Making Multiples of Small Parts

For the mini chest of drawers that I recently completed, I struggled to decide what type of drawer handles to use.  Eventually I chose small wood handles that angled on each side and had a tenon to fit into a mortise in the drawer face.  After making a couple of one-off prototypes to figure out sizing, I decided on a shape I liked.

The prototypes - one sticks out 3/8", the other 1/2"

I needed four of these handles.  Sometimes when making small parts like this, it makes more sense to make them all from one stick of wood, then cut them apart.  You can get greater consistency between parts, and it's far safer and easier to work with a longer stick of wood.

A 5/16" thick, 1" wide stick laid out for 6 handles.
Always good to make extras just in case.

Routed out dadoes which will be tenons later

Then removed the tenon sides

I used this "stop" to chisel a bevel on the wide sides (top and bottom of handle)

You can see the beveled shape with the piece on edge

Then I used the stop again to bevel the edges (sides) of each handle

The tenon areas were made longer than needed so that I could cut the tenons off at the length I wanted (1/4") and still have some cleanup to do on a shooting board to get the next handle ready.

Cutting off a handle with 1/4 long tenon

That leaves the next handle with extra material ...

... that I remove safely on the shooting board

The tenons were just over 3/16" thick and they fit nicely into mortises chopped into the drawer fronts.

One handle fitted

The moral of the story is this.  When making multiples of small parts, it makes sense to lay them out on a stick much larger than needed for a single piece.  It's easier to lay out this way, safer to do the cutting, and the repeatability is probably better, too.

Spreadsheet for Determining Radius of a Curve

This post will be considered by some as an intellectual exercise only.  Maybe it's for math geeks, like myself.  There are simpler methods to do what I write about.  I'm all for simplicity, but I like the math.

When I want to put a curve on the underside of a chair rail or a table apron, a simple method is to place a clamp at either end of the intended curvature and bend a stick (or ruler) to the desired "bulge" of the curve.  While this method will give attractive results, the resulting curve will not be exactly circular.  The stick bends more at it's center than at its ends.  And it may not bend equally both sides of center.

Marked lines 1" from ends, and placed clamps near those marks

Centerline marked and 1 inch "bulge" marked from lower edge

Bend a stick to the "bulge" mark, and draw the curve

Here's the resulting curve

There are times when I want to end up with a curve that is part of a circle.  And when this is the case, I'd like to know the radius of the circle that will give the desired curve so that I can lay it out on the workpiece.

Here's an example, same as in the above pics.  Suppose I have a 20" x 2 1/2" rail and I want to put a circular arc on the underside.  I want the arc to start 1" from each end of the rail and I want it to extend up into the rail's front face by 1".  So that's an arc 18" wide with a 1" bulge.

Using a little math, I can calculate that the radius of the circle that gives the appropriate arc is 41".  I can then use a stick (or piece of string) of that length and a pencil to lay out the curve.  I'll write more about the math below.

This stick will allow arcs up to approx. 48" radius

One end has a shallow slot to run a pencil in

You can put a screw anywhere on the stick for whatever radius you need.
The screw tip exits the bottom side, and it is used as a pivot point for drawing the arc.

Here, I'm using the radius jig to mark the arc on the workpiece.
Note that the screw has to be in line with the centerline of the workpiece.

The two methods give very similar results, so it really doesn't matter which you use.

You can see the slight difference in the two methods

But if you're doing a smaller piece, say an arc only 6" wide, then that bendy stick won't bend in that tight a space.  That's when it would be easier to use a radius stick.  For those smaller pieces, I can use my homemade trammel points to create the arc.  I'll show that below.

Another place where drawing a circular arc has come up is when laying out a camber on a scrub plane iron.  I recently made an iron that was 1 1/2" wide and I wanted a 1/16" bulge at the cutting edge.  For this iron, I simply filed a curve to get it close.  But if I wanted to be more precise, I could figure out what the radius of that curvature is and make a template.  Turns out it is about 4 1/2".

Cardboard template, 1 1/2" wide with centerline drawn

Homemade trammel points

Setting the distance to 4 1/2"

Drawing the appropriate arc on the template

Measuring the resulting bulge - it's about 1/16"

Comparing the template to the plane iron

OK, now here's the math part.  It has to do with right triangles and the Pythagorean theorem.  I'll start with the example of a table apron, which I want to put an arc on the bottom edge, starting some distance in from each end and rising up a certain amount.  In the picture below, points A and B are the ends of the arc, and C is a point centered between them.  The width of the arc I'll call "w", so the distance from C to B is 1/2 w.  The bulge of the arc is the distance from the lower edge of the board to the highest point on the arc.  I call it "b".

