Mathcad - P0843.xmcd - CBU

DESIGN OF MACHINERY SOLUTION MANUAL 8-43-1PROBLEM 8-43Statement:Design a cam to move a follower at a constant velocity of 100 mm/sec for 2 sec then return to itsstarting position with a total cycle time of 3 sec.Given: Constant velocity: v c 100mmsec 1Time duration of cv segment: t cv 2secCycle time: 3secSolution: See Mathcad file P0843.1. The camshaft turns 2rad during the time for one cycle. Thus, its speed is 2 rad2.094 radsec2. Use a two-segment polynomial as demonstrated in Example 8-12. The lift during the first segment and the svajequations for the first segment are:v cNormalized velocity: v cv v cv 47.746mmh cvt cv v c t cvh cv 200.000 mm 1 360deg 1 240 degs 1 v 1 v cva 1 0mmj 1 0mmh cv2. The boundary conditions for the second segment are: 1at = 1: s = h cv , v = v cv a = 0= 360 deg s = 0, v = v cv , a = 0This is a minimum set of 6 BCs. Define the total interval and the constant velocity interval, and the ratio ofconstant velocity interval to the total interval.Total interval: 360degCV interval: 1 240 deg 1A A 0.6673. Use the 6 BCs and equation 8.23 to write 6 equations in s, v, and a similar to those in example 8-9 but with 6 termsin the equation for s (the highest term will be fifth degree).For = 1: s = h cv , v = v cv a = 0h cv = c 0 c 1 A c 2 A 2 c 3 A 3 c 4 A 4 c 5 A 51v cv = c 1 2c 2 A 3 c 3 A 2 4c 4 A 3 5c 5 A 40 = 2c 2 6c 3 A 12c 4 A 2 20 c 5 A 3For = : s = 0, v = v cv , a = 00 = c 0 c 1 c 2 c 3 c 4 c 510 = c 1 2c 2 3c 3 4c 4 5c 50 = 2c 2 6c 3 12c 4 20c 5

DESIGN OF MACHINERY SOLUTION MANUAL 8-43-24. Solve for the unknown polynomial coefficients.10100C 0A10110A 22 A2122A 33A 26 A136A 44A 312A 21412A 55A 420A 31520H h cvv cv00v cv0c 0 c 1 c 2c 3 c 4 c 5C 1Hc 0 1.536 10 5 mm c 1 9.71710 5 mm c 2 2.430 10 6 mmc 3 2.99710 6 mm c 4 1.822 10 6 mm c 5 4.37410 5 mm5. Write the svaj equations for the second segment.2 3 4 5 c 2 c 3 c 4 c 5 s 2 c 0 c 1 v 2 1 c 1 2c 2 a 2 1 2 2 c 2j 2 1 3 6 c 3 3c 2 3 4c 3 4 5c 4 5 6c 3 12c 2 4 20c 3 5 24c 4 60c 2 5 4. To plot the SVAJ curves, first define a range function that has a value of one between the values of a and b andzero elsewhere. if a bR a b1 0S R 0 1 s 1 V R 0 1 v 1 R 1 R 1A R 0 1 a 1 J R 0 1 j 1 R 1 R 1 s 2 v 2 a 2 j 2 6. Plot the displacement, velocity, acceleration, and jerk over the interval 0

DESIGN OF MACHINERY SOLUTION MANUAL 8-43-41000JERK, J0Jerk, inJ( )mm100020000 60 120 180 240 300 360degCam Rotation Angle, deg