Velocity and Acceleration Analysis

Velocity and Acceleration Analysis - Chapter 3 Velocity and...

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Chapter 3 Velocity and Acceleration Analysis 3.1 Introduction The motion of a rigid body ( RB ) is deFned when the position vector, velocity and acceleration of all points of the rigid body are deFned as functions of time with respect to a Fxed reference frame with the origin at O 0 . Let ı 0 , j 0 , and k 0 , be the constant unit vectors of a Fxed orthogonal Cartesian reference frame x 0 y 0 z 0 and ı , j and k be the unit vectors of a body Fxed (mobile or rotating) orthogonal Cartesian reference frame xyz (±ig. 3.1). The unit vectors ı 0 , j 0 , and k 0 of the primary reference frame are constant with respect to time. r 1 r O x ı y j z k ( RB ) ω ı 0 j 0 k 0 x 0 y 0 z 0 O 0 O M α Fig. 3.1 ±ixed orthogonal Cartesian reference frame with the unit vectors [ ı 0 , j 0 , k 0 ]; body Fxed (or rotating) reference frame with the unit vectors [ ı , j , k ] ; the point M is an arbitrary point, M ( RB ) 43
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44 3 Velocity and Acceleration Analysis A reference frame that moves with the rigid body is a body-fxed (or rotating) reference frame. The unit vectors ı , j , and k of the body-Fxed reference frame are not constant, because they rotate with the body-Fxed reference frame. The location of the point O is arbitrary. The position vector of a point M , M ( RB ) , with respect to the Fxed reference frame x 0 y 0 z 0 is denoted by r 1 = r O 0 M and with respect to the rotating reference frame Oxyz is denoted by r = r OM . The location of the origin O of the rotating reference frame with respect to the Fxed point O 0 is deFned by the position vector r O = r O 0 O . Then, the relation between the vectors r 1 , r and r 0 is given by r 1 = r O + r = r O + x ı + y j + z k , (3.1) where x , y , and z represent the projections of the vector r = r OM on the rotating reference frame r = x ı + y j + z k . The magnitude of the vector r = r OM is a constant as the distance between the points O and M is constant, O ( RB ) , and M ( RB ) . Thus, the x , y and z compo- nents of the vector r with respect to the rotating reference frame are constant. The unit vectors ı , j , and k are time-dependent vector functions. The vectors ı , j and k are the unit vector of an orthogonal Cartesian reference frame, thus one can write ı · ı = 1 , j · j = 1 , k · k = 1 , (3.2) ı · j = 0 , j · k = 0 , k · ı = 0 . (3.3) 3.2 Velocity Field for a Rigid Body The velocity of an arbitrary point M of the rigid body with respect to the Fxed reference frame x 0 y 0 z 0 , is the derivative with respect to time of the position vector r 1 v = d r 1 dt = d r O 0 M = d r O + d r = v O + x d ı + y d j + z d k + dx ı + dy j + dz k , (3.4) where v O = ˙ r O represent the velocity of the origin of the rotating reference frame O 1 x 1 y 1 z 1 with respect to the Fxed reference frame Oxyz . Because all the points in the rigid body maintain their relative position, their velocity relative to the rotating reference frame xyz is zero, i.e., ˙ x = ˙ y = ˙ z = 0. The velocity of point M is v = v O + x d ı + y d j + z d k = v O + x i + y j + z ˙ k .
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Velocity and Acceleration Analysis - Chapter 3 Velocity and...

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