Frameless Motor Air Gap Eccentricity and UMP: Bearing Loads, Tolerances, and Assembly Guide

When engineering teams transition from traditional geared servo drives to direct-drive frameless torque motors, they gain zero backlash, extreme torsional stiffness, and compact packaging. However, they also inherit a critical mechanical responsibility: establishing, tolerancing, and preserving the motor air gap.

In a standard housed servo motor, the motor manufacturer grinds the shaft, seats the internal bearings, line-bores the end bells, and verifies the air gap before shipment. In contrast, a frameless motor is delivered as an unhoused stator lamination stack and a bare permanent-magnet rotor ring. The customer's housing structure, shaft machining, bearing selection, and assembly tooling dictate whether the rotor spins on a perfect magnetic center or deflects into destructive mechanical failure.

The frameless torque motor air gap must come from the selected motor drawing; there is no universal clearance for every frame. This guide uses a 0.50 mm nominal radial gap as an illustrative design case. Mechanical runout, housing deflection, or thermal growth can shift the rotor off-center and generate Unbalanced Magnetic Pull (UMP), a radial force that reduces the assembly's effective restoring stiffness.

This guide provides mechanical joint architects, robotics engineers, and procurement specialists with a comprehensive engineering framework for air gap tolerancing, UMP quantification, bearing fatigue life calculations, assembly fixture design, and RFQ specifications.

Published and technically reviewed: October 28, 2026. Scope: Global OEM selection and integration of radial-flux, inner-rotor permanent-magnet frameless motor kits. The calculations and tolerance allocations below are reproducible engineering examples, not measured product performance, customer cases, or certified acceptance limits. Outer-rotor and axial-flux machines need different geometry models. Supplier drawings, operating-temperature clearances, and validated electromagnetic and bearing models govern design release.

Design decision: Check the minimum operating gap and bearing reactions together. A geometrically positive gap does not establish acceptable bearing life. Use the RFQ checklist below or request an air gap integration review with your motor envelope, bearing arrangement, and duty cycle.

1. The Physics of Air Gap Eccentricity and UMP

To understand why air gap control is critical, consider what happens inside the magnetic circuit when a rotor shifts away from the stator bore geometric center.

The Maxwell Stress Tensor and Force Asymmetry

The magnetic attraction force per unit area (magnetic pressure) acting across the air gap between stator iron and rotor permanent magnets is governed by the Maxwell stress tensor:

sigma_mag = (B_g^2) / (2 * mu_0)

Where:

• sigma_mag is the radial magnetic attraction pressure (N/m^2 or Pa).
• B_g is the local radial air gap magnetic flux density; 0.85 T is an assumed example value, not a magnet-grade specification.
• mu_0 is the magnetic permeability of free space (4 * pi * 10^-7 H/m or 1.2566 * 10^-6 N/A^2).

At a typical operating flux density of B_g = 0.85 T, the baseline magnetic pull across the air gap is:

sigma_mag = (0.85^2) / (2 * 1.2566 * 10^-6) = 287,522 N/m^2 = 28.75 N/cm^2 (~2.93 kgf/cm^2)

In an ideal symmetric, concentric machine, opposing magnetic forces cancel and the net radial magnetic force is zero. Real slotting, winding asymmetry, saturation, and magnet variation can introduce additional force harmonics. The pressure expression neglects tangential flux components; a detailed force calculation requires the full field solution.

The Unbalanced Magnetic Pull Equation

When the rotor center is displaced radially by distance e from the stator bore center:

g(theta) = g_0 - e * cos(theta) = g_0 * (1 - epsilon * cos(theta))

Where:

• g_0 is the nominal radial air gap.
• e is the radial displacement (eccentricity).
• epsilon = e / g_0 is the relative eccentricity (0 <= epsilon < 1).

For an illustrative smooth-gap model, assume fixed magnetomotive force, inverse-gap flux variation, and a sinusoidal fundamental averaged over its electrical period. Then the mean squared flux is B_g^2 / 2, where B_g now denotes the centered fundamental peak. Circumferential integration of the averaged radial pressure gives this simplified estimate:

F_UMP approx (pi * R_rotor * L_stack * B_g^2) / (2 * mu_0) * (e / g_0) * (1 / (1 - epsilon^2)^(3/2))

This is a teaching model, not a validated UMP law for a particular PM kit. It neglects magnet reluctance, leakage, slotting, saturation, end effects, and pole-dependent harmonics. At epsilon = 0.20, its nonlinear correction is already about 6.3%; use the linear form only where that approximation is acceptable:

F_UMP approx (pi * R_rotor * L_stack * B_g^2) / (2 * mu_0 * g_0) * e = k_mag * e

Where:

• R_rotor is the outer radius of the rotor (m).
• L_stack is the active axial lamination stack length (m).
• B_g is the peak fundamental air gap flux density (T).
• k_mag is the positive magnitude of the destabilizing magnetic stiffness (N/m or N/mm); its contribution to restoring stiffness is negative.

