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0593_C16_fm Page 567 Tuesday, May 7, 2002 7:06 AM

Mechanical Components: Cams 567

Dwell

Dwell

FIGURE 16.14.1
Follower rise connection between
dwell positions.
4. Cycloidal rise function (Eq. (16.13.10)):

−
h θ () = h +  h − h  ( θθ ) −   h − h   sin  2 π θθ  (16.14.6)
−
2
1
1
1
2
1   θ − θ  1  2 π    θ − θ 1  
2
1
2
where the step function δ(θ – θ ) is defined in Eq. (16.10.2) as:
1
1(
−
δθ θ ) =  0  θ < θ 1 (16.14.7)
1
 1 θ > θ 1
Of interest (and concern) with these functions are those values of the acceleration and
jerk at the transition points. Table 16.14.1 provides a listing of these quantities. Observe
in Table 16.14.1 that the cycloidal rise function is the only function with finite values of
the jerk at the transition points. Observe further that, even though the accelerations for
the cycloidal rise functions at the transition points are zero, the acceleration is not zero
for all angles of the function. Indeed, the maximum value of the acceleration for the
cycloidal rise function occurs at cam rotation angles (θ – 7θ )/8 and (3θ – 5θ )/8 with
1
2
2
1
value 2πω (h – h )/(θ – θ ) (see Eq. (16.13.5)). Finally, observe that this maximum
2
2
1
1
2
2
acceleration value slightly exceeds the maximum acceleration values of the sinusoidal
rise function.
TABLE 16.14.1
Acceleration and Jerk at Transition Points for Various Follower Rise Functions
Follower
Rise
Function Acceleration Position Jerk Position
Linear ±∞ θ 1 , θ 2 ±∞ θ 1 , θ 2
Parabolic 2kω 2 θ 1 , θ 2 ±∞ θ 1 , θ 2
(ω = dθ/dt)
Sinusoidal − h   π  2 θ 1 , θ 2 ±∞ θ 1 , θ 2
±  h 2 2 1    θ 2 − θ 1  

Cycloidal 0 θ 1 , θ 2 3 h − h π  2 θ 1 , θ 2
1  

2
ω   θ − θ   θ − θ  
2
 
1
2
1
2