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where a = phase angle b = clamping factor R = resultant dynamic force F(t) = external driving force Fn = amplitude of nth harmonic of harmonic analysis of F(t) k = system elasticity m = effective mass of cam-follower system from equilibrium position z = de ection of cam-follower system wo = (k)1/2 = natural frequency of system, rad/sec wn = frequency of nth harmonic, rad/sec Examination of the equation shows that resonance occurs at speeds that are multiples of the natural frequency of the system With small b, the value under the square-root sign becomes very small when wo = wn; consequently, the contribution of that member which is in resonance to the complete solution is by far the largest With a given cam-follower system, the magnitude of the response at resonance will depend primarily on the magnitude of the component Fn By performing a harmonic analysis of the forces acting on the system, it is possible to predict the response with a given cam operating at a given speed The principal force acting on the system is the acceleration force Therefore, a harmonic analysis of the acceleration forces produced by a given cam design will give an indication of the performance to be expected In general, any periodic function, provided that it contains no more than a nite number of discontinuities, has a convergent Fourier series that represents it However, the number and extent of the discontinuities have an important bearing on the rapidity of the convergence, ie, the rate at which the harmonics approach zero as n becomes large If a function itself has discontinuities, the coef cients will decrease in general as 1/n, whereas if discontinuities do not exist in the function itself but only in its rst derivative, the coef cients will fall off approximately as 1/n2 Examination of the three acceleration forms considered, namely constant acceleration, simple harmonic, and cycloidal, shows that only the cycloidal is completely continuous It was stated in Mitchell (1950) that the cycloidal curve cam-follower system used has a natural frequency of 1625 cycles per sec At a cam speed of 140 rpm, resonance occurs with the harmonic 9750/140 = 70th harmonic Therefore, we would anticipate that the resonant response of the system to the cycloidal cam pro le would be on the order of 1/70 of the response to the other two pro les at this speed.

Determining the surface area of non-standard shapes is another uniquely calculus problem The technique for finding the surface area of a shape produced by rotating a curve about an axis is similar to finding volumes and additionally uses concepts from length of arc calculations Start with a parabola, y = x 2 , rotated about the y-axis and consider the surface of that parabola up to y = 4 The curve doesn't have to be a parabola A parabola is just convenient to visualize The surface area is viewed as a collection of strips wrapped around the parabola The area of these strips is 2n (radius), the length around, times the width, ds The differential piece of surface for a curve rotated about the y-axis is 2ltxds Refer to Fig 10-7 The total surface area is found by adding up, integrating, all of the 2nxds segments

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EXAMPLE A closed-track radial cam turning once every 4 sec operates a roller follower on a push rod This rod moves a rocking lever and, by means of a connecting link, a sliding table weighing 185 lb Some basic dimensions of the roller follower displacement from dwell to dwell are shown in Table 121 The strength of the connecting members are: under 100 lb of compressive load, the push rod de ects 00012 in, the connecting link 00165 in, and the rocking lever bends 0237 in The masses of these members are small in comparison with the table While the cam is operating, a troublesome chatter develops in the table Find the cause Note: it is recognized that the machine is very old and the cam surface is worn and has serious surface imperfections

9 4 1 0 1 4 9

Example of Harmonic Analysis 10 0080 100 2789 20 0215 110 2949 30 0439 120 3049 40 0730 130 3155 50 1112 140 3410 60 1500 150 3781 70 1898 160 4132 80 2250 170 4422 90 2561 180 4500

We plot these points on a single set of axes (Figure 119) Supposing that the curve we seek to draw is a smooth interpolation of these points (calculus will later show us that this supposition is correct), we find that our curve is as shown in Figure 120 This curve is called a parabola

The overall (equivalent) spring constant for a system in series is 1 1 1 1 = + + + k k1 k2 k3 1 1 1 = + + 100 00012 100 00165 100 0237 k = 392 lb in

1 392 2p 185 386

Fig 10-7

Sketch the graph of the curve {( x, y) : y = x 3 }

Plotting the displacement of the follower in Fig 128, we can nd the velocity and acceleration curves by employing the method of nite differences We nd a strong imposed frequency of acceleration of 41/2 cycles/sec It may be seen that the frequency ratio = 450 cycles sec = 18 1 rev sec 4

Thus, by observation of Fig 128, we see that the chatter is caused by (a) resonance between natural frequency and the strong eighteenth harmonic number and (b) excessive jerk values, especially at the dwell ends of the cam curve A new cam designed with a smoother acceleration curve with smaller maximum jerk values solved the problem by yielding weaker higher harmonics

It is convenient to make a table of values:

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