Cam and Follower Systems
J. Michael McCarthy
July 25, 2009
1 MAE 145: Machine Theory
Cam and Follower Systems
J. E. Shigley, C. R. Mischke and T. H. Brown, Standard Handbook of Machine Design, McGraw-Hill, 2004
Animations from K. J. Waldron and G. L. Kinzel, Kinematics, Dynamics and Design of Machinery, John Wiley, 2004
Select each cam-follower picture to run the animation.
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The Cam Joint
A connection between two links that is formed by general surfaces in contact is called a cam-joint. The input
link is called the cam and the output is called the follower.
Contact is defined as (i) a point A1 in the cam and a point A2 on the profiles of the cam and follower
positioned so they have the same coordinates Ac in the world frame W, and such that (ii) the tangent vectors
T1 and T2 of cam and follower profiles are the same Tc at Ac.
The cam joint has two degrees of freedom, because the relative configuration the cam B1 and follower B2 are
defined by specifying contact of a point A1 on the profile of B1 (one degree of freedom) with a point A2 on the
profile of B2 (the second degree of freedom).
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Mobility
Let c be the number cam joints in a collection of n rigid bodies and j the number of hinges and sliders, then the
mobility formula becomes
M=3N - 2j - c - 3,
where N=n+1 includes the ground frame.
This allows us to determine the number hinges and sliders in a mechanism as,
j = 3N - c - 3 - M.
For a 1 degree of freedom system, we have j = (3N - c - 4)/2, which yields,
N=3, j = 2, and c = 1.
Let C denote the cam-joint, then we have the combinations
• RCR--Radial cam and oscillating follower,
• RCP--Radial cam and translating (reciprocating) follower, and
• PCP--Translating cam and translating follower.
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Followers
The shape of the cam and follower combine to define the displacement function. Followers are usually selected to have
standard shapes so it is the cam that requires careful shaping operations.
The primary follower shapes are (i) the knife-edge, (ii) the flat-face, and (iii) the roller follower. The curved shoe is
spherical solid that has the circular cross-section of a roller follower.
The different shapes of these followers requires the cam profile to be different in order to define the same
displacement function.
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Radial Cam and Translating Followers
Displacement function:
s = {dwell from 0 to 90deg,
3in simple harmonic rise from 90 to 180deg,
3in simple harmonic return form 180 to 360deg.}
Select each the cam-follower to run the animation.
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Displacement Functions
The profiles of the cam and follower are shaped to provide a specific displacement function.
• RCR: ψ = f(θ),
• RCP: s = f(θ),
• PCP: t = f(s).
The displacement function also defines the
follower velocity and acceleration:
Assume the angular velocity of the cam is a constant ω,
then velocity and acceleration of the follower are seen in
the displacement function:
Displacement functions consists of three basic segments: the
rise, dwell and the return.
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Simple Harmonic Rise and Return
Simpler Harmonic Rise:
b is the radius of the base circle,
h is the amount of rise,
θ0 is the starting angle of the rise,
β= θ1 - θ0 is the angular range of the rise..
Simpler Harmonic Return:
h0 is the radius at the start of the return,
h is the amount of return,
θ1 is the starting angle of the return,
β= θ2 - θ1 is the angular range of the return..
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Cycloidal Rise and Return
Cycloidal Rise:
b is the radius of the base circle,
h is the amount of rise,
θ0 is the starting angle of the rise,
β= θ1 - θ0 is the angular range of the rise.
Cycloidal Return:
h0 is the radius at the start of the return,
h is the amount of return,
θ1 is the starting angle of the return,
β= θ2 - θ1 is the angular range of the return.
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Parabolic Rise
Parabolic Rise:
b is the radius of the base circle,
h is the amount of rise,
θ0 is the starting angle of the rise,
β= θ1 - θ0 is the angular range of the rise.
Accelerating segment:
Decelerating segment:
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Cam Profile for a Knife Edge Follower
The Cam Profile
The cam profile is a plane curve of the form A = R cosα i + R sinα j,
where i and j are the unit vectors in the x and y directions of the frame B attached to the cam.
Let Ai be the points of contact between the cam and follow at
different angular positions θi of the follower relative to the cam.
The coordinates of the points of contact define the cam:
Ai = R(si,θi) cosα(si,θi) i + R sinα(si,θi) j.
The functions R(s, θ) and α(s, θ) that define the cam profile are
derived from the follower geometry and the displacement function
s=f(θ).
For a knife-edge follower these functions are particularly simple.
They are
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R(s,θ) = s and α(s, θ) = θ.
Cam Profile for a Flat-Faced Follower
The point of contact A moves along the face of the follower the distance L as the cam rotates, which means the
angle α(s, θ) to the point of contact is not equal to the angle θ that defines the position of the follower relative to
the cam.
Let Ai be the points of contact between the cam and follow at
different angular positions θi of the follower relative to the cam.
The coordinates of these points of contact define the cam as
Ai = R(si,θi) cosα(si,θi) i + R sinα(si,θi) j.
or
Ai = X(si,θi) ex + Y(si,θi) ey,
where ex = cos θ i + sin θ j, and ey = -sin θ i + cos θ j.
The length L can be computed from the fact that the velocity of the
point of contact in the direction of the cam movement is s′ω =
(df(θ)/dθ)ω, and its velocity on the cam is L.
Therefore, L = s′ and the functions defining the cam profile for a
flat-faced follower are
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Cam Profile for a Roller Follower
The points of contact Ai with the roller follower define the cam profile. The angle φi
between the direction of the follower movement and the line PiAi perpendicular to the
common tangent of the cam and roller is the pressure angle.
The coordinates points on the cam profile are given by,
Ai = R(si,θi) cosα(si,θi) i + R sinα(si,θi) j,
or
Ai = X(si,θi) ex + Y(si,θi) ey,
where ex = cos θ i + sin θ j, and ey = -sin θ i + cos θ j.
The length L can be computed by determining the velocity vP = Lω of the point P1
which is equal to the follower velocity s′ω = (df(θ)/dθ)ω.
Thus, L=s′, and the pressure angle φ1 can be computed to be tan φ1 = L/s .
Let a = r cos φ1 and b = r sin φ1 then the functions defining the cam profile for a
roller follower are
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Summary
A cam-follower mechanism has a two degree of freedom cam joint that connects the input
and output links. The relative shape of the cam and follower define the displacement
function of the mechanism.
Displacement functions for radial cams are periodic functions consisting of sequences of
dwell, rise and return segments.
The geometry of the follower is usually simplified to a point, line or circle, and combines
with the displacement function to define the cam profile.
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