Rigid-Body mechanics Description of a Rigid-Body motion Rigid-body motion preserves distance Q Q(p +Q(p d A Define da= P三p-p lp=a-p+Q(p-a p a-p+Q(p-a)+ dA+(Q-1(p-a) edp=e dA+e(Q-1)(p-a)o P-Arbitrary Reference Left multiply QT. Take transpose Q e=e Q-1)=0 d dA三do Displacement of any point projected onto the rotation axis are same
Q O P P' A p a p' a' P – Arbitrary A - Reference Rigid-Body Mechanics • Description of a Rigid-Body Motion Rigid-Body motion preserves distance Define Left multiply Take transpose = 0 A' Displacement of any point projected onto the rotation axis are same
Rigid-Body mechanics General rigid-Body motion o 3.2.1 The component of the dis aoerments of all the points of a rigid body undergoing a general motion along the aris of the underlying rotation is a constant. e p dA三do A F Theorem 3.2.2(Mowzi, 1763; Chasles, 1830) Given a rigid body un- dergoing a general motion, a set of its points located on a line l undergo identical displacements of minimun magnitude. Moreover, line c and the minimum-magnitude displacement are parallel to the aris of the rotation nvolved
Rigid-Body Mechanics • General Rigid-Body Motion e
Rigid-Body mechanics General Rigid-body motion Geometric Interpretation /A A B Pitch 2-D case i 3-D case Any rigid body motion in 2-D can be Any rigid body motion in 3-D can be regarded as a pure rotation around regarded as a screw-like motion along one point O an axis. e
Rigid-Body Mechanics • General Rigid-Body Motion Geometric Interpretation A B A' B' O 2-D case 3-D case e Pitch p0 Any rigid body motion in 2-D can be regarded as a pure rotation around one point O Any rigid body motion in 3-D can be regarded as a Screw-like motion along an axis, e