Mechanical Engineering Chapter 8 Kinematics Of Gears Part 1

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CHAPTER 8

KINEMATICS OF GEARS

GearGeometry

ͳǤ ࡺ ൌ ૝૝Ǣࡼࢊൌ ૚૛

Ǥ ܦ ൌ ܰȀܲௗൌ ͶͶȀͳʹ ൌ ͵Ǥ͸͸͹

Ǥ ݐ ൌ ߨȀʹܲௗൌߨȀʹ

ሺͳʹሻൌ ͲǤͳ͵ͳ

Ǥ ܦ௢ൌሺܰ൅ʹ

ሻȀܲௗൌ Ͷ͸Ȁͳʹ ൌ ͵Ǥͺ͵͵

ʹǤ ࡺ ൌ ૜૝Ǣࡼࢊൌ ૛૝

Ǥ ܦ ൌ ͵ͶȀʹͶ ൌ ͳǤͶͳ͹

194

Ǥ ݐ ൌ ߨȀʹሺʹͶሻൌ ͲǤͲ͸ͷͶ

Ǥ ܦ௢ൌሺܰ൅ʹ

ሻȀܲௗൌ ͵͸ȀʹͶ ൌ ͳǤͷͲͲ

͵Ǥ ࡺ ൌ ૝૞Ǣࡼࢊൌ૛

Ǥ ܦ ൌ ͶͷȀʹ ൌ ʹʹǤͷͲͲ

ͶǤࡺ ൌ ૚ૡǢࡼࢊൌૡ

Ǥ ܦ ൌ ͳͺȀͺ ൌ ʹǤʹͷͲ

Ǥ ܲ௖ൌ ߨȀͺ ൌ ͲǤ͵ͻʹ͹

Ǥ ݉ ൌ ʹͷǤͶȀͺ ൌ ͵Ǥͳ͹ͷ

195

Ǥ ݉ ൌ ͵ǤͲ

Ǥ ܽ ൌ ͳȀͺ ൌ ͲǤͳʹͷ

Ǥ ܾ ൌ ͳǤʹͷȀͺ ൌ ͲǤͳͷ͸͵

Ǥ ܿ ൌ ͲǤʹͷȀͺ ൌ ͲǤͲ͵ͳ͵

Ǥ ݄௧ൌ ʹǤʹͷȀͺ ൌ ͲǤʹͺͳ͵

Ǥ ݄௞ൌ ʹȀͺ ൌ ͲǤʹͷͲ

Ǥ ݐ ൌ ߨȀʹሺͺሻൌ ͲǤͳͻ͸͵

Ǥ ܦ௢ൌ ʹͲȀͺ ൌ ʹǤͷͲͲ

ͷǤ ܰ ൌ ૛૛Ǣࡼࢊൌ૚Ǥૠ૞

Ǥ ܦ ൌ ʹʹȀͳǤ͹ͷ ൌ ͳʹǤͷ͹ͳ

Ǥ ܲ௖ൌ ߨȀͳǤ͹ͷ ൌ ͳǤ͹ͻͷ

Ǥ ݉ ൌ ʹͷǤͶȀͳǤ͹ͷ ൌ ͳͶǤͷͳͶ

Ǥ ݉ ൌ ͳ͸

Ǥ ܽ ൌ ͳȀͳǤ͹ͷ ൌ ͲǤͷ͹ͳͶ

Ǥ ܾ ൌ ͳǤʹͷȀͳǤ͹ͷ ൌ ͲǤ͹ͳͶ͵

͸Ǥ ࡺ ൌ ૛૙Ǣࡼࢊൌ ૟૝

 Ǥ ܦ ൌ ʹͲȀ͸Ͷ ൌ ͲǤ͵ͳʹͷ

Ǥܲ

௖ൌ ߨȀ͸Ͷ ൌ ͲǤͲͶͻͳ

 Ǥ ݉ ൌ ʹͷǤͶȀ͸Ͷ ൌ ͲǤ͵ͻ͹

 Ǥ ݉ ൌ ͲǤͶͲ

͹Ǥ ࡺ ൌ ૚ૡ૙Ǣࡼࢊൌૡ૙

 Ǥ ܦ ൌ ͳͺͲȀͺͲ ൌ ͲǤ͵ͳʹͷ

Ǥܲ

௖ൌ ߨȀͺͲ ൌ ͲǤͲ͵ͻ͵

