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= 464 ips in a positve direction Note that the distances ab and bc were arbitrarily chosen for reasonable measurement accuracy In the same manner, other velocity points may be found and the curve plotted as in Fig 21b Here is the acceleration at 008 sec: y = dy dt = distance de distance ef 158 008 = -198 in sec 2 =Again, the distances are chosen arbitrarily Note, the acceleration is negative because the follower is reversing its acceleration in a minus, downward direction Also, mathematically speaking, the slope is to the left (second quadrant), which means a negative value The jerk values may be approximated in the same manner by plotting the acceleration curve and nding the slopes

l ( )n2 9 0 - 2k = h 09) , k = - - 0053 ( As you were following along this problem and "punching the numbers," so you would be very proficient at this logarithm and exponent calculating for the test on this topic, you n09 may have noticed that your calculator displayed a negative number for l ( ) This is correct In the original statement of the problem, dA/dt = -kA so that the calculation of k should produce a positive number The reason for the In of numbers less than 1 being negative has to do with one of those other defintions of the ln and will be taken up shortlv

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In this chapter we present some basic curves of the DRD type and their kinematic relationships In design, the rst step is to sketch a time chart (discussed in Chap 1) Then the basic cam curve may be chosen to satisfy the cam-follower requirements The basic curves of the rise-fall displacement diagram are primarily of two families: the simple polynomial and the trigonometric

f (x j )

The displacement equations of simple polynomial curves y = Cq 2 (212)

f (x j )

where n = any number C = a constant In this polynomial family, we have the following popular curves with integer powers: straight line, n = 1; parabolic or constant acceleration, n = 2; cubic or constant jerk, n = 3 High degree polynomial curves are shown in 3

The curves of trigonometric form are: simple harmonic motion (SHM) or crank curve, which has a cosine acceleration curve; cycloidal, which has a sine acceleration curve; double harmonic; and elliptical (Appendix B presents tabulated values of the simple harmonic motion curve and the cycloidal curve as a direct approach to follower characteristics) In addition to these two families are the miscellaneous, little-used curves: modi ed straight-line circular arc and the circular arc curves (see 14) These are employed primarily as an improvement over the characteristics of the straightline curve and for special design requirements

This last is a Riemann sum for the integral (1/[b a]) the mesh of the partition go to zero, we declare average of f = = 1 b a

Step 5 : The specific equation for this situation is A = 2 0 , 0 0 0 ~ ~ ~ ~ ~ ~ The time for 10% remaining is the time for A to reach 2000 Substitute for A = 2000 and solve for t

A cam rotating at 120 rpm has the positive acceleration part of its rise of 3/8 in in 40 degrees of cam rotation A simple polynomial curve having n = 24 is used Find the velocity and acceleration values at the end of 30 degrees of cam rotation

In a tropical rain forest, the rainfall at time t is given by ( t) = 01 01t + 005t2 inches per hour, 0 t 10 What is the average rainfall for times 0 t 6

2p = 1257 rad sec The total cam 60 angle = 40p /180 = 0698 radian Substituting into Eq (212) gives the displacement

3 24 = C( 0698) 8

We need to only average the function : 1 6 0

Solving yields C = 0888 Therefore, the basic equation is y = 0888q 24 The velocity by differentiating Eq (212) is y = Cw nq n -1 ips Thus, the velocity after 30 degrees of rotation is p y = 0888(1257)(24) 30 180 The acceleration, differentiating again is y = Cw 2 n( n - 1)q n - 2 The acceleration after 30 degrees of rotation is p 2 y = 0888(1257) (24)(14) 30 180

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