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AEROSPACE FLUID COMPONENT DESIGNERS' HANDBOOKVOLUME 1 1 Revision D

TECHNICAL DOCUMENTARY REPORT NO. RPL-TDR-64-25

,---.FEBRUARY 1970

AIR FORCE ROCKET PROPULSION LABORATORY RESEARCH AND TECHNOLOGY DIVISION

AIR FORCE SYSTEMS COMMANDEdwards. California

PROJECT NO. 3058Prepared Under Contracts AFOq611)-8385, AF04(611)-11316, F04611-67-C-0062, and F04611-68-C-0065

bv

AEROSPACE FLUID COMPONENT DESIGNERS' HANDBOOKVOLUME IIRevision D

TRW SYSTEMS GROUP One Space Park Redondo Beach, California

Glen W. Howell Terry M. WeathersEditors

Notice: This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of AFRPL (RPPR-STINFO), Edwards, California 93523.

FEBRUARY 1970

Prepared forAIR FORCE ROCKET PROPULSION LABORATORY RESEARCH AND TECHNOLOGY DIVISION AIR FORCE SYSTEMS COMMAND Edwardc, California

Contract NumberA I R FORCE/56780/15 August 1980

- 500

AF 04(611)-8385 AF 04(611)-11316 Fo4611-67-c-0062 FO4611-68-C-0065

AEROSPACE FLUID COMPONENT DE5lGNCRS' HANOSOOK

TRW Systemse Group, TRW Inc. All rights rm"rv~d MAY 19640

T Copyriot 1970 by

Fiti

lim.ujte

Revisoda1

TOOERI 196S

Oualifiod requesters mayv obtain copies of this report from the Station Defense Documentation Center (DOC). Cameron Alexandria. Virginia

.

DYNAivlIC ANALYSIS

TABLE OF CONTENTS7.1 INTRODUCTION 7.2 CONTROL SYSTEM THEORY 7.2.1 An Introduction to Automatic Control Systems 7.2.2 A Basic Outline of Servo Mathematics 7.2.3 Criteria for Evaluating Servo 7.3.4 7.3.5 7.3.6 7.3.7 7.3.8 7.3.9

System Performance 7.2.3.1 Stability 7.2.3.2 Response 7.2.4 Analyzing a Servo System 7.2.5 Methods for Determining Transient Response of Servo Systems 7.25.1 Relation Between Transient Response and Frequency Response 7.2.5.2 Transient Response from Transfer Functions7.3 VIBRATION AND SHOCK ANALYSIS 7.3.1 General 7.3.2 Harmonic Motion 7.3.3 Natural Frequencies of Spring-Mass Systems

Elements of a Vibratory System Systems with One Degree of Freedom Vibration Isolation and Transmissibility Self-Excited Vibraticns Random Vibration Shock and Resulting Stresses

7.4 DYNAMIC PERFORMANCE ANALYSIS 7.4.1 Introduction 7.4.2 Methods of Component Design 7.4.3 Synthesis by Analysis 7.4.4 Performance Specifications for Closed-Loop

Systems7.4.5 Methods of Dynamic Performance Analysis 7.4.6 Advantages and Limitations in the Methods of

Analysis7.4.7 Analysis of a Hydraulic Servo-Actuator 7.4.8 Dynamic Behavior of a Simple Pneumatic

Pressure Regulator7.4.9 Dynamic Analysis of Pneumatic Dashpot for a

Regulator Control Element

TABLES(Sub-Topic 7.2.2) 1. Laplace Transfonns for S e w Analysis 2. Useful Th.orems for Lapklce Transforms (Detailed Topic 7.2.3.2) 3. Steady Following Errors of S a m 4. Whitely's Optimum Parameters (Sub-lopic 7.2.4) 1. Response to Unit Step Function Input 7.3.3. Equations to Calculate Natural Frequencies of Some Common Systems 7.3.9. Load Factors for Several Pulse Shapes 7.4.4. The Eleven Most Common Performance Spedfiutlons 7.4.5.2. Summary of Major Analytical Technlquer 7.4.6. Summary of Design Approaches (Sub-Topic 7.4.8) 1 Cornpatlson of Non-linear and Linearized . Solutions 7.4.9.2. Equations for Dynamics of Pneumatic Dashpot

ISSUED: MAY 1964

DYNAMIC ANALYSIS

CLOSED LOOP CONCPT

7.1 INTRODUCTIONComplex fluid components, such as hydraulic servoregulators, are actuators and propellant tank pure small-scale control systems. Thus, they are dynamic systems in which a fast response to a change in input or demand is -equiredL In addition, the units must be inherently stable in operation. The dynamic problems which are experienced in a complex fluid comnpourent can be divided into two groups: 1) Dynamic pe,)rmmnce-the problems involved in obtaining the required response rate and stability in the unit. 2) The effects of vibratlo andthk on the dynmic performance and structural integrity of the unit. The following sub-sections of this handbook deal with the dynamic problems in fluid components. SubSection 7.2 gives a general outline of control system theory; Sub-Section 7.3 gives a similar outline of vibration theory; and Sub-Section 7.4; covers dynamic performance analysis, which illustrates how the theory of Sub-Section 7.2 is applied to th design and performance analysis of fluid comp 7.2 CONTROL SYSTEM THEORY Sub-Section 7.2 conssof s n brprintedtfom 7Macinsist se ( vtm fyi f topiby Peprinted from Machine Desigjn (copyrighted by Penton Publishing Company) with perlmison of

human capabilities. Environmental conditions and

fatigue are but two of the factors which make human control unsatisfactory in many instainces. Not only do these facts establish a need for automatic control systems, they lead to a broad definition. Automatic control is the regulation of some variable-called the controlled variable-in accordance with a seqluence of desired conditions without human old. Since cantrol problems occur in many fields, the controlled variable may be of any physical nature. Displacement, speed, pressure, temperature, or voltage are but a few possibilities. Cloed-Loop Concept. Controlx mechanisms, too, may !o susceptible to certain almost human weaLiesses, these must, of course, be eliminated from a successful control system by proper design. To illustrate some of the weaknesses and how they may be overcome, the problem of regulating the rotati(mal speed of a turbinedriven pump wii be considered. Turbopumps are used in many liquid rocket engines to deliver proPellanLs to the engine thrust chamber. Figure 1(a) llustratsm a layout of a turbopump for a monopropet'nt engine. The centrifugal pump draws the liquid monopropellant from a tank at low pressure and pumps it to th&thrust chamber at high pressure. The pump is driven through a gear box by a turbine. The operating fluid in the turbine is a gas which is eupplied at high pressure by a gas generator. The flow rate from the gas generator is controlled by a valve. The gas flow rate sets the turbopump speed, which sets the flow rate of the propellant being pumped. la constant thrust rocket engines, iA is necessary to maintain the propellant flow rate, and thus the pump speed, at a constanm level. The iraplest method of achieving this is to use a valve with a fixed aetting in the location shown in Figure 1 (a). The valve setting is calibrated to provide a gas flow rate which gives the required turbopump speed. This system will give a fairly constant speed only if propellant tempernture and pressure, gas genrator efficiency, and other possible variables remain within tolerable

publisher. Tse sub-topis orginally appeared in Machine ')esign as Seven articles (Refrenm 1-272,1-273, 1-274, 1-275, 1-276, 1-277, 1-278) in an eighteen-part series written by J. M. Nightingale. The series was subsequently reprinted by the Penton Publiehing Company in three volumes, entitled "Hyeraulic Servo FL-Wamentais" (References ,-128, 1-129, 1-131). In adapting the material for me in this handbook, the first example in Sub-Topic 7.I was modifled from that appearing in the original reference (Reference 1-272) to conform with the hwunmO(k content, and an example which appeared in the ori-

ginal Machine Design series at the end of artile number five (Reference 1-276) wavs deleted in thehandbook adaptation. Any other changes in the ori-

limits, Since the valve is pre-set for only one set of operafg condistions.Greater accuracy in speed regulation can be obtained

gmnal articles consit of minor additions or deletio min order to condorm to the typographical style of the

by adding a governor or speed regulator to the systern, as shown in Figure 1 (b). In this case, the valve

handbook and do not affect content 7.2.1 An Inbvducton to Autonac Ctrol

sysem

Accuracy, sensitivity speed, and muscle needed forcontrol of many modern machines are often beyond ,ssUW: -Y 16ti

referred to above becomes a controllable throttle valve. The regulator senses the pump speed, compares it with the required value, and makes a orrection to the throttle valve setting if necessary. If the speed sensed is too low, for example, the regulator opensthe valve to increase speed and vice versa. With this 7.1 -1 7.2.1 -1

IVO S

M

DYNAMWU ANALYSIS

GASby GkNUMTOA

an automatic control system and a manual system. Figure 1(b), in which speed regulation is obtained a "-"co teby having an automatic governor or regulator in

THIOTTLUVALVE

_.Ssystem

GEAAWX TRIBINE

-xotrol loop, is an automatic system. A human operator, however, could theoretically take the place of the regulator. He would read the pump speed on a tachometer and wou.ld tLen adjust the gas valve setting by hand to obtain the correct speed Such a would be a manual control sltem. This systern, like the automatic, is a cloed-loop syq.em. The manual system, however, would have limitations due to The response and fatigue characteristics of the human operator. Automaatic control systems of the closed-loop type are usually dassified as either servo sytafes or reglators The difference between the two is primeml-y a matter of application. In semvo qystems, input v,'1e7 and often arbitrarily, avnd the prro,;c of the system is to follow the input clooely, an illustrated in Figure 2. In a regultor, the inj~ut ,, mnanmt for relativey long periods oi time, and the ourpose of the system is to maintain constant ous-ut deepite fluctuations in powr supply or extrnal load

Figse 1a. Twrbepunp wth Passiveocontinuously

G)

GEAR BOX

Time

FiWme lb. Trwbopump wt aSsed4.top aeuiafonmethod, changes in speed due to changes in gas and

Fwe 2. One ati'

pinmy

-uipnm da ofam

ya

propellant conditions are readily corrected for, so Htht a virtually constant speed is maintained. Figure 1 (b) is an eiample of a dosed-loop control system, which may be formally defined as a sydem in which the true state of the controlled variable (the output) is continuously compred with the desired state (the input), and a signal depending on ference between the two (the error) operates a controlling element which then act&an the~rest of the trolingclemnt ats Oi hic thei ~the I system to reduce the error to zero. Almost all practical, sensitive, control systems are based on this closed-loop principle. Mention should be made of the difference between 7.2.1 -2

k to vy pthut to dc* lopUt to the syWttm.

kflow a va.oft

Serv fytatm All closed-loop control systems with pox anplification around the loop are usually referre4 to a er t~nm or sros. The term ev to reserved for thoee servo systerm having ferrai a mechanical output. for eersubclasaification of a mec a is beupthe classfction of f elmcai outpuat means. sbsdupntelaifto a hydraulic servoFor example, t mehsniFmrussaaplotar hydraulicr m r vymeluaic m user a rotaryuhydraulic motorora hydraulic cylinde as the cutput device. However, oextain electrical or electronic devices might be used in a hydraulic servo iechmisn.

