Equivalent bearing load

Why an equivalent load is needed

A load rating is defined for one kind of load only: a pure radial load for radial bearings, a central axial load for thrust bearings. Real bearings carry combinations of radial load, axial load and sometimes moments, and the loads change over time. The dynamic equivalent load P is the constant load of the rated kind that would give the same life as the real loads. With it, the life equation $L_{10}={(C/P)}^p$ of ISO 281 applies to any loading.

The catalogue formula: X, Y and e

ISO 281 gives one linear formula for every bearing type (Eq. (3) radial ball, (10) thrust ball, (14) radial roller, (20) thrust roller):

Radial roller bearings with α = 0 take $P_r=F_r$ (Eq. (15)) and thrust bearings with α = 90° take $P_a=F_a$ (Eq. (11), (21)). X and Y come from Tables 3, 5, 8 and 11. Each table has a limit e: while Fa/Fr stays at or below e, a single-row bearing has X = 1 and Y = 0, so the axial load does not count; above e, X drops below 1 and Y takes over. For radial roller bearings $e=1.5\cdot \tan \alpha$ (Table 8).

For deep groove ball bearings the factors depend on the relative axial load f0·Fa/C0r (Table 3), because an axial load pushes the balls up the groove flanks and increases the contact angle. Intermediate values are interpolated linearly. The tables hold for groove radii within the limits of ISO 281, 5.1.1; for other radii ISO 281 refers to ISO/TR 8646.

ISO/TR 1281-1, Clause 5 shows where X, Y and e come from. The theoretical equivalent load follows from the load distribution of a bearing with rigid rings and no clearance (5.1.1 to 5.1.3), and practical straight-line approximations of it give the tabulated factors (5.1.4 to 5.1.6, 5.2). The tables are therefore a fit to an idealised bearing. They do not see:

• the operating clearance, which narrows the load zone;
• tilt and moment loads, and the inner ring held square or free;
• the change of contact angle under load, beyond the relative axial load;
• preload of matched pairs;
• the roller profile and edge loading of roller bearings.

What the app uses: the reference load

The app computes P the way ISO/TS 16281 does, from the element loads of the solved internal load distribution:

1. Equilibrium of the inner ring gives every element's load Qj (balls) or every lamina's load (rollers).

2. Each ring gets a dynamic equivalent element load. Each point of a ring that rotates relative to the load passes through the whole load zone and meets every load level in turn, so a cube mean applies: $Q_e={(\sum Q_j^3/Z)}^{1/3}$ (ISO/TS 16281, Eq. (25), (28)). On a ring that stands still relative to the load, each point of the raceway is over-rolled again and again at the load of its own position, and combining the survival of these points gives a 10/3 mean: $Q_e={(\sum Q_j^{10/3}/Z)}^{3/10}$ (Eq. (26), (27)).

3. Each ring has a rating element load Qci, Qce derived from C and the groove radii (Eq. (19) to (24)). The basic reference rating life combines both rings (Eq. (29)):

   $$L_{10r}={[{\left(\frac{Q_{ci}}{Q_{ei}}\right)}^{-10/3}+{\left(\frac{Q_{ce}}{Q_{ee}}\right)}^{-10/3}]}^{-9/10}$$

4. The dynamic equivalent reference load is the load that gives this life in the ISO 281 equation: $P_{\mathrm{ref}}=C/L_{10r}^{1/3}$ for ball bearings (Eq. (30), (31)).

Roller bearings follow the same path lamina by lamina (Eq. (47) to (67)), with $P_{\mathrm{ref}}=C/L_{10r}^{3/10}$.

Pref is what the results call equivalentLoad, and it is the P in eC·Cu/P for aISO. It agrees with the X/Y load for a ball bearing without clearance under pure radial load: the verification suite checks that ISO/TS 16281 reduces to ISO 281 (Pr = Fr) there within 2 %. Under combined load, clearance or tilt the two differ, and the reference load is the better estimate.

For the bearing-6207 template (Fr = 2 500 N, Fa = 800 N, radial clearance 0.015 mm, inner ring held square) the app gives Pref = 2 474.6 N. The same bearing under other conditions:

Fr and Fa are the same in every row, so the X/Y formula would give the same P each time. The table shows what it misses: with this axial load a small clearance helps (the balls turn to a larger contact angle and the load zone widens), a large one hurts, and a ring free to tilt lives about half as long.

Variable loads: the duty cycle

Loads, speeds and temperatures change in most machines. ISO 281 itself treats constant conditions. The classical way out is a mean equivalent load, $P_m={(\sum u_i\cdot P_i^p/\sum u_i)}^{1/p}$ with ui the revolutions in each case. That works for the basic life when C is the same in every case, but not for the modified life: aISO depends on P, on κ (speed and temperature) and on eC, so it differs from case to case.

The app therefore combines lives, not loads. Each load case is solved on its own (element loads, Pref, aISO, its own speed and temperature), and the Palmgren–Miner rule adds the fatigue damage of the cases:

where ti is the share of operating time of case i (shares are normalised, so percentages or hours both work) and Li that case's life in hours at its own speed. Each case's damage share is $(t_i/L_i)\cdot L$. The same rule combines basic and modified lives; the life checks use the combined life, the static safety the worst case.

A duty cycle on the template, with 30 % "Full load", 60 % "Part load" (0.6 × loads at 1 450 rpm) and 10 % "Peak" (1.5 × loads):

The peak case runs a tenth of the time and does almost three quarters of the damage, which is typical: life goes with the third power of load (ball bearings) and the aISO falls with load as well. The combined basic life is 8 555 h, and the static safety is that of the peak case, s0 = 5.02.

A case with speed 0 is standstill: the bearings are checked statically (contact stress, static safety) and collect no revolutions, so it adds no fatigue damage. Factors on individual loads (factors) let a case scale some loads and not others.

From the command line

axm apply p.axm.json '{"op":"setLoadCases","cases":[{"name":"Full load","share":30},{"name":"Part load","share":60,"scale":0.6,"speed":1450},{"name":"Peak","share":10,"scale":1.5}]}' --json
axm solve p.axm.json --json

Each bearing then reports duty with the per-case lives and damageShare, and the combined basicLifeHours and modifiedLifeHours.

Related

• Rating life (ISO 281)
• Internal load distribution (ISO/TS 16281)
• Static load rating and static safety
• Bearings
• Open the bearing-6207 example