 Ǥ ݉ ൌ ʹͷǤͶȀͺͲ ൌ ͲǤ͵ͳͺ

 Ǥ ݉ ൌ ͲǤ͵Ͳ

 Ǥ ܽ ൌ ͳȀͺͲ ൌ ͲǤͲͳʹͷ

197

ǤݐൌߨȀʹ

ሺͺͲሻൌ ͲǤͲͳͻ͸

Ǥܦ

௢ൌ ͳͺʹȀͺͲ ൌ ʹǤʹ͹ͷ

ͺǤ ࡺ ൌ ૛ૡǢࡼࢊൌ ૚ૡ

 Ǥ ܦ ൌ ʹͺȀͳͺ ൌ ͳǤͷͷ͸

Ǥܲ

௖ൌ ߨȀͳͺ ൌ ͲǤͳ͹Ͷͷ

 Ǥ ݉ ൌ ʹͷǤͶȀͳͺ ൌ ͳǤͶͳͳ

ͻǤ ࡺ ൌ ૛ૡǢࡼࢊൌ ૛૙

 Ǥ ܦ ൌ ʹͺȀʹͲ ൌ ͳǤͶͲͲ

Ǥܲ

௖ൌ ߨȀʹͲ ൌ ͲǤͳͷ͹ͳ

198

ͳͲǤ ࡺ ൌ ૜૝Ǣ࢓ ൌ ૜ ൌ ࡰȀࡺ

Ǥܦൌ݉ܰൌ͵

ሺ͵Ͷሻൌ ͳͲʹ

Ǥݐൌܲ

௖Ȁʹ ൌ గ

௠Ȁʹ ൌ ͶǤ͹ͳʹ

Ǥܦ

௢ൌ݉

ሺܰ൅ʹ

ሻൌ͵

ሺ͵͸ሻൌ ͳͲͺ

ͳͳǤࡺ ൌ ૝૞Ǣ࢓ ൌ ૚Ǥ૛૞

 Ǥ ܦ ൌ ݉ܰ ൌ ͳǤʹͷሺͶͷሻൌ ͷ͸Ǥʹͷ

Ǥܲ

௖ൌߨ݉ൌߨ

ሺͳǤʹͷሻൌ ͵Ǥͻʹ͹

199

௢ൌ݉

ͳʹǤ ࡺ ൌ ૚ૡǢ࢓ ൌ ૚૛

 Ǥ ܦ ൌ ݉ܰ ൌ ͳͺሺͳͺሻ ൌ ʹͳ͸

Ǥܲ

௖ൌߨ

ሺͳʹሻൌ ͵͹Ǥ͹Ͳ

 Ǥ ܧݍݑ݅ݒ݈ܽ݁݊ݐܲௗൌ ʹͷǤͶȀͳʹ ൌ ʹǤͳͳ͹

௢ൌ݉

ͳ͵Ǥࡺ ൌ ૛૛Ǣ࢓ ൌ ૛૙

 Ǥ ܦ ൌ ݉ܰ ൌ ʹͲሺʹʹሻ ൌ ͶͶͲ

Ǥܲ

௖ൌߨ

ሺʹͲሻൌ ͸ʹǤͺ͵

200

௢ൌʹͲ

ͳͶǤࡺ ൌ ૛૙Ǣ࢓ ൌ ૚

 Ǥ ܦ ൌ ݉ܰ ൌ ʹͲǤͲͲ

Ǥܲ

௖ൌߨ

ሺʹͲሻൌ ͵ǤͳͶ

 Ǥ ܧݍݑ݅ݒ݈ܽ݁݊ݐܲௗൌ ʹͷǤͶȀͳ ൌ ʹͷǤͶ

 Ǥ ܲௗൌʹͶ

௢ൌሺͳሻ

ͳͷǤࡺ ൌ ૚ૡ૙Ǣ࢓ ൌ ૙Ǥ૝

 Ǥ ܦ ൌ ͲǤͶሺͳͺͲሻ ൌ ͹ʹǤͲͲ

Ǥܲ

௖ൌߨ

ሺͲǤͶሻൌ ͳǤʹ͸

 Ǥ ܧݍݑ݅ݒ݈ܽ݁݊ݐܲௗൌ ʹͷǤͶȀͲǤͶ ൌ ͸͵Ǥͷ

201

 Ǥ ܾ ൌ ͳǤʹͷሺͲǤͶሻൌ ͲǤͷͲ

 Ǥ ܿ ൌ ͲǤʹͷሺͲǤͶሻൌ ͲǤͳͲ

Ǥ݄

௧ൌ ʹǤʹͷሺͲǤͶሻ ൌ ͲǤͻͲͲ

Ǥ݄

௞ൌʹ

ሺͲǤͶሻൌ ͲǤͺͲ

 Ǥ ݐ ൌ ߨሺͲǤͶሻȀʹ ൌ ͲǤ͸ʹͺ

Ǥܦ

௢ൌͲǤͶ