ISSUEo: MAY 1964

DYNAMIC ANALYSIS

Some examples of servomechanism applications are power steering of vehicles, auto-pilots for aircraft and misailes (including power controls for operating surfaces), machine tracing tools, automatic tracking radar, and remote gun control syst-ms. A hydraulic servomechanism for controlling angular displacement of a shaft (Figure 3) illustrates the components of a typical servomechanism. Additionfunctions ally, the circuit illustrates two important of a servomechanism: (1) remote control (usually of position), and (2) power amplification. Either may be the predominant requirement of a particular system, but often both are required to some extent. Here the error signal ultimately controls the output displacement by varying the speed of a final drive motor

tem, a measure of the erro.- can be ob-tained as a voltag- by simply 9ubtracting the output voltage from the command signal in the electronic amplifier. In some systems, however, some form of comparator or differential must be used (Figure 4).

Outputrota

jf

i.'Pnet wbeet

or servomotor. In this case, power supply for the aervomotor is a rotary hydraulic pump. Power flow is metered by a controlling element, such as hydraulic slide valve or servo valve. Since the power needed to operate the valve is negligibly small compared with that metered to the servomotor, the slide valve acts as a power amplifier Although the input is defined as the desired state of the controlled variable, which is an angular displacement in this example, the commanding signal is a voltage. The device supplying this information is called the input element. Thus, a measure of the output displacement has to be obtained as a voltage for comparison with the command signal. This is achieved by a potenltiometer measuring device, and it is the constant of this measuring device which relates the input to the command signal. In this sys-

Figure 4. In some servomechanisms a conparator or mechanical differential must be used to add or subtract the feedback signal from the input or command signal. Since the output signal can be transmitted by wires, the input and output stations can be quite remote in a mechanical sense, provided the output signal voltage does not deteriorate during transmission. Systems for transmitting signals from one place to another are called data transmission systems. Often in mechanical systems, rods and cables must be used to transmit the feedback signal. In this case, even with gears and levers the remoteness of the output station is limited.

Slide vlve amplifierZGeaor train Potenttionelier

Sa~,oleni

eay

Fir 3. A hydraitl

m vaomedianls

m for p"ition cmtro #lustrates the component

of

ISSUED: MAY 196*

7.2.1 -3

REGULATORS CONTROL THEORY

DYNAMIC ANALYSIS DYAI4C A LY,

++sin l I IIlm n n l:''''" _.

Figure 5. A block diagram of the servomechanism shown in Figure 3 Illustrates the general tenminodoe used for srvo system components.

Devices concerned with the measurement of the output and the transmission of a signal back to the differential are generally called feedback elements. Since the accuracy of the whole system depends upon accuracy of the signal arriving at the dillerential. feedback elements must be linear, accurate, and lightly loaded. Components of the typical system not yet discussed are the electronic amplifier and the solenoid. The purpose of the amplifier is to raise the power level of the error signal. The solenoid operates the slide valve. Such elements are called preamplifiers or signal amplifiers, and transducers, respectively. Using the general terms established for the specific elements of this typical servomechanism, a block diagram (Figure 5) showing at least the basic elements of nearly all servomechanisms can be constructed. In specific servomechanisms, some of the elements shown mqy not be p-esent, while other subsidiary elements. might be included. Sometimes two or more elements perform one of the functions described, and sometimes two oDr more functions are performed by a single element. Frequently the feedback path is purely virtual; that is, the input and output are directly compared, no feedback elements or differential being necessary (Figure6). Regulators. Typical aerospace applications of automatic control systems as regulators are found in the regulation of pressure in a missile propellant tank and in the control of thrust level in a constant-thrust rocket engine. In these cases, the objective is to maintain the controlled variable at a steady value over a period of time. Pressure regulators are described in detail in Sub-Section 5.4 of this handbook. An analysis of the dynamic performance of a pressure regulator or reducer is given in Sub-Topic 7.4.8. The

Teak Sui r___01t,01

Tonk

Re~otvevoe-eng=error

W-loo

0

Figure 6. A feedback path may be purely virtual. jin this hYdnIlc servomechanim, for example, .,he floating vallve directly sen.ss and corcts theerror.

regulation of thrust in a liquid rocket engine waz discussed previously in the present section. Figure 7 gives the block diagram of the pump speed regulation system of Figure 1 (b). This diagram is similar in principle to Figure 5, t&e block diagr-an of a servo-nechanism. Comtrol Theory: All types of seri'os can be treated by control theory, subject to certain mathematical limitations. At first, however, only a simple system in which an input 0,. causes an outPut 0. will be considered. First the effect of closed-loop operation on the static accuracy of control will be demonstrated by comparing it with open-loop control. In the simple open-loop system, Fig. 8, it is assumed that the applicarion of a constant input ,. will lead ultimately to a steady output 9.,, orA

7.2.1 -4

DYNAMIC ANALYSIS

CONTROL IHEORY

.rINU * SPEEUDPimp

rJLAOR_ 591(0rED' AnoSGNL VALVE NTOLR SPEW-

I(MAIN

TUMSNE

Isuch

(ANinte

where A depends on -the system components. Although It ',wodld be convenient for A to 'be ccn stant, this is impossible because of fluctuations As a result of An -the power supply and :load. fluctuation assume that A increas~es by some

AS

small amounta.bxif the new steady-state is 00 then(A

ROC

+ a) 61

(2)

.1

PUMP)

and the fractional change in output is', A

(3)

OUTPUT P`X SPED

flgW9'7. AloockSMSe.. &irutbapump'speed

Reg~itig5B'itfl1quite

Thus 8 gives a simple -measure of 'the inaccuracy f the airstem. If a/A = 0.1, the output -has -the same fractional error. Such an error would be unsuitakble in industrial controls. if a (closed-loop system, Fig. 9 , were sensitive to the same error, -then*eAe (4)

w~here By elimhinating e from Equations 4 and :5

Inputpu

outpu

(6ll lfubstitutionlofge, -sa A as0 thefore, andu ha showse -the fractionalchnei a n quatio 2:s,b .2O :a

I top

Inputa~

(dhmnge.'in the output is now only,0.01. 'This is ;a Mai~ked improvement upon the opnlopsystem. 16Makcing A large implies using 'a very sensitive' Nlow the dffpct of feedback elements on closedloop systems 'will, be considered. ;Suppose the signal fled %back to the 'differential is _B0.,, where B is ideally 1A. 'Then Equation 5 becomes

figure 9. AdtMd9mwAtdhmmt1ImnOftth IM 'vmpre.. w rtihisM111kdbpams s ffe Miuleh iTI ang~ mn@Wkqp .yilsme. urumcs amirii upm...- ~ M !d d -SI Ms MA tuIlevA MA M s

(8)9 :and from Equations 4 and,13

'ISSUED: IMY 3964

721-

INPUTOUTPUT RELATIONSHIPS

DYNAMIC ANALYSIS

If B now changes by some small amo-ant b, -'hen the fractional change in the outp-it isAb---

1diagram.

1 + A(B b--1I

(10)

If A - 100 as before, and b/B =0.1, then S 0.1; furthermore this inaccuracy increases if A is increased. In other words the accuracy of a cloed-loop control system is of the same order as the accuracy-of the feedback elemnints, no matter how rensitive the controller. This is a very Important point. This analysis has been q ialitative rather than quantitative. In practice the charazteristica of ayeterncomonens cn raelybe rpreente by corstants such as A and B. On esni ht Power amplification is always accompanied by time lags, and so a detailed analysis of servos must be bsed'n te dfferntil eqatins o moion which relate their input and output. From this analysis stem the standard techniques which make up control the,ry. Rasic theoretical L.echniqu.'r apply to those servos which are both continuous anct linear, that is, Pysiterns in which the error is m~easured continuously and acts on. the controlling elemrenit in..

Ispvt43atput RelaeaMoahps: A servo system can b~e represented an a abequence of elements In a block Each element has an input and an out-, put. Thus, in fig. 1 xt() is the input an(' t(t) the output. If the relationship between then. o the form y = kxr then P+any time the relatw~iship between x and y can be represented as a straight line, Fig. 2a. This is called a linear relat,,,nship, whereas y = 1czs is a nonlinear relationship. Here the relationship given a curved graph, Fig. 2b. Linear or proportion-i relations lead to differential equations whichi can be handled In a methodical and often simple manner. On the other hand nonlinear relationships lead to equations which are difficult, If not Impossible to solve. The general theory of control deals with linear systems. No general method of approach exigto for nonlinear servos, although considerable attention Is being given to cortain types of nonlinear systems. Ingerlaeem t vig nipus() and an output y(t), both varying with time t,

J-

r

pi portional manner. There are, however, two widely used types of discontinuous scrv,- on-off and sampling servcs. On, off servos are also known as relay or bang-.bang servos Here the error raust r-each a c~rtain 'nagnit' Ac before it, act4i on tLie conitroller. Then fullpower is app~ed to the servomotor through a 1AWROt or relay. There is a dead spot in the control for srnall errors, within which the system vi~n wandcr, The' magnitudt: of the dead spot is usually cnliical to the stability. Sampling serVIn

lmn

Figure 1. Any servo element may be represented by a box having an Inpu~t, x(t), and output y(t).

(a

y()

.

are

a1so

called

71I II II

pulsed data -,-d

definite coy--1

re~ct-', ser vos-. 11ere a measure of the ei.,or is obtained1 aC. r-scr.te intervals of tiie aiid Lhe cort~rol acts in a series of finite step:,but All servoa are nanlinehi- to some exten~t,buX0 very oftcr a good iLpproximation can be obtained by aassuming linearity. The justification for the as-aumoticni Aie in the accuracy of the predicted results.

r

7.2.2 A Bafic Outline of Servo MathematicsA cr,mprehensive investigation of control systemn petilormance requhxes a knowledge of certain mi,,Th'eratical techniques, basedi on differential -.1iuation analysis. These techniques are summarize, in the present articl-. Space limitations pre.'ent a rigorous treatmeiit.7.* 2-1

Figure 2. Relationship of Input and outpu of a servo element may be linear, a, or non-linear, b. General servo theoei deals with linear systems.

