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Implicit differentiation and a change of variable become essential when h c t i o n s become complicated and more than one rule is needed to perform a differentiation

as the peak acceleration is decreased Second, tuning a cam for higher speeds increases the tendency for the cam to excite vibrations at lower speeds Finally, we can verify that the vibration behavior of these tuned cams is better than that of a comparable cycloidal cam for speeds below and slightly above the designed speed if the designed operating point, F2d, is greater than the harmonic content of the tuned cam Tuned cams are appropriate for use in those systems that are intended to operate at a constant or nearly constant speed Where applicable, the method of tuned cam design may be the most powerful method available in that it allows the designer to specify entirely the output motion at the design speed However, the performance of the system away from the design speed may deteriorate depending on the designer s choice of optimization criteria and constraints In many applications, cam systems are required to operate over a wide range of operating speeds In the next section we develop methods for designing cams to have low vibration characteristics over a range of operating speeds A primary objective of these methods is the elimination of all unnecessary constraints to obtain the largest possible family of solutions 1344 Cam Design Based on Mean Squared Error Minimization Cams are often designed to operate over a wide range of speeds Since it is not possible to design a cam to be optimal at more than one speed at a time, the question becomes how best to design the cam for the best overall performance This section presents a method of designing cams for optimized performance over a range of operating speeds The cam performance index, which re ects the deviation of the system from its desired operating conditions at any one speed, is de ned as P = A1 ( X (Ti ) - Xi ) + A2.

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the non-trivial fact, explored in 5, that limx 0 x x = 1] t 2 5t + 6 = lim(t 3) = 1 (e) lim t 2 t 2 t 2 (f) lim s 2 2s 3 = lim(s + 1) = 4 s 3 s 3 s 3

( X (Tj ) - X j )2

(1322)

ln(x/2) = lim ln[(x/2)1/(x 2) ] = lim ln(1 + h/2)1/ h = x 2 x 2 x 2 h 0 lim ln[(1 + h/2)2/ h ]1/2 ln e 1/2 = 1/2 [Here we use the non-trivial

Minimizing P is an indication of optimal cam design, since this is a measure of the deviation in follower position, velocity, and acceleration In addition, it may also be desired to impose constraints on the cam pro le of the form y (Tk ) = yk , y (Tk ) = yk , k = 1, , l We want to optimize the performance over the operating range of speeds 1 1 1 F21 F2 F2 h (1324) (1323)

3- 13 Find the derivative of y = (x +3)25

fact, explored in s 5 and 6, that limh 0 (1 + h)1/ h = e, where e is Euler s number] x 2 16 = lim x 4 = 8 (h) lim x 4 x + 4 x 4 2 (a) The function is undefined at c = 2 (b) lim f (x) = 1/2 and f (1) = 1/2 so f is continuous at c = 1

To optimize the performance, we want to minimize the average value of the mean squared error P over this speed range Thus, we want to minimize P= Assuming that y = (a n cos( nT ) + bn sin( nT ))

(1325)

graph, then f is continuous at 0 (d) lim f (x) = 0 If we define f (0) = 0, which is plausible from the

(1326)

then the forced response of the follower for the worst case of an undamped system (x1 = x2 = 0) is, from Eq (137),

x 2

Solution: You could try to find someone to raise x+3 to the 25* power or you could view the x + 3 as a variable and apply the power rule Implicit differential also helps to simplify the problem

(1327)

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