Bearing Life: Method

Bearing Life: Rating Life, Reliability, Equivalent Load and the ISO 281 Modified Life

What the tool does

Rolling-element bearings do not wear out at a fixed time: a group of identical bearings under the same load fails by rolling-contact fatigue over a wide spread of lives, so a bearing is rated statistically. The tool takes the catalogue ratings of a bearing, the loads and speed it sees, and the life and reliability the design calls for, and reports the equivalent radial load, the basic rating life L₁₀ in revolutions and hours, the life at any reliability from the Weibull distribution of bearing lives, the reliability of reaching a given life, the catalogue rating a bearing must have for the design life and reliability (with the margin of the entered rating), the reliability and life of a two-bearing shaft, the induced-thrust rules of a tapered-roller pair, the cubic-mean load of a duty cycle, and the ISO 281:2007 modified rating life from lubrication, contamination and the fatigue load limit. The method is that of ISO 281 / ABMA 9 and 11 as presented in Chapter 11 of Shigley's Mechanical Engineering Design (10th ed.), whose equation numbers are quoted; the default inputs are its Example 11-3.

Basic rating life

The basic dynamic load rating C₁₀ (catalogue C or C_r) is the constant radial load under which 90 percent of a group of bearings reach the rating life L_R, one million revolutions for ISO 281 / ABMA catalogues and 90 million revolutions (3000 h at 500 rev/min) for Timken. Under any other load P the 90-percent life follows the Lundberg–Palmgren load-life relation

L₁₀ = L_R (C₁₀/P)^a, a = 3 for ball bearings, a = 10/3 for roller bearings,

and in hours L_10h = L₁₀/(60 n) with n in rev/min (Shigley Eq. 11-1 and 11-3). P is the design load a_f F_e: the equivalent radial load multiplied by an application factor a_f (Shigley Table 11-5: 1.0 to 1.1 for precision gearing, 1.1 to 1.3 commercial gearing, 1.2 poor seals, 1.0 to 1.2 machinery without impact, 1.2 to 1.5 light impact, 1.5 to 3.0 moderate impact) that stands in for the load uncertainty.

Equivalent radial load

A bearing carrying a radial load F_r and an axial load F_a at the same time is rated with the equivalent radial load F_e = X V F_r + Y F_a (Shigley Eq. 11-12), where V is 1.0 when the inner ring rotates relative to the load and 1.2 when the outer ring does. For ball bearings the factors come from the ABMA table (Shigley Table 11-1): the limiting ratio e and the axial factor Y₂ depend on F_a/C₀, where C₀ is the basic static load rating; the tool interpolates linearly between the table rows (0.014 ≤ F_a/C₀ ≤ 0.56; outside that range the end row is used). When F_a/(V F_r) ≤ e the axial load is neglected (X₁ = 1, Y₁ = 0); above it X₂ = 0.56 and Y₂ is the interpolated value. Manufacturers' current tables index e and Y by f₀ F_a/C₀ with a bearing-specific f₀ (ISO 281:2007 Table 2); Shigley's table corresponds to f₀ ≈ 12.3 and is the one used here.

Cylindrical roller bearings carry no thrust in the rating, so F_e = V F_r. Tapered (single-row) and spherical roller bearings take the catalogue X, Y and e of the bearing, entered by the analyst; without them the radial load alone is used and the page says so. Thrust bearings under pure axial load have F_e = F_a. In every radial case an equivalent load below V F_r is replaced by V F_r.

Tapered roller pairs

The inclined raceways of a tapered roller bearing turn a radial load into a thrust on the shaft, F_i = 0.47 F_r/K (Shigley Eq. 11-18), where K is the Timken factor (the ratio of the radial to the thrust rating, about 1.5 as a first estimate). Two bearings mounted against each other therefore load each other axially. Bearing A is the one the external thrust F_ae is directed toward. If F_iA ≤ F_iB + F_ae, bearing A carries the net thrust, F_eA = 0.4 F_rA + K_A(F_iB + F_ae) and F_eB = F_rB (Eq. 11-19); otherwise F_eB = 0.4 F_rB + K_B(F_iA − F_ae) and F_eA = F_rA (Eq. 11-20). An equivalent load below the bearing's radial load is replaced by it. The tool rates both bearings for the design life and reliability, at R_D each and at √R_D each (the usual practice when R_D is the goal for the pair), and gives their lives when both catalogue ratings are entered. Because a catalogue bearing brings its own K, which changes the induced thrust and the other bearing's equivalent load, the selection is iterated (Shigley Examples 11-7 and 11-8): re-enter the catalogue K values and run again.

Duty cycles

For piecewise steady loading (Shigley Section 11-7) each row's equivalent load F_ei is formed as above, the fraction of the total turns spent in row i is f_i = t_i n_i/Σ t_j n_j, and the single load that does the same fatigue damage in the same number of turns is the cubic mean F_e = [Σ f_i (a_fi F_ei)^a]^1/a (Eq. 11-14), with the row application factors inside the mean. The speed used for the hours is the turns-weighted mean Σ t_i n_i. The cubic mean is the Palmgren–Miner rule written on the loads: the life it gives equals 1/Σ (f_i/L_i) with L_i the life under each load alone.

