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Analytical Elements of Mechanics: Dynamics (Volume 2) by Thomas R. Kane

By Thomas R. Kane

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3a shows four representations of the same angular velocity. -2. rad sec" 2 rad sec FIG. 3a Kinematics] SEC. 1, draw a sketch of D showing two representations of €ωΌ for time t*. Solution: See Fig. 3b. FIG. 4 When a rigid body R moves in a reference frame Rf in such a way that there exists a unit vector k whose orientation in both R and R' is independent of time t, R is said to have an angular velocity of fixed orientation in Rf, and this angular velocity is given by (1) = ~n k at where 0 is the angular displacement of a line L fixed in Ä, relative to a line 1/ fixed in β', both lines being perpendicular to k, and 0 being regarded as positive when the displacement is generated by a k rotation of L relative to V (see Fig.

1) 1 ! dz\ h2 A cftiJ + 2! dz2\ + ... Problem: In Fig. 1, R and R' represent two reference frames (coplanar rectangles), ni and n2 represent unit vectors fixed R FIG. 1 in R, and n/ and n2' are unit vectors fixed in R'. A vector function v is defined as v = z2 n/ where z is the angular displacement of R relative to R'. 01 rad in (a) R and (b) Rr. 2 SEC. v 2 . 1 illustrates how Taylor's theorem may be used for purposes of computation. It should be noted, however, that the same results can be obtained, sometimes more conveniently, without the use of this theorem.

Di Χ 1 ( | ) · while Thus 'f-»x-)-*x-)-*x-)S-S di R R It follows from Sec. 2 that '

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