Load-to-Motor Inertia Ratio for Frameless Torque Motors: From “Stability Shackles” To A “Yardstick for Acceleration Capability”

In traditional servo selection, the inertia ratio is often treated as a red line: exceed the recommended value, and the system may oscillate, overshoot, or become difficult to stabilize. However, when the actuator is replaced by a frameless torque motor and the transmission is replaced by direct drive, the logic behind this red line changes fundamentally. The rigid connection of frameless direct drive shifts the inertia ratio from a stability question—“can it stay stable?”—to a dynamic-capability question—“can it accelerate quickly?” Understanding this shift is a prerequisite for correctly calculating and interpreting the load-to-motor inertia ratio of a frameless torque motor.

1. What Exactly Is the Inertia Matching Ratio?

The load-to-motor inertia ratio is a dimensionless number, defined as the total load inertia reflected to the motor shaft divided by the inertia of the motor rotor itself:

where J load，reflected is the equivalent rotational inertia of the load reflected to the motor shaft, and J motor  is the rotational inertia of the motor rotor about its own axis. For frameless torque motors, J motor  is usually taken from the “rotor inertia” given in the motor datasheet and does not include parts such as hubs or flanges that the user installs on the rotor.

The inertia ratio matters because it directly constrains the velocity-loop bandwidth and the system’s response speed to disturbances. The larger the inertia ratio, the “heavier” the load appears, the harder it is for the motor to change the load’s state of motion, and the narrower the range of gains that can be used stably. In conventional servo architectures with reducers, an excessively large inertia ratio can also couple with the elasticity and backlash of the transmission chain and excite oscillation.

2. Key Change Brought by Frameless Direct Drive: Rigid Connection Rewrites the “Safe Zone”

In traditional servo systems, one important reason the inertia ratio needs to be strictly kept around 5:1 to 10:1 is that elastic elements such as couplings, belts, and gears sit between the motor and the load. The larger the load inertia, the lower the resonant frequency of the transmission chain, and the more easily the system becomes unstable near the resonant frequency.

When a frameless torque motor directly drives a load, the rotor and load are usually rigidly or nearly rigidly connected—the rotor is bonded or press-fitted directly onto the load shaft, with no reducer or flexible transmission element in between. This eliminates the elasticity and backlash of a traditional transmission chain, significantly raises the mechanical resonant frequency of the system, and therefore greatly widens the tolerable inertia ratio range. A white paper from a direct-drive motor manufacturer notes that traditional servo applications usually allow a load inertia several times the motor inertia, whereas direct-drive applications can allow inertia ratios of hundreds or even thousands. Derui Precision’s direct-drive selection guide gives more specific references: in low-speed direct-drive applications, the inertia ratio can reach 1:150, while in high-speed direct-drive applications it is recommended to keep it within 1:10.

This means that for frameless torque motors, the calculation method for the inertia ratio has not changed, but its “passing threshold” and the way it is interpreted have changed. An inertia ratio that would be judged “too high” in a traditional servo system may be perfectly acceptable in a rigid direct drive. At this point, the main significance of the inertia ratio is no longer to judge “will it oscillate?” but to judge “is it fast enough to accelerate?”

3. Calculation Steps: From Components to Ratio

The calculation of the inertia matching ratio can be broken into five steps. There is only one core principle: the inertia of all forms of motion must ultimately be reflected to the motor shaft in a unified way before summing.

Step 1: Identify all moving components. For a rotary joint, this usually includes the motor rotor, the hub or flange mounted directly on the rotor, the load body, and any end mass connected through a reducer or transmission mechanism. For systems that include linear motion (such as lead screws or racks), the linearly moving mass must also be reflected as rotational inertia.

Step 2: Calculate the moment of inertia of each rotating component about its own axis. The formulas for common geometric shapes are as follows (mass m in kg, dimensions in m, result in kg·m²):

· Solid cylinder (about the central axis): 

· Hollow cylinder (about the central axis): 

.

· Rectangular prism (about an axis through the center of mass and perpendicular to one face): 

, where  a and b are the two side lengths perpendicular to that axis.

For loads with irregular shapes or those assembled from multiple parts, the load can be split into several regular geometric bodies, calculated separately, and then summed. Commercial CAD software can usually directly output the moment of inertia of a part about a specified axis; in engineering practice, this should be preferred.

Step 3: Reflect linearly moving mass. If the system contains linearly moving components driven by rotary motion (such as a slide stage driven by a lead screw), the equivalent inertia reflected to the motor shaft is:

where p is the linear displacement corresponding to one revolution of the motor (m/rev).

