AJrangement and scaling o/ equatmns 179

ARRANGEMENT AND SCALING OF EQUATIONS

SAMS

1. General. d The programme and method of scaling which are dt

recommended in this paper is applicable to most prob- lems There are however types of problems where it dO

- - //ca 2 can be more convenient to arrange and scale according dt to other principles, but an analysis of these is outside the scope of this recommandation, dv 2. Programme. d) = V

The programme can be divided into three steps: 2.1. Re-writing and simplifying of the equations. As we see one equation is given for every integra- 2.2. Choice of scale-factors, tion, which is necessary if the equations are to be time- 2.3. Scaling. scaled. If a variable is preceded by d/dt, this will 2.1. Re-writing and simplifying of the equations, indicate that the corresponding variable will be given

as output of an integrator Dot-derivatives (~0) are Usually it is not possible to start &rectly with the derivatives with respect to a physical independent

scaling of the equations for a given problem, but some variable. re-writing and simplifying has first to be done. The result should be a system of equatmns, where the highest-order derivatives of the different variables are 2.2. Choice o~ scale-factors, written as functions of lower-order derivatives and In choosing scale-factors the maximum value of the other terms. As an illustration of this and the follow- variables must first be determined. Dividing the refer- ing steps we choose the equation for a mathematical ence voltage (in the following examples 100 V is pendulum where the disturbing force is a sinusoid of assumed as reference voltage) by a maximum value of constant amplitude vo a variable gives the scale-factor, which together with

v the variable itself makes the computer-variable. To make scaling and recording easier we round off to convenient scale-factors.

2.3. Scaling.

An equation can either be only amplitude-scaled or both amplitude-scaled and time-scaled. In order to indicate the difference between these two alternatives, the previously used example will first be amplitude- scaled and then completed with time-scaling. In prac-

n~ tice the time-scaling is of course done simultantes with Fig 1 amplitude-scaling if the problem is to be both ampli-

tude- and time-scaled. V -~- Po COS oJ I

Provided that the angles are small the differential 2.3.1. Amplitude scaling. equation for 9 is Insertion of the computer variables into the equations,

}; -t- k~o + - - . 9 + V. g ~o - - 0 which are re-written for scaling, gives: 1 !

a' [J.00 ~;] 1 i where - [ , o o } 1 . , . - -

I = 0,5 m, k = 2 l/s, m = 3 rad/s, dt 100 100 Initial conditions 9 0,2 rad, v = 1 m/s. g I 30,V 1 100 1

After re-arrangement we have the following system - - [500 ~o]. l 500 [ 1---~] [500 ~0]. l 30 '500 of equations

g ~ d [500 ~;] 1 1 ~o" = - - k ~ - - T ' ~ - - + ' - l , b 500 - [lOO ~] . - - l o o

"/7 ----- - - V to 2

d [30 H I i In order to get the system in a more convenient form -- [100 v]. m -~ . 1{)0

before scaling, it can be expressed as follows: dt 30

d 9 g 9 d [i00 v] 1 1 - - - - - - - - [ 3 0 H . - -

dt k ~o 1 " ~o - - ~; . 1 dt loo 30

.180 Annales de l'Association internationale pour le Calcul analogique N ° 3 - - Juillet 1964

After simplifying and insertion of the numerical Pot No. 3 --- 1,0o00 values of the constants we have Pot No. 4 = 0,3924

Pot No. 8 = 0,3333 d [100 }] _ [100 }] 2 - - [500 ~o] 3,924 Pot No. 9 = 1,0000

dt ' Pot No. 12 = 0,2700 Pot No. 13 = o,1333

30~ [1-~] [500 ~o], 1,333 The brackets are not used m the diagram, and in

multiplication and division circuits the scale-factors are d [500 .~a] multiplied to make one number.

- [loo ~1.5 dt

2.3.2. Time.scaling. d [30 ~,]

- - [1o0 v]. 2,7 By time-scaling is meant that the problem is solved dt in a longer or shorter time T than the physically cor-

rect time t. d [loo v]

- - [30 ~]. 3,333 This can be written dt

T = f l t

On basis of these equations the circuit-diagram can where 3 > 1 corresponds to a longer solution time now be drawn: than the physically correct time and p < i corresponds

to a shorter solution time.

O A dme-scahng of the preceding example gives

- l~Ovi d [100,~b] __ [100 ~] 2 3,9'24

® d/e,/ 7 - 30b 1,333

/ ~[ d (/3 t) /3

I d [30 b] 2,7 ¢, - oo d (/3 t) [10o v]. - - /3

, [~ OOv d [loo v] 3,333

~ _ [ ~ -300 l t '~A_~l I~'~___ d (/3 t) - [30 ~ ] . - - /3 By this way of writing, the time-scale factor can be

included in the coefficients, which may be advisable I { ~ when several changes of the time-scale are planned.

We have so far discussed a case with time as the independent variable, but problems with another inde-

Fig. 2. pendent variable may be dealt with in the same way.

