Time-dependent Reynolds Equation Research Articles

Study the behaviour of transient elastohydrodynamic lubrication contact at motion start-up in the presence of surface roughness

PurposeThis study aims to present a numerical solution for the analysis of the influence of surface roughness as presented by a sinusoidal ripple of different amplitude and wavelength on the performance of transient elastohydrodynamic lubrication at motion start-up under different operational parameters of entraining speed and load as well as different acceleration rates.Design/methodology/approachA statistical asperity micro-contact model represented by a sinusoidal ripple expressed by two parameters (wavelength and undeformed amplitude) is considered. The ball equation of motion is used to calculate the force on the ball as it starts to move. The time-dependent Reynolds equation is solved together with surface deformation and statistical asperity models using the Newton–Raphson technique with the Gauss–Seidel iteration method.FindingsThe behaviour of the film thickness was found to be strongly influenced by the acceleration rate for different ripple amplitude and wavelength parameters. The effect of increasing the final entraining speed will eventually lead to rapid film thickness build-up and increase the film thickness jump at the moment of motion start-up. The effect of increasing applied load is to reduce the deviation of the minimum film thickness jump at the start-up of motion, making its value approximately equal to the steady-state value over the entire run-time period.Originality/valueInfluence of surface roughness for various wavelength and undeformed amplitude on the performance of transient elastohydrodynamic lubrication at motion start-up is presented at different acceleration rates as well as for different operating parameters of entraining speed and load. Ball equation of motion is used to calculate the force on the ball as it starts to move.

Load-Carrying Capacity of the Lubricating Film in the Gap Between Surfaces Textured by Hemispherical Pores

The load support of a lubricating film that separates the surfaces textured by identical equidistant spaced hemispherical pores was investigated. Two-dimensional time-dependent Reynolds equation is solved numerically for different pore-radius-to-gap and cell-dimension-to-pore-radius ratios and for different relative pore positions of opposite surfaces. The results are compared with the data obtained for the case when only one of the opposite surfaces is covered with pores. The obtained data show a maximum in the carrying capacity of the lubricating film when the cell-to-pore-radii ratio is approximately equal to two, in the case of two opposite surfaces with pores. At small pore radii and with increasing cell dimensions, the load support of two surfaces with pores is much greater than in the case of one surface with pores. This behavior reverses with increasing pore diameter. The presented analysis and the provided MATLAB programs are applicable for mechanisms having rubbing mechanical parts with surfaces covered with pores.

Effect of rotor misalignment on stability of journal bearings with finite width

The present work studies the dynamic response of fluid film bearing of finite length which is subjected to rotor misalignment. Vibration analysis of the rotor is used to obtain the limits of stability at different operating conditions. The lubricant flow in the bearing is considered laminar, incompressible and isoviscous. For a given small excitation of the bearing shaft, a time dependent Reynolds equation is introduced to investigate the dynamic response of the fluid film bearing. At a misalignment direction angle and misalignment degree, nonlinear perturbation dynamic equations are introduced and solved to determine the critical stability limit of motion for different values of eccentricity ratio. Vibration data is collected to obtain the transition from stable to unstable operating conditions (stability map). It is concluded here that increasing the value of misalignment degree yields to an increase in the value of critical bearing stability limit for a given value of misalignment direction angle. The critical stability number increases as the misalignment direction angle decreases and/or the steady state eccentricity ratio increases.

Numerical investigation of elastohydrodynamic contacts subjected to harmonic load variation

PurposeThe purpose of this paper is to study the behavior of elastohydrodynamic contacts subjected to forced harmonic vibrations including the effect of changing various working parameters such as frequency, load amplitude and entrainment speed.Design/methodology/approachThe time-dependent Reynolds equation is solved using the Newton–Raphson technique. The film thickness and pressure distribution are obtained at every time step by simultaneous solution of the Reynolds equation and film thickness equation including elastic deformation.FindingsThe frequency of vibration, load amplitude and entrainment speed are directly related to the film thickness perturbation, which is formed during load increasing phase of the cycle. The film thickness formed during load increasing phase is larger than that formed during load decreasing phase with larger deviation at a higher frequency or load amplitude and vice versa for lower frequency or load amplitude. The entrainment speed of the contact has an opposite effect to that of the frequency of vibration or load amplitude.Originality/valuePhysical explanations for the behavior of elastohydrodynamic contact subjected to forced harmonic vibration are presented in this paper for various working parameters of frequency, load amplitude and entrainment speed.

