ELEMENTS OF METRIC GEAR TECHNOLOGY I - SDP/SI · PDF fileThe governing equations become simple...

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ELEMENTS OF METRIC GEAR TECHNOLOGY

T-81

SECTION 10 TOOTH THICKNESS

There are direct and indirect methods for measuring tooth thickness. In general, there are three methods:

• Chordal Thickness Measurement • Span Measurement • Over Pin or Ball Measurement

10.1 Chordal Thickness Measurement

This method employs a tooth caliper that is referenced from the gear's outside diameter. Thickness is measured at the pitch circle. See Figure 10-1.

sj

hj

q

d

Fig. 10-1 Chordal Thickness Method 10.1.1 Spur Gears

Table 10-1 presents equations for each chordal thickness measurement.

No. Item Example

Circular Tooth Thickness

Half of Tooth Angleat Pitch Circle

Chordal Thickness

Chordal Addendum

Symbol Formula

s

sj

hj

p (–– + 2x tan a )m 290 360 x tan a –– + –––––––– z pz

zm sin q

zm –– (1– cos q) + ha 2

m = 10a = 20°z = 12x = +0.3ha = 13.000s = 17.8918q = 8.54270°sj = 17.8256hj = 13.6657

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Table 10-1 Equations for Spur Gear Chordal Thickness

10.1.2 Spur Racks And Helical Racks

The governing equations become simple since the rack tooth profile is trapezoid, as shown in Table 10-2.

No.

Chordal Thickness

Chordal Addendum

sj

hj

pm pmn ––– or ––– 2 2

ha

m = 3a = 20°sj = 4.7124ha = 3.0000

Item ExampleSymbol Formula

Table 10-2 Chordal Thickness of Racks

NOTE: These equations are also applicable to helical racks.

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10.1.3 Helical Gears

The chordal thickness of helical gears should be measured on the normal surface basis as shown in Table 10-3. Table 10-4 presents the equations for chordal thickness of helical gears in the radial system.

NOTE: Table 10-4 equations are also for the tooth profile of a Sunderland gear.

Table 10-4 Equations for Chordal Thickness of Helical Gears in the Radial System

No. Item Example

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FormulaSymbol

sn

zv

qv

sj

hj

Normal Circular Tooth Thickness

Number of Teeth in an EquivalentSpur Gear

Half of Tooth Angle at Pitch Circle

Chordal Thickness

Chordal Addendum

m = 4at = 20°b = 22° 30' 00"z = 20xt = +0.3ha = 4.7184sn = 6.6119zv = 25.3620qv = 4.04196°sj = 6.6065hj = 4.8350

p (–– + 2xt tan at )mt cos b 2 z––––– cos3 b

90 360xt tan at–– + ––––––––––– zv pzv zv mt cos b sin qv

zv mt cos b –––––––– (1– cos qv) + ha 2

Table 10-5 Equations for Chordal Thickness of Gleason Straight Bevel Gears

m = 4a = 20° S = 90°z1 = 16z2 = 40 z1–– = 0.4 z2 K = 0.0259ha1 = 5.5456 ha2 = 2.4544d1 = 21.8014° d2 = 68.1986°s1 = 7.5119 s2 = 5.0545sj1 = 7.4946 sj2 = 5.0536hj1 = 5.7502 hj2 = 2.4692

Obtain from Figure 10-2(on the following page)

pm – s2

pm––– – (ha1 – ha2) tan a – Km 2

s3

s – ––– 6d2

s2 cos d ha + –––––– 4d

No. Item Example

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Circular Tooth Thickness Factor(Coefficient of Horizontal Profile Shift)


Chordal Thickness

Chordal Addendum

Symbol

K

s1

s2

sj

hj

Formula

Table 10-3 Equations for Chordal Thickness of Helical Gears in the Normal System

No. Item Example

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FormulaSymbol

sn

zv

qv

sj

hj

Normal Circular Tooth Thickness

Number of Teeth of an EquivalentSpur Gear

Half of Tooth Angle at Pitch Circle

Chordal Thickness

Chordal Addendum

p (–– + 2xn tan an )mn 2 z–––––– cos3 b

90 360xn tan an –– + –––––––––– zv pzv

zv mn sin qv

zv mn –––– (1– cos qv ) + ha 2

mn = 5an = 20°b = 25° 00' 00"z = 16xn = +0.2ha = 6.0000sn = 8.5819zv = 21.4928qv = 4.57556°sj = 8.5728hj = 6.1712

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z1

0.7 0.6 0.5 0.4 0.3 0.2 0.1 z1Speed Ratio, –– z2

.110

.100

.090

.080

.070

.060

.050

.040

.030

.020

.010

.000

Fig. 10-2 Chart to Determine the Circular Tooth Thickness Factor K for Gleason Straight Bevel Gear (See Table 10-5)

Table 10-6 Equations for Chordal Thickness of Standard Straight Bevel Gears

No. Item ExampleFormulaSymbol

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6


Number of Teeth of an EquivalentSpur Gear

Back Cone Distance

Half of Tooth Angle at Pitch CircleChordal Thickness

Chordal Addendum

s

zv

Rv

qv

sj

hj

pm ––– 2 z–––– cos d d–––––2cos d 90 –– zv

zv m sin qv

ha + Rv (1 – cos qv)

m = 4a = 20° S = 90°z1 = 16 z2 = 40d1 = 64 d2 = 160ha = 4.0000d1 = 21.8014° d2 = 68.1986°s = 6.2832zv1 = 17.2325 zv2 = 107.7033Rv1 = 34.4650 Rv2 = 215.4066qv1 = 5.2227° qv2 = 0.83563°sj1 = 6.2745 sj2 = 6.2830hj1 = 4.1431 hj2 = 4.0229

Table 10-6 presents equations for chordal thickness of a standard straight bevel gear.

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If a standard straight bevel gear is cut by a Gleason straight bevel cutter, the tooth angle should be adjusted according to:

180° s tooth angle (°) = –––– (–– + hf tan a) (10-1) pRe 2

This angle is used as a reference in determining the circular tooth thickness, s, in setting up the gear cutting machine. Table 10-7 presents equations for chordal thickness of a Gleason spiral bevel gear.

Figure 10-3 is shown on the following page.

The calculations of circular thickness of a Gleason spiral bevel gear are so complicated that we do not intend to go further in this presentation.

10.1.5 Worms And Worm Gears

Table 10-8 presents equations for chordal thickness of axial module worms and worm gears.

No. Item Example

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2

Symbol

K

s1

s2

Obtain from Figure 10-3

p – s2

p tan an –– – (ha1 – ha2) –––––– – Km 2 cos bm

Formula

S = 90° m = 3 an = 20°z1 = 20 z2 = 40 bm = 35°ha1 = 3.4275 ha2 = 1.6725K = 0.060p = 9.4248s1 = 5.6722 s2 = 3.7526

Circular ToothThickness Factor


Table 10-8 Equations for Chordal Thickness of Axial Module Worms and Worm Gears

sx1

sx2

zv2

qv2

sj1

sj2

hj1

hj2

pmx –––– 2 p(–– + 2xx2 tan a x)mx 2

z2––––– cos3 g

90 360 xx2 tan ax––– + –––––––––– zv2 pzv2

sx1 cos g zv mx cos g sin qv2

(sx1 sin g cos g)2

ha1 + ––––––––––– 4d1

zv mx cos gha2 + –––––––– (1 – cos qv2) 2


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Axial Circular Tooth Thickness of WormRadial Circular Tooth Thickness of Worm Gear

No. of Teeth in an Equivalent Spur Gear (Worm Gear)

Half of Tooth Angle at Pitch Circle (Worm Gear)

Chordal Thickness

Chordal Addendum

mx = 3an = 20°zw = 2 z2 = 30d1 = 38 d2 = 90ax = 65 xx2 = +0.33333ha1 = 3.0000 ha2 = 4.0000g = 8.97263°ax = 20.22780°sx1 = 4.71239 sx2 = 5.44934 zv2 = 31.12885 qv2 = 3.34335°sj1 = 4.6547 sj2 = 5.3796hj1 = 3.0035 hj2 = 4.0785

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1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 z1 Speed Ratio, –– z2

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Fac

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K

0.260

0.240

0.220

0.200

0.180

0.160

0.140

0.120

0.100

0.080z = 15z = 16z = 170.060z = 20

0.040z = 25

Over 30

0.020

0

– 0.020

Number of Teeth of Pinion

50

12

1314

1516

45 40 35

1720

2530

50

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1516

45

40

35

17

20

25

30

Fig. 10-3 Chart to Determine the Circular Tooth Thickness Factor K for Gleason Spiral Bevel Gears

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Table 10-9 contains the equations for chordal thickness of normal module worms and worm gears.

Table 10-9 Equations for Chordal Thickness of Normal Module Worms and Worm Gears

sn1

sn2

zv2

qv2

sj1

sj2

hj1

hj2

pmn –––– 2 p(––– + 2xn2 tan an)mn 2

z2––––– cos3 g

90 360 xn2 tan an–– + –––––––––––zv2 pzv2

sn1 cos g zv mn cos g sin qv2

(sn1 sin g) 2

ha1 + –––––––– 4d1

zv mn cos gha2 + –––––––– (1 – cos qv2) 2


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Axial Circular Tooth Thickness of WormRadial Circular Tooth Thickness of Worm Gear

No. of Teeth in an Equivalent Spur Gear (Worm Gear)

Half of Tooth Angle at Pitch Circle (Worm Gear)

Chordal Thickness

Chordal Addendum

mn = 3an = 20°zw = 2 z2 = 30d1 = 38 d2 = 91.1433ax = 65 xn2 = 0.14278ha1 = 3.0000 ha2 = 3.42835g = 9.08472°sn1 = 4.71239 sn2 = 5.02419 zv2 = 31.15789 qv2 = 3.07964°sj1 = 4.7124 sj2 = 5.0218hj1 = 3.0036 hj2 = 3.4958

10.2 Span Measurement Of Teeth

Span measurement of teeth, sm, is a measure over a number of teeth, zm, made by means of a special tooth thickness micrometer. The value measured is the sum of normal circular tooth thickness on the base circle, sbn , and normal pitch, pen (zm – 1).