SSUED: MAY 1964

DYNAMIC ANALYSIS

INPUT-OUTPUT REIATIONSHIP6

will be related by an equation Involving their derivatives as well as x and y theaselves. Once again linearity implies proportionality between effectc. Thus for a simple mechanical netwoi . FIg. 8,,illit'

+

j-

dt

-f

: !1--

dt

+ kx

This is a linear differential equation with constant coefficients. Deshpt,f

that it is one which under steady conditions gives a sinusoldal output for a sinumoidal input of the mrme period. Although this is not a nmathematically precise definition, it permits treating certain nonlinear elements as linear ones when a airuttwhich, altl 'mguh soidal input causes an output of the same frenot sinusoidal, is periodic Pnd quency &v the input. Then only the first harmonic of the output is considered. i',e justification for thin lies only in the accuracy of the results it yields. If the input z(t) is known, then the right-hand side of Equation 2 is a known function of time, say f(t). Then the output curt be obtained bynolving(a. Du +... + a, D + a*) y

1(t)

(3)

To do thiu either the so-called classical or operatiorin methods of Afferential equation analysis may be used. Of the~e the latter is quicker Pndfar more suited tV servo work. Figure 3. In this simple mechanical network the output for a given Input depends upon mass m, damping e, spring constant k. and Nomeaceatur For any given input, the output will depend only, As

Residues of parial fraction expanion of

on the coefficients, such as m, f and k in Equation 1. Thus the element can be thought of anoperating on the input to give the output. Servo elements are therefore similar to the filters of the

a, b

=

Constaits yc

D = Differential operator, d1d1 e, = Steady-state position error f = Damping constant of mechanlcal system

communications engineer, and are sometimes giventhe same name. They arib also called transfer eletmts.

f1()

= Laplace transformation of 1(t)

The general relation between the input and

f(t) = Arbitrary function of time h. = Roots of characteristic equation SSquare root of --1 K = Scalar gain constant

cutput of a linear element can be written in theform

constant of mechanical system K = Scalargam = Mass constant of mechanical system r = Order of servo * = Laplace operator

(a,.D' + a,]brD+

D-

1

+

. . . +

a1

D

ao) ii (2)

-+ b, D + bo) x

T' T .= -... conantI . T= = Buildup time

where D is a shorthand notation for d/dt and

where the a and b factors are a~l constant. Any element governed by such an equation is

said to be linear. One importent property of suck elements is that if in input x, causes an output y3, and input x2 causes an output y2, then an Input (cxl +4 c x) causes an output (cly, + 2CW2Z), where C1 and C2 are. constants.

Td-= Decay time t = Time variable Unit step function t) --W(t) = Weighting function of servo

= nY(Jw) = = 1 ,(s) =

Input to transfer element Overall harmonic response function MCI Overall transfer function of servIfunction

This is known as liear superpoaitioa. It is sometimes given as the definition of a linear systemn, but since it holds god even If the constants are functions o1 time, it is not sufficiently prc se in this instance. A satisfactory definition of a linear system isISSUED: MAY 1964

Y 0 (Jo' ) = Loop harmonic response

= NeO* Y (a) = Transfer function of element Y,(s) = Loop transfer function of servo S--o Output of transfer element 1(t) = Unit impulse function a = 0.O1 W w-= Angul tr frequency, rad per sec

7.2.2 -2

I

-

LAPLACE TRANSFORMATION

DYtIAPt'

ANALYSIS

Lmime. TsnmstormattRProbably the best kwowt and most usetsil form of operatlocal calculus is Taplace traosformation. Even in moderately expertenmi hands Laplace transforms are pimerwful tools for solving differential equatios. brilay, Laplace transformation turns a differenta equation in which the variables ar funclions at tim,, t, Into an a'iebraie equation in which the varlbles are functio i of a new , iriable, a, called the Laplace Operator. Before a Laplace traftforrn is defined, two fur.tono which will be of int irest uIght first b*

.t

(g)

= LI(t). I 1 t)

,

To illustrate the applicatbfl of the theorems in Table 2 to the solution of differentiai equatioea,Theo,....4 'i and 2 are first applied to Equaeon 1 to give("w2 -+ la + k) y (a)

= (fa + k) X (1)

and since this equation can now be hw~dled algebralcally,,

(Fa+k) -( (a)

conuldeaed:1. Unit Step Punctio": This represents a uauiden change from sero to onc at time t = 0, ft. 4a. I' order for this function to be amenable to the mathomatical rulep of differentiation and integratlon, it Is de~inod as the limit oi a continuous fwsctlon, such as that shown dotted in Fig. 4a, &.a the build-up time r tends to zero. When definod in this way the function is called the eviL unit atcp function U(t). 2. Unit Impuic Function: This is defined as the limit as - 0 of the continuous function shown dotted ia Fig. 4b. The function is continuou2, equally spaced about the onrin kn~d its area zemains unity as r 0. o-- iefined in this way, the function is called the Dirac unit impulse 8 (t); it is the derivativc of U(t). Terms U(t - t.) and 6 (t - t,,) are respectively unit step and unit impulse functions at time t., The Laplace trpmzsform f(s) of a function 1(t) is defined as

"I'll =

~

b

Table 1-Laplace Transforms For Servo Analysis i U(t) 8(t) t" ... eat .. sin w t cos Wt

1/s 1 /..... .

....... ......

/(82 + W) a/(#2 + W2)

IU

(t)

7(e)

o-h(a

It)

e-,

df

(4)

.".

TI.{'.,

or as it is normally written (4a)A I.0

f(I)

2

1(t)

" dt

Making the lower limit of integ-ation 0-. Insttad Of simuply 0, insures that the full contribution of any impulse function at the origin is included. Many textbooks give comprehensive tables of "Laplace transforms. The mere important ones are listed in Table 1. In servo work, only functions which are zero for negative time are involved. The time origin, 0 is the time wher. an input is applied to t =for the system. Some ektremely useful theorems such finctions are given in Table 2. In connection with these theorems the following notation is used:

TOW,t (b)

Figure 4. The unit stop function Is defined as a sudden change from zero to one at time t - o, a. Mathematically It is defined as the limit of a continuotia function such as that shown dotted at a. The unAt Impulse tfnction is the limit, as i approaches zero, of the cotinuous funct -on shown dotted at 6.

7.2.2 -3

ISSUD: MAY 1964

DYNAMIC ANALYSS

LAPLACE TRANSFORMATIOW

Then if x(t) is specifled, x(s) can be obhtined and substitubd in Equatior 5. The o,;tput can then be obtained as a function of time, t, by Inverse transforming the riglit hand side of Equation 5. To do this a comprehensiv, table of transforms is very useful. However, the short list givenin Table 1 may be expanded by using the theorems in Table': As a simple example

Table 2--Jsof ;I Theoiems for Laplace Transforms Theatre No. al (s) + [Ia/ (t) 4- bg (t) jI where a. b are constants. d] -dt sf(s)bg

(s)

I

2

I (e-09 sin

-(

+

a)2

+

W2

[ f I[(t-

(t)dt

= f(a + a)to)

)

3 4

For any general element, by transforming Equation 2.

Ie,' f(t)J

to), U(t -

- eI . F(s)

5

X

Y (s Sb. \ . .... +... + . b+bo ) = .

a.sa-+

+.es.a

Y(*)

(a)

wheref(t)

to)

jI (t shifted forward by to.

where Y(s) is a property of the eler int orly and Is called its transfer fuwnlion. Obviously the transfer 'unction of an element governed by a linear differential equation is a rational function of a, as shown in Equation 6. As lomg as it is realizid that transformed quantities are being considered the i(s) notation can be discarded and x(s) or simply x can be used. It is possible to represent each element by simple block diagrarz. If two mu h elements are in series, the output of the first being the input to the second, Fig. 5a, and if they are governed by the respective equations, V=Yl

/

..

to 1o Time, P

Xz: yC(S)-

(7)

If

~f~ttt)u(tthen (a) = rJ

-

i)d----fog(i,)(t

-

,)d

6

'hen, since the equations can be handled algebraically z = Y, V_ X This shows that the two boxes in series can be replaced by a single bux containing the operator Y,Y 2 Fig. 5b. This can be extended to any number of elements in series. However, this is true only when the elements do not interact, that is, provided the output of any element depends only on its input and not upon the output of the sncceeding elements. This is only approximately true in practice. Serious interaction results where the succeeding elkients seriouslv overload the power source of the jystem. This technique can N extended to a servo syetern comprising a sequence of noninteracting elements and a feedback lo p. r.g. 5c. As shown here the system is a sintle-loop system. MoreISSUED: MAY 1964

(0) g-(a)

rI(t) lim

d

-

L.'m" (-ST0()

lim

provided this limit exUsts. Urn I (t) = lim IST(a)i t--*O + S--> 00 pro .4 this mit exists. If I (C) contains a term A 3 (t). then A = Urm 7 (a) and -+ 0lim 1(t) = lima Ieaf(s) -AilS--4

t -0

+

00

7.2.2

-4

SEMRV)

11PL T FUNCTI4OS

DYNAMIC ANALYSIS

Figure

5.Tow

dntti

oia.(my ) e

nkee

sa*

es~x.6

o

complicated muitiloop asth rlto differ sy myster only in 9.,MX where K sk a constant cald the c r O simplyba walc c astier Is wie therv ttemdowa, as rrr tail rather than principle. stmt of the "tm It Isometime -Med @imply Tih transfer function relating the output, 0., cle gn but this may lead to eomusloa with to the error, a, of this circuit, Fig. 5c, is called a simlatrly samed term. Irrom Equation 11 it foilows that the open-loop or the loop transfer function. This is

Y, (a)

(a)= YI Y, Y3

(9)

--

A.r(e)K(a) + a g (a) a

(14)

At the dI'erential we have the subtraction0 -,-,

This can be written in the mnre general brimIn(e)= () --

(10)

(

a-+....+&aS+ bo

where 0, in the input. Then by eliminating e from Equatiop- 9 and 10

' ea +...+4 1 +60. G(a) GOmparing this with Equation 6 sows that

_V, (a) =(a) =

/

)(11)

the eo

Is Itself a linear filter, operating on the

where Y.(s) which relates the trimnsforund output and input of the servo is called the cloedloop or overall transfer function. Although primary concern is vrith output-input relations, it is very convenient I work with the loop transfer function, Y, as will be shown. individual transfer functions of servo elements rt of the form ~ K KK (1t- a) .... S. K 1.+ Tja + T,.2

input to give the output. If f,(t) and hence Os(s) are known, then the output 9*(t) can be fond from (16) so M = 1-1 iyo (a) V,(a)j

SWVO input Fimetha: R is not possible to goneraie on the type of Input likely to be encountred in servo work. Indeed the kinds of Inputs normally encountered do not yield themselves to'

I

IA W--

-A

put functions upon which to base an analytical appomeh are choeon. They are: = 1. rj(a) 1. UxS Impulse Function, 3(t): Hem The output In this case is called the Weighting ftnetion, W(t), of the system. From Equation 16 (17) W (t) = f' y. (8)]or (5 18)

;2

and so on. If several of theft are compounded in the loop of a servo, as in 1ig. 5c, the loop transfar function will be of the form K t (a)Y. (a) (12)

Y, (a) =

W(t)

where J anC' g are finite polynomials in a which tend to 1 as :--0. Thus I and g are of the form(13)

This shows that the weighting functin is an Important property of the servo. From Equations 16 and 18 and Theorem 6, itcan be seen that if W(t) is known, the response to any Input 04(t)can be found from

g(s)

7- 1 + (T 1 ')a + (T

2

)1

2

+

...