The workpiece with arc drawn.  The arc has width "w" and height "b".

Now I'm going to zoom out so you can see the rest of the picture.  The arc drawn on the workpiece is part of a large circle, whose center is at point O.  The radius of the circle, given by distance R, is the distance from point O to any point on the circle.  I'm using point B here.

Zoomed out pic shows the complete circle with radius drawn

Another radius is from the center, O, to the point at the top center of the arc, drawn straight up from the center.  That radius is made up of two distances, the distance from O to C, and the distance from C to the top of the arc.  The latter of those I've already called "b", the bulge.  So the distance from O to C is a radius minus the bulge, or R-b.  Triangle OCB is a right triangle, so from Pythagoras, the sum of the squares of the two legs (legs are the sides of the triangle making the right angle) equals the square of the hypotenuse.

R^2 = (R-b)^2 + (w/2)^2       (the symbol ^2 means squared)

R^2 = R^2 - 2Rb + b^2 + (w^2)/4

And simplifying, we get:

2Rb = b^2 + (w^2)/4, and then

R = (b^2 + (w^2)/4) / (2b)

So to draw an arc with a certain width and bulge, you can calculate the radius of the circle that describes the arc and use a stick to draw the arc.  I've put this formula into an Excel spreadsheet.  If you want to do the same, type the following in an Excel sheet:

=(C4^2 + (C3^2)/4)/(2*C4)

In this formula, C3 is the location in the spreadsheet where I entered the arc width (not half width, the whole width).  C4 is the location in the spreadsheet where I entered the bulge.

Here's a picture of what my Excel sheet looks like (replace the word "Iron" with "Arc")

The example shown above has an arc width of 18" and a bulge of 1".  The resulting radius is 41".

Here are the details for the other example of this - shaping a plane iron with camber to make a scrub plane.  It works exactly the same way.

The plane iron drawn in gray, with markings like the earlier example.

Here's the upper end shown closer up

Using the spreadsheet for an iron that is 1 1/2" wide and giving a 1/16" bulge, I get a radius of 4 1/2".

The cambered iron example

For anyone who is actually interested in this stuff, if you can't figure out how to get the formula I wrote above (shaded in yellow) to work in your own Excel spreadsheet, contact me using the "contact me" gadget somewhere on this blog page.  Specify that you want the radius spreadsheet, because I've got another spreadsheet for a different application and don't want to send the wrong one.

Shop Tip: Vise Chop Markings

A few weeks ago, I replaced the worn out plywood "chops" on my front vise with hardwood chops lined with cork.  I find it helpful to mark a couple of lines on the top edges of the chops.  The lines mark the outside edges of the guide bars that are a few inches below.

Pointing to one of the marked lines

Closer view

How they're located: blade of the square is touching the guide bar

This way, when I need to clamp a workpiece vertically in the vise, I can butt the piece against the guide bar and align the piece with the marks on the chop.  This ensures the workpiece is (very nearly) perpendicular to the benchtop.

It's perpendicular to the bench, but not exactly vertical due to
the garage floor sloping at about a 1-2° angle

Here's another one.  On the cork that lines the vise chops I've marked a series of lines at 30° to the horizontal top of the vise.  For a tool that is challenging to sharpen, this can help.  When I line up the tool with the marked lines, the tool's bevel will be approximately horizontal, making it easier to hone the bevel.

Marking 30° lines on the rear chop

Clamped an incannel gouge along one of those lines

This makes sharpening the incannel gouge less difficult

I've found these to be helpful time-savers; hopefully you will too.

Calculating Radius of a Cambered Iron

So you want to add a camber to a plane iron and don't happen to have a round object with the curvature you're looking for.  In this post, I'll use geometry to figure out the radius so you can draw the arc with a compass.  I know, for some people I'm overcomplicating this, but for this analytically minded woodworker, it works.

This came up for me last week when I did some maintenance on a wooden jack plane.  After sharpening many, many times, the curvature of the iron wasn't as it should have been, so I set it right.  I needed to figure out the radius of curvature that would give me a 1/16" bulge on an iron that was 2 3/16" wide.  By the "bulge", I mean the amount the camber extends further than the corners of the iron.

For this blog post, I'll simplify the math by using a 2" iron width.

Artist's rendition of a 2" wide plane iron with 1/16" bulge

Closeup of the business end

I've labeled two points on this picture: Y is the right corner of the iron and Z is the point at the center of the iron directly over from Y.  The "bulge" is 1/16", the amount past Z that the iron extends.

In the next pic, I've drawn a circle whose curvature matches the camber of the iron.  The point labelled X is the center of the circle.  Also drawn is a line from X to Y, and since Y is on the circle, segment XY is a radius of the circle.

