Bearing examples you can open and explore

1. How nine balls share a radial load

Inputs: 100 lbf radial load; no thrust; zero initial clearance and contact angle; circular rings; inner ring at 60 rpm, outer ring stationary.

Only the loaded half of the bearing carries inner contact load. With nine equally spaced balls and one aligned with the force, five balls are loaded. The normal loads follow the Hertz/Stribeck distribution:

Q(φ) / Qmax = max(cos φ, 0)^3/2
Qmax = Fr / Σ max(cos φj, 0)^5/2

The finite nine-ball sum predicts a maximum inner load of 48.7245 lbf at 100 lbf total radial load. The development solver gives 49.0305 lbf, with a 0.628% resultant-force offset. The normalized load shape agrees within 0.000005 of peak load.

What to explore

Inspect the five loaded balls and compare their normalized loads. The finite-ball factor is 4.385208; NASA’s approximate 4.37 is a useful reference, rather than an exact factor for every ball count. The separate empirical factor 5 includes clearance and geometry allowances.

NASA SP-38, printed pp.165–166 / PDF pp.169–170, Eqs.6-52 through 6-60.

2. Equal ball loads under uniform thrust

Inputs: 100 lbf thrust; no radial load; nine balls; 30° initial contact angle; inner ring at 60 rpm, outer ring stationary.

Every ball carries the same normal load. Its axial contribution depends on the operating contact angle, which increases as the bearing deflects:

Fa = Σ Qj sin αj = Z Q sin α

The solved inner contact angle is approximately 30.7735°, and each ball carries approximately 21.8446 lbf. Projecting those normal loads gives a 0.590% difference from the requested thrust, retained as the legacy equilibrium offset.

What to explore

Compare the equal ball loads and the initial versus operating angle. Using 30° throughout would miss the change in force projection under load. This check establishes symmetry and projection; it does not qualify the reported fatigue life.

NASA SP-38, printed p.166 / PDF p.170, Eq.6-61.

3. Cage motion with a rotating outer ring

Inputs: stationary inner ring; outer ring at 600 rpm; 100 lbf thrust; nine balls and a 30° nominal contact angle.

Both ring speeds contribute to nominal cage motion. With ball diameter D, pitch diameter Dm and nominal angle α0:

k = (D / Dm) cos α0
nc = [ni(1 − k) + no(1 + k)] / 2

The ideal value is approximately 353.97238 rpm; the reported nominal cage speed is 353.97225 rpm. The tiny difference comes from the preserved input/output rpm conversion constants.

What to explore

Change which ring rotates and follow the change in cage speed. Level 1 reports ideal nominal-angle kinematics. It does not solve lubricated cage slip or traction, and this angular-contact teaching model is not a conveyor-idler design.

NASA SP-38, printed pp.159–160 / PDF pp.163–164, Eq.6-44.

4. Radial stiffness increases with load

A2 geometry: nine 3/16-inch balls; inner/outer groove ratios 0.570; 0° initial contact angle; bore 0.5906 inch; outside diameter 1.2598 inch.

The report’s program defines pitch diameter as the mean of bore and outside diameter: 0.9252 inch. This is its stated model rule, rather than a measured internal drawing.

At 100 and 700 lbf radial load, the source gives 288,000 and 555,000 lbf/in. The calculated tangents are 304,417 and 581,870 lbf/in. In a Hertz radial limit, deflection scales as F^2/3 and tangent stiffness as F^1/3. Increasing load sevenfold predicts a stiffness ratio of 1.91293; the calculated ratio is 1.91143.

What to explore

Compare the low- and high-load deflections, then estimate a tangent from small load changes. Multiplying force without accounting for nonlinear deflection does not give the incremental stiffness.

AFAPL Part IV, Table I, printed p.17 / PDF p.26; worked problem 1, printed p.59 / PDF p.68; Appendix A, Eqs.A-1 to A-5.

