Theodore R. (Theodore Rudolph) Running.

Empirical formulas online

. (page 7 of 8)
Online LibraryTheodore R. (Theodore Rudolph) RunningEmpirical formulas → online text (page 7 of 8)
Font size The volume is by (7)

^â€˘i'i(i49-oo4)=3i.2 cu. units.

Centroids
Let the coordinates of the centroid of an area be represented
by X and y. Then from the calculus

I xydx
x =

I ydx

- I y'^dx
ydx

y=

The integral in the numerator of the value of x may be
represented by the area bounded by the curve Y = xy, the :r-axis
and the two ordinates x = a and x = b. The original area is
bounded by the curve whose ordinates are represented by y,
the a;-axis and the two ordinates x = a and x = b. The integral
in the numerator of the value of y may be represented by the
area bounded by the curve Y=y^, the a:-axis and the two
ordinates x = a and x = b.

For a volume generated by revolving a given area about the
rc-axis

TT I y^xdx

TT I y'^dx

NUMERICAL INTEGRATION

When the volume is irregular

127

x =

Axdx

X

The process of finding the coordinates of the centroid of the
area in Fig. 28 is shown in the table:

y

xy

3,2

yH

.2

â€˘4

.6

.8

1.0

I.O

I-S

2.2

2.7

2.6

2-3

0.00

0.30

0.88

1.62

2.08

2.30

1. 00

2.25

4.84

7.29

6.76

5-29

0.000

0.450

1.936

4-374

5-408

5.290

1.2

2.1
2.52
4.41
5.292

The area under the curve Y = xy is

i^[o.oo+i.2o+i.76+6.48+4.i6+9.2o+2.52] = 1.688;
- 1.688

x =

2.58

= .654.

The area under the curve Y = ^y^ is
^[i.oo+9.cx>+9.68 + 29.i6 + 13.524-21.16+4.41] = 2.931

-_2-93i

2.58

1. 136.

As was pointed out before, large changes in the ordinates
must be avoided.

For the volume generated by revolving the area about
the x-sods

-_7ri^[o.ooo+ 1. 800+3.872 + 17.496+ 10. 816+ 21. 160+ 5. 292]
7r3^[i.oo+9.oo+9.68+29.i6+i3.52+2i.i6+4.4i]

60.436

87-93

.687.

128

EBiPIRICAL FORMULAS

Moments of Ineriia

The expression for the moment of inertia of an area about
the 3^-axis is

Iy= j x^ydx.

/*= I xfdy.

When the equation of the curve is known these integrals
can be calculated at once, but when this is not the case approxi-
mate methods must be resorted to.

I. The process of finding the approximate values of these
integrals is shown in the table below. The values of x and y
are taken from Fig. 28.

X

y

x'^y

\y'

.2

.4

.6

.8

1.0

I.O

1-5

2.2

2.7

2.6

2.3

O.CXXJ

0.060

0.335

0.972

1.664

2.300

0.333

I. 125

3 549

6.561

5.859

4.056

I. 2
2.1
3.024

3 087

If the values of x'^y be plotted to x we will have a curve
under which the area represents the moment of inertia of the
area in Fig. 28 about the 3/-axis.

Dividing this by the area found before, there results for the

Plotting iy^ to X and finding the area under the curve so
determined

7^ = 4.6136,
and

i?/ = 1.788.

NUMERICAL INTEGRATION

129

2. The form of a quarter section of a hollow pillar, Fig. 31,
is given by the following table. Find the moment of inertia
of the section about the axes of x and y.

y
.5

n

~~->

A

~^

\

^

K

N

\

.3

N

\

\

\

\

\

\

\

â€˘i

\

\

Fig. 31.

X

F

xW '

y

X

y^X

00

050

. 00000

00

100

. 00000

05

055

.00014

05

108

.00027

10

068

.00068

10

116

.00116

15

078

.00175

IS .

120

.00270

20

096

.00384

20

125

.00500

25

116

.00725

25

130

.00812

30

148

.01332

30

133

.01197

35

200

.02450

35

140

.01715

40

300

. 04800

40

150

.02400

45

215

â– 04354

45

215

â€˘04354

50

000

. 00000

50

000

.00000

In the above table X stands for the width of the area parallel
to the X-axis and Y for the width parallel to the ;y-axis. The
area is 0.066.

