Post on 12-Sep-2014
Design of Cam ProfileDesign of Cam Profile
Jiting Li, Mileta M. Tomovic
School of TechnologyMechanical Engineering Technology
Outline
�Task
�Basic Principle
� Graphical Layout of Cam Profiles
� Analytical Design of Cam Profiles
� Simulation
Task
The task is to determine the exact shape of a cam surface required to deliver a specified follower motion.
We assume here that the required motion has been completely determined------ graphically as well as
analytically.
We will only address the case of plate cams.
Basic Principle
In constructing the cam profile, we employ the principle of kinematicinversion, imagining the cam to be stationary and allowing the follower to rotate opposite to the direction of cam rotation.
Taking the cams with knife-edge follower for example, the locus generated by the trace point as the follower moves relative to the cam is identical to cam surface. By this way cam surface can be figured out.
inversion
Graphical Layout of Cam Profiles�For the case of reciprocating knife-edge follower
As shown in left figure, the
displacement diagram of the follower is given, s=s (φ). Construct the plate
cam profile.
Graphical Layout of Cam Profiles�For the case of reciprocating knife-edge follower
Step1: divide the displacement-diagram
abscissa into a number of segments.
Step2: divide the prime circle into
corresponding segments.
Step3: transfer distances, by means of
dividers, from the displacement diagram
directly onto the cam layout to locate the
corresponding positions of the trace
point.
Graphical Layout of Cam Profiles�For the case of reciprocating knife-edge follower
Step4: draw a smooth curve through
these points. The curve is just the
required cam profile.
Graphical Layout of Cam Profiles�For the case of reciprocating offset roller follower
As shown in left figure, the
displacement diagram of the follower is given, s=s (φ).
Construct the plate cam profile.
Graphical Layout of Cam Profiles�For the case of reciprocating offset roller follower
Step1: construct the prime circle with
radius r0.
Step2: construct the offset circle with
radius equal to the amount of offset e.
Graphical Layout of Cam Profiles�For the case of reciprocating offset roller follower
Step3: divide the displacement-diagram
abscissa into a number of segments.
Step4: divide the offset circle into
corresponding segments and assign
station numbers to the boundaries of
these segments.
Step5: construct lines tangent to the
offset circle from these station, dividing
the prime circle into corresponding
segments.
Graphical Layout of Cam Profiles�For the case of reciprocating offset roller follower
Step6: transfer distances, by means of
dividers, from the displacement diagram
directly onto the cam layout to locate the
corresponding positions of the trace
point, always measuring outward from
the prime circle.
Graphical Layout of Cam Profiles�For the case of reciprocating offset roller follower
Step7: draw a smooth curve
through these points. The curve
is just the required cam profile.
Step8: draw the roller in its proper
position at each station and then
construct the cam profile as a smooth
curve tangent to all these roller positions.
Graphical Layout of Cam Profiles�For the case of reciprocating flat-face follower
As shown in left figure, the
displacement diagram of the follower is given, s=s (φ).
Construct the plate cam profile.
Graphical Layout of Cam Profiles�For the case of reciprocating flat-face follower
Step1: divide the displacement-diagram
abscissa into a number of segments.
Step2: divide the prime circle into
corresponding segments.
Step3: transfer distances from the
displacement diagram directly onto the
cam layout.
Step4: construct a line representing the
flat face of the follower in each position.
Graphical Layout of Cam Profiles�For the case of reciprocating flat-face follower
Step5: construct a smooth curve tangent
to all the follower positions. This curve is
the required cam profile.
Graphical Layout of Cam Profiles�For the case of oscillating follower
As shown in left figure, the
displacement diagram of the
follower, radius of prime circle,
and follower length are given.
Construct the plate cam profile.
Graphical Layout of Cam Profiles�For the case of reciprocating flat-face follower
Step1: divide the displacement-diagram
abscissa into a number of segments.
Step2: draw a circle about camshaft
center O with radius OA0.
Step3: divide the circle and give the
station numbers to correspond to the
displacement diagram.
Graphical Layout of Cam Profiles�For the case of reciprocating flat-face follower
NOTE: in the case of an oscillating
follower, the ordinate values of the
displacement diagram represent angular
movements of the follower.
Step5: calculate the angular
displacement at each station traveled by
the follower.
Step6: measure outward along the arc
from the prime circle to locate the trace
point at each station.
Step4: draw arcs about each of these
centers, all with equal radii
corresponding to the length of follower.
Graphical Layout of Cam Profiles�For the case of reciprocating flat-face follower
Step7: construct a smooth curve
through these points. The curve is
just the required cam profile.
Analytical Design of Cam Profiles�For the case of reciprocating offset roller follower
As shown in left figure, the
displacement diagram of the
follower is given, s=s (φ). The
offset distance e, radius of
prime circle r0, and radius of
roller rr are also known.
Formulate the equation of plate
cam profile.
Analytical Design of Cam Profiles�For the case of reciprocating offset roller follower
• Step1: Equation of pitch curve
1) draw prime circle, offset circle, and
the initial position of the follower.
2) define the Cartesian coordinate
system O-xy.
3) rotate the follower backward an
arbitrary angle φ around the camshaft
center O.
