NSWC-11 Bearing Reliability Model

Rolling-element bearings are among the few mechanical parts designed for a finite life. Rolling-contact fatigue limits that life. The manufacturer states it as the L10 life. That is the number of revolutions that 90 percent of a group of identical bearings will complete or exceed before the first sign of fatigue. In service, though, wear, contamination, heat and shock usually end a bearing’s life before fatigue does.

This tool follows Chapter 7 of the Naval Surface Warfare Center Handbook of Reliability Prediction Procedures for Mechanical Equipment (NSWC-11). It takes the L10 life as the base failure rate. It then multiplies by seven factors: load, the reliability you need, lubricant viscosity, water in the oil, operating temperature, service condition and lubricant cleanliness. The result is a failure rate in failures per million hours, with FIT and MTBF, ready to drop into a FMECA.

You can start from the catalog dynamic load rating (the handbook’s Eq. 7-4) or from a published L10 life (Eq. 7-5). Every step of the working is available under “Show detailed calculation steps”, and the sections below explain each equation and chart.

NSWC-11 Chapter 7 Failure Rate Calculator
Ball & Roller Bearing Reliability (NSWC-11 Eq. 7-4 / 7-5)
Method & Bearing Type
Use the load rating when the catalog gives you C. Use the published life when the manufacturer quotes an L10 in hours or revolutions.
Load & Speed
The radial load for one million revolutions at 90 % reliability. Use one force unit for LS, FR and FA. Only their ratio matters.
Equivalent radial load LA = X·FR + Y·FA (Eq. 7-3). X and Y come from the bearing catalog. For a pure radial load leave X = 1, Y = 0 and FA = 0, or enter the equivalent load directly as FR.
Reliability
The handbook says to use L50 when the failure rate is combined with other MTBF-based components in an assembly. L10 is the catalog rating.
Lubricant & Environment
The viscosity the bearing was rated for, at operating temperature.
Same unit as νO (cSt, cP, or lbf·min/in²). Only the ratio matters.
Use 0 for water-based lubricants or dry oil.
The handbook figure is in °C and the factor starts above 183 °C (361 °F).
Service Condition & Cleanliness
Picks the column in Table 7-4.
Mission (optional)
8,760 h is one year of continuous operation. Leave blank to skip.
Failure Rate & Reliability Metrics
Predicted Failure Rate, λBE (failures / 106 h) iλBE = λBE,B × (CY) × CR × Cν × CCW × Ct × CSF × CC. The base failure rate after every correction factor. CY applies only when you start from a published L10 life.
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Failure Rate (FIT) iFailures In Time = failures per 109 hours. The same number as failures per million hours, multiplied by 1000.
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MTBF (hours) iMean time between failures = 106 / λBE. It assumes a constant failure rate, which is only a rough match for a fatigue wear-out part. Treat it as a way to compare designs and to feed a FMECA.
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MTBF (years) iMTBF in hours divided by 24 × 365.25.
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Expected Failures over the Mission iλ × mission hours / 106. It is the expected count of failures for one bearing, so a value above 1 means one failure is more likely than not.
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Survival Probability over the Mission iR(t) = e−λt, the chance one bearing survives the mission with no failure, assuming the constant failure rate above.
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Life & Load Intermediates
Equivalent Radial Load, LA (lbf)
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Load Ratio, LA / LS iThe handbook suggests LA is usually around 0.5 LS for a preliminary estimate. A lightly loaded bearing has a small ratio and a long life.
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Calculated L10 life (hours)
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Base Failure Rate, λBE,B (per 106 h) i1 / L10h, expressed per million hours.
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Combined Multiplier, Π factors iThe product of every multiplying factor. It is how many times worse than the base failure rate this bearing’s operating conditions are.
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Multiplying Factors
SymbolMeaningValue
Show detailed calculation steps
How the Model Works

NSWC-11 does not try to predict a bearing’s fatigue life from its internal geometry. That is hard to do and depends on dozens of design details. It starts from a number the bearing maker already gives you, the L10 life (or the dynamic load rating that produces it). It turns that life into a base failure rate and then corrects the rate for how the bearing is really used.

