Rotational inertia — the resistance of a rotating body to changes in its angular velocity — is a parameter that receives careful attention when sizing servo motors and gearboxes, but is frequently overlooked when selecting the coupling that sits between them. This oversight is understandable: the coupling is small, the numbers in the catalogue look small, and the component does not appear in most servo drive sizing spreadsheets as a line item. But coupling inertia contributes directly to the total inertia reflected at the motor shaft, and when it represents a significant fraction of the motor rotor inertia, it degrades servo dynamic performance in ways that cannot be compensated through tuning alone.
This article explains what rotational inertia is, how to calculate it for an Oldham coupling, why it matters for servo performance, and how to keep it within acceptable bounds during coupling selection.
Rotational inertia (also called moment of inertia) is the rotational analogue of linear mass. Just as a heavy object requires more force to accelerate linearly, a rotating object with high inertia requires more torque to accelerate angularly. The relationship is: Torque = Inertia × Angular Acceleration, which is the rotational equivalent of Newton’s second law.
For a servo motor system, the total inertia that the motor must accelerate consists of:
The coupling sits on the motor shaft (or very close to it), so its inertia contributes at a 1:1 ratio with no gearbox reduction to moderate it. A coupling with 50 g·cm² of inertia contributes the full 50 g·cm² to the motor’s total load. A load of the same mass on the far side of a 5:1 gearbox contributes only 50/25 = 2 g·cm² reflected to the motor. This geometric relationship — coupling inertia counts fully, load inertia is divided by the gear ratio squared — means coupling inertia can be disproportionately significant even when the coupling mass is small relative to the total system.
The inertia ratio — total load inertia divided by motor rotor inertia — is the key parameter for servo dynamic performance. When the inertia ratio is near 1:1 (load inertia equals motor inertia), the servo system is well matched and can be tuned to high bandwidth with good stability margins. As the inertia ratio increases, the achievable bandwidth decreases for a given stability margin, and the servo begins to exhibit characteristic problems:
If the coupling contributes 15 percent of the motor rotor inertia — not an uncommon situation when an oversized coupling is selected — and the rest of the reflected load is already at a 3:1 ratio, the actual inertia ratio becomes 3.15:1. This represents a 5 percent increase, which seems small but may be the difference between stable tuning at the required bandwidth and persistent instability that forces a gain reduction and degraded accuracy.
For a solid cylinder — the simplest approximation for a coupling hub — the moment of inertia about the rotational axis is:
J = ½ × m × r²
where m is mass in kg and r is the outer radius in metres. For an Oldham coupling assembly with a bore (hollow cylinder), the formula becomes:
J = ½ × m × (r_outer² + r_inner²)
In practice, the exact geometry of an Oldham coupling hub — with its slot, bore, and clamp features — makes an analytical calculation complex. The correct approach is to use the inertia value published in the manufacturer’s datasheet, which is calculated from the actual CAD geometry or measured on a physical sample. For hand calculations or preliminary estimates when datasheet values are not available, use the solid cylinder formula as an upper bound — the actual inertia will be somewhat lower due to material removed for the bore, slots, and clamp features.
The total coupling inertia is the sum of both hub inertias and the disc inertia. For polymer discs, the disc inertia is typically 5 to 15 percent of the total coupling inertia — small but not always negligible at high accuracy. Datasheet values for coupling inertia should include all three components.
Because inertia scales with mass (and therefore with material density), the choice between aluminium and stainless steel hubs has a direct and significant effect on coupling inertia. Stainless steel has approximately 2.9 times the density of aluminium, so stainless steel hubs of the same geometry contribute approximately 2.9 times the inertia of equivalent aluminium hubs.
This difference is important when stainless steel hubs are specified for environmental reasons (food industry, pharmaceutical, marine, washdown). The engineer must verify that the higher inertia of the stainless steel version does not push the servo system’s inertia ratio beyond the acceptable limit. In some cases, a larger stainless steel coupling may need to be replaced with a smaller one — accepting a reduced torque margin — to keep inertia within budget.
| Coupling OD (mm) | Aluminium Hubs (g·cm²) | Stainless Steel Hubs (g·cm²) | Inertia Ratio (SS / Al) |
|---|---|---|---|
| 20 | 0.35 | 1.0 | 2.9× |
| 25 | 0.85 | 2.5 | 2.9× |
| 32 | 2.8 | 8.1 | 2.9× |
| 40 | 8.5 | 24.7 | 2.9× |
| 50 | 22.0 | 63.8 | 2.9× |
A common misconception is that a shorter, wider coupling and a longer, narrower coupling of the same mass have the same inertia. They do not. Because inertia scales with the square of the radius, outer diameter has a disproportionately large effect on inertia relative to length.
