Spherical Thrust and Pivot Bearing Calculator

Estimate contact pressure, contact size, elastic indentation, spin-friction torque, and power loss for a spherical thrust or pivot bearing.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Contact screening result

Pressure is the canonical screening quantity; interpret it together with hardness and regime.

Mean contact pressure pₘ—GPa

Regime—

Maximum pressure p₀—GPa

Contact radius—mm

Friction torque M—N·mm

Power loss q—mW

Elastic indentation δ—µm

Effective modulus E′—GPa

Effective radius R′—mm

The indentation remains the Hertz elastic value if the pressure model moves into the transition or hardness-limited branch.

How to use

Enter both elastic moduli and Poisson ratios. Use representative values for the actual temperature and material condition.
Enter positive convex radii. For sphere on flat, set the flat body radius to 0; never set both radii to 0.
Enter the non-negative friction coefficient, positive weaker-body hardness, positive compressive load, and non-negative rotational speed in the displayed units.
Select Calculate or edit a value to update the result. Reset restores the steel–sapphire sphere-on-flat example.
Read mean and maximum pressure with the reported contact regime; use torque and power loss only as constant-friction screening estimates.
Correct any warning before using a result. Zero speed gives zero power but can still give nonzero friction torque.
Do not use this model for concave or conformal sockets, impact, severe misalignment, thermal equilibrium, lubrication-film prediction, wear, or fatigue-life qualification.

Positive radii denote convex surfaces; a zero radius denotes a flat mating body.

Worked example

1. Use E₁ = 213 GPa, ν₁ = 0.29, R₁ = 6 mm; E₂ = 440 GPa, ν₂ = 0.30, R₂ = 0; μ = 0.10, H = 7 GPa, F = 10 N, and n = 600 rpm.

2. The calculator reports E′ = 314.061 GPa, R′ = 3.000 mm, contact radius = 0.066 mm, mean pressure = 0.732 GPa, maximum pressure = 1.098 GPa, torque = 0.039 N·mm, and power loss = 2.440 mW.

3. Since pₘ < H/3, the result is in the elastic Hertz regime. The pressure and small contact size still require independent material, lubrication, thermal, and fatigue checks.

Equations used

All calculation equations use SI units internally. E₁ and E₂ are Young’s moduli; ν₁ and ν₂ are Poisson ratios; R₁ and R₂ are positive convex radii; F is positive compressive load; μ is a non-negative friction coefficient; H is positive weaker-body hardness; and n is a non-negative speed magnitude in rpm.

E′ = 1 / [(1 − ν₁²)/(2E₁) + (1 − ν₂²)/(2E₂)]

R′ = 1 / (2/R₁ + 2/R₂)

a = (3FR′/E′)^1/3

pₘ,Hertz = F/(πa²), p₀,Hertz = 1.5pₘ,Hertz

δ = a²/(2R′)

M_Hertz = (3F)^4/3(π/16)(R′/E′)^1/3μ

rₚ = √[F/(πH)], Mₚ = (2/3)μFrₚ

q = Mω, ω = 2πn/60

For a flat body, its reciprocal-curvature term is omitted. When pₘ,Hertz ≤ H / 3, the Hertz radius and torque are used. For H / 3 < pₘ,Hertz < H, define β = 1.5pₘ,Hertz/H − 0.5 and linearly interpolate r = a + β(rₚ − a) and M = M_Hertz + β(Mₚ − M_Hertz). At pₘ,Hertz ≥ H, use rₚ and Mₚ, set reported pₘ = H, and cap reported p₀ at H. The boundary convention is continuous.

Theory and method

The elastic branch treats the contact as a circular Hertz point contact between smooth, isotropic, nonconformal bodies. The implemented effective modulus and radius conventions are shown explicitly above. Mean pressure is load divided by the Hertz contact area, and the maximum semi-elliptic pressure is 1.5 times the mean.

Hardness H provides a simple limit for local pressure. Above H / 3 the model blends elastic and uniform-pressure torque/contact-radius estimates; at H it switches fully to the hardness-limited estimate. This is an engineering interpolation rather than a constitutive elastoplastic contact solution. The model also assumes constant μ and excludes lubrication regime, temperature rise, wear, adhesion, roughness, impact, misalignment, fatigue, and dynamic effects. Hertz use additionally requires small strains and a contact patch small relative to both curved-body radii.