KINEMATICS OF RIGID BODIES. y Copyright 1997 by The McGraw-Hill Companies, Inc. All rights reserved. KINEMATICS OF RIGID BODIES

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1 KINEMTICS F RIGID DIES Copight 1997 b The McGaw-Hill Companies, Inc ll ights eseved KINEMTICS F RIGID DIES In igid bod tanslation, all points of the bod have the same velocit and the same acceleation at an given instant Consideing the otation of a igid bod about a fied ais, the position of the bod is defined b the angle that the line, dawn fom the ais of otation to a point of the bod, foms with a fied plane The magnitude of the velocit of is ds whee is the time deivative of = sin

2 ds = sin The velocit of is epessed as d = whee the vecto = = is diected along the fied ais of otation and epesents the angula velocit of the bod d = = = Denoting b the deivative d/ of the angula velocit, we epess the acceleation of as a = + ( ) diffeentiating and ecalling that is constant in magnitude and diection, we find that = = = The vecto epesents the angula acceleation of the bod and is diected along the fied ais of otation

3 In thee dimensions, the most geneal displacement of a igid bod with a fied point is equivalent to a otation of the bod about an ais though The angula velocit and the instantaneous ais of otation of the bod at a given instant can be defined The velocit of a point of the bod can be epessed as d = Diffeentiating this epession, the acceleation is a = + ( ) Since the diection of changes fom instant to instant, the angula acceleation is, in geneal, not diected along the instanteneous ais of otation / The most geneal motion of a igid bod in space is equivalent, at an given instant, to the sum of a tanslation and a otation Consideing two paticles and of the bod v = v + v / whee v / is the velocit of elative to a fame attached to and of fied oientation Denoting b / the position vecto of elative to, we wite v = v + / whee is the angula velocit of the bod at the instant consideed The acceleation of is, b simila easoning a = a + a / o a = a + / + ( / )

4 The ate of change of a vecto is the same with espect to a fied Q fame of efeence and with espect to a fame in tanslation i The ate of change of a vecto with espect to a otating ti fame of efeence is diffeent The ate of change of a geneal vecto Q with espect a fied fame and with espect to a fame otating with an angula velocit is (Q) = (Q) + Q The fist pat epesents the ate of change of Q with espect to the otating fame and the second pat, Q, is induced b the otation of the fame i Conside the thee-dimensional motion of a paticle elative to a fame otating with an angula velocit with espect to fied fame The absolute velocit v of can be epessed as v = v + v /F whee v = absolute velocit of paticle v = velocit of point of moving fame F coinciding with v /F = velocit of elative to moving fame F

5 i The absolute acceleation a of is epessed as a = a + a /F + a c whee a = absolute acceleation of paticle a = acceleation of point of moving fame F coinciding with a /F = acceleation of elative to moving fame F a c = 2 () = 2 v /F = complementa (Coiolis) acceleation The magnitude a c of the Coiolis acceleation is not equal to 2v /F ecept in the special case when and v /F ae pependicula to each othe / The equations and v = v + v /F a = a + a /F + a c emain valid when the fame moves in a nown, but abita, fashion with espect to the fied fame, povided that the motion of is included in the tems v and a epesenting the absolute velocit and acceleation of the coinciding point Rotating fames of efeence ae paticulal useful in the stud of the thee-dimensional motion of igid bodies