Bearing friction and thermal speeds

Where bearing friction comes from

A rolling bearing loses power in several ways: elastic hysteresis and the pressure flow of the oil in the rolling contacts, sliding in the contacts (differential slip, spin) and at cage and roller ends, drag and churning of the lubricant, and the rubbing of contact seals. The app offers two models to estimate the sum: Palmgren's two-term model, on which DIN ISO 15312 and DIN 732 build, and a four-component model that separates rolling, sliding, seal and drag losses. Both are always computed; Friction model chooses which one drives frictionTorque, powerLoss and the heat balance.

All numbers come from the bearing-6207 template: Fr = 2 500 N, Fa = 800 N, 2 900 rpm, ν = 27.0 mm²/s at 70 °C, oil bath.

Palmgren: M = M0 + M1

Palmgren split the torque into a load-independent and a load-dependent part:

\begin{matrix} M_{0} & = 10^{- 7} \cdot f_{0} \cdot {(ν \cdot n)}^{2 / 3} \cdot d_{m}^{3} & for ν \cdot n \geq 2 000 \\ M_{0} & = 160 \cdot 10^{- 7} \cdot f_{0} \cdot d_{m}^{3} & below \\ M_{1} & = f_{1} \cdot P_{1} \cdot d_{m} \end{matrix}

with M in N·mm, ν in mm²/s, n in rpm and $d_{m} = (d + D) / 2$ in mm. The same terms appear in DIN ISO 15312, Eq. (8), (9) and DIN 732, Eq. (1). f0 depends on the bearing type and series and on the lubrication; f1 and the friction load P1 depend on the type, and for ball bearings f1 also on P0/C0 (DIN 732, Table A.1; DIN ISO 15312, Table A.1). The app reads the series from the designation and falls back on each type's common series.

DIN 732 gives f0 for oil bath with the level at the centre of the lowest element, notes that it can rise to three times that with a higher level, and that run-in grease gives about half to two thirds of it. Method (for friction) applies a factor to f0: 0.5 for "Grease or oil mist", 1 for "Oil bath" and 2 for "Oil jet or vertical shaft". A flanged cylindrical roller bearing that carries an axial load adds the rib friction.

For the template, f0 = 2 and f1 = 0.000364 with P1 = 2 500 N:

Term	Torque
M0 (idle)	56.1 N·mm
M1 (load)	48.7 N·mm
M	104.7 N·mm

On grease the same bearing has M = 76.7 N·mm.

The four-component model

Selected as "Four-component model (rolling, sliding, seals, drag)", the torque is

M = φ_{i s h} \cdot φ_{r s} \cdot M_{r r} + M_{s l} + M_{seal} + M_{drag}

Rolling: $M_{r r} = G_{r r} \cdot {(ν \cdot n)}^{0.6}$ , reduced by the inlet shear heating factor φish and the kinematic replenishment factor φrs, with $K_{r s} = 3 \cdot 10^{- 8}$ for oil and $6 \cdot 10^{- 8}$ for grease (grease starves sooner):
$\begin{matrix} φ_{i s h} & = \frac{1}{1 + 1.84 \cdot 10^{- 9} \cdot {(n \cdot d_{m})}^{1.28} \cdot ν^{0.64}} \\ φ_{r s} & = \exp (- K_{r s} \cdot ν \cdot n \cdot (d + D) \cdot {(\frac{K_{z}}{2 \cdot (D - d)})}^{0.5}) \end{matrix}$
Sliding: $M_{s l} = G_{s l} \cdot μ_{s l}$ , with the weighting φbl between boundary friction (0.12) and full film (μEHL 0.05; 0.02 cylindrical, 0.002 tapered roller bearings):
$\begin{matrix} μ_{s l} & = φ_{b l} \cdot 0.12 + (1 - φ_{b l}) \cdot μ_{E H L} \\ φ_{b l} & = \exp (- 2.6 \cdot 10^{- 8} \cdot {(n \cdot ν)}^{1.4} \cdot d_{m}) \end{matrix}$
Seals: the seal torque you enter.
Drag: in an oil bath up to the centre of the lowest element:
$1.093 \cdot 10^{- 7} \cdot n^{2} \cdot d_{m}^{3} \cdot {(\frac{n \cdot d_{m}^{2} \cdot f_{t}}{ν})}^{- 1.379} \cdot R_{s}$