7.2.2 -5

ISSUEO: MAY 1964

DYNAMIC ANALYSIS

SERVO INPUT FUNCTIONS "

()= o,

Ge(~*(8

#I (T) W(t-

db1)

oecteristic equation ame all distinct if, on the other" hAnd there are repeated ro*,4a such as (h-)j, the weightin funct.a n will contain terma

Thu th rspom t oThus e5Pwelghthig awmp~td 1pe.cmbe obtained. Impulses of duratior, AT, Equation 19 means that28 as

uc

an Bte'#. The moad general form) of the function is, therefore, written"'"

If the input can be thoucVt of

ana seies of( shown in Fig. 7._

A7 becomes very sml tho system in unable to

nguish between the series of Impulses and conUnuous input, Flig. 6. This concept is very helpful In assessing and improving the performance of existing systems, for If W(t) can be de-. tervdfr' experimentally, it Is possible to calculate how the system Will respond to any input. T.e diffieclty is to goerate an impulse of sufficiently short duration to approximate a 8-function. Generally, if the duration of the pulse is much smaller than iwy natural period of the system, very good results are obtained.Ifthe the form deoiaoroora)I

A typical weighting function of a linear servo is

-

W(f._.

I

.TI.E,

If the denominator of )(-2)

r,(s) Is expressedxresdi

InFi

ue 7 Yi ofa i e servo. v6 Figure 7. Typlcal iwig li function of a linear rs weighting f nto

S(8) -2- (a -

...

(S - ho)

(20)

when h, h,..., are the roots of 0(a) = O, called the cmaracteristic equation, Y,(a) can be ,splitinto partial fractions, thusAle.

-

1

--

+ +

(21)

2. Unit tep Fundim U (t): Hee .(s) = I/& and fro.mEq,,aUon !! ( (a) () -(24 FrorA Theorem 3, Table 2, it folows that. (t) = j w (t) dt

where Ap, A,,..

are the normal partial fractioa (25)

aonstanta. Then using Table 1 It has been assumed that the roots of the char-

It Is more likely that the response to function would be obtained directly from a step Equation 24. Thus expanding by partial fractions ,

00) CONMNUOLS

"N

/

A) o

+All. +

A.,-(26)

where, in general, n =(Ye- () A',()

1

T'areI

_i._

The general form of the output response, if there repeated roots in the chergcterisac equ.on is, therefore,*.

Figum . Ween sro hMput Is a sori of pulsas of sh0et dwaftn th snm Is unble to dslau" the pe frome cnnuods function. This coi a IsudIn 4proving assess and systw-

(t) = AoU(t) + ((A' + ft + 0'tz +...1-)

&j"

(27)

perfunnance.

ISSUED: MAY 1964

7.2.2 -6

SERV

INPUT FUNCTIONS

DYNAMIC ANALYSIS

A typkal rsmponse I shown in Fig. 8. Such a curve is very informative because it gives a simple plctcrial representation of the response to a sudden Jump in the Input. Thus, in the diagram, Ts gives a measure of the sensitivity, e. gives a meSure of the st-ady-state accurscy, and X and Ts give mome -es of the stabUity.

low rFuctio,s: cutput bere is & Bimo the Jr.eiwm rspoae. Istead of a real called sinuiAdl input, eg. sin ot, the complex form of a harmonic quantity will be considered, that is', = t (COS

I e + "in

ead)

(28)

wherm It i the frequency in radians per second.

Manipulation of Complex Quantities ) + (1 + SA060) (1I V0 0.513 Addition: Two response functions such as I--=1 ((I 4CU)' + 0.1912 (0)1 N~, and r'ae, must be added accordingto the parallelorpam law of vectors,.Sketch L = Nreft, x Nom, Multiplication: If NMe then N = ,IN; that In, modull are multiplied, and = *1 + 01, the phase angler are j, = a Y. (a) tan-' (2.252) + tan-' (200) 0.435 tan-_.43"-

-

'1-4.0I

2

added, Sketch 2.Division: It Ned* =X-j1 ,/NKe, then N = NV and * = 1- 021 f/2N eththesc To illustrate the application of *, particular

where 2 = O.o1lo. The loop response vector has been plotted for these values in Fig. 16.The overall transfer function in given by 50.4.2+ 2490. + 113,000 5

a, cri)pose Y.(jf) is known for frequercy. Then 1 + Y.(06) can be obtained by addition, Sketch 3a, and y,(I.) can then V nhta-ned by divison from Equation 36. is shown In Sketch 3b. N"rmeical Ezamp.e: Suppose a servo has the loop t-anafer function a.7(I++20.7-0 = 51.3 (1 + 0.02251 Y. (a) a (1 + 0.00435. + 0.00045.') The- the loop response function Is51.3 (1 + 0.0225 jo) (1 + 0.2j) J o d (I - 0.00004 6 + 0.00435 4)

F,(a)

83 + 59.6 s2 + 4690 # + 113.300

The denom]inator can be factored into (3+ 29.6) (is' + 30s + 3800). Then for a unit stp input ( = a +29.6sponse aso (t) = U(t) - 0.71 e-".Gt +

a'+30a+3800

20.70.29

29.71

nveirsm transforming gives the transient re-

Thus the modulus and phase are given. To

0.4,-151 (sin 60t - 0.725 cog 60t)

change to a more convenient frequency scale,Sketch IA'~iz Sh.tch

This is plotted in Sketch 4.

iiol

Al I

70

-

l

l

o

oaxis0I19 step functon iO)1+03

Red axis

YO Ploe

respo-w 7uIpu2

Sketch 4

C

0

OD)5

0.10

0.15

0.20

Tkne, I (scd

Ske"Ch 3

YC Plane

7.2.2 -7

ISSUED: MAY 1964

DYNAMIC ANALYSIS

SERVO INPUT FUNCTIONS

Time,

I

Fus.

T

nu "t tpfnto AAM cf oti~put Outputl Ospn coharsgep fIn spondkqi to any sadden ehmng in lnput

Figure 9. Servo respnse to pure /armonic Is s-nsInput o~daM and of dw~ samne frequency. The amplitude, however, is Increased in the ratio U:1 a" phase nl.o i h~e ya

(9)Iand I is the symbolic quantity for V-/Ci This is a dodge whl."- greatly simplifies the mathematics.. It Is justified because of the pinciple of lines ronperpostition, since the reel pert of the output ca be considered as the response of ecm wt and the imnaglnary part as the response to siD mt. From Table 1 01(s) = 1 (s-loe) and, therefore,E)=Ne~' X low (wt + ~)+

--

-- . -r'_

aWT (*G~(v] and called the ov~erall phase ansgle. Thus Equation 33 can be writtenJ sin (wt + )J(35) SeParating the real and imaginary parts shows that the response to the real inputs cosn w., and(a) ., --t W v rJ" V,-&-1%04 -

Ye~ ~Je Zd te A 1a(30)-(]

(a

~

~

~

- Al AM

)if4

P" a(a)

to + #) MTat is, sinuscal sin (-t input to also the responseand any puire har)monic of the sa==frequency, but the amplitude is increated in the

This may be expanded in partial fracties giving

fratiodwith

X: 1, and the phase ir shifted by an anglerespect to the input, M.g. 9. In pr.ct3cal

atkuF.

systems the output will lag the Input; that is,will be negative. It is poesible to draw Yand as a vector in the complex plane. If this vector it drawn for all

The time variation of the output Is, thereore,

it)

+ Bet

-

(e31)

frequencies between is, and 0

then its end point will

The first term, M4.", represents a bwom which ultimately disappears if the seto is The remainder Je'~ in the atesdyett6 freqmessc respose. The value of B is

trame out a continuous curve in the Y.-ptLne, as shown dotted in Fig. 10a. In practice, however, itahis. it is more usual to plot the overall frequency s reSpOwe charavteristics as separate Curves of X

(8))= "in I(a - T,(40)= Un 1"

(32)$I funfunc

and # Plotted a~alngt w,FIg. 10b.work Just as it is powible to(with

Y, (j)

ne, s te Ae- + l

tion funti e.es rs i taed smt

an aso et

s e dra

Je for e to a corIn YT(). Thus the steady-state reopons Y.(O= Xm e .) (l. (W ) ,,) plex harmonic input of frequency e is t. (Ui)

It is obtained simply by substituting

putt0ng ca ats bytcton s b puttingo(30

tn be =urvin

the loop fonssn Thin ill d Y.-aneThe (.6) (6

dotted

0. Mt = Y, 00oe ef"'

(33)

1 + Y, (Joe) It is usual to plot Y, (je)

Term Y*(J-e)

is in general a complex quantity as a vector in tre comn To do thistis ao must be expressed in the form; Yo(j,) = ut(e) + ivr(w) wherespo)is the real part and is plotte along tbe horisontal wris and v(s ) is the imaginary part and is plotted

which can be written oe. The) aueI of Bplex-plane. where M(io) = IY,(Joe), sometimes writttn an (0./8 1) (o-)1. and called the osterai! amptude ratio; 4(w) = t ars Y.0ed )] sometimes written

vertically, Fig. 11.

ISSUED: WAY 1964

7.2.2 -

SMAUMTY

DYNAMIC ANALYSiS

jUW Ln1Oe~(

0

0 0 - ,, (a) -

ii

i

....

!sd lwv. topo.l. OMA

NPkne (b) rqecyrsos __s asIpaoami.otm fig dm

m~ It th P F, I OWMo thero R t cwv plo be~l a1 honmr.

I

dogd beyd ame I, I - w &anm it is . -o arm AL-s

-',Is by

a VactndIn

4

N

Joop gaO or loop anpltvd ratio and #.(.) = z [MO.()] = t~-1 (v/)') An 1, the IM loPphaft aglse. the tipof the a As - invaried from 0too, Ir. (Ow) vector will trace out a continous cuem in the Y.-plane. FI. 12. This curve is called the oop mvetor Was or the Nyquit plot. Its very great value in servo anl ysis will be discussed incaed the

_ #P gadsk

he next article, which will deal with performanee criteria.

7.2.3SFiurm 11. 'te Iop humonic funcmon Y.(I.) Is umfy plotted as a vector In the complex p=as The mal part., u(), is platted on tih houzota axsny. t ti p

riteria for Evluaft Perfoalace

Swvo System

Pqrformance can be described generally in tena at two qualities: (1) stability and (2) rae.P Sthailty describee the ability of a servo to settle down after a disturbance has been removed. It Is closely related to the response of the system. m is the term used to describe the accuracy ad ty of the system when responding o s-me input or command slgnL 7.21l STABILIT

Alternatively, the loop harmonic response funetion may be expressed as Y O) (1) =

(

(37)

The formal definition of a stable servo is very dear-cut. It is a system In which the output Is always finite, or limited. for any finite Input. An unstable servo Is one in which the output drifts away from the Input without limit. This dom not nessarily happen for all InpLte, but if it will

7.2.3 -1

ISSUED: MAY 1964

DYNAMIC M4MLYSqS

STAOIU

SIW (14t t must exist and be finite, where W(t) is the weltgtInlg function.' In practical servos a suffiient orm_-4) 0_* R"o ones v_

~oo/

dit~lon In that W(f)--*O as t-noo. Physitally thin, meens that the output must return to fts initial position If fThe wyaten: is g~ive a sudden impuLvive kick at the input.