ሺͳͺʹሻൌ ͹ʹǤͺͲ

ͳ͸Ǥࡺ ൌ ૛ૡǢ࢓ ൌ ૚Ǥ૞

 Ǥ ܦ ൌ ݉ܰ ൌ ͳǤͷሺʹͺሻ ൌ ʹʹǤͶͲ

Ǥܲ

௖ൌߨ݉ൌߨ

ሺͳǤͷሻൌ ͶǤ͹ͳ

Ǥܦ

௢ൌͳǤͷ

ሺ͵Ͳሻൌ ͶͷǤͲͲ

ͳ͹Ǥࡺ ൌ ૛ૡǢ࢓ ൌ ૙Ǥૡ

 Ǥ ܦ ൌ ͲǤͺሺʹͺሻ ൌ ʹʹǤͶͲ

202

 Ǥ ܽ ൌ ݉ ൌ ͲǤͺͲ

 Ǥ ܾ ൌ ͳǤʹͷሺͲǤͺሻൌ ͳǤͲͲ

 Ǥ ܿ ൌ ͲǤʹͷሺͲǤͺሻൌ ͲǤʹͲ

Ǥ݄

௧ൌ ʹǤʹͷሺͲǤͺሻ ൌ ͳǤͺͲ

Ǥ݄

௞ൌʹ

ሺͲǤͺሻൌ ͳǤ͸Ͳ

 Ǥ ݐ ൌ ߨሺͲǤͺሻȀʹ ൌ ͳǤʹͷ͹

Ǥܦ

௢ൌͲǤͺ

ሺ͵Ͳሻൌ ʹͶǤͲͲ

ͳͺǤ ȂͺǦͶͺǦͷ

ͳͻǤ ͳǣܲௗൌ ͳʹǣͲǤͲͲ͸ͲǤͲͲͻ



VelocityRatio

ʹͲǤ Ǥ ܥ ൌ ே೛ାே೒

ଶ௉೏ൌଵ଼ା଺ସ

ଶሺ଼ሻ ൌ ૞Ǥ૚૛૞ܑܖ

ʹͳǤ Ǥ ܥ ൌ ሺʹͲ ൅ ͻʹሻȀʹሺͶሻൌ ૚૝Ǥ૙૙૙ܑܖ

 Ǥ ܸܴ ൌ ͻʹȀʹͲ ൌ ૝Ǥ૟૙

203

ʹʹǤ Ǥ ܥ ൌ ሺ͵Ͳ ൅ ͸ͺሻȀʹሺʹͲሻൌ ૛Ǥ૝૞૙ܑܖ

 Ǥ ܸܴ ൌ ͸ͺȀ͵Ͳ ൌ ૛Ǥ૛૟ૠ

ʹ͵Ǥ Ǥ ܥ ൌ ሺͶͲ ൅ ʹͷͲሻȀʹሺ͸Ͷሻൌ ૛Ǥ૛૟૟ܑܖ

 Ǥ ܸܴ ൌ ʹͷͲȀͶͲ ൌ ૟Ǥ૛૞

ʹͶǤ Ǥ ܥ ൌ ሺʹͶ ൅ ͺͺሻȀʹሺͳʹሻൌ ૝Ǥ૟૟ૠܑܖ

 Ǥ ܸܴ ൌ ͺͺȀʹͶ ൌ ૜Ǥ૟૟ૠ

ʹͷǤ Ǥ ܥ ൌ ൫ܰீ൅ܰ

௣൯݉Ȁʹ ൌ ሺ͸ͺ ൅ ʹʹሻሺʹሻȀʹ ൌ ૢ૙Ǥ૙૙ܕܕ

ଵଽ଴ଽଽ ୫

ୱൌሺଶሻሺଶଶሻሺଵ଻ହ଴ሻ

ଵଽ଴ଽଽ ൌ ૝Ǥ૙૜ܕȀܛ

ʹ͸Ǥ Ǥ ܥ ൌ ሺͶͺ ൅ ͳͺሻሺͲǤͺሻȀʹ ൌ ૛૟Ǥ૝૙ܕܕ

 Ǥ ܸܴ ൌ ͶͺȀͳͺ ൌ ૛Ǥ૟૟ૠ

204

ʹ͹Ǥ Ǥ ܥ ൌ ሺͶͷ ൅ ͵͸ሻሺͶሻȀʹ ൌ ૚૟૛ܕܕ

 Ǥ ܸܴ ൌ ͶͷȀ͵͸ ൌ ૚Ǥ૛૞

ʹͺǤ Ǥ ܥ ൌ ሺ͸͵ ൅ ͳͷሻሺͳʹሻȀʹ ൌ ૚૟૛ܕܕ

 Ǥ ܸܴ ൌ ͵͸Ȁͳͷ ൌ ૛Ǥ૝૙

ErrorsinstatementsforProblems29Ǧ32:

ʹͻǤ ǯǤ



HousingDimensions

͵͵Ǥ ͲǤͳͲȀ

ܽൌͳȀܲ

ௗൌ ͳȀͺ ൌ ͲǤͳʹͷǢܦ௢ಸൌሺܰீ൅ʹ

ሻȀܲௗൌ ͸͸Ȁͺ ൌ ͺǤʹͷ



205

͵ͶǤ ܦ௢ீൌሺܰீ൅ʹ

ሻȀܲௗൌ ʹͷʹȀ͸Ͷ ൌ ͵Ǥͻ͵ͺǣܻ ൌ ͵Ǥͻ͵ͺ ൅ ͲǤʹ ൌ ૝Ǥ૚૜ૡܑܖ ൌ ࢅ

͵ͷǤ ܦ௢ಸൌሺܰீ൅ʹ

ሻ݉ൌͷͲ

ሺͲǤͺሻൌ ͶͲǤͲǢܻ ൌ ͶͲǤͲ ൅ ʹሺʹ݉݉ሻൌ ૝૝Ǥ૙૙ܕܕ ൌ ࢅ

͵͸Ǥ ܦ௢ಸൌͶ͹

ሺͶሻൌ ͳͺͺǣܻ ൌ ͳͺͺ ൅ ʹሺʹሻൌ ૚ૢ૛ܕܕ ൌ ࢅ

GearTrains–Analysis

͵͹Ǥ ܸܶ ൌ െேಳ

ேಲൈேವ

ே಴ൈேಷ

ேಶൌെସଶ

ଵ଼ ൈହସ

ଶସ ൌ െͳͷǤ͹ͷ ൌ ௡೔೙

௡೚ೠ೟

͵ͺǤ ܸܶ ൌ െேಳ

ேಲൈே಴

ேಳൈேಶ

ேವൌെ଺଼

ଶଶ ൈ଺଼

ଶହ ൌ െͺǤͶͲ͹ ൌ ௡೔೙

௡೚ೠ೟

͵ͻǤ ܸܶ ൌ ஽ಳ

஽ಲൈ஽ವ

஽಴ൈ஽ಷ

஽ಶൈேಹ

ேಸൌଶǤ଼଻ହ

ଵǤଶହ଴ ൈଶǤଷ଻ହ

ଵǤଵଶହ ൈଶǤଶହ଴

ଵǤହ଴଴ ൈଷ଴

ଵ଼ ൌ ͳʹǤͳ͵ͻ

ܦ

஺ൌܰ

஺Ȁܲௗൌ ʹͲȀͳ͸ ൌ ͳǤʹͷͲ

ͶͲǤ ܸܶ ൌ ேಳ

ேಲൈேವ

ே಴ൌଶସ

଼଴ ൈଵ଼

଺଴ ൌ ൅ͲǤͲͻͲͲ



206

HelicalGearing

ͶͳǤ ܲௗൌͺǢ߶

௧ൌͳͶଵ

ଶιǢܰ ൌ ͶͷǢܨ ൌ ʹǤͲͲ

 ൌ ߰ ൌ ͵Ͳι

ൌ݌ൌߨȀܲ

ௗൌ ߨȀͺ ൌ ૙Ǥ૜ૢ૛ૠܑܖ ൌ ࢖

 ൌ ݌௡ൌ݌ܿ݋ݏ߰ൌሺͲǤ͵ͻʹ͹ሻܿ݋ݏሺ͵Ͳιሻൌ ૙Ǥ૜૝૙࢏࢔ൌ࢖࢔

ͶʹǤ ܰ ൌ ͶͺǢܲ௡ௗ ൌ ͳʹǢ߶௡ൌ ʹͲιǢܨ ൌ ͳǤͷͲǢ߰ ൌ Ͷͷι

݌ൌߨȀܲ

ௗെܲ

ௗൌܲ

௡ௗ ܿ݋ݏ߰ ൌ ͳʹ ൈ ܿ݋ݏሺͳͶǤͷιሻ ൌ ͺǤͶͺͷ