5. Thrust stiffness depends on contact-angle change

Inputs: A2 dimensions above; 10° initial contact angle; pure thrust of 100 or 700 lbf.

The chart discussion assigns 0° to the radial family and 10° to the thrust family. The operating angle then increases under load. Reusing the radial case’s 0° initial angle would compare different bearing states.

The source gives 91,000 and 322,000 lbf/in. Whole-bearing reaction/displacement derivatives give 96,009 and 336,782 lbf/in, approximately 5.50% and 4.59% higher.

What to explore

Follow the operating angle and axial displacement as thrust increases. Calculate a central derivative from small force/displacement changes. In these uniform-thrust cases, the final legacy report’s axial stiffness value is about one ninth of the whole-bearing derivative. The independent comparison uses the derivative; the report discrepancy remains to be investigated.

AFAPL Part IV, chart scope printed p.14 / PDF p.23; Fig.7, printed p.28 / PDF p.37; worked problem 2, printed p.59 / PDF p.68.

6. Two opposed preloaded bearings add their tangents

Inputs: two opposed A2 bearings; 10° initial angle; 200 lbf axial preload per member.

A small axial shift increases load on one member and reduces it on the other. Their restoring-force derivatives add. At symmetric preload, the pair tangent is 2 × the single-member tangent.

The calculated member tangent is approximately 148,149 lbf/in, giving 296,298 lbf/in for the pair. The source gives about 280,000 from summed tangents. It also gives 267,000 from rounded chart deflections; those two approximations are retained separately.

What the input represents

The downloadable document models one 200-lbf member. The comparison combines two member derivatives analytically. It does not infer an unspecified duplex span, mounting arrangement or assembly moment stiffness.

AFAPL Part IV, worked problem 3, printed pp.59–60 / PDF pp.68–69; Figs.7–8.

7. Preload is a displacement constraint

PA2 geometry: thirteen 3/16-inch balls; groove ratios 0.570; 15° initial angle; pitch diameter 0.9252 inch; 50-lbf medium axial preload.

The source establishes axial displacement under pure preload, then holds that displacement fixed while radial load changes. Holding a 50-lbf axial force instead would solve a different boundary condition.

The development-solver baseline axial displacement is approximately 0.000770606 inch. Reaching that displacement at 100 lbf radial load requires a mapped requested thrust of approximately 56.1533 lbf; at 700 lbf radial load, approximately 255.9801 lbf. The audit remaps thrust for every radial perturbation.

The resulting radial tangents, 531,526 and 701,917 lbf/in, remain about 19% above the source’s 445,000 and 590,000. The documents are therefore research comparisons. Current material treatment and the finite thirteen-ball solution differ from the historical model and its continuous circumferential integration; their individual contributions have not been established.

What to explore

Compare fixed-displacement and fixed-force behavior. Each saved input is one mapped equilibrium point. Changing its radial load while retaining the saved thrust changes the boundary; remap thrust before making another source comparison.

AFAPL Part IV, Table II, printed p.18 / PDF p.27; Fig.30, printed p.52 / PDF p.61; worked problem 4, printed pp.60–61 / PDF pp.69–70; fixed axial displacement in Appendix A, printed p.66 / PDF p.75, and the program loop, printed p.73 / PDF p.82.

8. A published 218 bearing under centered thrust

Ricci (2009) inputs: 16 balls; ball diameter 22.23 mm; pitch diameter 125.26 mm; both groove radii 11.63 mm; free diametral clearance 0.48 mm; 17,800 N axial force; zero eccentric moment.

The free clearance gives an initial contact angle of approximately 39.9156°. In this document that angle represents the source clearance, so the additional clearance adjustment is zero. The input uses the app’s English-unit document format and a 60-rpm reporting surrogate.

Ricci states a centered-thrust axial deflection of 36.0111 µm. An independent implementation of the paper’s static contact equations reproduces 36.0111 µm. The development solver gives 35.7780 µm, approximately 0.647% lower.