1. The two equations

\[ \lambda_{BE} = \lambda_{BE,B}\cdot C_R \cdot C_\nu \cdot C_{CW} \cdot C_t \cdot C_{SF} \cdot C_C \qquad\text{(Eq. 7-4, from the load rating)} \] \[ \lambda_{BE} = \lambda_{BE,B}\cdot C_Y \cdot C_R \cdot C_\nu \cdot C_{CW} \cdot C_t \cdot C_{SF} \cdot C_C \qquad\text{(Eq. 7-5, from a published life)} \]
SymbolMeaning
λBEPredicted bearing failure rate, failures per million operating hours
λBE,BBase failure rate, 1 / L10h, expressed per million hours
CYApplied-load factor, (LA/LS)y. Only in Eq. 7-5
CRLife adjustment for the reliability level you need
CνLubricant viscosity factor
CCWWater-in-lubricant factor
CtOperating temperature factor
CSFService factor for shock and vibration
CCLubricant contamination factor

2. Why there are two equations

The base failure rate is 1 / L10h, so it depends on which L10 you hold.

  • Eq. 7-4 (load rating). You know LS and the real load LA. The L10 life is computed at the real load (Eq. 7-1 and 7-2), so the load is already inside the base rate. No separate load factor is needed.
  • Eq. 7-5 (published life). You hold an L10 that the manufacturer measured at the rated load. The base rate is for that rated condition, so CY scales it to the load you actually apply.

The two agree when the published life is the life at rated load. The calculator’s test suite checks this: feeding Eq. 7-5 the rated-load life at the same speed gives the same answer as Eq. 7-4.

3. Step by step

  1. Combine the radial and thrust loads into the equivalent radial load LA = X·FR + Y·FA.
  2. Find the L10 life in millions of revolutions, (LS/LA)y, and convert it to hours at your speed.
  3. Take the reciprocal, scaled to per million hours, as the base failure rate.
  4. Multiply by each factor in turn. Each equals 1.0 when its condition matches what the bearing was rated for.
  5. Convert the result to FIT (×1000), MTBF (106/λ) and, if you give a mission time, the expected failures and survival probability.
Read the result with care. A bearing is a wear-out part, so a constant failure rate (and an MTBF) describes it only roughly. The model is an engineering estimate for comparing designs and populating a FMECA, not a life guarantee. The handbook warns that limited funding prevented full validation of every prediction equation.
Factor Reference Guide
NSWC-11 Chapter 7: Bearings
λB

L10 Life & Base Failure Rate

Equations 7-1, 7-2 and the base rate
Eq. 7-1 & 7-2 \[ L_{10} = \left(\frac{L_S}{L_A}\right)^{y}\ \text{million rev} \qquad L_{10h} = \frac{10^6}{60\,n}\left(\frac{L_S}{L_A}\right)^{y}\ \text{h} \] \[ \lambda_{BE,B} = \frac{10^6}{L_{10h}} = 60\,n\left(\frac{L_A}{L_S}\right)^{y}\ \text{failures}/10^6\,\text{h} \]

LS is the dynamic load rating: the constant radial load a group of bearings can carry for one million revolutions with 90 % surviving. LA is the equivalent radial load in service. Life falls with the cube of the load for ball bearings (y = 3.0) and with the 3.3 power for roller bearings. A modest rise in load costs a large share of the life. The exponent difference reflects line contact in roller bearings against point contact in ball bearings.

Dividing by 60 n converts revolutions to hours. The base failure rate is the reciprocal of that life. The handbook writes it in failures per million hours, so the calculator multiplies by 106. The second form shows that the base rate rises linearly with speed and with the cube of the load ratio.