EFFECT OF LUBRICANT TEMPERATURE ON THE DYNAMIC AND STABILITY BEHAVIOR OF NANO-LUBRICATED JOURNAL BEARING

The effects of lubricant temperature on the dynamic behaviour of Nano-lubricated finite length journal bearings has been investigated in the present work. Modified time dependent Reynolds equation include the effect of oil film temperature was perturbed in order to calculate the eight dynamic coefficients (four stiffness, and four damping) required to evaluate the dynamic characteristics of the journal bearing, the oil film temperature was obtained by solving numerically the energy and heat conduction equations simultaneously with the Reynolds equation using appropriate boundary conditions. Suitable viscosity temperature model has been used to consider the effect of oil film temperature. The bearing lubricated with oil containing Titanium dioxide (TiO2 ) nanoparticles dispersed in to the base oil with various particle concentration. The effect of adding TiO2 Nano-particles with various particle concentrations (0.1%, 0.5%, 1%, 1.5% and 2%) in order to enhance its viscosity has been discussed. A validation to the mathematical model and the computer program written in Fortran90 prepared to solving the governing equation has been carried out by comparing the result for the dynamic coefficients (stiffness and damping) obtained in the present work with that obtained by Sheeja and Prabhu . The results seen to be in a good agreement with percentage of error less than 2%. The results obtained in the present study show that the bearing stiffness coefficients increases by 30% while the damping coefficients increased by 27% when the bearing lubricated with Nano lubricant that contains TiO2 Nano particles with particle concentration of (1.5%).

Nonlinear Dynamic Analysis of a Rigid Rotor Supported By a Spiral-Grooved Opposed-Hemisphere Gas Bearing

ABSTRACTThe nonlinear dynamic behavior of a rigid rotor supported by a spiral-grooved opposed-hemisphere gas bearing is investigated in this article, focusing particular attention on its whirl motion. The finite element method combined with the finite difference method is employed to solve the time-dependent Reynolds equation that is coupled with the rotor motion considering five degrees of freedom. The rotor responses to the initial disturbance and synchronous and nonsynchronous excitations are investigated. To analyze the complicated dynamic behavior of the rotor–bearing system, the trajectories of the rotor centerline, time responses, phase portraits, power spectra, Poincare maps, and bifurcation diagrams are obtained from the numerical procedure. The results show that the conical whirl instability appears earlier than the cylindrical whirl instability with increasing rotational speed for the rotor–bearing system with no unbalance mass. Moreover, it reveals that the complex dynamic behavior of the system excited by unbalance mass varies with rotational speed and rotor mass. In addition, bifurcation diagrams employing the rotating speed and rotor mass as bifurcation parameters are obtained. Finally, the nonsynchronous excitation responses are presented, which behave in a different way than the synchronous excitation responses. The results of this study offer a further understanding of the nonlinear characteristics of spiral-grooved opposed-hemisphere gas bearings.

플로팅 링 베어링으로 지지된 터보차저 로터의 안정성 해석

The use of turbocharger in internal combustion engines has increased as it is a key components for improving system efficiency without increasing engine size. Because of increasing demand, many studies have evaluated rotordynamic performance so as to increase rotation speed. This paper presents a linear and nonlinear analysis model for a turbocharger rotor supported by a floating ring bearing. We constructed rotor model by using the finite element method and approximated bearings as being infinitely short. In the linear model, we considered fluid film force as stiffness and damping element. In nonlinear analysis, calculation of the fluid film force involved solving the time dependent Reynolds equation. We verified the developed model by comparing the results to those of previous research. The analysis results show that there are four unstable modes, which are rigid body modes combining ring and rotor motion. As the rotating speed increases, the logarithmic decrement shows that certain unstable modes goes into the stable area or the stable mode goes into the unstable area. These unstable modes appear as sub-synchronous vibrations in nonlinear analysis. In nonlinear analysis frequency jump phenomenon demonstrated in several experimental studies appears. The analysis results also showed that frequency jump phenomenon occurs when the vibration mode changes and the sequence of unstable mode matches the linear analysis result. However, the natural frequency predicted using linear analysis differs from those obtained using nonlinear analysis.

Numerical study on tribological performance of the floating valve-plate pair in axial piston pump

A numerical model for studying the time-dependent tribological performance of the floating valve-plate pair in axial piston pump is proposed. The lubrication analysis and the dynamics of the floating valve plate are taken into consideration. An additional source term is adopted into time-dependent Reynolds equation so that the model to analyze the auxiliary balance effects is set up. The boundary pressure of the kidney ports is dynamically coupled with the flow rate towards the loading. It is found that all the performance parameters fluctuate periodically and the valve-plate interface tilt has a vital impact on the tribological performances. The results obtained by the proposed model are highly detailed, and help to improve the understanding of floating valve-plate pair.