The independent calculation gives an operating angle of 41.4180° and 1,681.66 N normal load per ball. Those are calculated reference quantities, rather than separately quoted experimental measurements. The app’s projected axial reaction is 17,913.15 N, 0.636% above the requested force.

What to explore

Inspect equal ball loads and the increase in contact angle under thrust. Compare deflection with the stated source value. Ricci uses E = 207.5 GPa and ν = 0.3; the active inherited contact mode uses approximately 209.618 GPa and 0.295973. The force offset and material difference are both retained; their separate contributions to the deflection difference have not been established.

The independent check also covers quadrature refinement and the circular Hertz limit. Re-solving the input at 6 rpm changes axial displacement by less than 0.00001%. Eccentric loading, fatigue life and thermal behavior remain outside this checked case.

Mário César Ricci, Internal Loading Distribution in Statically Loaded Ball Bearings Subjected to an Eccentric Thrust Load (2009). Geometry: Section 5, p.13; thrust: Section 5.1, p.14; exact centered deflection: Figure 12 discussion, p.17. Read the paper.

Cylindrical rollers: compare methods at the same imposed motion

Krantz (2011), Case B: 13 straight rollers; 7.5 mm diameter; 8.6 mm length; 39 mm pitch diameter; zero clearance; E = 207 GPa; ν = 0.3. The outer ring is displaced radially by 0.010 mm.

This is a useful displacement-controlled benchmark: the paper reports different restoring forces and stiffnesses for analytical and finite-element treatments of the same case. Keep each method’s target separate.

Swipe the table horizontally to see every column.

What remains to check

Reproduce the paper’s finite-ring geometry, mounting and two-dimensional assumptions before comparing against its finite-element results. A rigid-ring Hertz calculation alone does not reproduce those boundaries. No program agreement or downloadable whole-bearing case is claimed yet.

Timothy L. Krantz, On Calculation Methods and Results for Straight Cylindrical Roller Bearing Deflection, Stiffness, and Stress (2011), Tables 7–8, pp.8–9. Read the NASA PDF.

Temperature and fit: how mounting changes ball loads

Ricci (2010): 218 bearing, 90 mm bore and 160 mm outside diameter; hollow steel shaft with 63.5 mm bore and k6 fit; titanium housing with 203.2 mm effective outside diameter and M6 fit.

The paper compares unfitted, fitted and hot conditions under combined loading. Assembly temperature is 21.1°C; the hot inner and outer rings are 148.9°C and 121.1°C. It offers a useful example of clearance changing with interference and thermal expansion.

What remains to check

The internal geometry, exact fit values, expansion properties and thermal definitions rely on supporting references. Recover and reconcile them before building the thermal case. The 2009 paper’s 218 geometry must not be assumed identical solely because the designation matches.

M. Ricci, Static Load Distribution in Ball Bearings Including the Effects of Temperature and Fit (2010), Section 3, p.228. Read the paper.

Matched angular-contact bearings: preload and arrangement

Gu and colleagues (2022), Table 3: NSK7206B; 13 balls of 8.8 mm diameter; 46.0855 mm pitch diameter; 40° contact angle; both groove radii 4.6041 mm; 480 N preload.

The paper compares back-to-back and face-to-face arrangements under combined loads and reports numerical stiffness comparisons. It can extend the simple symmetric pair example into radial, axial and tilting behavior.

What remains to check

Verify the pair constraints, preload convention, spacing and load units. Table 1’s separate validation geometry must be kept apart from Table 3’s NSK7206B study. A complete pair model is needed before offering its combined-load cases.

Jianguo Gu and colleagues, Modeling and Mechanical Characteristics Analysis of Matched Angular Contact Ball Bearings under Combined Loads (2022), Tables 1–3. The link opens the publisher’s article.

Spherical rollers: published geometry and defect response

Ghalamchi, Sopanen and Mikkola (2016), Table 1: 22216-EK; two rows of 21 rollers; 15.544 mm roller diameter; 132 mm pitch diameter; 8.25° contact angle; 41 µm clearance; E = 206 GPa; ν = 0.3.

