cams synthesis

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.


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.


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


displacement diagram of the follower is given, s=s (φ).

Construct the plate cam profile.


Step1: construct the prime circle with

radius r0.

Step2: construct the offset circle with

radius equal to the amount of offset e.




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.


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.


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


displacement diagram of the follower is given, s=s (φ).





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.


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


displacement diagram of the

follower, radius of prime circle,

and follower length are given.





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.


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.


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


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.


• 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.


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)


• 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).


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.


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.


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.


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.


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.”