The moment of inertia about the y-3Lxis is

X

x^Ydx = .ooj;^6;

i?-:52M = o.iiis.

.060

130 EMPIRICAL FORMULAS

The moment of inertia about the nc-axis is

I y^Xdy = .oo6ig;
Jo

^'^ = ^' = -^38,

where R stands for the radius of gyration.

The values of the above integrals were computed by for-
mula (7).

APPENDIX

If a chart could be constructed with all the different forms
of curves together with their equations which may arise in
representing different sets of data it would be a comparatively-
simple matter to select from the curves so constructed the one
best suited for any particular set. Useful as such a chart would
be its construction is clearly out of the question. The most
that can be done of such a nature is to draw a number of curves
represented by each one of the simpler equations.

A word of caution is, however, necessary here. A particular
curve may seem to the eye to be the one best suited for a given
set of data, and yet, when the test is applied, it may be found
to be a very poor fit. It is of some aid, nevertheless, to have
before the eye a few of the curves represented by a given formula.

The purpose of the following figures is to illustrate the
changes in the form of curves produced by shght changes in
the constants. Figs. I, II, III, and IV show changes produced
by the addition of terms. Figs. V to XIX changes in form
produced by changes in the values of the constants, and Fig.
XX the changes in form brought about by varying both the
values of the constants and the number of terms.

A discussion of all the figures is unnecessary. A few words
in regard to one will suffice. Formula XIV, for example,
y = a-\-bx'', an equation which can be made to express fairly
well the quantity of water flowing in many streams if x
stands for mean depth and y for the discharge per second,
represents a family of triply infinite number of curves. Fixing
the values of b and c and varying the value of a does not
change the form of the curve, but only moves it up or down

131

132

EMPIRICAL FORMULAS

along the >'-axis. Keeping the values of a and b constant and
varying the value of c, the formula will represent an infinite
number of curves all cutting the ^'-axis in the same point. In
the same way, keeping the value of a and c constant and vary-
ing the value of b, an infinite number of curves is obtained,
all of which cut the ^'-axis in a fixed point. In Fig. XIV the
quantity a is constant and equal to unity, while b and c vary.

To one trained in the theory of curves the illustrations are,
of course, of no essential value, but to one not so trained they
may be of considerable help.

The text should be consulted in connection with the curves
in any figure. The figures are designated to correspond to the
formulas discussed in the first five chapters.

(1) y=l-.lx

(2) 2/=l-.lx+.01x2

(3) 2/=l-.lx+.01a;2_.001xS

(4) j/=l-.lx+.01x2-.001ar34-

.OOOlx*

(5) 2/=l-.lx+.01x2-.001x'+

.OOOlx^-.OOOOlx^

(6) 2/=-l-.lx+.01a:2-.001a;'+

.OOOlx^-.OOOOlx^+.OOOOOlx^

See formula I, page 13

Fig. I.

APPENDIX

133

V
1.4

ll

1.2

4-

'

4-

\'2)

OX

%

^

^

"/

/

0.4
0.2

' /

/

f

-0.2
-0.4

i

-0.8

-1,

Jul

(1) y=l-l/x

(2) 2/=l-l/x+l/x2

(3) y=l-l/x+l/x^-l/x^

(4) y=l-l/x+l/x'^-l/x^+l/x*

(5) y=l-l/x+l/x^-l/x^+

l/x*-l/x^

(6) y=l-l/x+l/x^-l/x^+

1/x^-l/x^+l/x^

See formula II, page 22

10 a u X

Fig. II.

