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71867 bugbear@c... (Paul Womack) 1999‑12‑03 maths and planing straight
To those of you who hate analysis, engineering and
especially maths, but prefer honest sweat and shavings,
I say: Move along, nothing to see here.

Hmmm. Gone kinda quiet. Anyway...

I write concerning jointers, lengths and planing straight.
To start with, some breathtakingly poor ASCII art.

        ___----^^^^^^----___
    ..--         |h         --..
   /_____________|______________
  A      L      |X
                |
                |
                |
                |
        
       |r
                |
                |
                |
                |
                |
                |
                |
                |
                 O

This is intended to show an arc of a circle of radius r,
with a chord of length l. The distance from the centre of
the chord to the apex of the arc is h.
There are radii of the arc at the end and centre of the chord.

This diagram (and the analysis reliant on it)
represents planing under the DODGY assumptions that:
* planes are perfectly flat
* do not deflect under load
* have blades in the middle of their soles
* make contact at toe, heel and blade whilst cutting a circular arc
(the diagram is also upside down as far as planing concerned)

On with the show (from bad ASCII diagrams, to ASCII maths - yech)

Examining the right angled triangle O, X, A
we have:
(r - h)^2 + (l/2)^2 = r^2 (pythagoras)
expanding, and simplifying in stages:
r^2 -2hr + h^2 + (l^2)/4 = r^2
-2hr + h^2 + (l^2)/4 = 0
h^2 + (l^2)/4 = 2hr
r = (h^2 + (l^2)/4)/2h
r = h/2 + (l^2)/8h
Assuming h is much less than l (and we're comparing blade setting
with plane length here)
r = (l^2)/8h

This equation is also the one need to work out radius versus centre camber
for scrub and jack irons. Example: for a 2" iron, desirous of a camber of
1/32 we have radius = 2^2/(8(1/32)) = 16"

Breaking from algebra into numbers for a moment:
A 14" Jack plane taking 4 thou shavings gives a radius of 49000 inches. Which
sounds a lot,
until you move to a well tuned jointer (#7 jeff)
22", 2 thou shavings, 242000 inches radius. Wow!

But what do these radii mean? Well, we have a formula for moving between arc
heights,
chord lengths and radii, so let use it. More algebra.
(you might want to transcribe this to paper, and use subscripts - it's
much clearer)
h(board) means h on the board. l(plane) means length of plane etc.

h(board) = (l(board)^2)/8r     (1)
r = (l(plane)^2)/8h(plane)     (2)
substituting (2) into (1)
h(board) = (l(board)^2)/8((l(plane)^2)/8h(plane))

(take my word for it, or use paper, but this boils down to...)
h(board) = (l(board)^2)/(l(plane)^2)) * h(plane)

switching symbols, where P is length of plane, B is length of board, S is set
of plane
and G is gap between the board and a straight line we have the "jointing
equation"

G = S(B^2/P^2)

Which is almost interesting. Feeding in numbers, and our 2 planes:

4 foot board:
jack plane: .041 inch
jointer .01 inch

8 foot board:
jack plane: .164 inch
jointer .038 inch

I actually worked all this out for my own curiousity, which may speak volumes
about me. If it interests, intrigues or amuses fellow galoots, my cup
will be full indeed.

        BugBear.


71873 "Paul Pedersen" <wouldbejoyner@m...> 1999‑12‑03 re: maths and planing straight
Paul Womack calculates...

I must admit I did not do the math myself (my brain is too worn out for that...
)
but I get the feeling that it doesn't take into account what happens at the
edges
of the board.  As the plane's nose passes over the end the plane effectively
gets shorter and shorter until it turns into a chisel plane at the very end and
knocks off a chunk of the board, destroying the nice arc.  What happens
at the beginning of the stroke is even worse in that the blade digs in
immediately
to the depth of the set.

I wonder if the oft-repeated technique of shifting pressure on the plane from
toe to heel is not in fact to get the plane to bend just a bit, changing its
geometry,
thus overcoming the problems at the edges.  I can't see that it would help
otherwise.