Step 4: Handle the reduction ratio. If there is no reducer between the frameless torque motor and the load (direct drive), then i=1 and no reflection is needed. If there is a transmission mechanism with reduction ratio i in between, the load-side inertia must be divided by i² when reflected to the motor shaft:

The larger the reduction ratio, the “smaller” the load inertia appears; this is the fundamental reason a reducer can relax inertia matching. The typical use of a frameless torque motor is direct drive, so this step can usually be skipped, but precisely because of this, the load inertia appears unchanged at the motor shaft end.

Step 5: Sum and divide by the motor rotor inertia. Add all the inertias reflected to the motor shaft to obtain J load，reflected, then divide by J motor  from the motor datasheet to obtain the inertia matching ratio.

4. A Practical Boundary in Frameless Motor Calculations: Where Does the “Motor Rotor” End?

A frameless torque motor has no housing or finished shaft; the rotor is a component that the user must integrate into the mechanical structure. The “rotor inertia” given by the motor manufacturer usually covers only the rotor body (permanent magnets, rotor yoke, etc.). In principle, the adapter flange, clamping sleeve, hub, and any mass directly fixed to the rotor that the user adds to connect the load to the rotor should all be counted as load-side inertia, not as part of the “motor inertia.”

Mathematically, this does not change the ratio—whether the flange is counted as “motor” or “load,” the result of total inertia divided by motor rotor inertia is the same, because the denominator is always the rotor inertia of the motor body. But it is meaningful for engineering judgment: when you find the inertia ratio is on the high side, the first thing to examine is the additional mass rigidly connected directly to the rotor, because it is accelerated together with the rotor, contributes most directly to the inertia ratio, and is often the part that can most easily be reduced through lightweight design.

5. Recommended Ranges: Look at Scenarios, Not a One-Size-Fits-All Number

The “appropriate value” of the inertia matching ratio depends heavily on the transmission form and dynamic requirements. The following table consolidates engineering references from different sources for positioning during selection:

The “≤5:1” and “5:1–10:1” values for frameless direct drive in the table come from manufacturers’ engineering recommendations and apply to joints or rotary tables with clear dynamic performance requirements. When the load inertia is far greater than the motor inertia and the application itself does not require high bandwidth, a rigid direct drive can operate stably at a very high inertia ratio—the cost is that acceleration capability is limited by the motor’s peak torque rather than by stability. In this case, the inertia ratio is more like a “yardstick for acceleration capability”: the closer the reciprocal of the ratio (J motor /J total ) is to 1, the higher the share of motor inertia in total inertia, and the greater the angular acceleration that the same peak torque can produce.

6. A Simplified Calculation Example

Consider a direct-drive rotary joint: the rotor inertia of a frameless torque motor is 

. An aluminum alloy hub with mass 0.5 kg and radius 0.04 m is mounted on the rotor; a load fixed to the hub can be approximated as a solid cylinder with mass 3 kg and radius 0.08 m. It is direct drive, with no reducer.

Hub inertia: 

Load inertia: 

Total reflected inertia: 

Inertia matching ratio: 

If this ratio appeared in a traditional servo system with a coupling, it would almost certainly mean difficult commissioning. However, in a rigid direct drive, whether it is acceptable depends on the application requirements: if the joint only needs to perform medium- and low-speed positioning with relatively low angular acceleration, 51:1 does not necessarily lead to instability; if the application requires millisecond-level settling or high-frequency reciprocating motion, the ratio should be reduced to a level closer to 10:1 or even 5:1 by reducing the load mass, reducing the load radius, or selecting a motor model with a larger rotor inertia. The calculation itself only answers “what is the ratio”; whether “this ratio can be used” must be answered in combination with transmission stiffness and dynamic performance requirements.

Conclusion

The calculation path for the load inertia matching ratio of a frameless torque motor is clear: list the inertia sources → obtain values using geometric formulas or CAD → reflect linear motion and reduction ratio → sum → divide by the motor rotor inertia. What really needs to change is the way the calculation result is interpreted. Under a rigid direct-drive architecture, the inertia ratio is no longer the “stability red line” that must be strictly observed in traditional servo systems; instead, it is a ruler for measuring how much say motor inertia has in total inertia. It does not directly tell you “will it oscillate?” but tells you “how much of the acceleration task must be borne directly by motor torque.” The correct way to use inertia matching for frameless torque motors is to first calculate the ratio clearly, then return to T=Jα to verify acceleration capability.