3

2.8

y-

/

/

j

/

/

/

/

2.2

2

1.8

1.6

(1)/

//

â€˘y

i

/

//

/

/

/;

/

/J

^

/^..^

N,

1.4

â€” .

y

C^

^

^

X

sA

^

1.2

a

0.8
0.6
0.4
0.2

^

^

^

\

s\

N

8

>

I

J

3

K

J

) 1

1

i 1

d -x

(1) -=l-.la;

V

(2) -=l-.lx+.01x2

L3) -=l-.lx+.01x2-.001a;3
(4) -=l-.lx+.01x2-.001.3f

y

.000 Ix*

(5) -=l-.lx+.01x2-.00U3+
V

.0001x4-. OOOOlx^

(6) -=l-.lx+.01x2-.001x3+

y

.OOOlx*-. 00001x6+. OOOOOlxÂ®
See formula III, page 25

Fig. in.

134

EldPIRICAL FORMULAS

1

w

1JI

/.

A

,y^

"^

^

=^

^

^

J2L

^

^

^

â– ^

^

<Â«

>

s^

:^

^

N

s\

^

Jl

//j

â€˘<w

1

1

{

J

*

5

e

8

\

1 u

I h

r-t

(1) yÂ«=l-.lx

(2) i/Â«=l-.lx+.01x2

(3) vÂ»=l-.li+.01x*-.001xÂ»

(4) yÂ»-l-.lx+.01xÂ«-.001xÂ«+

.000 Ix*

(5) 1/2=1-. Ix+.01x2-.001xÂ»+

.0001x*-.00001x'

(G) 2/2=1-. Ix+.01x2_.001a:'' +

.0001x*-.00001x*+.000001xÂ«

See formula IV, page 25

Fig. IV.

/

/

^

(1) l/=(.5)^

(2) l/=(.6)*

(3) 2/= (.7)*

(4) l/=(.8)*

(5) y=(.9)*

(6) y=(.95)*

(7) y=.99)*

(8) j/=(1.01)*

(9) i/=(1.05)*

(10) 2/ = (l.l)*

(11) 2/=(1.2)*

See formula, "^

a

y

/

/

/

/

^^

/

/

/

/

/

f

^

/

/

^

^

/

/

^

^

/

y

^

/ ^

y

^^

A-r

(S)

â€”

v

r~~"

â€”I

Sl

\>

c

^^

â–  -

â€˘^

^V

\

"^

-^

^

â€”

\

S:

^

-^

â– ^

â– ^

â€”-

.

â€”

>^, page

1

^

^

^

â€” â–

â€”

!

Â«

s

e

3

e

Â«

1

1

I \.

8 1

J 14

U

Fig. V.

APPENDIX

135

V

1 8

(li.

â– i)^

-^^^^

â– ::z^

^=

,,^

_^

/

K

(4)^

^

;::::;

^^

1.4

y/

/
^

^
^

^

[6)^

^-â–

^

y.

\^

(7)

9)

-â€”

nj

â€”

1

^>^

===:

â€”

â€”

R

^^

â€”

i::

.(11)

â€”

â€”

0.8
0.6

â– "

'^

:>N

â– ^

(12)

â€”

\

"S.

\

^^

-02

n

\

â– n

^-v

\

\

\

N

\

\

â€” s
S

\

%

I

\

ÂŁ

Â«

t

s

1

) 1

I 1

I Â»

(1) 2/ = 2-(.5)*

(2) |/=2-(.6)*

(3) y=2-(.7)*

(4) j/=2-(.8)*

(5) y-2-(.85)*

(6) y=2-(.9)*^

(7) 2/ = 2-(.95)*

(8) 2/=2-(.97)*

(9) 2/=2-(.99)*

(10) 2/=2-(1.01)*

(11) y=2-(1.03)^

(12) y=2-(1.05)*
(I3)y=2-(1.07)*
(14) j/=2-(1.08)*

See formula VI, page 28

Fig. VI.

^

^

'^^

^

â–

^

J

c-

y

^^

^

^^

-^

/

//

>

y

^

^

//

v

/

^i.

^

r1

//

y

(5)^

- ^

//

^

(6)

,-^

^

X

^(7);

-(-8^

â–

â€”

\^

\^

"-^

s^

v^

^^

U^

__

"

â€”

â€˘

(1) log2/=l-.5(.5)*

(2) logy=l-.5(.6)^

(3) logy=l-.5(.7)^

(4) log2/=l-.5(.8)^

(5) logy=l-.5(.9)*

(6) log2/=l-.5(.95)*

(7) logy=l-.5(.98)*

(8) log2/=l-.5(1.02)'^

(9) logy=l-.5(l.l)*

(10) logy=l-.5(1.2)3J

(11) log2/=l-.5(1.3)a^

(12) log2/=l-.5(1.5)a'

(13) log2/=l-.5(2)a?
base=10

See formula VII, page 32

Fig. VII.

