Acoustic noise: linking stator tooth forces to rotor vibration modes

Most “mystery tones” in e-machines are just a bookkeeping failure: the tooth-force harmonic is indexed one way, the structural modes another, and the rotor path gets ignored because the stator is easier to blame. Put everything on the same axes—frequency, spatial order, and where the load actually closes—and the link usually shows itself.

What “stator tooth force” needs to mean (or your link breaks)

If your “tooth force” is really airgap pressure on a convenient surface one day, then nodal forces on tooth tips the next, you can make any mode look responsible. The force definition is the contract. Break it and your modal correlation becomes a storytelling exercise.

Also, the boring detail that matters: force accuracy is mesh-sensitive, and people quietly ship bad force maps into structural solvers. One pragmatic check is to compare torque from the mapped forces against the solver’s torque; when they match, you’re at least not violating conservation in a loud way.

Separate note. If you’re comparing methods (virtual work vs Maxwell stress on different surfaces), you’re not being academic. You’re trying to stop a 3 dB argument from turning into a 15 dB mistake.

Keep two indices: frequency and circumferential wavenumber

Force frequency alone is half a label. The other half is the spatial pattern around the airgap: circumferential wavenumber (often written as r). If you don’t carry r through the pipeline, you’ll “match” a peak to the wrong mode because a lot of modes sit near the same frequency band.

The tooth-FRF community is blunt about this: magnetic forces are distinguished by frequency and spatial distribution, and that spatial distribution is the circumferential wavenumber. They even give the sanity anchors: r = 0 is a pulsating wave, r = 1 corresponds to unbalanced magnetic pull (UMP).

There’s an old rule that keeps surviving because it’s true: strong radiation happens when the excitation frequency is near a natural frequency and the spatial order lines up with the mode shape. Not optional. Two locks.

How tooth forces end up exciting rotor modes

The jump from “tooth force” to “rotor mode” is not mystical coupling. It’s load closure.

A big chunk of tooth force content is radial and lives in the stator. That’s the standard story and it’s often correct: stator vibrations driven by electromagnetic forces in the airgap radiate as the exterior surface moves, and resonances occur when force harmonics sit near vibration modes.

But some harmonics don’t just shake teeth. They create a net resultant on the rotor, or a net moment, or they modulate bearing reactions. The cleanest example is r = 1: UMP acts like a lateral force vector rotating with the field/eccentricity pattern, and it routes straight into rotor bending dynamics through the bearings.

Burakov’s UMP framing is useful because it’s a spectral statement, not a vibe: rotor eccentricity produces additional field harmonics shifted by ±1 in spatial order, and UMP comes from interactions meeting that ±1 relationship. That’s the rotor path announcing itself in the index math.

Then there’s the tangential side. People underweight it. A recent eAxle study correlates torsional vibration/noise with the tangential electromagnetic force array and treats radial vs tangential contributions explicitly. If your “acoustic noise” peak is tied to a torsional mode and you only carried radial tooth forces, you already decided the answer.

A mapping pipeline that doesn’t hide the math

You don’t need a grand coupled model to do the linking. You need disciplined artifacts and consistent indexing.

That table is the whole game. Not every row is needed every time, but when a link is disputed, the missing row is usually the one that would settle it.

Modal matching without pretending it’s exact

Treat “match” as a filter, not a proof.

You look for a force component at frequency ( f_k ) with circumferential wavenumber ( r_k ). You look for a rotor mode with natural frequency near ( f_k ) and a compatible circumferential order (nodal diameter / lobes) that can accept ( r_k ). Sometimes the rotor mode label is $(n, m)$ or $(\text{ND}, \text{axial})$. Different dialect, same idea.

If you skip spatial order, you’ll correlate the 2.9 kHz tone to whichever stator mode is closest. If you carry spatial order, you’ll notice the tone is picky about where it radiates, where it’s sensed, which bearing sees it first.

Two test-side diagnostics that expose rotor participation

Order tracking can do a lot with almost no modeling. The NASA rotor-vibration report uses the fundamental electrical frequency relation $f_1 = f_{\text{mot}} P / 120$ and marks harmonics; peaks clustering around a rotor eigenfrequency while following an electrical harmonic order is a strong hint that EM excitation is feeding a rotor resonance.

Tooth FRF concepts also scale better than people admit. The ISMA paper’s move—characterize structural response by exciting stator teeth, convert tooth FRF to wave FRF, then analyze rotating Maxwell stress waves—makes “spatial order” measurable, not just simulated. They even discuss extension toward rotor FRF thinking, which is where rotor-mode blame stops being speculative.

A concrete example where the rotor mode owns the acoustic peak

NASA’s “Scorpion” motor data is a clean case because it names the rotor modes and shows the acoustics. Peak acoustic radiation sits near 3000 Hz; it peaks at specific motor speeds (6292 and 6441 rpm in that report), the peak corresponds to the fourth electrical harmonic at those speeds, and the tone aligns with the rotor ((2,1)) mode from FEA (with experimental frequency nearby). That’s “tooth/EM forcing” meeting “rotor resonance” in public, with numbers.

If your competitor content stops at “stator vibrations cause noise,” this is the missing chapter: the forcing can be electromagnetic and still be routed through rotor modes, with directivity patterns and sensor-to-sensor deltas that a stator-only story can’t match.

Mitigation choices, framed by the link you just proved

If the link is “force harmonic exists” → “rotor mode accepts it” → “bearing path transmits it,” then you have three levers, and mixing them blindly wastes time.

You can reduce the harmonic content in the tooth forces. Slot/pole choices and tooth-modulation effects shift what harmonics exist and how strong they are; one recent PMSM study explicitly compares slot counts and shows major harmonic components aligning with natural frequencies in NVH results, which is basically a resonance map in disguise.

You can move the rotor mode. Stiffness, mass distribution, end-bell participation—whatever changes the rotor eigenfrequency or damping—works when the excitation order is hard to eliminate. That NASA report leans on exactly this logic: peak radiation occurs when operating speeds excite rotor resonance modes.

You can weaken the transmission path. Bearings and supports are not neutral; they decide whether rotor vibration shows up on the radiating surfaces. And when the root driver is UMP-ish content, remember UMP is sensitive to eccentricity-driven harmonic interactions; Burakov also notes parallel paths and rotor cage effects can reduce UMP in some configurations, which is an electromagnetic-and-circuit-side “path” lever people forget exists.

Interpretation table: how the tone behaves tells you which link is active

One last remark, because it saves weeks: if your force spectrum and your modal database don’t share the same spatial language, the “link” will look random. It isn’t random. It’s mislabeled. Carry ( f ) and ( r ), keep phase, and make the bearings part of the story from the start.