\magnification=\magstep1
\noindent
Melody and Zak,
\bigskip
\noindent
Your presentation to the class on the project was quite good.  I know
that talking about mathematics in a language that is not your first 
language is hard.  But I thought you got across the main points in
a way that was very understandable for the class.  Because you didn't
have a way to show the animations in the paper, though, the paper
was not quite as good.  Instead of the animations, it would have
been better to try to give a more complete description of what was
happening in your examples, and maybe include graphs of one ``frame''
of the animations to illustrate what you were saying.  The main things
I would have liked to see more of was an analysis showing how the
particular examples fit into the general sort of classification 
from Grashof's theorem, and then some different types of examples
showing how the ``end-effector'' does not always reach all points on 
the coupler (Watt) curve.  In other words, I think you didn't have
enough time to do more experimentation and the limited examples you
had did not really exhaust what the possibilities are.  
\bigskip
\noindent
{\it Specific Comments}
\bigskip
\item{1.}  The terminology should be explained more clearly here.
A side rod is called a {\it crank} if it can rotate through a 
full circle relative to the fixed rod (i.e. around the joint
where it connects to the fixed rod).  Similarly, a side rod
is called a {\it rocker} if it can only rotate through a range
of angles less than a full circle around the joint where it
connects to the fixed rod.  I think I know what you meant in 
the sentence ``If all four links become connected, then there is 
a change point.''  But this is not clear and it calls for further
explanation.  I think you are referring to the configurations 
from which it is possible to move the mechanism in more than
one way.  These are examples of the {\it kinematic singularities}
studied in Section 3 of Chapter 6 of IVA.  (I think you could have used some
of the ideas in the first three sections of Chapter 6 more than 
you did!)  Also, where did the table on page 6 come from?  If you 
were taking this from another source, you should cite it (and put it
in the list of references).  
\item{2.}  Since you were using a different naming system 
for the lengths of the rods in the four-bar mechanism, 
Figure 2 is rather confusing.  It would have been much 
better to make a figure yourself showing the naming you 
were going to use.
\item{3.}  The discussion of the 3RPR parallel manipulator
does not really relate to the rest of what you were doing
(except as an example of a different type of mechanism).  It
would have been OK to leave that out, since it was not 
adding anything to your main points.
\item{4.}  As I indicated above, even though you can't show
the animations in the paper, it would have been really good
to show one or two frames of the animations to show how 
the different solutions of the quadratic equation for $n$
do give motions tracing out the whole coupler curve in the 
cases $a = 1, b = 3, c = 2, d = 4$ and also in the case
$a = 4, b = 3, c = 4, d = 3$.  Both of these are the 
``Case 4'' of the table on page 6.  With the ``changepoints''
you can trace out the whole coupler curve (both components)
with different motions.  In that case, both siderods are
also cranks, so both choices of $n$ from the quadratic equation
give double-crank motions.  It would have been good to 
try more different possibilities guided by the table(!)
For example, 
$a = 1, b = 3, c = 4, d = 3$ is ``Case 2'' in the table. 
If you plot the coupler curve, you'll see that it splits
into two disconnected pieces.  Depending on the choice 
of $n$ from the quadratic equation, you can reach every
point on either part, but there are no ``changepoints''
and both of those motions are rocker-crank motions
(the side of length 4 cannot rotate through a full circle).  
See the attached figures.  It would have also been really
interesting to work out one of the ``double rocker'' motions
from case 3 of the table.  
\bigskip
\noindent
Final Project Presentation:  92 (A-)
\bigskip
\noindent
Final Project Paper: 85 (B)
\bye



