Analytical Elements of Mechanics. Dynamics by Thomas R. Kane

By Thomas R. Kane

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1, p' = J k X n + k Diß. of Vectors] SEC. 1 If the orientation of a rigid body R in a reference frame R' depends on only a single scalar variable z) there exists for each value of z a vector R'œzR such that the derivative with respect to z in R' of every vector c fixed in R (see Fig. 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.

1). Proof: Let nt, i =? 1, 2, 3, be unit vectors (not parallel to the same plane) fixed in i£, and Vi the n t measure number of v. 3, and letting v be a vector given by v = ni ft sec 1 z determine the derivative of v with respect to z in P, for z = 2 ft. Solution : R dv _ Hence Now P dz p dv\ dz\ 2 = R 2 dv dz\ G[V dz + W Xv - W | . _ 2 X v| 2 = 2 y\i=2 = —n~ ni = 180ni ft sec - 1 54 SECS. 3) ωζρ\ζ κ Thus p [Chapter 2 1 = — ( - 6 η ! 1) with respect to z in R and Rr are equal to each other, as may be seen by letting η'ωζκ play the part of v in the expression given in Sec.

3a). 1) at c = — 5(k X n)i5lcsec- dp\ = — 15(k X n)*ft sec" dt\ 26 SEC. 1 is nevertheless used to evaluate cdn/dt. ) Result: See Fig. 3b. FIG. 5) of v at z* + h and at 2*, where z* is a particular value of z and h a scalar having the same dimensions as 2, can be expressed in terms of values of derivatives of v with respect to z in R at z*, as follows : R h Rdv\ h? RcN\ *+A "~ V U* 1! dz\ 2* 2 ! dz2 + 1, 2, 3, be unit vectors (not parallel to the Proof: Let nt, i same plane) fixed in R. 3) and, by Taylor's theorem for scalar functions, WdhJ , « _ h^dvj\ v i\z*+h ~~ ΌΑζ* ~ Wfa 2!

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