Check yourself. A ball bearing with LS = 9000 lbf carrying 1000 lbf has L10 = 93 = 729 million revolutions. At 1750 rpm that is 729×106/(60×1750) = 6,943 hours.
Using the published life instead

Many catalogs quote an L10 life directly. When you have it, use Eq. 7-5. The base rate is 106/L10h of the published life, and CY (below) accounts for the load you really apply versus the rated load. If the life is given in millions of revolutions, the calculator converts it to hours at your operating speed.

LA

Equivalent Radial Load

Equation 7-3
Eq. 7-3 \[ L_A = X\,F_R + Y\,F_A \]

Catalog ratings are for pure radial load. When a thrust load FA is present, it is converted to the radial load that would give the same theoretical fatigue life. X is the radial factor and Y the thrust factor. Both depend on the contact angle. Y also depends on the thrust load and on the number and size of the rolling elements. Take both from the manufacturer for your bearing. With a pure radial load, LA equals FR.

The handbook says FA should not exceed 30 % of FR. The calculator warns when it does.
CY

Applied Load Factor: CY

Equation: NSWC-11 Figure 7.2
Eq. CY \[ C_Y = \left(\frac{L_A}{L_S}\right)^{y},\qquad y = \begin{cases}3.0 & \text{ball}\\ 3.3 & \text{roller}\end{cases} \]

This is the same load dependence that sets the L10 life, applied as a failure-rate multiplier. At the rated load (LA = LS) it equals 1. Run at half the rated load and a ball bearing’s failure rate drops to one eighth, since 0.53 = 0.125. CY applies only when you start from a published L10 life (Eq. 7-5).

CY vs load ratio (NSWC-11 Fig. 7.2)
Log scale. The red dot is your load ratio on the curve for your bearing type.
CR

Reliability Factor: CR

Equation: NSWC-11 Table 7-2
Eq. CR \[ C_R = \frac{0.223}{\left[\ln\!\left(\dfrac{100}{R}\right)\right]^{2/3}} \]

The L10 life is the life that 90 % of bearings survive. You may need a higher survival rate, such as 99 % for a safety-critical application. The usable life is then shorter, so the failure rate is multiplied up. If you want the median (L50), the life is longer and CR falls below 1.

Interpretation (not stated in the handbook). The formula equals L10/LR for a Weibull life distribution with a shape (slope) of 1.5. The check is that 1/1.5 = 2/3 and 0.223 = [ln(100/90)]2/3. That is a useful way to remember it, but the calculator uses the handbook formula as printed.
Table 7-2
Reliability R %LaCR (printed)
90L101.00
95L51.62
96L41.88
97L32.29
98L23.01
99L14.79
50L500.29
The calculator evaluates the formula, so values between table rows are available. It gives 0.285 at 50 %, which the handbook rounds to 0.29.
When to use which level
  • L10 (90 %). The standard catalog rating. CR = 1.
  • L50 (50 %). The handbook’s choice when combining with other MTBF-based components in an assembly, because MTBF is an average.
  • L5 to L1. For applications such as aircraft engines where safety is an issue and the reliability must exceed 90 %.
Cν

Lubricant Factor: Cν

Equation: NSWC-11 Eq. 7-6, Figure 7.3
Eq. 7-6 \[ C_\nu = \left(\frac{\nu_O}{\nu_L}\right)^{0.54} \]

The oil film that separates the rolling elements from the races gets thinner as viscosity falls. νO is the viscosity of the specification lubricant the bearing was rated with. νL is the viscosity of what you actually use. Both are at operating temperature. If your oil is thinner than the specification, the ratio exceeds 1 and so does the factor. A thicker oil gives a factor below 1.

Only the ratio matters, so enter both viscosities in any one unit. The handbook prints the unit as lb-min/in². Centistokes or centipoise work as long as both entries match. Remember to use the viscosity at the bearing’s operating temperature, not at 40 °C, unless that is where it runs.