The published Maxwell-stress analysis supports the integration method and eccentricity-dependent force mechanisms. Its rotor study does not validate the assumed flux or numerical results for a commercial frameless kit.

Why UMP Acts as a Negative Spring

In conventional mechanics, a positive spring resists deflection: push a shaft sideways, and the spring pushes back (F = -k * x).

UMP operates in reverse. As the rotor moves off-center by distance e, the magnetic attraction in the direction of displacement increases. The mechanical system experiences a destabilizing force that actively pulls the shaft further off-center:

dF_UMP / de = +k_mag > 0  ==>  Mechanical spring constant K_eff = K_mechanical - k_mag

If the mechanical support stiffness K_mechanical provided by the bearings and housing is lower than k_mag, the rotor undergoes pull-over instability—the rotor snaps instantaneously against the stator lamination wall, causing catastrophic contact, chipped NdFeB magnets, and severed winding insulation.

Even when K_mechanical > k_mag, magnetic attraction reduces the effective stiffness:

K_net = K_bearing_housing - k_mag

Evaluate this condition at the operating equilibrium using the local force slope dF_UMP/de. Manufacturing offset, external load, bearing compliance, and magnetic pull jointly determine the running eccentricity; an unloaded runout measurement alone does not establish stability.

2. Quantitative Sizing Example: 110 mm Frameless Joint

To demonstrate the model's load sensitivity, use a hypothetical 110 mm stator outside diameter (OD) kit. These are assumed inputs, not a catalog model or a measured robot-joint result.

Motor Baseline Parameters

• Stator OD: 110 mm
• Rotor OD (2 * R_rotor): 70 mm (R_rotor = 0.035 m)
• Active Stack Length (L_stack): 30 mm (0.030 m)
• Nominal Radial Air Gap (g_0): 0.50 mm (0.0005 m)
• Peak Air Gap Flux Density (B_g): 0.85 Tesla
• Magnetic Permeability (mu_0): 4 * pi * 10^-7 H/m

Calculating Negative Magnetic Stiffness

Calculate the base coefficient:

C_base = (pi * R_rotor * L_stack * B_g^2) / (2 * mu_0)
C_base = (3.14159 * 0.035 * 0.030 * (0.85)^2) / (2 * 1.2566 * 10^-6)
C_base approx 948.3 N

Now, calculate the negative magnetic stiffness k_mag:

k_mag = C_base / g_0 approx 1,896,600 N/m = 1.897 kN/mm (~190 N per 0.1 mm)

Force Calculation at 10%, 20%, and 30% Eccentricity

1. At 10% Eccentricity (e = 0.05 mm = 50 um):

   epsilon = 0.05 / 0.50 = 0.10
   F_UMP = C_base * 0.10 / (1 - 0.10^2)^1.5 approx 96.3 N (~9.8 kgf)

2. At 20% Eccentricity (e = 0.10 mm = 100 um):

   epsilon = 0.10 / 0.50 = 0.20
   F_UMP = C_base * 0.20 / (1 - 0.20^2)^1.5 approx 201.6 N (~20.6 kgf)

3. At 30% Eccentricity (e = 0.15 mm = 150 um):

   epsilon = 0.15 / 0.50 = 0.30
   F_UMP = C_base * 0.30 / (1 - 0.30^2)^1.5 approx 327.7 N (~33.4 kgf)

Within this model, increasing eccentricity from 10% to 30% multiplies side pull by about 3.4x. Treat this as sensitivity analysis: request supplier FEA or force-versus-offset measurements before allocating bearing loads. Split the total motor force into bearing reactions using the actual support span and motor force location.

3. The Three Topologies of Air Gap Eccentricity

The three geometries below are useful assembly checks. Static and dynamic eccentricity can coexist, and angular tilt can combine with either. Force spectra and wear patterns depend on the motor and support structure; none of these signatures alone proves a particular fault.

