Gears Differential

A differential gear train

General solution for differential gears

Gear 3 is fixed to the carrier of the planetary drive. Considering gears 5

and 6,

Relative to the carrier,

Gears 2 and 3 rotate relative to the frame. Therefore

Solution

1 15 3 6

1 16 3 5

1 1 1 15 5 5 3 6 6 6 3

1 1 1 15 5 6 6 6 3 5 3

NN

N N N N

N N N N

1 1

1 5 5 6 63

5 6

N ω + N ωω =N + N

Solution

1 11 5 5 6 6

35 6

N ω + N ωω =N + N

This defines the relation between the ring gear speed and This defines the relation between the ring gear speed and the wheel speeds. The ring gear speed is in turn related to the wheel speeds. The ring gear speed is in turn related to the driveshaft speed which comes from the engine. the driveshaft speed which comes from the engine.

NormallyNormally

5 6

1 11 5 5 5 6

35

1 11 5 6

3

N = N

N ω + N ωω =2N

ω + ωω =2

Ring gear speed is the Ring gear speed is the mean of the two wheel mean of the two wheel

speedsspeeds

Solution

1 1

1 1 1 15 63 5 6 3

ω + ωω = ω = - ω + 2 ω

2

For straight line motion For straight line motion both wheels must have both wheels must have

the same speedthe same speed. .

2

i e

1 15 6

1 1 16 6 3

1 16 3

1 1 15 6 3

ω = ω

ω = - ω + 2 ω

ω = 2 ω

ω = ω = ω

For straight line motion ring gear must rotate at the same For straight line motion ring gear must rotate at the same speed as the wheelsspeed as the wheels

Solution

1 14 3 6

1 16 3 4

1 1 1 14 4 4 3 6 6 6 3

11 14 4 6 4 3 6 6

1 16 4 3 6 61

44

NN

N N N N

N N N N

N N NN

364

36 4

NN

1 1 3

4 3 4

Solution

1 16 4 3 6 61

44

11 4 3

44

1 14 3

N N NN

NN

1 1 15 6 3ω = ω = ωFor straight line For straight line

motionmotion

For straight line motion all gears must rotate at the same For straight line motion all gears must rotate at the same absolute speed.absolute speed.

Hence for straight line motion the differential moves like Hence for straight line motion the differential moves like a rigid shafta rigid shaft

Sample Problem involving differential gear train

The differential for a rear wheel-driven vehicle is shown schematically. If the drive shaft turns at 900 rpm, what is the speed of the vehicle if neither wheel slips and the outside diameter of the wheels is 24 in?

Solution to sample Problem involving differential gear train

The differential for a rear wheel-driven vehicle is shown schematically. If the drive shaft turns at 900 rpm, what is the speed of the vehicle if neither wheel slips and the outside diameter of the wheels is 24 in?

12

1 1 23 2

3

1 1 13 5 6

900 28900 273.9192

For straight line motion

273.91 rpm

rpmN (ignoring sign)N

1 1 13 5 6ω = ω ω


Assume that the vehicle is stopped so that the right wheel sits on a small icy patch and can spin freely while the left wheel does not spin. Determine the angular velocity of the right wheel if the angular speed of the drive shaft is 500 rpm.


Assume that the vehicle is stopped so that the right wheel sits on a small icy patch and can spin freely while the left wheel does not spin. Determine the angular velocity of the right wheel if the angular speed of the drive shaft is 500 rpm.

12

1 1 23 2

3

15

1 11 1 16 6

3 6 3

13

500 28500 152.74 92

0

For general motion

02 2 152.74

2 2304.35

rpmN

(ignoring sign)N

rpm

1 11 5 6

3

ω + ωω =

2


Assume that the vehicle is traveling at 35 mph and turns around a curve with a radius of 50 ft from the centerline of the vehicle. The center-to-center distance between the treads of the right and left wheels is 60 in. Compute the rotational speed of each rear wheel, the rotational speed of the ring gear, and the rotational speed of the drive shaft.


154Speed of center of vehicle=35 35 5280 / 3600 51.333

Assuming a left turn,

1Radius of circle traversed by left wheel = 50 60 /12 47.52

Radius of circle traversed

mile mile ft s ft fthr hr mile hr s s

inin ftft

1by right wheel = 50 60 /12 52.52

For no slip for both wheels154 47.5Linear velocity of left wheel = 48.77

3 50154 52.5Linear velocity of right wheel = 53.9

3 50wheel radius = 1

inin ftft

ft fts s

ft fts s

f

1 15 6

13

1 1 132 3 2

2

48.77 / , 53.9 / ,

48.77 53.9For general motion 51.335 /2

168.67 / 10120 1610.65

t rad s rad s

rad s

N(ignoring sign) rad s rad/min = rpm

N

1 11 5 6

3

ω + ωω =

2

Assume that the vehicle is traveling at 35 mph and turns around a curve with a radius of 50 ft from the centerline of the vehicle. The center-to-center distance between the treads of the right and left wheels is 60 in. Compute the rotational speed of each rear wheel, the rotational speed of the ring gear, and the rotational speed of the drive shaft.

Gears Differential

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