Ball Slide Load, Stiffness and Friction Calculator

Content

Screen contact pressure, stiffness, displacement, and rolling resistance for a symmetric limited-travel precision ball slide.

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

Materials, geometry and load

Use consistent elastic properties and the number of balls that actually carry the positive downward payload.

Model domain: E>0, 0≤ν<0.5, db>0, whole n≥1, μs≥0, P>0.

Guide screening result

Symmetric 45-degree load paths · elastic circular Hertz contacts

Vertical slide displacement, δslide

—µm
Slide stiffness— ×10⁷ N/m
Effective slide friction—
Peak pressure— GPa
Mean pressure— GPa
Friction force— N
Ball-contact load— N
Contact radius— mm
Contact indentation— µm
Contact stiffness— MN/m
Effective modulus— GPa
Effective radius— mm

P45° contacts

This idealized screen omits preload, clearance, groove conformity, recirculation, lubrication, cage/seal losses, misalignment, fatigue, and dynamic loading.

How to use

  1. Enter Young’s moduli E₁ and E₂ in GPa and Poisson ratios ν₁ and ν₂ for the two contacting bodies.
  2. Enter positive ball diameter db in mm and the positive whole count n of balls that carry the load.
  3. Enter nonnegative local sliding coefficient μs and positive downward payload P in N. This calculator has one symmetric-guide mode and does not infer preload.
  4. Select Calculate. Select Reset to restore the worked example and recalculate.
  5. Read vertical displacement first, then inspect guide stiffness, Hertz pressures, contact size, and estimated friction. Correct any red validation warning.
  6. Do not use this screen for recirculating-guide rating/life, preload selection, clearance, race conformity, lubrication/cage/seal losses, shock, misalignment, edge loading, or certification.

Worked example

1. Enter E₁=E₂=213 GPa, ν₁=ν₂=0.29, db=5 mm, n=10, μs=0.15, and P=400 N. 2. Calculate. Expected values are E′=232.6 GPa, R′=1.25 mm, F=28.3 N, a=0.08 mm, δ=2.37 µm, S=17.90 MN/m, pm=1.52 GPa, pmax=2.28 GPa, δslide=3.35 µm, Sslide=17.90×10⁷ N/m, Ff,slide=2.18 N, and μslide=0.0054. 3. Interpret these as ideal elastic screening values, not a bearing rating or preload recommendation.

Equations used

SI base units are used internally. E₁ and E₂ are converted from GPa to Pa; db is converted from mm to m. Loads and friction are positive magnitudes.

E′ = 1 / [(1−ν₁²)/(2E₁) + (1−ν₂²)/(2E₂)]
R₁=db/2; R′=1/(2/R₁)=db/4; F=P/(n√2)
a=(3FR′/E′)^(1/3); δ=a²/(2R′)
F⁺=1.001F; a⁺=(3F⁺R′/E′)^(1/3); δ⁺=(a⁺)²/(2R′); S=(F⁺−F)/(δ⁺−δ)
pm=F/(πa²); pmax=1.5pm
δslide=√2δ; P⁺=n√2F⁺; Sslide=(P⁺−P)/[√2(δ⁺−δ)]
Ff,slide=(√2π/4)(μs/db)(3F)^(4/3)(R′/E′)^(1/3)n; μslide=Ff,slide/P

Symbols: E₁ and E₂ are body Young’s moduli; ν₁ and ν₂ are Poisson ratios; db is ball diameter; n is load-carrying ball count; μs is local sliding friction coefficient; P is vertical payload; E′ is effective modulus; R₁ is ball radius; R′ is reduced contact radius; F is load per contact; a is Hertz contact radius; δ is local indentation; superscript + marks the retained 0.1% load-increment state; S is local incremental stiffness; pm and pmax are mean and peak contact pressure; δslide is vertical guide displacement; P⁺ is incremented payload; Sslide is incremental guide stiffness; Ff,slide is estimated slide friction force; and μslide is effective guide friction coefficient.

Theory and method

The model applies circular Hertz elastic contact to identical balls between locally flat race surfaces. Two symmetric 45-degree load paths give F=P/(n√2). The contact radius, indentation, and pressures follow the sphere-on-flat Hertz solution with the implemented effective modulus and reduced radius. Stiffness deliberately retains a 0.1% finite difference rather than replacing it with an analytical tangent.

Rtec

The friction expression estimates rolling resistance from local sliding friction and elastic contact geometry. It is not a manufacturer bearing-friction model. Equal load sharing, isotropic elasticity, small deformation, identical geometry, and full contact are assumed. Real guides may be controlled by preload, clearance, groove conformity, finite race geometry, recirculation, cage/seal/lubrication losses, manufacturing tolerances, misalignment, fatigue, shock, or dynamics.

References: H. Hertz, classical elastic-contact theory; K. L. Johnson, Contact Mechanics, Cambridge University Press, 1985.

Continue in TriboSolver to refine contact geometry, pressure, material, and load-sharing assumptions with a higher-fidelity analysis.