Week 7 : Number of teeth for the sun and ring gear. (DHEV)

Q1. Consider 2 planetary gear sets as shown in the figure. Number of teeth for the sun and ring gear for both the sets are mentioned below.

Consider the following connections:

1. Input – S1; Output – C2; Ground – R1, R2; C1 connected to S2
   b. Input – S1; Output – C2; Ground – R2; C1 connected to S2; R1 connected to C2

Draw lever diagrams. Calculate the ratios (Input Speed / Output Speed) for both cases. Calculate steady state speed and torque for each element (S1, C1, R1, S2, C2, R2) for the following input condition: Input
speed = 1000rpm, input torque = 100Nm.

Given :

Input speed = 1000 rpm

Input Torque = 100 Nm

`N_(S1)` = 38

`N_(R1)` = 94

`N_(S2)` = 40

`N_(R2)` = 75

Condition :

1. Input - S1 ; Output - C2 ; Ground - R1,R2 ; C1 connected to S2.
2. Input - S1 ; Output - C2 ; Ground - R2 ; C1 connected to S2 ; R1 Connected to C2.

To Find :

1. Speed Ratio for Both Condition.
2. Steady State Speed (S1,C1,R1,S2,C2,R2)
3. Torque (S1,C1,R1,S2,C2,R2).

Calculation :

`N_(S1)` = 38

`N_(R1)` = 94

`N_(C1)`= ?

`N_(C1) = (N_(S1) + N_(R1))/N_(S1)`

`N_(C1) = 132/38`

`N_(C1) = 3.474`

`W_(S1) = ((N_(S1) + N_(R1))/N_(S1))*W_(C1)`

`W_(C1) = ((N_(S1))/(N_(S1)+ N_(R1)))*W_(S1)`

`= 1000 * (38/132)`

`W_(C1) = 287.9` rpm.

`W_(C1) * T_(C1) = W_(S1) * T_(S1) + W_(R1) * T_(R1)` ( there fore `W_(R1)` = 0 due to ring is grounded )

`W_(C1) * T_(C1) = W_(S1) * T_(S1)`

`T_(C1) = (W_(S1) * T_(S1))/W_(C1)`

`= ((1000*100)/287.9)`

`T_(C1) = 347.3` N-m

`W_(C1) = W_(S2)`

`W_(S2) = ((N_(S2) + N_(R2))/N_(S2))*W_(C2)`

`W_(C2) = ((N_(S2))/(N_(S2)+ N_(R2)))*W_(S2)`

`= 287.9 * (40/115)`

`W_(C2) = 100.1` rpm.

`W_(C2) * T_(C2) = W_(S2) * T_(S2) + W_(R2) * T_(R2)` ( there fore `W_(R2)` = 0 due to ring is grounded )

`W_(C2) * T_(C2) = W_(S2) * T_(S2)`

`T_(C2) = (W_(S2) * T_(S2))/W_(C2)`

`= ((287.9*247.3)/100.1)`

`T_(C2) = 711.3` N-m

`Speed Ratio = W_(S1)/W_(C2)`

`Speed Ratio = 1000/100.1`

`Speed Ratio = 9.99`

Q2. Using the following torque converter specifications, calculate input and output speeds and torques, speed ratio, torque ratio, capacity factor, input and output powers, power lost in the torque converter and efficiency. Calculate for the following conditions:

1. Turbine Speed = 0 rpm; Turbine torque = 300 Nm
   ii. Pump Speed = 2000 rpm; Turbine Speed = 1000 rpm
   iii. SR = 0.9, Pump torque = 100 Nm

Torque Converter Specification

1. Turbine Speed`omega_t = 0`rpm; Turbine torque`T_t = 300`Nm

Formula :

