Charles Proteus Steinmetz. # Engineering mathematics; a series of lectures delivered at Union college online

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n

kv.

120

1001.00

80 0.-8Q

0.60

0.40

0.20

100 $)

FIG. 86. Transmission Line Characteristics.

For instance, when investigating the short-circuit current of an

electric generating system, it is of importance to know whether

this current is 3 or 4 times normal current, or whether it is

40 to 50 times normal current, but it is immaterial whether

it is 45 to 46 or 50 times normal. In studying lightning

phenomena, and, in general, abnormal voltages in electric

systems, calculating the discharge capacity of lightning arres-

ters, etc., the magnitude of the quantity is often sufficient. In

254 ENGINEERING MATHEMATICS.

calculating the critical speed of turbine alternators, or the

natural period of oscillation of synchronous machines, the

same applies, since it is of importance only to see that these

speeds are sufficiently remote from the normal operating speed

to give no trouble in operation.

(6) Approximate calculation, requiring an accuracy of one

or a few per cent only; a large part of engineering calcu-

lations fall in this class, especially calculations in the realm of

design. Although, frequently, a higher accuracy could be

reached in the calculation proper, it would be of no value,

since the data on which the calculations are based are sus-

ceptible to variations beyond control, due to variation in the

material, in the mechanical dimensions, etc.

Thus, for instance, the exciting current of induction motors

may vary by several per cent, due to variations of the length

of air gap, so small as to be beyond the limits of constructive

accuracy, and a calculation exact to a fraction of one per cent,

while theoretically possible, thus would be practically useless,

The calculation of the ampere-turns required for the shunt

field excitation, or for the series field of a direct-current

generator needs only moderate exactness, as variations in the

magnetic material, in the speed regulation of the driving

power, etc., produce differences amounting to several per

cent.

(c) Exact engineering calculations, as, for instance, the

calculations of the efficiency of apparatus, the regulation of

transformers, the characteristic curves of induction motors,

etc. These are determined with an accuracy frequently amount-

ing to one-tenth of one per cent and even greater.

Even for most exact engineering calculations, the accuracy

of the slide rule is usually sufficient, if intelligently used, that

is, used so as to get the greatest accuracy. Thus, in dividing,

for instance, 297 by 283 by the slide rule, the proper way is

to divide 297-283 = 14 by 283, and to add the result to 1.

This gives a greater accuracy than direct division. For accu-

rate calculations, preferably the glass slide should not be used,

but the result interpolated by the eye.

163. While the calculations are unsatisfactory, if not carried

out with the degree of exactness which is feasible and desirable,

it is equally wrong to give numerical values with a number of

NUMERICAL CALCULATIONS. 255

ciphers greater than the method or the purpose of the calcula-

tion warrants. For instance, if in the design of a direct-current

generator, the calculated field ampere-turns are given as 9738,

such a numerical value destroys the confidence in the work of

the calculator or designer, as it implies an accuracy greater

than possible, and thereby shows a lack of judgment.

The number of ciphers in which the result of calculation is

given should signify the exactness, In this respect two

systems are in use:

(a) Numerical values are given with one more decimal

than warranted by the probable error of the result; that is,

the decimal before the last is correct, but the last decimal may

be wrong by several units. This method is usually employed

in astronomy, physics, etc.

(6) Numerical values are given with as many decimals as

the accuracy of the calculation warrants; that is, the last

decimal is probably correct within half a unit. For instance,

an efficiency of 86 per cent means an efficiency between 85.5

and 86.5 per cent; an efficiency of 97.3 per cent means an

efficiency between 97.25 and 97.35 per cent, etc. This system

is generally used in engineering calculations. To get accuracy

of the last decimal of the result, the calculations then must

be carried out for one more decimal than given in the result.

For instance, when calculating the efficiency by adding the

various percentages of losses, data like the following may be

given :

Core loss 2.73 per cent

i 2 r 1.06 ,"

Friction. . 0.93 "

Total 4.72

Efficiency 100-4.72 = 95.38

4pproximately 95.4

It is obvious that throughout the same calculation the

same degree of accuracy must be observed.

It follows herefrom that the values :

2i; 2.5; 2.50; 2.500,

256 ENGINEERING MATHEMATICS.

while mathematically equal, are not equal in their meaning as

an engineering result :

2.5 means between 2.45 and 2.55;

2.50 means between 2.495 and 2.505;

2.500 means between 2.4995 and 2.5005;

while 2J gives no clue to the accuracy of the value.

Thus it is not permissible to add zeros, or drop, zeros at

the end of numerical values, nor is it permissible, for instance,

to replace fractions as 1/16 by 0.0625, without changing the

meaning of the numerical value, as regards its accuracy.

This is not always realized, and especially in the reduction of

common fractions to decimals an unjustified laxness exists

which impairs the reliability of the results. For instance, if

in an arc lamp the arc length, for which the mechanism is

adjusted, is stated to be 0.8125 inch, such a statement is

ridiculous, as no arc lamp mechanism can control for one-tenth

as great an accuracy as implied in this numerical value: the

value is an unjustified translation from 13/16 inch.