Grr and Gsl come from each bearing's loads (the resultant of its element forces), dm and representative constants of one series per type (62, 72 B, 33 A, 12, QJ 2, 511, NU 2, 811, 302, 222). The churning term of a deep oil bath is not included.

Term (template)	Torque
Rolling, unreduced Mrr	112.2 N·mm
Rolling, φish · φrs · Mrr (0.937 · 0.950)	99.9 N·mm
Sliding, μsl = 0.050	132.4 N·mm
Drag	1.0 N·mm
Total	233.3 N·mm

Seal torque and power loss

Contact seals add a torque of their own, from the seal maker; roughly 20–100 N·mm per lip at 30–60 mm shaft diameter is typical. Enter it as Seal friction (N·mm). It is added in both models and in the permissible speed.

The power loss is $P = M \cdot ω = M \cdot π \cdot n / 30$ , in W with M in N·m. The template loses 31.8 W with Palmgren and 70.8 W with the four-component model. The project's Power loss table lists every bearing with its torque and loss, and the total.

Heat balance

Without a given temperature, a bearing settles where the heat it makes equals the heat it loses. With "Compute the temperature from a heat balance" the app finds the temperature T at which

friction power = K \cdot (T - T_{ambient}) + 28.5 \cdot V \cdot (T - T_{inlet})

with K the heat transfer through housing and shaft in W/K (Heat transfer) and V the circulating oil flow in l/min (Oil flow, Oil inlet). The 28.5 W per l/min and K is DIN 732, Eq. (12) for common mineral oils. Left empty, K comes from the ISO 15312 reference: the reference heat flow density qr = 0.016 W/mm² on the area $A_{r} = π \cdot B \cdot (D + d)$ at a 50 K temperature rise, reduced above Ar = 50 000 mm² (DIN ISO 15312, 5.3, Eq. (1), (5)).

For the template, Ar = 5 715 mm², the reference heat flow is 91.4 W and K = 1.83 W/K. With 20 °C ambient the bearing settles at 46.2 °C, where the thicker oil raises the friction to 157.5 N·mm and 47.8 W. A bearing whose friction outruns the heat removal at every temperature fails the check heatBalance: it overheats.

Thermal reference speed nθr (DIN ISO 15312)

The thermal reference speed is a rating that allows bearings to be compared, like C for life. It is the speed at which a bearing under fixed reference conditions reaches a heat balance with its outer ring at 70 °C in 20 °C surroundings (DIN ISO 15312:2004, Clause 5):

load P1r = 0.05·C0, radial, for radial bearings; 0.02·C0, central axial, for thrust bearings;
mineral oil without EP additives, νr = 12 mm²/s (radial) or 24 mm²/s (thrust) at 70 °C, oil bath to the centre of the lowest element;
horizontal axis, outer ring stationary, normal clearance, no contact seals;
heat removed through the reference area Ar at the reference heat flow density qr.

The speed solves the balance of Eq. (11), friction power $π \cdot n_{θ r} / (30 \cdot 10^{3}) \cdot [M_{0 r} + M_{1 r}]$ against qr·Ar, by iteration, with f0r and f1r of Table A.1. For the template, P1r = 765 N and nθr = 10 345 rpm.

