Cusp Height Calculator

What Is Cusp Height in Ball End Mill Finishing?

When a ball end mill makes parallel finishing passes, it leaves scallop-shaped ridges (cusps) between adjacent passes. The cusp height is the maximum peak-to-valley height of these ridges and directly determines the achievable surface roughness. Smaller stepover produces lower cusps but increases machining time.

The cusp height is a geometric function of the tool radius and stepover distance: h = r − √(r² − (stepover/2)²). For small stepovers relative to tool radius, this approximates to h ≈ stepover²/(8r). The theoretical surface roughness Ra is approximately 0.25 × cusp height for a scallop profile.

In practice, surface finish is worse than the geometric calculation due to tool runout, machine vibration, workpiece deflection, and BUE (built-up edge). A safety factor of 1.5-2× should be applied to the theoretical cusp height when estimating actual surface roughness. Five-axis machining can maintain a more consistent cusp height on freeform surfaces.

Formula: Cusp Height: h = r − √(r² − (s/2)²) Approx: h ≈ s² / (8r) for s << r Surface Roughness: Ra ≈ h / 4 Stepover for target h: s = 2 × √(2rh − h²)

Example Calculation

Ball end mill R5 (Ø10) with 0.3 mm stepover: h = 5 − √(25 − 0.0225) = 5 − 4.99775 = 0.00225 mm = 2.25 μm. Theoretical Ra ≈ 2.25/4 = 0.56 μm. For target Ra 1.6 μm (h ≈ 6.4 μm): stepover = 2 × √(2×5×0.0064) = 2 × 0.253 = 0.506 mm.

When to Use This Calculator

• Programming 3D surface finishing passes with a ball end mill and needing to determine the stepover for a target surface roughness
• Estimating machining time for 3D finishing — the stepover directly determines the number of passes and total cycle time
• Evaluating whether a larger tool radius could reduce finishing time while still meeting the surface finish specification
• Setting up CAM software surface finishing parameters and verifying the theoretical finish quality before running the program

Common Mistakes to Avoid

• Assuming the theoretical cusp height equals the actual surface roughness — tool runout, vibration, and material effects make actual finish 1.5-2× rougher than the geometric calculation
• Using the same stepover for all surface orientations on a 3D part — steep walls produce much smaller cusps than flat/shallow surfaces for the same stepover in 3-axis machining
• Neglecting tool runout — even 0.01 mm runout on a ball end mill adds directly to the cusp height and can double the theoretical Ra value
• Choosing extremely small stepover to achieve mirror finish — beyond a point, surface quality is limited by machine rigidity and tool condition, not stepover; diminishing returns set in below 0.1 mm stepover

How to Interpret Results

• The Ra output is the theoretical geometric value (≈ cusp height / 4) — measured roughness typically runs 1.5-2× higher due to runout and vibration, so target a theoretical Ra of about half the drawing specification
• Cusp height grows with the square of stepover (h ≈ s²/8r) — halving the stepover cuts the cusp to a quarter but doubles machining time; balance finish against cycle time
• If the result is marginal against the specification, a larger ball radius is often cheaper than a smaller stepover — it lowers cusp height at the same stepover and pass count
• The values apply to surfaces roughly normal to the tool axis in 3-axis work — steep walls present a different effective geometry and will measure differently from this prediction

Related Standards & References

• ISO 4287 — Surface texture parameters (Ra, Rz definitions for verifying achieved finish)
• ISO 12085 — Geometrical product specifications — Surface texture: Motif method
• VDI 3400 — Electrical discharge machining (EDM) surface finish classification (for comparison with milled surfaces)

Frequently Asked Questions

How do I choose the optimal stepover for 3D finishing?

Start with the required surface finish (Ra). Calculate the maximum allowable cusp height (h ≈ 4 × Ra). Then calculate stepover from the cusp height and tool radius. Apply a 50% safety factor for the first test cut. For Ra 1.6 μm with R5 ball end mill: h = 6.4 μm, theoretical stepover = 0.51 mm, practical starting point = 0.35 mm. Adjust based on actual surface measurement.

Does surface inclination affect cusp height?

Yes, significantly. On steep walls (near vertical), the effective tool radius increases and cusps become smaller for the same stepover. On flat/shallow surfaces (near horizontal), the effective radius decreases and cusps grow larger. Five-axis machining maintains a consistent tool contact angle (typically 10-15° tilt) to produce uniform cusp height across complex surfaces. Three-axis machines produce variable surface quality depending on surface orientation.