Transient Reynolds Equation Research Articles

Numerical Computation and Experimental Research for Dynamic Properties of Ultra-High-Speed Rotor System Supported by Helium Hydrostatic Gas Bearings

This study delves into the dynamic behavior of ultra-high-speed rotor systems underpinned by helium hydrostatic gas bearings, with a focus on the impact of rotational velocity on system performance. We have formulated an integrative dynamic model that harmonizes the rotor motion equation with the transient Reynolds equation. This model has been meticulously resolved via the Finite Difference Method (FDM) and the Wilson-Θ technique. Our findings unveil intricate nonlinear dynamics, including 2T-periodic and multi-periodic oscillations, and underscore the pivotal role of first-order temporal fluctuations, which account for over 20% of the transient pressure at rotational speeds exceeding 95.0 krpm. Further, we have executed empirical studies to evaluate the system’s performance in practical settings. It is observed that when the ratio of low-frequency to fundamental frequency approaches 0.3 and the amplitude ratio exceeds 3, the vigilant monitoring of system stability and reliability is imperative. Collective insights from both computational simulations and experimental studies have enriched our understanding of the dynamic attributes of ultra-high-speed rotor systems. These revelations are crucial for the advancement of more efficacious and resilient rotor systems designed for high-speed applications.

A new efficient flow continuity lubrication model for the cylinder liner‐piston ring based on the locally oriented roughness

AbstractThe locally oriented roughness is a common feature of surface finished by a machining process, such as the surface of the cylinder liner, and may significantly influence lubrication characteristic. A homogenised transient Reynolds equation for the locally oriented roughness is derived. The Jacobsson–Floberg–Olsson (JFO) boundary conditions was implicitly incorporated into the transient Reynolds equation to ensure the mass flow conservation. On this foundation, a model for the transient lubrication in the cylinder liner‐piston ring considering the effects of the surface roughness and cavitation is established. Then, the results of model developed here are compared with those from the stochastic model by Patir and Cheng. and the effect of the direction of roughness on the lubrication properties is analysed. The homogenised transient Reynolds equation and the lubrication model developed here have great potential for understanding and optimising the transient lubrication considering the surface local roughness orientation, cavitation and starved lubrication.

Mathematical Model and Theoretical Investigation of the Performance of Journal Bearing using a discretized Reynolds Lubrication Equation with Finite Width

Hydrodynamic journal bearings are widely used for a large number of applications such as rotating turbo machinery, axial flow compressors, axial flow turbines, and internal combustion engines. These types of journals are used to support high loads high speed rotating turbo machinery elements mainly shafts and attached components and also for enhancing the quality of these machines. In this work, the importance of journal bearing and an overview of the Reynolds hydrodynamic equation in three dimensions have been studied using the Boundary Value finite-difference method. The second-order nonlinear partial differential equation has been solved using numerical iterations approach with MATLAB software. The numerical solution of the transient Reynolds equation has been studied to investigate the Pressure distribution, pressure gradient as well film lubricant thickness on journal bearings. A numerical solution using ANSYS FLUENT has been applied to investigate the main parameters that have the main influence on the performance of the journal bearing and bearing performance.

Thermo-elastohydrodynamic lubrication performance study of step seal under transient condition

Purpose This paper aims to study the impact of transient velocity changes on sealing performance during reciprocating sealing processes. Design/methodology/approach Establish a model of transient mixed lubrication, solve the transient Reynolds equation, consider the effect of temperature rise at the seal interfaces, and determine the behavior of the seal interfaces, such as film thickness and fluid pressure. Evaluation with friction and leakage rate, calculate the variation of sealing performance with reciprocating velocity under different working conditions, and verify it through bench experiments. Findings Within a reciprocating stroke, the frictional force decreases with increasing velocity, and the frictional force of the outstroke is greater than that of the instroke; at the time of the stroke transition, the fluid pressure is smallest and the rough peak contact pressure is greatest. At present, the dynamic pressure effect of fluids is the largest, and the friction force also increases, which increases the risk of material wear and failure. Friction and leakage increase with increasing pressure and root mean square roughness. As temperature increases, friction increases and leakage decreases. In studying the performance variations of seal components through a reciprocating sealing experiment, it was found that the friction force decreases with increasing velocity, which is consistent with the calculated results and more similar to the calculated results considering the temperature rise. Originality/value This study provides a reference for the study of transient sealing performance.

