Reynolds Equation Research Articles

The Influence of Non-Gaussian Surface Topography, and Contact Models on Mixed-Lubrication Parameters for Water-Lubricated Journal Bearings (WLJB)

Abstract Greenwood and Williamson (GW) and Greenwood and Tripp (GT) elastic contact models have extensively been used previously for mixed-lubrication analysis of water-lubricated journal bearings (WLJBs). The approximate expressions of parabolic cylinder function involved in these contact models are available in the literature which makes these models easy to implement in mixed-lubrication analysis. However, approximate expressions available so far are valid only for Gaussian distribution of asperity heights. Moreover, elastic-plastic contact models with few exceptions have rarely been used in mixed-lubrication analysis of WLJBs. The present work demonstrates a simple numerical procedure to predict the mixed lubrication parameters for WLJBs. The average Reynolds equation considering non-Gaussian flow factors is solved along with film thickness and load balance equations in a coupled manner. A routine is developed to predict the pressure curve (a plot between average asperity pressure and rigid body displacement) for different contact models and non-gaussian surface topography, and the asperity pressure for a particular surface topography is predicted utilizing the concept of macro-micro approach. The effect of non-Gaussian surface topography on tribological performance of WLJBs is discussed in detail. The influence of different contact models on the performance of WLJBs is also presented. It is shown that the GT elastic contact model fails to simulate the effect of kurtosis and skewness properly. However, the KE elastic-plastic contact model which is based on FEM-based analysis excellently simulates the non-Gaussian effect on performance of the WLJB.

Dynamic Behavior Analysis of Journal Bearings Lubricated by Non-Newtonian Fluid During Start-Up

Abstract The dynamic behavior of journal bearing lubricated by non-Newtonian fluid (conforms to the Carreau model) during start-up is investigated. A mixed lubrication model consisting of a GT (Greenwood-Tripp) contact equation and transient modified average Reynolds equation is proposed. The transient modified average Reynolds equation is established with the average flow method. The effect of non-Newtonian lubricant, surface roughness, external load, relative clearance, and accelerating time on the dynamic behavior of journal bearings during start-up is analyzed. The results show that increasing the relaxation time, external load, and relative clearance, as well as decreasing the power-law exponent, composite root-mean-square roughness and accelerating time can reduce the contact time and facilitate the rapid formation of hydrodynamic lubrication during start-up.

Turbulent flushing of internal pipe-type surfaces in hydraulic components

The cleanliness of hydraulic systems has long been regarded as a crucial indicator of their performance and longevity. This study aims to elevate the cleanliness level of hydraulic components from 500 μm to 300 μm. To achieve this goal, this study integrates the flushing process and establishes a physical and mathematical model of wired solid pollutants on internal pipe-type surfaces during flushing based on Reynolds equation. This paper proposes the introduction of a cleanliness coefficient as a quantitative measure for evaluating cleanliness. The cleanliness coefficient serves as a threshold to determine whether particles can be effectively removed or not. The results of the orthogonal experiment demonstrate that the error between theoretical models and actual outcomes is merely 7.4%, affirming the effectiveness of the physical and mathematical models. The cleanliness coefficient provides guidance for setting flushing process parameters. The innovation of this study lies in the introduction of the cleanliness coefficient concept, which offers a more quantitative approach compared to traditional qualitative assessment methods.

Extrapolation of cavitation and hydrodynamic pressure in lubricated contacts: a physics-informed neural network approach

A comprehensive understanding of the dynamics of tribological interactions is essential for enhancing efficiency and durability in a multitude of technical domains. Conventional experimental techniques in tribology are frequently costly and time-consuming. In contrast, elastohydrodynamic lubrication (EHL) simulation models present a viable alternative for calculating frictional forces in sealing contacts. These calculations are based on the hydrodynamics within the sealing contact, as defined by the Reynolds equation, the deformation of the seal, and the contact mechanics. However, a significant drawback of these simulations is the time-consuming calculation process. To overcome these experimental and computational limitations, machine learning algorithms offer a promising solution. Physics-informed machine learning (PIML) improves on traditional data-driven models by incorporating physical principles. In particular, physics-informed neural networks (PINNs) are as effective hybrid solvers that combine data-driven and physics-based methods to solve the partial differential equations that drive EHL simulations. By integrating physical laws into the parameter optimization of the neural network (NN), PINNs provide accurate and fast solutions. Thus, unlike traditional NNs, PINNs have the potential to make accurate predictions beyond the limited training domain. The objective of this study is to demonstrate the feasibility of spatial and temporal extrapolation of the PINN and to analyze its reliability, both with and without consideration of cavitation. Two test cases are employed to examine the pressure and cavitation distribution within a sealing contact that extends beyond the spatial and temporal training range. The findings indicate that PINNs can surmount the typical constraints associated with NNs in the extrapolation of solution spaces, which represents a notable advancement in terms of computational efficiency and model flexibility.

