Mechanical Engineering Chapter 8 Kinematics Of Gears Part 2

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209

Piniondedendumangle:ߜ

௉ൌݐܽ݊

ିଵሺܾ௉Ȁܣ௠ீሻൌݐܽ݊

ିଵሺͲǤͲͷͳȀʹǤ͸ͻʹሻൌ ͳǤͲͻι

Gearouteraddendum:ܽ

௢ீ ൌܽ

ீ൅ͲǤͷܨݐܽ݊ߜ

௉

ͲǤͳ͵ͳ͵݅݊

Gearoutsidediameter:ܦ

௢ൌܦ൅ʹܽ

௢ீ ܿ݋ݏ߁

210



PROBLEM : 8-49 EXA M PLE PROBLEM : 8-3

No of teeth in pinion = 18 No of teeth in pinion = 16

No of teeth in gear = 72 No of teeth in gear = 48

Diametral pitch = 12 Diametral pitch = 8

Maximum face width (a) 1.031 in Maximum face width (a) 1.054 in

Maximum face width (b) 0.833 in Maximum face width (b) 1.250 in

Max face width = smallest of (a) or (b) Max face width = smallest of (a) or (b)

INPUT Face width 0.800 in INPUT Face width 1.000 in

Pinion mean dedendum 0.051 in Pinion mean dedendum 0.077 in

Gear dedendum angle 2.767 degrees Gear dedendum angle 3.992 degrees

Pinion dedendum angle 1.090 degrees Pinion dedendum angle 1.663 degrees

Gear outer addendum 0.041 in Gear outer addendum 0.065 in

Pinion outer addendum 0.131 in Pinion outer addendum 0.194 in

Gear outside diameter 6.020 in Gear outside diameter 6.041 in

Pinion outside diameter 1.755 in Pinion outside diameter 2.369 in

COMPUTED VALUES COMPUTED VALUES

BEVEL GEAR GEOMETRY

Computes values for features described in Table 8–8 in Section 8-8 of the text

GIVEN DATA GIVEN DATA

211



PROBLEM : 8-45 PROBLEM : 8-47

No of teeth in pinion = 15 No of teeth in pinion = 25

No of teeth in gear = 45 No of teeth in gear = 50

Diametral pitch = 6 Diametral pitch = 10

Maximum face width (a) 1.318 in Maximum face width (a) 0.932 in

Maximum face width (b) 1.667 in Maximum face width (b) 1.000 in

Max face width = smallest of (a) or (b) Max face width = smallest of (a) or (b)

INPUT Face width 1.250 in INPUT Face width 0.900 in

Pinion mean dedendum 0.103 in Pinion mean dedendum 0.068 in

Gear dedendum angle 4.257 degrees Gear dedendum angle 3.450 degrees

Pinion dedendum angle 1.774 degrees Pinion dedendum angle 1.670 degrees

COMPUTED VALUES COMPUTED VALUES

GIVEN DATA GIVEN DATA

BEVEL GEAR GEOMETRY

Computes values for features described in Table 8–8 in Section 8-8 of the text

PROBLEM : 8-50 PROBLEM : 8-51

No of teeth in pinion = 16 No of teeth in pinion = 12

No of teeth in gear = 64 No of teeth in gear = 36

Diametral pitch = 32 Diametral pitch = 48

Maximum face width (a) 0.344 in Maximum face width (a) 0.132 in

Maximum face width (b) 0.313 in Maximum face width (b) 0.208 in

Max face width = smallest of (a) or (b) Max face width = smallest of (a) or (b)

INPUT Face width 0.300 in INPUT Face width 0.125 in

Pinion mean dedendum 0.019 in Pinion mean dedendum 0.013 in

Gear dedendum angle 3.113 degrees Gear dedendum angle 5.316 degrees

Pinion dedendum angle 1.227 degrees Pinion dedendum angle 2.217 degrees

Gear outer addendum 0.015 in Gear outer addendum 0.011 in

Pinion outer addendum 0.049 in Pinion outer addendum 0.032 in

BEVEL GEAR GEOMETRY

Computes values for features described in Table 8–8 in Section 8–8 of the text

COMPUTED VALUES

GIVEN DATA GIVEN DATA

COMPUTED VALUES

Wormgearing

ͷʹǤ  WormGearing–Givendata:ܦ௪ൌ ͳǤʹͷͲǢܰ௪ൌ ͳǢܲௗൌ ͳͲǢ߶௡ൌ

ͳͶǤͷι

Addendumൌࢇൌ ଵ

௉೏ൌଵ

ଵ଴ ൌ ૙Ǥ૚૙૙ܑܖǢ۲܍܌܍ܖ܌ܝܕ ൌ ࢈ ൌଵǤଵହ଻

௉೏ൌ ͲǤͳͳͷ͹

Ǥ

ͷʹǤͺǦͳͲǡͺǦʹͲ

ͺǦʹͷǤͺǦͶͺǦͷǡGeneralGuidelinesforWormandWormgear

Dimensions.



