR DriveAmpliers Using an Inner Current Loop

A linear analysis for the static stiffness at the motor of a DC electric driveusing a three-phase, half-wave silicon controlled rectier (SCR) amplierand an inner current loop can be made starting with the basic equations fora DC motor, referring to Figures 28 and 29 and Eq. (12.3-70) to (12.3-75).

When a current loop is added to the drive amplier, the motor andcurrent loop will appear as the block diagram shown in Figure 32.

To facilitate solving the loop equations, the block diagram of Figure 30can be modied as shown in Figure 33.

The DC electric motor with a current feedback loop can then beincorporated into a position servo loop with an inner velocity servo loop asshown in Figure 34.

The system block diagram can be redrawn for a position output and atorque disturbance input in Figure 35.


Solving for the outer loop of Figure 29 yields

ysTLs

Vo

A

1

s6

1

1 VoA

1s

KTK1tAs1R2K2t1s1KDKfbtBs1 LaRas1 RaK1KietAs1 t2s1Rin

(12.3-103)

ysTLs

Vo

A

61

s VoA

KTK1KDKfbtAs1T1s1R2K2tBs1 LaRas1 RaK1KietAs1 t2s1Rin

(12.3-104)

For the steady-state solution,

yTL

VoA6

1VoA 6

KTK1KDKfbR2K2RaK1KieRin

(12.3-105)

as s?0

yTL

Ra K1KieRinKTK1KDKfbR2K2

steady-state compliance (12.3-106)

TL

y K1K2KDKTR2KfbRa K1KieRin steady-state stiffness (12.3-107)

Solving for the closed current loop of Figure 34,

iasvis

K1tAs 1tBs 1 LaRa s 1

Ra K1tAs 1Kie(12.3-108)

For the current closed loop steady state,

ia

vi

s!0 K1Ra K1Kie (12.3-109)

Solving for the motor closed loop from Figure 34 yields the following:

Vosvrs

K1tAs 1KTtBs 1 LaRa s 1

Ra K1tAs 1Kieh i

JTs

61

1 K1tAs1KTtBs1 LaRas1 RaK1tAs1Kie JT stBs1KeK1tAs1h i (12.3-110)


Simplifying Eq. (13.2-110) yields

Vosvrs

K1tAs 1KTtBs1 LaRa s1

Ra K1tAs1Kieh i

JTs K1tAs 1KT tBs1KeK1tAs1(12.3-111)

Vosvrs

s!0 K1KTK1KT

KeK1

K1Ke

steady state for motor loop (12.3-112)

Solving for the open velocity loop of Figure 34 yields

Kvos KTARf

R2K2t1s 1t2s 1

6K1tAs 1KT

tBs 1 LaRa s 1

Ra K1KietAs 1h i

JTs K1tAs 1KT tBs1KetAs1K1(12.3-113)

as s !0

The steady-state open velocity loop gain is

Kvo KTAR2K2Rf

KeK1

KTAK1K2R2RfKe

(12.3-114)

Solving for the closed tachometer loop of Figure 34 yields

Vosv2s

R2K2t1s 1t2s 1 6

Vosvrs 6

1

1 R2K2 t1s1t2s1Vosvrs

KTARf

(12.3-115)

Substituting Eq. (12.3-112) into (12.3-115) and reducing Eq. (12.3-115)yields

Vosv2s

R2K2t1s 1K1KTt2s 1K1KT KeK1 R2K2t1s 1K1KT KTARf

(12.3-116)

Vo

v2

s!0 R2K2K1KTKTKe R2K2K1KT KTARf

For the steady-state closed velocity loop,

Vo

v2 K1K2R2RfKeRf K1K2R2KTA (12.3-117)


The position-loop velocity constant is the steady state from Figure 34 andEq. (12.3-117):

Kv KDKfbRin

6Vo

v2 KDKfb

Rin6

K1K2R2Rf

KeRf K1K2KTR2KTA (12.3-118)