Neural network based prediction of lemon bore journal bearing performance with micropolar lubricants

The research paper aims to predict the static characteristics of lemon bore journal bearings lubricated with micropolar lubricant using an artificial neural network (ANN) and compare the results using traditional numerical methods. It commences by employing the Finite Difference Method (FDM) to solve the Reynolds equation for micropolar lubricants. Swift-Stieber boundary conditions were functional to evaluate pressure distribution, which enables the examination of the static characteristics by varying the eccentricity ratio (ε) = 0.25, 0.3, and 0.35, length of micropolar (lm) = 1 to 50, and coupling number (N2) = 0.1 to 0.9. Thereafter, the results from the FDM approach are used to train artificial neural network models. The study utilizes a dataset comprised of over 1560 data points with corresponding inputs and outputs. Three distinct feed-forward ANN architectures were considered: Levenberg-Marquardt optimization technique, Bayesian regularization technique, and Scaled Conjugate Gradient, and their performance was compared with results obtained through FDM. Results indicate a substantial reduction in computational effort and time when employing the trained ANN models compared to the FDM approach. Furthermore, it is found that the difference between the results given by the ANN models and the FDM results is not above 5%; the agreement between these two approaches thus appears good. This study reveals the effectiveness of ANNs in reducing computational time and effort while maintaining accuracy in evaluating the performance of lemon bore journal bearings lubricated with micropolar lubricant, and provides a new approach to predict the performance of fluid film bearings.

Analysis of the dynamic response of aerostatic bearings based on the dynamic equilibrium position method

To fully consider the influence of time-varying pressure and film thickness, the transient Reynolds equation is typically employed for numerical computations in the analysis of gas bearing-rotor systems. Solving the transient Reynolds equation is complex and time-intensive. To enhance computational efficiency, this paper presents an optimized algorithm, termed as the equilibrium position (EP) method, which leverages the solution of the steady-state Reynolds equation to analyze the axis shift induced by rotor imbalance. By neglecting certain transient terms while retaining the essential physics, the proposed method achieves high computational accuracy and significantly reduces computational time. The computational errors are evaluated, and comparisons with orbit method simulations demonstrate the method's efficiency and accuracy. In applications such as predicting rotor vibration amplitude or estimating unbalanced forces during ultra-precision machining and rotor design, the EP method offers a streamlined computation process without compromising on result fidelity.

The Impact of Pad Elasticity on Frequency-Dependent Linear Dynamic Coefficients of Tilting-Pad Journal Bearings

Abstract The frequency-dependent dynamic coefficients of tilting-pad bearings are significantly influenced by the pad pivot elasticity. In addition to the elasticity of the pivot, there is also the flexibility of the structure that predominantly leads to a bending of the pad. For the calculation of the dynamic coefficients, it is common to describe the structural stiffness by a support stiffness in series with the lubricant film. For this purpose, the stiffness according to Hertz's theory is widely used. This approach only allows the rigid body motion of the pad in the theoretical model. However, the elasticity of the pad leads to a film thickness characteristic, which cannot be represented with a pure rigid body motion. Additionally, the structural elasticity already includes the stiffness of the contact between the pad and the support, and therefore the elasticity according to Hertz cannot simply be added. This paper investigates the influence of dynamic pad deformation on dynamic bearing coefficients considering the contact boundary conditions and examines whether it can be replaced by a suitable support stiffness. A three-dimensional finite element model provides the eigenmodes for calculating the dynamic coefficients by perturbed Reynolds equations within a thermo-elasto-hydrodynamic bearing analysis. The model is validated with measurement data.

Fast Computation of Hydrodynamic Pressure in Lubricated Contacts: Which Lp Loss is Most Suitable for Physics-Informed Neural Networks Solving the Reynolds Equation?