136

EMPIRICAL FORMULAS

V

(1)

â€”

____

u

i

^

"

J3),

â– ^^

â€”

1 1

-^

"^

^-

N

\

^

Sw.

^

^^

1

r

\

â€˘^

s.

^

^

-^

^

N

N

X

\

^

^

\

N

V

N,

1

\

\

N

\

N

\

â€˘OS

\

\

(1) i/=2-.01x-(.6)*

(2) v=2-.03a;-(.5)*

(3) v = 2-.05i-(.5)*

(4) v-2-.08x-(.5)*

(5) v=2-.lx-(.5)*

(6) i/=2-.12x-(.5)*

(7) j/=2-.15x-(.5)*

(8) i/=2-.2x -(.5)*

See formula VIII, page

15 6 7 8 9

Fig. VIII

V

1

1

\

\

/

\\

//

\

u>

/

/

f

\

\

/

/

L

\\

^

/

1

-ar

^

^

,

^

(St,

^

^

â€”'

^

\

I

i

I

i

Â»

r

}

) 1

Q 1

1 1

2 Â»

(1) j/=10-81-.36x + .03xÂ«

(2) j/= 10-54 -.24x + . 02x2

(3) y=10-27-.12x + .01x2

(4) y= 10-135 -.0&r + .005xÂ«

(5) j/=10 - 135 + .06x- 005x2

(6) y = 10 - 54 + -24x-.02x2

See formula IX, page 37

Fig. IX.

APPENDIX

137

V.

\

\

i'

1.6

\

(1)^

^

[(2)^

1.4

\

^

^

?^

^

(3)

V-

â–

""^

\

/ i

1

\

y

/

N

5s

â€˘/

/

^

^

^

.y

: â€” â–

(6)

5

i

6

r

i

9 1

1

I J

^ *

(1) y=(1.01)* (1.05)(l-2)^

(2) j/=(1.01)* (1.05)<1-16)^

(3) y=(1.01)* (1.05)(1-15)^

(4) 2/= (.5)== (2) (1.24)^

(5) j/=(.5)^ (2) (1-23)^

(6) j/=(.5)* (2) (1-2)''

See formula X, page 37

Fig. X.

f

\

/

\

\

/

\

1

/

\

â– \

\^

1

\,

\

<<-

V

\

\

^->

\

V

.

â€”

L

>,

\^.

\

X

\

^

/-

\

â– ^<j

\

.^^

^

.^^

r

N

-^

-~

â€” .

^

â€”

1

r

"~â€”

' â€”

â€”

4 5 6 7

Fig. XI.

9 10 U 12 Â«

(1) y=

(2) V-

(3) y-

(4) y=

(5) y

.2-

.lx+.05x2

X

2â€”

.lx+.07x2

X

.2-

.lx+.la;2

X

.2-

.la;+.2x2

X

.2-.lx+.4i2
See formula XI, page 38

138

EMPIRICAL FORMULAS

Â» -

y

y

,

f

/

" 1

/

^^

' \

/

K^

.

'\

^''

J2-

Jl

::;;-

-JiL

1

f \

v^

jw__

__

]\

^

__

olâ€”

1 i

f

i

\ (

\

) I

1

i> 1

1 1

2 Â»

(1) y=5x-*

(2) i/=5x-2

(3) V-5X-1

(4) |/=5x - l

(5) j/=5x - 2

(6) y^Sx - *

See formula XII, page 42

Fig. XII.

y

t

18

te

14
It

â€”

.

-fc

^

::^

u

1

u=

^

I

y

^

__^

w

\

/

/

^

^

â–

\

^

lO,

"

y

\

.