Circle drawn that matches curvature of the camber

Zooming in again, we see a triangle formed by XYZ.

Triangle XYZ

We have enough information now to calculate the radius of the circle, and from that, we can draw a template of the camber to be transferred to the iron to guide grinding and sharpening.

Triangle XYZ is a right triangle.  The Pythagorean theorem tells us that YZ^2 + ZX^2 = XY^2.  (Read "^2" as "squared".)

XY is the radius of the circle: I'll call it R.

YZ is half the width of the iron, in this example 1".

ZX is slightly less than a radius, by 1/16", so ZX = R-(1/16).

So we get:   1^2 + (R - 1/16)^2 = R^2

Simplifying: 1 + R^2 - (1/16) R - (1/16) R + (1/16)^2 = R^2

Simplifying: 1 -(1/8) R + (1/256) = 0

Simplifying: 1+(1/256) = (1/8) R

And finally: R = 8 * (1 + 1/256) = 8  1/32"

In the case of my jack plane, the iron is 2 3/16" wide and the calculation came out to give a radius R =  9.6" ( I rounded to 9 5/8").

If one wanted to extend this to any bulge (call it B) and any width iron (call it W), then the formula comes down to:

R = ((W/2)^2 + B^2)/2B

I know this is not for everybody.  In fact, it's probably a lot easier to simply draw an arc using a nail, a string and a pencil, changing the string length until you get the right bulge on the arc.  But I like the analytical approach.  Math has served me so well over the years.

Don't Panic!

Yesterday (11-Jan-22), I laid out and cut four mortises for a stool project I'm working on.  This morning, the moment I woke up, I realized I had done something wrong.  Funny how that can happen - I wasn't dreaming about it or even thinking about it.  All of the sudden I realized the mistake.  I'll admit that I was in a bit of a just-woke-up fog, but when my wife asked me if the error was recoverable, I said "no", thinking I'd have to remake the parts.

But I went for a long walk that morning and let it slosh around for a while in my head.  I wasn't really thinking about it during the walk.  But at some point I came up with a solution.

The mortise was supposed to be angled, like the sliding bevel,
but I made the end walls 90° like the combo square

The solution was to angle the end walls, going past my mortise end layout line
(note how the mortise goes beyond the right-most layout line)

On the opposite side, I want past the layout line on the other end of the mortise

The outer layout lines in the above pics show the extents of the rail that is to fit into the mortise.  If I kept to those lines, the end shoulders would have to be different. Fortunately I hadn't cut the tenons yet, and I still wanted the tenons to have equal shoulders.  The solution was to cut the tenon with equal edge shoulders (a little smaller than originally planned).  This resulted in the rail being about 1/16" higher on the part shown above than where I wanted it.  I just had to make sure to adjust the mortise and tenon on the other side the same way.

And here's the moral of the story.  We all make mistakes in our work.  Don't panic!  Step back, think about the problem, evaluate options.  A solution just might present itself.  It might force you to make small changes in your design, but ask yourself if that really matters.  It might not.

And by the way, this project has a lot of angles and I made another boo boo in these same mortises.  But I thought about it and came up with a solution that I believe won't affect the function of the joint.

Stack The Odds In Your Favor

On the desk-side table that I just built, I added a bead detail to the lower edges of the aprons and shelf rails.

Bead detail on a shelf rail

This one came out great, but it wasn't just dumb luck.  I hadn't put a bead on anything for a while, so I took the time to make sure it would come out right.

First, I touched up the plane iron with a strop.

The strop for a 5/16" beading iron is leather with compound on it, wrapped around a dowel

Then I practiced a few beads with some scrap.  This not only helped me get more comfortable with the plane, it also allowed me to dial in the iron's depth setting before using it on a project part.  Further, I got to see how it would perform with grain running in different directions.

In the pic below, you can see the wood grain runs away from the bead as the plane is pushed forward.  That's a sure-fire way to rag out the far rim of the quirk.

A practice bead on scrap

So I used the test bead to set a marking gauge to the extent of the quirk and ran a gauge line.

Running a gauge line

This really helped get a sharp edge at the extent of the quirk.  I started the bead cuts at the end of the board, moving back little by little before I took full length shavings.  This helps to minimize tear out in the event the wood grain is diving as you push the plane forward.

In the end, these few little things stacked the odds in my favor of getting a nicely defined bead.

The beads came out very nice with minimal sanding

If you haven't done a particular operation in a while, take the time to sharpen up, practice on scrap, and do whatever else is necessary to ensure a good result on the keeper wood.