Consider two couplings with the same mass: one with OD 40 mm and length 30 mm, and another with OD 32 mm and length 47 mm. The 40 mm coupling will have approximately 56 percent higher inertia than the 32 mm coupling, despite identical mass, because its mass is distributed at a greater radius. The practical implication: when minimising inertia is important, choose the smallest OD that meets the torque requirement, even if this means accepting a longer coupling to accommodate the required bore length. Inertia reduction through OD reduction is always more effective than length reduction.
The following guidelines cover the most common scenarios in precision motion and servo drive design:
High-bandwidth servo axes (CNC, robotics, pick-and-place): Keep coupling inertia below 5 percent of motor rotor inertia. These applications require rapid acceleration and deceleration with tight positioning accuracy. Any unnecessary inertia directly limits achievable bandwidth. Use the smallest coupling OD that meets the torque and bore requirements, with aluminium hubs.
Standard servo axes (general automation, packaging, conveyor indexing): Coupling inertia up to 10 percent of motor rotor inertia is generally acceptable. The servo bandwidth required for these applications is moderate, and a slightly higher inertia ratio can be accommodated by servo tuning adjustments.
Stepper motor drives: Coupling inertia up to 15 percent of stepper rotor inertia is often acceptable, because stepper motors are typically operated well below their dynamic torque limit and can tolerate higher inertia ratios than servo motors before step loss becomes a concern. However, check that the coupling inertia does not push the total reflected inertia above the stepper motor’s recommended maximum inertia load.
Encoder connections: Coupling inertia should be below 1 percent of the encoder shaft and rotor inertia. Miniature Oldham couplings in the 16 to 20 mm OD range with aluminium hubs typically satisfy this criterion easily — disc and hub inertia values in the range of 0.1 to 0.35 g·cm² are achievable.
Given: A servo motor with rotor inertia J_motor = 150 g·cm² drives a ballscrew through a 32 mm OD aluminium Oldham coupling. The catalogue states coupling inertia J_coupling = 2.8 g·cm². The reflected load inertia at the motor shaft (ballscrew + nut + carriage) is J_load = 280 g·cm².
Total reflected inertia: J_total = J_motor + J_coupling + J_load = 150 + 2.8 + 280 = 432.8 g·cm²
Inertia ratio: J_load_total / J_motor = (J_coupling + J_load) / J_motor = (2.8 + 280) / 150 = 1.89:1
Coupling contribution: J_coupling / J_motor = 2.8 / 150 = 1.87% — well within the 10% guideline. This coupling is correctly sized for inertia in this application.
If the same application required stainless steel hubs for a washdown environment, J_coupling would increase to approximately 8.1 g·cm² (2.9× the aluminium value). The coupling’s contribution rises to 5.4% of motor inertia — still within the 10% guideline, but worth confirming before finalising the specification.
Coupling inertia is a small number that can have a disproportionate effect on servo system performance when it is ignored. Because the coupling sits directly on the motor shaft with no gear ratio to reduce its reflected inertia, it contributes fully to the total inertia ratio. Keeping coupling inertia below 5 to 10 percent of motor rotor inertia — by selecting the smallest OD that meets torque and bore requirements, using aluminium rather than stainless steel hubs where the environment permits, and including the coupling term in every servo inertia calculation — eliminates a source of servo performance degradation that is easy to prevent at the design stage and difficult to diagnose after commissioning.
Browse our Oldham coupling range with full inertia specifications, or contact our engineering team for inertia calculation assistance specific to your servo drive system.
Solar tracking systems are among the most thermally dynamic mechanical environments that a coupling will…
Textile machinery operates at the intersection of high speed, continuous duty, and precision synchronisation —…
Oldham coupling catalogues present a dense matrix of numbers — outer diameters, bores, torque ratings,…
Flexible coupling selection looks straightforward from the outside — find something with the right bore…
Semiconductor manufacturing is arguably the most demanding environment in which any mechanical component can be…
Most Oldham coupling applications involve horizontal shaft connections — a motor mounted beside or above…