Introduction to Gear TrainsIntroduction to Gear Trains

Most of the pictures and animations are from Norton “Design of Machinery”

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Most of the pictures and animations are from Norton, Design of Machinery McGrawHill, 2004

GEAR TRAINS

PinionPinion

RackRack & Pinion W G

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Rack & Pinion Worm Gear

quieter than spur gearp g

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Very high gear ratio is possible in small package. Allow one directional drive: worm →worm wheel

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Pitch circlevP

T 2

d2 3P

d2 3 d2

3P

d

vP

T2T3

V V r r= = =ω ω

3P

T3V V r rP P3 2 13 3 12 2= = =ω ω

13 13 2 223

n r dRn r d

ωω

= = = − = −2

12 12 3 3n r dω

Two gears can only mesh if they have the same “circular pitch”, “diametral pitch” or “module”

13 13 2 223

12 12 3 3

n r dRn r d

ωω

= = = + = +

Circular pitch, cp: 2πr = cpT (T= no of tooth on the circumference)cp= the distance between a point on one gear tooth and the same point on the next gear tooth (cp is measured in inches).Diametral Pitch PD= T/D

Module, m: πD = m T (m is measured in mm)n r d T13 13 2 2 2ω

The + sign is used here to take into account the direction of rotation.

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Hence: Rnn

rr

dd

TT23

13

12

13

12

2

3

2

3

2

3

= = = ± = ± = ±ωω

Simple Gear Trains⎛ ⎞⎛ ⎞

R R RTT

TT

TT24 23 34

2

3

3

4

2

4

= = −⎛⎝⎜

⎞⎠⎟ −⎛⎝⎜

⎞⎠⎟ =

Gear 3 is “idler” Used to change theGear 3 is idler . Used to change the direction of rotation or to connect two shafts at a distance.

Simple “Compound”Gear Train

n n n n T T T T/ /

( )Rn n n nn n n n

R R R RT T T TT T T T26

16 15 14 13

15 14 13 1223 34 45 56

4 2 3 4 5

3 4 5 6

1= = = −

In general:

( )Rnnij

j

i

k= = −1

1

1 Product of driving gear tooth numbersProduct of driven gear tooth numbers

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k = number of external gear meshes.

Reverted GearUsed in automotiveUsed in automotive transmission:

- compact, save space

Revert = go back to a previous stateRevert = go back to a previous state

Speed change gear box

0-4-5-6-10-12

2 22 23 11( 1) 0.32334 46 34

output

input

ωω

•= − = =

•

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Important: Gears are numbered. Not the links!!

Input Shaft1

Intermidiate Shaft10.80640.6470.514

1-92-84-53-7

Intermidiate Shaft0.514

7-116-10

Output Shaft

1.156

0.932

0.748

0.594

0.5000.4030.3240.257

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Automotive Transmission Low gear: Gear 3 meshes with gear 6, power flows 1-4-6-3

14 18 0.30131 27

out

in

ωω

= • =

Second Gear: gear 2 meshes with gear 5, power flows 1-4-5-2

14 25

High Gear: gear 2 is shifted so that

14 25 0.56431 20

out

in

ωω

= • =

High Gear: gear 2 is shifted so that the clutch teeth on the end of gear 2 mesh with the clutch on gear 1 (direct drive)drive)

Reverse gear: gear 3 is shifted to

1out

in

ωω

=http://auto.howstuffworks.com/transmission.htm

Reverse gear: gear 3 is shifted to mesh with gear 8, power flows 1-4-7-8-3.

14 14 0 234outω

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0.23431 27

out

inω= − • =

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Dual Clutch Transmission

From: How stuffs work: http://auto howstuffworks com/dual clutch transmission1 htm

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From: How stuffs work: http://auto.howstuffworks.com/dual-clutch-transmission1.htm

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PLANETARY GEAR TRAINSThe axis of one of the gear (planet) is not fixed

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Planetary Gear Trains

APlanet

i i ij

iA

k k k kArmArm

jj

jA0

Sun Gear

AA0

0

Sun Gear

Ring Gear

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VP/A P

VA

VP =VP

AVA

ω

Aω1i j

1i

i

AP/AV

PP=VVP

ωω 1k 1k

Aj

i

j

ii

ωA A

1j00j

∓ω 1j

∓

VPi = VPj = VA + VP/A

1

( )Pj Pi j jv v r

v r r

ω

ω

= =

= ∓(- if external, + if internal mesh)

1

/ 1

1 1 1

( )

( )

A k j i

P A i i

j j k j i i i

v r r

v rr r r r

ω

ωω ω ω

=

= ±= ±

∓

∓ d T

± =−ri j kω ω1 1

1 1 1( )j j k j i i ir r r rω ω ω±∓± = ± = ± =

rr

dd

TT

Ri

j

i

j

i

jij Gear Ratio

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−rj i kω ω1 1 Gear Ratio is not equal to the speed ratio

ωarm=ω14=40s-1 (CW)

ωsun=ω12=?

