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Unformatted text preview: 4. Kinematics of Rigid Bodies Introduction • Kinematics of rigid bodies: relations between time and the positions, velocities, and accelerations of the particles forming a rigid body. • Classification of rigid body motions: general motion motion about a fixed point general plane motion rotation about a fixed axis curvilinear translation rectilinear translation translation: Translation • Consider rigid body in translation: direction of any straight line inside the body is constant, all particles forming the body move in parallel lines. • For any two particles in the body, A B A B r r r G G G + = • Differentiating with respect to time, A B A A B A B v v r r r r G G ¡ G ¡ G ¡ G ¡ G = = + = All particles have the same velocity. A B A A B A B a a r r r r G G ¡ ¡ G ¡ ¡ G ¡ ¡ G ¡ ¡ G = = + = • Differentiating with respect to time again, All particles have the same acceleration. Rotation About a Fixed Axis. Velocity • Consider rotation of rigid body about a fixed axis AA’ • Velocity vector of the particle P is tangent to the path with magnitude dt r d v G G = dt ds v = ( ) ( ) ( ) φ θ θ φ θ φ θ sin sin lim sin ¡ r t r dt ds v r BP s t = Δ Δ = = Δ = Δ = Δ → Δ locity angular ve k k r dt r d v = = = × = = G ¡ G G G G G G θ ω ω ω • The same result is obtained from Rotation About a Fixed Axis. Acceleration • Differentiating to determine the acceleration, ( ) v r dt d dt r d r dt d r dt d dt v d a G G G G G G G G G K G G × + × = × + × = × = = ω ω ω ω ω • k k k celeration angular ac dt d G ¡ ¡ G ¡ G G G θ ω α α ω = = = = = component on accelerati radial component on accelerati l tangentia = × × = × × × + × = r r r r a G G G G G G G G G G G ω ω α ω ω α • Acceleration of P is combination of two vectors, Rotation About a Fixed Axis. Representative Slab • Consider the motion of a representative slab in a plane perpendicular to the axis of rotation. • Velocity of any point P of the slab, ω ω ω r v r k r v = × = × = G G G G G • Acceleration of any point P of the slab, r r k r r a G G G G G G G G G 2 ω α ω ω α − × = × × + × = • Resolving the acceleration into tangential and normal components, 2 2 ω ω α α r a r a r a r k a n n t t = − = = × = G G G G G Equations Defining the Rotation of a Rigid Body About a Fixed Axis • Motion of a rigid body rotating around a fixed axis is often specified by the type of angular acceleration. θ ω ω θ ω α ω θ θ ω d d dt d dt d d dt dt d = = = = = 2 2 or • Recall • Uniform Rotation, α = 0: t ω θ θ + = • Uniformly Accelerated Rotation, α = constant: ( ) 2 2 2 2 1 2 θ θ α ω ω α ω θ θ α ω ω − + = + + = + = t t t Sample Problem 5.1 Cable C has a constant acceleration of 9 in/s 2 and an initial velocity of 12 in/s, both directed to the right....
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This note was uploaded on 11/09/2009 for the course ENG 91301 taught by Professor Lui during the Spring '08 term at Hong Kong Institute of Vocational Education.
 Spring '08
 LUI

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