The frictional behavior of pneumatic seals significantly impacts the functionality of fluid power systems, particularly in fast-switching applications where precision and responsiveness are critical. However, the complex relationship between component properties and friction often makes experimental characterization infeasible or prohibitively expensive. To address this challenge, the Institute for Fluid Power Drives and Systems (ifas) developed the ifas Dynamic Seal Simulation (DDS), an iterative elastohydrodynamic lubrication (EHL) simulation capable of accurately solving the relevant partial differential equations (PDEs) [3]. Since iterative solvers are computationally intensive, neural networks have been explored as a more efficient alternative. While traditional neural networks offer computational advantages, they often lack the ability to understand the physical context of the systems they model, potentially limiting their accuracy and reliability. Physics-Informed Neural Networks (PINNs) have been introduced to overcome these limitations. PINNs integrate the governing physical laws directly into the training process, allowing them to grasp the system’s physical context. This approach opens up new possibilities, including more robust training and the ability to extrapolate beyond the training domain, thereby providing a more reliable and efficient tool for modeling the friction of seals in fluid power systems. In this paper, a previously validated hydrodynamic PINN framework [8, 9] is utilized to solve a variant of the averaged Reynolds equation across three scenarios: divergent, convergent, and curved gaps. The investigation focuses on four p-norm training metrics: L1, L2, L22, and L∞. The results indicate that the commonly used L22 metric is the most suitable for the scenarios examined.

Magnetohydrodynamic squeeze film characteristics of micropolar fluids with piezo-viscous dependency between wide parallel rectangular plates

The study introduces a novel approach to squeeze film behavior by combining Micropolar fluid, its piezo-viscous dependency, and MHD effects on wide parallel rectangular plates. The Reynolds equation is developed analyticaly by utilising the principles of hydromagnetic flow, Eringen’s microcontinuum theory, the micropolar fluid model and the influence of variable viscosity.The current study seeks to maximize the bearing performance by scrutinizing the pressure, load carrying capacity, and squeezing time. The influence of Hartmann number M, coupling number N, fluid gap interacting number L, and Piezo-Viscous Dependency parameter G are numerically calculated and depicted graphically. Notably, the pressure, the load carrying capability, and squeezing time are enhanced by the presence of the micropolar fluid and the magnetic field compared to their Newtonian and non-magnetic counterparts. Additionally, the piezo-viscous dependency, characterized by the viscosity variation factor, also increases the pressure, load and squeezing time. The study highlights the potential of utilizing couple stress fluid, viscosity variation, and MHD as a means to improve load carrying capability and extend the operational life of squeeze film bearings in various engineering applications. The Reynolds equation is derived by applying theory of hydromagnetic flow and Eringen’s micro continuum theory along with micro polar fluid model.

Physics-Informed Deep Learning for Lubricated Contacts with Surface Roughness as Parameter

Physics-informed neural networks (PINNs) are developed to solve variants of averaged Reynolds equations for accurately and time-efficiently modeling pressure build-up and cavitation in sealing contacts and journal bearings with rough surfaces. We use microscale coefficients provided through Patir and Cheng’s average flow model or homogenization to integrate roughness or texture height into these macroscale equations. Based on these equations we implement parameter-dependent PINNs to solve multi-case scenarios with varying roughness or texture heights, thus investigating the adaptability and generalizability of PINNs for modeling rough lubricated interfaces. The results demonstrate the promising potential of PINNs to accelerate tribological system computations.

Efficient simulation of hydrodynamic bearings using the SBFEM with eigenvalue problem derivatives

AbstractThis work presents a semi-analytical solution of the Reynolds equation under Gümbel conditions for hydrodynamic bearings in rotordynamic simulations. The algorithm is based on the SBFEM (scaled boundary finite element method) in combination with eigenvalue problem derivatives. The pressure field is discretized by standard finite elements along the circumferential coordinate, while an exact analytical formulation is used in the axial direction. This transforms the partial differential equation into a system of ordinary differential equations and leads to an eigenvalue problem. The computation of the continuously changing eigenvalues and eigenvectors is achieved by accurate Taylor approximations so that the repeated call of a numerically expensive eigensolver can be avoided. The algorithm is incorporated into a time integration scheme for the dynamic analysis of a simple system, consisting of a rotor with two degrees of freedom in a hydrodynamic bearing. Thereby, the method is verified, and its numerical efficiency is demonstrated. For a bearing with a slenderness ratio (length-to-diameter ratio) of 1, the computational time is reduced to $$8.8\%$$ 8.8 % compared to a standard FVM (finite volume method) solution.

