The student Pello Alberdi Quevedo obtained an EXCELLENT CUM LAUDE

The student Pello Alberdi Quevedo obtained an EXCELLENT CUM LAUDE

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• Thesis title: Towards a unified minimum-energy framework for ball kinematics in lubricated Rolling Machine Elements considering contact and helix angle

Court:

• Presidency: Guillermo Morales-Espejel (SKF e INSA Lyon)
• Vocal: Amaya Igartua Aranzabal (Tekniker)
• Vocal: Enrique Chacón Tanarro (Universidad Politécnica de Madrid)
• Vocal: Carlos Fernandes (Universidade do Porto)
• Secretary: Aurea Iñurritegui Marroquin (Mondragon Unibertsitatea)

Abstract:

Rolling machine elements, such as ball bearings, ball screws, or linear guides, are widely used in precision machinery and motion systems to transmit and support loads while enabling efficient relative motion between components. Ball bearings guide rotating parts and carry loads, whereas ball screws convert rotational motion into linear displacement (and vice versa), and linear guides support and guide translational motion. Geometrically, these mechanisms can be regarded as equivalent components distinguished by the helix angle of the raceways: ball bearings correspond to  GrSc/A= 0º, ball screws to  0º < GrSc/A< 90º, and linear guideways to  GrSc/A= 90º. The mechanical performance of these machine elements relies on the balls, which transmit load through rolling contact while permitting relative motion and thereby reducing friction. In practice, these elements are typically lubricated to further improve efficiency, durability, and reliability.

The kinematic behaviour of the balls plays a fundamental role at both the global and contact interface scales, as it governs key performance aspects such as frictional power losses, wear processes, lubrication performance, and overall efficiency. Consequently, an accurate description of ball kinematics is essential for the reliable analysis and improved design of rolling machine elements.

However, accurately predicting ball motion remains challenging. The raceway geometry imposes a fully three-dimensional trajectory, and the contact conditions generate coupled rolling, spinning, and side sliding. Furthermore, ball motion is governed by frictional interactions occurring ideally under elastohydrodynamic lubrication conditions, where friction depends on both load and local sliding. As a result, determining ball motion requires solving a tightly coupled problem involving load distribution, contact kinematics, and friction behaviour.

However, existing modelling approaches do not provide a method for determining ball motion accounting for this coupled interaction. Instead, simplified assumptions, such as the raceway control assumption, are typically imposed. In addition, while the conventional vector formulation for ball bearing and linear guide kinematics is consistent, the classical formulations for ball screws reveal inconsistencies, indicating a misunderstanding of their kinematic description.  Consequently, the current literature lacks a valid and unified framework capable of describing ball kinematics at both global and contact scales, compatible with such rolling machine elements.

To overcome these limitations, this thesis redefines the theoretical kinematic framework for ball screws and derives new expressions for both the global motion of the ball and the associated contact kinematics. The revised formulation leads to a fundamental reinterpretation of ball screw kinematics and provides a generalised vector description that is valid across the full spectrum of rolling-element systems: from ball bearings to ball screws and linear guides. Building on this revised formulation, a physically consistent quasi-static model is developed to analyse ball kinematics. The model assumes that the ball moves so as to minimise energy dissipation, thereby enabling a consistent coupling among contact load, kinematics, and frictional behaviour under lubricated conditions.

The model is then employed to analyse ball kinematics in various rolling machine elements. First, the framework is implemented at the angular-contact ball bearing scale, establishing the basis for predicting ball motion under lubricated conditions. Next, the influence of the nominal contact angle is investigated, covering the full range of ball bearing configurations, from deep-groove to thrust bearings. Finally, the framework is extended to non-zero helix angles, enabling the analysis of ball motion in helical geometries, such as ball screw mechanisms, ultimately reaching the limiting case of  GrSc/A= 90º , corresponding to linear guides. In this way, the proposed approach provides a unified method for predicting ball kinematics across rolling machine elements.