Re: 2 dimensional engineering problem

• To: mathgroup at smc.vnet.net
• Subject: [mg112165] Re: 2 dimensional engineering problem
• From: Chris Pemberton <cjpembo at gmail.com>
• Date: Fri, 3 Sep 2010 06:08:13 -0400 (EDT)

```On 09/02/2010 01:31 AM, Joseph Gwinn wrote:
> In article<i5l9qg\$7ge\$1 at smc.vnet.net>,
>   Dave Francis<suilvenassociates at googlemail.com>  wrote:
>
>
>> Hi all,
>>
>> I have a friend in a manufacturing business who, I think, needs
>> Mathematica to solve a problem. Could anyone here tell me if the
>> following is possible and perhaps if they would be interested in
>> taking on the project for a fee?
>>
>> Here's the problem... It is purely 2 dimensional cam-follower type
>> puzzle.
>>
>> Imagine a cartoon heart shape rotating about a fixed point at its
>> centre (x).  As the heart shape rotates, a small diameter wheel, which
>> is attached to an arm of fixed length pivoted at point y, follows the
>> circumference of the heart (like a cam follower). The distance xy is
>> greater than the greatest radius of the heart shape. Point y lies at
>> 12 o'clock to point x and the wheel touches the heart at about 10
>> o'clock.
>> The arm which is pivoted at point y has a 90 degree bend at that point
>> and this shorter arm caries another wheel at its end (z). This arm
>> extends downwards from point y at about 4 o'clock.
>> My friend needs to define a shape that also rotates about x at the
>> same speed as the heart shape, and is always in contact with the
>> second wheel on the arm at point z.
>> The heart shape, or, of course, any closed loop shape, would be
>> defined by a set of x,y coordinates or polar coors wrt x. The new
>> shape would need to be defined in the same way.
>> NB Please don't be misled by the "heart", the profile is such that the
>> wheel that follows it, only touches the shape at a single point at any
>> time - so pure cam-following.
>>
>> I would love to dive into Mathematica and try this for myself, but
>> time does not allow that I'm afraid.
>>
>> TIA  Dave Francis
>>
> This is a classic problem in the design of cams, the cartoon heart being the cam
> and the little wheel (roller) being the cam follower.
>
> One can certainly use Mathematica for cam design, but unless your friend
> understands the mathematics of cam design, or wants to learn, he may be happier
> with commercial cam-design software.
>
> Joe Gwinn
>
>
Back when I studied machine design, we would derive the equations of and
then code the positions, velocities, accelerations, and the forces of
4-bar linkages, cams, etc.  It was mostly trig; with a few tricks thrown
in to take care of special cases.  The entire class used C or C++ to do
the coding, then imported their data into excel to plot the coupler
curves, velocities, etc.  I was the only student who chose Mathematica
(this was back in 1997).

Mathematica gave me a few advantages over C or C++ coupled with Excel:

1.  I could easily "animate" my homework;  you could see the linkages
moving, rotating.  If there was an error in my derivation of the coupler
equations, my linkages would literally fly apart on the screen.  It was
quite comical to see a linkage come apart, fly upwards like a baton, and
land right back in place the next go around.

2.  If a picture is worth a thousand words, an animation is worth a
million.  My classmates had to hope their static graphs were correct,
but the graphical capabilities of Mathematica allowed me to prove I was
correct.

In my opinion, you'll need to solve whatever problem you've described
using 3-dimensional trig techniques "by hand".  Then, code it in
Mathematica, using the graphic abilities of the software to see if your
solution is correct.  I have no idea if there exist software that will
do what you've described; probably so.

I know I've got my old homework sitting around here somewhere; and I'd
be happy to send it to you. I can send a scan of all the hand-written
derivations and the Mathematica notebooks as well.  It ran in
Mathematica 3.0; so it may need a tweak or two to get going in anything
more up-to-date.  Just let me know.

Chris

```

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