Flow field computation and sealing performance analysis of high-speed micro-grooved pumping seal for new energy vehicles

A micro-grooved pumping seal for electric vehicles was investigated. In this regard, the real gas effect and viscosity were described using the virial and Lucas equations, respectively, and the oil–air ratio was used to determine the equivalent density and viscosity of the two-phase fluid formed in the mixture of the lubricant and air in the seal. The compressible steady-state Reynolds equation was solved using the finite difference method. The pumping mechanisms of the seal and the effects of operating parameters (rotational velocity, oil–air ratio, inlet pressure, and temperature) on the steady-state sealing performance were analysed. The seal produces a significant hydrodynamic effect on the sealing face. The opening force of the seal increases with rotational velocity, oil–air ratio, and inlet pressure. However, the force decreases as temperature increases. The oil–air ratio has the most significant effect on the opening force. The pumping rate first increases and then decreases as the rotational velocity increases. It increases with the oil–air ratio and temperature but decreases as the inlet pressure increases. In particular, the pumping rate reaches a maximum when the rotational velocity or temperature increases to a certain value. This phenomenon requires special attention during seal design. Our results provide input for the design of micro-grooved pumping seals for new energy vehicles.

Wear Simulations of Involute Harmonic Gear Under Mixed Lubrication Condition

ABSTRACTHarmonic gears are widely used in precise space technology, robotic, medical equipment and other fields, while the magnitude of surface topography changes due to wear is usually comparable to or larger than the original surface roughness and elastic deformation, leading to severe transmission failures. This paper reports a numerical approach to simulate the lubrication status considering wear evolution based on mixed elastohydrodynamic lubrication (EHL) and Archard models, in which the Reynolds equation is solved with finite difference method and surface deformation is calculated by the discrete convolution‐fast Fourier transform (DC‐FFT) algorithm. The interfacial pressure and film thickness distributions are validated by comparison with available results from literature. The harmonic gear lubrication and wear performances are calculated, including effects of machined surface, velocity, load, wear time and material properties, and the results suggest that avoiding long‐term and high‐torque working with a large wear coefficient can effectively prevent surface wear failure, which is beneficial for increasing the harmonic gears' lifespan.

Influence of vortex generator dimensions on film cooling efficiency

Enhancements in gas turbine blade cooling techniques, such as film cooling, have significantly advanced the aerothermal effi-ciency of turbines, especially in transportation sectors like aeronautics and automotive industries. This study aims to enhance turbine blade cooling by incorporating an obstruction at the jet exit. The vortex generator angle has been modified to 25°, 45°, 60°, 90° and 110°. These five designs were assessed in comparison to the conventional cylindrical hole configurations. Two injection ratios (M = 0.25, and M = 0.5) were studied within the ANSYS CFX 16 software, utilizing the finite volume method to solve the average Reynolds equations and the energy equation. The findings show qualitative alignment with experimental data for the base scenario, indicating that the vortex generator angle notably amplifies film cooling effectiveness. Typically, the vortex generator configured at 90° exhibits a stronger mixing capability compared to the other cases and cylindrical design.

Simulation and theoretical studies on the influence of removable panel deformation on the seal characteristics of ceramic wafer seal

PurposeThe purpose of this study is to investigate the influence of geometric parameters of removable panels on the sealing characteristics of ceramic wafer seal structure subjected to high-temperature gas flow.Design/methodology/approachBased on the laminar flow Reynolds equation, the theoretical and numerical calculation models were constructed to investigate the influence of external convex deformation of removable panel on leakage rate. The theoretical formula for leakage rate after deformation of the removable panel was derived, and the flow field and leakage characteristics of ceramic wafer seal under different operating parameters were studied.FindingsThe leakage rate exhibits consistent trends between theoretical calculation, numerical simulation and experimental value, with a maximum discrepancy of 8.9%. This validates the accuracy of both the theoretical model and numerical simulation. As the deformation angle of the removable panel increases, the sealing gap gradually widens, resulting in a compromised sealing effect. Moreover, the leakage rate in the central region of the sealing area is lower compared to that at both ends.Originality/valueThe leakage of the ceramic wafer seal structure under the removable panel with different deformation angles can be monitored based on Reynolds equation. The pseudo-transient numerical calculation method can be used to determine the leakage value of the micro-state ceramic wafer seal structure. These research findings provide a theoretical foundation and numerical investigation approach for studying ceramic wafer seal structures.