Numerical Investigations of Tribological Characteristics of Biomimetic-Textured Surfaces

Rail transportation has dramatically improved travel convenience, but it has also led to environmental pollution and energy consumption issues. These challenges can be partially addressed by reducing friction loss in the mechanical transmission of rail systems. This paper examines the tribological properties of bionic-textured surfaces inspired by snake- and sharkskin. This study focuses on generating bionic textured surfaces with randomly distributed peaks through numerical simulation and connecting them to a transient Reynolds equation and friction fatigue model. The bionic surface wear lubrication model considers the lubricating film’s thickness and contact pressure obtained from the GT model. The results reveal that the existence of a bionic texture can reduce the friction coefficient and wear amount on the contact surface. The findings of this study not only offer a potential solution for reducing energy consumption and emissions in intelligent rail transit systems but also hold promise for providing further insights into the numerical simulation of bionic weaving and the investigation of tribological characteristics.

Analysis on the dynamic characteristics of spiral groove dry gas seal based on the gas film adaptive adjustment model

PurposeThis study aims to acquire the influence mechanism of gas film adaptive adjustment (GFAA) acted on the dynamic characteristics of spiral groove dry gas seal (S-DGS) and then propose a sealing stability enhancement measure.Design/methodology/approachThe gas film dynamic stiffness and damping of S-DGS are obtained by numerically solving the transient Reynolds equation based on perturbation method and finite difference method. The dynamic coefficients in GFAA model and constant gas film thickness (CGFT) model are compared and analyzed.FindingsThere is the risk to misestimate the instability of DGS with rotational speed or medium pressure grows under the condition of CGFT assumption. Based on GFAA model, increasing balance ratio B properly is an effective measure to improve the stability of DGS. The balance ratio can stimulate the sensitivity of gas film dynamic coefficients to the variation of rotational speed. Increasing medium pressure in small balance ratio range will be conducive to reducing the risk of angular instability.Originality/valueThe influence mechanism of GFAA on S-DGS dynamic characteristics is analyzed. The interactions between rotational speed and balance ratio, medium pressure and balance ratio acted on gas film dynamic characteristics are explored based on the GFAA model.

Nonlinear Dynamic Analysis of Gas Bearing-Rotor System by the Hybrid Method Which Combines Finite Difference Method and Differential Transform Method

Gas bearings have been widely applied to high-speed rotating machines due to their low friction and high rotational speed advantages. Nevertheless, gas lubrication is low viscosity and compressible. It causes the gas bearing-rotor system easy to produce self-excited vibration, which leads to instability of the rotor system and hinders the increase of rotor system speed. It is necessary to study the nonlinear behaviors of the aerostatic bearing-rotor system and the nonlinear vibration of the gas bearing-rotor system, especially considering the distribution mass and flexible and gyroscopic effects of the real rotor. In this paper, the nonlinear behavior of the gas bearing-rotor system is investigated from the viewpoint of nonlinear dynamics. Firstly, the dynamics model of a gas bearing rotor is established by combining the transient Reynolds equation and rotor dynamic equation obtained by finite element method (FEM). The transient Reynolds equation is solved using a hybrid method combining the differential transform method (DTM) and finite difference method (FDM). Then the transient gas force is substituted into the FEM rotor dynamic equation. In the end, based on the bifurcation diagram, the orbit of the rotor center, the frequency spectrum diagram and Poincaré map, the rotor system’s nonlinear behaviors are studied using a solution for the rotor dynamic equation with the Newmark method. Results show that there exists a limited cycle motion in the autonomous rotor system and half-speed whirl in the nonautonomous rotor system.