C H E C K P R O C E D U R E S

S A M S

1. General. 2. Static Problem Check.

To make sure that the scaled equations have been The most general method is the static problem check. written correctly, that the circuit diagram has been It is based upon control of static values measured on drawn and patched correctly and that the equipment the computer against the same values calculated from is functioning we must carry out a check, the original equations. The check values are in this

By this means we will verify that the necessary con- case easy to calculate arid the method exposes all errors ditions for obtaining correct solutions are satisfied, m scaling, patching and malfunctioning equipment.

Check procedm'es 181

To demonstrate the stat:c check we will work an 2.7. easy example. The figures are assigned to the block diagram, d [10 k]

- - -- 8 0 - - 4 0 + 0,4 ~ . 4 0 . 6,25 = - - 8 0 d (2 t)

2,2. Let us assume that the physical system is de- (4) scribed by the following differential equation d [20 x] - 80

d (2 t) d" x dx

q - 2 ~ q - 4 x - x a = 0 (1)

dr"- dt 2,8. From equations (4) and the values according to 2.5 we can easily calculate

2 4. With scale factors according to max. values in dx dUx table 1 and with dx/dt = x we obtain equations (2) x, fl = 2 gives us equations (3) which are adapted for dt ' dt 2 generation of the circuit diagram, dx d~x

x = 2; - - 8; ' = - - 1 6 Table 1. dt dt 2

Variables Max. values Scale factor Computer If equations (3) represent equation (1), the values in variables equations (4) will obviously satisfy equation (1). In-

x 10 10 10 k troduce the values in equation (1) and we obtain

k 5 20 20X - - 1 6 -I- 2 . 8 -I- 4 . 2 - - 2 a : 0

which shows that the scaled equations have been d _ 2 2 d (fl t) - - - - [lOk] ~ - - - [20 x] - - written correctly. We can obviously detect errors in

fl scaling without the computer. Any faults that may occur will also be exposed and can be corrected before the circuit diagram has been drawn and the potentio-

20 x 12,5 q- [ - ~ [ " . [20 x] meter set-up sheet filled in,

fl (2) 2.11. The circuit diagram will be based upon equa- d 20 X 2 tlons (3) as shown in figure 2.

= [ l O . q - d (/9 t)' fl 2.12. List the check parameters on a potentiometer

set-up sheet.

d [10~] 2.15. Then measure the static check values on the d (2 t) = - - [10 k.] - - [20 x] computer and compare them with the calculated values.

20 x 2.20. If errors exist go back to 2.11 and start all q- [-70--~] 2 . [20 x] . 6,25 over again. It will be noted that we do not need to

check the scaling, hence we have done the particular d [20 x] (3 ' ) check. Conceivable errors are accordingly faulty patch-

- [10 k] ing or incorrectly chosen gains. d (2 t)

2.21. Exchange any malfunctioning components that To investigate the scaling we will carry out a part- have been located. Check from 2.16.

icular check that later will also be used as a check of the patched problem.

3. Other Check Methods.

2.5. Choose for this particular check special test 3.1. Comparisons of solution from the analog corn- initial values. T.hey should be values convenient for puler with solutions obtained by other means.

the calculations, which produce reasonable outputs Occasionally, some important characteristics of a (10-100 V) of all other components. Use also con- system of differential equations that is to be solved venient values for the parameters. For this example on an analog computer are known, (i.e. damping, natur- the following test initial values are chosen, al frequency, period time, or roots of the characteristic

10 k = 80 V equation). This is frequently true in the case of analysis of servomechamsm systems for which a stability analysis

20 x = 40 V has prevmusly been made. Knowledge of such charac- teristics allows checking to be performed by analysis of the computer results in those respects. This method

2.6 Introduce the test values in equations (3), which is in a certain sense more complete then the static check, are scaled equations containing computer variables, and but can not be used to locate the source of any faults we get that may occur.

182 Anna/es de l'Assoclation mternat~onale pour le Calcul analogique N ° 3 - - ]uillet 1964

i ,,,i

i

, , j '! ~-~. I J -~ Physical MathematLca~ Scaling of ~ Computer System I 1 Model 1 Equations 2.~ I Eq uati°ns F

Introduction of values 2.6. L I ChSckvalues for all 2.5.

from 2. in 2.4. ~ parameters and computer , va r i ~bl . e

"l Calculate the voltage I;o Calculate the vglues of h represent the highest physical varxables from .

l

_ Wrong Result - In~roduetlon of values 1 l 1 fr°m 2"8" ~n 2"2"

, i, m i i

~ Right I ',, " ' I ~.~'. I ~.~4 I

Circuit diagram from ~I meterP°tenti°-set-u])~'l Patching R ..... equations in 2.4. I sh@et ] I I I

Setting of tes t initial Oompater ,values from 2.5. , . . . . . set-up ,,

I ~ ...................... 2 . 1 6 . / 2.1~ I Measurement of c~Iculated check valueo,(hi&~es% derivs)i " Result OT ~omn.wa'r'. I , l

L I

Comparison of values 2.18. I according to 2.7. and 2.17. I , i

J)ynamic c h e c f " Check2.11.from ~1

Mal funct ~ on i ng Ilnits ] Exchan 2.21[

"I unit' I