Cν vs νO/νL (NSWC-11 Fig. 7.3)
CCW

Water Contamination Factor: CCW

Equation: NSWC-11 Eq. 7-7, Figure 7.4
Eq. 7-7 \[ C_{CW} = \begin{cases} 1 + 25.5\,C_W - 16.25\,C_W^{2} & C_W \le 0.8\,\% \\[4pt] 11.0 & C_W > 0.8\,\% \end{cases} \]

Water leaking into oil lubrication cuts fatigue life sharply. CW is the percentage of water in the lubricant. Even 0.1 % water raises the failure rate by about 3.4 times, and 0.5 % by about 9.7 times. The handbook derived the curve from published life data and caps it at 11.0 above 0.8 %. The quadratic reaches exactly 11.0 at 0.8 %, so the curve is continuous.

For bearings designed for water-based lubricants, use CW = 0 so that CCW = 1.00.

CCW vs water content (NSWC-11 Fig. 7.4)
Ct

Temperature Factor: Ct

Equation: NSWC-11 Figure 7.5
Eq. Ct \[ C_t = \begin{cases} 1.0 & T_O < 183\,^\circ\text{C} \\[4pt] \left(\dfrac{T_O}{183}\right)^{3} & T_O \ge 183\,^\circ\text{C} \end{cases} \]

Heat thins the lubricant, which makes more heat as it loses the ability to support the load. Residue on the bearing parts can harden and stop the grease or oil from lubricating, and can add solid particles. Below 183 °C (361 °F) the handbook applies no penalty. Above it the factor grows with the cube of the temperature ratio, reaching about 2.9 at 260 °C.

The equation uses temperature in degrees Celsius, as printed. If you enter °F the calculator converts first.
Ct vs operating temperature (NSWC-11 Fig. 7.5)
CSF

Service Factor: CSF

Table 7-3: Bearing Service Factors

The real load on a bearing can exceed the calculated load because of vibration and shock. The service factor adjusts the failure rate for the type of application. The handbook gives separate columns for ball and roller bearings, and leaves some applications blank for one type. The calculator only offers the rows that are tabulated for your bearing type.

Type of applicationBallRoller
Uniform and steady load, free from shock1.01.0
Normal operation, light shock load1.51.0
Moderate shock load2.01.3
Heavy shock load2.51.7
Extreme and indeterminate shock load3.02.0
Precision gearing1.2–
Commercial gearing1.3–
Toothed belts–1.2
Vee belts–1.8
Flat belts–3.0
Handbook references 57 and 119. A dash means the handbook leaves the cell blank.
Choosing a row
  • Pick the row that matches the equipment the bearing sits in, not the bearing itself.
  • In the handbook, the roller column is lower than the ball column for the shock-load rows.
  • If your application is not listed, choose “Custom CSF value” in the calculator and enter your own, for example from the bearing maker.
CC

Lubricant Contamination Factor: CC

Table 7-4: Bearing Contamination Level

Hard particles in the lubricant dent the smooth bearing surfaces. The rough surfaces raise contact stress and shorten life, so the quality of the filtration system has a real effect. The handbook’s factors are lower for bearings over 100 mm, so the table has two columns by bearing diameter.

Contamination condition< 100 mm> 100 mm
Extreme cleanliness: particle size about the lubricant film thickness (laboratory conditions)1.01.0
High cleanliness: oil filtered through a fine filter, 10 micron or finer1.41.2
Normal cleanliness: slight contamination in the lubricant1.81.4
Slight contamination: hard particles larger than 10 micron2.52.0
Severe contamination: coarse filtering, no integral seals5.03.3
Handbook reference 112 (NSK).
Practical reading
  • A bearing with integral seals, in clean oil through a fine filter, sits near the top of the table.
  • A bearing sharing an oil system with gears, with only a coarse screen, is in the bottom two rows.
  • The factor can move the failure rate more than any other single choice, so it is worth being honest about this input.
T 7-1

Typical Modes of Bearing Failure

The two main failure modes are wear and fatigue. A well-lubricated, well-sealed bearing at moderate load and speed ends its life by fatigue, which is what the L10 life describes. Real operating conditions are rarely that kind, so wear has to be considered too. The multiplying factors above are how the model accounts for it.

