E V A L U A T I N G S Y S T E M A T I C ERRORS IN Q U A D R A N T S C A L E S

A. L. S h n e i d e r m a n UDC 681.26.088

In making the balancing and recording of weighed articles automatic and using remote-measur ing systems it becomes often necessary to evaluate systematic measurement errors.

Let us assume we are dealing with a weighing device (Fig. 1) which consists of a quadrant balancing mecha- nism I with lever system 2 at tached to it. An in te rmedia te lever carries inductive transducer 3 used for remote e l ec -

t r i ca l measurements with an output signal proportional to the displacement of the transducer plunger.

Taking into consideration that commerc i a l models of secondary remote-measur ing instruments have a uniform scale, it is obvious that they wil l produce a systematic measurement error.

Let us determine the maximum value of the error contributed to the system by remote measurements. By de- noting the current value of the measured quantity with P and the corresponding displacement of the plunger with y,

let us find relationship P = fly).

The quadrant 's condit ion of balance has the form of

Pl o cos r = ORo sin (qo + %). (1)

where ~ is the current va lue of the quadrant 's rotation angle, R 0 is the distance from the bearing of the quadrant to its center of gravity, ~o 0 is the angle of the quadrant 's in i t i a l position, G is the weight of the quadrant, l 0 is the weight-

carrying arm of the quadrant.

From (1) we obtain

6Ro P = -io - ( tan ~p cos % + sin %). (2)

By adopting notations

6Ro A ~ ~oo COS % ,

B = 6RQ sin %, lo

(3)

we obtain

sin q~

P = a V i - - sir,~ ~ ~ S . (4)

The current value of the plunger 's displacement in transducer 3 (Fig. 1) is represented by relationship

l~ (5) y =

where y~ is the displacement of the weight-bear ing lever 's knife edge for a rotation of the quadrant through angle ~o.

Taking into consideration that y~o = I0 sin 9 we obtain:

y -~- k sin r (6)

where

l_3_~ k = l0 l l �9

Translated from Izmer i t e l ' naya Tekhnika, No, i, pp, 24-26, January, 1967, Original ar t ic le submitted

October 15, 1965.

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, ~ , ' : ~ !

<1 , / - 7-2 ~ '~-~ ,

' ~ D r a w bar of the weight-

bearing mechanism

Fig. 1.

H

i ma~

' ~Pma~

--252 f i +2. 2 -# ' r am

Fig. 2.

and for P = Pmax we have

By substituting in (4) for the sine its value from (6) we obtain:

A y P - - k .~ / y2 + B . (7)

V l--k- 7

The above relationship shows that the measured quan- t i ty has a nonlinear relationship to the displacement of the

transducer's plunger. Therefore, in observing or recording the readings on a uniform scale of a secondary instrument

we obtain a systematic error.

Let us i l h s t r a t e the method of evaluat ing this error on par t icular examples.

Let us first note the following. Coeff icient A and B in

equation (7) incorporate quadrant parameters G and R 0 which in prac t ice are not always avai lab le for c o m p u t a t i o n . There-

fore, it is advisable to express these values by the max imum effort P -- Pmax on the quadrant, which is normal ly known.

Let us derive a balance equation for the quadrant in

two of its positions, namely for P = I/2Pma x we obtain

I -~- Pmax lo = GRo sin q)o, (8)

Pmax lo ---- GRo {.tom qemaxCOS q~o q sin ~Po).

By comparing (8) and (9) and taking into consideration (3) it is easy to find that

1 1 I , 8 = P ax' A = ] - Pmax

tanqamax 2 l , *

(9)

(10)

By substituting in (7) for A and B their values thus found we obtain

P = -2- max ktanq)ma x " g=' " (11)

By assuming for a numer ica l computat ion that ~max = 23~ l0 = 23 ram, l , = 80 ram, g2 = 23.3 mm we find k = 6.7 and tan ~Oma x = 0.424.

By substituting in (11) for these quantities their values given above we have

_1 p ( 1 + 0 . 3 5 2 Y ) P = 2 max V l -o .0223y~ " (12)

The graph of this function is shown in Fig. 2 (curve 1). It is known that the mechanism of a quadrant with the a t tached lever system is assembled and adjusted so that in the middle position of the indicat ing instrument 's pointer the lever knife edges are in a horizontal plane. Thus, the measured weight value Pmax is made to correspond to a plunger displacement Ymax which according to (6) is equal to Ymax = 6 .7 .0 .391 m m = 2.62 ram.

In order to evaluate the nonlineari ty of the sys temat ic-er ror function (12) it is necessary to compare it with the l inear relationship which corresponds to straight l ine2 in Fig. 2 and has the form of:

1 Plin = 2 - Pmax -I ag. (13)

Coeff icient a,which represents the slope angle of the straight l ine ,can be evaluated from conditions y = Ymax = 2.62 mm and Plin = Pmax as a = 0.382. I/2 Pmax'

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Finally (13) assumes the form of

I P l i n = ~ " Pmax (1 + 0.382 g). (14)

The difference between (12) and (14) represents the systematic error's current value 6 (y) = P -P l in , i .e. ,

I ( Y - - 0 . 3 8 2 y ) . (15) 8(y) ~- -~- Pmax 0 .352V1 _ 0.0223y 2

it is advisable to evaluate the systematic error in the middle of the interval, since it is difficult to find analyt- ically the extremal value of 5 (y) and the corresponding value of the independent variable y,

By substituting in (12) and (14) for y and Ymax their values 0.5 and 1,81 mm we obtain respectively:

1 1 P = -~- Pmax" 1,47U20, Plin = - 2 Pmax" 1.50042.

From the above we find the maximum measurement error on a linear scale of a secondary instrument to be equal to

A ~ 2 % .

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