For a vertical hemispherical ball nose end mill cutting a locally flat surface, the effective cutting diameter at axial depth ap is the diameter of the cutter’s circular cross-section at that height. It is smaller than the nominal diameter below the ball radius, and equals the nominal diameter at the ball radius. Use it to calculate cutting speed at that contact height.
Ball Nose End Mill Effective Diameter Formula
The effective cutting diameter is the width of the engaged circular section at the cutting height. For a vertical ball nose end mill on a flat surface, effective diameter equals the square root of [D² − (D − 2 × ap)²]. Use this diameter when calculating cutting speed.
The ball nose end mill formula gives the maximum cutting diameter at a specific depth.
Here, D is the nominal cutter diameter and ap is the axial depth measured from the tip. This calculation applies over the ball portion for 0 < ap ≤ D/2. At ap = D/2, the engaged section reaches the nominal diameter.
As ap approaches zero, the effective diameter approaches zero. At the exact tip, tangential cutting speed is zero for any finite spindle speed.
Worked Example
With an illustrative D = 10 mm and ap = 0.5 mm:
- Effective diameter = sqrt(10² − (10 − 2 × 0.5)²)
- Effective diameter = sqrt(100 − 81) = sqrt(19) ≈ 4.36 mm
At this depth, the cutting edge’s circular path has an effective diameter of about 4.36 mm, even though the tool’s nominal diameter is 10 mm.
Adjust Spindle Speed for Cutting Diameter
Cutting speed depends on the diameter at the contact height. If you calculate RPM using the full nominal diameter while cutting at a shallower height, the actual cutting speed there will be lower than the target. For any nonzero effective diameter, a finite RPM can produce nonzero cutting speed.
For the 4.36 mm effective diameter in the example, suppose the illustrative target cutting speed is 100 m/min. Using RPM = (cutting speed × 1000) / (π × effective diameter) gives about 7,300 RPM when rounded to the nearest hundred. Treat 100 m/min as an example only. Select a target appropriate to the work and check it against published machine and tooling limits.
For tilted tools or non-flat surfaces, use CAM contact geometry or the toolmaker’s method to determine the contact diameter.
Keep Diameter Separate From Feed and Engagement
Effective diameter changes the cutting-speed calculation. It does not replace feed per tooth or side engagement: those describe different aspects of the cut and should be set using the relevant machining guidance for the operation.
If you’re selecting a cutter for this type of work, our Ballnose Endmill HRC50 page provides product details to review alongside your machining requirements.
FAQ
Is Effective Diameter the Same as Nominal Diameter?
For a vertical hemispherical cutter over a locally flat surface, effective diameter is smaller than nominal diameter when axial depth is below the ball radius, and equals nominal diameter when depth equals the radius.
What Happens at the Exact Tip?
At the exact tip, effective diameter and tangential cutting speed are zero for any finite spindle speed.
Does This Formula Apply to a Tilted Cutter?
The formula assumes a vertical hemispherical ball nose end mill cutting a locally flat surface. For tilted or non-flat cuts, use CAM contact geometry or the toolmaker’s method.