Numerical investigation of slider dynamics induced by contacts with disk asperities

A time dependent Reynolds equation simulator combined with a finite element-based transient contact model between a slider and a disk asperity is used to study slider dynamics induced by contacts with disk asperities. The flying height change at the trailing edge of the slider is investigated as a function of asperity height, asperity diameter, and the spacing between the thermal protrusion of a thermal-flying control slider and a disk asperity. The effect of material properties of the disk asperities is studied. Slider vibrations corresponding to the first and the second pitch modes are excited by disk asperities.

Numerical Analysis for the Stability of Cylinder Gas Film Seal

Non-contacting gas film seal applies to the high speed working condition and a numerical method was presented for analyzing the effect of speed on the stability of cylinder gas film seal. The dynamics analysis model was established, solving the time-dependent Reynolds equation coupling with the dynamic equations. Through numerical simulation, the critical speed of cylinder gas film seal system and the diagram of critical mass versus rotor speed were obtained. The influence of the speed on dynamic stability was studied. The results show that the system stability becomes worse as rotor speed increases.

A rigorous derivation of the time-dependent Reynolds equation

We study the asymptotic behavior of solutions of the evolution Stokes equation in a thin three-dimensional domain bounded by two moving surfaces in the limit as the distance between the surfaces ap ...

Theoretical Investigation in Transient Elastohydrodynamic Lubrication of Reciprocating Motion in Air Compressor Piston Pin

This paper presents the effect of squeeze motion to elastohydrodynamic lubrication (EHL) in piston pin of air compressor with non-Newtonian lubricants based on a Carreau viscosity model. The time dependent modified Reynolds equation and elasticity with initial conditions were formulated and solved numerically using a multigrid multilevel with full approximation technique for a piston pin. Effects of bearing materials and liquid properties were examined. The results show that minimum film thickness and friction coefficient increases when lubricant viscosity increases. At the same time, for increasing of elastic modulus of bearing bushing the friction coefficient decreases.

A Method for Solving Stiffness of Finite Length Self-Acting Gas-Lubricated Journal Bearing

Based on the differential transformation method, an efficient method is proposed to calculate stiffness of finite length self-acting gas-lubricated journal bearing. Time dependent Reynolds equation in gas lubrication of the self-acting journal bearing is solved, gas film pressure distribution and stiffness of the bearing are obtained. In comparison with the partial derivative method, the proposed method for solving stiffness of self-acting gas-lubricated journal bearing is verified. The proposed method can provide the theoretical reference to the dynamic design of self-acting gas-lubricated journal bearing-rotor system.

Effect of Liquid-Solid Lubricant on Mixed Lubrication in Line Contact

This paper presents the performance characteristics of two surfaces in line contact under isothermal mixed lubrication with non-Newtonian liquid–solid lubricant base on Power law viscosity model. The time dependent Reynolds equation, elastic equation and viscosity equation were formulated for compressible fluid. Newton-Raphson method and multigrid technique were implemented to obtain film thickness profiles, friction coefficient and load carrying in the contact region at various roughness amplitudes, applied loads, speeds and the concentration of solid lubricant. The simulation results showed that roughness amplitude has a significant effect on the film pressure, film thickness and surface contact pressure in the contact region. The film thickness decrease but friction coefficient and asperities load rapidly increases when surface roughness amplitude increases or surface speed decreases. When the concentration of solid lubricant increased, friction coefficient and asperities load decrease but traction and film thickness increase.

Numerical Analysis Method and Bifurcation of Gas-Lubricated Journal Bearing-Rotor System with Gyroscopic Effect

Based on the nonlinear theory, the unbalanced responses of the gas-lubricated journal bearing-rotor system are investigated. A time-dependent mathematical model is established to describe the pressure distribution of gas-lubricated journal bearing with nonlinearity. The rigid rotor with gyroscopic effect supported by self-acting gas journal bearing with three axial grooves is modeled. The differential transformation method is employed to solve the time-dependent gas-lubricated Reynolds equation, and the dynamic motion equation is solved by Newmark-β method. The unbalanced responses of the rotor system supported by finite gas-lubricated journal bearings are analyzed by bifurcation diagram, orbit diagram, Poincaré map. The numerical results reveal periodic, period-4 motion of nonlinear behaviors of the system.