Wi *K ito

The most generai expression weighting funcUon In, **OO

for W(t).

the

~~W(t)vP

-'- (A, + Fljt 4 CIOt (At 4 Bit 4 .. )'(1)

rh-.1(, +

Lmo d

~I AJ

O

Time.

I

Ftewo 12.

Tt*

w

mlea vialer lews,,

N

aiie IoI

-

of

tveM In sne ensdyaitIs ten almue ft~ b"n A_._f r4SW~~ fiZ PuS,14 for Vaue of w fmn wei ho kiffnty.4-

,,I 0,(?0

Sfosainputbapa.,

then the system is obvi-.o;ly

V

uisatlsfactoy. The kde of output Increasing without limit Is only a matheatical concept. WhatI pm -tics is that output will only In-t~

-'Tn

1. -- -1

zatti

h

yimh

damw,

or until some nonlinearity Intervenes to conFle

(b)

strain the output.AltbovA this definitlou gives a definite divistablS lid unstIle m the term stability Is gally used In a relative &a. Adit bet

1. A sewo eydy m wth tised rWoatve ablflty characWtsm a. may be defned as one hevk a T.. equal ha Owlt decay tn,. lb. A rlative abm* tour time the budup yti. unltabl syaetn, b. has ovoid ofo.2 a".d t vrn.The teld 10 tabes *UM@l the des . howew.Imm r syst posis bsolut bad e

system with good relative ,tsta

ty ca

CtrustIcs.

owwiftet o 0. and

WiV 1a6 might Ihave a maxzium overshoot at 0.3 and Its oedllatio"w would decay in a comparatively short time such a four times the buildup tine.

0a the other hand. a system having a maximumovevhoot of C.8 amd a decay time equal to ter times buildup time, Fig. ib, although stable in an absolutsens would be said to have poor relative stawlity characteristcs. *e mathematical definition of stability is thatISSUED: MAY 1964

stablty sice oeaatlons dkl

7.2.3 -2

SUASOUT "WIQUI CRTE*MOare all thle values of a whlichyk~ msele O(s), the denominator of' the overall tramo for function. Y.t#). set . Each h may be either real. imsaginary. or In the most general caseco--_ p'~ea Any complex root can be mrtten in the foram A + PQ. Presence of mach a root 4'adicates a damped sinusoid in the weightiag ftua*ion. Only It a is negative will this omeeflatiom decay as thnie t increages. Thus. a neressary earnditban for .blllfy imthat .1* the roft, of Of a) =pones.t 0 Ptuat poass" a sWehtv~a redl pert.Where h,, N', owt

DYNAMIC ANALYSIS

-O,() --

i.0 go, that when the Input f requency a, is equal to U. the partia fraction expansion for 9. (sa) will cmntain the termn. C/(s PIPl. T'his results in the tarn Ofe" in the weighting functionu. T'his cmof the response is an oacillation whose sucocessive amplitude. increase linearly without(w - b,) 11)(0

(4

is aloaroot. The presence ot a purely imaginary root. say a 1. is to be deploreid. It does not satisfy the above condition for stability sad meanse that there Is an uvalamped oncillatlom in the weighting function. With a periodic functiont input of f requency ii., the output can Increase without Ulmit, at least in theory. It is therefore. Posible to it vestigrate the stability of a servo by finding the roots of the characteristic eq~uaton(a) +A.I 6 o n-1

fal rutlsome nonlinoarity, such as saturation of the power source, intervenes to limit the ampltude. A self-maintained oedllatiun is thesme.t Iup. This phwromrroon, called hunting or limit "cvling. will only ocwrur in practice where a dlomed-loop aequemvn&x.m~uors a power source. Self-maintaitnsc oscillation in other spheres (for example, aircraft flutter vibrations) can be traced to the same. cause.

It is possible to plot Y,(jai) agaii

'

rrquerwy.

+at

+ e

0(2)

Thus.I. r,.f=M. r (k I___

This rAn be very tedious ift '> 3. am it probably will be in moet servos. Further on rapid wasthods for investigating the absolute and the relative stability of systems will be discussed. There ars certain helpful rules regarding stability based itipon the transfer function. +-

b, + NThus,(2)

)__ =(

.(a

6. s +

..

+ at 8 + 00

of a linear servo. These rules are: 1. if ?m2.If

the

to

,.1VI141

una

ony of the a coefficients In the denomninator is negative. then Ume system is in genera ur-

stable.3. If a. exists and any ol other coefficients %._.

ss eg .

zero, then the system is unstable,

It must be realised that although these rules cama reveal an unstable servo, they cannot prove that a system is stable. In other wordis they ae inot suffirient tests for rtability. Frequsucy Respume arni Stabilty: Suppose a - 1 is an imaginary root of G(.a 7 0. Then for a complex "iusoidal input of frequenacy .,Nyquist the transformed output is given by

I;FI'..(~+ G V~i' l'berefo, if A )a in an ima~ginary root of Ui(s) =0, W(i.) will become infinite when u 0=1, ViU. 2. if the overall amplitsdoe retponse curve becomes infinite at %ny frequency. it indicates the presence of an undamped oicillation in tse weighting function, and therstore Instab~lity. A servo will also be unstables if there is a root of ior--um~ ~~ o' + J4 mKfJWU case the amplitude plot would be the sane if we replaced the u otable root by -- a' + 1'r This method does not give oonclusive proof of stability. although as will be shown later, once absolute stability has been ewtablished, M(we) and #(w) give useful information on r-lative stability. Obviously, somec ximple and conclusive tests for stability would be very helplul. Two approachem to this problem will be outlined. They are: (1) The Nyquist criterion mid (2) algebraic criteria.-16 kT-V-L h

Criterion: Thia utilizes the open-loop harmonic response fuliction Y.(jw), and is based

7.2.3 -3

ISSUED

MAY

1964

DYNAMIC ANALYSIS

NW1UWST

*TADX1

WV

N

po/ -

Rsg~mntLom%

'

dl cof~rwh'a

+

+

t neots!. *o OS comutan to

0-00

Pone 2i

__

__

-resPOnd

Figure 3. Line of constant . and ,' in M s plane owrto simnllar oonkurs In the Y. plaiewhich depend on th, funvtn Y.(s). This is known as coetormal napp , The small shaded "square in the . pane cwv4spLvs in the thnR to '. iem. ma.ll shaded area in t

Frequer'y, w e 2. e eIasv ri oththe

se'. frawe4m. a, Otars a is ressNa PeW ssem would be enqmwp of the Isrv So as hcy the s pa" en and fallet n al -pdeao

PsIeow types of 5 shOfto.l IN Fita" N Sintoary rest of One donemkoner so the oper&- serv bassim flw*"n, it hidilafr that at amft" udeVA OW ravhtoied

nfin ifty.

+IO

an oothper upon the Properties of functions of i variable. Consider first the loop transfor tunetion Y(a)oo where in pelerul a isa compnt number orrosp.onding to each + j.. of the form a . value of a ther is particular value of Y(*), This can be shown by showing the value of a as a point in a comple plane called the a plane, and the corresponding value of Y.(#) as a point an another compex plun, called the Y. plane. Ctrresponding to a coutour In the a plane there in a contour in tht on the s Y, plane. The shape of the 1-tte. ,.calle crA function Y.(#), and hence on the plrase.rthe servo it epre eor Thus, if the a plane is divided into a net of a and constant -, paraleol trnsf lines of costnt toamal imi saoxsodn ~.3 br eorralm inthe qartn M~W'he of I lines in the Yr rof e. This is called coafomal oin tlltha th l~ta uipwi uWr Formtio ThI- imortnt zJ. 16 wtaU-. I-w.Fistil

v0a.I_-

t

o

,y 0 il at

5 4 The t sa

to.)

r Pl0"

(b)

th4xs

must ceNokm %or stsablit Is that the on the Y. plans obtlned by conformal in the io s and of raon ai htods b bouneshaded egion in the)s plane. The true P1rovde all sy"te ta) Movemevt .s)hadO. eqatin wihi thame themgym ftelee. ipuit notl in 0 shal ne~ot falsihown thhaded i

function, and it cextzinly to for the linear aervoo being zonsiderad, then anuill squares in the s plane correspxond in the limit to small squares in theY.F&sormatioss.

In otlm

words (S1I

+ J-41)=

is a root of the 1 + Y, (a) = 0.

characteristic equation G (a)

The the same ame traversed inImportant The point 1- + 10)) Y. plane corineponds to the a ph.e. That in,

plane, +

This Is called a co--foa true point inas will be sshown. sense, that the

For stability al must be negative, or f aT + ,ut t must 4s. Corresponding to this region there ng. not lie in the region shown shaded in

written 1-, 0ov in the a point (,x, + j.) in

r. (a,+ jol)(6)

In a shaded region in the Y. plane as shown in Fig. 4b. Because of the previously mentioned cOwformial transformation, thi region in b ounded by the contour Y*(,e) and lies to the right of it as the contour is traveraed from#a:I au thrWA ats 0toe.w 4- a. The condition

ISSUEO:

MAt 1964

7.2.3

-4

STABILITY ALGEBRAIC cRtrERIA

DYNAMIC ANALYSIS

for stabilty is therefore that the point (-1, 0) shall not lie In this "haded region of Y. plane. The condition stated -iolds if all the elements_ in the systen ame themselves stable. Very occasionadly systems do contain uinsatble componen~ts,

usually due to some icxal positive feedback loop

I ,)

R--

Ra

A--I

(-10)

0

040Ra

__0e

( ,G ). In determining the stability of thws-WO Y lae WO Y ln cale onmlinimum-phase systems the exact form of (b) (a) teloop transfer function must Arest be obtained. However, they are sufficiently rare in mechanical k Figure 5. Application of the NyqLIst Prltsro forstblt servos to be neglected in this discussion. They will can be skipffl~lfd In practttigcth be discussed lu. a later article, Th 0 to I'. (j)e) contour only frorte Li the couditlon for stability Just sti~ed it would conditon for stablity then becomes that the, point (- 1,0) must always be to the left of the be necessary to draw the whole of the Y, (I&) contour inludng lare crcuar rc. he wee ofcontour when It Is travenied in the direction of tourincluing larg cireularam. Te swep ofIncreasing w. Thej* for anuntbestm Is shown at a. The contour atb flilh this arc depends on the power r In the denominaconditionsfor stability. tor of the loop transfer function (Equation 1.2, R.ef. 2). But in practical strvos it Is uninecessary Wo go to all this complic tion. lf the (*j)contour from w=0 to.,+ ao is plotted, then above which the servo becomes unstable. In the condition for stability is: T'4e point ( -1, 0) patcfrgo eaiesaiiy utb reaiesbltyKmute prdcfrgo must alw~ysiae to the left of t~ie _ontour when somewhat less than this critkal value, as will st fi trversd 1 th diecttn c' inreaingset ~it ittaesd5 h ieto fices~ be shown. From Equation 7 it can be seen that Fg5.A contour passing tbikough the point changing K merely alters the scae of the Y*(j..) (1, 0) represents the critical s'abllity boundary. ptasiisfeenlcled cotuoN.qa The Nyquist crit'-rion can be given a simple physical explanation. Where Y, (jio) crosses the Algebraic CrItei.a: These are expressed in terms negative real axis, the output lags the error by 180 Of relations between tse coefficients of the powers degres.Thusanysinuoidl puse ntrouce as qain ntecaatrsi o an error passes .hrough the loop to the output and is reintroduced as an error 180 degrees behind 0 (a) -=-a an + a, a + ao(8.

the initial pulse, as shown in Fig. 6a. The amplitude of this pulse will be iY,, times the amplitude 1 at thi of the initial pulse. Thua, if IY.) frequency, a continuous QuC!uilleUU can be maln taiiied, since thia second puise will cause in equal and opposite one to be introduced, and so on. If jY~t>1 at 180-degree phase lag, the oscillation b Obviously the will increase irtapiueFg desired condjijon for stability is IY.I 1, the osilto Increas In ampltude, b. Obvkmft for as tabis systern iYoA must be less tban one. R Is this condition which the Nyquist plot serems to evt~blish,

passe Uthrugh tC

Impl the output and to

and the condition for stability insiainmycm 41 a= :> ao 43 As a further ilhmtration, If ( &4 then J A= ,al Go 0 as % ad[ 0 a4 a384:.