The following engineering matrix compares the three misalignment topologies:

4. Including UMP in Bearing Fatigue Life (ISO 281)

Bearing selection must include motor-generated forces as well as external payload, gravity moments, tool thrust, and belt tension. The example below isolates load sensitivity; it is not a field-failure report.

ISO 281 Basic Rating Life Formula

Per ISO 281:2007, the basic rating life in operating hours (L_10h) for a rolling element bearing is:

L_10h = (1,000,000 / (60 * n)) * (C / P)^p

Where:

• n is rotational speed in RPM.
• C is the basic dynamic radial load rating from the bearing manufacturer (N).
• P is the dynamic equivalent bearing load (N), determined from bearing reactions and the manufacturer's radial/axial load factors.
• p is the life exponent: p = 3 for ball bearings, p = 10/3 approx 3.333 for roller bearings (including crossed-roller rings).

Illustrative Comparison: Sizing With vs. Without UMP

Consider a robotic actuator joint operating at n = 120 RPM:

• Assumed ball-bearing dynamic capacity: C = 1,250 N.
• Calculated external payload: F_payload = 180 N.

Scenario A: External Load Only

The engineer assumes the only radial load is payload: P_A = 180 N.

L_10h(A) = (1,000,000 / (60 * 120)) * (1,250 / 180)^3
L_10h(A) approx 46,514 hours

This is basic rating life at 90% reliability under the model assumptions, not a service-life guarantee.

Scenario B: Adding the Example UMP at 20% Eccentricity

Assume a total running offset e = 0.10 mm (20% relative eccentricity). Section 2 estimates:

F_UMP approx 201.6 N

For this simplified radial-load comparison, assume the entire force reacts at the bearing being evaluated and aligns with its 180 N external reaction:

P_B = F_payload + F_UMP = 180 N + 201.6 N = 381.6 N

Recalculate L_10h:

Load Ratio = P_B / P_A = 381.6 / 180 = 2.120
Life Reduction Factor = (Load Ratio)^3 approx 9.53
L_10h(B) = 46,514 / 9.53 approx 4,882 hours

For real multi-bearing axes, resolve force directions and moments at each support before calculating equivalent load. Preload, oscillatory motion, small stroke, lubrication, contamination, and duty-cycle load spectra can govern life. Check static safety and contact stress separately; a brief shock is not a continuous fatigue load.

If support reactions or hot running clearances are still unknown, send the bearing layout for an integration review before finalizing the RFQ.

5. Mechanical GD&T Stack-Up and Tolerancing Budget

For this example, allocate a 10% of nominal gap target to total running eccentricity. This is an illustrative engineering budget, not a universal safe limit or an ISO acceptance requirement. The approved limit must satisfy minimum clearance, supplier UMP data, bearing reactions, and structural stiffness across temperature and load.

For a motor with a 0.50 mm air gap, the maximum allowable total radial displacement is:

e_max_total <= 0.10 * 0.50 mm = 0.050 mm (50 micrometers)

The following seven allocations sum to 50 micrometers. Treat each as an independent contribution to loss of minimum radial clearance, not seven catalog tolerance classes. For a pure eccentric circle, TIR is twice center offset; form errors require their own surface-envelope treatment. Do not count the same assembled runout again as a component error. ISO size fits do not specify concentricity, and bearing precision classes do not specify operating clearance.

6. Shop-Floor Assembly Protocols & Tooling Design

Permanent-magnet rotors can pull sharply toward the stator during insertion, including with the windings unpowered. The insertion force depends on motor size and overlap; the steady-state UMP example above does not specify fixture design loads. Follow the selected motor manufacturer's assembly procedure and use a guided carrier rated for the expected forces. An uncontrolled sideways movement can cause:

• Technician pinch or crush injuries.
• Chipped or fractured permanent magnets (rare-earth NdFeB is brittle ceramic-like material).
• Gouged stator lamination teeth, severing slot paper insulation and causing ground faults.

Three Assembly Fixture Options to Review With the Supplier

1. The Removable Polymer Shim Wrap Method (Low-to-Medium Volume):
   • Select polymer film material and thickness from the supplier-approved tooling design and available minimum gap.
   • Leave a generous tab extending axially past the rotor.
   • A thin protective film does not fill a 0.50 mm gap or carry the full radial attraction. Keep the rotor constrained by the carrier throughout travel.
   • After the bearing arrangement supports the rotor, withdraw the film and verify complete removal before rotation.
2. Precision Non-Magnetic Guide Rods (Medium-to-High Volume):
   • Provide approved tooling interfaces in the housing or carrier; do not drill the supplied motor components without drawing approval.
   • Select guide material for low magnetic attraction and adequate stiffness. Check the finished material; a stainless grade label alone does not establish magnetic behavior after processing.
   • Guide bushings on the rotor installation carrier slide along the non-magnetic rods, constraining the rotor to pure coaxial travel.
3. Controlled Threaded Jackscrew / Arbor Press Fixture (Production Cells):
   • Mount the stator housing to a dedicated assembly base.
   • Advance the rotor using a precision mechanical lead screw or servo-electric press with built-in load cell monitoring.
   • Set validated force and travel limits for misalignment or contamination. An axial load cell cannot replace lateral restraint or guarantee detection of radial contact.