^

/

^

' â–

/

â€”

(S)

1

,

( 1

]

^IG

X

II]

1

1

I 1

2 1

3 1

1 1

\

(l)y=l+Iogx+.llog2x;

y=â€” 1.5 (min.) when log x= â€”5

(2)|/=l+logx+.011og2x;

2/= â€”24 (min.) when log x= â€”50

(3) y=l+.21ogx+.31og2x

(4) y=lâ€” log x+log^ X

(5) v=l-logx+.51og*x

See formula XIII, page 44

APPENDIX

139

^

^^

^^

â– ^

â€”

^

^

=^

^

^

- -

(i)

^

><^>

X

N

^

N^

\

I i

i

.

> (

r

3

) 1

I

1 1

2 â– Â»

(1) y=l+.008xl-7

(2) y=l+.007a;l-6

(3) y=l+.006xl-5

(4) 2/= 1-. 002x2

(5) y=l-.003a;2.1

(6) 2/= 1-. 004x2.2

See formula XIV, page 45

Fig. XIV.

^

\

^

\

\

\

^\

^

\

\

\

\

k

\

\

\

V

^.

^

K

N

N,

\

\

N

^

\

~~^,

:x.

\

s.

b:

-<v.

^

i

i

4

f

(

]

1

^ 1

, 1

r-t

(1) y=(2.0) 10 - 01^^

(2) 2/= (1.6) 10- â€˘02x1-''

(3) 2/= (1.2) 10 - 03i^-^

(4) 2/= (1.0) 10 - 04x^-^^

(5) 2/= (0.8) 10 - 05a;l-24

(6) 2/= (0.6) 10- â€˘06x1-12

See formula XV, page 49

Fig. XV.

140

EMPIRICAL FORMULAS

^

5

â€”

^

^
%

:^

â–

^

^

w

7\

n1

s

\,

K

\

/''

V

\

V

^

^

j^

" - ^

^

â€”

â– -^

"^

___;

Â«^

i

a

\

8

t

6

t

Y 1

1

1

e X

(1) (v-2) (xf.S) 1

(2) (v-2) (x+.75) = -1.5

(3) (v-2) (x+l) = -2

(4) (1/+.1) (x+4)=8.2

(5) (1/+.1) (x+3) = 6.3

(6) (1/+.1) (x+2)=4.2

See formula XVI, page 53

Fig. XVI.

V

~

I 1

1

^

y

^

\

^

-^

y

\

^.1

3-

^^

â€˘

.^

-f2h;

1

"^

J4)

â€ž.

i

J5L

â€˘8

(

1

\

S

!

} \

9

) )

1

1 1

Â» s>

24

(1) y=\ 10a;+24

-12

(2) j/=J lCa;-24

2

(3) 2/=i 10a;+2

2_

(4) 2/=i35 10^

â– 5

(5) 2/=-10a: + l

See formula XVIo, page 56

Fig. XVIa.

APPENDIX

141

v.

C

1

^

^

^

,^

^^

._^

J2-

^rf*

^

^

â€” -

(3)

.

1.6

rÂ«=^

i^

iS^

\

â–

1.4

-^

\

^

~"~~~-

fy^

1.2

\

rs,

^^

^

^^

- v^

.fe;

\

N,

â–

"^

s,,^

^

^

" â€” .

^

fe

.^

1

^

r^

^

^

i^

0.2

^^

'

-^

^

^

=*"

=â€”

^

^

-0.4

/

/

/

/

-1

/

(1) y=.5e-01a:+e.05a;

(2) y=2e-05a;_.5e.lx

(3) y=2.25e-05a;_.75e.Ix

(4) y=1.8e-01a;_.3e.li

(5) 2/=1.92e - l^-.42e - 01a;

(6) y=2e - 05a:_e-.01a;

(7) y=4.2e - 2a;-3.5e - 25a;

(8) y=4.5e - 2a;-4.1e - 25a:

(9) y=.25e - 01aJ-.]Ce - 152

(10) y=e- lx-i,u-.2x

(11) y=.27e - 01^-.77e - 25a;

(12) y=e - la;-2e - 25a;

See formula XVII, page 58

Fig. XVII.

p^

\

\,

\

^

^

\

\&)