Paul Pedersen       my real email address : perrons@c...
Montreal (Quebec)

Get your FREE Email at http://mailcity.lycos.com
Get your PERSONALIZED START PAGE at http://my.lycos.com


71872 "Tom Dugan" <tom_dugan@h...> 1999‑12‑03 re: maths and planing straight
>To those of you who hate analysis, engineering and
>especially maths, but prefer honest sweat and shavings,
>I say: Move along, nothing to see here.
>
>Hmmm. Gone kinda quiet. Anyway...

Not in this crowd, Paul.

[Long treatise on calculating the sagitta beneath a chord
snipped, sadly.]

>I actually worked all this out for my own curiousity, which may speak
>volumes about me. If it interests, intrigues or amuses fellow galoots, my
>cup will be full indeed.
>
>        BugBear.

Enjoy your full cup, Paul. But I can't resist making a few
comments. After all, how often do I get to delurk?

First, the assumption here is that the mouth is at the center
of the plane, and all of mine are rather towards the front of
the sole. This doesn't really significantly affect Paul's
calculations, so let us press onward.

Second, the 'stute galoot will observe while planing that his
boards become crowned rather more than the calculations show.
As the plane approaches the end of the board it begins to
overhang, and the apparent length decreases until the mouth
reaches the end of the board. At this point, the iron is at the
very front of the apparent length and is taking a full depth
shaving. The reverse is happenening at the start of a stroke,
of course. Texts say to keep pressure on the toe at the start
and on the heel at the end of a stroke, which I think is to
counter this tendancy. That's never quite been enough for me,
though, and I always joint the middle until the board eyeballs
straight, then take a full-length pass and call it flat.

Comments?

Tom Dugan
Still Living in Lower Marlboro, but just changed jobs and
am working in the deepest suburbs of Baltimore. Temporarily,
thankfully. I need to cut down from my current 57-mile one-
way commute. That should happen once they move me south another
15 +/- miles.

And I beg pardon if I'm rehashing someone else's reply. It's
tough staying current when receiving digests.

Now back under the porch...

______________________________________________________
Get Your Private, Free Email at http://www.hotmail.com


71885 JPagona@a... 1999‑12‑03 Re: maths and planing straight
In a message dated 12/3/99 8:16:48 AM Eastern Standard Time,
bugbear@c... writes:

<< To those of you who hate analysis, engineering and
 especially maths, but prefer honest sweat and shavings,
 I say: Move along, nothing to see here.

 Hmmm. Gone kinda quiet. Anyway...


  >>
Ahh, but the analysis is faulty for another reason.  The surface of the wodd
ahaed of the iron is coing to be higher than the suface of the wood behind
the iron.

Let's make some unrealistic assumptions.
1.   the sole of the plane is perfectly flat.
2.   the board is perfectly straight.

If the iron is set at .05mm (that's a hypothetical example equal to .002",
everyone but Jeff), and the cut starts with the sole from the toe to the iron
flat on the board, then the cut starts at .05mm deep.  According to Paul's
figuring, by the time the heel is over the board, the plane will be cutting a
concave arc.  However, if the shaving stays at .05mm, and it can't become
greater than that, then there are three points on the board that are at the
toe, the iron and the heel that describe an arc.  This arc is the exact same
radius as the one that Paul calculated, but it is convex instead of concave.
Keep in mind that the board is not actually arced, it is two flat planes
offset by the depth of the cut.  This counters the tendancy to make an arc if
the cut was started in the middle of the board.  Therefore, it is extremely
feasable to plane a board and make it flat.

David

who is getting called away to light candles


71890 Tom Holloway <thh1@c...> 1999‑12‑04 Re: maths and planing straight
At 6:51 PM -0400 12/3/99, David wrote:
>Ahh, but the analysis is faulty for another reason.  The surface of the wood
>ahead of the iron is going to be higher than the suface of the wood behind
>the iron.
        [more snippage]
the board is not actually arced, it is two flat planes
>offset by the depth of the cut.