214

ͳͲǡͳͲǦͳͲͳͲǦͳͳǤ

Ǥ



WORMGEARING PROBLEM: 52 WORMGEARING PROBLEM: 53A

Worm pitch diameter = 1.250 in Worm pitch diameter = 1.000 in

Diametral pitch = 10 Diametral pitch = 12

No. of worm threads = 1 No. of worm threads = 1

No. of gear teeth = 40 No. of gear teeth = 20

Gear pitch diameter = 4.000 in Gear pitch diameter = 1.667 in

Center distance = 2.625 in Center distance = 1.333 in

Velocity ratio = 40.00 Velocity ratio = 20.00

WORMGEARING PROBLEM: 53B WORMGEARING PROBLEM: 53C

INPUT DATA

Addendum = 0.083 in Addendum = 0.083 in

Dedendum = 0.096 in Dedendum = 0.096 in

Worm outside diameter = 1.167 in Worm outside diameter = 1.167 in

INPUT DATA INPUT DATA

INPUT DATA

215



WORMGEARING PROBLEM: 54 WORMGEARING PROBLEM: 55

Worm pitch diameter = 0.625 in Worm pitch diameter = 2.000 in

Diametral pitch = 16 Diametral pitch = 6

Worm root diameter = 3.229 in Worm root diameter = 0.285 in

Gear pitch diameter = 18.000 in Gear pitch diameter = 1.667 in

Center distance = 11.000 in Center distance = 1.000 in

Velocity ratio = 54.00 Velocity ratio = 20.00

INPUT DATA

216

GearTrains–Analysis

ǡǤ

ͷͺǤ  ൌ ܸܶ ൌ ݊ଵȀ݊଺Ǣ݊ଵൌ ͵ͶͷͲ

ିேಳ

ͷͻǤ  ݊ଵൌ ͳʹʹͲͲǢ݊ହǣܸܶ ൌ ݊ଵȀ݊ହ

ேಳ

்௏ ൌଵଶଶ଴଴ோ௉ெ

ଷ଴଴଴଴ ൌ ૙Ǥ૝૙૟ૠܚܘܕ

͸ͲǤ  ݊ଵൌ ͸ͺͶͲǢ݊ସǣܸܶ ൌ ݊ଵȀ݊ସ

ேಳ

͸ͳǤ  ݊ଵൌ ʹͺ͹ͷǢ݊ସǣܸܶ ൌ ݊ଵȀ݊ସ

ேಳ

217

۹ܑܖ܍ܕ܉ܜܑ܋۲܍ܛܑ܏ܖܗ܎܉܁ܑܖ܏ܔ܍۵܍܉ܚ۾܉ܑܚ

VELOCITY RATIO FOR GEARS PROBLEM 62 VELOCITY RATIO FOR GEARS PROBLEM 63

17 53.41 53 3.1176 0.02395 17 29.44 29 1.7059 0.02617

19 59.69 60 3.1579 0.01630 19 32.91 33 1.7368 0.00479

22 69.12 69 3.1364 0.00523 XX 22 38.11 38 1.7273 0.00478 XX

24 75.40 75 3.1250 0.01659 24 41.57 42 1.7500 0.01795

0.00126 0.00478

VELOCITY RATIO FOR GEARS PROBLEM 64 VELOCITY RATIO FOR GEARS PROBLEM 65

17 104.80 105 6.1765 0.01206 17 126.14 126 7.4118 0.00824

XX 18 110.96 111 6.1667 0.00225 XX 18 133.56 134 7.4444 0.02444

23 141.78 142 6.1739 0.00950 23 170.66 171 7.4348 0.01478

0.00225 0.00105Minimum differential = Minimum differential =

Kinematic Design of Gear Trains: Solutions to Problems 62-65 at the end of Chapter 8.

Enter data in blue-shaded cells.

XX shows combination of NP and NG with minimum differential.

Minimum differential = Minimum differential =

218

out =

1800 rpm 2.00 rpm

1AE 5

۹ܑܖ܍ܕ܉ܜܑ܋۲܍ܛܑ܏ܖܗ܎۵܍܉ܚ܂ܚ܉ܑܖܛ

[Ensurethatnointerferenceoccursforanypairofgears.]

