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The curve y = x 3 , 0 x 3, is rotated about the y-axis What volume does the resulting surface enclose

FIGURE 98 Example of surface failure by pitting and spalling (gear teeth) (Source: Graham, JD, Pitting of Gear Teeth, Handbook of Mechanical Wear, C Lipson, Ed, U Mich Press, pp 138 43, 1961, with permission)

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It is convenient in this problem to treat y as the independent variable and x as the dependent variable So we write the curve as x = y1/3 Then, at position y, the curve is distanced y1/3 from the axis so the disc generated under rotation will have radius y1/3 (Figure 817) Thus, the disc will have area A( y) = [y1/3 ]2 Also, since x ranges from 0 to 3 we see that y ranges from 0 to 27 As a result, the volume enclosed is

.

On your hand calculator raise e to a power, then take the In of that number to return to the original power (number) [ e3 = 20; ln20 = 3 ] As you go through the problems in this chapter keep your calculator handy and practice "punching the numbers"

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Rolling-element fatigue can be initiated by hard inclusions in the material; corrosion; surface stress raisers such as dents, grinding imperfections, or geometric stress concentrations; microspalling; and surface interactions dependent on roughness and lubrication lm thickness Sometimes thin layers of case hardening, surface checks from heat treatment and high sliding velocities, excessive temperature, type of lubricant, or contamination can accelerate the action of fatigue If the surface is stressed in a corrosive environment, a phenomenon called corrosion fatigue or stress corrosion occurs Corrosion on either the cam or roller follower will initially roughen the smooth rolling surface and form minute pits from which a fatigue crack will start due to the local stress concentration The use of a lubricant has another function besides the reduction of friction of the surfaces, which is to prevent corrosion and ensure longer fatigue life 9442 Surface Fatigue Design As mentioned previously cams, rolling-element bearings, and gearing are machine elements that are similar in their performance, and they fail most frequently by surface fatigue Unfortunately statistical wear data have been developed for rolling-element bearings and gearing but not for cam-follower systems The diversity and complexity of cam-follower systems has prevented a complete wear design methodology to evolve Nevertheless, this section presents a simpli ed design procedure for the cam engineer The data presented will be of initial value as a guide in selecting compatible material combinations for the cam and follower Ultimately, the proper materials, wear performance, and lubricant must be con rmed in the eld under actual operating conditions Test data are presented for the comparison of cam and follower fatigue under pure rolling and rolling with some sliding conditions Experience has shown that some sliding may occur, even at low speeds, affecting the wear life of the contacting surfaces Life surface fatigue tests were conducted by Talbourdet (1950), Morrison (1968), and Cram (1956) utilizing radially loaded contacting cylinders with (a) pure rolling and (b) rolling with 9 percent sliding Three-inch diameter mating rollers were used; one roller had a hardened steel surface (60-62 RC) and the other, softer roller was cast iron, steel (of different hardnesses), bronze, and nonmetallic materials The weaker roller material wore out under the cyclical loading which established the life of the materials in combination The stress test data algorithm relates smax to the hertzian compressive stress Eq (911) for dissimilar metals in surface contact For a Poisson s ratio m = 03 and the normal force per cylinder length P = K lb in 1 1 + r1 r2 (913).

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y5/3 5/3

(914)

729 5

which is called the load-stress factor for cylinders in contact Table 93 lists values of K for 100 million stress repetitions for pure rolling and rolling plus 9 percent sliding action One roll is of hardened steel 60-62RC while the other is of various selected materials Values of K have been determined as a function of the number of stress cycles necessary for surface failure The test lubricant was mineral oil 280-320 SSU and 100 F at a surface

TABLE 93 Pure Rolling Rolling & 9% sliding A B K1 s max* A B K1 s max*

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