10.2.1 Spur And Internal Gears

The applicable equations are presented in Table 10-10.

NOTE: 1 K(f) = –– [sec a √(1 + 2f )2 – cos2 a – inv a – 2f tan a] (10-2) p x where f = z

Table 10-10 Span Measurement of Spur and Internal Gear Teeth

No. Item Example

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2

Span Number of Teeth

Span Measurement

Symbol Formula

m = 3a = 20°z = 24x = +0.4zmth = 3.78787zm = 4sm = 32.8266

zm

sm

zmth = zK(f) + 0.5 See NOTESelect the nearest natural number of zmth as zm.

m cos a [p (zm – 0.5) +z inv a ] + 2xm sin a

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Figure 10-4 shows the span measurement of a spur gear. This measurement is on the outside of the teeth. For internal gears the tooth profile is opposite to that of the external spur gear. Therefore, the measurement is between the inside of the tooth pro-files.


Tables 10-11 and 10-12 present equations for span measurement of the normal and the radial systems, re-spectively, of helical gears.

sm

d

Fig. 10-4 Span Measurement of Teeth (Spur Gear)

NOTE: 1 sin2 b K(f, b) = –– [(1 + –––––––––––––)√(cos2 b + tan2 an)(sec b + 2f )2 – 1 – inv at – 2f tan an ] (10-3) p cos2 b + tan2 an

xn where f = z

Table 10-11 Equations for Span Measurement of the Normal System Helical Gears

No. Item Example

1

2


Span Measurement

Symbol Formula

mn = 3, an = 20°, z = 24b = 25° 00' 00"xn = +0.4as = 21.88023°zmth = 4.63009zm = 5sm = 42.0085

zm

sm

zmth = zK (f, b) + 0.5 See NOTESelect the nearest natural number of zmth as zm.

mn cos an [p (zm – 0.5) + z inv at] + 2xnmn sin an

NOTE: 1 sin2 b ––––––––––––––––––––––––– K(f, b) = –– [(1 + ––––––––––––) √(cos2 b + tan2 an)(sec b + 2f )2 – 1 – inv at – 2f tan an ] (10-4) p cos2 b + tan2 an

xt where f = –––––– z cos b

Table 10-12 Equations for Span Measurement of the Radial System Helical Gears

No. Item Example

1

2


Span Measurement

Symbol Formula

mt = 3, at = 20°, z = 24b = 22° 30' 00"xt = +0.4an = 18.58597°zmth = 4.31728zm = 4sm = 30.5910

zm

sm

zmth = zK(f, b) + 0.5 See NOTESelect the nearest natural number of zmth as zm.

mt cos b cos an [p (zm – 0.5) + z inv at ] + 2xt mt sin an

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There is a requirement of a minimum blank width to make a helical gear span measurement. Let bmin be the minimum value for blank width. Then

bmin = sm sin bb + Db (10-5)

where bb is the helix angle at the base cylinder,

bb = tan–1 (tan b cos at) = sin–1 (sin b cos an) (10-6)

From the above, we can determine that at least 3mm of Db is required to make a stable measurement of sm.

10.3 Over Pin (Ball) Measurement

As shown in Figures 10-6 and 10-7, measure-ment is made over the outside of two pins that are inserted in diametrically opposite tooth spaces, for even tooth number gears; and as close as possible for odd tooth number gears.

Fig. 10-5 Blank Width of Helical Gear

bmin bb

sm

Fig. 10-7 Odd Number of Teeth

dp

dm

Fig. 10-6 Even Number of Teeth

dp

dm

d

The procedure for measuring a rack with a pin or a ball is as shown in Figure 10-9 by putting pin or ball in the tooth space and using a micrometer between it and a reference surface. Internal gears are similarly measured, except that the measurement is between the pins. See Figure 10-10. Helical gears can only be measured with balls. In the case of a worm, three pins are used, as shown in Figure 10-11. This is similar to the procedure of measuring a screw thread. All these cases are discussed in detail in the following sections. Note that gear literature uses “over pins” and “over wires” terminology interchangeably. The “over wires” term is often associated with very fine pitch gears because the diameters are accordingly small.

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10.3.1 Spur Gears

In measuring a gear, the size of the pin must be such that the over pins measurement is larger than the gear's outside diameter. An ideal value is one that would place the point of contact (tangent point) of pin and tooth profile at the pitch radius. However, this is not a necessary requirement. Referring to Figure 10-8, following are the equations for calculating the over pins measurement for a specific tooth thickness, s, regardless of where the pin contacts the tooth profile: For even number of teeth: d cos f dm = ––––––– + dp (10-7) cos f1

For odd number of teeth: d cos f 90° dm = ––––––– cos (––––) + dp (10-8) cos f1 z

where the value of f1 is obtained from: s dp p inv f1 = ––– + inv f + –––––– – –– (10-9) d d cos f z

When tooth thickness, s, is to be calculated from a known over pins measurement, dm, the above equations can be manipulated to yield:

p dp s = d (––– + inv fc – inv f + ––––––) (10-10) z d cos f

where d cos f cos fc = –––––– (10-11) 2Rc

For even number of teeth:

dm – dp Rc = ––––– (10-12) 2

For odd number of teeth: dm – dp Rc = ––––––––– (10-13) 90° 2 cos (–––) z

In measuring a standard gear, the size of the pin must meet the condition that its surface should have the tangent point at the standard pitch circle. While, in measuring a shifted gear, the surface of the pin should have thetangent point at the d + 2xm circle.

Fig. 10-8 Over Pins Diameter of Spur Gear

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The ideal diameters of pins when calculated from the equations of Table 10-13 may not be practical. So, in practice, we select a standard pin diameter close to the ideal value. After the actual diameter of pin dp is determined, the over pin measurement dm can be calculated from Table 10-14.

Table 10-13 Equations for Calculating Ideal Pin Diameters

NOTE: The units of angles ψ/2 and f are radians.


ψ –2

ap

f

dp

p 2x tan a (–– – inv a ) – –––––– 2z z zm cos acos–1 [–––––––––] (z + 2x)m

ψ tan ap + 2

ψ zm cos a (inv f + ––) 2

m = 1a = 20°z = 20x = 0 ψ –– = 0.0636354 2 ap = 20°f = 0.4276057dp = 1.7245

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Half Tooth Space Angle at Base Circle

The Pressure Angle at the Point Pin is Tangent to Tooth Surface

The Pressure Angle atPin Center

Ideal Pin Diameter

See NOTE dp p 2x tan a––––––– – ––– + inv a + ––––––– mz cos a 2z z

Find from Involute Function Table zm cos aEven Teeth –––––––– + dp cos f zm cos a 90°Odd Teeth –––––––– cos –– + dp cos f z

dp

inv f

f

dm


Let dp = 1.7, then:

inv f = 0.0268197f = 24.1350°dm = 22.2941

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Actual Diameter of Pin

Involute Function f

The Pressure Angle at Pin Center

Over PinsMeasurement

Table 10-14 Equations for Over Pins Measurement for Spur Gears

NOTE: The value of the ideal pin diameter from Table 10-13, or its approximate value, is applied as the actual diameter of pin dp here.

Table 10-15 is a dimensional table under the condition of module m = 1 and pressure angle a = 20° with which the pin has the tangent point at d + 2xm circle.

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Numberof Teeth

z

Coefficient of Profile Shift, x m = 1, a = 20°

––1.62311.64181.65001.6547

1.65771.65981.66141.66251.6635

1.66421.66491.66541.66591.6663

1.66661.66691.66721.66741.6676

10 20 30 40 50

60 70 80 90100

110120130140150

160170180190200

1.63481.65991.66491.66691.6680

1.66871.66921.66951.66981.6700

1.67011.67031.67041.67051.6706

1.67061.67071.67081.67081.6708

1.78861.72451.70571.69671.6915

1.68811.68571.68391.68251.6814

1.68051.67971.67911.67851.6781

1.67771.67731.67701.67671.6764

1.99791.81491.76321.73891.7248

1.71551.70901.70421.70051.6975

1.69511.69311.69141.69001.6887

1.68771.68671.68581.68511.6844

2.26871.93061.83691.79301.7675

1.75091.73921.73051.72371.7184

1.71401.71041.70741.70481.7025

1.70061.69891.69731.69601.6947

2.60792.07181.92671.85891.8196

1.79401.77591.76251.75211.7439

1.73721.73161.72691.72291.7195

1.71641.71381.71141.70931.7074

3.02482.23892.03241.93651.8810

1.84481.81931.80031.78571.7740

1.76451.75671.75001.74441.7394

1.73511.73141.72801.72501.7223

3.53152.43292.15422.02571.9516

1.90321.86911.84381.82421.8087

1.79601.78551.77661.76901.7625

1.75671.75171.74721.74321.7396

– 0.4 – 0.2 0 0.2 0.4 0.6 0.8 1.0

Table 10-15 The Size of Pin which Has the Tangent Point at d + 2xm Circle of Spur Gears

10.3.2 Spur Racks And Helical Racks

In measu ring a rack, the pin is ide-ally tangent with the tooth flank at the pitch line. The equa tions in Table 10-16 can, thus, be derived. In the case of a he-lical rack, module m, and pressure angle a, in Table 10-16, can be sub stituted by normal module, mn, and normal pressure angle, an , resulting in Table 10-16A.

Fig. 10-9 Over Pins Measurement for a Rack Using a Pin or a Ball

dpsj

dmH

pm pm – sj––––––– 2 tan a

No. Item ExampleSymbol Formula pm – sj–––––––– cos a pm – sj dp 1H – ––––––– + ––– ( 1 + –––– ) 2 tan a 2 sin a

dp'

dm

Ideal Pin Diameter


1

2

Table 10-16 Equations for Over Pins Measurement of Spur Racks

m = 1 a = 20°sj = 1.5708Ideal Pin Diameter dp' = 1.6716Actual Pin Diameter dp = 1.7H = 14.0000 dm = 15.1774

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Table 10-16A Equations for Over Pins Measurement of Helical Racks

No. Item ExampleSymbolmn = 1 an = 20°sj = 1.5708 b = 15°Ideal Pin Diameter dp' = 1.6716Actual Pin Diameter dp = 1.7 H = 14.0000 dm = 15.1774

dp'

dm

Ideal Pin Diameter


1

2

Formula pmn – sj–––––––– cos an

pmn – sj dp 1H – –––––––– + ––– (1 + –––––) 2 tan an 2 sin an

10.3.3 Internal Gears

As shown in Figure 10-10, measuring an internal gear needs a proper pin which has its tangent point at d + 2xm cir cle. The equations are in Table 10-17 for obtaining the ideal pin diameter. The equations for calculating the between pin measurement, dm , are given in Table 10-18.

