# Gear Trains

IIT Kanpur
9. Oct 2015
1 von 17

### Gear Trains

• 1. THOERY OF MECHANISMS AND MACHINES Gears Prepared by: Abhishek Attal Final Year Dual Degree Student Department of Mechanical Engineering Indian Institute of Technology Kanpur attalab@iitk.ac.in NL-312
• 2. Contact Ratio • To transmit rotational motion continuous there must be at least one pair of contacting teeth at all times • Typically, there are more than one pair in contact, hence overlapping of teeth • Contact Ratio is used to provide quantitative measure of the amount of overlap 𝐶𝑜𝑛𝑡𝑎𝑐𝑡 𝑅𝑎𝑡𝑖𝑜 𝑚 𝑐 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑎𝑐𝑡𝑖𝑜𝑛 𝑏𝑎𝑠𝑒 𝑝𝑖𝑡𝑐ℎ 𝑏𝑎𝑠𝑒 𝑝𝑖𝑡𝑐ℎ(𝑝 𝑏) = 2𝜋𝑟𝑏2 𝑁2
• 3. 𝑚 𝑐 = 𝐸𝐹 𝑏𝑎𝑠𝑒 𝑝𝑖𝑡𝑐ℎ = 𝑆𝑇 𝑏𝑎𝑠𝑒 𝑝𝑖𝑡𝑐ℎ = 𝐸𝐹 𝑝 𝑏 𝐸𝐹 = 𝐸𝐹 + 𝐴𝐸 + 𝐸𝐹 + 𝐵𝐹 − 𝐴𝐸 + 𝐵𝐹 + 𝐸𝐹 𝐸𝐹 = 𝐴𝐹 + 𝐵𝐸 − 𝐴𝐵 𝐴𝐵 = 𝐶𝑠𝑖𝑛𝛼 𝐶 = 𝑟𝑝1 + 𝑟𝑝2 𝑚 𝑐 = ( 𝑟2 𝑂1 − 𝑟2 𝑏1 ) 1 2+( 𝑟2 𝑂2 − 𝑟2 𝑏2 ) 1 2−𝐶𝑠𝑖𝑛𝛼 2𝜋𝑟𝑏2 𝑁2 Contact Ratio
• 4. Question The pitch circle radii of two involute spur gears in mesh are 51.5mm and 64.2mm. The outer circle radii are 57.5mm and 71.2mm, respectively, the operating pressure angle being 20 degrees. Determine 1) Length of the path of contact 2) Contact ratio if the number of teeth on the larger gear is 20 𝑟𝑏 = 𝑟𝑝 𝑐𝑜𝑠𝛼
• 5. Primary Gear Characteristics • Pressure angle/ tooth profile • Face Width • Gear ratio or number of teeth on both gears • Centre to centre distance • Module
• 6. Gear Trains • Combination of gears that is used for transmitting motion from one shaft to another is known as a Gear Train • 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
• 7. 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 𝑁𝐴, 𝑁 𝐵 and 𝑁𝐶 • 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
• 8. 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
• 9. Gear Trains ))()(( 5 4 4 3 2 1 1 5 N N N N N N n n 
• 10. 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
• 11. A Arm B O 𝜃 P T R 𝜃 ∅ S Q 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 𝐷𝐴 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
• 12. Planetary Gear Trains
• 13. 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
• 14. Solution-1 𝐺𝑒𝑎𝑟 𝑅𝑎𝑡𝑖𝑜 = 𝐼𝑛𝑝𝑢𝑡 𝑆𝑝𝑒𝑒𝑑 𝑂𝑢𝑡𝑝𝑢𝑡 𝑆𝑝𝑒𝑒𝑑 = 𝑁1 𝑁2 = 1500 300 = 5 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 = 2𝜋𝑁1 𝑇1 60 η = 𝑃𝑜𝑤𝑒𝑟 𝑂𝑢𝑡𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝑂𝑢𝑡𝑝𝑢𝑡 = 2𝜋𝑁2 𝑇2 60 𝑇1 + 𝑇2 + 𝑇3 = 0 𝑇3 = 𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑇𝑜𝑟𝑞𝑢𝑒
• 15. Example-2
• 16. Extras • Continuously variable transmission (CVT) • Internal and external gears • Harmonic gears- https://www.youtube.com/watch?v=bzRh672peNk • Falkrik Wheel (Boats lift in Scotland) https://www.youtube.com/watch?v=k27gOmrDgpo • Involute Gear Profile https://www.youtube.com/watch?v=4QM0juVXW54 • Involute Gear profile Draw https://www.youtube.com/watch?v=3bkw7C5z60E • Differential https://www.youtube.com/watch?v=yYAw79386WI • Compound Gear Train http://www.technologystudent.com/gears1/gears3.htm
• 17. Thank you