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

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...

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

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

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

Gear Trains

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

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

Planetary Gear Trains

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

Solution-1 𝐺𝑒𝑎𝑟 𝑅𝑎𝑡𝑖𝑜= 𝐼𝑛𝑝𝑢𝑡 𝑆𝑝𝑒𝑒𝑑 𝑂𝑢𝑡𝑝𝑢𝑡 𝑆𝑝𝑒𝑒𝑑 = 𝑁1 𝑁2 = 1500 300 =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

Example-2

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 Differential https://www.youtube.com/watch?v=yYAw79386WI Compound Gear Train http://www.technologystudent.com/gears1/gears3.htm

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