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Evaluation of the Follower Response The physical parameters of the cam-follower system used are: M = 075 lb, Ks = 200 lb/in, Kf = 10000 lb/in, wd = 1000 rpm, b = p /3 rad h = 05 in and Fp = 70 lb A damping ratio of 05(Cs + Cf)/M/wn = 02 is used (Rothbart, 1958) and the damping coef cients Cs and Cf are set equal The performance of the spline-based motion as compared to the optimized polydyne motion is shown in the following gures Table 57 shows the peak values of vibrations and contact forces for these two output motions Figure 529 compares the amplitudes of primary and residual vibrations of the two motions with the damping values given above Finding the Cam Motion To solve differential Eq (518) for the cam displacement and its derivatives, the spline collocation method is applied again Here the seven boundary conditions cited earlier must be satis ed Also the left side of Eq (518) at the 64 collocation points must be satis ed Application of these constraints yields a total of 71 conditions requiring 71 B-splines Splines of order k = 11 are used, requiring the knots at t = 0 and 1 to be repeated eleven times Again, at mesh points of t = 1/16, 2/16, , and 15/16 the knots are repeated four times each As before a system of linear equations is assembled and solved for the unknown coef cients in Eq (56) Once this is done, the

ConsoleWriteLine("{0:###;(###);000}", num);

If num is positive, the value is displayed with two decimal places If num is negative, the value is displayed with two decimal places and is between a set of parentheses If num is zero, the string 000 is displayed When using the separators, you don t need to supply all parts If you just want to specify how positive and negative values will look, omit the zero format To use the default for negative values, omit the negative format In this case, the positive format and the zero format will be separated by two semicolons The following program demonstrates just a few of the many possible custom formats that you can create:

1 x ln a 1 f (x) = 2 x ln a f (1) = 0 f (x) =

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// Using custom formats using System; class PictureFormatDemo { static void Main() { double num = 643542345; ConsoleWriteLine("Default format: " + num); // Display with 2 decimal places ConsoleWriteLine("Value with two decimal places: " + "{0:###}", num); // Display with commas and 2 decimal places ConsoleWriteLine("Add commas: {0:#,#####}", num); // Display using scientific notation ConsoleWriteLine("Use scientific notation: " + "{0:####e+00}", num); // Scale the value by 1000 ConsoleWriteLine("Value in 1,000s: " + "{0:#0,}", num); /* Display positive, negative, and zero values differently */ ConsoleWriteLine("Display positive, negative, " + "and zero values differently"); ConsoleWriteLine("{0:##;(###);000}", num); num = -num; ConsoleWriteLine("{0:###;(###);000}", num); num = 00; ConsoleWriteLine("{0:###;(###);000}", num); // Display a percentage num = 017; ConsoleWriteLine("Display a percentage: {0:#%}", num); } }

Peak Values of Dynamic Response with Damping Ratio = 02 Residual vibration peak (in) 001181 001220 Primary vibration peak (in) 001539 001604 Interior peak contact force (lb) 21342 21638

22:

Placeholder # , % 0 ; E0 E+0 E-0 e0 e+0 e-0

cam motion is easily determined as described earlier Figures 524 to 532 provide a convenient means of interpreting the outcome Figures 530 to 532 show the cam displacements, velocities, and accelerations found from Eq (518) for the two cases being compared It can be seen from Figure 532 that the cam accelerations of these output motions appear to have small discontinuities at t = 1 However, the motion carried on into the dwell period is continuous

Figure 610 shows the graphs of loga x for several different values of a > 1 Figure 611 shows the graphs of a x for several different values of a > 1 You Try It: Sketch the graph of y = 4x and y = log4 x on the same set of axes

Meaning Digit Decimal point Thousands separator Percentage, which is the value being formatted multiplied by 100 Pads with leading and trailing zeros Separates sections that describe the format for positive, negative, and zero values

The output is shown here:

144 1500

Default format: 643542345 Value with two decimal places: 6435423 Add commas: 64,35423 Use scientific notation: 6435e+04 Value in 1,000s: 64 Display positive, negative, and zero values differently 643542 (6435423) 000 Display a percentage: 17%

We next show how to use the logarithm as an aid to differentiation The key idea is that if F is a function taking positive values then we can exploit the formula [ln F ] = EXAMPLE 628

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