Heat Generation Bearing⚠ unverified
Mechanical / Bearings · Compute the frictional heat generation rate of a bearing
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| T | T | N*m | 1.0 | Frictional torque |
| omega | ω | rad/s | 1.0 | Angular velocity |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Q | W | Heat generation rate, in watts (W) |
The science & history
Understanding the Parameters
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Frictional torque $T$ — the drag torque the bearing exerts against rotation. It is not the torque the shaft transmits to its load; it is the parasitic loss, made up of rolling resistance, sliding at the contacts and cage, seal drag, and lubricant churning/shear. Manufacturers estimate it from a friction model (classically $T = \tfrac{1}{2}\mu F d$ with an equivalent friction coefficient $\mu$, bearing load $F$, and bore $d$; SKF's modern model splits it into rolling, sliding, seal, and drag terms).
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Angular velocity $\omega$ — the shaft speed in radians per second ($\omega = 2\pi n/60$ for $n$ in rev/min). Heat scales linearly with speed for a fixed torque, but in practice $T$ itself grows with speed (more churning, more lubricant shear), so high-speed bearings heat up faster than linear.
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Heat rate $Q$ — power, in watts. It is the same $T\omega$ that defines any rotational mechanical power; the point is that for the frictional torque this power does no useful work and appears entirely as heat that must be removed.
Derivation (Approaching a Proof)
The instantaneous power delivered by a torque acting through an angular displacement is the rotational analogue of $P = F v$. For a force $F$ at radius $r$ moving at tangential speed $v = r\omega$, the power is
$$P = F v = F r \omega = T\omega,$$
since $T = F r$. When the torque in question is the frictional drag torque, this power is not transmitted onward — by the first law of thermodynamics it is converted, with essentially 100 % efficiency, into internal (thermal) energy at the sliding and rolling contacts:
$$Q = T_{\text{friction}}\,\omega.$$
There is no lossy intermediary and no stored-energy term in steady rotation, so the frictional power is the heat-generation rate. This $Q$ then enters the steady-state thermal balance $Q = h A (T_{\text{brg}} - T_{\text{amb}})$ (plus lubricant enthalpy flow) that sets the equilibrium bearing temperature.
Dimensional check. $[T\omega] = \text{N}\cdot\text{m} \times \text{s}^{-1} = \text{J/s} = \text{W}$ (radians are dimensionless). ✓
History and Development
Frictional heating has been central to bearing design since Osborne Reynolds' 1886 theory of hydrodynamic lubrication explained the viscous origin of much of the drag, and since Petroff's 1883 analysis of journal-bearing friction (see Petroff Friction). The $Q = T\omega$ balance underlies every bearing thermal rating and lubricant-cooling calculation. Modern manufacturer models (e.g. SKF's frictional-moment model) refine the torque term into rolling, sliding, seal, and drag components, but they all feed the same $T\omega$ heat-generation identity used here.
Related Concepts: Bearing Power Loss, Petroff Friction, Friction, Viscosity Required, Stribeck Curve
Notes: Requires the frictional (drag) torque as input, not the transmitted load torque. Combine $Q$ with a cooling model ($Q = hA\,\Delta T$ plus oil-flow enthalpy) to estimate steady-state bearing temperature and required lubricant flow.