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[ 3/(3 + h)2 ] [ 3/32 ] h h 0 9 + (3 + h)2 2 h 0 3h(3 + h) 6h + h2 2 h 0 3h(3 + h) 6+h 2 h 0 3(3 + h) 2 9

The required integration of Eq (1322) and Eq (1325) can now be performed exactly The 2m + 1 coef cients can then be determined from the two equations, Eq (1323), and the 2m + 1 - 2l extremum conditions dP dP dP = = = 0, i = 1, , m - l d a0 d ai d bi (1328)

Using this approach, one can design an entire cam pro le at once, thereby obtaining maximum control of the motion and its harmonic content Alternatively, for greater generality, one can design segments of low vibration motion by requiring that y (Tk ) = Xk , y (Tk ) = Xk , for 0 Tk p , (1329)

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The derivative is therefore equal to 2/9 4 (a) x 2 2 (x 2 2) 1 x 2x d x = 2 = dx x 2 2 (x 2 2)2 (x 2)2 d d 2 d cos(x 2 ) = cos (x 2 ) x = [ sin(x 2 )] 2x (b) dx dx dx d d d cot (t 3 t 2 ) (t 3 t 2 ) = (c) [cot(t 3 t 2 )] = dt dt dt 2 csc (t 3 t 2 ) (3t 2 2t) d dx x2 + 1 x2 1 = 4x (x 2 1) (2x) (x 2 + 1) (2x) = 2 (x 2 1)2 (x 1)2

in order to minimize R, see Eq (135) These segments can then be pieced together to give a desired motion To obtain an optimum cam pro le design to satisfy any particular case requires performing several designs using different values of A2/A1 and A3/A1 to nd the design that appears to give the best compromise between optimizing response for accuracy of displacement and velocity responses and minimaxed acceleration As an example we consider the design of a unit rise dwell-rise-dwell motion designed to operate over the range F2 9 Three motions designed in this way are tabulated in Table 132 and their associated acceleration and residual vibration characteristics are shown in Figs 139 and 1310, respectively These motions were obtained using values of A1 = A2 = 1 and A3 = 00, 5 10-8, and 7 10-7 in Eq (1322) for examples 4, 5, and 6, respectively We again observe that for large values of F2, the residual vibration of a family of cam pro les increases as the peak acceleration is decreased In this section we have not imposed the constraints on the magnitude of jerk (ie, y ) or continuity of y which are generally suggested by rules of thumb In the absence of these constraints we observe that the peak value of jerk is relatively high and that the acceleration begins and ends with de nite discontinuities that are small but increasing as A3 increases Nonetheless the high-speed vibration characteristics of these motions represent a signi cant improvement over those of the cycloidal motion, and the accuracy of this predicted performance has been assured by controlling the harmonic content of the motion The existence of such motions is contrary to

Second Solution: Instead of just thinking of the x + 3 as the variable you can define a new variable, U = (x+3) so the function reads y = u25 with implicit derivative

(d) (e) (f) (g) (h) 5 (a)

Jerk y'" =

Displacement, y Velocity, y'

sin x d = ln(cos x) x tan x [x ln(cos x)] = 1 ln(cos x) + x dx cos x d s(s 3) e = e s(s 3) [1 (s 3) + s 1] = e s(s 3) [2s 3] ds d d cos(x2 ) 2 2 e [cos(x 2 )] = e cos(x ) [ sin(x 2 ) 2x] = e cos(x ) dx dx 1 ex 1 (e x 1) = x [ln(e x x)] = x e x e x Since the ball is dropped, v0 = 0 The initial height is h0 = 64 Therefore the position of the body at time t is given by p(t) = 16t 2 + 0 t + 64

y, y', y", y'"

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