Reducing Eq. (12.3-118) yields

Kv K1K2R2KDKfbRin Ke K1K2R2KTARf

(12.3-119)


KeKvo K1K2R2KTARf

(12.3-120)

Then substituting Eq. (12.3-120) into Eq. (12.3-119) yields

Kv K1K2R2KDKfbRinKe KeKvo

K1K2R2KDKfb

RinKe1 Kvo (12.3-121)

Rearranging yields

Kv1 Kvo K1K2R2KDKfbRinKe

(12.3-122)

TL

yo K1K2KDKTR2Kfb

RinRa K1Kie Eq:12:3 107

Substituting Eq. (12.3-122) into (12.3-107) yields

TL

yo Kv1 KvoKeKT 1Ra K1Kie

TL

yo Kv1 KvoKeKT

Ra 1 K1KieRa (12.3-123)

Applying the equation for drive stiffness at the motor of a three-phase,half-wave SCR amplier with a current and velocity loop to a commercialGettys drive, Eq. (12.3-123) can be simplied since

Current open-loop gain K1KieRa

275A=secA

Velocity open-loop gain Kvo 35,965V=secV


Therefore, for a specic commercial SCR drive amplier, Eq. (12.3-123) willreduce to

TL

yo Kv1 35,965KeKT

Ra1 275in.-lb

rad

(12.3-124)

Applying the equation drive stiffness at the motor of a three-phase,half-wave SCR amplier with a current and velocity loop to a commercialGettys drive, Eq. (12.3-124) can be used for an example:

Motor 1000 rpm 30-2Ke 1:15V=rad=secKT 10:1 in.-lb=AKieK1

Ra open current loop gain 275A=sec

A

Rmotor 0:25 ohmRinductor 0:016 ohmRtransformer 0:1 ohmRa 0:25 0:016 0:1 0:36 ohmKv 1 ipm=mil 16:8 secKvo 35,965V=VTL

yo 16:863596561:15610:1

0:366275 70,888 in.-lb/rad

12.4 DRIVE RESOLUTION

One of the most important I.P.s for machine slides that have a very lowfeed-rate requirement is the drive resolution, which is dened as thedifference between the breakaway position error and the run error:

Resolution breakaway error run error (12.4-1)Drive resolution can also be described as the least amount of position errorrequired to cause motion to take place. It is directly related to the quality ofsurface nish on a contouring controlled milling machine. The units of driveresolution are inches. The drive resolution for hydraulic and electric drives isdiscussed in this section.

A graphical representation of drive resolution is shown in Figure 36.The difference between the breakaway error and the run error is the drive


resolution. For machines having a signicant static friction above thecoulomb friction, the drive resolution can be unacceptable.

Drive Resolution for Hydraulic Drives

The drive resolution for a hydraulic drive is a function of the port-to-portleakage at breakaway. Thus

Breakaway leakage LBADPBA6KL friction torque of slideBAKTH

6100KL

(12.4-2)

LBA KL DPBA FBAKTH

6100

in:3

sec

(12.4-3)

In Eq. (12.4-1) the breakaway characteristics of the motor andmachine slide must be added to obtain a true value for the overallbreakaway characteristics. A drive resolution representation is shown inFigure 36. Likewise the run leakage characteristics can be equated as

Run leakage Lr KL no-load motor psi run friction torqueKTH

6100

(12.4-4)

Having computed the breakaway and run leakage characteristics, it ispossible to compute the breakaway and run velocity from

Breakaway velocity VBA LBA6 1Dm6

L

N6

1

2p(12.4-5)

Run velocity Vr Lr6 1Dm6

L

N6

1

2p(12.4-6)

The contouring control position loop can be modeled as in Figure 37.

yos EsKvs

(12.4-7)

EsKv yos Vos (12.4-8)

Position following error Es VosKv

(12.4-9)


Therefore, the breakaway and run velocities divided by the Kv yield thebreakaway and run errors.

