Predicting and Minimising Tooth Friction in Gear Transmissions: A Closed-Form Model of Load, Temperature, Speed, and Roughness

Accurate modeling of the tooth friction coefficient is central to analyses of efficiency, vibration, and durability in enclosed gear drives. Physics-based models describe these contacts using a large set of coupled thermal, contact, and lubrication laws whose individual parameters have uncertainties that are difficult to propagate. This study proposes a compact alternative: the HAF model, a three-parameter phenomenological description of the friction coefficient as a function of the slide-to-roll ratio (SRR). The three parameters have distinct tribological interpretations: h (hysteresis) governs the steepness of the sigmoidal transition, a (attrition) governs the slope of the plateau, and ν (friction level) governs the overall magnitude. The model is calibrated using nonlinear regression with fourteen two-disc traction curves acquired with a one-factor-at-a-time (star) design around a reference operating point (1.6 GPa, 80 °C, and 20 m/s): the contact pressure (1.2, 1.6, and 1.9 GPa), the injection temperature (40, 80, and 100 °C), and the mean speed (10, 20, and 30 m/s) are each varied in turn, for both smooth and rough discs. Parameter uncertainty is quantified using the residual-resampling bootstrap (BIG) established in a companion paper and applied over 105 iterations; it yields near-Gaussian, weakly correlated parameter distributions. The three HAF parameters are then expressed as linear functions of load, temperature, speed, and roughness; least-squares inference across the fourteen conditions shows that eleven of the fifteen regression coefficients differ significantly from zero at the 5% level, with roughness having the strongest effect on the friction level (t = 7.98). Substituting these laws into the HAF equation and reoptimizing the resulting expression globally yields a single closed-form model that reproduces the measured friction coefficient with a residual spread of σ ≈ 0.001 in friction-coefficient units (R2 ≈ 0.996) and approximately Gaussian, zero-mean, and homoscedastic residuals with no evident systematic structure. Being differentiable and equipped with bootstrap confidence intervals, the model predicts friction throughout the tested operating envelope—across which the maximum friction coefficient varies by a factor of eight, from 0.0049 to 0.0388—and supports gradient-based optimization of low-friction operating conditions. The contribution of this study is a compact, interpretable, and statistically characterized predictive law for tooth friction, expressed in closed form as a function of the operating conditions and of the surface state.