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As the companion paper of Part I, this paper aims to get more insight into the essence of lambda and to reveal its nature and role in the transition of lubrication states. Mixed lubrication (ML) model with micro-asperities contacts has been discussed in details in Part I.

Mimetic algorithm is used to get numerical solutions. Relationships between film thickness ratios and lubrication states transition with different external loads, rotating speeds, radial clearances, elastic modulus, surface hardness and roughness parameters are obtained.

The characteristic parameters of transitions from boundary lubrication (BL) to ML and ML to hydrodynamic lubrication (HL) are studied to determine how these parameters change with above factors. Finally, the essence and major influencing factors of lambda are summarized for such bearings.

In Part II, the authors believe that the paper presents for the first time: further insight into the essence of the lambda ratio, and its role in the lubrication states transition are given; the determinations of the characteristic parameters of transition from BL to ML and ML to HL are investigated for the first time; the characteristic parameters of transitions from BL to ML and ML to HL are also studied to determine how parameters (external load, rotating speed, radial clearance, elastic modulus, surface hardness and roughness parameter) change with above factors; a summary of the essence and major influencing factors of lambda for such bearings is given.

This paper aims to provide efficient methods to calculate the friction coefficients and film thickness ratios in mixed lubrication (ML) regime for water lubricated bearings. Mathematical models consider influence of micro-asperities contacts which is based on the Gauss random distribution.

Effects of external loads, rotating speeds and radial clearances are obtained. Algorithm shown here is applied to a class of common industrial problems. Calculated Stribeck values are given and evaluated. The calculated and experimental results agree well which proves the correctness of the model.

In Part I, the authors believe that the paper presents the following for the first time: universal methods are developed for the calculation of friction coefficients and film thickness ratios (lambda) in ML regime; effects of different external loads, rotating speeds and radial clearances on friction coefficients and film thickness ratios are presented in detail; comparisons are made between the results predicted by the model and experimental results, and they agree rather well which proves the correctness of the model.

Present work successfully develops universal methods for predicting the friction coefficients and film thickness ratios.

This paper aim to take the ship shaft stern bearing as the research object, and studies the influence of journal axial vibration on bearing dynamic characteristics under different misaligned angles and rotation speeds.

Computational fluid dynamics (CFD) and harmonic excitation method were used to build bearing unstable lubrication model, and the dynamic mesh technology was used in calculation.

The results indicate that journal axial vibration has a significant effect on bearing dynamic characteristics, like maximum oil film pressure, bearing stiffness and damping coefficients, and the effect is positively correlated with journal misaligned angle. The effect of shaft rotation speed and journal axial vibration on bearing dynamics characteristics are independent; they have no coupling. Bearing axial stiffness is mainly affected by the journal axial displacement, bearing axial damping is mainly affected by journal axial velocity and they are positively correlated with the misaligned angle. The influence of rotational speed on bearing axial stiffness and axial damping is not obvious.

This paper establishes the bearing dynamic model by CFD and harmonic excitation method with consideration of cavitation effect and analyzing the influence of journal axial vibration on the dynamic characteristics. The results are benefit to the design of ship propulsion shaft and the selection of stern bearing. Also, they are of great significance to improve the operation stability of the shaft bearing system and the vitality of the ship.

The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-11-2022-0337/

In this study, the dynamic characteristics of the water-lubricated rubber bearing considering asperity contact are numerically studied, including water-film stiffness and damping coefficients and plastic-elastic contact stiffness coefficient.

The Kogut-Etsion elastic-plastic contact model is applied to calculate the contact stiffness coefficient at the bearing-bush interface and the perturbed method is used to calculate the stiffness and damping coefficients of water-film. In addition, the rubber deformation is determined by the finite element method (FEM) during the simulation. Parametric studies are conducted to assess the effects of the radial clearance, rubber thickness and elastic modulus on the dynamic characteristic of water-lubricated rubber bearing.

Numerical results indicate that stiffness and damping coefficients of water film and the contact stiffness of asperity are increased with the decreasing of the radial clearance and the dynamic coefficients are less sensitive to the rubber thickness compared with the elastic modulus of rubber. Furthermore, due to the existed groove, a sudden change of the water film direct stiffness and damping coefficients is observed when the eccentricity ratio ranges from 0.6 to 1.0.

It is expected this study can provide more information to establish a dynamic equation of water-lubricated rubber bearings exposed to mixed lubrication conditions.