For the division of mounting and bearing responsibilities, see Kollmorgen's mounting and installation guidelines, printed page 34. The fixture above is a conceptual illustration, not a supplier-approved tooling drawing.

7. Metrology: How to Inspect Air Gap Concentricity in Blind Enclosures

An enclosed actuator may prevent direct access after assembly. Plan inspection datums and access before releasing the housing drawing. These three approaches have different capabilities and limitations:

Method 1: Non-Magnetic Feeler Gauges

For accessible stator ends, use a calibrated gauge with verified low magnetic attraction and a surface compatible with the magnet coating. Do not force a blade into the gap. Four points spaced 90 degrees apart provide an initial screen; sample additional angular positions, both axial ends, and multiple rotor positions to detect offset, tilt, and runout:

Eccentricity Ratio = (Gap_max - Gap_min) / (2 * g_nominal)

The formula assumes circular, parallel surfaces and that the true gap extrema have been found. In the example budget, a gap spread above 0.10 mm exceeds the 50 um offset target. Release or rejection must follow approved drawing limits, including absolute minimum gap, temperature, load condition, and measurement uncertainty; four samples alone cannot certify the full gap envelope.

Method 2: High-Frequency Eddy Current / Capacitive Probes

Where inspection ports are designed in, calibrated eddy-current or capacitive probes can record displacement during slow rotation. Validate target material, coating, curvature, mounting stability, and temperature effects. Measuring an accessible reference surface establishes the air gap only if its relationship to the active rotor and stator surfaces is known; sensor resolution alone does not establish measurement accuracy.

Method 3: Electrical Back-EMF Phase Voltage Symmetry Testing

Back-EMF testing is a supplementary electrical screen, not proof of minimum mechanical clearance. At an approved, controlled speed, compare open-circuit line-to-line voltage amplitudes and waveforms (U-V, V-W, W-U) against supplier limits and a known-good baseline. Magnet variation, winding errors, speed ripple, and instrumentation can produce similar deviations. Some eccentricities produce little phase imbalance. Correlate anomalies with mechanical measurements; do not apply a universal +/-0.5% threshold or release a blind assembly on voltage symmetry alone.

8. Sourcing & Engineering RFQ Checklist for Air Gap Integration

When sourcing frameless torque motors, engineering and procurement teams should require suppliers to furnish complete magnetic and mechanical interface data. Use this checklist during your next RFQ:

• Nominal and Minimum Operating Air Gap: Request radial versus diametral definitions, drawing tolerances, and minimum clearance at cold/hot conditions and maximum load. Keep size tolerance separate from permissible eccentricity.
• Stator Bore & Rotor OD Runout Limits: Confirm maximum permissible Total Indicated Runout (TIR) for the stator ID and rotor magnet OD.
• Unbalanced Magnetic Pull Gradient (k_ump): Request force-versus-offset curves and local slope in N/mm, identifying current, speed, temperature, geometry, and whether results are analytical, FEA, or measured.
• Rotor Dynamic Balance Requirement: Agree the applicable ISO 21940-11 procedure, operating speed, residual unbalance, and assembled rotor boundary with the supplier. Do not assign a balance grade by application name alone; mass balancing does not correct geometric eccentricity.
• Magnetic Centering Axial Datum: Ensure the motor drawing specifies an unambiguous axial datum surface for magnetic alignment between rotor magnets and stator laminations to prevent axial magnetic centering thrust loads.
• Magnet Retention Verification: Confirm the magnet retention architecture (structural adhesive class, glass-fiber banding, or carbon fiber composite sleeve) and maximum overspeed limit.
• Assembly Fixture Guidelines: Request the motor manufacturer's recommended assembly tooling drawings, including guide rod patterns and shim thickness recommendations.
• Incoming Inspection Criteria: Agree gap/runout datum, sampling positions, uncertainty, temperature, and pass/fail limits. Treat phase resistance/inductance and Back-EMF balance as supplementary checks.