Performance of the Compliant Foil Gas Seal with Surface Micro-Textured Top Foil

In various fields, micro-textures have been successfully applied to the surface of friction pairs to effectively improve flow field and friction performance. This paper aims to investigate how different textures affect the sealing performance of compliant foil gas film seals. In theoretical simulations, a facile method for characterizing the shape of micro-textures is proposed, and the equilibrium relationship between the gas film pressure, gas film thickness, and foil deformation is established. The transient Reynolds equation considering the eccentric convergence problem and abrupt Rayleigh step changes is solved to analyze the static and dynamic characteristics. The results show that (i) compared with the directionality of the texture, the gas volume accommodated by the texture has a greater impact on the sealing performance, and a convergent texture can effectively control the leakage rate; (ii) when the texture depth exceeds 9 μm, the sealing system may be unstable; (iii) the compliant foil seal is well suited to higher-speed service conditions, and the inverted triangular texture shows the best comprehensive sealing performance.

The effects of couple stress lubricants and surface roughness on squeeze EHL motion between porous medium layer and elastic ball

This paper investigated the effects of porous media layer (PML) and couple stress (CS) lubricants at pure squeeze action in an isothermal elastohydrodynamic lubrication (EHL) point contact with surface roughness (SRN) under constant load condition. The modified transient stochastic Reynolds equation was derived in polar coordinates by means of the Christensen’s stochastic theory and the Stokes’s CS fluid theory. The Gauss-Seidel method (GSM) and the finite difference method (FDM) were both used to solve simultaneously modified transient stochastic Reynolds, force equilibrium, elasticity deformation, and rheology equations. The simulation results revealed that the greater permeability ( K) and porous layer thickness (Φ) are, the smaller central pressures ( Pc) are, and the smaller film thicknesses ( H) are. The times required for achieving maximum central pressures (Pcmax) decrease with increasing K and Φ. Due to squeeze and elastic deformation effects, the slope values of central velocity ( Vc) change from negative to positive. The magnitude of Vc and Vmin decreased with decreasing K and Φ. The Hcmin of circular type RN are greater than that of radial type RN. The Pcmax of radial type RN are slightly greater than that of circular type RN.

The influence of 3-DOF film thickness disturbance on transient performance of typical spiral groove dry gas seals

PurposeThe purpose of this paper is to investigate the influence rule and mechanism of three degrees of freedom film thickness disturbance on the transient performance of spiral groove, upstream pumping spiral groove dry gas seal (UP-SDGS) and double-row spiral groove dry gas seal (DR-SDGS).Design/methodology/approachThe transient performance of spiral groove, UP-SDGS and DR-SDGS are obtained by solving the transient Reynolds equation under different axial and angular disturbance coefficients. The transient and steady performance of the above-mentioned DGSs are compared and analyzed.FindingsThe film thickness disturbance has a remarkable impact on the sealing performance of DGS with different structures and the calculation deviations of the leakage rate of the UP-DGS will increase significantly if the film thickness disturbance is ignored. The axial and angular disturbance jointly affect the film thickness distribution of DGS, but there is no significant interaction between them on the transient sealing performance.Originality/valueThe influence mechanism of axial disturbance and angular disturbance on the transient performance of typical SDGSs behavior has been explained by theory. Considering small and large disturbance, the interaction between axial disturbance and angular disturbance on the transient performance have been studied.

HD-lubricated high-speed small reduction rolling of hard steel strips with elastically deformable work rolls

This research investigates the effects of elastic work rolls on the hydrodynamic (HD)-lubricated high-speed rolling of hard steel strips with a small reduction. An innovative numerical method was established to couple the interactions across deformable rolls, pressurised lubricant and strips undergoing elastoplastic deformation. A dynamic explicit finite element analysis (FEA) was used to obtain the deformed solid bodies while a transient Reynolds equation was utilised to govern the hydrodynamic pressure. A non-linear lubricant shearing stress was integrated in obtaining the interface friction. The analysis demonstrates that the hydrodynamic pressure can be more affected by the yield strength of the metal strip and roll radius for a fixed rolling gap in lubricated strip rolling.

