2.000 Planetary Gear Application & Derivation Cordless Screw Driver: SCREW DRIVER SHAFT TGT-2, ωGT-2 TGT-1, ωGT-1 MOTOR SHAFT TEM, ωEM Battery Pack Switch Electric Motor GEAR TRAIN # 1 GT-1 GT-2 Screw Driver Shaft GEAR TRAIN # 2 Cross Section of Power Flow Through Cordless Screw Driver Planetary Gear Components Planetary gear trains consist of the following: • 1 Ring gear 1 or more planet gear(s) 1 Sun gear 1 Arm Legend: RING GEAR PLANET GEAR PLANET GEAR ARM ω2 SUN GEAR PLANET GEAR Cordless Screw Driver Gear Trains Special Gear Set: Planetary gear trains consist of the following: • 1 Ring gear 1 or more planet gear(s) 1 Sun gear 1 Arm How a planetary gear train functions (analogous to solar system) • • • • Planetary gear systems have VERY small/large train ratios and are very compact The sun gear is located at the center of the gear train “solar system” The arm carries the planet gears in “orbit” around the sun gear The planet gears “orbit” on the inside of the ring gear Screw driver gear trains Motor Gear (Sun Gear-1) Arm 1 Gear (Sun Gear-2) + + = GT-1 + + = GT-2 GEAR TRAIN # 1 GEAR TRAIN # 2 Screw Driver Planetary Gear Characteristics Nomenclature • Ni = Number of teeth on gear i • ωi = Rotational speed of gear i Subscripts • 1 = Component in planetary gear system # 1 • 2 = Component in planetary gear system # 2 • s = Sun gear • p = Planet gear • r = Ring gear • a = Arm ARM PLANET GEAR RING GEAR Example: Ns1 = Number of teeth on sun gear in planetary gear train #1 This is a complicated mechanism to solve. The easiest way is to work in terms of relative rotary velocities. ωri ωsi = Speed of ring gear with respect to the arm Speed of sun gear with respect to the arm = ωri - ωai ωsi - ωai Now we consider the sun gear as the input and the ring gear as the output. To determine the train ratio between the two gears, we use the following: Because sun gear and ring gears rotate in opposite directions ei = ωri ωsi = Product of Driving Teeth Product of Driven Teeth - Nsi Npi = - Nsi = Npi Nsi = Nri ωri - ωai ωsi - ωai ωai = ωri Nri - ωsi Nsi Nri + Nsi We can use the result for the speed of the arms to solve for combined planetary gear trains. We begin by solving for the speed of the arm in the first gear train. Note that the speed of the first arm = speed of the second sun gear (they are both part of the same piece). We then use the same equation to solve for the second arm velocity. Note the arm is connected to the screw driver shaft, so the second arm velocity = the screw driver shaft velocity. This procedure is more clearly outlined on the following page. Solving For the Screw Driver Train Ratio GIVEN: Gear Teeth Np1 = 19 Ns1 = 6 Train Equation(s) ωai = Nr1 = 48 Ns2 = 6 Np2 = 19 Nr2 = 48 ωri Nri - ωsi Nsi Nri + Nsi PROCEDURE: Write equations Solve for ωa2 for Trains 1 & 2 ωa2 = ωSD-SHAFT Plug ωa1 into eqxn. for ωa2 Train Ratio = HINTS: ωs1 = ωmotor ωr1 = ωs2 = ωr2 ωa1 Motor gear is sun gear for first gear train = 0 ωSD-SHAFT Ring gears do not move Sun gear for 2nd train sits on arm of first train ωmotor