9. Frequently Asked Questions (FAQ)

Why is air gap eccentricity more dangerous in frameless motors than standard housed servo motors?

In a housed servo motor, the manufacturer supplies the shaft, housing, and bearing system that establishes the running gap. A frameless kit leaves those interfaces to the OEM. The OEM must therefore verify gap geometry and bearing reactions under load and temperature; the kit alone does not establish an assembled-axis tolerance margin.

What is Unbalanced Magnetic Pull (UMP), and why is it called a negative spring?

Eccentricity can create a net radial magnetic force toward the smaller gap. Its local force slope is destabilizing: defining k_mag = dF_UMP/de > 0, the restoring stiffness becomes K_effective = K_mechanical - k_mag. This sign convention distinguishes the positive force slope from its negative contribution to mechanical restoring stiffness. The slope is motor- and operating-point-dependent.

Can standard internal radial clearance (CN) ball bearings be used with frameless torque motors?

Yes, if the mounted bearing arrangement maintains the required minimum gap and stiffness throughout the load and temperature range. CN is a clearance class, not a universal displacement value; fits, bearing size, preload, and thermal growth affect operating clearance. Preloaded angular-contact or crossed-roller arrangements may help, but excess preload increases friction and heat. Confirm the arrangement with the bearing supplier rather than rejecting CN or specifying C2 by default.

How does motor field weakening or heavy phase current affect UMP?

Armature reaction and saturation change air gap flux and radial-force harmonics. Field weakening may reduce one flux component while changing others, so neither high current nor field weakening establishes a universal increase or decrease in UMP. Request supplier force-versus-offset results at the required d-axis and q-axis currents, speed, and temperature; do not extrapolate the open-circuit example to the full duty cycle.

How can mechanical assembly teams install high-energy permanent magnet rotors without crashing into the stator?

Use supplier-approved guided tooling that constrains lateral motion and supports the rotor throughout insertion. A controlled lead-screw carrier and verified low-attraction guides are possible approaches. Protective polymer film is optional and model-specific; it does not replace a load-bearing guide. Verify fixture forces, complete film removal, and minimum clearance before rotation or energizing.

10. Next Steps & Engineering Consultation

Integrating frameless torque motor kits into high-performance robotic joints, gimbals, and rotary tables requires balancing electromagnetic design with precision mechanical tolerancing, bearing stiffness, and safe assembly procedures.

To prepare a model-specific integration review:

• Explore our standard Frameless Torque Motors catalog for stator and rotor kit specifications.
• Review Compact Robot Joint Motors for motor kits that must be coordinated with your bearing and encoder package.
• Inspect Large Hollow Shaft Torque Motors designed for optical pass-through and large-diameter bearings.
• Contact our engineering team for Custom OEM Motor Assemblies with tailored lamination geometries, custom air gap clearances, and pre-engineered assembly tooling fixtures.

Submit your joint envelope drawings, bearing layout, operating temperatures, speed, and duty cycle to [email protected] or via our online contact form. Ask which drawing tolerances and force-versus-offset data are available for the proposed kit, and whether additional simulation or testing is needed before design release.

11. References and Technical Standards

Reviewed October 28, 2026. These sources support the physical method and integration responsibilities; none certifies this article's assumed motor, 10% example budget, or fixture concept.

1. Calculation and Analysis of Unbalanced Magnetic Pull of Rotor under Motor Air Gap Eccentricity Fault (2023), Sustainability, 15(11), 8537. Sections 2.1–2.2 cover eccentricity and Maxwell-stress integration; Section 3 compares analytical and finite-element results for the study's own rotor. Read the research paper.
2. Kollmorgen — Mounting and Installation Guidelines, printed page 34. Explains the user's responsibility for bearing stiffness, model-specific concentricity, mounting interfaces, and preload selection. Read the manufacturer guidelines (PDF).
3. Kollmorgen — Frameless Motor Mounting Practices: Clamping and Bolting (February 3, 2017). Describes user-supplied bearings and shafting, radial running clearance, and mounting options. It supports drawing-specific gap selection, not a universal gap value. Read the mounting overview.

Standards used by name: ISO 281 provides the basic bearing rating-life framework; ISO 21940-11 addresses balancing of rigid rotors. Obtain the applicable edition and the selected bearing manufacturer's calculation guidance for design release. These standards do not prescribe this article's illustrative eccentricity budget, fit classes, or assembly film thickness.