௧௔௡ట ൌ ૙Ǥ૜ૠ૙ܑܖ ൌ ࡼ࢞Ǣܦ

ࡰࡳ

ିଵ ቀ௧௔௡థ೙

ିଵ ቀ௧௔௡ଶ଴ι

Ͷ͵Ǥ ܰ ൌ ͵͸Ǣܲௗൌ͸Ǣ߶

௧ൌͳͶଵ

ଶιǢ߰ ൌ ͶͷιǢܨ ൌ ͳǤͲͲ

207

ିଵሺݐܽ݊߶௧ܿ݋ݏ߰ሻ ൌ ૚૙Ǥ૜૟ι ൌ ࣘ࢔

ͶͶǤ ܰ ൌ ͹ʹǢܲ௡ௗ ൌ ʹͶǢ߶௡ൌͳͶଵ

ଶιǢܨ ൌ ͲǤʹͷǢ߰ ൌ Ͷͷι

݌ ൌ ߨȀͳ͸Ǥͻ͹ ൌ ૙Ǥ૚ૡ૞૚ܑܖ ൌ ࢖Ǣ݌௡ൌ ݌ܿ݋ݏ ߰ ൌ ͲǤͳͺͷͳܿ݋ݏͶͷι ൌ ૙Ǥ૚૜૙ૢܑܖ ൌ࢖࢔



BevelGears

ͶͷͷͳǤͺǦͺͺǦͺ

ͶͻǤ Given:ࡺ࢖ൌ ૚ૡǢࡺࡳൌ ૠ૛Ǣࡼࢊൌ ૚૛Ǣ૛૙ιܘܚ܍ܛܛܝܚ܍܉ܖ܏ܔ܍

ComputedValues:

208

Pitchdiameter–Gear:ܦൌܰ

ீȀܲௗൌ ͹ʹȀͳʹ ൌ ͸ǤͲͲͲ

Pitchconeangle–Pinion:ߛ ൌݐܽ݊ିଵሺܰ௉Ȁܰீሻൌݐܽ݊

ିଵሺͳͺȀ͹ʹሻ ൌ ͳͶǤͲ͵ι

Pitchconeangle–Gear:Ȟൌݐܽ݊

ିଵሺܰீȀܰ௉ሻൌݐܽ݊

ିଵሺ͹ʹȀͳͺሻ ൌ ͹ͷǤͻ͸ι

Outerconedistance:ܣ

௢ൌͲǤͷܦȀ

ሺ߁ሻ

ܣ

௢ൌͲǤͷ

ሺ͸ǤͲͲ݅݊ሻȀሺ͹ͷǤͻ͸ιሻൌ ͵ǤͲͻʹ

Facewidthmustbespecifiedǣࡲ ൌ ૙Ǥૡ૙૙ܑܖǡǣ

Nominalfacewidth:ܨ௡௢௠ ൌ ͲǤ͵Ͳܣ௢ൌͲǤ͵Ͳ

ሺ͵ǤͲͻʹ݅݊ሻൌ ͲǤͻʹͺ

Clearance:   ܿ ൌ ͲǤͳʹͷ݄ ൌ ͲǤͳʹͷሺͲǤͳͶͷ݅݊ሻൌ ͲǤͲͳͺ

Meanwholedepth: ݄௠ൌ ݄ ൅ ܿ ൌ ͲǤͳͶͷ ൅ ͲǤͲͳͺ ൌ ͲǤͳ͸͵

Meanaddendumfactor:ܿ

ଵൌ ͲǤʹͳͲ ൅ ͲǤʹͻͲȀሺ݉ீሻଶ

Geardedendumangle:ߜ

ீൌݐܽ݊