EBA VBAKv

LBAL2pKvDmN

(12.4-10)

EBA LKL2pKvDmN

DPBA of motor FBA of slideKTH

6100

(12.4-11)

Er VrKv

LrL2pKvDmN

(12.4-12)

Er LKL2pKvDmN

NL motor psi FrKTH

6100

(12.4-13)

The drive resolution can therefore be equated as:

Resolution LKL2pKvDmN

6 DPBA FBAKTH

6100

NL psi FrKTH

6100

(12.4-14)

Motor DPBA NL psi

Vickers MFA5B 600 psi 300

Hartman 12 psi 10

Nutron 50 psi 25

For antifriction ways:

Friction breakaway FBA run friction FrResolution LKL

2pKvDmNDPBA NL psi (12.4-15)

Drive Resolution for Electric Drives

The drive resolution for an electric drive is a function of the static frictionbreakaway of the drive motor and machine slide. The derivation of theelectric drive resolution can be developed similar to the case for a hydraulic


drive as follows:

Breakaway current IBA ImBA TFBAKT

(12.4-16)

motor breakaway slide breakawayRun current Ir Imr TFr

KT(12.4-17)

Breakaway velocity VBA IBA6 RaL2pKeN

(12.4-18)

Run velocity Vr Ir6 RaL2pKeN

(12.4-19)

For a position loop as shown in Figure 36, the position error is velocity/Kv(Eq. [12.4-9]):

Breakaway error EBA VBAKv

(12.4-20)

IBARaLKv2pKeN

RaLKv2pKeN

ImBA TFBAKT

Run error Er VrKv

(12.4-21)

IrRaLKv2pKeN

RaLKv2pKeN

Imr TFrKT

Resolution RaL2pKvKeN

ImBA TFBAKT

RaL2pKvKeN

Imr TFrKT

(12:4-22)


ImBA TFBAKT

Imr TFrKT

(12.4-23)

For antifriction ways:


ImBA Imr (12.4-24)

Motor breakaway currents are given for some examples:


Motor ImBA A Imr AGettys

10-3 1.28 0.85

20-1 3.0

20-2 1.48 0.99

30-1 3.6

30-2 1.3 0.87

Inland

4020 0.8 0.535

4023 0.8 0.535

5301 2.2 1.46

5302 1.9 1.3

4501A 1.8 0.36

4501B 2.4 1.5

4503A 1.36 0.89

4503B 1.81 1.19

Allen-Bradley

1326-ABC4C 0.832 0.665

1326-ABC3E 0.949 0.712

12.5 DRIVE ACCELERATION

Feed drive acceleration is a critical factor with large industrial machines. Ifthe acceleration is too high, the resulting excessive forces could bedestructive to mechanical components in the drive. Three types ofacceleration have been used with industrial feed servo drives: ramp invelocity, exponential increase in velocity, and S curve type of acceleration.

These are shown graphically in Figure 38. The velocity ramp (a) hasconstant acceleration as velocity increases. At the time where accelerationceases and the velocity is a constant a small visible mark will be evident(because of a small overshoot) on a contouring-type machine. This small butobjectionable mark can be eliminated by using an exponential increase invelocity (b), which rounds the high point in velocity to become a constantvelocity. The objection to this type of acceleration is that the acceleration attime zero is a maximum and is sometimes referred to as jerk. To furtherimprove the acceleration characteristics, the S type of acceleration can beused. The advantage of the S curve acceleration is that the acceleration iszero at time zero and becomes a maximum at a time when the velocity isincreasing at its maximum rate.


For a worst-case condition of a step input,

_yyis VFs

(12.5-4)

Substituting Eq. (12.5-4) into (12.5-3),

Vos VFKvss Kv (12.5-5)

The inverse transform for Eq. (12.5-5) is

Vot VF VF eKvt VF 1 eKvt (12.5-6)

Differentiating Eq. (12.5-6) yields acceleration:

_VVot at VFKveKvt (12.5-7)

The greatest acceleration occurs at t 0. Thus, the motor angularacceleration is

atmax VFKv (rad/sec2 (12.5-8)

where:

VF rad/secKv 1/sec

Fig. 39 Positioning servo block diagram.


