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Unformatted text preview: Chapter 8 Kinematics and Kinetics of Rigid Bodies in Threedimensional Motion 8.1 Spherical coordinates 8.2 Angular Velocity of Rigid Bodies in ThreeDimensional Mo tion 8.3 Angular Acceleration of Rigid Bodies in ThreeDimension Mo tion 8.4 General Motion Of and On ThreeDimensional Bodies 87 8.4.1 GOAL: Determine the angular velocity and acceleration of one arm of the illustrated mechanism. GIVEN: Constant angular velocity of inner arm, and constant angular velocity of outer arm with respect to inner arm DRAW: The figure shows the mechanism with the original coordinate axes and some newly at tached unit vectors. Unit vectors * , * , * k are aligned with the groundfixed X, Y, Z axes. Unit vectors * b 1 , * b 2 , * b 3 are attached to the inner arm OA . FORMULATE EQUATIONS: Well use the expressions for angular velocity and acceleration on a rotating body. SOLVE: The angular velocity of arm AB is equal to the angular velocity of arm OA plus the relative angular velocity of arm AB with respect to arm OA . The angular velocity of OA is * k , and the relative angular velocity of AB with respect to OA is * b 2 . Thus: * AB = * k * b 2 (1) This can be written is the * bframe as: * AB = * b 3 * b 2 To determine the angular acceleration of AB we can differentiate (1): * AB = d dt * AB = d dt * k * b 2 = * k {z} =0 + d dt * k  {z } =0 * b 2 {z} =0 d dt * b 2 Since the angular speeds are constant, and * k is fixed in space, the only term remaining is d dt * b 2 . The tip of unit vector * b 2 sweeps in the * b 1 direction with speed . So we have: * AB = d dt * b 2 = ( * b 1 ) * AB = * b 1 88 Alternatively, we could have used the expression: d dt * AB = d dt S * AB + * OA * AB = 0 + * k ( * k * b 2 ) = * b 1 89 8.4.2 GOAL: Determine the angular velocity of a rotating disk. GIVEN: Inner shafts angular velocity and the angular velocity of the disk with respect to the shaft. DRAW: FORMULATE EQUATIONS: Well use the expression for angular velocity on a rotating body. SOLVE: The angular velocity of the disk D is equal to the angular velocity of inner shaft AB plus the relative angular velocity of disk D with respect to shaft AB . The angular velocity of AB is 1 * , and the relative angular velocity of disk to shaft is 2 * . Thus: * D = 2 *  1 * 90 8.4.3 GOAL: Determine the angular acceleration of a rotating caster. GIVEN: Angular velocity of the casters frame is 1 * b 3 and the angular velocity of the caster with respect to the frame is 2 * b 1 ....
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This note was uploaded on 09/19/2009 for the course ME 242 taught by Professor Kam during the Spring '06 term at Nevada.
 Spring '06
 KAM

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