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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.

Notice how the ThreeD objects are used to control if statements and a while loop In the case of the if statements, the ThreeD object is evaluated using true If the result of this operation is true, then the if statement succeeds In the case of the do-while loop, each iteration of the loop decrements b The loop repeats as long as b evaluates as true (that is, it contains at least one non-zero coordinate) When b contains all zero coordinates, it evaluates as false when the true operator is applied and the loop stops

l H pital s Rule applies one more time to nally yield 6 = 0 x + ex lim (b) limx + ln x = limx + x = + so l H pital s Rule applies Thus

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As you know, C# defines the following logical operators: &, |, !, &&, and || Of these, only the &, |, and ! can be overloaded By following certain rules, however, the benefits of the short-circuit && and || can still be obtained Each situation is examined here

( X (Tj ) - X j )2

Let s begin with the simplest situation If you will not be making use of the short-circuit logical operators, then you can overload & and | as you would intuitively think, with each returning a bool result An overloaded ! will also usually return a bool result Here is an example that overloads the !, &, and | logical operators for objects of type ThreeD As before, each assumes that a ThreeD object is true if at least one coordinate is non-zero If all three coordinates are zero, then the object is false

// A simple way to overload !, |, and & for ThreeD using System; // A three-dimensional coordinate class class ThreeD { int x, y, z; // 3-D coordinates public ThreeD() { x = y = z = 0; } public ThreeD(int i, int j, int k) { x = i; y = j; z = k; }

(1322)

limx + e x = limx + ln[x/(x + 1)] = 0 so l H pital s Rule applies Thus e x e x = lim x + 1/x 1/[x + 1] x + ln[x/(x + 1)] lim It is convenient to rewrite this expression as x2 + x x + ex lim Now l H pital s Rule applies once more to yield

// Overload | public static bool operator |(ThreeD op1, ThreeD op2) { if( ((op1x != 0) || (op1y != 0) || (op1z != 0)) | ((op2x != 0) || (op2y != 0) || (op2z != 0)) ) return true; else return false; } // Overload & public static bool operator &(ThreeD op1, ThreeD op2) {

Part I:

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)

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