Fig. 10-10 Between Pin Dimension of Internal Gears

dm

db

d

d + 2xmf

inv ap

inv f

ap

ψ––2

tan ap

Table 10-17 Equations for Calculating Pin Size for Internal Gears

NOTE: The units of angles ψ/2 and f are radians.

No. Item Example

1

2

3

4

Half of Tooth Space Angle at Base Circle

The Pressure Angle at the Point Pin is Tangent to Tooth Surface

The Pressure Angle atPin Center

Ideal Pin Diameter

FormulaSymbol

ψ –2

ap

f

dp

m = 1a = 20°z = 40x = 0 ψ –– = 0.054174 2 ap = 20°f = 0.309796dp = 1.6489

p 2x tan a (––– + inv a) + –––––– 2z z

zm cos acos–1 [–––––––––] (z + 2x)m ψ tan ap – ––

ψzm cos a (–– – inv f)

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Table 10-18 Equations for Between Pins Measurement of Internal Gears

See NOTE p dp 2x tan a( –– + inv a) – ––––––– + ––––––– 2z zm cos a z

Find from Involute Function Table zm cos aEven Teeth –––––––– – dp cos f zm cos a 90°Odd Teeth –––––––– cos ––– – dp cos f z

dp

inv f

f

dm


Let dp = 1.7, then:

inv f = 0.0089467f = 16.9521°dm = 37.5951

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2

3

4

Actual Diameter of Pin

Involute Function f

The Pressure Angle at Pin Center

Between PinsMeasurement

NOTE: First, calculate the ideal pin diameter. Then, choose the nearest practical actual pin size.

Table 10-19 lists ideal pin diameters for standard and profile shifted gears under the condition of module m = 1 and pressure angle a = 20°, which makes the pin tangent to the pitch circle d + 2xm .

Table 10-19 The Size of Pin that is Tangent at Pitch Circle d + 2xm of Internal GearsNumberof Teeth

z

Coefficient of Profile Shift, x m = 1, a = 20°

––1.46871.53091.56401.5845

1.59851.60861.61621.62221.6270

1.63101.63431.63711.63961.6417

1.64351.64511.64661.64791.6491

10 20 30 40 50

60 70 80 90100

110120130140150

160170180190200

1.47891.56041.59421.61231.6236

1.63121.63681.64101.64431.6470

1.64921.65101.65251.65391.6550

1.65611.65701.65781.65851.6591

1.59361.62841.64181.64891.6533

1.65621.65831.66001.66121.6622

1.66311.66381.66441.66491.6653

1.66571.66611.66641.66661.6669

1.67581.67591.67511.67451.6740

1.67371.67341.67321.67311.6729

1.67281.67271.67271.67261.6725

1.67251.67241.67241.67241.6723

1.72831.70471.69491.68951.6862

1.68391.68221.68101.68001.6792

1.67851.67791.67751.67711.6767

1.67641.67611.67591.67571.6755

1.75191.71541.70161.69441.6900

1.68701.68491.68331.68201.6810

1.68011.67941.67881.67831.6779

1.67751.67721.67681.67661.6763

1.74601.70841.69561.68931.6856

1.68321.68151.68021.67921.6784

1.67781.67721.67681.67641.6761

1.67581.67551.67531.67511.6749

1.70921.68371.67711.67441.6732

1.67251.67211.67181.67171.6716

1.67151.67141.67141.67141.6713

1.67131.67131.67131.67131.6713

– 0.4 – 0.2 0 0.2 0.4 0.6 0.8 1.0

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The ideal pin that makes contact at the d + 2xn mn pitch circle of a helical gear can be obtained from the same above equations, but with the teeth number z substituted by the equivalent (virtual) teeth number zv. Table 10-20 presents equations for deriving over pin diameters.

Table 10-20 Equations for Calculating Pin Size for Helical Gears in the Normal System

NOTE: The units of angles ψv /2 and fv are radians.


1

2

3

4

5

zv

ψv ––2

av

fv

dp

z––––– cos3 b

p 2xn tan an––– – inv an – –––––––– 2zv zv zv cos ancos–1 (–––––––) zv + 2xn

ψv tan av + –– 2

ψvzv mn cos an (inv fv + ––) 2

mn = 1an = 20°z = 20b = 15° 00' 00"xn = +0.4zv = 22.19211ψv –– = 0.0427566 2 av = 24.90647°fv = 0.507078dp = 1.9020

Number of Teeth of an Equivalent Spur Gear


Pressure Angle at the Point Pin is Tangent to Tooth Surface

Pressure Angle at Pin Center

Ideal Pin Diameter

Table 10-21 presents equations for calculating over pin measurements for helical gears in the normal system.

Table 10-21 Equations for Calculating Over Pins Measurement for Helical Gears in the Normal System

NOTE: The ideal pin diameter of Table 10-20, or its approximate value, is entered as the actual diameter of dp.

No. Item Example

1

2

3

4

Actual Pin Diameter

Involute Function f


Over Pins Measurement

FormulaSymbol

dp

inv f

f

dm

Let dp = 2, then

at = 20.646896°inv f = 0.058890f = 30.8534dm = 24.5696

See NOTE dp p 2xn tan an–––––––– – –– + inv at + ––––––– mnz cos an 2z z

Find from Involute Function Table

zmn cos atEven Teeth: ––––––––– + dp cos b cos f zmn cos at 90°Odd Teeth: ––––––––– cos –– + dp cos b cos f z

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Table 10-22 and Table 10-23 present equations for calculating pin measurements for helical gears in the radial (perpendicular to axis) system.


Table 10-22 Equations for Calculating Pin Size for Helical Gears in the Radial System

No. Item Example

1

2

3

4

5

Symbol

zv

ψv ––2

av

fv

dp

Formula

mt = 3at = 20°z = 36b = 33° 33' 26.3"an = 16.87300°xt = + 0.2zv = 62.20800ψv –– = 0.014091 2 av = 18.26390fv = 0.34411inv fv = 0.014258dp = 4.2190

z––––– cos3 b

p 2xt tan at––– – inv an – ––––––– 2zv zv

zv cos an –––––––– cos–1 ( xt ) zv + 2 ––––– cos b ψv tan av + –– 2

ψvzv mt cos b cos an (inv fv + ––) 2

Number of Teeth of an Equivalent Spur Gear


Pressure Angle at thePoint Pin is tangent toTooth Surface


Ideal Pin Diameter

NOTE: The ideal pin diameter of Table 10-22, or its approximate value, is applied as the actual diameter of pin dp here.

See NOTE

dp p 2xt tan at–––––––––––– – –– + inv at + ––––––– mt z cos b cos an 2z z


zmt cos atEven Teeth: ––––––––– + dp cos f zmt cos at 90°Odd Teeth: ––––––––– cos –– + dp cos f z

No. Item Example

1

2

3

4

Actual Pin Diameter

Involute Function f

Pressure Angleat Pin Center

Over Pins Measurement

FormulaSymbol

dp

inv f

f

dm

Table 10-23 Equations for Calculating Over Pins Measurement for Helical Gears in the Radial System

dp = 4.2190inv f = 0.024302f = 23.3910dm = 114.793

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10.3.5 Three Wire Method Of Worm Measurement

The teeth profile of Type III worms which are most popular are cut by standard cutters with a pressure angle ac = 20°. This results in the normal pressure angle of the worm being a bit smaller than 20°.The equation below shows how to calculate a Type III worm in an AGMA sys-tem. 90° r an = ac – ––– –––––––––– sin3 g (10-14) zw rc cos2 g + r

where: r = Worm Pitch Radius rc = Cutter Radius zw = Number of Threads g = Lead Angle of Worm

The exact equation for a three wire method of Type III worm is not only difficult to comprehend, but also hard to calculate precisely. We will introduce two approximate calculation methods here:

(a) Regard the tooth profile of the worm as a linear tooth profile of a rack and apply its equations. Using this system, the three wire method of a worm can be calculated by Table 10-24.

Fig. 10-11 Three Wire Method of a Worm

dp

dmd

Table 10-24 Equations for Three Wire Method of Worm Measurement, (a)-1

No. Item ExampleSymbol

dp'

dm

Ideal Pin Diameter

Three WireMeasurement

1

2

Formulamx = 2 an = 20°zw = 1 d1 = 31g = 3.691386° ax = 20.03827° dp' = 3.3440; let dp = 3.3dm = 35.3173

pmx––––––2 cos ax

pmx 1d1 – –––––– + dp(1 + ––––) 2 tan ax sin ax

These equations presume the worm lead angle to be very small and can be neglected. Of course, as the lead angle gets larger, the equations' error gets correspondingly larger. If the lead angle is considered as a factor, the equations are as in Table 10-25.

Table 10-25 Equations for Three Wire Method of Worm Measurement, (a)-2

mx = 2 an = 20°zw = 1 d1 = 31g = 3.691386°mn = 1.99585dp' = 3.3363; let dp = 3.3dm = 35.3344

pmn––––––– 2 cos an

pmn 1d1 – –––––– + dp(1 + –––––) 2 tan an sin an

(dp cos an sin g )2

– ––––––––––– 2d1

Ideal Pin Diameter

Three WireMeasurement

No. Item ExampleSymbol

dp'

dm

Formula

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2

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(b) Consider a worm to be a helical gear. This means applying the equations for calculating over pins measurement of helical gears to the case of three wire method of a worm. Because the tooth profile of Type III worm is not an involute curve, the method yields an approximation. However, the accuracy is adequate in practice.