스러스트 볼 베어링이 적용된 왕복동형 압축기의 마찰손실 해석

In this paper, a study on the frictional losses and dynamic behaviors of a reciprocating compression mechanism used in small refrigeration compressor is performed. In the problem formulation of the compressor dynamics, the viscous frictional force between piston and cylinder wall is considered in order to determine the coupled dynamic behaviors of piston and crankshaft supported on a thrust ball bearing. The solutions of the equations of motion of the reciprocating mechanism along with the time dependent Reynolds equations for the lubricating film between piston and cylinder wall and lubricant films of the journal bearings are obtained simultaneously. The hydrodynamic forces of journal bearings are calculated using finite bearing model and G<TEX>$\hat{u}$</TEX>m-bel boundary condition. And, a Newton-Raphson procedure was employed in solving the nonlinear equations of piston and crankshaft with a thrust ball bearing. The results explored the effects of design parameters on the frictional losses and dynamic stability of the compression mechanism.

Nonlinear Dynamic Behaviors and Bifurcation of Symmetrical Rotor System Supported by Self-Acting Gas Jounal Bearings

The unbalanced response and corresponding bifurcation behavior of the rotor dynamic system supported by gas journal bearings are investigated. A time-dependent mathematical model is used to describe the pressure distribution of gas journal bearing with nonlinearity. The rigid Jeffcott rotor with self-acting gas journal bearing supports is modeled. The finite difference method and the Successive Over Relaxation (S.O.R.) method are employed to solve the time-dependent Reynolds equation of gas journal bearings. The bifurcation of unbalanced responses of the rotor is analyzed by a Poincaré map. The numerical results reveal periodic, period-doubling, quasi-periodic, and chaotic motion of rich and complex non-linear behaviors of the system.

Transient elastohydrodynamic lubrication in artificial knee joint with non-Newtonian fluids

This paper presents the transient analysis of a human artificial knee joint under elastohydrodynamic lubrication (EHL) for point contact with non-Newtonian lubricants. The artificial knee joints use ultra high molecular weight polyethylene (UHMWPE) against metal with time-varying speed and load during walking. This numerical simulation employed a perturbation method, Newton Raphson method and multigrid method with full approximation technique to solve simultaneously both the time-dependent Reynolds equation, with non-Newtonian fluid based on a Carreau model, and the elasticity equation. The general numerical schemes are implemented to investigate the characteristics of elastohydrodynamic lubrication in human artificial knee joints; profiles of pressure and film thickness are determined, with varying material and lubricant properties, applied loads and speeds. The results show that the elastohydrodynamic fluid film thickness between the metallic component of the artificial knee joint and the soft polyethylene bearing becomes larger as the contact area increases and the fluid film pressure decreases. At the beginning of the first walking cycle, the film thickness is lower than in subsequent cycles because of the time required to develop the fluid film; after the first cycle, the fluid film is similar for every cycle and is dependent on transient applied load and speed during human movement.

Experimental and numerical investigation of separated flows in subsonic diffusers

The results of a special investigation of the diffuser flowfield are presented for two models of curvilinear diffuser channels with annular and rectangular cross-sections. The flow is visualized and the total pressure fields are measured by means of low-inertia transducers. At the same time, the flows are numerically calculated using commercial programs, together with codes developed by the authors. In these calculations the stationary and time-dependent Reynolds equations closed by different turbulence models, as well as the time-dependent Navier-Stokes equations, were integrated. A considerable difference between the measured data and the results of the numerical calculations in the stationary formulation is found to exist. At the same time, it has been possible to describe the occurrence of spatial inhomogeneities, the flow pattern, and the level of the experimentally observed aerodynamic losses on the basis of the solution of time-dependent problems.

Six-DOF vibrations of partial contact sliders in hard disk drives

A six-degree-of-freedom slider dynamic simulator is developed to analyze the slider’s motion in the vertical, pitch, roll, yaw, length and width directions. The modified time-dependent Reynolds equation is used to model the air bearing and a new second order slip model is used for a bounded contact air bearing pressure. The simulator considers the air bearing shear acting on the air bearing surface and the slider–disk contact and adhesion. Simulation results are analyzed for the effects of the disk surface micro-waviness and roughness, skew angle, slider–disk friction and micro-trailing pad width on the vertical bouncing, down-track and off-track vibrations of a micro-trailing pad partial contact slider.