This posting is being made as continuation to my own response of
20.20.11 (Ref. 2), to John A. Tuttle's posting of 20.11.03 (Ref. 1),
titled "Force Versus Distance of a Pneumatic."

Pneumatic Forces, Part 1 -- First Case

(Begin.)

There seems existing much mystery surrounding the behavior of
pneumatics (a.k.a. 'bellows'), found as used in very many of our
favorite mechanical musical instruments.

As I think a fair attempt to dispel some of the unknowns attending
pneumatic actions observed within such devices, I am posting by
sections what I've learned from observation and study on-and-off
(mostly that last), over a period now nearing half a century. Just
as with this present one, each will be accompanied by a drawing.

Initially, I have to stress that in no way am I a physicist,
a mathematician or an arithmetician or mechanical engineer even
but, after having pursued that notorious bête noire of those four
 -- 'Perpetual Motion' -- (or, more importantly and to the very point
per se, where energy might be found where it should not, such that
might serve to empower that scientifically reviled Chimeric Fiend,
ever threatening!), one has no choice but to learn some bits, at least,
concerning the physical principles that can aid one to understandings
of 'how things work' actually.

Such learning has proved of great help to myself, what with the coming
to at least some comprehension answering for the behavior of things
generally and rationally, as distinguished say from notionally or
mystically even.

Even though, with that now said, Don Quixote-like and for those very
same reasons, the Wonderful Fruitless Pursuit proceeds forth unabated!
(Well, one just never knows for sure, does one?)

From early on, just post having discovered the beguiling existence of
the Reproducing Piano (and, very many other types of self-playing
musical miracles that so tickle our fancies), my first understanding
of pneumatics was expectantly simplistic -- two rigid hinged boards
separated by some distance covered all round by leakage impervious
cloth material and so -- input suction and voila -- action!

And so it remained to myself for a fair while, until I'd suspected that
there just might be more to the story than that simplistic, initial
view. This, of course, has since proved the very case.

Excepting for that obviousness as I first understood it, the action of
pneumatics seemed not much susceptible to intuitive understanding. This
being because of the fact, as I was to discover, that many interactions
of force upon every single part of pneumatics did occur simultaneously,
and these (excepting for the primary observed motion itself) were not
at all apparent, being invisible utterly -- seemingly almost as
physical mystery afoot!

Much of this that was mysterious, if not all, is to now be dispelled.

As aids to the impartation of what I have learned (or, at least some
meaningful aspects of it here) of the behavior-and-why of pneumatics,
I have created drawings as help in the accomplishment of the trick.
A good deal of time and attention I've put into this work, so as to
make them intuitively comprehensible to any, and with but only little
study as likely required.

Now, to address our No. 1 drawing having to it six parts. (Ref. 3)

As noticed right off, I'm sure, will be the many colored arrows both
small and larger placed just so, here and there, these so as to make
clear where balanced or unbalanced forces are at. These I have borrowed
from the mathematicians and physicists. They are what they term
vectors. These possess two most useful qualities for our present
purpose, namely direction and magnitude, and serve nicely so as to
make clear what otherwise without, would likely not be. For our purpose
I have made dissimilar ones of different sizes rather than lengths as
their usual.

The first five show views of a pneumatic having four equal sized sides
with no hinge, so a side view would appear as just identical to it.
Also, it's maximum displacement I have chosen so as to be the same
as it's width. The reason for these preconditions will become soon
apparent, as in for the lessening of complexity with our first early
considerations.

The 'a' view is of it with no vacuum applied, the 'b' with this.
As to the first, all of the vector arrows indicating for direction
and magnitude are identical in their directions and relative sizes,
and therefore are balanced, with no displacing forces whatever being
present. (We are not here taking into consideration any second-order
constructional/mechanical ones, such as covering folds or hinge binds,
etc.)

With the second, 'b', the case has changed as there is now within
existing within a condition of fewer gaseous molecules, and thus a
tug-of-war exists where force is produced by these greaters vs. the
lessers at all points. This, then, producing forces that act so as
to effect displacements where possible, as Nature surely does abhor
a vacuum. ("Natura abhorret vacuum" - Rabelais)

To these always are found two discrete parts (as here analyzed and
considered statically only), the first being those described by black
vector arrows, and the other as this by ones red. The black ones I have
named for clearest understanding "direct forces" and the others
"leverage forces." Upon any displacement, both of these as contributors
to the total sums of forces, are ever changing in their portions of it.

A careful study of the drawings will make clear and with ease, the how
of this working. As our pneumatic continues to collapse while under
vacuum, the working widths/areas of the direct forces narrow, while
the leverage forces continue to widen and dominate. In the enlarged 'c'
view, it will be seen how the forces redistribute and weaken with ever
increasing closure. As all of these continue their progress towards
full closure, they weaken to the point of nothingness. This sad result
view 'd' shows. View 'e' gives us a top X-ray image of our square
pneumatic and shows how little direct area remains after full folding.

If, say, our example pneumatic I had designed with greater width as
compared to it's full closure and hinged at one end, then some portion
of direct force would have remained at closure and thus of use but, in
this configuration all sides are to meet. This very thing is made clear
by X-ray top view 'f' where a fair amount of direct force remains for
use. From any carefully analyzed pneumatic as graphed, this area can be
subtracted as it is constant -- never diminishing.

In the next posting of the Second Case, we shall examine the effects
produced by stiffening of the leverage force sides. (The Third Case
gets really interesting!!!)

(End.)

Corrections and/or additions to what I have said and present here,
I heartily solicit from any physics-aware authority, professional or
amateur.

Also, if any who are interested here might have questions, please do
not hesitate to state them but, I ask that they best be put publicly
where possible, so that all might benefit from any answers.

That said, any otherwise private communications, of course, I'll be
happy to respond to as well. (I enjoy correspondence and could do it
all day, never tiring of it! Drawing too.)

Jim Miller
Las Vegas, Nevada
akruzam@gmx.com.geentroep [delete ".geentroep" to reply]

Ref. 1 https://www.mmdigest.com/Archives/Digests/202011/2020.11.03.01.html 

Ref. 2 https://www.mmdigest.com/Archives/Digests/202011/2020.11.11.05.html 

Ref. 3 Six Views of First Case Striker-Pneumatic Forces
https://www.mmdigest.com/Attachments/20/11/22/201122_095345_No.2RollingSides6Views.png