Transcript 22.322 Mechanical Design II Spring 2013 Learning with Purpose January 28, 2013 Lecture 3 S V A J Diagrams Ultimately we would like to choose a.

22.322 Mechanical Design II
Spring 2013
Learning with Purpose
January 28, 2013
Lecture 3
S V A J Diagrams
Ultimately we would like to choose a mathematical
expression that will allow the follower to exhibit a desired
motion.
Easiest approach is to “unwrap” the cam from its circular
shape and consider it as a function plotted on Cartesian
axes.
The motion of the cam is analyzed using a S V A J diagram
•
•
•
•
S = displacement
V = velocity (first derivative)
A = acceleration (second derivative)
J = jerk (third derivative)
Learning with Purpose
January 28, 2013
Lecture 3
Consider the specifications for a four-dwell cam that has
eight segments:
These function
characteristics can be easily
investigated with program
DYNACAM
• Generate data and plots
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January 28, 2013
Lecture 3
y=4/5x
parabolic
3
4/5
-2/5
+∞
3/2
zero
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-∞
January 28, 2013
Lecture 3
Consider the following cam timing diagram. There are two
dwells and we would like to design the cam such that there
is good motion in the rise and the return.
Learning with Purpose
January 28, 2013
Lecture 3
Because we’re new at
designing cams, let’s just
try a linear function
between the low and high
dwells:
The cam designer should
be more concerned with
the higher derivatives!
Infinite acceleration requires an infinite
force (F=ma)
Infinite forces are not possible to
achieve, but the dynamic forces will be
very large at the boundaries and will
cause high stresses and rapid wear.
Separation between the cam and
follower may occur.
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January 28, 2013
Lecture 3
Fundamental Law of Cam Design
Any cam designed for operation at other than very low
speeds must be designed such that:
• a) The cam function must be continuous through the first and
second derivatives of displacement across the entire 360o interval
of motion
• b) The jerk must be finite across the entire 360o interval. The
displacement, velocity, and acceleration function must have no
discontinuities in them.
Large or infinite jerks cause noise and vibration. To obey
the Fundamental Law of Cam Design, the displacement
function needs to be at least a fifth order polynomial.
• V=4th order, A=3rd order, J=2nd order and finite
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January 28, 2013
Lecture 3
Simple Harmonic Motion
Sinusoids are continuously differentiable:
On repeated differentiation, sine becomes cosine, which
becomes negative sine, which becomes negative cosine,
etc.
If we apply a simple harmonic motion rise and return to our
cam timing diagram:
continuous
Piecewise
continuous
Discontinuous  leads to
infinite jerk (bad cam)
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January 28, 2013
Lecture 3
Cycloidal Displacement
Better approach is to start with consideration of higher
derivatives (acceleration)
Cycloidal displacement = sinusoidal acceleration
If we apply the cycloidal displacement function to our cam
timing diagram:
Disadvantage: high level of peak acceleration
Cam motion is continuous through acceleration;
jerks are finite = acceptable cam
Learning with Purpose
January 28, 2013
Lecture 3
Square wave (constant acceleration) function best minimizes
peak magnitude of acceleration
However, this function is not continuous  unacceptable
Square wave’s discontinuities can be removed by simply
“knocking the corners off” the square wave function 
trapezoidal acceleration
Disadvantage: discontinuous jerk function
Solution: Modified Trapezoidal Acceleration
Finite jerk
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January 28, 2013
Lecture 3
Learning with Purpose
January 28, 2013
Lecture 3
Modified Sinusoidal Acceleration
Modified trapezoidal function is one of many combined
functions created for cams by piecing together various
functions, while being careful to match the values of the s, v,
and a curves at all the interfaces between the joined
functions.
The modified sinusoidal acceleration function is a
combination of cycloidal displacement (smoothness) and
modified trapezoidal (minimize peak acceleration).
The combination results in lower peak velocity.
Made up of two sinusoids with two different frequencies.
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January 28, 2013