Nonlinear behaviors analysis of high-speed rotor system supported by aerostatic bearings

In the paper, the dynamic model of aerostatic bearing-rotor system combining rotor motion equation and transient Reynolds equation is established to study influences of rotational speed, unbalance, rotor mass and supply pressure on the nonlinear behaviors of rotor system. The results reveal abundant nonlinear phenomenon consisting of 3 T-periodic, 5 T-periodic and quasi-periodic motions. Furthermore, chosen proper unbalance, smaller rotor mass and larger supple pressure can control the nonlinear vibration and improve the system stability. The results provide a theoretical basis for a simple and feasible method to control the nonlinear vibration of rotor system by adjusting supply pressure.

Multi-Field Coupling Dynamics Modeling of Aerostatic Spindle

The aerostatic spindle in the ultra-precision machine tool shows the complex multi-field coupling dynamics behavior under working condition. The numerical investigation helps to better understand the dynamic characteristics of the aerostatic spindle and improve its structure and performance with low cost. A multi-field coupling 5-DOF dynamics model for the aerostatic spindle is proposed in this paper, which considers the interaction between the air film, spindle shaft and the motor. The restoring force method is employed to deal with the times varying air film force, the transient Reynolds equation of the aerostatic journal bearing and the aerostatic thrust bearing is solved using ADI method and Thomas method. The transient air film pressure of aerostatic bearings is obtained which clearly presents the influence induced by the tilt motion of the spindle shaft. The motion trajectory of the spindle shaft is obtained which shows different stability of the shaft under different external forces. The dynamics model shows good performance on simulating the multi-field coupling behavior of the aerostatic spindle under external force. which is quite meaningful and useful for the further research on the dynamic characteristics of the aerostatic spindle.

Nonlinear Analysis of Stability and Rotational Accuracy of an Unbalanced Rotor Supported by Aerostatic Journal Bearings

This paper presents the nonlinear analysis of stability and dynamic rotational accuracy of an unbalanced rotor supported by aerostatic journal bearings. A finite element method is utilized with the Runge-Kutta fourth order method to solve the transient Reynolds equation and the rotor dynamics equations simultaneously for the dynamic response analysis of the rotor. The dynamic behavior of the rotor center is analyzed under different rotor masses. It is shown that the dynamic responses of the rotor strongly depend on the rotor mass. The periodic, multi-periodic or quasi-periodic motions are observed as the rotor mass changes. Under a given operating speed, the mass at which the resonance occurs is studied and its relationship with the mass of the rotor at the threshold of instability is found for the first time. The influences of supply pressure, bearing clearance, orifice diameter and eccentric distance on the rotational accuracy, the resonance and instability threshold are also investigated. The result of this study can provide guidance for designing aerostatic bearing rotor systems with required running accuracy and stability.

Analysis of Dynamic Engaged Characteristics of Wet Clutch in Variable Speed Transmission of a Helicopter

The working principle and motion process of an aviation wet clutch are analyzed. The initial velocity before the friction pair engaged is solved. The transient Reynolds equation is modified, and an oil film bearing capacity model and a micro-convex bearing capacity model are derived. The film thickness equation between N friction pairs and a pressure-plate is derived. A dynamic engaged model of springs, pistons, friction pairs, and pressure plates are established. The torque balance equation is established of two pairs of friction pairs. The friction torque, rate of change in the oil film, and law of relative change in speed are obtained. The results demonstrate that the spring preload and the viscosity of the lubricating oil have a significant influence on the engagement characteristics. Increasing the quality of the friction plate will reduce the time of engagement, whereas the quality of the friction plate has slight effect on the friction torque characteristics and oil film thickness. The initial speed generated by the collision process will reduce the output speed, sharply increase the torque peak at the lock, and increase the shift shock.

Thermo-elastohydrodynamic lubrication simulation of reciprocating rod seals under transient condition

A thermo-elastohydrodynamic lubrication transient simulation model for reciprocating rod seals was developed. In the model, the viscoelastic response of rubber materials under transient operating conditions was considered, and the transient Reynolds equation was solved. The stiffness matrix method, which is more stable and effective than the traditional influence coefficient method, was used to solve the problem of fluid - solid coupling in reciprocating sealing. The transient temperature distribution of the sealing system was calculated, and the viscosity of the fluid film was corrected according to the calculation result.