The principle thus should be adhered to, that all calcula-

tions are carried out for one decimal more than the exactness

required or feasible, and in the result the last decimal dropped;

that is, the result given so that the last decimal is probably

correct within half a unit.

c. Intelligibility of Engineering Data.

164. In engineering calculations the value of the results

mainly depends on the information derived from them, that is,

on their intelligibility. To make the numerical results and

their meaning as intelligible as possible, it thus is desirable,

whenever a series of values are calculated, to carefully arrange

them in tables and plot them in a curve or in curses. The

latter is necessary, since for most engineers the plotted curve

gives a much better conception of the shape and the variation

of a quantity than numerical tables.

Even where only a single point is required, as the core

loss at full load, or the excitation of an electric generator at

rated voltage, it is generally preferable to calculate a few

NUMERICAL CALCULATIONS.

257

02

01

08

Volts

400

500

200

100

10

FIG. 87. Compounding Curve.

points near the desired value, so as to get at least a short piece

of curve including the desired point.

The main advantage, and foremost purpose of curve plotting

thus is to show the shape of the function, and thereby give

a clearer conception of it ;

but for recording numerical

values, and deriving numer-

ical values from it, the plotted

curve is inferior to the table,

due to the limited accuracy

possible in a plotted curve,

and the further inaccuracy

resulting when drawing a

curve through the plotted cal-

culated points. To some

extent, the numerical values

as taken from a plotted curve,

depend on the particular

kind of curve rule used in

plotting the curve.

In general, curves are used for two different purposes, and

on the purpose for which the curve is plotted, should depend

the method of plotting, as the scale, the zero values, etc.

^^ When curves are used to

illustrate the shape of the

function, so as to show how

much and in what manner a

quantity varies as function

of another, large divisions of

inconspicuous cross-section-

ing are desirable, but it is

essential that the cross-

sectioning should extend to

the zero values of the func-

tion, even if the numerical

values do not extend so

far, since otherwise a wrong

impression would be con-

ferred. As illustrations are plotted in Figs. 87 and 88, the

compounding curve of a direct-current generator. The arrange-

02

06

08

Volts

-580^

510

-490

FIG. 88. Compounding Curve.

258

ENGINEERING MATHEMATICS.

ment in Fig. 87 is correct ; it shows the relative variation

of voltage as function of the load. Fig. 88, in which the

cross-sectioning does not begin at the scale zero, confers the

FIG. 89. Curve Plotted to show Characteristic Shape.

FIG. 90. Curve Plotted for Use as Design Data.

wrong impression that the variation of voltage is far greater

than it really is.

When curves are used to record numerical values and

derive them from the curve, as, for instance, is commonly the

NUMERICAL CALCULATIONS.

259

case with magnetization curves, it is unnecessary to have the zero

of the function coincide with the zero of the cross-sectioning, but

rather preferable not to have it so, if thereby a better scale of

the curve can be secured. It is desirable, however, to use suffi-

ciently small cross-sectioning to make it possible to take

numerical values from the curve with good accuracy. This is

illustrated by Figs. 89 and 90. Both show the magnetic charac-

teristic of soft steel, for the range above (B = 8000, in which it is

usually employed. Fig. 89 shows the proper way of plotting

for showing the shape of the function, Fig. 90 the proper way of

plotting for use of the curve to derive numerical values therefrom.

\

\

ii

\

FIG. 91. Same Function Plotted to Different Scales; I is correct.

165. Curves should be plotted in such a manner as to show

the quantity which they represent, and its variation, as well as

possible. Two features are desirable herefor:

1. To use such a scale that the average slope of the curve,

or at least of the more important part of it, does not differ

much from 45 cleg. Hereby variations of curvature are best

shown. To illustrate this, the exponential function y = e~ x is

plotted in three different scales, as curves I, II, III, in Fig. 91.

Curve I has the proper scale.

2. To use such a scale, that the total range of ordinates is

not much different from the total range of abscissas. Thus

when plotting the power-factor of an induction motor, in

Fig. 92, curve I is preferable to curves II or III.

260

ENGINEERING MATHEMATICS.

These two requirements frequently are at variance with

each other, and then a compromise has to be made between

them, that is, such a scale chosen that the total ranges of the

two coordinates do not differ much, and at the same time

the average slope of the curve is not far from 45 deg. This

usually leads to a somewhat rectangular area covered by the

curve, as shown, for instance, by curve I, in Fig. 91.

In curve plotting, a scale should be used which is easily

read. Hence, only full scale, double scale, and half scale

should be used. Triple scale and on,e-third scale are practically

unreadable, arid should therefore never be used. Quadruple

FIG. 92. Same Function Plotted to Different Scales; I is Correct.

scale and quarter scale are difficult to read and therefore unde-

sirable, and are generally unnecessary, since quadruple scale

is not much different from half scale with a ten times smaller

unit, and quarter scale not much different from double scale

of a ten times larger unit.