Reliability

Bearing lives follow a three-parameter Weibull distribution in the life multiple x = L/L₁₀: R = exp{−[(x − x₀)/(θ − x₀)]^b} (Shigley Eq. 11-5), with x₀ the guaranteed life, θ the characteristic life and b the shape parameter, all in multiples of L₁₀. Shigley's Table 11-6 gives x₀ = 0.02, θ = 4.459, b = 1.483 for the ISO / ABMA manufacturers (the parameters that make R = 0.90 at x = 1) and x₀ = 0, θ = 4.48, b = 1.5 for Timken; choosing the rating basis fills them, and they may be edited. The life at a required reliability is x_R = x₀ + (θ − x₀)[ln(1/R)]^1/b (Eq. 11-6); at 95 percent this gives 0.62 L₁₀, at 99 percent 0.22 L₁₀, close to the a₁ factors of ISO 281:2007 (0.64 and 0.25), which the page prints for comparison. The median life (R = 0.5) is about 3.5 L₁₀; ISO quotes "about five times" from its two-parameter slope.

The catalogue rating required for a design life L_D at reliability R_D follows by writing the design life in rating lives, x_D = 60 L_D n/L_R, and solving the load-life relation for the rating: C₁₀ = a_f F_e [x_D/x_R]^1/a (Eq. 11-9). With x_R = 1 (90 percent reliability) this is Eq. 11-3. Shigley's Eq. 11-10 replaces ln(1/R) by 1 − R, an approximation good above R = 0.90; the tool uses the exact logarithm, so its ratings differ from the book's by a fraction of a percent at R = 0.95. The margin of safety printed is C₁₀/C_10,required − 1 on the rating, not on the life: because of the exponent a, a 10 percent margin on the rating is a 33 percent margin on life for a ball bearing.

Two bearings on one shaft

A shaft runs only while both of its bearings run, so the system reliability at a life L is the product R₁(L) R₂(L) of the two Weibull reliabilities, each at its own life multiple. The tool gives the system reliability at the design life and the life at which the system reliability falls to R_D (bisection on the product), and beside it the ISO / JIS closed form L = (L₁^−e + L₂^−e)^−1/e with the two-parameter Weibull slope e = 10/9 for ball and 9/8 for roller bearings, which is what the catalogues print; the two differ because the three-parameter fit has a guaranteed life. When a pair is to reach R_D, each bearing is rated at √R_D (Shigley Examples 11-7, 11-8).

ISO 281:2007 modified rating life

Since 2007 the standard writes L_nm = a₁ a_ISO L₁₀, where a₁ is the reliability factor (1 at 90 percent, 0.64 at 95, 0.25 at 99, down to 0.077 at 99.95) and a_ISO is the "systems approach" factor that joins three effects: the lubricant film through the viscosity ratio κ = ν/ν₁, where ν is the oil's kinematic viscosity at the operating temperature and ν₁ = 45 000 n^−0.83 D_pw^−0.5 (n < 1000 rev/min) or 4500 n^−0.5 D_pw^−0.5 (n ≥ 1000) mm²/s is the viscosity that just gives adequate separation at the mean diameter D_pw = (d + D)/2 in mm; the contamination factor e_C for the cleanliness of the lubricant (Table 13 of the standard, from 1 for laboratory cleanliness to 0 for very severe contamination, in two size classes; the tool uses the middle of the band and prints the band); and the fatigue load limit C_u (SKF: P_u), the load at which the most heavily loaded contact reaches the fatigue stress limit of about 1.5 GPa. The factor rises steeply as e_C C_u/P grows and κ approaches 1 to 4, and falls toward 0.1 for poor lubrication or heavy loads. The standard's diagrams (Figures 1 to 4) are reproduced by its approximating equations, which the tool evaluates: for radial ball bearings a_ISO = 0.1 [1 − (2.5671 − A/κ^B)^0.83 (e_C C_u/P)^1/3]^−9.3 with (A, B) = (2.2649, 0.054381) for 0.1 ≤ κ < 0.4, (1.9987, 0.19087) for 0.4 ≤ κ < 1 and (1.9987, 0.071739) for 1 ≤ κ ≤ 4; for radial roller bearings 0.1 [1 − (1.5859 − A/κ^B)^0.4 (e_C C_u/P)^0.4]^−9.185 with (1.3993, 0.054381), (1.2348, 0.19087) and (1.2348, 0.071739); thrust ball bearings use C_u/(3P) in the ball form and thrust roller bearings C_u/(2.5P) in the roller form. A κ above 4 is taken as 4, a κ below 0.1 is outside the method, and a_ISO is limited to 50. The ball-bearing equation and its constants are as printed in Mitrović, Lazović and Marinković (2020) from the standard; the roller constants were checked against the JIS B 1518 / ISO 281 diagrams reproduced in the Koyo and NTN catalogues and for continuity across the κ bands.

Limits

The ratings are for through-hardened bearing steel of standard quality at moderate temperature; no temperature, material or speed factors are applied beyond a_ISO. Static capacity (C₀ against the maximum load), minimum load, misalignment, preload, grease life, and the catalogue's own f₀-indexed factor tables are not covered; a modified life above about 100 000 h is governed by failure modes the fatigue model does not contain. The table values are Shigley's, which reproduce the ABMA and Timken material; current catalogue values of C, C₀, C_u, K, X, Y and e should be taken from the manufacturer.