By Thomas R. Kane
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Additional info for Analytical Elements of Mechanics. Dynamics
1) is given by FIG. 1 (1) The vector R'œzR is called the rate of change of orientation of R in R' with respect to z} and is given by R 'd< dz R, do dz 49 l 'db dz b (2) 50 SEC. 1 [Chapter 2 where a and b are any two nonparallel vectors fixed in R. R,wzR is SL free vector; that is, it is in no way associated with any particular point of either R or R'. Proof : If a and b are fixed in Ä, their magnitudes and the angle between them are independent of z. Hence da2 dhr n d . ,. or, carrying out the indicated differentiations and using primes to denote differentiation with respect to z in R', a' · a = 0, b' · b = 0 a' · b + a · b' = 0 (3) (4) From these expressions it follows that the relationships α/ = °' X b' wX a ^7~b and b' = * ^ £ a ·b are identities.
4, dz dz Again from Sec. 1) is called the angular velocity of R in Rr and is denoted by R'wR. 1). 1 SEC. 3, determine the angular velocity αωΌ of the disc D in the track C for the instant i*. Solution: Let n and n' be unit vectors parallel to the plane of D, and k a unit vector perpendicular to Z), all three unit vectors being fixed in D, as shown in Fig. 1. 1 FIG. 1) at Hence D = ° dW . i) at (k X n) X (k X nQ άθ = (k X n) - i ^ k M (k X n) · n' dt (k X n) · n' dt or Thus c
Kinematics] SEC. 5 59 Problem : The driver D of a Geneva Stop Mechanism rotates counterclockwise at 3 rpm, thereby causing intermittent rotation of the follower F (see Fig. 5a). V2> FIG. 5a Determine the angular speed of the follower at the instant when point P crosses line AB (between A and B). Solution: It is implied that F and D rotate about axes which are normal to the middle planes of the discs, pass through points A and B, and are fixed in a reference frame R in which line AB is fixed. In accordance with Sec.
Analytical Elements of Mechanics. Dynamics by Thomas R. Kane