166. Any engineering calculation on which it is worth

while to devote any time, is worth being recorded with suffi-

cient completeness to be generally intelligible. Very often in

making calculations the data on which the calculation is based,

the subject and the purpose of the calculation are given incom-

pletely or not at all, since they are familiar to the calculator at

the time of calculation. The calculation thus would be unin-

NUMERICAL CALCULATIONS. 261

telligible to any other engineer, and usually becomes unintelli-

gible even to the calculator in a few weeks.

In addition to the name and the date, all calculations should

be accompanied 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., in short., they should include all the information required

to make 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 work required to do this is negligible compared with the

increased utility 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. Reliability of Numerical Calculations.

167. The most important and essential requirement of

numerical engineering calculations is their absolute reliability.

When making a calculation, the most brilliant ability, theo-

retical 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 an error in the numerical calculation of an

example which illustrates a general proposition, does not detract

from the interest and value of the latter, which is the main

purpose; in physics, the general law which is the subject of

the' investigation 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 calcu-

lated wrong, the rotor ot the machine flies to pieces by centrifugal

forces, or the bridge collapses, and with it the reputation of the

engineer. The essential difference between engineering and

purely scientific caclulations is the rapid check on the correct-

ness of the calculation, which is usually afforded by the per-

262 ENGINEERING MATHEMATICS.

formance of the calculated structure but too late to correct

errors.

Thus rapidity of calculation, while by itself useful, is of no

value whatever compared with reliability that is, correct-

ness.

One of the first and most important requirements to secure

reliability is neatness and care in the execution of the calcula-

tion. If the calculation is made on any kind of a sheet of

paper, with lead pencil, with frequent striking out and correct-

ing of figures, etc., it is practically hopeless to expect correct

results from any more extensive calculations. Thus the work

should be done with pen and ink, on white ruled paper; if

changes have to be made, they should preferably be made by

erasing, and not by striking out. In general, the appearance of

the work is one of the best indications of its reliability. The

arrangement in tabular form, where a series of values are calcu-

lated, offers considerable assistance in improving the reliability.

168. Essential in all extensive calculations is a complete

system of checking the results, to insure correctness.

One way is to have the same calculation made independently

by two different calculators, and then compare the results.

Another way is to have a few points of the calculation checked

by somebody else. Neither way is satisfactory, as it is not

always possible for an engineer to have the assistance of another

engineer to check his work, and besides this, an engineer should

and must be able to make numerical calculations so that he can

absolutely rely on their correctness without somebody else

assisting him.

In any more important calculations every operation thus

should be performed twice, preferably in a different manner.

Thus, when multiplying or dividing by the slide rule, the multi-

plication or division should be repeated mentally, approxi-

mately, as check; when adding a column of figures, it should be

added first downward, then as check upward, etc.

Where an exact calculation is required, first the magnitude

of the quantity should be estimated, if not already known,

then an approximate calculation made, which can frequently

be done mentally, and then the exact calculation; or, inversely,

after the exact calculation, the result may be checked by an

approximate mental calculation.

NUMERICAL CALCULATIONS. 263

Where a series of values is to be calculated, it is advisable

first to calculate a few individual points, and then, entirely

independently, calculate in tabular form the series of values,

and then use the previously calculated values as check. Or,

inversely, after calculating the series of values a few points

should independently be calculated as check.

When a series of values is calculated, it is usually easier to

secure reliability than when calculating a single value, since

in the former case the different values check each other. There-

fore it is always advisable to calculate a number of values,

that is, a short curve branch, even if only a single point is

required. After calculating a series of values, they are plotted

as a curve to see whether they give a smooth curve. If the

entire curve is irregular, the calculation should be thrown away,

and the entire work done anew, and if this happens repeatedly

with the same calculator, the calculator is advised to find

another position more in agreement with his mental capacity.

If a single point of the curve appears irregular, this points to

an error in its calculation, and the calculation of the point is

checked; if the error is not found, this point is calculated

entirely separately, since it is much more difficult to find an

error which has been made than it is to avoid making an

error.

169. Some of the most frequent numerical errors are :

1. The decimal error, that is, a misplaced decimal point.

This should not be possible in the final result, since the magni-

tude of the latter should by judgment or approximate calcula-

tion be known sufficiently to exclude a mistake by a factor 10.

However, under a square root or higher root, in the exponent

of a decreasing exponential function, etc., a decimal error may

occur without affecting the result so much as to be immediately

noticed. The same is the case if the decimal error occurs in a

term which is relatively small compared with the other terms,

and thereby does not affect the result very much. For instance,

in the calculation of the induction motor characteristics, the

quantity ri 2 +s 2 Xi 2 appears, and for small values of the slip s,

the second term s 2 xi 2 is small compared with ri 2 , so that a

decimal error in it would affect the total value sufficiently to

make it seriously wrong, but not sufficiently to be obvious.