Tables 10-26 and 10-27 contain equations based on the axial system. Tables 10-28 and 10-29 are based on the normal system.


See NOTE 1 dp p–––––––––––––– – ––– + inv at mx zw cos g cos an 2zw


zw mx cos at –––––––––– + dp tan g cos f

Actual Pin Size

Involute Function f

Pressure Angleat Pin Center

Three Wire Measurement

Let dp = 3.3at = 79.96878°inv at = 4.257549inv f = 4.446297f = 80.2959°dm = 35.3345

dp

inv f

f

dm

1

2

3

4

Table 10-27 Equation for Three Wire Method for Worms in the Axial System, (b)-2

NOTE: 1. The value of ideal pin diameter from Table 10-26, or its approximate value, is to be used as the actual pin diameter, dp. tan an 2. at = tan–1(––––––) sin g

1

2

3

4

5

zv

ψv ––2

av

fv

dp

zw–––––––––– cos3 (90 – g )

p––– – inv an 2 zv

zv cos ancos–1 (–––––––) zv


ψvzv mx cos g cos an(inv fv + ––) 2

Number of Teeth of anEquivalent Spur Gear


Pressure Angle at the Point Pin is Tangent to Tooth SurfacePressure Angle at Pin Center

Ideal Pin Diameter

mx = 2an = 20°zw = 1d1 = 31g = 3.691386°zv = 3747.1491ψv –– = -0.014485 2 av = 20°fv = 0.349485inv fv = 0.014960dp = 3.3382

No. Item Symbol Formula Example

Table 10-26 Equation for Calculating Pin Size for Worms in the Axial System, (b)-1

NOTE: The units of angles ψv / 2 and fv are radians.

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Table 10-28 shows the calculation of a worm in the normal module system. Basically, the normal module system and the axial module system have the same form of equations. Only the notations of module make them different.

Table 10-28 Equation for Calculating Pin Size for Worms in the Normal System, (b)-3



1

2

3

4

5

zv

ψv ––2

av

fv

dp

zw–––––––––– cos3 (90 – g )

p––– – inv an 2zv

zv cos ancos–1 (–––––––) zv


ψvzv mn cos an(inv fv + ––) 2

Number of Teeth of anEquivalent Spur Gear

Half of Tooth Space Angle at Base CirclePressure Angle at the Point Pin is Tangent to Tooth Surface


Ideal Pin Diameter

mn = 2.5an = 20°zw = 1d1 = 37g = 3.874288°zv = 3241.792ψv –– = – 0.014420 2 av = 20°fv = 0.349550inv fv = 0.0149687dp = 4.1785

Table 10-29 Equations for Three Wire Method for Worms in the Normal System, (b)-4


See NOTE 1 dp p––––––––– – ––– + inv atmn zw cos an 2 zw


zw mn cos at ––––––––– + dp sin g cos f

Actual Pin Size

Involute Function f


Three Wire Measurement

dp = 4.2at = 79.48331°inv at = 3.999514inv f = 4.216536f = 79.8947°dm = 42.6897

dp

inv f

f

dm

1

2

3

4

NOTE: 1. The value of ideal pin diameter from Table 10-28, or its approximate value, is to be used as the actual pin diameter, dp. tan an 2. at = tan–1(––––––) sin g

10.4 Over Pins Measurements For Fine Pitch Gears With Specific Numbers Of Teeth

Table 10-30 presents measurements for metric gears. These are for standard ideal tooth thicknesses. Measurements can be adjusted accordingly to backlash allowance and tolerance; i.e., tooth thinning.

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mm Inch mm InchMeas. Over WirePitch Diameter

No.of

Teethmm Inch mm Inch

Meas. Over WirePitch Diameter

No.of

Teeth

TABLE 10-30 METRIC GEAR OVER PINS MEASUREMENTPitch Diameter and Measurement Over Wires for External, Module Type Gears, 20-Degree Pressure Angle

Module 0.30Wire Size = 0.5184mm; 0.0204 Inch


56789

1011121314

1516171819

2021222324

2526272829

3031323334

3536373839

4041424344

4546474849

5051525354

5556575859

6061626364

6566676869

7071727374

0.07870.09450.11020.12600.1417

0.15750.17320.18900.20470.2205

0.23620.25200.26770.28350.2992

0.31500.33070.34650.36220.3780

0.39370.40940.42520.44090.4567

0.47240.48820.50390.51970.5354

0.55120.56690.58270.59840.6142

0.62990.64570.66140.67720.6929

0.70870.72440.74020.75590.7717

0.78740.80310.81890.83460.8504

0.86610.88190.89760.91340.9291

0.94490.96060.97640.99211.0079

1.02361.03941.05511.07091.0866

1.10241.11811.13391.14961.1654

56789

1011121314

1516171819

2021222324

2526272829

3031323334

3536373839

4041424344

4546474849

5051525354

5556575859

6061626364

6566676869

7071727374

0.32100.3357

0.35260.36740.38420.39910.4157

0.43070.44730.46230.47890.4939

0.51040.52550.54190.55710.5735

0.58870.60500.62020.63650.6518

0.66810.68330.69960.71490.7311

0.74640.76260.77800.79410.8095

0.82570.84100.85720.87260.8887

0.90410.92020.93560.95170.9671

0.98320.99871.01471.03021.0462

1.06171.07771.09321.10931.1247

1.14081.15621.17231.18771.2038

8.154 8.528

8.956 9.333 9.75810.13710.560

10.94011.36111.74312.16312.546

12.96413.34813.76514.15014.566

14.95215.36715.75416.16816.555

16.96917.35717.76918.15818.570

18.95919.37119.76020.17120.561

20.97221.36221.77222.16322.573

22.96423.37323.76424.17324.565

24.97425.36625.77426.16626.574

26.96727.37527.76728.17528.568

28.97529.36829.77630.16930.576

6.115 6.396

6.717 7.000 7.319 7.603 7.920

8.205 8.521 8.808 9.122 9.410

9.72310.01110.32410.61310.925

11.21411.52511.81512.12612.417

12.72713.01813.32713.61913.927

14.21914.52814.82015.12815.421

15.72916.02216.32916.62216.929

17.22317.53017.82318.13018.424

18.73019.02419.33119.62519.931

20.22520.53120.82621.13121.426

21.73122.02622.33222.62722.932

0.24080.2518

0.26440.27560.28810.29930.3118

0.32300.33550.34680.35910.3705

0.38280.39410.40650.41780.4301

0.44150.45380.46520.47740.4888

0.50100.51250.52470.53620.5483

0.55980.57200.58350.59560.6071

0.61920.63080.64290.65440.6665

0.67810.69010.70170.71380.7253

0.73740.74900.76100.77260.7847

0.79630.80830.81990.83190.8435

0.85560.86720.87920.89080.9028

2.000 2.400 2.800 3.200 3.600

4.000 4.400 4.800 5.200 5.600

6.000 6.400 6.800 7.200 7.600

8.000 8.400 8.800 9.200 9.600

10.00010.40010.80011.20011.600

12.00012.40012.80013.20013.600

14.00014.40014.80015.20015.600

16.00016.40016.80017.20017.600

18.00018.40018.80019.20019.600

20.00020.40020.80021.20021.600

22.00022.40022.80023.20023.600

24.00024.40024.80025.20025.600

26.00026.40026.80027.20027.600

28.00028.40028.80029.20029.600

1.500 1.800 2.100 2.400 2.700

3.000 3.300 3.600 3.900 4.200

4.500 4.800 5.100 5.400 5.700

6.000 6.300 6.600 6.900 7.200

7.500 7.800 8.100 8.400 8.700

9.000 9.300 9.600 9.90010.200

10.50010.80011.10011.40011.700

12.00012.30012.60012.90013.200

13.50013.80014.10014.40014.700

15.00015.30015.60015.90016.200

16.50016.80017.10017.40017.700

18.00018.30018.60018.90019.200

19.50019.80020.10020.40020.700

21.00021.30021.60021.90022.200

0.05910.07090.08270.09450.1063

0.11810.12990.14170.15350.1654

0.17720.18900.20080.21260.2244

0.23620.24800.25980.27170.2835

0.29530.30710.31890.33070.3425

0.35430.36610.37800.38980.4016

0.41340.42520.43700.44880.4606

0.47240.48430.49610.50790.5197

0.53150.54330.55510.56690.5787

0.59060.60240.61420.62600.6378

0.64960.66140.67320.68500.6969

0.70870.72050.73230.74410.7559

0.76770.77950.79130.80310.8150

0.82680.83860.85040.86220.8740

Continued on the next page

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No.of



No.of

Teeth

TABLE 10-30 (Cont.) METRIC GEAR OVER PINS MEASUREMENTPitch Diameter and Measurement Over Wires for External, Module Type Gears, 20-Degree Pressure Angle



