Machinery's Handbook, 31st Edition
CAMS AND CAM DESIGN
2379
Based upon polar coordinates, the radius of curvature is:
2 2 3
d m c m dr 2 φ − E
r
2
;
c
+
(10)
ρ
=
dr
r d d r 2
r
2 2 +
c
m
d
2
φ
φ
*Positive values (+) indicate convex curve; negative values ( − ), concave. In Equation (10), r = ( R min + y ), where R min is the smallest radius to the pitch curve (see Fig. 12 ) and y is the displacement of the follower from its lowest position given in terms of φ , the angle of cam rotation. The following formulas for r , dr / d φ , and d 2 r / d φ 2 may be substituted into Equation (10) to calculate the radius of curvature at any point of the cam pitch curve; however, to determine the possibility of undercutting of the convex portion of the cam, it is the minimum radius of curvature on the convex portion, ρ min , that is needed. The minimum radius of curvature occurs, generally, at the point of maximum negative acceleration. Parabolic motion: (11a) r R h h 2 1 min 2 β φ = + − − c m
h 720 1 °
dr
β φ
2 # # β
− c m
=
(11b)
d
φ β
φ πβ
h 4 180 2 °
d d r 2 φ
− ^ h
=
(11c) These equations are for the deceleration portion of the curve as explained in the footnote to Table 1. The minimum radius of curvature can occur at either φ = β /2 or at φ = β , depending on the magnitudes of h , R min , and β . Therefore, to determine which is the case, make two calcula tions using Formula (10), one for φ = β/ 2, and the other for φ = β . Simple harmonic motion: (12a) 2 2 2 π β cos r R h 2 1 min ; 180 ° β φ = + − c m E
180
2 180
dr
h
φ
°
°
sin
c
m
=
0 ≤ φ ≤ β
(12b)
d
φ
β
β
180
2 180
2 2 ° h
h
d d r 2
= ^
°
φ
cos
c
m
(12c) The minimum radius of curvature can occur at either φ = β /2 or at φ = β , depending on the magnitudes of h , R min , and β . Therefore, to determine which is the case, make two calcula tions using Formula (10), one for φ = β/ 2, and the other for φ = β . Cycloidal motion: (13a) 2 φ β β sin r R h 1 360 min ° π φ = + − c m ; E
β φ
2
β
360
h 180 1 °
dr
°
φ
cos
;
E
c
m
=
−
(13b)
0 ≤ φ ≤ β
d
φ πβ
β
360
h 2 180 2 2 ° πβ h
d d r 2 φ
= ^
°
φ
sin
c
m
(13c)
2
β
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