Machinery's Handbook, 31st Edition
CAMS AND CAM DESIGN
2367
4. Cycloidal Motion: (Fig. 7 )
360
sin 2 1 360 T π a
1
y h T t :
t
β φ
φ
°
°
or
sin
y h
k
= − ;
E
c
m
D
= −
(4a) } 0 ≤ i ≤ T (4b) (4c)
2
π
β
360
360
v T h
t
φ
°
°
ω
1 = − :
cos
or
1
cos
v h =
a
k
;
E
c
m
D
−
T
β
β
360
a T h 2 2 π =
360
2
t
h β π ω 2
2
φ
°
°
sin
or
sin
a
a
k
c
m
=
T
β
Acceleration
Velocity
B
h
V
A
T
Fig. 7. Cam Displacement, Velocity, and Acceleration Curves for Cycloidal Motion This time-displacement curve has excellent acceleration characteristics; there are no abrupt changes in its associated acceleration curve. The maximum value of the acceleration of the follower for a given rise and time is somewhat higher than that of the simple harmonic mo- tion curve. In spite of this, the cycloidal curve is used often as a basis for designing cams for high-speed machinery because it results in low levels of noise, vibration, and wear. Displacement Diagram Synthesis.— The straight-line graph shown in Fig. 3 has the im- portant advantage of uniform velocity. This is so desirable that many cams based on this graph are used. To avoid impact at the beginning and end of the stroke, a modification is introduced at these points. There are many different types of modifications possible, ranging from a simple circular arc to much more complicated curves. One of the better curves used for this purpose is the parabolic curve given by Equation (2a). As seen from the derived time graphs, this curve causes the follower to begin a stroke with zero velocity but having a finite and constant acceleration. We must accept the necessity of acceleration, but effort should be made to hold it to a minimum. Matching of Constant Velocity and Parabolic Motion Curves: By matching a parabolic cam curve to the beginning and end of a straight-line cam displacement diagram it is possible to reduce the acceleration from infinity to a finite constant value to avoid impact loads. As illustrated in Fig. 8, it can be shown that for any parabola the vertex of which is at O , the tangent to the curve at the point P intersects the line OQ at its midpoint. This means that the tangent at P represents the velocity of the follower at time X 0 as shown in Fig. 8. Since the tangent also represents the velocity of the follower over the constant velocity portion of the stroke, the transition from rest to the maximum velocity is accomplished with smoothness.
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