2. Omission of the factor or divisor 2.

264 ENGINEERING MATHEMATICS.

3. Error in the sign, that is, using the plus sign instead of

the minus sign, and inversely. Here again, the danger is

especially great, if the quantity on which the wrong sign is

used combines with a larger quantity, and so does not affect

the result sufficiently to become obvious.

4. Omitting entire terms of smaller magnitude, etc.

APPENDIX A.

NOTES ON THE THEORY OF FUNCTIONS.

A. General Functions.

170. The most general algebraic expression of powers of

x and y,

)2T = 0, . . . . (1)

is the implicit analytic function. It relates y and x so that to

every value of x there correspond n values of y, and to every

value of y there correspond m values of x, if m is the exponent

of the highest power of x in (1).

Assuming expression (1) solved for y (which usually cannot

be carried out in final form, as it requires the solution of an

equation of the nth order in y, with coefficients which are

expressions of x), the explicit analytic function,

y=f(x), ........ (2)

is obtained. Inversely, solving the implicit function (1) for

x, that is, from the explicit function (2), expressing x as

function of y, gives the reverse function of (2); that is

z=A(2/) ........ (3)

In the general algebraic function, in its implicit form (1),

or the explicit form (2), or the reverse function (3), x and y

are assumed as general numbers; that is, as complex quan-

tities; thus,

}

........ (4)

J

and likewise are the coefficients aoo, oi <W.

265

266 ENGINEERING MATHEMATICS.

If all the coefficients a are real, and x is real, the corre-

sponding n values of y are either real, or pairs of conjugate

complex imaginary quantities: y\ +jy 2 and yi jyz-

171. For n = l, the implicit function (1), solved for y, gives

the rational function,

and if in this function (5) the denominator contains no x } the

integer function,

y = ao+aix+a 2 x 2 -} -. . .+a m x m , , . . . (6)

is obtained.

For n = 2, the implicit function (1) can be solved for y as a

quadratic equation, and thereby gives

1 22 ...

that is, the explicit form (2) of equation (1) contains in this

case a square root.

For n>2, the explicit form y=f(x) either becomes very

complicated, for n = 3 and n = 4, or cannot be produced in

finite form, as it requires the solution of an equation of more

than the fourth order. Nevertheless, y is still a function of

x, and can as such be calculated by approximation, etc.

To find the value yi, which by function (1) corresponds to

x = xi, Taylor's theorem offers a rapid approximation. Sub-

stituting Xi in function (1) gives an expression which is of

the nth order in y, thus: Ffay), and the problem now is to

find a value 2/1, which makes F(xi,yi) = 0.

However,

where h = y\ y is the difference between the correct value y\

and any chosen value y.

APPENDIX A.

267

Neglecting the higher orders of the small quantity h,

(8), and considering that F(x\,yi) =0, gives

in

F(x l ,y)

'dF(xi,y)'

dy

(9)

and herefrom is obtained yi = y+h, as first approximation.

Using this value of y\ as y in (9) gives a second approximation,

which usually is sufficiently close.

172. New functions are defined by the integrals of the

analytic functions (1) or (2), and by their reverse functions.

They are called Abelian integrals and Abelian functions.

Thus in the most general case (1), the explicit function

corresponding to (1) being

the integral,

(2)

then is the general Abelian integral, and its reverse function,

the general Abelian function.

(a) In the case, n = l, function (2) gives the rational function

(5), and its special case, the integer function (6).

Function (6) can be integrated by powers of z. (5) can be

resolved into partial fractions, and thereby leads to integrals

of the following forms :

(1)

(2)

(10)

268 ENGINEERING MATHEMATICS.

Integrals (10), (1), and (3) integrated give rational functions,

(10), (2) gives the logarithmic function log (x&), and (10), (4)

the arc function arc tan x.

As the arc functions are logarithmic functions with complex

imaginary argument, this case of the integral of the rational

function thus leads to the logarithmic function, or the loga-

rithmic integral, which in its simplest form is

= lQgx, ...... (11)

and gives as its reverse function the exponential function,

x=e* .......... (12)

It is expressed by the infinite series,

as seen in Chapter II, paragraph 53.

173. b. In the case, n = 2, function (2) appears as the expres-

sion (7), which contains a square root of some power of x. It.s

first part is a rational function, and as such has already been

discussed in a. There remains thus the integral function,

_ C

. . . .

; ; 5- dx. . . . (14)

2

This expression (14) leads to a series of important functions.

(1) Forp = l or 2,

dx (15)

Co+CiX+C 2 X 2 +. .

By substitution, resolution into partial fractions, and

separation of rational functions, this integral (11) can be

reduced to the standard form,

. . ' (16)

In the case of the minus sign, this gives

z= I = = = arc sin x, (17)

APPENDIX A.

and as reverse functions thereof, there are obtained the trigo-

nometric functions.

x = sin z, ]

(18)

Vl x 2 = cos z. j

In the case of the plus sign, integral (16) gives

z= ( dx =-log{Vl+x 2 -o:i-arc sinh x, (19)

J vl+x 2

and reverse functions thereof are the hyperbolic functions,

e"

X =

; = cosh 2.

