Motor Inertia⚠ unverified
Electrical / Motors · Compute the effective rotational inertia including a coupled mass
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| J | J | kg.m**2 | 1.0 | Rotor (base) moment of inertia |
| m | m | kg | 1.0 | Coupled mass treated as a solid disc |
| r | r | m | 1.0 | Radius of the coupled mass |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Jeff | kg.m**2 | Effective moment of inertia, in kilogram square metres (kg.m**2) |
The science & history
Understanding the Parameters
- $J$ — the rotor's inertia, a fixed motor property.
-
$m$, $r$ — the coupled load; here modelled as a solid disc, whose inertia about its axis is $\tfrac12 m r^2$ (a hoop would be $mr^2$, a point mass at radius $r$ also $mr^2$).
-
$J_{eff}$ — the total inertia the motor must accelerate; larger $J_{eff}$ means slower dynamics for a given torque.
Derivation (Approaching a Proof)
Moment of inertia is additive for bodies sharing the same rotation axis, so the effective inertia is the rotor's plus the load's:
$$J_{eff} = J + J_{load}.$$
For a solid disc/cylinder of mass $m$ and radius $r$ rotating about its axis, integrating $r'^2\,dm$ over the disc gives $J_{load} = \tfrac12 m r^2$, hence
$$J_{eff} = J + \tfrac{1}{2} m r^2.$$
Newton's rotational law then sets the acceleration for a given net torque: $T = J_{eff}\,\alpha$. When a load is coupled through gearing of ratio $n$, its inertia reflects to the motor shaft divided by $n^2$ — a key fact in drive sizing (not included in this simple direct‑coupled form).
History
Rotational inertia dates to Euler's 18th‑century rigid‑body dynamics. In drives, matching motor and (reflected) load inertia — the "inertia ratio" — is a standard servo‑tuning rule for responsive, stable motion.
Related Concepts: Motor Torque, Motor Power
Notes: Registry calculator motor-inertia (unverified). Models the coupled mass as a solid disc; use
the correct $J$ expression for other shapes, and reflect geared loads by $1/n^2$.