Thermally permissible speed nθ (DIN 732)

The thermally permissible operating speed applies the same balance to the actual case: the speed at which the bearing's mean temperature reaches the permissible value under the real load, lubricant and cooling (DIN 732:2010, 3.1, Eq. (1)). DIN 732 writes it with a lubricant parameter KL (Eq. (5)), a load parameter KP (Eq. (6)) and the speed ratio $f_{n} = n_{θ} / n_{θ r}$ :

$K_{L} \cdot f_{n}^{5 / 3} + K_{P} \cdot f_{n} = 1$ (Eq. (2)), $n_{θ} = f_{n} \cdot n_{θ r}$ (Eq. (4)).

The heat removed is the sum of the heat through the seats, $Φ_{S} = k_{q} \cdot A_{S} \cdot (T - T_{A})$ (Eq. (8)), and the circulating oil's (Eq. (9) to (12)). The app takes:

the permissible temperature from Permissible temperature (nθ) (limits.permissibleTemperature), else 70 °C, and the viscosity at that temperature;
f0 times the lubrication method's factor, and f1·P1 of Table A.1 with P0/C0;
the seal torque;
the heat transfer of the heat balance, else the ISO 15312 reference, plus circulating oil.

It solves Eq. (2) exactly and reports the approximation of Eq. (3) alongside. For the template KL = 1.551, KP = 0.577 and fn = 0.5967 (Eq. (3): 0.5950), so nθ = 6 173 rpm at 70 °C. On grease, with the lower f0, nθ = 8 301 rpm.

The check thermalSpeed holds the operating speed against nθ. It is a warning only, unless a permissible temperature is set; then it can fail. nθ is a thermal limit only: cage strength, centrifugal forces and other kinematic limits are not included (DIN 732, 3.1). DIN 732 does not cover thrust ball bearings (Clause 1), and Table A.1 has no values for double-row deep groove and double-row caged cylindrical roller bearings; these get no thermal speeds, and thermalSpeedsNote says why. A matched pair of angular contact ball bearings is computed as its paired rows.

Minimum load

Rolling elements need enough load to roll. Below a minimum, the friction of the lubricant and the cage outweighs the traction in the contacts, the elements skid, and the surfaces smear. The app applies the bearing makers' rules:

radial ball bearings: $F_{r m} = k_{r} \cdot {(ν \cdot n / 1000)}^{2 / 3} \cdot {(d_{m} / 100)}^{2}$ kN, with the operating viscosity;
cylindrical roller bearings: $F_{r m} = k_{r} \cdot (6 + 4 \cdot n / n_{r}) \cdot {(d_{m} / 100)}^{2}$ kN, nr the reference speed (Reference speed nr (minimum load), else estimated as 550 000/dm);
needle and tapered roller bearings: 0.02·C; spherical roller bearings: 0.01·C0;
thrust bearings: P ≥ 0.01·C (ball) or 0.02·C (roller).

kr is typical of the type (deep groove 0.025, cylindrical 0.15, among others) unless you enter the catalogue's as Minimum load factor kr. The check minimumLoad compares the radial load (thrust bearings: P) with the minimum, for every bearing of a pair or set. A bearing held by a preload, or by an axial load of at least the minimum on raceways that take it, passes with a warning instead.

The template needs 131 N and carries 2 500 N. With the heat balance the oil at 46.2 °C is thicker and the minimum rises to 254 N. The NU 207 preset at 2 900 rpm needs 312 N.

From the command line

axm bearing update p.axm.json BR1 --patch '{"thermal":{"ambient":20},"lubricationDetails":{"frictionModel":"fourComponent"}}' --json
axm solve p.axm.json --json

bearings[].lubrication holds palmgrenTorque (idleTorque, loadTorque), fourComponent and powerLoss; thermal the heat balance; thermalSpeeds holds referenceSpeed, permissibleSpeed, lubricantParameter, loadParameter, speedRatio and speedRatioApproximation; maintenance the minimum load.

Where bearing friction comes from

Palmgren: M = M0 + M1

The four-component model

Seal torque and power loss

Heat balance

Thermal reference speed nθr (DIN ISO 15312)

Thermally permissible speed nθ (DIN 732)

Minimum load

From the command line

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