Analytical Design of Cam Profiles�For the case of reciprocating offset roller follower
4) determine the coordinates (x,y) of
trace point B.
0
0
( )sin cos
( )cos sin
x s s e
y s s e
ϕ ϕ
ϕ ϕ
= + +
= + −
where2 2
0 0s r e= −
s is the displacement of the follower
when cam rotates angle φ.
(1)
Analytical Design of Cam Profiles�For the case of reciprocating offset roller follower
• Step2: Equation of cam profile
We know that the point B’ on the cam
profile corresponding to the trace
point B must lie on its normal n-n.
The slope at point B is
dxdx d
tgdydy
d
ϕβ
ϕ
= =− −
where
dx/dφ and dy/dφ can be
calculated from Eq. (1).
Analytical Design of Cam Profiles�For the case of reciprocating offset roller follower
Therefore the coordinates of point B’ is
' cos
' sin
r
r
x x r
y y r
β
β
=
=
m
m
Analytical Design of Cam Profiles�For the case of reciprocating flat-face follower
The method is similar with that
of design the cam profile with
roller follower.
Analytical Design of Cam Profiles�For the case of reciprocating flat-face follower
Step1: draw prime circle and the
initial position of the follower.
Step2: define the Cartesian
coordinate system O-xy.
Step3: rotate the follower backward
an arbitrary angle φ around the
camshaft center O.
Analytical Design of Cam Profiles�For the case of reciprocating flat-face follower
0
0
BPsin OP cos ( )sin cos
BP cos OPsin ( ) cos sin
dsx r s
d
dsy r s
d
ϕ ϕ ϕ ϕϕ
ϕ ϕ ϕ ϕϕ
= + = + +
= − = + −
Step4: determine the coordinates (x,y)
of point B.
Analytical Design of Cam Profiles�For the case of oscillating roller follower
The method is similar with that
of design the cam profile with
reciprocating roller follower.
Analytical Design of Cam Profiles�For the case of oscillating roller follower
1) draw prime circle and the initial
position of the follower.
2) define the Cartesian coordinate
system O-xy.
3) rotate A0 backward an arbitrary
angle φ around the camshaft center O.
• Step1: Equation of pitch curve
4) Locate the follower according to its
angular displacement.
Analytical Design of Cam Profiles�For the case of oscillating roller follower
5) determine the coordinates (x,y) of
trace point B.
0
0
OAsin ABsin( )
OA cos ABcos( )
x
y
ϕ ϕ ψ ψ
ϕ ϕ ψ ψ
= − + +
= − + +
ie.
0
0
sin sin( )
cos cos( )
x a l
y a l
ϕ ϕ ψ ψ
ϕ ϕ ψ ψ
= − + +
= − + +
• Step2: Equation of cam profile
The method is same as that of design
the cam profile with reciprocating roller
follower.
Simulation
The synthesis results can be validated by simulation. Here is an example. The simulation is done with software ADAMS/VIEW.
Example: design a plate cam profile, as shown in below.
Knowing: the cam rotates with constant angular
velocity in clockwise. The radius of prime circle
r0=30mm. The knife-edge follower rises with
uniform motion, and the lift is 50mm during which the cam rotates 180°. Then the follower dwells
during which the cam rotates 60°. With cam
rotating 120 ° to complete the work cycle, the
follower returns to its initial position with parabolic
motion.
Simulation
The design result is shown as following table which gives the coordinates of points on the
cam profile.
-76.0413.4170
-69.9525.46160
-62.0635.83150
-52.7744.28140
-42.4950.64130
-31.6654.84120
-20.7156.90110
-10.0356.90100
0.0055.0090
9.0651.4280
16.9146.4670
23.3340.4160
28.2133.6250
31.4926.4240
33.1919.1630
33.4112.1620
32.275.6910
30.000.000
y (mm)x (mm)Cam rotating angle (°°°°)
30.22-5.33350
30.80-11.21340
31.39-18.12330
31.49-26.42320
30.44-36.28310
27.50-47.63300
21.42-58.86290
11.96-67.84280
0.00-73.75270
-13.40-76.04260
-27.12-74.52250
-40.00-69.28240
-51.42-61.28230
-61.28-51.42220
-69.28-40.00210
-75.17-27.36200
-78.78-13.89190
-80.000.00180
y (mm)x (mm)Cam rotating angle(°°°°)
Simulation
Virtual Prototype
Follower displacement:
Solid red line ---- actual displacement
Dash blue line---- given displacement
Simulation result
Simulation
Simulation shows that the error between the actual follower displacement and
given follower displacement varies, but the maximum absolute error is 0.3747mm.
The error is brought by step length of programming and simulation and is
acceptable. Therefore the synthesis result is proved to be correct.
Reference
1. Wu Ruixiang et al.,Theory of Machines and Mechanisms. BeihangUniversity, 2005.
2. Joseph Edward Shigley and John Joseph Uicker, Jr., Theory of Machines and Mechanisms, second edition. McGraw-Hill, Inc., 1995.
Acknowledgments
The author wishes to acknowledge the support from the Society for Manufacturing Engineers - Education Foundation,
SME-EF Grant #5004 for “Curriculum Modules in Product
Lifecycle Management.”