75 76 77 78 79

80 81 82 83 84

85 86 87 88 89

90 91 92 93 94

95 96 97 98 99

100101102103104

105106107108109

110111112113114

115116117118119

120121122123124

125126127128129

130131132133134

135136137138139

140141142143144

1.18111.19691.21261.22831.2441

1.25981.27561.29131.30711.3228

1.33861.35431.37011.38581.4016

1.41731.43311.44881.46461.4803

1.49611.51181.52761.54331.5591

1.57481.59061.60631.62201.6378

1.65351.66931.68501.70081.7165

1.73231.74801.76381.77951.7953

1.81101.82681.84251.85831.8740

1.88981.90551.92131.93701.9528

1.96851.98432.00002.01572.0315

2.04722.06302.07872.09452.1102

2.12602.14172.15752.17322.1890

2.20472.22052.23622.25202.2677

75 76 77 78 79

80 81 82 83 84

85 86 87 88 89

90 91 92 93 94

95 96 97 98 99

100101102103104

105106107108109

110111112113114

115116117118119

120121122123124

125126127128129

130131132133134

135136137138139

140141142143144

1.21931.23531.25081.26681.2823

1.29831.31381.32981.34531.3613

1.37681.39281.40831.42431.4398

1.45581.47131.48731.50291.5188

1.53441.55031.56591.58181.5974

1.61331.62891.64481.66041.6763

1.69191.70781.72341.73931.7549

1.77081.78641.80231.81791.8338

1.84941.86531.88091.89681.9124

1.92831.94391.95981.97541.9913

2.00692.02282.03842.05432.0699

2.08582.10142.11732.13292.1488

2.16442.18032.19592.21182.2274

2.24332.25892.27482.29042.3063

30.96931.37631.77032.17632.570

32.97733.37033.77734.17134.577

34.97135.37735.77136.17736.572

36.97737.37237.77838.17238.578

38.97339.37839.77340.17840.573

40.97841.37341.77842.17442.579

42.97443.37943.77444.17944.574

44.97945.37445.77946.17546.579

46.97547.37947.77548.17948.575

48.97949.37549.78050.17650.580

50.97651.38051.77652.18052.576

52.98053.37653.78054.17654.580

54.97655.38055.77756.18056.577

56.98057.37757.78058.17758.580

23.22723.53223.82724.13224.428

24.73225.02825.33325.62825.933

26.22826.53326.82927.13327.429

27.73328.02928.33328.62928.933

29.23029.53329.83030.13430.430

30.73431.03031.33431.63031.934

32.23032.53432.83133.13433.431

33.73434.03134.33434.63134.934

35.23135.53435.83136.13536.431

36.73537.03237.33537.63237.935

38.23238.53538.83239.13539.432

39.73540.03240.33540.63240.935

41.23241.53541.83242.13542.433

42.73543.03343.33543.63343.935

0.91440.92650.93810.95010.9617

0.97370.98530.99731.00901.0210

1.03261.04461.05621.06821.0799

1.09191.10351.11551.12711.1391

1.15081.16271.17441.18641.1980

1.21001.22171.23361.24531.2572

1.26891.28091.29251.30451.3162

1.32811.33981.35171.36341.3754

1.38711.39901.41071.42261.4343

1.44621.45791.46991.48161.4935

1.50521.51711.52881.54071.5524

1.56441.57611.58801.59971.6116

1.62331.63521.64691.65891.6706

1.68251.69421.70611.71781.7297

30.00030.40030.80031.20031.600

32.00032.40032.80033.20033.600

34.00034.40034.80035.20035.600

36.00036.40036.80037.20037.600

38.00038.40038.80039.20039.600

40.00040.40040.80041.20041.600

42.00042.40042.80043.20043.600

44.00044.40044.80045.20045.600

46.00046.40046.80047.20047.600

48.00048.40048.80049.20049.600

50.00050.40050.80051.20051.600

52.00052.40052.80053.20053.600

54.00054.40054.80055.20055.600

56.00056.40056.80057.20057.600

22.50022.80023.10023.40023.700

24.00024.30024.60024.90025.200

25.50025.80026.10026.40026.700

27.00027.30027.60027.90028.200

28.50028.80029.10029.40029.700

30.00030.30030.60030.90031.200

31.50031.80032.10032.40032.700

33.00033.30033.60033.90034.200

34.50034.80035.10035.40035.700

36.00036.30036.60036.90037.200

37.50037.80038.10038.40038.700

39.00039.30039.60039.90040.200

40.50040.80041.10041.40041.700

42.00042.30042.60042.90043.200

0.88580.89760.90940.92130.9331

0.94490.95670.96850.98030.9921

1.00391.01571.02761.03941.0512

1.06301.07481.08661.09841.1102

1.12201.13391.14571.15751.1693

1.18111.19291.20471.21651.2283

1.24021.25201.26381.27561.2874

1.29921.31101.32281.33461.3465

1.35831.37011.38191.39371.4055

1.41731.42911.44091.45281.4646

1.47641.48821.50001.51181.5236

1.53541.54721.55911.57091.5827

1.59451.60631.61811.62991.6417

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145146147148149

150151152153154

155156157158159

160161162163164

165166167168169

170171172173174

175176177178179

180181182183184

185186187188189

190191192193194

195196197198199

200201202203204

205240280300340

380400440480500

2.28352.29922.31502.33072.3465

2.36222.37802.39372.40942.4252

2.44092.45672.47242.48822.5039

2.51972.53542.55122.56692.5827

2.59842.61422.62992.64572.6614

2.67722.69292.70872.72442.7402

2.75592.77172.78742.80312.8189

2.83462.85042.86612.88192.8976

2.91342.92912.94492.96062.9764

2.99213.00793.02363.03943.0551

3.07093.08663.10243.11813.1339

3.14963.16543.18113.19693.2126

3.22833.77954.40944.72445.3543

5.98436.29926.92917.55917.8740

145146147148149

150151152153154

155156157158159

160161162163164

165166167168169

170171172173174

175176177178179

180181182183184

185186187188189

190191192193194

195196197198199

200201202203204

205240280300340

380400440480500

2.32192.33782.35342.36932.3849

2.40082.41642.43232.44792.4638

2.47942.49532.51092.52682.5424

2.55832.57392.58982.60542.6213

2.63692.65282.66842.68432.6999

2.71582.73142.74732.76292.7788

2.79442.81032.82592.84182.8574

2.87332.88892.90482.92042.9363

2.95192.96782.98342.99933.0149

3.03083.04653.06233.07803.0938

3.10953.12533.14103.15683.1725

3.18833.20393.21973.23543.2512

3.26693.81824.44814.76315.3930

6.02306.33796.96797.59787.9128

58.97759.38159.77760.18160.577

60.98161.37761.78162.17862.581

62.97863.38163.77864.18164.578

64.98165.37865.78166.17866.581

66.97867.38167.77868.18168.578

68.98169.37869.78170.17870.581

70.979 71.381 71.779 72.181 72.579

72.981 73.379 73.782 74.179 74.582

74.979 75.382 75.779 76.182 76.579

76.982 77.382 77.782 78.182 78.582

78.982 79.382 79.782 80.182 80.582

80.982 81.379 81.780 82.180 82.580

82.980 96.982112.983120.983136.983

152.984160.984176.984192.984200.984

44.233 44.535 44.833 45.135 45.433

45.735 46.033 46.336 46.633 46.936

47.233 47.536 47.833 48.136 48.433

48.736 49.033 49.336 49.633 49.936

50.234 50.536 50.834 51.136 51.434

51.736 52.034 52.336 52.634 52.936

53.234 53.536 53.834 54.136 54.434

54.736 55.034 55.336 55.634 55.936

56.234 56.536 56.834 57.136 57.434

57.736 58.036 58.336 58.636 58.936

59.236 59.536 59.836 60.136 60.436

60.736 61.035 61.335 61.635 61.935

62.235 72.737 84.737 90.737102.738

114.738120.738132.738144.738150.738

1.74141.75341.76511.77701.7887

1.80061.81231.82421.83601.8479

1.85961.87151.88321.89511.9068

1.91871.93051.94241.95411.9660

1.97771.98962.00132.01322.0249

2.03682.04862.06052.07222.0841

2.09582.10772.11942.13132.1431

2.15502.16672.17862.19032.2022

2.21392.22582.23762.24952.2612

2.27312.28492.29672.30852.3203

2.33212.34402.35582.36762.3794

2.39122.40292.41472.42662.4384

2.45022.86373.33613.57234.0448

4.51724.75355.22595.69845.9346

58.000 58.400 58.800 59.200 59.600

60.000 60.400 60.800 61.200 61.600

62.000 62.400 62.800 63.200 63.600

64.000 64.400 64.800 65.200 65.600

66.000 66.400 66.800 67.200 67.600

68.000 68.400 68.800 69.200 69.600

70.000 70.400 70.800 71.200 71.600

72.000 72.400 72.800 73.200 73.600

74.000 74.400 74.800 75.200 75.600

76.000 76.400 76.800 77.200 77.600

78.000 78.400 78.800 79.200 79.600

80.000 80.400 80.800 81.200 81.600

82.000 96.000112.000120.000136.000

152.000160.000176.000192.000200.000

43.50043.80044.10044.40044.700

45.00045.30045.60045.90046.200

46.50046.80047.10047.40047.700

48.00048.30048.60048.90049.200

49.50049.80050.10050.40050.700

51.00051.30051.60051.90052.200

52.50052.80053.10053.40053.700

54.000 54.300 54.600 54.900 55.200

55.500 55.800 56.100 56.400 56.700

57.000 57.300 57.600 57.900 58.200

58.500 58.800 59.100 59.400 59.700

60.000 60.300 60.600 60.900 61.200

61.500 72.000 84.000 90.000102.000

114.000120.000132.000144.000150.000

1.71261.72441.73621.74801.7598

1.77171.78351.79531.80711.8189

1.83071.84251.85431.86611.8780

1.88981.90161.91341.92521.9370

1.94881.96061.97241.98431.9961

2.00792.01972.03152.04332.0551

2.06692.07872.09062.10242.1142

2.12602.13782.14962.16142.1732

2.18502.19692.20872.22052.2323

2.24412.25592.26772.27952.2913

2.30312.31502.32682.33862.3504

2.36222.37402.38582.39762.4094

2.42132.83463.30713.54334.0157

4.48824.72445.19695.66935.9055


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5 6 7 8 9

