A correct methodology to evaluate the friction coefficient in lubricated gear pairs is paramount for both the estimation of energy losses and the prediction of wear. In the first part of the paper, a methodology for estimating the coefficient of friction with a semi-empirical formulation is presented, and its results are also employed to analyze mechanical efficiency losses in a hypoid gearset. Hypoid gears have complex tooth surface geometries, and the entraining kinematics of the lubricant is quite involved. The second part of the paper showcases a simulated wear investigation based on the Archard model. The main focus is on the impact of the frequency adopted for updating the worn geometry of the gear and pinion teeth on the fidelity and consistency of the tribological outcomes. These are measured in terms of overall quantity of material removed and characteristics of the loaded contact pattern. More in detail, a sensitivity analysis is presented that compares the total wear of a hypoid gearset after 30 million cycles estimated using different geometry update steps. Contact pressures, which are necessary to perform the aforementioned analyses, are calculated through an accurate, state-of-the-art loaded tooth contact analysis solver.

This paper derives the dynamic equations of a reduced-order race-car model using Lie-group methods. While these methods are familiar to computational dynamicists and roboticists, their adoption in the vehicle dynamics community is limited. We address this gap by demonstrating how this framework integrates smoothly with the Articulated-Body Algorithm (ABA) and provides a fresh and systematic formulation of vehicle dynamics. For the first time, we model the car body as the end effector of a serial robot with a floating base connected to the track via virtual revolute and prismatic joints. Our formulation also accounts for the effects of 3D track geometry, providing a natural embedding of the car into the 3D track. We rigorously reconcile the ABA steps with key aspects of vehicle dynamics, including road-tire interactions, aerodynamic forces, and load transfers. The resulting model, simple yet accurate, is a powerful tool to efficiently solve Minimum-Lap-Time Planning problems. To demonstrate the effectiveness of our approach, we show numerical results obtained on the Nürburgring circuit. Our optimization problem is formulated via a direct collocation method and solved using the CasADi optimization suite. To validate the results, we test our reduced-order model against a full-fledged multi-body model recently developed by the same authors. The comparison confirms the validity of our reduced-order model, proving both the accuracy of the solution and the computational efficiency achieved.

Minimizing NVH and friction-induced power losses is becoming paramount in the design of geared transmissions. The aim of this paper is to present an automatic methodology to explore Pareto-optimal designs of bevel gears when minimization of noise and frictional losses is essential. In the first part, a semi-empirical model to estimate frictional power losses under elasto-hydrodynamic lubrication is described. The model has been validated against experimental data available in the literature in previous works by the authors. The efficiency calculation is coupled with a state-of-the-art loaded tooth contact analysis (LTCA) tool to obtain accurate predictions of the instantaneous load shared by the mating tooth pairs during the meshing cycle. In the second part, an automatic framework based on multi-objective optimization (MOO) is presented where the tooth micro-geometry is systematically designed. The design variables are represented by few coefficients of a polynomial basis that embodies the tooth flank ease-off topography. To ensure manufacturability, the polynomial modifications are projected onto the feasible set of the machine-tool envelopes. This step is achieved through a state-of-the-art identification algorithm that the authors have developed in previous work. Frictional losses are estimated with the aforementioned model, whereas the NVH level is measured by the loaded transmission error (LTE), directly available from the simulation tool. The maximum contact pressures are limited by the material properties, thus proper nonlinear constraints are prescribed. Application to a test case involving the design of a spiral bevel gearset reveals that the methodology presented allows the designer to obtain Pareto-optimal solutions in a systematic and automatic manner.

Minimum lap-time planning (MLTP) is a well-established problem in the race car industry to provide guidelines for drivers and optimize the vehicle’s setup. In this paper, we tackle the 3D nature of the problem in its full extension, making no simplifying assumptions on the mechanics of the system. We propose a multibody vehicle model, described by rigorous dynamical equations. To effectively handle the resulting complexity, we devised an efficient direct dynamics computational method based on Featherstone’s articulated-body algorithm (ABA). To solve the MLTP, we employed a direct-collocation technique, discretizing the problem so that all information of the 3D track is pre-processed and directly embedded into the discrete problem. This discretization approach turns out to be perfectly compatible with our vehicle model, leading to a solution in accessible computational time frames. The high level of detail of the model makes the proposed approach most useful for in-depth vehicle dynamics analyses on complex tracks. To substantiate the analysis, we provide a comparison with the results obtained by a double-track model on the Nürburgring Nordschleife circuit. Consistently with the average trend defined by the double track, the proposed model features a more dynamically rich behavior, realistically capturing the higher-order effects elicited by the sharp corners and the highly variable slope of the track.

The aim of this paper is to present a holistic framework to design optimized spiral bevel and hypoid gearsets with accurate finite element simulations in the loop. Starting from the basic transmission data, we first size gear and pinion blanks, and then we synthesize the basic machine-tool settings required to generate the two toothed members. This first step represents the macro-geometry design phase and its outcome is a conjugate spiral bevel or hypoid gearset. The second design phase is represented by the definition of the optimal pinion micro-geometry. This is formulated as a multi-objective optimization problem (MOOP) where the obtained optimal ease-off is guaranteed to be manufacturable. To this end, an original strategy is proposed where the search for the pinion optimal tooth surface happens in the space of the coefficients of a polynomial representation of its micro-topography. However, thanks to a fast identification algorithm that can handle all the higher-order motions, the ideal ease-off is projected onto set of machine-tool settings, thus ensuring manufacturability from the outset. It is worth remarking that the objective functions in the MOOP are evaluated by calling as a back-end solver one of the most accurate loaded tooth contact analysis software available on the market. A dedicated parallel implementation of such MOOP allows to maintain computation times within very reasonable limits. A fully worked out numerical test case clearly demonstrates that the whole procedure far surpasses the current state of the art.