80 20ω ω12 14

11 14

80 20 220 40

ω ωω ω

−= − = −

−i

0ω11=0

12 14 14 142( ) 3ω ω ω ω= − − =

ωsun = ω12 =120 s-1 (CW)

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T'

k

T

p

j

T

j Rijki

ki

j k

i k

= =−

−ωω

ω ω

ω ω1 1

1 1

k ip

Tj R pjkj

kp

j k

p k

= =−

−

ω

ω

ω ω

ω ω1 1

1 1

1R

Rpj

jp=

RRRip

ij

pj

p k

i k

= =−

−

ω ω

ω ω1 1

1 1

Tp ( )R R RT TT Tip ij jp

k j i

j p

= = −1 /

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90T2-3 and 2-4 are simple gear trains

30T

5 6

n14= - 3624

n12 n14= - 32

n12

20T 324T

5

7

70T20T

6

n13= - 4020

n12 n13= -2 n12

3

n12=100rpm 30T

4

2

7Now consider links 3,4,5,6 and 7.

Link 5 is the planet, link 3 is the arm

40T 36Tn17-n13

n14-n13

30*30*20

90*20*70= - = - 1

7

n17= 87

n14n13- 1

7 297 7

n17= -8*2 n12n - -3

n17= - 2914

n12

207 1418

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n17 7

n12n127*2 n17= - 207.14 rpm

91T

3 5

91T 92T 92T

PlanetPlanet

n12= 2000 rpm

n16=?n16 ?

First planet (arm red- link2)n14-n12 90*92

2 64Arm Arm

14 12

n11-n12

= 90 92

91*91

Since n 0 :

90T 90T91T 91T

Since n11=0 :

n14= n12-90*92

91*91 90T 90T91T 91T91*91 n12

n14= 8281-82808281

n12

Second planet (arm blue- link4)

n16-n14 = 90*92 n16= n14-90*92

8281

n14= 18281

n12

n11-n14

=91*91

16 1491*91

n = 8281-8280 n14

n14

n16= 1 n148281 n16=8281

148281

14

n16=1 n12 n16= 1 n n16= 0.0000292 rpm

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n1682812

n12 1668,574,961

n12

Example: Model T Ford gearboxGearbox : Integral with the engine. 2: On g gFoot operated 2 speed and reverse epicyclic transmission foot-brake, 1908 for 19 yrs9 million were made!

P1P2

2: On

http://www.t-ford.co.uk/car.htm

I/PO/P

S1S2

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Low gear for the model T Ford

nout-nin

ns1-nin

= 21*2733*27

= 7/11

ωin =0.3636 ninnout=(1-7/11) nin =4/11nin

ωout

Fixed

Reverse gear for the model T Ford

nout-nin

ns1-nin

= 30*2724*27

= 5/4

= - 0.25 ninnout=(1-5/4) nin =-1/4nin

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Fixed

GEAR TRAINS WITH BEVEL GEARST3r3r3

+

θ β

P

r2

O

θ

α

β

T2

( )ω13 2 2 2r T r OP/( )( )ω

13

12

2

3

2

3

2

3

= = =r T r OP/

= =siniαβ

R 23sinβ

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P1

80T

60T

4+

+

3

O

56T3 +

2+

24T4

P2

2Input Output 2

541

P2

18T

20T

Simple compound gear train ( f ll fi d )

+

20T35T

76T

+

Planetary Gear Train(axes of all gears are fixed axes)

R 2313

12

2080

= =ωω

and

R n nn n12

12 15

11 15

76 5656 20

195

=−−

= − = −**

5R 34

14

13

1860

= =ωω

R 14 20 18 3= = = =ω * Product of driving gear tooth number

or

n n15 125

24=Since n11=0:

Rn n

1414 15 76 24

56 35228245

=−

= + = +**R 24

12 80 60 40= = = =ω * Product of driven gear tooth number

075.0403R

12

1424 +=+=

ωω

=

n n1411 15 56 35 245− *

Since n11=0:

1 228 17 17 5⎛⎜

⎞⎟ *n n n n14 15 15 121

245 245 245 24= −⎛⎝⎜

⎞⎠⎟

= = *

Nn

2414 0 0145= = .

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n2412

28T 56 3

92TMotion from 2 to 3 and 2 to 4 are simple gear trains (axes fixed):

90T

30T

4n14= 5020

n12

50 50T20T

INPUT

n13= 5020

n12

Links 3 and 4 rotate in different directions

2

20T

INPUT

n13= - n14= 2.5 n12

Considering links 3,4,5 and 6; link 6 is the planet and link 5 is the arm (output)

O

planet and link 5 is the arm (output)

n13-n15

n -n90*28

30*92= +

OUTPUTn14-n15 30*92

n13-n15 = 2123

(n14-n15)232

23n15 = n13 - 21

23 n14

2 5n15 = 55 n12

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2

23n15 = 44

23n12

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