1011121314

1516171819

2021222324

2526272829

3031323334

3536373839

4041424344

4546474849

5051525354

5556575859

6061626364

6566676869

7071727374

0.14760.17720.20670.23620.2657

0.29530.32480.35430.38390.4134

0.44290.47240.50200.53150.5610

0.59060.62010.64960.67910.7087

0.73820.76770.79720.82680.8563

0.88580.91540.94490.97441.0039

1.03351.06301.09251.12201.1516

1.18111.21061.24021.26971.2992

1.32871.35831.38781.41731.4469

1.47641.50591.53541.56501.5945

1.62401.65351.68311.71261.7421

1.77171.80121.83071.86021.8898

1.91931.94881.97832.00792.0374

2.06692.09652.12602.15552.1850

5 6 7 8 9

1011121314

1516171819

2021222324

2526272829

3031323334

3536373839

4041424344

4546474849

5051525354

5556575859

6061626364

6566676869

7071727374

0.60190.6295

0.66110.68890.72030.74830.7795

0.80760.83870.86690.89780.9261

0.95700.98541.01611.04461.0753

1.10381.13441.16291.19351.2221

1.25261.28131.31171.34041.3708

1.39951.42991.45871.48901.5178

1.54811.57691.60721.63601.6663

1.69521.72541.75431.78451.8134

1.84351.87251.90261.93161.9617

1.99072.02082.04982.07992.1089

2.13892.16792.19802.22702.2571

15.28815.990

16.79217.49918.29619.00719.800

20.51321.30322.01922.80523.524

24.30825.02825.81026.53227.312

28.03628.81329.53930.31531.041

31.81632.54433.31834.04634.819

35.54836.32037.05137.82138.552

39.32240.05440.82341.55642.324

43.05743.82444.55845.32546.060

46.82647.56148.32649.06249.827

50.56351.32852.06452.82853.565

54.32955.06655.82956.56757.330

10.19210.660

11.19511.66612.19812.67113.200

13.67614.20214.67915.20415.683

16.20516.68517.20617.68818.208

18.69019.20919.69220.21020.694

21.21121.69622.21222.69823.212

23.69924.21324.70025.21425.702

26.21526.70327.21527.70428.216

28.70529.21629.70630.21730.706

31.21731.70732.21832.70833.218

33.70934.21834.70935.21935.710

36.21936.71037.21937.71138.220

0.40130.4197

0.44070.45930.48020.49890.5197

0.53840.55910.57790.59860.6174

0.63800.65690.67740.69640.7168

0.73580.75630.77530.79570.8147

0.83510.85420.87450.89360.9139

0.93300.95330.97250.99271.0119

1.03211.05131.07151.09071.1109

1.13011.15021.16951.18961.2089

1.22901.24831.26841.28771.3078

1.32711.34721.36651.38661.4059

1.42601.44531.46531.48471.5047

3.750 4.500 5.250 6.000 6.750

7.500 8.250 9.000 9.75010.500

11.25012.00012.75013.50014.250

15.00015.75016.50017.25018.000

18.75019.50020.25021.00021.750

22.50023.25024.00024.75025.500

26.25027.00027.75028.50029.250

30.00030.75031.50032.25033.000

33.75034.50035.25036.00036.750

37.50038.25039.00039.75040.500

41.25042.00042.75043.50044.250

45.00045.75046.50047.25048.000

48.75049.50050.25051.00051.750

52.50053.25054.00054.75055.500

2.500 3.000 3.500 4.000 4.500

5.000 5.500 6.000 6.500 7.000

7.500 8.000 8.500 9.000 9.500

10.00010.50011.00011.50012.000

12.50013.00013.50014.00014.500

15.00015.50016.00016.50017.000

17.50018.00018.50019.00019.500

20.00020.50021.00021.50022.000

22.50023.00023.50024.00024.500

25.00025.50026.00026.50027.000

27.50028.00028.50029.00029.500

30.00030.50031.00031.50032.000

32.50033.00033.50034.00034.500

35.00035.50036.00036.50037.000

0.09840.11810.13780.15750.1772

0.19690.21650.23620.25590.2756

0.29530.31500.33460.35430.3740

0.39370.41340.43310.45280.4724

0.49210.51180.53150.55120.5709

0.59060.61020.62990.64960.6693

0.68900.70870.72830.74800.7677

0.78740.80710.82680.84650.8661

0.88580.90550.92520.94490.9646

0.98431.00391.02361.04331.0630

1.08271.10241.12201.14171.1614

1.18111.20081.22051.24021.2598

1.27951.29921.31891.33861.3583

1.37801.39761.41731.43701.4567


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75 76 77 78 79

80 81 82 83 84

85 86 87 88 89

90 91 92 93 94

95 96 97 98 99

100101102103104

105106107108109

110111112113114

115116117118119

120121122123124

125126127128129

130131132133134

135136137138139

140141142143144

2.21462.24412.27362.30312.3327

2.36222.39172.42132.45082.4803

2.50982.53942.56892.59842.6280

2.65752.68702.71652.74612.7756

2.80512.83462.86422.89372.9232

2.95282.98233.01183.04133.0709

3.10043.12993.15943.18903.2185

3.24803.27763.30713.33663.3661

3.39573.42523.45473.48433.5138

3.54333.57283.60243.63193.6614

3.69093.72053.75003.77953.8091

3.83863.86813.89763.92723.9567

3.98624.01574.04534.07484.1043

4.13394.16344.19294.22244.2520

75 76 77 78 79

80 81 82 83 84

85 86 87 88 89

90 91 92 93 94

95 96 97 98 99

100101102103104

105106107108109

110111112113114

115116117118119

120121122123124

125126127128129

130131132133134

135136137138139

140141142143144

2.28612.31612.34522.37522.4043

2.43432.46342.49342.52252.5524

2.58152.61152.64062.67062.6997

2.72962.75882.78872.81782.8478

2.87692.90682.93602.96592.9951

3.02503.05413.08403.11323.1431

3.17233.20223.23143.26123.2904

3.32033.34953.37943.40863.4384

3.46763.49753.52673.55653.5858

3.61563.64483.67473.70393.7337

3.76303.79283.82203.85193.8811

3.91093.94023.97003.99924.0290

4.05834.08814.11744.14724.1764

4.20624.23554.26534.29464.3243

58.067 58.830 59.568 60.331 61.069

61.831 62.570 63.331 64.070 64.832

65.571 66.332 67.072 67.832 68.572

69.333 70.073 70.833 71.573 72.333

73.074 73.834 74.574 75.334 76.075

76.834 77.575 78.334 79.076 79.835

80.576 81.335 82.076 82.835 83.577

84.335 85.077 85.836 86.578 87.336

88.078 88.836 89.578 90.336 91.078

91.836 92.579 93.337 94.079 94.837

95.579 96.337 97.080 97.837 98.580

99.337100.080100.837101.581102.338

103.081103.838104.581105.338106.081

106.838107.582108.338109.082109.838

38.71239.22039.71240.22040.713

41.22141.71342.22142.71443.221

43.71444.22144.71445.22245.715

46.22246.71547.22247.71548.222

48.71649.22249.71650.22350.716

51.22351.71752.22352.71753.223

53.71754.22354.71855.22355.718

56.22456.71857.22457.71858.224

58.71959.22459.71960.22460.719

61.22461.71962.22462.71963.225

63.72064.22564.72065.22565.720

66.22566.72067.22567.72068.225

68.72169.22569.72170.22570.721

71.22571.72172.22572.72173.226

1.52411.54411.56351.58351.6029

1.62291.64221.66221.68161.7016

1.72101.74101.76041.78041.7998

1.81981.83921.85911.87861.8985

1.91791.93791.95731.97731.9967

2.01662.03612.05602.07552.0954

2.11492.13482.15422.17422.1936

2.21352.23302.25292.27242.2923

2.31182.33172.35112.37102.3905

2.41042.42992.44982.46932.4892

2.50862.52852.54802.56792.5874

2.60732.62682.64672.66622.6860

2.70552.72542.74492.76482.7843

2.80412.82372.84352.86302.8829

56.250 57.000 57.750 58.500 59.250

60.000 60.750 61.500 62.250 63.000

63.750 64.500 65.250 66.000 66.750

67.500 68.250 69.000 69.750 70.500

71.250 72.000 72.750 73.500 74.250

75.000 75.750 76.500 77.250 78.000

78.750 79.500 80.250 81.000 81.750

82.500 83.250 84.000 84.750 85.500

86.250 87.000 87.750 88.500 89.250

90.000 90.750 91.500 92.250 93.000

93.750 94.500 95.250 96.000 96.750

97.500 98.250 99.000 99.750100.500

101.250102.000102.750103.500104.250

105.000105.750106.500107.250108.000

37.50038.00038.50039.00039.500

40.00040.50041.00041.50042.000

42.50043.00043.50044.00044.500

45.00045.50046.00046.50047.000

47.50048.00048.50049.00049.500

50.00050.50051.00051.50052.000

52.50053.00053.50054.00054.500

55.00055.50056.00056.50057.000

57.50058.00058.50059.00059.500

60.00060.50061.00061.50062.000

62.50063.00063.50064.00064.500

65.00065.50066.00066.50067.000

67.50068.00068.50069.00069.500

70.00070.50071.00071.50072.000

1.47641.49611.51571.53541.5551

1.57481.59451.61421.63391.6535

1.67321.69291.71261.73231.7520

1.77171.79131.81101.83071.8504

1.87011.88981.90941.92911.9488

1.96851.98822.00792.02762.0472

2.06692.08662.10632.12602.1457

2.16542.18502.20472.22442.2441

2.26382.28352.30312.32282.3425

2.36222.38192.40162.42132.4409

2.46062.48032.50002.51972.5394

2.55912.57872.59842.61812.6378

2.65752.67722.69692.71652.7362

2.75592.77562.79532.81502.8346


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145146147148149

150151152153154

155156157158159

160161162163164

165166167168169

170171172173174

175176177178179

180181182183184

185186187188189

190191192193194

195196197198199

200201202203204

205240280300340

380400440480500

4.2815 4.3110 4.3406 4.3701 4.3996

4.4291 4.4587 4.4882 4.5177 4.5472

4.5768 4.6063 4.6358 4.6654 4.6949

4.7244 4.7539 4.7835 4.8130 4.8425

4.8720 4.9016 4.9311 4.9606 4.9902

5.0197 5.0492 5.0787 5.1083 5.1378

5.1673 5.1969 5.2264 5.2559 5.2854

