NUMERICAL CALCULATIONS.

addition to the name and the date, all calculations shouldaccompanied by a complete record of the object and purpose

of

the calculation, the apparatus, the assumptions made, the

data

used, reference to other calculations or data employed,

etc.,

to

in short, they should include all the information requiredmake the calculation intelligible to another engineer without

further

information besides that contained in the calculations,

or in the references

given therein. The small amount of time

and

increased

work required to do this is negligible compared with theutility of the calculation.

Tables

and curves belonging to the calculation should in

the

same way be completely identified with it and contain

sufficient

data to be intelligible.

d.

167.

Reliability of Numerical Calculations.The most important and essential requirement of

numerical

engineering calculations is their absolute reliability.

When

making a calculation, the most brilliant ability, theoretical

knowledge and

practical experience of an engineer are

made

useless, and even worse than useless, by a single error in

an

important calculation.

Reliability

of the numerical calculation is of vastly greater

importance

in engineering than in any other field. In pure

mathematics

example which

an error in the numerical calculation of anillustrates a general proposition, does not detract

from the

interest and value of the latter, which is the main

purpose; in physics, the general

the'

law which is the subject ofinvestigation remains true, and the investigation of interest

and

use, even if in the numerical illustration of the law an

error

is made. With the most brilliant engineering design,

however,

if in the numerical calculation of a single structural

member

an error has been made, and its strength thereby calculated

wrong,

the rotor of the machine flies to pieces by centrifugal

forces,

engineer.

or the bridge collapses, and with it the reputation of theThe essential difference between engineering and

purely

scientific caclulations is the rapid check on the correctnessof the calculation, which is usually afforded by the per

In

be

ENGINEERING MATHEMATICS.

be used visually also, in determining the frequency

of

hunting of synchronous machines, etc. In the phenomenon

of

frequency,

hunting, frequently two periods are superimposed, forcedresulting from the speed of generator, etc., and the

natural

frequency of the machine. Counting the number of

impulses,

/, per minute, and the number of nodes, n, gives the

two

frequencies :/+- and/ ; and as one of these frequencies

2i 2i

is

the impressed engine frequency, this affords a check.

Not

infrequently wave-shape distortions are met, which

are

but

not due to higher harmonics of the fundamental wave,are incommensurable therewith. In this case there are

two

entirely unrelated frequencies. This, for instance, occurs

in

the secondary circuit of the single-phase induction motor;

two

sets of currents, of the frequencies / and (2ffs) exist

(where

/ is the primary frequency and / the frequency of

slip).

Of this nature, frequently, is the distortion produced by

surges,

oscillations, arcing grounds, etc., in electric circuits;

it is

a combination of the natural frequency of the circuit

with the impressed

frequency. Telephonic currents commonly

show

such multiple frequencies, which are not harmonics ofeach

Engineering

work leads to more or less extensive

numerical

investigation

calculations, when applying the general theoreticalto the specific cases which are under consideration.

Of importance

in such engineering calculations are;

(a)

The method of calculation.

(6)

The degree of exactness required in the calculation.

(c)

(d)

a.

Method of Calculation.

Before

carefully

beginning a more extensive calculation, it is desirableto scrutinize and to investigate the method, to find

the simplest

system

way, since frequently by a suitable method andof calculation the work can be reduced to a small fraction

of

what it would otherwise be, and what appear to be

hopelessly

complex calculations may thus be carried out

quickly

and expeditiously by a proper arrangement of the

work.

The most convenient way usually is the arrangement

in

tabular form.

As

example, consider the problem of calculating the regulation

of

a 60,000-volt transmission line, of r=60 ohms resistance,

x

= 135 ohms inductive reactance, and fe 0.0012 condensive

susceptance,

for various values of non-inductive, inductive,

and

condensive load.

Starting

transmission

with the complete equations of the long-distanceline, as given in "Theory and Calculation of

Transient

Electric Phenomena and Oscillations," Section III,

paragraph

power-factors,

2.

90 PER CENT POWER-FACTOR, LAG.

cos

0=09; sin0=Vl-0.92

=0.436;

j

sin 0)

=

i (0 9+0.436j);

Si

= (0.919- 0.03Gj> + (56.8- 131.8/K0.9 +0.436j>o

=

(0.919- 0.036j>o + (108.5

-

93.8/H

=

4 + '

:

/i

=

(0.919-0.036j)(0.9

+0.436j)io- (0.0144 +U

'

=

(0.843

+0.366j>

-

(0.0144

+1.168j> 10-3 =C/ -D,

and

now the table is calculated in the same manner as under 1.

Then

manner,

corresponding tables are calculated, in the samefor power-factor, =0.8 and =0.7, respectively, lag,

and

for power-factor -0.9, 0.8, 0,7, lead; that is, for

cos

0+] sin 0=0.8 +0.6]';

0.7+0.714]';

0.9-0.436]';

0.8-0.6]';

0.7-0.714].

Then

curves are plotted for all seven values of power-factor,

from

0.7 lag to 0.7 lead.

From

these curves, for a number of values of i

,

for instance,

to

taken,

=20, 40, 60, 80, 100, numerical values of ii, e^ cos Q, aroand plotted as curves, which, for the same voltage

ei

= 60 at the step-up end, give i\ } eo, and cos 6, for the same

value

IQ, that is, give the regulation of the line at constantcurrent output for varying power-factor.9; and considering that for every one of the variouslag, and lead, a sufficient number of valuesThe intelligibility of the results,The reliability of the calculation.other.

frequently