The article supplies internal geometry and considers localized defects, contact stiffness and measured vibration frequencies. Its examples include 750-rpm operation and defect depths from 0.05 to 0.15 mm.

What remains to check

Implement and verify spherical contact and transient defect response, then recover the rotor and support conditions needed for the experiments. These dimensions belong to the published 22216-EK case and must not be reused as the internal geometry of an SKF 230/900.

Behnam Ghalamchi, Jussi Sopanen and Aki Mikkola, Modeling and Dynamic Analysis of Spherical Roller Bearing with Localized Defects: Analytical Formulation to Calculate Defect Depth and Stiffness (2016), Table 1 and Sections 4–5.

Clearance changes how many balls carry the load

Hou, Yin and Wang (2024), 6209: nine 12.7 mm balls; race diameters 52.292/77.706 mm; groove radii 6.6 mm; one ball aligned with the radial load.

Compare clearance with interference at the same force. At 1,000 N, three balls carry load with +0.015 mm diametral clearance, while seven carry load with −0.010 mm interference. The interference case includes contacts whose radial force projections oppose the applied load.

Swipe the table horizontally to see every column.

What to explore and check

Project each ball load along the force direction. The positive-clearance columns balance within 0.0001%; the interference columns retain residuals up to about 0.009875%. These source-table residuals remain distinct from Maven’s results. The repaired static calculation uses half the diametral clearance at each radial gap and reactivates contacts as the rings move. All six loaded-ball counts agree, and independently projected Maven force residuals are below one part per million.

Peak agreement does not establish every contact. At +0.015 mm / 5,000 N, the 80° load is 0.297 N versus the source’s 0.215 N. At −0.010 mm / 1,000 N, the 120° load is 33.955 N versus 34.469 N. Maven’s steel law, elliptic iteration and deformed pitch differ from the source’s fixed contact coefficient. Life and thermal predictions remain outside this comparison.

Yu Hou, Yi Yin and Xi Wang, A Discrete Calculation Method for Radial Load Distribution Integral of Radial Bearings (2024), Tables 1–3. The paper also gives NU209 roller cases; their remaining geometry requires recovery.

Two worked NASA cases: clearance and maximum element load

Oswald, Zaretsky and Poplawski (2012): 210-size ball and cylindrical-roller bearings; steel E = 205.9 GPa and ν = 0.3. Appendix D works through approximate load-distribution calculations.

Swipe the table horizontally to see every column.

The ball case uses 12.7 mm balls, 70 mm pitch diameter and groove ratio 0.52. The roller case uses 13 mm diameter and nominal length, with 70.65 mm pitch diameter. Its crowned-roller approximation uses an effective length of 11.05 mm, 85% of nominal length.

What to explore and check

Use St = Z Qmax / Fr to connect maximum element load with bearing load, then compare the source’s approximate method with its separate analysis-code result. The ball approximation is stated to be 2.5% higher; the roller approximation agrees within 0.01%. Keep those methods separate.

The ten-ball input now solves with three loaded balls: Maven gives peak 335.679 N and St 5.353737. Independent finite-ball equilibrium with the source’s rounded contact coefficient gives 335.352 N and St 5.348525. The printed continuous approximation, St = 5.499, instead gives 344.787 N. These are different methods. Maven’s material/contact conventions differ from the source; the roller case and separate life-factor model remain unqualified.

Fred B. Oswald, Erwin V. Zaretsky and Joseph V. Poplawski, Effect of Internal Clearance on Load Distribution and Life of Radially Loaded Ball and Roller Bearings, NASA/TM-2012-217115 (2012), Appendix D, printed pp.27–28 / PDF pp.31–32. Read the complete NASA PDF.