5.3150 5.3445 5.3740 5.4035 5.4331

5.4626 5.4921 5.5217 5.5512 5.5807

5.6102 5.6398 5.6693 5.6988 5.7283

5.7579 5.7874 5.8169 5.8465 5.8760

5.9055 5.9350 5.9646 5.9941 6.0236

6.0531 7.0866 8.2677 8.858310.0394

11.220511.811012.992114.173214.7638

145146147148149

150151152153154

155156157158159

160161162163164

165166167168169

170171172173174

175176177178179

180181182183184

185186187188189

190191192193194

195196197198199

200201202203204

205240280300340

380400440480500

4.3536 4.3834 4.4127 4.4425 4.4718

4.5015 4.5308 4.5606 4.5899 4.6196

4.6489 4.6787 4.7080 4.7378 4.7671

4.7968 4.8261 4.8559 4.8852 4.9149

4.9443 4.9740 5.0033 5.0331 5.0624

5.0921 5.1214 5.1512 5.1805 5.2102

5.2396 5.2693 5.2986 5.3284 5.3577

5.3874 5.4167 5.4465 5.4758 5.5055

5.5349 5.5646 5.5939 5.6236 5.6530

5.6827 5.7122 5.7418 5.7713 5.8008

5.8303 5.8599 5.8894 5.9189 5.9485

5.9780 6.0073 6.0369 6.0664 6.0959

6.1255 7.1591 8.3403 8.930810.1120

11.293111.883613.064814.245914.8364

110.582111.338112.082112.839113.582

114.339115.083115.839116.583117.339

118.083118.839119.583120.339121.083

121.839122.584123.339124.084124.840

125.584126.340127.084127.840128.584

129.340130.084130.840131.585132.340

133.085133.840134.585135.340136.085

136.840137.585138.340139.085139.840

140.585141.340142.086142.841143.586

144.341145.091145.841146.591147.341

148.091148.841149.591150.341151.091

151.841152.587153.337154.087154.837

155.587181.842211.843226.843256.844

286.844301.845331.845361.845376.845

73.72174.22674.72175.22675.722

76.22676.72277.22677.72278.226

78.72279.22679.72280.22680.722

81.22681.72282.22682.72283.226

83.72384.22684.72385.22685.723

86.22786.72387.22787.72388.227

88.72389.22789.72390.22790.723

91.227 91.723 92.227 92.724 93.227

93.724 94.227 94.724 95.227 95.724

96.227 96.727 97.227 97.727 98.227

98.727 99.227 99.727100.227100.727

101.227101.724102.224102.724103.224

103.725121.228141.229151.229171.229

191.230201.230221.230241.230251.230

2.90242.92232.94182.96162.9812

3.00103.02053.04043.05993.0798

3.09933.11913.13873.15853.1780

3.19793.21743.23733.25683.2766

3.29623.31603.33553.35543.3749

3.39473.41433.43413.45373.4735

3.49303.51293.53243.55223.5718

3.59163.61123.63103.65053.6704

3.68993.70973.72933.74913.7687

3.78853.80823.82783.84753.8672

3.88693.90663.92633.94603.9656

3.98534.00494.02464.04434.0640

4.08374.77285.56025.95396.7413

7.52877.92248.70989.49739.8910

108.750109.500110.250111.000111.750

112.500113.250114.000114.750115.500

116.250117.000117.750118.500119.250

120.000120.750121.500122.250123.000

123.750124.500125.250126.000126.750

127.500128.250129.000129.750130.500

131.250132.000132.750133.500134.250

135.000135.750136.500137.250138.000

138.750139.500140.250141.000141.750

142.500143.250144.000144.750145.500

146.250147.000147.750148.500149.250

150.000150.750151.500152.250153.000

153.750180.000210.000225.000255.000

285.000300.000330.000360.000375.000

72.500 73.000 73.500 74.000 74.500

75.000 75.500 76.000 76.500 77.000

77.500 78.000 78.500 79.000 79.500

80.000 80.500 81.000 81.500 82.000

82.500 83.000 83.500 84.000 84.500

85.000 85.500 86.000 86.500 87.000

87.500 88.000 88.500 89.000 89.500

90.000 90.500 91.000 91.500 92.000

92.500 93.000 93.500 94.000 94.500

95.000 95.500 96.000 96.500 97.000

97.500 98.000 98.500 99.000 99.500

100.000100.500101.000101.500102.000

102.500120.000140.000150.000170.000

190.000200.000220.000240.000250.000

2.85432.87402.89372.91342.9331

2.95282.97242.99213.01183.0315

3.05123.07093.09063.11023.1299

3.14963.16933.18903.20873.2283

3.24803.26773.28743.30713.3268

3.34653.36613.38583.40553.4252

3.44493.46463.48433.50393.5236

3.54333.56303.58273.60243.6220

3.64173.66143.68113.70083.7205

3.74023.75983.77953.79923.8189

3.83863.85833.87803.89763.9173

3.93703.95673.97643.99614.0157

4.03544.72445.51185.90556.6929

7.48037.87408.66149.44889.8425


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5 6 7 8 9

1011121314

1516171819

2021222324

2526272829

3031323334

3536373839

4041424344

4546474849

5051525354

5556575859

6061626364

6566676869

7071727374

0.19690.23620.27560.31500.3543

0.39370.43310.47240.51180.5512

0.59060.62990.66930.70870.7480

0.78740.82680.86610.90550.9449

0.98431.02361.06301.10241.1417

1.18111.22051.25981.29921.3386

1.37801.41731.45671.49611.5354

1.57481.61421.65351.69291.7323

1.77171.81101.85041.88981.9291

1.96852.00792.04722.08662.1260

2.16542.20472.24412.28352.3228

2.36222.40162.44092.48032.5197

2.55912.59842.63782.67722.7165

2.75592.79532.83462.87402.9134

5 6 7 8 9

1011121314

1516171819

2021222324

2526272829

3031323334

3536373839

4041424344

4546474849

5051525354

5556575859

6061626364

6566676869

7071727374

0.80250.8394

0.88150.91860.96040.99771.0394

1.07681.11831.15591.19711.2349

1.27601.31381.35481.39281.4337

1.47171.51251.55061.59131.6295

1.67011.70831.74901.78721.8278

1.86611.90661.94491.98542.0237

2.06412.10262.14292.18142.2217

2.26022.30052.33902.37932.4178

2.45802.49662.53682.57542.6156

2.65422.69442.73302.77312.8118

2.85192.89062.93072.96943.0094

20.38421.320

22.39023.33224.39525.34226.400

27.35128.40429.35930.40731.365

32.41033.37134.41335.37636.415

37.38138.41839.38540.42041.389

42.42243.39244.42345.39546.425

47.39848.42649.40150.42851.403

52.42953.40554.43055.40756.431

57.40958.43259.41160.43361.413

62.43463.41464.43565.41666.436

67.41768.43769.41970.43871.420

72.43873.42174.43975.42276.440

16.30717.056

17.91218.66619.51620.27421.120

21.88122.72323.48724.32625.092

25.92826.69727.53028.30129.132

29.90530.73431.50832.33633.111

33.93734.71435.53936.31637.140

37.91838.74139.52140.34241.122

41.94342.72443.54444.32645.145

45.92746.74647.52948.34749.130

49.94850.73251.54852.33353.149

53.93454.75055.53556.35057.136

57.95158.73759.55160.33861.152

0.64200.6715

0.70520.73490.76840.79820.8315

0.86150.89460.92470.95770.9879

1.02081.05111.08391.11421.1469

1.17731.21001.24051.27311.3036

1.33611.36671.39921.42981.4622

1.49291.52521.55591.58831.6190

1.65131.68211.71431.74511.7774

1.80821.84041.87121.90341.9343

1.96641.99732.02952.06032.0925

2.12342.15552.18642.21852.2494

2.28152.31252.34452.37552.4075

5.000 6.000 7.000 8.000 9.000

10.00011.00012.00013.00014.000

15.00016.00017.00018.00019.000

20.00021.00022.00023.00024.000

25.00026.00027.00028.00029.000

30.00031.00032.00033.00034.000

35.00036.00037.00038.00039.000

40.00041.00042.00043.00044.000

45.00046.00047.00048.00049.000

50.00051.00052.00053.00054.000

55.00056.00057.00058.00059.000

60.00061.00062.00063.00064.000

65.00066.00067.00068.00069.000

70.00071.00072.00073.00074.000

4.000 4.800 5.600 6.400 7.200

8.000 8.800 9.60010.40011.200

12.00012.80013.60014.40015.200

16.00016.80017.60018.40019.200

20.00020.80021.60022.40023.200

24.00024.80025.60026.40027.200

28.00028.80029.60030.40031.200

32.00032.80033.60034.40035.200

36.00036.80037.60038.40039.200

40.00040.80041.60042.40043.200

44.00044.80045.60046.40047.200

48.00048.80049.60050.40051.200

52.00052.80053.60054.40055.200

56.00056.80057.60058.40059.200

0.15750.18900.22050.25200.2835

0.31500.34650.37800.40940.4409

0.47240.50390.53540.56690.5984

0.62990.66140.69290.72440.7559

0.78740.81890.85040.88190.9134

0.94490.97641.00791.03941.0709

1.10241.13391.16541.19691.2283

1.25981.29131.32281.35431.3858

1.41731.44881.48031.51181.5433

1.57481.60631.63781.66931.7008

1.73231.76381.79531.82681.8583

1.88981.92131.95281.98432.0157

2.04722.07872.11022.14172.1732

2.20472.23622.26772.29922.3307


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75 76 77 78 79

80 81 82 83 84

85 86 87 88 89

90 91 92 93 94

95 96 97 98 99

100101102103104

105106107108109

110111112113114

115116117118119

120121122123124

125126127128129

130131132133134

135136137138139

140141142143144

2.95282.99213.03153.07093.1102

3.14963.18903.22833.26773.3071

3.34653.38583.42523.46463.5039

3.54333.58273.62203.66143.7008

3.74023.77953.81893.85833.8976

3.93703.97644.01574.05514.0945

4.13394.17324.21264.25204.2913

4.33074.37014.40944.44884.4882

4.52764.56694.60634.64574.6850

4.72444.76384.80314.84254.8819

4.92134.96065.00005.03945.0787

5.11815.15755.19695.23625.2756

5.31505.35435.39375.43315.4724

5.51185.55125.59065.62995.6693

75 76 77 78 79

80 81 82 83 84

