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Therefore, the area of the acceleration curve for any cam has an area of positive action (+) equal to the area of the negative action (-) This observation could aid the designer in developing the optimum cam acceleration curve shape The follower jerk y or shall have the smallest values Note that with industrial cams y the jerk curve is not frequently of design control since the production tolerance is about 0001 inch The jerk curve is sensitive to much smaller deviations However, in the automotive cam eld the cam operates over a very small distance, requiring higher tolerance of about 00001 inch or better In this case the jerk curve data is pertinent Given the displacement diagram of a pitch curve in Fig 21a, in which the follower has a total rise of 1/2 in in 01 sec of cam rotation and falls in an additional 008 sec, plot the velocity curve and determine the velocity and acceleration after the initial 008 sec of cam rotation by applying the graphical-slope differentiating method

EXAMPLE Solution At the 008 sec point, we draw a tangent to the displacement curve Caution is suggested This is the instantaneous slope of the curve which equals the velocity

This rate of separation is to be evaluated at t = 300s (Fig 6-4)

x f (x1 )

(Continued)

x + f (x2 )

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CAMSHAFT ANGLE (DEGREES) PRESSURE ANGLE (DEGREES) 355085 350206 344975 339399 333483 327236 320664 313774 306573 299067 291264 283171 274797 266150 257239 248074 238664 229022 219158 209086 198818 188370 177757 166994 156098 145087 133980 122795 111552 100272 88975 21430 21696 21960 22222 22482 22739 22992 23241 23486 23727 23963 24193 24418 24636 24848 25054 25253 25444 25628 25804 25972 26132 26283 26426 26560 26685 26801 26907 27005 27093 27172 96072 95524 94822 93966 92960 91805 90503 89058 87472 85749 83893 81907 79797 77566 75219 72762 70200 67538 64783 61939 59014 56013 52943 49812 46624 43388 40111 36799 33459 30100 26728 -169255 -225103 -280464 -335217 -389243 -442427 -494653 -545807 -595780 -644462 -691748 -737536 -781727 -824224 -864937 -903776 -940658 -975503 -1008235 -1038783 -1067081 -1093069 -1116689 -1137891 -1156628 -1172860 -1186552 -1197674 -1206203 -1212119 -1215410

FOLLOWER ANGLE (DEGREES)

x + + f (xk )

These next two problems utilize similar triangles to write the d e f ~ equation for the g problem The first problem, concerning the rate a shadow of something is moving, is in nearly every calculus book The following problem concerning a conical-shaped container is also in nearly every calculus book in one form or another If you know how to use similar triangles to "get started" on a problem you will have mastered yet another category of related rate problems

CAMCO B-12345 (SAMPLE) FOLLOWER RADIUS FOLLOWER VELOC FOLLOWER ACCEL (INCHES) (INCHES/SEC) (INCH/SEC/SEC)

RADIUS OF CURV (INCHES) 17909 17596 17297 17010 16735 16470 16215 15969 15732 15503 15283 15071 14868 14673 14486 14309 14141 13982 13833 13694 13565 13447 13339 13243 13159 13087 13027 12979 12944 12922 12913

Here the symbol k denotes the sum of the expression to its right for each j=1 of the instances j = 1 to j = k The reasoning just presented suggests that the true area A is given by lim R( f, P)

930000 940000 950000 960000 970000 980000 990000 1000000 1010000 1020000 1030000 1040000 1050000 1060000 1070000 1080000 1090000 1100000 1110000 1120000 1130000 1140000 1150000 1160000 1170000 1180000 1190000 1200000 1210000 1220000 1230000

y = f (x)

930000 940000 950000 960000 970000 980000 990000 1000000 1010000 1020000 1030000 1040000 1050000 1060000 1070000 1080000 1090000 1100000 1110000 1120000 1130000 1140000 1150000 1160000 1170000 1180000 1190000 1200000 1210000 1220000 1230000

away from a 12-fi tall penguin way light What is the length of her shadow and how fast is the tip of her shadow moving when she is 40 Et away from the light

(Continued)

y = f (x)

CAMSHAFT ANGLE (DEGREES) PRESSURE ANGLE (DEGREES) 77683 66478 55560 45138 35412 26570 18781 12195 06937 03108 00781 00000 00000 00781 03108 06937 12195 18781 26570 35412 45138 55560 66478 77683 88975 100272 111552 122795 133980 145087 156098 27242 27302 27353 27395 27429 27454 27473 27486 27494 27498 27500 27500 27500 27500 27498 27494 27486 27473 27454 27429 27395 27353 27302 27242 27172 27093 27005 26907 26801 26685 26560 23350 19993 16718 13588 10665 08005 05660 03676 02092 00937 00235 00000 00000 -00235 -00937 -02092 -03676 -05660 -08005 -10665 -13588 -16718 -19993 -23350 -26728 -30100 -33459 -36799 -40111 -43388 -46624 -1215410 -1197674 -1156628 -1093069 -1008235 -903776 -781727 -644462 -494653 -335217 -169255 00000 00000 -169255 -335217 -494653 -644462 -781727 -903776 -1008235 -1093069 -1156628 -1197674 -1215410 -1215410 -1212119 -1206203 -1197674 -1186552 -1172860 -1156628

is an elongated S , coming from summation The dx is an historical artifact, coming partly from traditional methods of developing the integral, and partly

FOLLOWER ANGLE (DEGREES)

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