McCarthyNotes-Cams2

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1 MAE 145: Machine Theory

Cam and Follower Systems

J. Michael McCarthy

July 25, 2009


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Cam and Follower Systems

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.

J. E. Shigley, C. R. Mischke and T. H. Brown, Standard Handbook of Machine Design, McGraw-Hill, 2004


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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 thefollower.

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 vectorsT1 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 camB1 and followerB2are

defined by specifying contact of a point A1 on the profile ofB1 (one degree of freedom) with a point A2 on the

profile ofB2 (the second degree of freedom).


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Mobility

Let c be the number cam joints in a collection ofn rigid bodies andjthe number of hinges and sliders, then the

mobility formula becomes M=3N - 2j - c - 3,whereN=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 havestandard 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: = f(),

RCP: s = f(),

PCP: t = f(s).

Displacement functions consists of three basic segments: the

rise, dwell and the return.

The displacement function also defines the

follower velocity and acceleration:

Assume the angular velocity of the cam is a constant,

then velocity and acceleration of the follower are seen in

the displacement function:


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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,

0 is the starting angle of the rise,

= 1 - 0 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,

1 is the starting angle of the return,

= 2 - 1 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,


= 1 - 0is the angular range of the rise.

Cycloidal Return:

h0 is the radius at the start of the return,

h is the amount of return,

1 is the starting angle of the return,

= 2 - 1is 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,


= 1 - 0is the angular range of the rise.

Accelerating segment:

Decelerating segment:


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Cam Profile for a Knife Edge Follower

The Cam ProfileThe cam profile is a plane curve of the form A =R cos i +R sin j,

where i andj are the unit vectors in the x and y directions of the frameB attached to the cam.

Let Ai be the points of contact between the cam and follow at

different angular positions iof the follower relative to the cam.

The coordinates of the points of contact define the cam:

Ai=R(si,i) cos(si,i) i +R sin(si,i)j.

The functionsR(s, ) and (s, ) that define the cam profile are

derived from the follower geometry and the displacement function

s=f().

For a knife-edge follower these functions are particularly simple.

They are

R(s,) = s and (s, ) = .


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Cam Profile for a Flat-Faced Follower

The point of contact A moves along the face of the follower the distanceL as the cam rotates, which means the

angle (s, ) to the point of contact is not equal to the angle that defines the position of the follower relative tothe cam. Let Ai be the points of contact between the cam and follow at

different angular positions iof the follower relative to the cam.

The coordinates of these points of contact define the cam as

Ai=R(si,i) cos(si,i) i +R sin(si,i)j.

or

Ai=X(si,i) ex+ Y(si,i) ey,where ex = cos i + sin j, and ey = -sin i + cos j.

The lengthL can be computed from the fact that the velocity of the

point of contact in the direction of the cam movement is s =

(df()/d), andits velocity on the cam isL.

Therefore,L =sand the functions defining the cam profile for aflat-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 i

between the direction of the follower movement and the line PiAi perpendicular to thecommon tangent of the cam and roller is thepressure angle.

The coordinates points on the cam profile are given by, Ai=R(si,i) cos(si,i) i +R sin(si,i)j,

or Ai=X(si,i) ex+ Y(si,i) ey,where ex = cos i + sin j, and ey = -sin i + cos j.

The lengthL can be computed by determining the velocity vP =L of the point P1

which is equal to the follower velocity s = (df()/d).

Thus,L=s, and thepressure angle 1can be computed to be tan

1= L/s .

Let a = r cos 1and b = r sin 1thenthe 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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