Failure modeFailure mechanismFailure cause
Fatigue damage, noisy bearing, bearing seizure, bearing vibration, presence of electric currents
  • Spalling of ball/roller raceway
  • Brinelling
  • Smearing
  • Surface fatigue
  • Glazing
  • Microspalling of stressed surfaces
  • Crack formation on rings and balls or rollers
  • Skidding
  • Scuffing
  • Fretting
  • Pitting of surfaces
  • Pits on raceways and balls, corrosion
  • Heavy, prolonged load
  • Excessive speed
  • Shock load
  • Excessive vibration
  • Loss of lubricant
  • Housing bore out of round
  • Corrosive agents
  • Distorted bearing seals
  • Inadequate heat removal capability
  • High temperature
  • Misalignment
  • Unbalanced/excessive load
  • Inadequate housing support
  • Extensive pitting caused by electric current
Condensed from handbook Table 7-1 (reference 121). Bearing failure can also be caused by excessive shaft bending. Chapter 20 covers shaft deflection.
Worked Example

A ball bearing in a conveyor gearbox has a catalog dynamic load rating LS = 9,000 lbf. It carries a pure radial load of 1,000 lbf at 1,750 rpm. The unit has light shock loading. The specification oil is 32 cSt but the oil in use is 25 cSt at operating temperature, with 0.05 % water and normal cleanliness. The bearing is under 100 mm and runs at 90 °C. The failure rate is wanted for a FMECA, so the L50 level is used. These are the calculator’s default inputs.

Step-by-step
  • LA = 1×1000 + 0×0 = 1,000 lbf (pure radial)
  • L10 = (9000/1000)3 = 729 million revolutions
  • L10h = 106/(60×1750) × 729 = 6,943 h
  • λBE,B = 106/6943 = 144.0 failures per 106 h
  • CR (R = 50 %) = 0.223/[ln 2]2/3 = 0.2847
  • Cν = (32/25)0.54 = 1.1426
  • CCW = 1 + 25.5(0.05) − 16.25(0.05)2 = 2.2344
  • Ct = 1.0 (90 °C < 183 °C)
  • CSF = 1.5 (ball bearing, light shock, Table 7-3)
  • CC = 1.8 (normal cleanliness, under 100 mm, Table 7-4)

λBE = 144.0 × 0.2847 × 1.1426 × 2.2344 × 1.0 × 1.5 × 1.8 ≈ 282.7 failures / 106 h.

That is 282,681 FIT and an MTBF of 3,538 h (0.40 years). Over a year of continuous running (8,760 h) the expected failure count is 2.48 and the survival probability is 8.4 %.

The combined multiplier is 1.96. The water allowance is the biggest single factor here, because it more than doubles the rate. Moving to 0.01 % water and high cleanliness would cut the multiplier from 1.96 to about 0.85 without changing the bearing.

Important Notices
  • Not an official DoD document. NSWC-11 is the product of a Naval Surface Warfare Center research program, approved for public release. The handbook cautions that limited funding prevented full validation of every prediction equation. It should not be treated as an official Department of Defense standard.
  • No Navy affiliation or endorsement. The Naval Surface Warfare Center, Carderock Division and the U.S. Navy have not participated in the development of this calculator and do not approve or endorse it.
  • Use with the full procedure. NSWC-11 warns against extracting equations without regard to application procedures and parameter limits. Results are a design screening tool, not a substitute for the bearing manufacturer’s life calculation, testing, or the judgment of a qualified engineer.
  • Rolling-element bearings only. This chapter covers ball and roller bearings. Plain (sliding) bearings and bushings are covered by other parts of the handbook.
  • Constant failure rate. An MTBF assumes a constant failure rate. Fatigue is a wear-out mechanism, so the estimate is only a rough guide over long missions.
  • Catalog values. The dynamic load rating and the X and Y factors should come from the manufacturer’s catalog for the exact bearing. Different catalogs rate bearings on different bases, so check that the rating is for one million revolutions.

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