͸͸Ǥ Design:݊௜௡ ൌ ͳͺͲͲǢ݊௢௨௧ ൌ ʹǢǤ

 ʹͲιǤǤǡܰ௠௜௡ ൌ ͳ͸ͳ͹

ܸܴ

௠௔௫Ǧൌ ͳͷͲȀͳ͹ ൌ ͺǤͺʹ

ൌ

ሺͺǤͺʹሻଶൌ ͹͹Ǥͺ

 ǣܸܴଵൌ ͸Ǣܸܴଶൌ͸Ǣܸܴ

ଷൌͷǢܸܴ

ସൌͷ

ܰ

஺ൌ ͳ͸  ܰாൌͳ͸  

ܰ

஻ൌͻ͸  ܰ

ிൌ ͺͲ

͸͹Ǥ ǣ݊௜௡ ൌ ͳͺͲͲǢʹͳ ൏ ୭୳୲ ൏ ʹʹǢܰ௠௔௫ ൌ ͳͷͲǢʹͲι

ܸܶ

௡௢௠ ൌ ͳͺͲͲȀʹͳǤͷ ൌ ͺ͵Ǥ͹Ǣܸܶ௠௜௡ ൌ ͳͺͲͲȀʹʹ ൌ ͺͳǤͺǢܸܶ

௠௔௫ ൌ ͳͺͲͲȀʹͳ ൌ ͺͷǤ͹

 ͺǦ͹Ǧܰ௣൒ ͳ͹ʹͲιǤǤǤ

ܸܴ

௠௔௫݌ ൌ ͳͷͲȀͳ͹ ൌ ͺǤͺʹǣܸܴ௠௔௫ ൌሺͺǤͺʹሻଶൌ ͹͹Ǥͻ

 ͺǦͶͺȂǤ

Remaining

͸

ͷ

219

యൌ ͶǤ͵͹

͸ͺǤ ǣ݊௜௡ ൌ ͵͵͸Ͳ െ Ǣ݊௢௨௧ ൌ ͳʹǤͲ െ Ǣܰ௠௔௫ ൌ ͳͷͲ

 ʹͲιǤǤǤͺǦ͹ǡܰ௠௜௡ ൌ ͳ͹Ǥ

ܸܴ

ଷൌ ͷǣܰாൌ ͳ͹Ǣܰிൌ ͺͷ

ଵଷ଺

͸ͻǤ ǣ݊௜௡ ൌ ͶʹͲͲǣͳ͵ǤͲ ൏ ݊௢௨௧ ൏ ͳ͵Ǥͷǣܸܶ

ܸܶ

௠௜௡ ൌସଶ଴଴

ଵଷǤହ ൌ ͵ͳͳǤͳǢܸܶ

௡௢௠ ൌସଶ଴଴

ଵଷǤଶହ ൌ ͵ͳ͸ǤͻͺǢܸܶ

௠௔௫ ൌସଶ଴଴

ଵଷǤ଴ ൌ ͵ʹ͵ǤͲͺ

 ͸ͺǡ͵ǤͺǦͶͺTVǤ

 Ǥ

 Ǥܸܴ ൌ ξ͵ͳ͹

యൌ ͸Ǥͺʹ

ܸܴ

ଵൌ͹Ǣܸܴ

ଶൌ ͸Ǥܸܴଷ؆ ͵ͳ͹ȀͶʹ ؆ ͹Ǥͷͷǣܸܴଷൌ͹ǤͷͲ

ܸܶൌܸܴ

ଵൈܸܴ

ଶൈܸܴ

ଷൌሺ͹ሻሺ͸ሻሺ͹ǤͷͲሻൌ ͵ͳͷ

 Ǥ

ܸܴ

ଵൌ ͹ǤͷǢܸܴଶൌ ͹Ǣܸܴଷൌ͸

570

214

220

Shaft 1 A

5IE

5500 rpm

out =

222.78 rpm

 ǣ

͹ͲǤ ǣ݊௜௡ ൌ ͷͷͲͲ

 ʹʹͳ ൏ ݊௢௨௧ ൏ ʹʹͷ

 ǣǤ

ܸܶ

௠௔௫ ൌ ͷͷͲͲȀʹʹͳ ൌ ʹͶǤͺͺǣܸܶ௠௜௡ ൌ ͷͷͲͲȀʹʹͷ ൌ ʹͶǤͶͶ

்௏ ൌ ૛૛૛Ǥૠૡ܀۾ۻOk

͹ͳǤ ǣ݊௜௡ ൌ ͷͷͲͲǢͳ͵ǤͲ ൏ ݊௢௨௧ ൏ ͳͶǤͲ

ܸܶ

