Dynamics_Part62

Dynamics_Part62 - Problem Set 9 Problem 1 Problem A...

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Unformatted text preview: Problem Set 9: Problem 1. Problem: A gear-reduction system consists of three gears A, B and C. Gear A rotates in the clockwise direction with constant angular velocity A . (a) Determine the angular velocities of Gears B and C, B and C , respectively. (b) Determine the accelerations of the points on Gears B and C that are in contact, aB and aC , respectively. Solution: (a) Denote the contact point between gears A and B by D. Then, the velocity at Point D is vD = vA + A rD/A Gear A is not translating so that vA = 0. Also, A = -A k and rD/A = R i. Hence, vD = (-A k) (R i) = -A R j So, to find the angular velocity of gear B, B , we note that vD = B rD/B = - A R j = (B k) (-2R i) = -2B R j 1 A 2 Therefore, the angular velocity of gear B is B = Because B is positive, gear B rotates counterclockwise. Now, denote the contact point between gears B and C by E. Then, the velocity at Point E is vE = vB + B rE/B Since gear B is not translating, necessarily vB = 0. Also, B = B k while rE/B = R i. Thus, vE = (B k) (R i) = B R j Consequently, the velocity at Point E is 1 A R j 2 To determine the angular velocity of gear C, C , we can express vE as the sum of a translation of Point C and a rotation about C, viz., vE = vC + C rE/C vE = ...
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