Non-linear stability analysis of two-layered porous journal bearings with velocity slip and percolation effect of additives of coupled-stress lubricant

In this paper a non-linear stability analysis of the two-layered porous journal bearing under coupled-stress lubricant has been presented with velocity slip phenomenon and additive’s percolation effect. In this non-linear transient analysis, system stability is determined by tracing the locus of the journal center and various trajectories of journal center locus have been represented in graphical form for different operating conditions. Furthermore, stability characteristics in respect of critical mass parameter and whirl ratio have been studied under various parametric conditions and a comparison between the linear and non-linear stability analysis have been demonstrated. To acquire the non-dimensional pressure values, non-dimensional transient Reynolds equation has been solved and with these pressure values, bearing load carrying capacity are derived. Fourth order Runge-Kutta method is used to solve the second order equations of motion for journal bearing system to obtain the stability characteristics. Results of this analysis may be helpful for designing such bearings.

Effects of surface roughness on transient squeeze EHL motion of circular contacts with micropolar fluids

A numerical method was developed to study the effects of the micropolar (MP) lubricants at pure squeeze action in an isothermal elastohydrodynamic lubrication point contact with surface roughness under constant load condition, but without asperities contact. The modified transient stochastic Reynolds equation was derived in polar coordinates by means of the Christensen’s stochastic theory and the Eringen’s MP fluid theory. The Gauss–Seidel method and the finite difference method were both used to solve simultaneously the modified transient stochastic Reynolds, the load balance, the lubricant rheology and the elasticity deformation equations. The simulation results revealed that the effect of the MP lubricant was equivalent to enhancing the lubricant viscosity. Therefore, the central pressure and film thickness for MP lubricants were larger than those of Newtonian fluids under the same load in the elastic deformation stage. The maximum central pressure of the radial type roughness (RN) was greater than that of the circular type RN. The film thickness of the radial type RN was smaller than that of the circular type RN.

Experimental and numerical study of wavy mechanical face seals operating under pressure inversions

This paper investigates the impact of the face waviness and pressure inversions on the leakage and on the outer fluid entry of mechanical face seals using a numerical model and an experimental setup. The numerical model couples a transient Reynolds equation, an analytical contact model, a force balance solver, and a solver for the thermo-mechanical deformations. The experimental tests on a face seal with low waviness and on a face seal with high waviness provide leakage and outer fluid entry data, which are reproduced by the model. Contrary to the face seal with low waviness, the face seal with high waviness has poor performance and the pressure inversions increase significantly the ingression of outer fluid. The parametric study shows a decrease of leakage with increasing spring force, and an increase of leakage and outer fluid entry with increasing values of waviness amplitude. The higher leakage observed for wavy seals is shown to be due to the higher average film thickness, and to some extent due to the mechanisms associated with waviness: hydrodynamic pressure generation, film squeeze and stretching.

Non-linear stability analysis of micropolar fluid lubricated journal bearings with turbulent effect

PurposeThe purpose of this paper is to investigate the effect of turbulence on the stability characteristics of finite hydrodynamic journal bearing lubricated with micropolar fluid.Design/methodology/approachThe non-dimensional transient Reynolds equation has been solved to obtain the non-dimensional pressure field which in turn used to obtain the load carrying capacity of the bearing. The second-order equations of motion applicable for journal bearing system have been solved using fourth-order Runge–Kutta method to obtain the stability characteristics.FindingsIt has been observed that turbulence has adverse effect on stability and the whirl ratio at laminar flow condition has the lowest value.Practical implicationsThe paper provides the stability characteristics of the finite journal bearing lubricated with micropolar fluid operating in turbulent regime which is very common in practical applications.Originality/valueNon-linear stability analysis of micropolar fluid lubricated journal bearing operating in turbulent regime has not been reported in literatures so far. This paper is an effort to address the problem of non-linear stability of journal bearings under micropolar lubrication with turbulent effect. The results obtained provide useful information for designing the journal bearing system for high speed applications.