(12)

+

all

8

+ a2 as + a,

+ a,

(13)

from that of servomtechanisms.

Here there mnay

(14)

be very large time lags, especially in the plant itself. Since inputs are so variable, the analysis presented here will be performed by considering the response to certain idealized input functions. The choice of which method to use for design ]

and the condition for stability in a ( >a3=ao : a) as- I)

Thiere are other similar algebraic criteria-fzor exIample, Routh'-,4 criterion. Although they differ in S~method they give the mmxe results.

posem is purely optional and depends ultimately on the preferences of the desgner. Each met~hod has certain advan.'ages and disadvantages which will be briefly outlined.

ISSUED: MAY 1964

7.2.3 .

RESP(WfSE CRITERIA

DYNAMIC ANALYSIS

The

mnlet method in ususl'.y based sponro

disappeared at rimie

td

V/at., where at is ftile

to thes Heaviaidu unit otep f-wetlon. oft res'f IV(t). Rsaults ame mW to interpret when plotted graphtci)ly, h'ut they are difficult and tedious to Oitain because the chametter~stf equation has to ~.solved, and then the final txpronslon plotted in graphictol form. Another big disadvantage is

magnitude of the smallest rcal component of "~ the rn-ote. The number of omxcllationm depends on the ratio a/01 for 2,.wb of the roots. IL value of about 0.5 Is usually quoted as satisfactory forthis ratio.

A measure of sensitivity to given by the build-up

that If anly paranaetet, is changed or if addlitonaelermets ars put. into the loop, the whole proces has to be reworked. It in also very difficult to aseseclate any characte-Utlc iin the response with

time Tj,. This has been variously defined as*i. T1me to pass through 1,0 for first Urn.. 2. 'rime to get within a steady 2 per cent of 1.0. S. Time to &Wi-g through 1.0 at malxlifuMl rate of responsa.

particular elements in the loop. Thus, while transileat response canl be used o identify a ;,.ood or bad system, it does not often suggest bow to modify the system so as to improve its response. These ftults become very much worse when the degree, a, of the characteristic equatiola is greater thin three. With frequency response methods, mathematical labor in shorter and simpler. tliso In this dir~ction some simple aids exist. Thjese wi'll be discussed in a later section, The 16Teat advantage of frequency response methoois is that the effect of modifying the elements in the system, or adding new components, can be eassly accounted for. The disadvantage is that the response vector curves do not Irive a physical picture of system behavior. That means that a set of rutes inust, be available to correlate frequcncy response curves with the transient behavior of the system. No concrete met of such k- es exiats, unfortunately, but there are some approximate riles which will shortly be gven.This Successful use of either of these design techniques, therefore, depends Ia gely on the skill of the enigineer. Only with experience can he weigh ry alu crt the esin of 'Ia.2. In practice it is convenien. !:o lo the initial desirn work usinic frequency response methods. Oncethe design has meen more or waes fIuMals-cU A LXA way, then a check can b, 'nade by plotting its transient response..*A..

Based on the overall response function Y,(jse),I the requirement for no steady-state positional errorlInthatiX I when =Oor 0, that a = be,. where b ,j 0Y'(w -- Neio = (

(16 In practical servos %> m, so that N --) 0 and is negative answ -- co. Thus a typical response Is of the form shown in ftg. . The amplitude or Mf(m) curve Is verly informgtive. High resnant peaks currespond to lightly domped roots in the characteristic equation; that is, a/fl is about 0.2 or lses. An ideal type of characteristic in shown in 1ig. 7. If the maximum vaue of M is finrited to 1.3 or 1.5, then in general a grod transient response in obtained without tomn vrhos-

o +1a, j +

..

+ a.(j,~ 4

Sestvt

is~ de~terindbyte

SniiiyI eemndb h adit is variously defined as: 1. N(ajjb= 1.0 beyond resonant peak, V. responsle in of typ shownl in 1'. 7. A( d.'-f

adwdh5

1.0 holds for curve with no reso-

~

p~S ndwidiih,_W&

4 o-

Response Criteria: i;9as#" on transient responseto the unit step function, U ( t) response of a stable system will in general invoi e an overshoot, followed by a decayinrf oscil~ation. The response ratio

Imliw_______

g0Fyeque--,y

W

is generally considereL satisfactory if the 'naximu .i overshoot is about 30 per cent cot the step, with only two or three large overswinga following0it. Fig. Ia. Less than 10 per cent overshoot in some'times necessary. Fiue7Lmtngheaxumvleo to13o1. Fg resf iiin 7 tae aimumealutasen rsons ctor1.3toe1. N

Decay of the oscillations depends on the values of the roots, ( - z + jQ), ol the characteristicequation.7.2.3-7

istic without too many oversihoots.

All oscillations will have .4ubstantialiyISSUED: MAY 1964

DYNAMIC ANALYSIS

MUIW

STT

IEfff

3. X (mb)

=1/2 beyond any resonant peak.

Since both re'r~te to sevaitivity a relationship between the bandwidth ab cud the build-up trnme T& might be erpected. There in an approximate nslationship between the two, but goner %lly nothing more can be said except that increasing___ the bardwidth reduces the build-up time and hence improves the sensitivity of the servorincehnlam. An approximate relationship between the two can be established if an idealised frequency response, ftg. 8, is coy sidered. Here M up to the ba- -dwidtka frequency w,, akd itszero for all1 higher frequencies, while the phase angle in linear in bandwidth. The response of a system, haviag such a char".teristtic, to a step function in r'%own ir #1g. 9. This response hau a *zmall value when t =0,ao the mystem is not physically realizable. Using the third of the de(Wationw of r, previc a. ly Wveu, it can be shown that

1 Feny

Lfw

Figure 8. With the Idesktad frequency rep ns shown hers, N = 1 ove the enitheabsndwldth an m Ow4*hou th erW

This fripplort,, the previous remarik on lvz~ceaslug the bandwidth. Gtnerally to hcreaae w. to chieve amorerai repnse theacalsr gi " le.4 to inta~bility, ard saotinvartsbly means a more oscillatory reaponse. Theref~ore. a compromiss value for K~ must be achia-ed. One of the fundamnental problems of servo design Is to ge t the tnaximum possible bandwidth for a given scala gainK.

~~-

0

-

Tm~

ii

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_p

Fire9Terspn oassemwttmciacrFiur TheN i.. ofasy Soposo the 1U#aIwk, a (. Iiuthstes the re~tok nswm between bendekth and buildup time. Response of such a system ks much asdesired; howvr, the visbm is h y si ally u nre a l aub le si ncs r e s pon e is sc,

4, Ri= %Ant = Square rod. of mnus .e (symbolic) lm K = Sa~a~ ~86sady-State ah~

r

Ord~er opsra*or

Zmrrs: Apart from sensitiveity and stability another important favntor in afsewasng per-

U (0= bW'.Aep uncionin W~t) = Weighting

the response to a unit step function liaput Ut). Fig. 10s bmccon

STMM-STATIE ERSORS

DYNAMIC ANALYSIS

0lQ. FMMMrI

Tinis,1

0

Tinte,

1

ke ti acnumeY of a saerm~w IystidlGs by Via eeystel~ OW In resome to a unit dIkwielon bup.*. a. Dpewim aeueycI be asomani by finOWa thu obeeit. obe at In uepnw e I* a tiit imi Wvsetsc hipui b.

can be ossily found. In the case of positlos mince this Input U (C)t is csall a mnkstoa~p "Owy Mrg. 10b_ Occasionally. in some pos.ition cop-tiol servos, tt a output must be able to follow, with & mnall steady error, a caustant acceleamtion input. such an inPut Is the wagi steP .ccelerstlos U(t)ts/2.r t' Tho difftculties involved here will shortly be dilbs relatleusbip between the error and ingut, is

Obviously, for save uteady-etate error, the require1a ment is that r g '_ If r = 0, thunis Asttic error of e, 1/ (1 + K). For aunit veloelty step input 06(t) U M Et(a) = 1,*. "jbjmIojre. tt, -

I..+~

3

susitutlndj tram squation 7 givas(s)T_

Thum for gero steady foll~wlag error r ;' 2 is

##(I.) Kj(a)__eg(&

(19)vg~~ha)b

required. If r = 1, there is a steady tolLWlg 0flM rO=t0,thenfte fllowing 42102is a,= 11K. If creases without it-it. in other words. the servo is IeOf following the input. ic -%----.-. I-A- 6w 110.1 Oft h bm axtended at will and is symseetrical apart fo thet first term. An the table shows, a flist Order serVO (r= 1) has a astv static error, but a Itinite steady following error to a step velocity input- A secondorder servo has a nero steady following error for step velocity input. For this reason secomd'order

Now Theorem 8. Sub-Topic 7.2.2, is used to obtain the steady-state error. This ist-Ur

[OMt)

t

=

UM [as(*)) Pooa

#*'it0.Probably since Bg 1 as Tim U10 ceaf of a unit #AV fucto I/*,, X0sha t, a

servos ane frequently called sero-veloc~y-*fYfWf h most mechanical servos anr first-drder kind. Where high dynamic accuracy in required, for example in pr i-control systems, s and and even thire? order systems ane somotimes ~used. Here there are Inhemet stability problems simple example.ISSUED: MAY 1964

-

I. ~~J(21) orTp.2.3 .9

r

to be solyed This will be illustrated with a very4

DYNAMIC ANALYSIS

PERFORHAhCE CHARACTERSTIC

Table 2--ke*Wy Following km's

:1

lip

5 1

1I/K

-

201

0

FPurSuppose a ascomoed-w-e for funtion, mmr ha. the loop trams-

1. Th'e wsebr OP In this ~Idk

oweet the Voilue of YvZw'p

NqltPlot repman reuny

X, (0)

(2)

