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Although cam synthesis problems in which the cam is constrained by the application to satisfy 20 kinematic conditions are rare, the capacity of the spline method to accommodate a large number of constraints is still important It enables the designer to introduce constraints, in addition to those that are prescribed by the application, to obtain desirable qualitative characteristics in the follower motion For example, it might be desired to ensure continuity of higher derivatives at the boundaries of a rise to improve the dynamic behavior (Wiederrich, 1981; Chew et al, 1983) In such a case constraints on jerk and the derivatives of jerk could be imposed at the boundary Splines of a suitable order would have to be used, but the number of added constraints would not pose a problem The addition of constraints to tune the motion to a speci c need can easily yield a large array of constraints that cannot be reliably handled by other techniques As illustrated by the series of examples above, the spline approach to cam synthesis affords a systematic procedure that is versatile enough to accomplish many tasks that ordinarily require a variety of techniques This approach can be conveniently used to satisfy the requirements of routine cam synthesis problems while remaining versatile enough to handle some cases that are very dif cult by any other means Implementation

A convenient, intuitive way to think about the logarithm function is as the inverse to the exponentiation function Proceeding intuitively, let us consider the function

ConsoleWrite(i + " "); ConsoleWriteLine(); } }

The output is shown here:

Displacement (cm) Velocity (cm/rad) Acceleration (cm/rad/rad) Constraint 1 0 45 90 Cam rotation angle (deg)

Original order: 1 2 3 4 5 Reversed order: 5 4 3 2 1 Range reversed: 5 2 3 4 1

To operate with this f, we choose an x and take 3 to the power x For example, f (4) = 34 = 3 3 3 3 = 81

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Copying all or part of one array to another is another common array operation To copy an array, use Copy( ) Copy( ) can put elements at the start of the destination array or in the middle, depending upon which version of Copy( ) you use Copy( ) is demonstrated by the following program:

FIGURE 56 Cam follower displacement, velocity, and acceleration obtained by spline synthesis with uniform knot spacing in Example 3

// Copy an array using System; class CopyDemo { static void Main() { int[] source = { 1, 2, 3, 4, 5 }; int[] target = { 11, 12, 13, 14, 15 }; int[] source2 = { -1, -2, -3, -4, -5 }; // Display source ConsoleWrite("source: "); foreach(int i in source) ConsoleWrite(i + " "); ConsoleWriteLine(); // Display original target ConsoleWrite("Original contents of target: "); foreach(int i in target) ConsoleWrite(i + " "); ConsoleWriteLine(); // Copy the entire array ArrayCopy(source, target, sourceLength); // Display copy ConsoleWrite("target after copy: foreach(int i in target) ConsoleWrite(i + " "); ConsoleWriteLine();

// Copy into middle of target ArrayCopy(source2, 2, target, 3, 2); // Display copy ConsoleWrite("target after copy: foreach(int i in target)

of the computational aspects of the approach is largely procedural and can be accomplished using commonly available software to handle the major computational burden

Part II:

f ( 2) = 3 2 =

ConsoleWrite(i + " "); ConsoleWriteLine(); } }

Rational B-spline functions permit greater exibility in re ning motion programs than the method described above As noted earlier, the spline method is suf ciently exible and general to permit the motion programs to be re ned or optimized while still satisfying motion constraints by adjusting certain spline function parameters that are independent of the kinematic constraints It is with regard to this last feature of the generalized spline approach that rational B-splines offer an advantage over the methods described earlier Rational B-splines incorporate additional function parameters and consequently offer additional degrees of design freedom As was the case with nonrational B-splines, these parameters are independent of the motion constraints and can be controlled to re ne and

The output is shown here:

source: 1 2 3 4 5 Original contents of target: 11 12 13 14 15 target after copy: 1 2 3 4 5 target after copy: 1 2 3 -3 -4

1 9 0 f (0) = 3 = 1

118 1

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