. . . . . (20)

The trigonometric functions are expressed by the series :

kv.

120

1001.00

80 0.-8Q

0.60

0.40

0.20

100 $)

FIG. 86. Transmission Line Characteristics.

For instance, when investigating the short-circuit current of an

electric generating system, it is of importance to know whether

this current is 3 or 4 times normal current, or whether it is

40 to 50 times normal current, but it is immaterial whether

it is 45 to 46 or 50 times normal. In studying lightning

phenomena, and, in general, abnormal voltages in electric

systems, calculating the discharge capacity of lightning arres-

ters, etc., the magnitude of the quantity is often sufficient. In

254 ENGINEERING MATHEMATICS.

calculating the critical speed of turbine alternators, or the

natural period of oscillation of synchronous machines, the

same applies, since it is of importance only to see that these

speeds are sufficiently remote from the normal operating speed

to give no trouble in operation.

(6) Approximate calculation, requiring an accuracy of one

or a few per cent only; a large part of engineering calcu-

lations fall in this class, especially calculations in the realm of

design. Although, frequently, a higher accuracy could be

reached in the calculation proper, it would be of no value,

since the data on which the calculations are based are sus-

ceptible to variations beyond control, due to variation in the

material, in the mechanical dimensions, etc.

Thus, for instance, the exciting current of induction motors

may vary by several per cent, due to variations of the length

of air gap, so small as to be beyond the limits of constructive

accuracy, and a calculation exact to a fraction of one per cent,

while theoretically possible, thus would be practically useless,

The calculation of the ampere-turns required for the shunt

field excitation, or for the series field of a direct-current

generator needs only moderate exactness, as variations in the

magnetic material, in the speed regulation of the driving

power, etc., produce differences amounting to several per

cent.

(c) Exact engineering calculations, as, for instance, the

calculations of the efficiency of apparatus, the regulation of

transformers, the characteristic curves of induction motors,

etc. These are determined with an accuracy frequently amount-

ing to one-tenth of one per cent and even greater.

Even for most exact engineering calculations, the accuracy

of the slide rule is usually sufficient, if intelligently used, that

is, used so as to get the greatest accuracy. Thus, in dividing,

for instance, 297 by 283 by the slide rule, the proper way is

to divide 297-283 = 14 by 283, and to add the result to 1.

This gives a greater accuracy than direct division. For accu-

rate calculations, preferably the glass slide should not be used,

but the result interpolated by the eye.

163. While the calculations are unsatisfactory, if not carried

out with the degree of exactness which is feasible and desirable,

it is equally wrong to give numerical values with a number of

NUMERICAL CALCULATIONS. 255

ciphers greater than the method or the purpose of the calcula-

tion warrants. For instance, if in the design of a direct-current

generator, the calculated field ampere-turns are given as 9738,

such a numerical value destroys the confidence in the work of

the calculator or designer, as it implies an accuracy greater

than possible, and thereby shows a lack of judgment.

The number of ciphers in which the result of calculation is

given should signify the exactness, In this respect two

systems are in use:

(a) Numerical values are given with one more decimal

than warranted by the probable error of the result; that is,

the decimal before the last is correct, but the last decimal may

be wrong by several units. This method is usually employed

in astronomy, physics, etc.

(6) Numerical values are given with as many decimals as

the accuracy of the calculation warrants; that is, the last

decimal is probably correct within half a unit. For instance,

an efficiency of 86 per cent means an efficiency between 85.5

and 86.5 per cent; an efficiency of 97.3 per cent means an

efficiency between 97.25 and 97.35 per cent, etc. This system

is generally used in engineering calculations. To get accuracy

of the last decimal of the result, the calculations then must

be carried out for one more decimal than given in the result.

For instance, when calculating the efficiency by adding the

various percentages of losses, data like the following may be

given :

Core loss 2.73 per cent

i 2 r 1.06 ,"

Friction. . 0.93 "

Total 4.72

Efficiency 100-4.72 = 95.38

4pproximately 95.4

It is obvious that throughout the same calculation the

same degree of accuracy must be observed.

It follows herefrom that the values :

2i; 2.5; 2.50; 2.500,

256 ENGINEERING MATHEMATICS.

while mathematically equal, are not equal in their meaning as

an engineering result :

2.5 means between 2.45 and 2.55;

2.50 means between 2.495 and 2.505;

2.500 means between 2.4995 and 2.5005;

while 2J gives no clue to the accuracy of the value.

Thus it is not permissible to add zeros, or drop, zeros at

the end of numerical values, nor is it permissible, for instance,

to replace fractions as 1/16 by 0.0625, without changing the

meaning of the numerical value, as regards its accuracy.

This is not always realized, and especially in the reduction of

common fractions to decimals an unjustified laxness exists

which impairs the reliability of the results. For instance, if

in an arc lamp the arc length, for which the mechanism is

adjusted, is stated to be 0.8125 inch, such a statement is

ridiculous, as no arc lamp mechanism can control for one-tenth

as great an accuracy as implied in this numerical value: the

value is an unjustified translation from 13/16 inch.