It is also possible to calculate the time required to accelerate to a givenvelocity from Eq. (12.5-6) by rearranging terms:

eKvt 1 VtVF

(12.5-9)

Kvt loge 1Vt

VF

(12.5-10)

t 1Kv

loge 1Vt

VF

(12.5-11)

For example, with a Kv 25=sec to reach to velocity 98% of nalvelocity for a step input, the time to reach this velocity is

t 125

loge 1 0:98 (12.5-12)

t 125

log 0:02 1256 4:0173 0:16 sec (12.5-13)

The acceleration and velocity versus time characteristics are shown inFigure 40.

The model of Figure 37 can be used to determine the velocitycharacteristics and time to reach velocity when there is an initial velocity.Eq. (12.5-6) can be rearranged as

Vos s KvKv

_yyis (12.5-14)

with initial conditions

1

KvsVos V0

Kv Vos _yyis (12.5-15)

For a step input,

_yyis VFs

(12.5-16)

VossKv

VinitialKv

Vos VFs

(12.5-17)

s

Kv 1

Vos VFs Vinitial

Kv(12.5-18)


The time to reach a given velocity from an initial velocity can be solved byrearranging Eq. (12.5-22):

Vt VFVinitial VF e

Kvt

t 1Kv

logeVt VFVinitial VF

(12.5-22)

It is sometimes desirable to know how far the machine axis has movedfor a step input in velocity. The distance traveled can be found byintegrating Eq. (12.5-22):

Z t

0

Vt dt Z t

0

VFdtZ t

0

VinitialeKvtdtZ t

0

VF eKvtdt

Xt VFt VinitialKv

eKvt VinitialKv

VFKv

eKvt VFKv

(12.5-23)

Xt VFt Vinitial VFKv

VF VinitialKv

eKvt (12.5-24)

If the model of Figure 39 were to be considered for a ramp input, the inputof Eq. (12.5-4) would change to

_yyis AIs2

(12.5-25)

where AI is the initial acceleration.Similar equations for displacement, velocity, and acceleration with the

step input can be derived for a ramp input summarized as follows:

Xt AI t2

2 tKv

1K2v

eKvt

K2v

(in:) (12.5-26)

Vot AI t 1Kv

eKvt

Kv

(ips) (12.5-27)

Aot AI 1 eKvt (in:=sec2) (12.5-28)

Unfortunately the maximum acceleration at time equals zero produces anobjectionable mechanical shock on the drive mechanics (gears, lead screw,etc.). Another problem relates from a desire on the part of the user to callfor high acceleration rates to meet productivity requirements. Thus therequired drive torque/current is sufciently high to cause the drive to go intocurrent limit condition. As long as the drive remains in current limit, thevelocity increases linearly. This linearly increasing velocity will continue


until the required target velocity is reached and exceeded. As the velocityovershoots its target the drive will come out of current limit and the velocitywill drop back to its target value with a resulting severe shock on the driveand machine components.

S curve type of acceleration

The predominant advantage in using an S curve type of acceleration isthat the initial acceleration will be zero at time equals zero. Thus the initialshock or jerk on the drive and machine will be eliminated. However, themaximum acceleration can be larger than the initial acceleration withexponential acceleration depending on how the S curve is generated. Incommercial servo drive applications the type of S curve is a function ofthe path control generated in the computer control. There are numerousways to generate S curve accelerations. One method is the use of asinusoid to generate an S curve (Figure 41) represented as the following:

Velocity Vpeaksin wt (12.5-29)The velocity prole is a zero-shifted negative cosine wave (Figure 42)

with double the peak of the sine wave.

Where: y angle of motor shaft (rad)T period of the motor shaft rotation (sec)

VF desired final velocity y=T (rad=sec)

Fig. 41 Sinusoidal curve.

Copyright 2003 by Marcel Dek

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