ch08 - Chapter 8 Kinematics and Kinetics of Rigid Bodies in...

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Unformatted text preview: Chapter 8 Kinematics and Kinetics of Rigid Bodies in Three-dimensional Motion 8.1 Spherical coordinates 8.2 Angular Velocity of Rigid Bodies in Three-Dimensional Mo- tion 8.3 Angular Acceleration of Rigid Bodies in Three-Dimension Mo- tion 8.4 General Motion Of and On Three-Dimensional 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 ground-fixed 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 * b-frame 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.

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ch08 - Chapter 8 Kinematics and Kinetics of Rigid Bodies in...

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