Transient Line-Contact Lubrication Under Harmonic Load

Transient line-contact lubrication

Estimate central-film amplitude and phase under sinusoidal normal load using the Sasaki analytical response, with Martin and Grubin stationary references.

Sasaki harmonic responseMartin quasi-steadyGrubin stationary EHL

Contact and load conditions

Final-cycle response

Sasaki responseMartin quasi-steady

How to use

Review the three model surfaces: the canonical result and blue curve use the Sasaki harmonic response, the grey curve is the instantaneous Martin reference, and the metric rail includes a stationary Grubin EHL estimate.
Enter reduced modulus E′, effective radius R, and loaded contact length L in the displayed units.
Enter dynamic viscosity η₀—not kinematic viscosity—and entrainment speed U at the operating condition. Enter α for the stationary Grubin comparison.
Enter baseline load W₀, sinusoidal peak amplitude Wₐ, and frequency f. Wₐ must be below W₀ and no more than 80% of it.
Choose 2–20 integers for displayed cycles and 60–2000 integers for plot samples per cycle.
Select Calculate to update the result, metrics, and plot. Select Reset to restore the worked example.
Interpret minimum and maximum film, phase lag φ, response amplitude a′/a, and damping parameter Dₚ together. Values of Dₚ outside 0.007–1.3 are beyond the cases tabulated by Bedewi.
Do not use this approximation for starvation, thermal effects, non-Newtonian lubricants, rough-contact load sharing, mixed lubrication, impact, load reversal, pressure prediction, or final component design. Use a validated transient Reynolds/EHL solver and experimental evidence for those cases.

Equations used

The imposed total load, amplitude ratio, angular frequency, and baseline line load are:

W(t) = W₀[1 + a sin(ωt)], a = Wₐ/W₀, ω = 2πf, w₀ = W₀/L

Martin’s rigid-isoviscous baseline central film and the Sasaki damping parameter are:

h_MR = 4.8936 η₀UR / w₀

D_p = ωR√[η₀/(Uw₀)]

The Sasaki response amplitude, phase angle, and transient film history are:

a′ = 2a / √[16 + 18π²D_p²]

φ = tan⁻¹(3π√2 D_p/4)

h(t)/h_MR = [1 + a′ sin(ωt − φ)]⁻²

The stationary elastic-piezoviscous comparison is:

h_G = {2.3664 η₀Uα√R / [w₀/(2E′)]^1/8}^8/11

Symbols and units: W₀ and Wₐ are total loads in N; L, R, h, h_MR, and h_G are converted to metres; w₀ is N/m; η₀ is Pa·s; U is m/s; f is Hz; α is Pa⁻¹; E′ is Pa; a, a′, and Dₚ are dimensionless; φ is displayed in degrees. Sign convention: compressive normal load is positive and increasing separation is positive.

Assumptions and limits: fully flooded smooth rigid line contact, positive sinusoidal load, and isothermal Newtonian isoviscous transient behavior. Reduced modulus, radius, contact length, dynamic viscosity, entrainment speed, baseline load, frequency, and pressure–viscosity coefficient must be positive. Load amplitude must be non-negative and no more than 80% of baseline load. Displayed cycles must be 2–20 integers and plot resolution must be 60–2000 integers. The Sasaki approximation does not solve the pressure distribution or downstream cavitation boundary. Grubin is a separate stationary EHL comparison.

References: T. Sasaki, H. Mori, and N. Okino, “Fluid Lubrication Theory of Roller Bearing—Part I,” DOI 10.1115/1.3657240; M. A. A. Bedewi, Non-Steady State Lubrication of Counterformal Contacts, Chapter 6, equations 6.9–6.13 and tables 6.2–6.4; H. M. Martin, “Lubrication of Gear Teeth,” Engineering 102 (1916), 199; A. N. Grubin, Fundamentals of the Hydrodynamic Theory of Lubrication of Heavily Loaded Cylindrical Surfaces (1949).