Thusmoo Y, (in)

where the effectt at tiam lags have been neglected for s1impicty. The cbicsrsh equation Is therefoe(neatv 44 + x (26)

autofhidvsonreA(r)

OP

(A)

as shown).PON frmn thi process for a number of frequemlew permits blottiog o. K(.) sad #(w). This le a rather tedious tsask it can however be avoided superposing curves of constant X sand # on the IY. plame These contours are orthogona circles, / Fg.12 N cositouro, have their centers at [(OP - 4) + 10] a&W radii a~IN/(N -1)1. The contours have their centers at C>1,-W cot as previously suggested the =Ladmun aniplitiade ratio is limited to 1z, then the regona shown shaded in FIM. 12 is prohibited& The servo having tao 100op Ie5POMe shown plotted in Fig. L"2 obviously has a mwaimum overall amplitude raU, of Since changing the scalar gain constant it changes the scale of the Nyquist plot it obviously must be met so that the curve does not enter the region. The best way to do this in first to plot a curve off~OW'

*

Thishastwotmaivay rotsby ad "Act the goV tw syistmi uns ble.u 57551 1 UtbI Now ""a"e the synte= loop mrants, funtio,

is aUinU

a tor: (())K(1+

?eIf

The chrce~tcequation is now The system is now stabW sad awl retain its se-o velocity-error chareeteristic. Thin type of problems wil be investigated =or %woughly in a atr articleprshblted Nqmktnod: Use fen in a typca NyquWs plot, FIrg. U1, the vector 7,00r for mome peaticular twoMP top t quency w- Then to the @am scale the vector A P tset1 + T#(Jw). Then the overall re "powe r6Pre funlction can be found by division.ISSUED: MAY 1964

W O)1 which is just V. (j.) with Kr om0itted. Themn the critical stability polLt is (-/Kr, 0) Instead of (1, 0). Fig. 13. Thus instead of altering the con-

7.2.3 -10

P NOAAMCARACTENOTM

DYNAMIC ANALYSIS

towr whom A Is changed, the critical stability point Ws noe until the contour is Ine the rHot position relative to It. UO, riumately, cuanging the critical stability polut lir olves changing the scale and location Of the N end # constant contours. Thene awe coaructions for ignuring tb-it the critical stability point (-/K, 0) Is ponltloned so that the T*(Jw) locus Just touches the reqluired At contour, thus fixing the optimum value of K.

(__A___

a

0

Aeo

Ark

Figure 13. One wany of InnsshW estlsebaen of the Nyqulst caftesen Is to mom the qdhleso ~b~yPoint by dwrnging theveeeltesceran cenetant. Cenenasblen u~~ tat h 4 kmnc just teuches the -nq-1e value to ftx the ephlimw vauen of K.spouse shown in Fig. 13. Although this satisfies

00,ro QW)ThisAl,1.0low

Figure 12. Maotting a sewim of orthogoal cirie repredwsetn constant M and # In the same plane as theObw Pk* Ny plot lie d tIn- s* of henmnumsiofwerat.sN.Th het telsp repve pie 'Iw has ama-IMt nvmm~~ naft

vat

go1

COl the optimum values of 0 and P, the curve comes Ams very clone to the crit ical point and has a hi191 maximum N. Thus the value of K would have to be much less than that predicted by the above method, unless the locus in modified to give better characteristics in the neighborhood of the critical point. That is probably what would happen. The order of the servo and therefore its steadystate errors are also revealed by the Nyquist plot is because Y*(jo) behaves like K/(ju)' at frequencies. Thus for r = 1, the curve approaches the negative iwiglnary axis asymptotically. Fig. 14. While for r =2 the loop response locus approaches the negative real axle asymptotically, and so on. This is particularly useful if only an experimental Nyquist plot is available. Thenif he order' can be found and K JIMknown. the

steady-state errors can be obtained from Table 3

Ttese constructions are somewhat complicated aid some prefer a simpler method involving two figures of merit known as the gain margin, 0, and phase mar gin p, Fig. 13& Deslired values are: 0 from 0.5 to 0.8, and #? from 35 to 45 degrees. Thus once the point A and hence the value K possible to plot Y*(jie) to the correct scale on a graph coritaining contours of constant M and #. Value, of the gain margin and phase margin is purh her seinobtinngvey imlyan apurl' inobtinig vry impyY. hei usreliable figures of merit to assess performance, although some have used them as -ich. The danger of doing this is demonstrated by the dotted re7.2.3 -11

As previously stated, second sand higher-order omwenituire e(a) = Tanformed error 0 =Gain marginX(. I-)=odulus

have been fixed to agr". withx those _figures,

Ii

Iq

K =Scalar gain constant of Y0 (jw) r mz Order of servo T =Time constant

proximate best gain constant K. They are not

U (t) = Unit step function Ye (Jw.) =Overall harmt iic rexponse function Y(a) =oo transfer function

0=Phase margin

c,(a) =zTransformed input *.(s) =Transformed output (w Phase or argument of Y, (jii)ISSUEU: MAY 1964

DYNAMIC ANALYSIS

TRANSIENT RESPONSE CRITEMA RELATIVE DAMPING CRITERIO

(V -21

ti.1

t" )~

FiurPIloop

18.4 qu otes h sefgrve. forth e coeffW6Icnt&A k *~ to aendsith buadu Itime.e mscod-ore thn usde ondlf tale a e the 1w~ h iespora+umbe large . + &Wmoif

a irst orderstyu the ~

ta..creFg

15Icr

~ cnb lomohc y eresete by thtrhomeuntale moifin thefth

tnaimum ovrsoo an iluyrteuss 1-

the/fbuldu time. al, eon-r

ore

r(1

+f sev awT

V-0

ther

arSem

nber

l

u

onditianall Cum

ablWI If

as (as +anfatr

-si

-* '2

s*4- 3 11 03 ec h

abow am sevtod ordeiogerm with onot rcticme to n Tesaa athep toop ehich an bhe transienterbspone iepn s telrumieso=

.50

.25\I

mae

t6) giv

reltins bewe4h prmtr-ooso thi I nt osibe.Howve, n ttmp hs ben

toa

te

W haraterisOticmeuato Paramet0. fncio is* largel deeriedlkii by ter

Thieeys aregivns

figen oeurestable. 4, armormilyi frngo the for uyuutom arodn to thei orde

form60 prvoul

sttd C3th1mgntudf 0s

ie

fction of~te fomd thneis. er hkrra-'t-, gh As "+m zttit)

