mech_4_rigid_body_kinematics

mech_4_rigid_body_kinematics - Rigid body is defined as a...

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Rigid body is defined as a system of particles such that the distance between any two particles is always fixed in magnitude. The number of co-ordinates required to determine the position and orientation of a body in plane motion is three : the system is said to have three degrees of freedom. x y y a A B x G A B path θ 1 2 When the body as above moves from one position to another position with angle unchanged, the motion is described as general plane motion . * If the path is a straight line and unchanged, the translation is rectilinear. If the path is not a straight line and unchanged, the translation is curvilinear. Consider the general case with changed from 1 to 2 within time change ( t 2 - t 1 ), then the average angular speed = = ω 21 tt average . As ( ) ( ) 2 1 0 −→ − → , θθ 0 , the angular speed is defined as :- == lim t t d dt 0 The angular velocity vector is defined as having a magnitude equal to the angular sped and a direction perpendicular to the plane of rotation, the positive sense is given by the usual right hand screw rule. Therefore in this case (1) = || k It should be noted that infinitesimal rotations and angular velocity are vector quantities, but finite angular displacement is not . Angular speed is independent of translation i.e. independent of position. Relative motion between two points on a rigid body can be expressed in terms of rotation and vector product form x r 1
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Referring to the diagram, v B/A has a magnitude of | ω r | and in the direction e θ . A B y x r= | r | e B/A r | r|e = || k e e r In order to find v B/A , we apply vector product to 2 vectors, we have v B/A = x r B/A ( always before r ) = | | k x | r | e r = | r | e (1) ( since k x e r = e ) Consider point A and B are not on the same xy-plane, suppose r B/A = | r | e r + | z | k therefore v B/A = | | k x ( | r | e r + | z | k ) = | r | k x e r + |
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This note was uploaded on 05/10/2009 for the course MXXM 2XX9 taught by Professor Gxxy during the Spring '09 term at City University of Hong Kong.

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mech_4_rigid_body_kinematics - Rigid body is defined as a...

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