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Example 8: Synthesis of Cam Motion with Nonrigid Follower An example application is provided here to illustrate the mechanics of applying the procedure described earlier A DRD motion is synthesized using the spline interpolation procedure The dynamic behavior of the follower system is then investigated and the cam motion is found using the spline collocation procedure In preparing this example a case was selected from the literature for which a cam had been designed using an optimized polynomial (Peisakh, 1966; Chen, 1981) The solution obtained using the optimized polynomial gives a convenient basis for evaluating the performance of the methods described here, at least for one case The results shown were re ned iteratively by trying a total of ten different combinations of spline parameters and sets of constraints Space limitations prevent the presentation of all the iterations; however, it should be clear that the procedures being used allow great freedom in specifying motion constraints and can easily evaluate the effects of changing these constraints and of altering the spline parameters In this example, each step of the process is illustrated even though some of the operations can be easily reduced to an algorithm and eliminated as a concern to the

ConsoleWriteLine("{0:#,####}", 34213);

SOLUTION For clarity we set (x) = cot x, (x) = csc2 x Then our integral becomes 3 (x) (x) dx = = 1 ln 3 1 ln 3 3 (x) (x) ln 3 dx

3,4213

designer For instance, selection of the collocation points is readily automated Such operations are described in the example, however, for the sake of thoroughness

It is not necessary to specify each comma for each position Specifying one comma causes it to be inserted into the value every third digit from the left of the decimal point For example,

ConsoleWriteLine("{0:#,####}", 87634213);

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Synthesis of the Output Motion In this illustration the basic goal is to satisfy the ten basic constraints at t = 0 and at t = 1 in the list of constraints that follows These constraints are the primary design constraints The two additional constraints at t = 05 were added in the iterative process to re ne the motion somewhat Since the degree of the spline curve is k - 1, splines of order k = 6 are required for S(4) to be continuous It is necessary to achieve this continuity if the constraints on S(4) are to be satis ed, as they must be if continuous cam acceleration is to be obtained The output motion that is synthesized using spline functions is shown in Figures 524 to 528 and compared to the motion produced by the optimized polynomial Collocation Solution of the Differential Equations To solve the differential Eqs (518) and (531) for the cam displacements and for vibrational responses, the normalized time domain (t) is divided into sixteen equal elements (e = 16) and in each element four Gaussian points (p = 4) are located Both the number of elements used and the choice of

produces this output:

Resubstituting the expression for (x) now gives that 3cot x ( csc2 x) dx = You Try It: Evaluate (log6 (x 3 )/x) dx You Try It: Calculate the integral x 3x dx Our new ideas about arbitrary exponents and bases now allow us to formulate a general result about derivatives of powers: For any real exponent a we have d a x = a x a 1 dx EXAMPLE 627

8,763,4213

FIGURE 524 Normalized displacements of output motions for spline (k = 10) and optimized polydyne in Example 8

Commas have a second meaning When they occur on the immediate left of the decimal point, they act as a scaling factor Each comma causes the value to be divided by 1,000 For example,

ConsoleWriteLine("Value in thousands: {0:#,###,#}", 87634213);

d x = x 1 , dx d 3 x = 3 x 3 1 , dx d e x = e x e 1 dx You Try It: Calculate (d/dx)5sin x x Calculate (d/dx)x 4

FIGURE 525 Normalized velocities of output motions for spline (k = 10) and optimized polydyne in Example 8

produces this output:

Value in thousands: 8,7634

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