The principle thus should be adhered to, that all calcula-

tions are carried out for one decimal more than the exactness

required or feasible, and in the result the last decimal dropped;

that is, the result given so that the last decimal is probably

correct within half a unit.

c. Intelligibility of Engineering Data.

164. In engineering calculations the value of the results

mainly depends on the information derived from them, that is,

on their intelligibility. To make the numerical results and

their meaning as intelligible as possible, it thus is desirable,

whenever a series of values are calculated, to carefully arrange

them in tables and plot them in a curve or in curses. The

latter is necessary, since for most engineers the plotted curve

gives a much better conception of the shape and the variation

of a quantity than numerical tables.

Even where only a single point is required, as the core

loss at full load, or the excitation of an electric generator at

rated voltage, it is generally preferable to calculate a few

NUMERICAL CALCULATIONS.

257

02

01

08

Volts

400

500

200

100

10

FIG. 87. Compounding Curve.

points near the desired value, so as to get at least a short piece

of curve including the desired point.

The main advantage, and foremost purpose of curve plotting

thus is to show the shape of the function, and thereby give

a clearer conception of it ;

but for recording numerical

values, and deriving numer-

ical values from it, the plotted

curve is inferior to the table,

due to the limited accuracy

possible in a plotted curve,

and the further inaccuracy

resulting when drawing a

curve through the plotted cal-

culated points. To some

extent, the numerical values

as taken from a plotted curve,

depend on the particular

kind of curve rule used in

plotting the curve.

In general, curves are used for two different purposes, and

on the purpose for which the curve is plotted, should depend

the method of plotting, as the scale, the zero values, etc.

^^ When curves are used to

illustrate the shape of the

function, so as to show how

much and in what manner a

quantity varies as function

of another, large divisions of

inconspicuous cross-section-

ing are desirable, but it is

essential that the cross-

sectioning should extend to

the zero values of the func-

tion, even if the numerical

values do not extend so

far, since otherwise a wrong

impression would be con-

ferred. As illustrations are plotted in Figs. 87 and 88, the

compounding curve of a direct-current generator. The arrange-

02

06

08

Volts

-580^

510

-490

FIG. 88. Compounding Curve.

258

ENGINEERING MATHEMATICS.

ment in Fig. 87 is correct ; it shows the relative variation

of voltage as function of the load. Fig. 88, in which the

cross-sectioning does not begin at the scale zero, confers the

FIG. 89. Curve Plotted to show Characteristic Shape.

FIG. 90. Curve Plotted for Use as Design Data.

wrong impression that the variation of voltage is far greater

than it really is.

When curves are used to record numerical values and

derive them from the curve, as, for instance, is commonly the

NUMERICAL CALCULATIONS.

259

case with magnetization curves, it is unnecessary to have the zero

of the function coincide with the zero of the cross-sectioning, but

rather preferable not to have it so, if thereby a better scale of

the curve can be secured. It is desirable, however, to use suffi-

ciently small cross-sectioning to make it possible to take

numerical values from the curve with good accuracy. This is

illustrated by Figs. 89 and 90. Both show the magnetic charac-

teristic of soft steel, for the range above (B = 8000, in which it is

usually employed. Fig. 89 shows the proper way of plotting

for showing the shape of the function, Fig. 90 the proper way of

plotting for use of the curve to derive numerical values therefrom.

\

\

ii

\

FIG. 91. Same Function Plotted to Different Scales; I is correct.

165. Curves should be plotted in such a manner as to show

the quantity which they represent, and its variation, as well as

possible. Two features are desirable herefor:

1. To use such a scale that the average slope of the curve,

or at least of the more important part of it, does not differ

much from 45 cleg. Hereby variations of curvature are best

shown. To illustrate this, the exponential function y = e~ x is

plotted in three different scales, as curves I, II, III, in Fig. 91.

Curve I has the proper scale.

2. To use such a scale, that the total range of ordinates is

not much different from the total range of abscissas. Thus

when plotting the power-factor of an induction motor, in

Fig. 92, curve I is preferable to curves II or III.

260

ENGINEERING MATHEMATICS.

These two requirements frequently are at variance with

each other, and then a compromise has to be made between

them, that is, such a scale chosen that the total ranges of the

two coordinates do not differ much, and at the same time

the average slope of the curve is not far from 45 deg. This

usually leads to a somewhat rectangular area covered by the

curve, as shown, for instance, by curve I, in Fig. 91.

In curve plotting, a scale should be used which is easily

read. Hence, only full scale, double scale, and half scale

should be used. Triple scale and on,e-third scale are practically

unreadable, arid should therefore never be used. Quadruple

FIG. 92. Same Function Plotted to Different Scales; I is Correct.

scale and quarter scale are difficult to read and therefore unde-

sirable, and are generally unnecessary, since quadruple scale

is not much different from half scale with a ten times smaller

unit, and quarter scale not much different from double scale

of a ten times larger unit.

