Non-Standard Radius Calculations

Steven Horace

If you tilt an endmill off vertical and then use it to cut a shallow groove in a workpiece, it will cut a radiused groove with a radius different from the half-diameter of the endmill. Machinists use this trick to cut odd-sized radii when a cutter of the required diameter is not to hand. A typical application might be recessing the bottom of rifle scope mounts to fit a barrel/receiver radius.

The calculations that relate groove radius and depth, endmill diameter and tilt off vertical are complicated and non-intuitive. Thankfully, Steve Horace has beaten the problem to death. In his writeup below, Steve supplies three methods of making the required calculations. I've programmed the most complex (and most accurate) third method. As a check it calculates, internally, the width of the cut and solves Steve's second relation. The first method is trivial enough to carry out on a scientific calculator and requires no program.

Questions regarding the technique should be addressed to Steve. He's the expert. Questions regarding the program itself should be addressed to me (mklotz@alum.mit.edu).

For those of you who have back issues of HSM (Home Shop Machinist magazine), pg.10 of the May/June 1988 issue has a treatise on deriving Steve's second relation.

Enter radius, error, and either depth or width of cut Desired Radius
Allowable Error
Depth of Cut
Width of Cut

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Depth of Cut (in) =
Width of Cut (in) =
Minimum inclination angle (deg) =
Cutter diameter at minimum angle (in) =
Cutter diameter to use must be

Cutter diameter you plan to use [within above range] (in) =

Angle from vertical to cutter axis (deg) =
Using Steve's second method, we get, as a check:
Steve's first method yields (deg):
Using a cutter of diameter results in the following error:

Listed below are three methods for getting a non-standard radius from a mill cutter.

The first is for a rough approximation, the second for a closer approximation, and the third for setting the error you are willing to accept.

* denotes multiplication, / denotes division, [] denotes expotientals (squares and square roots), () and {} denote groupings of operators

Method 1

R=radius desired, r=radius of cutter, R must be greater than r

Formula: r/R = sine of angle from vertical to cutter axis

Method 2

R = radius desired, r = radius of cutter, W = half-width of cut, R must be greater than r

Sine of angle from vertical to cutter axis =

\frac{R - \sqrt{(R^{2} - W^{2})}}{r - \sqrt{(r^{2} - W^{2})}}

Method 3

First the deviation from the ideal radius must be defined, and then the depth of the cut, and the ideal radius desired.

Let E = error allowable in cut, D = depth of cut, R = radius desired, alpha = angle from vertical to set cutter, and r = radius of cutter. Find alpha and r with D,R and E known. All dimensions in inches and decimal inches.

Calculate D from the geometry of the job.

Formula 1 provides the sine of the angle from vertical to cutter axis

\sin (alpha) = \frac{\frac{D}{r}}{1 - \sqrt{(1 - \frac{2 R D}{r^{2}} + {(\frac{D}{r})}^{2})}}

Formula 2 gives the exact radius of cutter at minimum angle alpha.

r = R \times \frac{2 + \frac{D}{R} \times ((\frac{1}{\sin^{2} (alpha)}) - 1)}{\frac{2}{\sin (alpha)}}

Formula 3 is the error in the radius.

E = \frac{R}{8} \times \frac{D^{2}}{R^{2}} \times (\frac{1}{\sin^{2} (alpha)} - 1)

Formula 4 gives the sine of the minimum angle of inclination at the specified error

\sin (alpha) = \frac{1}{\sqrt{1 + 8 \times \frac{E}{R} \times \frac{R^{2}}{D^{2}}}}

Procedure: Select the D, R and E for the job. E is calculated from the width and radius of the cut.

Next, calculate an angle using formula 4.

Then, use Formula 2 to find the cutter radius for that angle. It will normally not be a standard size.

Now, select an available cutter larger than the size found, but smaller than the radius desired.

Compute the new angle alpha for the available cutter using Formula 1.

Formula 3 will calculate the exact error if needed.

When to use which method depends on the accuracy required and tools available. If a tool close to the desired size is available, either method 1 or 2 will give good results. If the need is for a 6" radius and the largest cutter is 3.5" radius, method 3 will give best results.

Set up the work in the mill at the angle required and cut until the full surface is radiused. Be careful to be on center before cutting, and to have the work level left to right.

Horace K. B. "Steve" Steven, Jr.

Engineer, Surface Ship Test Section

Code 223, Bldg 19, NNSY

SUPSHIP Portsmouth, VA 23709

email: StevenHK@supship.navy.mil, Phone: 757 396 4001 x 2113

FAX 757 396 4055, efax 208 379 9699