1. THOERY OF MECHANISMS AND MACHINES
Module-09
Gears
Instructed by:
Dr. Anupam Saxena
Associate Professor
Department of Mechanical Engineering
Indian Institute of Technology Kanpur
FB 361
Prepared by:
Abhishek Attal, Abhishek Sharma
Final Year Dual Degree Student
Department of Mechanical Engineering
Indian Institute of Technology Kanpur
NL-312, FB369

2. Contact Ratio La = A1B1 = A1B1 + A1A + A1B1 + B1B – A1A – B1B – A1B1
= (AB1) + (A1B) – (AB)
Contact ratio = La/(base pitch) = La/(2*pi*r/N);
r/N is the same for the two gears...

3. Gear Trains
Combination of gears used to transmit motion from one shaft to another
Two or more gears working in a sequence (train) are called a gear train
Idler gear:
 A single idler gear is used to reverse the direction
Gear Trains
1. Simple gear train
2. Compound gear train
3. Epicyclic gear train

4. Simple Gear Train
Axes of rotation of each gear is fixed, each shaft has only one gear
Addition of each intermediate gear reverses direction of rotation of the final gear
In a sequence of gears chained together, the ratio depends only on the number
of teeth on the first and last gear.
Consider gear train with 3 gears A,B,C. Let speed of rotation be 𝑛 𝐴 , 𝑛 𝐵 and 𝑛 𝐶 , a
and let number of teeth in each gear be 𝑁 𝐴 , 𝑁 𝐵 and 𝑁 𝐶
𝑛 𝐴 𝑛 𝐵 = 𝑁 𝐵 𝑁 𝐴 𝑛 𝐵 𝑛 𝐶 = 𝑁 𝐶 𝑁 𝐵 𝑛 𝐴 𝑛 𝐶 = 𝑁 𝐶 𝑁 𝐴
The intermediate gears, regardless of their size,
do not alter the overall gear ratio of the chain

5. Compound Gear Train If any one of the intermediate shafts carries more
than one gear, then it is compound gear train
Number of teeth on the intermediate gears will
influence the speed ratio
𝑛 𝐴 𝑛 𝐵 =− 𝑁 𝐵 𝑁 𝐴 , 𝑛 𝐶 𝑛 𝐷 =− 𝑁 𝐷 𝑁 𝐶 , 𝑛 𝐵 𝑛 𝐷 =− 𝑁 𝐷 𝑁 𝐶
𝑛 𝐴 𝑛 𝐷 = 𝑁 𝐷 𝑁 𝐶 𝑁 𝐵 𝑁 𝐴

6. Gear Trains

7. Epicyclic Gear Train In epicyclic gear train one or
more of the gear axes moves
Term epicyclic comes from the
fact that points on gears with
moving axes of rotation will
describe epicycloid paths
Falkrik Wheel: Lifts boats in
Scotland

8. 𝑛 𝐵 − 𝑛 𝑎𝑟𝑚 𝑛 𝐴 − 𝑛 𝑎𝑟𝑚 = 𝜃 ∅ =− 𝑁 𝐴 𝑁 𝐵 =train value from A to B
Analytical Method
Consider Simple Epicyclic gear train, where gear A can rotate about an arm which itself rotates about the axis of gear B
Degree of Freedom of this System ??
𝑛=4, 𝑗=3, ℎ=1
𝐹=3 𝑛−1 −2𝑗 −ℎ
Consider 𝑛 𝐵 =0 , 𝑛 𝑎𝑟𝑚 =𝑔𝑖𝑣𝑒𝑛 𝑛 𝐴 = ??
Arc RT = Arc RP
∅ 𝜃 = 𝐷 𝐵 𝐷 𝐴 = 𝑁 𝐵 𝑁 𝐴 𝐷 𝐵 and 𝐷 𝐴 are respective pitch diameters
Relative rotation of B with respect to arm = 𝜃 (CCW)
Relative rotation of A with respect to arm = ∅= 𝑁 𝐵 𝑁 𝐴 𝜃 (CW)
𝑛 𝐵 − 𝑛 𝑎𝑟𝑚 𝑛 𝐴 − 𝑛 𝑎𝑟𝑚 = 𝜃 ∅ =− 𝑁 𝐴 𝑁 𝐵 =train value from A to B A Q ∅ S T
Arm 𝜃 P R 𝜃 B O

9. Planetary Gear Trains

10. Example-1
A gearbox has an input speed of 1500 rev/min clockwise and an output speed of 300 rev/min anticlockwise. The input power is 20kW and the efficiency is 70%
Determine the following
Gear Ratio
Input Torque
Output Power
Output Torque
Holding Torque

11. Solution-1
𝐺𝑒𝑎𝑟 𝑅𝑎𝑡𝑖𝑜= 𝐼𝑛𝑝𝑢𝑡 𝑆𝑝𝑒𝑒𝑑 𝑂𝑢𝑡𝑝𝑢𝑡 𝑆𝑝𝑒𝑒𝑑 = 𝑁1 𝑁2 = =5 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡= 2𝜋 𝑁 1 𝑇 1 60 η= 𝑃𝑜𝑤𝑒𝑟 𝑂𝑢𝑡𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝑂𝑢𝑡𝑝𝑢𝑡= 2𝜋 𝑁 2 𝑇 2 60 𝑇 1 + 𝑇 2 + 𝑇 3 =0 𝑇 3 =𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑇𝑜𝑟𝑞𝑢𝑒

12. Example-2

13. Extras Continuously variable transmission (CVT)
Internal and external gears
Harmonic gears-
Falkrik Wheel (Boats lift in Scotland)
Involute Gear Profile
Differential
Compound Gear Train

14. Thank you