166. Any engineering calculation on which it is worth

while to devote any time, is worth being recorded with suffi-

cient completeness to be generally intelligible. Very often in

making calculations the data on which the calculation is based,

the subject and the purpose of the calculation are given incom-

pletely or not at all, since they are familiar to the calculator at

the time of calculation. The calculation thus would be unin-

NUMERICAL CALCULATIONS. 261

telligible to any other engineer, and usually becomes unintelli-

gible even to the calculator in a few weeks.

In addition to the name and the date, all calculations should

be accompanied 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., in short., they should include all the information required

to make 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 work required to do this is negligible compared with the

increased utility 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. Reliability of Numerical Calculations.

167. The most important and essential requirement of

numerical engineering calculations is their absolute reliability.

When making a calculation, the most brilliant ability, theo-

retical 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 an error in the numerical calculation of an

example which illustrates a general proposition, does not detract

from the interest and value of the latter, which is the main

purpose; in physics, the general law which is the subject of

the' investigation 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 calcu-

lated wrong, the rotor ot the machine flies to pieces by centrifugal

forces, or the bridge collapses, and with it the reputation of the

engineer. The essential difference between engineering and

purely scientific caclulations is the rapid check on the correct-

ness of the calculation, which is usually afforded by the per-

262 ENGINEERING MATHEMATICS.

formance of the calculated structure but too late to correct

errors.

Thus rapidity of calculation, while by itself useful, is of no

value whatever compared with reliability that is, correct-

ness.

One of the first and most important requirements to secure

reliability is neatness and care in the execution of the calcula-

tion. If the calculation is made on any kind of a sheet of

paper, with lead pencil, with frequent striking out and correct-

ing of figures, etc., it is practically hopeless to expect correct

results from any more extensive calculations. Thus the work

should be done with pen and ink, on white ruled paper; if

changes have to be made, they should preferably be made by

erasing, and not by striking out. In general, the appearance of

the work is one of the best indications of its reliability. The

arrangement in tabular form, where a series of values are calcu-

lated, offers considerable assistance in improving the reliability.

168. Essential in all extensive calculations is a complete

system of checking the results, to insure correctness.

One way is to have the same calculation made independently

by two different calculators, and then compare the results.

Another way is to have a few points of the calculation checked

by somebody else. Neither way is satisfactory, as it is not

always possible for an engineer to have the assistance of another

engineer to check his work, and besides this, an engineer should

and must be able to make numerical calculations so that he can

absolutely rely on their correctness without somebody else

assisting him.

In any more important calculations every operation thus

should be performed twice, preferably in a different manner.

Thus, when multiplying or dividing by the slide rule, the multi-

plication or division should be repeated mentally, approxi-

mately, as check; when adding a column of figures, it should be

added first downward, then as check upward, etc.

Where an exact calculation is required, first the magnitude

of the quantity should be estimated, if not already known,

then an approximate calculation made, which can frequently

be done mentally, and then the exact calculation; or, inversely,

after the exact calculation, the result may be checked by an

approximate mental calculation.

NUMERICAL CALCULATIONS. 263

Where a series of values is to be calculated, it is advisable

first to calculate a few individual points, and then, entirely

independently, calculate in tabular form the series of values,

and then use the previously calculated values as check. Or,

inversely, after calculating the series of values a few points

should independently be calculated as check.

When a series of values is calculated, it is usually easier to

secure reliability than when calculating a single value, since

in the former case the different values check each other. There-

fore it is always advisable to calculate a number of values,

that is, a short curve branch, even if only a single point is

required. After calculating a series of values, they are plotted

as a curve to see whether they give a smooth curve. If the

entire curve is irregular, the calculation should be thrown away,

and the entire work done anew, and if this happens repeatedly

with the same calculator, the calculator is advised to find

another position more in agreement with his mental capacity.

If a single point of the curve appears irregular, this points to

an error in its calculation, and the calculation of the point is

checked; if the error is not found, this point is calculated

entirely separately, since it is much more difficult to find an

error which has been made than it is to avoid making an

error.

169. Some of the most frequent numerical errors are :

1. The decimal error, that is, a misplaced decimal point.

This should not be possible in the final result, since the magni-

tude of the latter should by judgment or approximate calcula-

tion be known sufficiently to exclude a mistake by a factor 10.

However, under a square root or higher root, in the exponent

of a decreasing exponential function, etc., a decimal error may

occur without affecting the result so much as to be immediately

noticed. The same is the case if the decimal error occurs in a

term which is relatively small compared with the other terms,

and thereby does not affect the result very much. For instance,

in the calculation of the induction motor characteristics, the

quantity ri 2 +s 2 Xi 2 appears, and for small values of the slip s,

the second term s 2 xi 2 is small compared with ri 2 , so that a

decimal error in it would affect the total value sufficiently to

make it seriously wrong, but not sufficiently to be obvious.

2. Omission of the factor or divisor 2.

264 ENGINEERING MATHEMATICS.

3. Error in the sign, that is, using the plus sign instead of

the minus sign, and inversely. Here again, the danger is

especially great, if the quantity on which the wrong sign is

used combines with a larger quantity, and so does not affect

the result sufficiently to become obvious.