\

â€˘n

\

(3)

\

\

(5)

' â€”

^

-J

^\

/

~"^

::3

N

s

/-

^^'^

(S)^

'

N

'^

x:

:N;

\

v/

/

)

/ \

^^

_^

\:

A

V

/

\

^

y

"^ - - _

^

(1) j/=c.01a;(i.5 cos .lx-.5 sin .Ix)

(2) j/=e~'2a^(1.5cos.5xâ€” .5sin.5x)

(3) 2/=e - la:(.6cos .lx+.8sin .Ix)

(4) j/=e-1^(.2 cos .3x-.l sin .3x)

(5) j/=e-02a;(.4cos.l6x+.17sin.l6x)

(6) j/-=.5e~'^^ sin X

See formula XVIII, page 61

Fig. XVIII

142

EMPIRICAL FORMULAS

Â»

â€”

%

â€”

^

^

â€”

/

y

N

/

/

V'h

/

/

1

Ul_

/

^^

^

^

7^

^^

\

â€”â–

=

/

"5

>^'

-^

t

/

^

(3)

,

a4

A

P*

^

â€˘^

A

l

Fig. XIX.

(1) i/=2x-l-i-2

(2) v=3i-5-2.2x-8

(3) y-2.3x-8-2x.85

(4) y=.lx-l+.5x-2

(5) t/=.33x-.0012x3 ^

(6) i/=.25x-5+.05x-8

See formula XIX, page 65

(1) y=15xl-5(.4)'^

(2) y=3x2(.5)* ^

(3) j/=3x-2(i.5)*

(4) y=-.5xl-5(.75)*

See formula XlXa, page 72

Fig. XlXa.

APPENDIX

143

^

(2)

rv

//

'^

\

7^

?-^

,. â€” 1

(3)

>^^

if

^

^

^i

.^=*'

*^

s.

r^

X

s-^

'

1

^

'^

-^^142

y

s

\

N

\

^

/

^

A

Vi

-^

\

!^.

^

^

r^

y"

?

Fig. XX.

(1) 2/=166.25â€” 14.5 cos x-2.75 cos 2xâ€” 10 sin x

(2) y=167.83-20 cos x-4.33 cos 2x+5.5 coa 3x-13.28 sin x-17.32 sin 2x

(3) i/=167.62-17.5 cos x-2.75 cos 2x+3 cos 3x-1.38 cos 4x-12.42 sin x-18 sin 2x-

2.42 sin 3x

(4) y= 167.08-17.22 cos x-3. 5 cos 2x+5.5 cos 3x-0.83 cos 4x-2.78 cos 5x+

0.75 cos 6xâ€” 12.14 sin Xâ€” 19.05 sin 2xâ€” sin 3xâ€” 1.73 sin 4x+ 1.14 sin 5x

See formula XX, page 74

INDEX

Approximation, Accuracy of, 114, 118,

119, 121
Area, 11, 114, 121, 124, 125, 126, 128

Graphical Determination of, 121
Arithmetical Series, 11, 13, 15, 19, 22,

2SÂ» 27, 28, 32, 33, 37, 38, 44, 58, 61, 104

Centroids, 11, 126, 127
Check, 46, 92

Common Difference, 11, 16
Compound Interest Law, 27
Constants, 11, 92, 94, 131, 132

Graphical Determination of, 13, 15,
21, 23, 24, 25, 28, 31, 32, 36, 40,
44, 47, 48, 51, 55, 60, 64, 65, 67 ,70

Changes in, 131
Curves, 10, 11, 114, 121, 122, 131, 132

Damped Vibration, 65
Derivative, 95

First, 108

Second, 108
Differences, 11, 12, 16, 33, loi, 115,
n6

First Order, 12, 18

Higher Order, 12, 16, 18, 19, 22, loi
Differential Equation, 68, 71, in
Differentiation of Tabulated Functions,

100, 108

Elastic Limit, 14

Empirical Formulas, 9, n, 68, 90, 100
Errors of Observation, 11, 68
Fourier Series, 74

Online LibraryTheodore R. (Theodore Rudolph) RunningEmpirical formulas → online text (page 7 of 8)