Contact stiffness and whole-bearing stiffness

Gabrielli, Battarra and Mucchi (2022): NU202 ECP cylindrical rollers and a 6210 ball bearing; steel E = 210 GPa, ν = 0.3 and density = 7,800 kg/m³.

Swipe the table horizontally to see every column.

The paper compares contact coefficients, mesh convergence and complete-bearing radial stiffness. A separate study introduces 0.040 mm diametral clearance and examines its effect alongside load direction.

What to explore and check

Separate the coefficient K in a contact law such as Q = K δ^3/2 from the bearing’s incremental stiffness dF/dx. Reproduce finite-ring deformation, mounting and cage orientation before comparing whole-bearing curves. Numerical targets read from graphs need their own uncertainty.

Alberto Gabrielli, Mattia Battarra and Emiliano Mucchi, A Numerical Finite-Element Method for Radial Bearing Stiffness Estimation Based on Load Dependent Meshing, ISMA 2022, pp.1527–1541; Table 1, Section 2.1, Tables 4–7 and Sections 3–4.

Angular-contact stiffness versus speed and preload

Luo and colleagues (2018), NSK 60TAC120B: twenty 10 mm balls; 93 mm pitch diameter; 60° contact angle. Ring E = 206 GPa and ν = 0.3; ball E = 320 GPa and ν = 0.25.

Tables 3–4 supply loading and unloading sweeps: 400–2,400 rpm at 1,500 N preload, and 0–2,500 N preload at 1,200 rpm. The separate finite-element case uses 2,000 N preload, 1,500 N radial load and 3,000 rpm.

What to explore and check

Compare the loading and unloading trends, retaining the different ball and ring materials. The source prints stiffness in 10⁶ N/mm; those units and the measured equivalent force/displacement definition require reconciliation before using the values as numerical targets. Recover groove geometry, preload constraints and fixture/shaft corrections. Static bearing stiffness alone does not reproduce the measured dynamic quantity.

Haitao Luo and colleagues, Numerical Simulation and Experimental Testing of Dynamic Stiffness of Angular Contact Ball Bearing (2018), Table 1 and experimental Tables 3–4.

When cylindrical rollers lose traction

Zhang and colleagues (2020): a thirty-roller experiment with 12 mm roller diameter, monitoring friction torque, temperature and vibration as speed rises.

Reviewed operating descriptions include a 0.2 kN radial load with 2.166 L/min oil delivery, and a 1.0 kN load with 4.332 L/min delivery. Both specify acceleration of 360,000 rpm/hour.

What to explore and check

Study how load and oil supply influence skid-damage onset, rather than treating nominal cage speed as a skid prediction. This assessment uses indexed primary experiment descriptions; full bearing geometry, lubricant properties, fixture and measurements still need a complete source audit. A calibrated transient traction and thermal model is needed for numerical comparison.

Qing Zhang and colleagues, Experimental Study on the Skidding Damage of a Cylindrical Roller Bearing, Materials 13(18), 4075 (2020), Figures 3–6.

Spherical-roller geometry: capacity and lubricant film

Jat and Tiwari (2018 online / 2020 volume), 22317 envelope: 85 mm bore; 180 mm outside diameter; 60 mm width; 900 rpm; source dynamic capacity 399.3 kN.

The optimization varies pitch diameter, roller diameter, roller count, effective length and contact angle. Its objectives compare dynamic capacity with minimum lubricant film thickness.

What to explore and check

Select one complete optimized row and retain its matching operating conditions, load convention and viscosity definitions. Each row represents a numerical design; its dimensions are not measured manufacturer internals. Film-thickness objectives do not establish experimental wear life, and these geometries do not establish SKF 230/900 internals. Spherical contact and lubricant-film comparisons remain pending.

Ashish Jat and Rajiv Tiwari, Multi-Objective Optimization of Spherical Roller Bearings Based on Fatigue and Wear Using Evolutionary Algorithm, Journal of King Saud University – Engineering Sciences 32 (2020), pp.58–68, Tables 2–9.