how to solve 16

Question

6.1. a pinion with 40 teeth is meshing with ring gear. The centre distances between the shafts is approximately 75mm and the module of the teeth is 3mm. Determine the pitch-circle diameter of the ring gear and the number of teeth on the ring gear. 6.2. a simple gear train consists of a gear wheel and pinion with centre distance of approximately 500mm. assume the circular pitch is 35,5mm and the gears must have a velocity ration of 4,5:1 Determine the following: 6.2.1. the number of teeth on each gear 6.2.2. the true centre distances between the two shafts

Answer 1

6.1)

Simple gear train have the following specifications

Module of  (m) is 3 mm .

Number of teeth on pinion is 40 .

The centre distance is (a) = 75 mm .

The pitch circle diameter of ring gear can be calculated using the Centre distance formula .

a = ( d _g + d _p) / 2    

Where d _g = gear pitch circle diameter & d _p = pinion pitch circle diameter 

pitch circle diameter of pinion is  d _p = z . m   

d _p = 40*3

d _p = 120 .

So pitch circle diameter of pinion is  d _p = 120  .

Now calculate pitch circle diameter of ring gear .

a = ( d _g + d _p) / 2  

75 = ( d _g + 120) / 2 

d _g + 120 = 75*2

d _g + 120 = 150

d _g  = 150 - 120 

d _g  = 30 .

So the pitch circle diameter of ring gear is 30 .

Now calculate for number teeths of ring gear :

Number teeth of ring gear can be calculated using the formula  z = d _g/m .

 z =d _g/m

 z = 30/3

z = 10 

So the number of teeth on the ring gear is 10 .

Answer 2

6.2.1)

Simple gear train have the following specifications

circular pitch is (p) 35.5mm .

Pitch circle is (p) = mπ

So the module m = p/π 

m = 35.5/π 

m = 11.30

So module  (m) is 11.3 mm .

Gear speed ratio is 4.5:1 .

So gear ratio is 4.5:1 .

Central distance is 500 mm .

The number of teeth of gear and pinion can be evaluated using the formula of Centre distance .

a = ( d _g + d _p) / 2    ----------------(1)

Where d _g = gear pitch circle diameter & d _p = pinion pitch circle diameter 

pitch circle diameter  d = z . m   --------------------(2)

From (1) and (2)

a = ( Z₁ m + Z₂ m) / 2  

The Gear ratio (Z₁:Z₂) is 4.5:1  .

Let the number of teeth of  pinion is Z₂ = x .

Then the number of teeth of  ring gear is Z₁ = 4.5x . 

a = ( 4.5x m + x m) / 2

a = m (4.5x  + x) / 2

500 = 11.3 (5.5 x) / 2

11.3 (5.5 x)= 1000

5.5 x = 1000/11.3

5.5x = 88.49

x = 88.49/5.5

x = 16.090

So the number of teeth on  pinion is ≈ 16 .

The number of teeth on  ring gear is 4.5*16.090 = 72.40 .

So the number of teeth on ring gear is ≈ 72 .

Answer 3

6.2.2)

Simple gear train have the following specifications

circular pitch is (p) 35.5mm .

Pitch circle is (p) = mπ

So the module m = p/π 

m = 35.5/π 

m = 11.30

So module  (m) is 11.3 mm .

Gear speed ratio is 4.5:1 .

So gear ratio is 4.5:1 .

Central distance is 500 mm .

The number of teeth of gear and pinion can be evaluated using the formula of Centre distance .

a = ( d _g + d _p) / 2    ----------------(1)

Where d _g = gear pitch circle diameter & d _p = pinion pitch circle diameter 

pitch circle diameter  d = z . m   --------------------(2)

From (1) and (2)

a = ( Z₁ m + Z₂ m) / 2  

The Gear ratio (Z₁:Z₂) is 4.5:1  .

Let the number of teeth of  pinion is Z₂ = x .

Then the number of teeth of  ring gear is Z₁ = 4.5x . 

a = ( 4.5x m + x m) / 2

a = m (4.5x  + x) / 2

500 = 11.3 (5.5 x) / 2

11.3 (5.5 x)= 1000

5.5 x = 1000/11.3

5.5x = 88.49

x = 88.49/5.5

x = 16.090

So the number of teeth on  pinion is Z₂ ≈ 16 .

The number of teeth on  ring gear is 4.5*16.090 = 72.40 .

So the number of teeth on ring gear is Z₁ ≈ 72 .

The true  Centre distance can be evaluated using the formula 

a = ( Z₁ m + Z₂ m) / 2  

a = m ( Z₁ + Z₂ ) / 2  

a = 11.3 ( 72 + 16 ) / 2  

a = 11.3 (88 ) / 2  

a = 11.3 * 44

a = 497.2 mm

So the true Centre distance is 497.2 mm .