Effect of rotational speed on transient thermoelastohydrodynamic tribo-dynamic characteristics of journal bearings

In journal bearings operating under high-speed and heavy-load conditions, the loads borne by the bearings and the rotational speeds of the journals are the key parameters affecting their working characteristics. In this paper, taking journal bearings as the research object, the thermoelastic deformation of the bearing shell, the asperity contact between the journal and the bearing shell, the density-temperature-pressure and viscosity-temperature-pressure equations of lubricating oil, and the motion equation of the journal are comprehensively considered. Based on the generalized Reynolds equation of the average mass flow model, a transient thermoelastohydrodynamic (TTEHD) model of journal bearings coupled with dynamics and tribology is established. The established model is mainly aimed at the effect on the characteristics of bearings at different initial speeds and different lift speeds. The results show that: in the process of journal lift speeds, a larger lift speed will have a very obvious effect on the eccentricity and the maximum bearing temperature, and increase the possibility of journal vortexing; a larger initial speed will greatly reduce journal running stability; the increase of the load will to a certain extent strengthen journal stability.

Dynamic Analysis of Radial Journal Bearing-Rotor System Based on the Meshless Barycentric Rational Interpolation Collocation Method

This study focuses on a rigid rotor supported by radial journal bearings. Initially, models for the unsteady oil film force in bearing lubrication and the dynamics of the bearing-rotor system are established. Subsequently, the Reynolds equation for dynamic lubricating oil films is discretely solved using the meshless barycentric rational interpolation collocation method. By combining this with the equation of motion for the axis orbit, the oil film pressure distribution, the dynamic response of the rotor, and the axis orbit are calculated. Furthermore, the study investigates the dynamic response of the rotor at different rotational speeds, both with and without considering unbalanced loads. Finally, the influence of step load on the stability of rotor motion is analyzed, revealing that applying an appropriate step load to the rotor can effectively mitigate the lubricating oil films oscillation conditions. The findings of this study hold significant reference value and practical utility for engineering applications.

Influence of roughness patterns and slip velocity on the lubrication of MHD-affected secant curved annular plates

The purpose of this article is to look at how well an electrically conducting fluid forms a squeeze film when a magnetic field is applied as well as the slip velocity and couple stress lubricants for secant curved annular plates. We derive a modified Reynolds equation based on Christensen stochastic theory for rough surfaces, which takes into account two types of roughness patterns radial and azimuthal. We obtain expressions for pressure, load carrying capacity and squeeze film time for both types of roughness patterns. Compared to a smooth surface, the azimuthal roughness pattern provides the higher squeeze film pressure, load carrying capacity, and response time, whereas the radial roughness pattern has adverse effects when increase of roughness parameter. The presence of an applied magnetic field, slip velocity and couple stress lubricant improves bearing performance.

Study of offset factor and depths of dimple on the performance of two-lobe journal-bearings

This study investigates the combined effect of offset factor and depths of a dimple on the performance parameters of hydrodynamic two-lobe journal-bearings. The finite element method (FEM) technique is employed to solve the governing Reynolds equation, facilitating the computation of oil-film pressure within the bearing. Three distinct texture configurations on the bearing surface are analyzed. The performance parameters of textured bearing configurations are calculated by varying the offset factor and depths of a dimple. The results indicate that the performance enhancement ratio of textured-II configuration of two-lobe journal-bearing is higher than that of other bearings. Furthermore, textured-II configuration shows the better performance in terms of static, dynamic and stability parameters. This superior performance is observed with an increase in the offset factor up to 0.5. Moreover, a comparative analysis of spherical and kite-shaped textures reveals that spherical textures generally outperform kite-shaped textures in terms of critical mass and threshold speed, while kite-shaped textures exhibit better damping characteristics.

Dynamic Modeling of 5-DOF Aerostatic Bearing Rotor System with Adjustable Gas Film Gap

In the application of an aerostatic motorized spindle, given the different requirements for the optimal gas film thickness of gas bearing under various processing conditions, this paper puts forward the tapered aerostatic bearing as the radial support element of the spindle and realizes the adjustability of gas film gap in a particular range through the axial fine-tuning mechanism. A 5-DOF dynamic model of the bearing rotor system is established, and the transient Reynolds equation is solved using the finite difference method to obtain the pressure distribution characteristics of the gas film. Based on this, the spindle’s translation and angular displacement responses are determined by solving the spindle’s motion equation. The simulation results show that the tilting motion of the spindle significantly affects the pressure distribution of the gas film, and the nonlinear gas film force will lead to nonlinear severe vibration of the spindle. The study also reveals that reducing the gas film thickness under low-speed and heavy-load conditions effectively decreases the amplitude and offset of the spindle. However, increasing the gas film thickness enhances the system’s speed and stability under high-speed and light-load conditions.