        Ain't gonna get into the math(s), because I can barely do enough
cipherin' to balance my checkbook.  BUT, I have a, um, practical question
prompted by David's seemingly logical comments on these arcane matters.
I've thought about it when these issues have come up before, as when
discussing the difference between removing wood with a p*w*r jointer vs. a
hand plane.  But I can't let it pass this time.
        The standard bench plane configuration, as I contemplate it, is a
(nominally) flat surface or sort of platform with the cutting edge of the
iron protruding from it. Again speaking "nominally," the plane won't cut
unless the cutting edge protrudes from the (nominally) flat sole. I would
like to focus our attention on those little strips of the sole that flank
the cutting edge in either side, preventing the body of the plane from
falling into a lower *plane* behind the cut than in front of the cut.
        If I'm right about this, then all surfacing is like a delicate
version of a scrub, in which the shaving is a slice of wood scalloped out
from between those little strips of sole on each side of the mouth.
        I started thinking about the process in this way when I once
grabbed my Stanley #10 1/2, a rabbet plane in which the cutting edge of the
iron extends through openings in the cheeks clear to the edge of the sole,
and tried to plane a flat surface with it.  It is ground as straight across
as I can make it, no radius, which is essential for a rabbet plane.  It
grabbed.  I backed off the iron.  It grabbed.  I backed off some more.  It
didn't cut.  A #3 smoother with a straight-across edge will take a
U-section shaving off, but its sole will slide along supported on both
sides of the cutting edge by those little runner-like pieces of sole on
each side of the mouth.
        So maybe a bench plane *needs* to be a sort of a platform or
4-sided frame with a slot through which the cutting iron protrudes.  And
maybe a board, after a plane has made a cut, is still (nominally) flat,
with a very shallow gouge down the path where the plane passed.
        Of course, I don't think much of this arcane speculation has much
to do with dimensioning and smoothing wood in the real world, but since
BugBear decided to get theoretical on us, I thought about my experience
trying to plane a flat surface with a rabbet plane, and decided to join the
fun.
        Am I crazy, or just silly?
                Tom Holloway


71892 "Jeff Gorman" <Jeff@m...> 1999‑12‑04 RE: maths and planing straight
~  -----Original Message-----
~  From: owner-oldtools@l...
~  [mailto:owner-oldtools@l...]On Behalf Of Paul
~  Pedersen
~  Sent: Friday, December 03, 1999 2:05 PM
~  To: oldtools
~  Subject: re: maths and planing straight
~
~
~  I wonder if the oft-repeated technique of shifting pressure
~  on the plane from
~  toe to heel is not in fact to get the plane to bend just a
~  bit, changing its geometry,
~  thus overcoming the problems at the edges.  I can't see that
~  it would help
~  otherwise.

This standard advice is, I think, principally aimed at beginners who
tend to let the heel of the plane drop at the beginning of the stroke
and the toe at the end.

Jeff


71893 "Jeff Gorman" <Jeff@m...> 1999‑12‑04 RE: maths and planing straight

~  -----Original Message-----
~  From: owner-oldtools@l...
~  [mailto:owner-oldtools@l...]On Behalf Of
~  Tom Dugan
~  Sent: Friday, December 03, 1999 2:26 PM
~  To: oldtools@l...
~  Subject: re: maths and planing straight
~
~
~  [Long treatise on calculating the sagitta beneath a chord
~  snipped, sadly.]

Thanks for the word, Tom.

~  Second, the 'stute galoot will observe while planing that his
~  boards become crowned rather more than the calculations show.
~  As the plane approaches the end of the board it begins to
~  overhang, and the apparent length decreases until the mouth
~  reaches the end of the board. At this point, the iron is at the
~  very front of the apparent length and is taking a full depth
~  shaving. The reverse is happenening at the start of a stroke,
~  of course.

This suggests that, assuming that the plane is always trying to plane
a concave curve, that the chord of the arc progressively increases
from the start of the stroke and decreases towards the ends, ie the
radius (proportional to the square of the chord's length) is less at
each end, thus making the curve 'deeper' at each end.

Although the maths relevance is distinctly galoot, there is a
practical value to the information since if one starts planing at the
mid-area of a surface/edge until the plane will not cut any more, one
knows that that section of the the surface is not convex (ie, tha's
taken 't bumps aht). Subsequently, one increases the length of the
stroke until it is working full-length. If the job involves glueing
two surfaces together, concave surfaces can be desirable, of course.