`SR = omega_t/omega_p`

`TR = T_t/T_p`

`SR = 0/omega_p`

`0 = 0/omega_p`

`omega_p = 0/0`

`omega_p=0`

`TR = T_t/T_p`

`T_p = T_t/(TR)`

`T_p = 300/2`

`T_p = 150`

Capacity Factor`= 150`

Input Power (kW) = `(T_t * omega_t)/9.5488`

`= (300*0)/9.5488`

`= 0/9.5488`

Input Power (kW)`= 0`

Output Power (kW) = `(T_p*omega_p)/9.5488`

`= (150 * 0 ) /9.5488`

Output Power (kW) `= 0`

Power Loss = input power - output power

Power Loss =0

Efficiency = output power /input power

= 0

1. Taking SR = 0.5; TR = 1.5; K = 170 as input.

`SR = omega_t/omega_p`

`TR = T_t/T_p`

`SR = 0/omega_p`

`0.5 = 0/omega_p`

`omega_p = 0/0.5`

`omega_p=0`

`TR = T_t/T_p`

`T_p = T_t/(TR)`

`T_p = 300/1.5`

`T_p = 200`

Capacity Factor`= 170`

`Input Power (kW) = `(T_t * omega_p)/9.5488`

`= (300*0)/9.5488`

`= 0/9.5488`

Input Power (kW)`= 0`

Output Power (kW) = `(T_p*omega_p)/9.5488`

`= (200 * 0 ) /9.5488`

Output Power (kW) `= 0`

Power Loss = input power - output power

= 0-0

Power Loss =0

Efficiency = output power /input power

= 0/0

= 0

1. Taking SR = 0.9; TR = 1; K = 250 as input.

`SR = omega_t/omega_p`

`TR = T_t/T_p`

`SR = 0/omega_p`

`0.9 = 0/omega_p`

`omega_p = 0/0.9`

`omega_p=0`

`TR = T_t/T_p`

`T_p = T_t/(TR)`

`T_p = 300/1`

`T_p = 300`

Capacity Factor`= 250`

`Input Power (kW) = `(T_t * omega_p)/9.5488`

`= (300*0)/9.5488`

`= 0/9.5488`

Input Power (kW)`= 0`

Output Power (kW) = `(T_p*omega_p)/9.5488`

`= (300 * 0 ) /9.5488`

Output Power (kW) `= 0`

Power Loss = input power - output power

= 0-0

Power Loss =0

Efficiency = output power /input power

= 0/0

= 0

1. Pump Speed = 2000 rpm; Turbine Speed = 1000 rpm
2. Taking SR= 0;TR= 2; K = 150 as input

Formula :

`SR = omega_t/omega_p`

`TR = T_t/T_p`

`SR = omega_t/omega_p`

`SR = 1000/2000`

`SR = 0.5`

Capacity factor = `omega_p /root 2 T_p`

`150 = 2000/root2 T_p`

`T_p= 177.8`

`TR = T_t/T_p`

`T_t = T_p*(TR)`

`T_t = 177.8*2`

`T_t = 355.6`

Input Power (kW) = `(T_t * omega_t)/9.5488`

`= (355.6*1000)/9.5488`

`= 37240.3`

Input Power (kW)`= 0`

Output Power (kW) = `(T_p*omega_p)/9.5488`

`= (177.8 * 2000 ) /9.5488`

Output Power (kW) `= 37240.3`

Power Loss = input power - output power

= 37240.3-37240.3

Power Loss =0

Efficiency = output power /input power

= 37240.3/37240.3

= 1

1. Taking SR = 0.5; TR = 1.5; K = 170 as input.

Formula :

`SR = omega_t/omega_p`

`TR = T_t/T_p`

`SR = omega_t/omega_p`

`SR = 1000/2000`

`SR = 0.5`

Capacity factor = `omega_p /root 2 T_p`

`170 = 2000/root2 T_p`

`T_p= 138.4`

`TR = T_t/T_p`

`T_t = T_p*(TR)`

`T_t = 138.4*2`

`T_t = 276.8`

Input Power (kW) = `(T_t * omega_t)/9.5488`

`= (276.8*1000)/9.5488`

`= 28987.9`

Input Power (kW)`= 0`

Output Power (kW) = `(T_p*omega_p)/9.5488`

`= (177.8 * 2000 ) /9.5488`

Output Power (kW) `= 28987.9`

Power Loss = input power - output power

= 28987.9-28987.9

Power Loss =0

Efficiency = output power /input power

= 28987.9/28987.9

= 1

1. Taking SR = 0.9; TR = 1; K = 250 as input.

Formula :

