Hertzian formulas are commonly used for the evaluation of deformation and pressure distribution of non-conformal and slightly conformal mechanical pairs to estimate component stiffness and durability. For the sake of simplicity, their use is extended even to those cases in which Hertz’s hypotheses do not hold. This paper summarizes Hertz’s theory and compares the results obtained with theoretical and finite element analysis of the point contact of non-conformal and conformal pairs made of spheres, caps, and spherical seats. This study was motivated by the non-Hertzian behavior of a tilting pad bearing ball-and-socket pivot conforming contact observed by the authors in previous experiments. In particular, the displacement and force relation were investigated by varying the geometrical parameters, the materials, the boundary conditions, and the friction coefficient. In the case of non-conformal contact, the parameter variations had negligible effect in agreement with Hertz’s theory while for conformal contact, the cap and seat height and width and the relative clearance were the most influential parameters on the non-Hertzian behavior. These novel results indicate that in conformal pairs, such as for tilting pad bearing ball-and-socket pivots, whenever Hertz’s hypotheses are not satisfied and the assessment of contact stiffness is crucial, Hertzian formulas should not be applied as done in common practice, instead more accurate numerical or experimental evaluation should be made.

This paper reviews the current knowledge on turbulence effects in tilting pad journal bearings. Turbulence is becoming increasingly important in the design of hydrodynamic bearings due to the trend to increase power density in turbomachines and consequently the operating speeds of the hydrodynamic bearings. Turbulence has a series of effects on the bearing performance which may be beneficial or detrimental, depending on the operating conditions. The main turbulence models are recalled and a historical overview on the evolution of numerical simulations of turbulent flow in tilting pad journal bearings is presented. The two broad simulation strategies used are the generalized Reynolds equation and computational fluid dynamics. The main experimental works are then reviewed, and a unified comparison of these works is provided. Novel results on the critical Reynolds number in a tilting pad journal bearing are reported. Much emphasis is given on the experimental evidence for laminar-to-turbulent transition. The evidence used in the literature is reviewed and its reliability is discussed. Lastly, some future research directions are suggested.

The rolling motion of the pads with rocker back (RB) and ball and socket pivots is normally neglected in common software programs for the study of the rotor dynamic behavior of tilting pad journal bearings (TPJB). In other words, the theoretical contact point of the pivot is considered fixed. The aim of this work is to provide a novel way to implement in commercial software the effect of the variation of the circumferential coordinate of the theoretical contact point due to the pad rolling motion in RB TPJB. This is done by introducing an equivalent pivot rotational stiffness evaluated with an analytically derived formula, validated through finite element analysis. Such a stiffness is a function of the pad load and the radii of the contact pair, increasing with the load, the radii, and the degree of conformity of the contact. The static and dynamic characteristics of a five pad RB TPJB are then evaluated with a commercial software with and without the rotational stiffness contribution for two different pivot geometries. Non-negligible differences were found, particularly regarding the cross-coupled dynamic coefficients that show the higher sensitivity to the rotational stiffness. The inclusion of a pivot rotational stiffness among the data of commercial software for simulation of RB TPJB could contribute to fill the gap between numerical and experimental results.

Hertz’s classical theory of contact requires the surfaces to be non-conformal. Despite of this, Hertzian formulas are often used also for conformal contacts as for instance for the evaluation of pivot stiffness in tilting pad journal bearings. In this paper, finite element simulations of conformal contacts between spherical elastic bodies are performed for different materials and geometry, in particular by varying the clearance. A first result is the introduction of a novel normalization which allows to calculate stiffness as a clearance-invariant function. Then, a novel model for stiffness is introduced. The model reduces back to Hertz’s theory in the non-conformal limit. The model requires fitting of three empirical parameters which depend on the boundary conditions and on the material properties. Analytical expressions for the parameters are provided for a subset of contact problems with a simple geometry and given material properties. More general formulas for the parameters will be developed in a future work.