Naturally, if one wants a dead straight/flat surface, one then has to
start snipping the ends a bit.

Jeff


71894 TomPrice@a... 1999‑12‑04 Re: maths and planing straight
Tom H. wrote:

>   The standard bench plane configuration, as I contemplate it, is a
>(nominally) flat surface or sort of platform with the cutting edge of the
>iron protruding from it. Again speaking "nominally," the plane won't cut
>unless the cutting edge protrudes from the (nominally) flat sole. I would
>like to focus our attention on those little strips of the sole that flank
>the cutting edge in either side, preventing the body of the plane from
>falling into a lower *plane* behind the cut than in front of the cut.
>   If I'm right about this, then all surfacing is like a delicate
>version of a scrub, in which the shaving is a slice of wood scalloped out
>from between those little strips of sole on each side of the mouth.

Yes! And a revelation trumpets across the world in these millenial times.
This is why bench planes are NOT like p*wer planers or jointers and why
trying to think of them as such is a fallacy. If we could visually
exaggerate the differences in height on the board surface as we plane we
would see a series of broad shallow grooves flanked by narrow twin
plateaus. If you are smoothing a surface with a bench plane (or a scraper
plane, for that matter) you are ridin' the rails. You are NOT simply
picking up a non-powered version of a tailed jointer, turning it over and
applying it to the wood. Nope.

Even when the wood is narrower than the width of the mouth, we often skew
the plane body somewhat and the 'rails' adjacent to the mouth are brought
into play to some extent.

In fact, I submit that for a bench plane to behave properly, the edges of
the sole flanking the mouth should be included in the 'co-planar' regions
oft-mentioned previously, i.e. the toe, front of the mouth, and rear of
the sole.

>   Am I crazy, or just silly?

Heck, no, you ain't any of the above. You just one very observant Galoot.
****************************
Tom Price (TomPrice@a...)
Will Work For Tools
The Galoot's Progress Old Tools site is at:
http://members.aol.com/tomprice/galootp/galtprog.html


71934 bugbear@c... (Paul Womack) 1999‑12‑06 Re: maths and planing straight
>>
> Tom H. wrote:
>
<
> >   If I'm right about this, then all surfacing is like a delicate
> >version of a scrub, in which the shaving is a slice of wood scalloped out
> >from between those little strips of sole on each side of the mouth.
>
> Yes! And a revelation trumpets across the world in these millenial times.
> This is why bench planes are NOT like p*wer planers or jointers and why
> trying to think of them as such is a fallacy.

And Tom Price agreed...

>
> Even when the wood is narrower than the width of the mouth, we often skew
> the plane body somewhat and the 'rails' adjacent to the mouth are brought
> into play to some extent.
>

I'm not trying to defend the model my maths was based on. It's poor.

But I can't agree with this one either. (so we're still looking
for the right model IMHO).

If the side strips were critical, and performing important "work"
surely they would wear. And being small, surely they would
wear A LOT?

        BugBear.

(who still wants to know how planes work. Next stop: a careful reading
of George Langford's post)


71949 Tom Holloway <thh1@c...> 1999‑12‑06 Re: maths and planing straight
At 6:09 AM -0400 12/6/99, Paul Womack wrote:
        [snip of snips]
>If the side strips were critical, and performing important "work"
>surely they would wear. And being small, surely they would
>wear A LOT?

        Wull, lessee.  Since those "lateral margin" parts of the sole are
(nominally) co-planar with the rest of the sole, wouldn't they have to wear
down only as much as the rest of the sole?  I mean, I don't see how those
spots, in normal use in dimensioning and smoothing, could develop dimples
that would make them recessed from the rest of the sole.
        I've seen some iron bench planes that were used so much, over much
of a century and probably several (human) working lifetimes that the edges
all around were dubbed over from wear, but can't recall ever seeing the
side strips in question worn so much as to form a depression in the sole on
each side of the mouth--nor do I see how it could happen.
                Tom Holloway,
who will never be able to put than much wear on an iron plane, in this
lifetime.



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