stfl

Trs"Isat~ ~nrIfsvUreainbtwn

~

C.a~+ Sam'

~~~~~yceprcsca the

,,

~%

decayonll amliud 3ome cycl of oilaton pe fi2 the minmu va 2 4. ofti2rtoas"! 1 0.5.: Tisgie a deca I8 h rai of 0.20 per alf ado Thatesi~trafoaTase 0.0 ini Fig. 16.u epns to/A 0.5 is used thtiiumrtoofoo eacheer of the characteristic equation moot 0i=0.Al ,oma +jivsrs oatr

+aet givforED

MeraYn optimu

type3

ofrsos.ro-12h

POSiitIO

CONTROL SEMV

ANALYSES

DYNAMIC ANALYhSI

eatAtyiekWa

Al A

As Uses came by a Fle &Thime requlvford th e Fue1.banelss to dle &9aM Is dwessmhts wokse* of a. The raft *~ -.-I----s ft=ee t-nok-f P er ot seulleis.. to thlw au of &/a a stbe'ewv Is OX

Equation 32 Ie a pob~s.mla in a with real xWuf&iLvnts. so H4itrorts' criterin cam be applied. This deslrea relstlouIvAlps between the "ystom coefficlaents. Th. method nay Involve some tedious numeiewcal worik sice the degree of tlhe charactertlc equation is doublied. Some simplifying techniques have been developed, Referenco 436-1 but theme are too lengthy to discuss here. Others have developed siminlar criteria locating th roots in Gther restricted reitom ot the a pians For inatance. to insure that all roots ame In the hwdsd region in Fig. 17b, mnagnitude of all the real parts of the roots is mades greater than a certain value e,. This means the total oscillations will decay within a time determined by .

the roots =*At lie in the region of th 1'k shown shade in fit. 17., Makin trig restraint permits the Nyquist and Algebraic stability erltarts to be modified so that they become relative as Well as absolute criteria. This can be mast easily done with the algebraic criteria. The modilfled characteristic equation is0'(a 0

7.2.4 AIelyZI a Seam SysftmuThusiioint response criteria provide methods which ar particularly suited to the analysis of relatively simfple servo systea. Conrmspmdiingly. the @quapsbip . be relativiely tin (bir" th ytmms w' I be A simple position control semuhair analyzed in this Sub-Topic. Sub-Topics 1.2.1 through 7.2.3 have outlined tke fundamuital concepts of closed-loop control, brWeL-' diactuemd the mathemastics of contro systms and oudijpM performancip criteria. This Sub-Topic illustrates the

(a) Q1 (a) = 0

(32)

where+ d1(') + &'j 01 so-'~ .as0,()..

ai,-

61h on-,

+

+ aor(111 -0, a +so at ('IW a + of$i s*

appLicat in of this material.ftrJtles Central Sol Ve: Function of the poeftken proportional to the input and output displace-

R* ,O is ini0

"

n ro0. V.Z~o

and the difference between them gives a measurej o error, orV. V1v.=pe1

-.

p

(a)(h)where Figure 17. Hf0.5 is acepted as a mlinimumi ftr uj. all roots of the characteirfstic equaftio must faleh S, widthi the shaded zone at a. if al fall wfthin the shaded zone at b, total oscuflbions will decay In thms det mIsed by a.

the voltage-displacement ratios of the two

-,otentlometers are taken to be equal and constant. This errosr voltage Is then fed to an electronit amplifier. mlfe otg, nte ple ste ple Th.rpiidvlae to a dc motor which gives a roughly nroportional-KV

torque, thusT (21

=K~,V

7.2.4 -1ISSUED: MAY 1%b4

DYNAMIC ANALYSIS

POSIMJON CONTROL SERVO ANALYSIS

_I--

I

gee'

4"the atA*d Oh'aft to the

,0

AVICOUS amIP: /t Visc om dt~ Wo s qo

Figum 1. This sinyms poation-contvi swivanechenism Lz u*sd to Muefrate Ow application

o

I

twoy Go an actual sy"m. Functio of the systm Is snmply to roatesamse psmMin as tt of the Input shaft.

This torque in driving the output shaft is op. posed by a load, which in this case is the re-ult of an Inertia J ad a damper 1. This load is inclusive of the Inertia and mechanical resistance of the motor Itself. Relationship between the output displacemenk, an.. tor'que. 11. is, theefor. ad J + - 4 1 7 (at(It

rather than the quantities themselvs. if. howe"or. the constants of the potentiometers are equal. then It is pomsble to redraw the diagram in the convetional mannr. Fig. 2b. Here the pote, tUometer constant is included in the loop tramsfer function. This change is made purely to conf0o. frym wfth normal practice In representing ser.os by" bloc-k diagrams. can ie asn thnt the loop transfer function In given by

Laplace transformatdon of Equation 3. toin -*ro Initial conditions, results It the transfer fusetion. *D.I

-

(a) = Vo (s)

+,

,6 i

7

JIi-.o41

,

It In now possible to construct a block diagram for the complete system. eg. 21. This diffle from the conventional block diagram of o closedloop system I4 that quantiUes proportional to the output and Input are subtracted at the differential,

wbere K = K.K%8/f. Then the overall transfer function is given by .() ()452 +

fa +KI

Flguve 2. sowk diepems afthw V simulate ac

sysden layout or b, in line with conventional servo practice.

simple position-controi system may be constructed. a.

ISSUED: MAY 1964

7.2.4 -2

Poorlww

CONTROl SERIO ANALYSIS

DYNAMIC ANALYSIS

In dealing with quadratic factors much as thd denominator of the transfer function in quation G. It is very helpful to adopt a well-known not.tio". Then Erqustion 6 cnn be written as0" 1 )-

to

*! e : 14,

4

CIA-0 TVIw

-

where

0 0

syl

41

TTX;-Z;Timeservo system to absolutely stabia provided I Is positive: In order to determine its rolatlve located Figure 4.

ft4wvtmchmatoec eqateton to a unit s_ p-Wflen kint for various vskn o .

oae

wo mnes

wthi a qwith rC

stability by transient response me~thods it is necessary to Ietsrmine the roots of the character-

istic equation. In this cae the roots ar

in the left half of the a plane as shown in FMS. 3.is Owe the roots have been found, the response of the system to's unit-step function Input can be found. The type of response depends on the valu, of C. If C>1 the response is purely exponential. but if C< 1. the response also contains oscillatory components. Demarcation between the two types of responsible exists when C 7= 1 and in called the criticaallv damped case. Expressions for the responst to unit step function input for various values of C are given in Table I. while Mig. 4 plots them. responses for numerical values of C. In this application. K.. the gain of the amplifier.

may be not aMW an easily adjusia paramet, to gli Optimum 1"somie. OCtaa C = 0.5 is takm as *-m xmm* desirabe cam If this value is aubtibecms tUtod in Eqiuaion 8a&. r-(K 1n J In some P4aplcatlons, however, it may he uaenary to have a aoe heavfiy damped iuapoam. For instance by choosing = 0, C I very little oveishoot or oscillaton is obtained, fig. 4. This lacnrened danpt tmfortnately resuits in a mome sluarish response with a loner uild-up time.

S .s."-AwlI

Thim mervo is of the drat odwer as sahown by Equation 5. "TaerefaM It has mad smtdy-etatf positional ernw. Her, bowever, positional &ecuracy really depeod on the accuracy of the potIitiometo-,ut-*t

IA repowe to a unit-velocity input,rit _.rin a w ei_ yv _n wrmin

ar

r a

ii"K

----.,

Thus the so-called velocity error can be redmled by increasing K., but here again improvememt is

1

,

Table 1-lResponse to Unit Step Function inputOf

o

ResoneuestW Ieapee.,. Equatdi.1 ()e.1

-- hf y I+(

sinh y r

Figure 3.

Rootu of the charactertic equation ars located

in the loft half of tle s lane. ReOiNi stsbilitY of the servo is d etermined b y th e roo s.

.-.

con 9. They also introduce the idea where ~ on a logarithmic basis, similar of plotting Y.*() to logarithmic frequency-response curves.1' 15 Reference 21 gives plots of magnitude And phase curv,& veisus log R for given values of C for simple

.____

/:V4S Of

lag and q'iadrafic lag terms. These may be added

CON;TANT(

to give loop plots, for example, Fig. 18. For cer-/ tain values of C. 0 db gain and - 180 deg phase occur at the samne value of R, such an C. and R.l SfA in Fig. 18, and them~e vLlues substitulted in Equation 48 give thie roots of the characteristic equaFigure 17. The s phine divided by radial lines rve an tion. approach to solution of chat-acteristi-. From~iegai an phse urvs i isals ~-other Fromhe ninandphae n aso os-equationi. cuvesit sib'e to ca~culate the coefficiect 9,. of the comnponemit of response to unit-step inplit due to a ean'h R there is a given Z for which "-g ~For a . eal. 80 deg +- k3&o deg, For these I - values LitiT,,8 cae atJY*s LhPthe cas, i at3 gain curve can be lifted by adjusting K ! cut th. 0 db line. Thus the root-locus conditions have B 20 ~ gaii urv. i~ pe decdeFor wher.f P= 31pe giu urv, dbperdecdE-tionis, wher f P ~ aope umplexrootLwith1-

P 20 2.3

0.)

where P slope of gain curve, db per decade, for particular C, R at s :=p,, and Q slope of phase curve, rad per des' ide, for s R:'~ . at

fine a curve in th" s pl~iie, satisfying these condiyet another way of constructing root-loci, K r-i a parameter, is available. LIo -iting A1osed-loop poles by the method just outlined can be tedious since a wide range~ uf val~ues of C and R rauat be cover"d to iocate all thie poles;. In order to reduce the amnoInt of *ork it would be convenient to locate the .-oles approximately as a first step- Diernsonv' suggests a good approach to tUne task. First step is a ey ude api~roximation giving three locations for poles:k. Reference 127-2

a. Reference 436-1

ISSUED: MAY

1964

7.2.5 -13

OTHER METHODS

DYNAMIC ANALYSIS

1.Apole at a ~--~ where in frequency at which BodA gain plot, 20 logjlY0 (jw)J, cuts the "&-b iin,. !e. Ipole at 2~, z,... zetos of Y. (a) for which1XII< ."..-

0.-OF0

CONSI

NTC

!Z2

1 (overdamped), the motion is rot periodic and no vibration takes place (aperiodic motion) ; thus (Eq 7.3.51)

Differentiating Equation (7.3.5q) for i and xi, and substituting into Equation (7.3.5r) (Eq 7.3.5u)

x -- AeIf

lw,

Be +-fY

[

,-

x XVThe,

t kXsin (w,,,,t

-'

(s2/

2)

. < 1 (light damping), the radical of S,,2 is imaginary (Eq 7.3.5m) x =

and m,;tion is oscillatory; thus

0) 1 F, in ,,,,t

o0

e

-+,,

(Ce

V'

&,,

-I- De -Vi

,) .andt"i and

If ,"

1 (critical daraping), the body "eturns to the equi-

x 1

,,

2

.

(Eq 7.3.5v)'.

)

tIP

librium position in the shortest time without oscillation; thus x - (E + Ft) e ' (Eq 7.3.5n)

(Eq 1#.3

tatt q,

k

(Eq,/.3

iSSUED: MAY

1964

7.3.5 -4

r!

VISPATON 'MTN CW4LOMB DAMPING

D)YNAMIC ANALYS1`1

Let

('AS

IV'

F'rr Vi',rat on wfith ('oifomb flampi,0, 0consitantt

#'(

X.

zero frecianry deflection of maiss k under *rnpreri4'd force, F., eady-state) it(Eq 73.5m)

T'hen the magnification fokctor'(

(E ~

~ ~

Coulomb (f-iction) damping is due to rictional forceB and is connidered independent of displacrment, velocity, and ac celerhtion. The sign ( !the force' caninot 1wetaken into account for a cemplete cycle. llowever, en( "gy methods may j sed . ki.M) for conducting the t,nalysis.R-

x

I......--------

_

!~d\~

~.

~the

the work done by (c .- ing to Figure 7.3-' iuequaintig ,.pring, the work of . ciction to the kinetic energy per

(lEq 7.3.5z)

k

k (x,,

b)

V ('2x, b

tan(v

7..5)

vi')

rh* magnification (actor i-t 4he ratio of the amplitude of steady oscillation ti the deffiettior 4,f masQ under thos stati force, FThe equation for the ina nifirsioo factor ip ;lotted in Figure 71.5b. For relatively small va'tues of ,resonance occurs when the driving frequency is near the undampeO natural frequency of the system. For large. -,alues of r. resonance occurs at ritios of driving frequency to undamped natural frequency which approach 6ero as C approaches 0.707F0

Ahere

VA

Vt

0

x.

initial spring displacement

b - decrease in amplitude per half cycle =2F/k The amplitude decrease is constant for each half cycle, and the decay pe- cycle. n thus

2

4F*-

(Eq 7.3.satl- FRICTION FOS.2V,-INITIAL VELOCITY.

FINAL VILOC ITY

50

0(

0

0.0,

C

t1A

rVI)

r-

oA

yV

+ IL2 I11,0O 3.00 4.0 5.0

FRU~ YC &AK "'.10~

Figure 7.3.5c~. Vibrating System wsith Coulomb Damping raste of decaiy is .:ho~wn i-a Figure 7.3.5d. The rnotion will cease when the spring force is insufficient to overcomestatic frijction force.

of qU~i~ns7 Figue 73.5b No 3Sx Pd 73.5yiorThe Damped Syste nl Of a 73.b.?Ith oViEqations ViS .~ F~ur the a Vscojly umpei Sytemthe o ibraon

7.3.6 Vibratit -i Isolation and TransmissibilityAn elemecnt rigidly attsebed to ,i foundation or supporting structure will transmit tu that support any vibration orivinri.ting from it. Conversely, arty vibration of the sur~port-.

7.3.6 -1

ISSUC:D: MAY 1964

DYNAMIC ANALYSIS

SELF-EXCITE.D VIBRATIONS

in& structure is transmitted to the element. Vibration isnIntors Are a means of mhirnizing the transmitted vibration These isolators may take the form of rubber mounts, springs, padding, deshpot dampers, etc. Aisuming that isolators can be represented,? by the spring and dashpot shown in Figure 7.3.ba, the m..,,nitude of the transmitted force is given by

The TR equation is plotted in Figure 7.3.6. When TR- 1, all the curves pass throughthe point where Figltn uie 7.3.6 also shows that for values of < )/Y-, the trans40" mitte" force is greater than the value for rigid mounting. -2-. Vibration isolation then is possible ,nly when -W 7> If damping is negligible, 1 0, then

TR

1

(Eq 7.3.6d)

where

ton

_.

7

(-0.

J..--0.050.'103.0 -[

0 ;5 ?50_0

t.

.

Figure 7.3.5d. Rate of Decay of Amplitude with Coulomb Damping Is Unear with Time Figure 7.3.6. Transmlssibillty (Eq 7.3.6s)FTR

_ 1.0

0---

1.0

V5-

2.0

3.0- ----

-

.0

5 .0

/edX .- (kX)2 +

(C+nX)2

2

PEQ~fNCYR ATIO

--where X is the amplitude of strady oscillation, given in Sub-Topic 7.3.5, Case III. Then A

Figure 7.3.6. Transmissibility Versus Frequency Ratio

7.3.7 Self-Excited VibrationsA self-excited vibration may occo, when the exciting fl.rce is a function of the displacement, velocity, or acceleration.

F0FTR I-t--

1-+h2

-

T JE I~-~-

If a system is excited by

force proport