Page 414 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 402

402 CAM DESIGN HANDBOOK

the stiffness of the spring, the calculations are repeated to define a curve called a response
spectrum. We can calculate these curves over a range of damping ratios to obtain a set of
curves, each applicable for a different damping ratio. The normalized displacement, veloc-
ity, and acceleration DRSs are defined, respectively, as
x
S = m
ND
y
m
w x
S NV = nm
y ˙ m
w 2 x
S = n m . (13.1)
NA
y ˙˙
m
Nomenclature for these equations and the figures:
w n = cam-follower system natural frequency
x m = maximum displacement of output
y m = cam maximum displacement
.
y m = cam maximum velocity
y ¨ m = cam maximum acceleration
T 1/T N = ratio of pulse width to system natural period
Figure 13.2 shows the acceleration DRS of a modified trapezoidal cam with 5 percent
critical damping. Curves with other damping values can be added if so desired. The curves
peak at several values of w n . From the spectra a designer can ascertain how systems of
different periods relate to specific inputs. Also, it is found that damping has a greater effect
on the residual vibration than on the primary response. In establishing the acceptable
damping factor it is usually conservative to use a value near the low end of the choices.

13.2.1 Example
For illustration, consider a cam-follower system used in a high-speed automatic machine
modeled with a single DOF, Chen (1982). The follower is actuated by a dwell-rise-dwell
2
cam with a modified trapezoidal profile. The peak input acceleration is 500m/sec , and
the duration of the excitation is 0.015sec. If the follower linkage of the system is such that
it has a natural frequency of 100Hz, what will be the peak acceleration response of the
follower during the lift stroke and during the dwell period? Let us assume the damping
factor is 0.05.

13.2.2 Solution

With reference to Fig. 13.2 with 5 percent damping and at time ratio T 1 /T N = 0.015
¥ 100 = 1.5, the primary acceleration amplification is 2.98, and the residual accelera-
tion amplication is 1.88. Therefore, the primary acceleration response will be 2.98 times
2
the input or 1490m/sec , and the residual acceleration response will be 1.88 times the
2
input or 940m/sec . If the effective mass of the follower is 40N, the corresponding iner-
tial load of the follower is 6075N due to primary vibration and 3833N due to residual
vibration.
Next, a quantitative comparison (Chen, 1981) of the dynamic characteristics of various
types of dwell-rise-dwell cam profiles will be shown. Based on a single-DOF model, a