4. Omitting entire terms of smaller magnitude, etc.

APPENDIX A.

NOTES ON THE THEORY OF FUNCTIONS.

A. General Functions.

170. The most general algebraic expression of powers of

x and y,

)2T = 0, . . . . (1)

is the implicit analytic function. It relates y and x so that to

every value of x there correspond n values of y, and to every

value of y there correspond m values of x, if m is the exponent

of the highest power of x in (1).

Assuming expression (1) solved for y (which usually cannot

be carried out in final form, as it requires the solution of an

equation of the nth order in y, with coefficients which are

expressions of x), the explicit analytic function,

y=f(x), ........ (2)

is obtained. Inversely, solving the implicit function (1) for

x, that is, from the explicit function (2), expressing x as

function of y, gives the reverse function of (2); that is

z=A(2/) ........ (3)

In the general algebraic function, in its implicit form (1),

or the explicit form (2), or the reverse function (3), x and y

are assumed as general numbers; that is, as complex quan-

tities; thus,

}

........ (4)

J

and likewise are the coefficients aoo, oi <W.

265

266 ENGINEERING MATHEMATICS.

If all the coefficients a are real, and x is real, the corre-

sponding n values of y are either real, or pairs of conjugate

complex imaginary quantities: y\ +jy 2 and yi jyz-

171. For n = l, the implicit function (1), solved for y, gives

the rational function,

and if in this function (5) the denominator contains no x } the

integer function,

y = ao+aix+a 2 x 2 -} -. . .+a m x m , , . . . (6)

is obtained.

For n = 2, the implicit function (1) can be solved for y as a

quadratic equation, and thereby gives

1 22 ...

that is, the explicit form (2) of equation (1) contains in this

case a square root.

For n>2, the explicit form y=f(x) either becomes very

complicated, for n = 3 and n = 4, or cannot be produced in

finite form, as it requires the solution of an equation of more

than the fourth order. Nevertheless, y is still a function of

x, and can as such be calculated by approximation, etc.

To find the value yi, which by function (1) corresponds to

x = xi, Taylor's theorem offers a rapid approximation. Sub-

stituting Xi in function (1) gives an expression which is of

the nth order in y, thus: Ffay), and the problem now is to

find a value 2/1, which makes F(xi,yi) = 0.

However,

where h = y\ y is the difference between the correct value y\

and any chosen value y.

APPENDIX A.

267

Neglecting the higher orders of the small quantity h,

(8), and considering that F(x\,yi) =0, gives

in

F(x l ,y)

'dF(xi,y)'

dy

(9)

and herefrom is obtained yi = y+h, as first approximation.

Using this value of y\ as y in (9) gives a second approximation,

which usually is sufficiently close.

172. New functions are defined by the integrals of the

analytic functions (1) or (2), and by their reverse functions.

They are called Abelian integrals and Abelian functions.

Thus in the most general case (1), the explicit function

corresponding to (1) being

the integral,

(2)

then is the general Abelian integral, and its reverse function,

the general Abelian function.

(a) In the case, n = l, function (2) gives the rational function

(5), and its special case, the integer function (6).

Function (6) can be integrated by powers of z. (5) can be

resolved into partial fractions, and thereby leads to integrals

of the following forms :

(1)

(2)

(10)

268 ENGINEERING MATHEMATICS.

Integrals (10), (1), and (3) integrated give rational functions,

(10), (2) gives the logarithmic function log (x&), and (10), (4)

the arc function arc tan x.

As the arc functions are logarithmic functions with complex

imaginary argument, this case of the integral of the rational

function thus leads to the logarithmic function, or the loga-

rithmic integral, which in its simplest form is

= lQgx, ...... (11)

and gives as its reverse function the exponential function,

x=e* .......... (12)

It is expressed by the infinite series,

as seen in Chapter II, paragraph 53.

173. b. In the case, n = 2, function (2) appears as the expres-

sion (7), which contains a square root of some power of x. It.s

first part is a rational function, and as such has already been

discussed in a. There remains thus the integral function,

_ C

. . . .

; ; 5- dx. . . . (14)

2

This expression (14) leads to a series of important functions.

(1) Forp = l or 2,

dx (15)

Co+CiX+C 2 X 2 +. .

By substitution, resolution into partial fractions, and

separation of rational functions, this integral (11) can be

reduced to the standard form,

. . ' (16)

In the case of the minus sign, this gives

z= I = = = arc sin x, (17)

APPENDIX A.

and as reverse functions thereof, there are obtained the trigo-

nometric functions.

x = sin z, ]

(18)

Vl x 2 = cos z. j

In the case of the plus sign, integral (16) gives

z= ( dx =-log{Vl+x 2 -o:i-arc sinh x, (19)

J vl+x 2

and reverse functions thereof are the hyperbolic functions,

e"

X =

; = cosh 2.

. . . . . (20)

The trigonometric functions are expressed by the series :

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