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The Fundamental Theorem of Cutting Acoustics: Deriving the Tooth-Pass Frequency Formula
Category: Acoustic AI • Published: 2026-08-12 • By Muhammad Ali
In industrial manufacturing, structural sound is not mere background noise—it is the direct acoustic signature of mechanical shear stress. The fundamental harmonic of any rotary cutting tool is governed by the Tooth-Pass Frequency ($f_{tp}$).
### The Mathematical Derivation
Let a spindle rotate at frequency $n$ in revolutions per minute (RPM). The rotational frequency in Hertz (revolutions per second) is:
$f_{rot} = \frac{\text{RPM}}{60}$
If an endmill has $Z$ equally spaced cutting flutes, each revolution produces $Z$ distinct shear events as the cutting edges engage the workpiece. The fundamental tooth-pass frequency is therefore:
$f_{tp} = Z \times f_{rot} = \frac{\text{RPM} \times Z}{60}$
### Worked Production Examples
1. **Standard 2-Flute Endmill at 18,000 RPM:**
$f_{tp} = \frac{18000 \times 2}{60} = 600\text{ Hz}$
2. **Single-Flute Acrylic 'O' Flute at 24,000 RPM:**
$f_{tp} = \frac{24000 \times 1}{60} = 400\text{ Hz}$
3. **4-Flute Finishing Mill at 12,000 RPM:**
$f_{tp} = \frac{12000 \times 4}{60} = 800\text{ Hz}$
### Separating Signal from Ambient Chaos
A factory floor is inundated with ambient noise: dust collector impellers ($120\text{ Hz}$), stepper motor pulse frequencies ($1.2\text{ kHz}$), and air compressors ($50\text{ Hz}$).
By synchronizing our Fast Fourier Transform (FFT) analysis window directly to the commanded spindle RPM, **Forge AI** applies a dynamic tracking bandpass filter centered exactly on $f_{tp}$ and its first harmonic ($2f_{tp}$). When energy suddenly shifts away from $f_{tp}$ into non-harmonic resonant sidebands, the system flags regenerative chatter in under 30 milliseconds.