`SR = omega_t/omega_p`

`TR = T_t/T_p`

`SR = omega_t/omega_p`

`SR = 1000/2000`

`SR = 0.5`

Capacity factor = `omega_p /root 2 T_p`

`250 = 2000/root2 T_p`

`T_p= 64`

`TR = T_t/T_p`

`T_t = T_p*(TR)`

`T_t = 64*2`

`T_t = 128`

Input Power (kW) = `(T_t * omega_t)/9.5488`

`= (128*1000)/9.5488`

`= 37240.3`

Input Power (kW)`= 13404.8`

Output Power (kW) = `(T_p*omega_p)/9.5488`

`= (128 * 2000 ) /9.5488`

Output Power (kW) `= 13404.8`

Power Loss = input power - output power

= 13404.3-13404.8

Power Loss = 0

Efficiency = output power /input power

= 13404.8/13404.8

= 1

iii. SR = 0.9, Pump torque = 100 Nm

1. Taking SR= 0;TR= 2; K = 150 as input

Formula :

`SR = omega_t/omega_p`

`TR = T_t/T_p`

Capacity factor = `omega_p /root 2 T_p`

`150 = omega_p/root2 100`

`omega_p= 150 * root2 100`

`omega_p = 1500`

`SR = omega_t/omega_p`

`0.9 = omega_t/1500`

`omega_p = 1350`

`TR = T_t/T_p`

`T_t = T_p*(TR)`

`T_t = 100*2`

`T_t = 200`

Input Power (kW) = `(T_t * omega_t)/9.5488`

`= (200*1350)/9.5488`

`= 28275.8`

Input Power (kW)`= 28275.8`

Output Power (kW) = `(T_p*omega_p)/9.5488`

`= (100 * 1500 ) /9.5488`

Output Power (kW) `= 15708.8`

Power Loss = input power - output power

= 28275.8-15708.8

Power Loss = 12567

Efficiency = output power /input power

= 15708.8/28275.8

= 1.8

1. Taking SR = 0.5; TR = 1.5; K = 170 as input.

Formula :

`SR = omega_t/omega_p`

`TR = T_t/T_p`

Capacity factor = `omega_p /root 2 T_p`

`170 = omega_p/root2 100`

`omega_p= 170 * root2 100`

`omega_p = 1700`

`SR = omega_t/omega_p`

`0.9 = omega_t/1700`

`omega_p = 1530`

`TR = T_t/T_p`

`T_t = T_p*(TR)`

`T_t = 100*1.5`

`T_t = 150`

Input Power (kW) = `(T_t * omega_t)/9.5488`

`= (150*1530)/9.5488`

`= 24034.4`

Input Power (kW)`= 24034.4`

Output Power (kW) = `(T_p*omega_p)/9.5488`

`= (100 * 1700 ) /9.5488`

Output Power (kW) `= 17803.3`

Power Loss = input power - output power

= 24034.4-17803.3

Power Loss = 6231.1

Efficiency = output power /input power

= 24034.4/17803.3

= 1.3

1. Taking SR = 0.9; TR = 1; K = 250 as input.

Formula :

`SR = omega_t/omega_p`

`TR = T_t/T_p`

Capacity factor = `omega_p /root 2 T_p`

`250 = omega_p/root2 100`

`omega_p= 250 * root2 100`

`omega_p = 2500`

`SR = omega_t/omega_p`

`0.9 = omega_t/2500`

`omega_t = 2250`

`TR = T_t/T_p`

`T_t = T_p*(TR)`

`T_t = 100*1`

`T_t = 100`

Input Power (kW) = `(T_t * omega_t)/9.5488`

`= (100*2250)/9.5488`

Input Power (kW)`= 23563.2`

Output Power (kW) = `(T_p*omega_p)/9.5488`

`= (100 * 2500 ) /9.5488`

Output Power (kW) `= 26181.3`

Power Loss = input power - output power

= 23563.2-26181.3

Power Loss = - 2618.1

Efficiency = output power /input power

= 23563.2/26181.3

= 0.9