85 86 87 88 89

90 91 92 93 94

95 96 97 98 99

100101102103104

105106107108109

110111112113114

115116117118119

120121122123124

125126127128129

130131132133134

135136137138139

140141142143144

3.04823.08823.12693.16703.2057

3.24573.28453.32453.36333.4032

3.44203.48203.52083.56083.5996

3.63953.67843.71833.75713.7970

3.83593.87583.91473.95453.9934

4.03334.07224.11204.15094.1908

4.22974.26964.30854.34834.3872

4.4271 4.4660 4.5058 4.5448 4.5846

4.6235 4.6633 4.7023 4.7421 4.7810

4.8208 4.8598 4.8996 4.9385 4.9783

5.0173 5.0571 5.0960 5.1358 5.1748

5.2146 5.2536 5.2933 5.3323 5.3721

5.4111 5.4508 5.4898 5.5296 5.5686

5.6083 5.6473 5.6870 5.7261 5.7658

77.423 78.440 79.424 80.441 81.425

82.441 83.426 84.442 85.427 86.442

87.428 88.443 89.429 90.443 91.429

92.444 93.430 94.444 95.431 96.444

97.432 98.445 99.432100.445101.433

102.446103.433104.446105.434106.446

107.435108.447109.435110.447111.436

112.447113.436114.447115.437116.448

117.437118.448119.438120.448121.438

122.449123.438124.449125.439126.449

127.439128.449129.440130.450131.440

132.450133.440134.450135.441136.450

137.441138.450139.441140.451141.442

142.451143.442144.451145.442146.451

61.93962.75263.53964.35365.140

65.95366.74167.55368.34269.154

69.94270.75471.54372.35573.144

73.95574.74475.55576.34577.156

77.94578.75679.54680.35681.146

81.95682.74783.55784.34785.157

85.94886.75787.54888.35889.149

89.95890.74991.55892.34993.158

93.95094.75895.55096.35997.150

97.95998.75199.559100.351101.159

101.951102.759103.552104.360105.152

105.960106.752107.560108.353109.160

109.953110.760111.553112.360113.153

113.961114.754115.561116.354117.161

2.43852.47062.50152.53362.5646

2.59662.62762.65962.69062.7226

2.75362.78562.81672.84862.8797

2.91162.94272.97463.00573.0376

3.06873.10063.13173.16363.1947

3.22663.25773.28963.32083.3526

3.38383.41563.44683.47863.5098

3.54163.57283.60463.63583.6676

3.69883.73063.76183.79373.8248

3.85663.88783.91973.95083.9826

4.01384.04564.07684.10864.1398

4.17164.20284.23464.26594.2976

4.32894.36064.39194.42364.4549

4.48664.51794.54964.58094.6126

75.000 76.000 77.000 78.000 79.000

80.000 81.000 82.000 83.000 84.000

85.000 86.000 87.000 88.000 89.000

90.000 91.000 92.000 93.000 94.000

95.000 96.000 97.000 98.000 99.000

100.000101.000102.000103.000104.000

105.000106.000107.000108.000109.000

110.000111.000112.000113.000114.000

115.000116.000117.000118.000119.000

120.000121.000122.000123.000124.000

125.000126.000127.000128.000129.000

130.000131.000132.000133.000134.000

135.000136.000137.000138.000139.000

140.000141.000142.000143.000144.000

60.000 60.800 61.600 62.400 63.200

64.000 64.800 65.600 66.400 67.200

68.000 68.800 69.600 70.400 71.200

72.000 72.800 73.600 74.400 75.200

76.000 76.800 77.600 78.400 79.200

80.000 80.800 81.600 82.400 83.200

84.000 84.800 85.600 86.400 87.200

88.000 88.800 89.600 90.400 91.200

92.000 92.800 93.600 94.400 95.200

96.000 96.800 97.600 98.400 99.200

100.000100.800101.600102.400103.200

104.000104.800105.600106.400107.200

108.000108.800109.600110.400111.200

112.000112.800113.600114.400115.200

2.36222.39372.42522.45672.4882

2.51972.55122.58272.61422.6457

2.67722.70872.74022.77172.8031

2.83462.86612.89762.92912.9606

2.99213.02363.05513.08663.1181

3.14963.18113.21263.24413.2756

3.30713.33863.37013.40163.4331

3.46463.49613.52763.55913.5906

3.62203.65353.68503.71653.7480

3.77953.81103.84253.87403.9055

3.93703.96854.00004.03154.0630

4.09454.12604.15754.18904.2205

4.25204.28354.31504.34654.3780

4.40944.44094.47244.50394.5354


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145146147148149

150151152153154

155156157158159

160161162163164

165166167168169

170171172173174

175176177178179

180181182183184

185186187188189

190191192193194

195196197198199

200201202203204

205240280300340

380400440480500

5.7087 5.7480 5.7874 5.8268 5.8661

5.9055 5.9449 5.9843 6.0236 6.0630

6.1024 6.1417 6.1811 6.2205 6.2598

6.2992 6.3386 6.3780 6.4173 6.4567

6.4961 6.5354 6.5748 6.6142 6.6535

6.6929 6.7323 6.7717 6.8110 6.8504

6.8898 6.9291 6.9685 7.0079 7.0472

7.0866 7.1260 7.1654 7.2047 7.2441

7.2835 7.3228 7.3622 7.4016 7.4409

7.4803 7.5197 7.5591 7.5984 7.6378

7.6772 7.7165 7.7559 7.7953 7.8346

7.8740 7.9134 7.9528 7.9921 8.0315

8.0709 9.448811.023611.811013.3858

14.960615.748017.322818.897619.6850

145146147148149

150151152153154

155156157158159

160161162163164

165166167168169

170171172173174

175176177178179

180181182183184

185186187188189

190191192193194

195196197198199

200201202203204

205240280300340

380400440480500

5.8048 5.8445 5.8836 5.9233 5.9623

6.0020 6.0411 6.0808 6.1198 6.1595

6.1986 6.2383 6.2773 6.3170 6.3561

6.3958 6.4348 6.4745 6.5136 6.5533

6.5923 6.6320 6.6711 6.7107 6.7498

6.7895 6.8286 6.8682 6.9073 6.9470

6.9861 7.0257 7.0648 7.1045 7.1436

7.1832 7.2223 7.2620 7.3011 7.3407

7.3798 7.4194 7.4586 7.4982 7.5373

7.5769 7.6163 7.6557 7.6951 7.7344

7.7738 7.8132 7.8525 7.8919 7.9313

7.9707 8.0098 8.0492 8.0885 8.1279

8.1673 9.545511.120411.907813.4826

15.057515.844917.419718.994519.7819

147.443148.451149.443150.451151.443

152.452153.443154.452155.444156.452

157.444158.452159.444160.452161.444

162.452163.445164.453165.445166.453

167.445168.453169.445170.453171.446

172.453173.446174.453175.446176.453

177.446178.453179.446180.454181.447

182.454183.447184.454185.447186.454

187.447188.454189.447190.454191.448

192.454193.454194.454195.454196.454

197.454198.455199.455200.455201.455

202.455203.449204.449205.449206.449

207.449242.456282.457302.458342.459

382.459402.460442.460482.460502.461

117.954118.761119.554120.361121.155

121.961122.755123.561124.355125.162

125.955126.762127.555128.362129.156

129.962130.756131.562132.356133.162

133.956134.762135.556136.362137.157

137.962138.757139.563140.357141.163

141.957142.763143.557144.363145.157

145.963146.758147.563148.358149.163

149.958150.763151.558152.363153.158

153.963154.763155.563156.364157.164

157.964158.764159.564160.364161.164

161.964162.759163.559164.359165.159

165.959193.965225.966241.966273.967

305.967321.968353.968385.968401.968

4.64394.67564.70694.73864.7699

4.80164.83294.86464.89594.9276

4.95894.99065.02195.05365.0849

5.11665.14795.17965.21095.2426

5.27395.30565.33695.36865.3999

5.43165.46295.49465.52595.5576

5.58895.62065.65195.68365.7149

5.74665.77795.80965.84095.8726

5.90395.93565.96685.99866.0298

6.06156.09306.12456.15606.1875

6.21906.25056.28206.31356.3450

6.37656.40786.43936.47086.5023

6.53387.63648.89639.5262

10.7861

12.046012.675913.935715.195615.8255

145.000146.000147.000148.000149.000

150.000151.000152.000153.000154.000

155.000156.000157.000158.000159.000

160.000161.000162.000163.000164.000

165.000166.000167.000168.000169.000

170.000171.000172.000173.000174.000

175.000176.000177.000178.000179.000

180.000181.000182.000183.000184.000

185.000186.000187.000188.000189.000

190.000191.000192.000193.000194.000

195.000196.000197.000198.000199.000

200.000201.000202.000203.000204.000

205.000240.000280.000300.000340.000

380.000400.000440.000480.000500.000

116.000116.800117.600118.400119.200

120.000120.800121.600122.400123.200

124.000124.800125.600126.400127.200

128.000128.800129.600130.400131.200

132.000132.800133.600134.400135.200

136.000136.800137.600138.400139.200

140.000140.800141.600142.400143.200

144.000144.800145.600146.400147.200

148.000148.800149.600150.400151.200

152.000152.800153.600154.400155.200

156.000156.800157.600158.400159.200

160.000160.800161.600162.400163.200

164.000192.000224.000240.000272.000

304.000320.000352.000384.000400.000

4.5669 4.5984 4.6299 4.6614 4.6929

4.7244 4.7559 4.7874 4.8189 4.8504

4.8819 4.9134 4.9449 4.9764 5.0079

5.0394 5.0709 5.1024 5.1339 5.1654

5.1969 5.2283 5.2598 5.2913 5.3228

5.3543 5.3858 5.4173 5.4488 5.4803

5.5118 5.5433 5.5748 5.6063 5.6378

5.6693 5.7008 5.7323 5.7638 5.7953

5.8268 5.8583 5.8898 5.9213 5.9528

5.9843 6.0157 6.0472 6.0787 6.1102

6.1417 6.1732 6.2047 6.2362 6.2677

6.2992 6.3307 6.3622 6.3937 6.4252

6.4567 7.5591 8.8189 9.448810.7087

11.968512.598413.858315.118115.7480

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ELEMENTS OF METRIC GEAR TECHNOLOGY I - SDP/SI · PDF fileThe governing equations become simple...

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