௡௢௠ ൌ ͷͷͲͲȀͳ͵Ǥͷ ൌ ͶͲ͹ǤͶ

 ȂTVα͹͹ǤͺͷǢȂTVα͸ͺ͹Ȃ

 ǣξͶͲ͹ǤͶ

యൌ͹ǤͶͳ

221

Shaft 1 AE

nin =

1750 rpm

ሺܸܴଵሻሺܸܴଶሻൌቀ

ଵଷହ

ଶൌ ͸͵ǤͲ͸

͹ʹǤ ǣ݊௜௡ ൌ ͳ͹ͷͲǢͳͶ͸ ൏ ݊௢௨௧ ൏ ͳͷͲ

ܸܶ

௡௢௠ ൌ ͳ͹ͷͲȀͳͶͺ ൌ ͳͳǤͺʹ

ܸܴ

ଵൌܰ

஻Ȁܰ஺ൌ ͹ͷȀͳͺ ൌ ͶǤͳ͸͹

222

= H n

out =

3600 rpm 3.984 rpm

Shaft 1 AE 5

͹ͶǤ ǣǣ݊௜௡ ൌ ͵ͲͲͲǣͷͶͺ ൏ ݊௢௨௧ ൏ ͷͷʹ

͹ͷǤ ǣ݊௜௡ ൌ ͵͸ͲͲǢ͵ǤͲ ൏ ݊௢௨௧ ൏ͷǤͲ

ܸܶ

௡௢௠ ൌ ͵͸ͲͲȀͶǤͲ ൌ ͻͲͲǣͶ

ܸܴ

ଷൌ ͷ ൌ ͺͲȀͳ͸ ൌ ܰிȀܰா

ܸܴ

ସൌ ͷ ൌ ͺͲȀͳ͸ ൌ ܰுȀܰீ

 ǣ

ܰ

஺ൌܰ

஼ൌܰ

ாൌܰ

ீൌ ͳ͸Ǥܰ஻ൌܰ

஽ൌ ͻͷǤܰிൌͺͳ



Remaining

Factor Ratio

223

Output shaft

= A n

out =

͹͸Ǥ ǣ݊௜௡ ൌ ͵͸ͲͲǢ͵ǤͲ ൏ ݊௢௨௧ ൏ͷǤͲ

͹͹Ǥ ǣ݊௜௡ ൌ ͳͺͲͲǢ݊௢௨௧ ൌ ͺǤͲ

͹ͺǤ ǣ݊௜௡ ൌ ͵͵͸ͲǢ݊௢௨௧ ൌ ͳʹǤͲǢTVα͵͵͸ͲȀͳʹαʹͺͲαʹͲuͳͶ

͹ͻǤ ǣ݊௜௡ ൌ ͶʹͲͲǢͳ͵ǤͲ ൏ ݊௢௨௧ ൏ ͳ͵Ǥͷ

 Ȁ͹͹Ǥ

ܸܴ

ଶൌ ͷͲǤǡܰ஼ൌͳǡܰ

஽ൌ ͷͲ

ܸܶ

௡௢௠ ൌ ͶʹͲͲȀͳ͵Ǥʹͷ ൌ ͵ͳ͸ǤͻͺǢܸܴଵൌ ͵ͳ͸ǤͻͺȀͷͲ ൌ ͸Ǥ͵Ͷ

ܰ

஺ൌ ͳͺǤܰ஻ൌͳͺ

ሺ͸Ǥ͵Ͷሻൌ ͳͳͶǤͳ ՜ ͳͳͶ ൌ ܰ஻

 ൌ ͶʹͲͲ ൈ ଵ଼

ଵଵସ ൈଵ

ହ଴ ൌ ૚૜Ǥ૛૟ܚܘܕOk



ͺͲǤ ǣ݊௜௡ ൌ ͷͷͲͲͳ͵ǤͲ ൏ ݊௢௨௧ ൏ ͳͶǤͲ

 ͹͸Ǥ

ܸܶ

௡௢௠ ൌ ͷͷͲͲȀͳ͵Ǥͷ ൌ ͶͲ͹ǤͶ

ͺͳǤ ǣnpαͷͲǢPdαͳʹǢNPα͵ͲǢα͹ǤͲͲαͺͶǤ

 ǤDPαNPȀPdα͵ͲȀͳʹα2.500in

 ǤαB=0.917in.ͺǦͳͲPdαͳʹ