A generalized Reynolds equation based on non-Newtonian flow in lubrication mechanics

A generalized Reynolds equation based on non-Newtonian flow is derived in this paper. This equation is suitable for a number of non-Newtonian flow models and can be solved numerically to obtain pressure fields in thermalhydrodynamically or elastohydrodynamically lubricated fluid films. A mathematical approach is given for solving simultaneously the shearing stress, shearing rate, velocity and equivalent viscosity. To show the application of this equation, two rheological models which have been widely used in lubrication mechanics are incorporated into this equation to obtain numerical solutions to the line contact thermal elastohydrodynamic lubrication problem.     

Using adjoint error estimation techniques for elastohydrodynamic lubrication line contact problems

The use of an adjoint technique for goal‐based error estimation described by Hartit et al. (Int. J. Numer. Meth. Fluids 2005; 47 :1069–1074) is extended to the numerical solution of free boundary problems that arise in elastohydrodynamic lubrication (EHL). EHL systems are highly nonlinear and consist of a thin‐film approximation of the flow of a non‐Newtonian lubricant which separates two bodies that are forced together by an applied load, coupled with a linear elastic model for the deformation of the bodies. A finite difference discretization of the line contact flow problem is presented, along with the numerical evaluation of an exact solution for the elastic deformation, and a moving grid representation of the free boundary that models cavitation at the outflow in this one‐dimensional case. The application of a goal‐based error estimate for this problem is then described. This estimate relies on the solution of an adjoint problem; its effectiveness is demonstrated for the physically important goal of the total friction through the contact. Finally, the application of this error estimate to drive local mesh refinement is demonstrated. Copyright © 2010 John Wiley & Sons, Ltd.     

渐开线齿轮传动非牛顿润滑介质的线弹流数值分析研究

采用适合各种流变模型的广义Reynolds方程，通过数值联立求解非牛顿介质的线弹流润滑基本方程组，获得了渐开线齿轮啮合过程的油膜压力、膜厚、表面剪应力分布，并分析了啮合过程中非牛顿效应对齿轮传动最小油膜厚度的影响。在数值计算方向引入延拓方法，使表面煎应力迭代具有大范围收敛性。     

A solution to the hydrodynamic lubrication of a circular point contact sliding over a flat surface with cavitation

This letter presents an analytical solution to the hydrodynamic lubrication of a circular point contact sliding over a flat surface with cavitation. The solution is found by solving the Reynolds equation with Reynolds boundary condition for cavitation. The cavitation boundary is shown to be straight lines directed 108.4° against the sliding direction. The result is experimentally verified in the limit of large values of viscosity, sliding velocity and radius of a spherical ball. The solution raises questions about the coupling between cavitation and film rupture and can be used as an independent check on the validity of numerical solutions.     

A numerical scheme for solving a periodically forced Reynolds equation

A modified Reynolds equation is used to model the air‐film in a high‐speed squeeze‐film bearing. The axial position of the bearing stator is prescribed as a finite amplitude periodic oscillation. A numerical approach is considered for solving the uncoupled and coupled periodic problems associated with this model. The uncoupled problem requires the computation of the squeeze‐film dynamics when the rotor is held at a fixed axial position and the coupled problem incorporates the additional air–rotor interaction since the rotor position is unknown and modelled as a spring‐mass‐damper system. The details of a Fourier spectral collocation scheme are provided for the reduction of the modified Reynolds equation to a system of non‐linear, first‐order ordinary differential equations in space. Using the Matlab boundary value problem solver bvp4c this system of equations is solved to give the periodic pressure distributions and rotor heights. The high degree of accuracy in the spectral collocation scheme is demonstrated through comparison with an appropriate analytical solution. Further analysis indicates that the direct periodic solver is at least 10 times faster than the equivalent Crank–Nicholson finite‐difference scheme. For changing values of a selected physical parameter the method of arc‐length continuation is employed to track branches of solutions computed using the spectral collocation scheme. A selection of results is presented to demonstrate the range of accessible solutions and the robust nature of the numerical scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

Jacobian-free Newton-Krylov subspace method with wavelet-based preconditioner for analysis of transient elastohydrodynamic lubrication problems with surface asperities

This paper presents an investigation into the effect of surface asperities on the over-rolling of bearing surfaces in transient elastohydrodynamic lubrication(EHL) line contact. The governing equations are discretized by the finite difference method. The resulting nonlinear system of algebraic equations is solved by the Jacobian-free Newtongeneralized minimal residual(GMRES) from the Krylov subspace method(KSM). The acceleration of the GMRES iteration is accomplished by a wavelet-based precondit...     

On the lubrication approximation for a class of viscoelastic fluids

We present a rigorous derivation of a dimensionally reduced Reynolds type equation for thin film flow lubrication of a class of viscoelastic fluids by employing a perturbation analysis on the upper-convected Maxwell model in natural orthogonal coordinates. This approximation accounts for the viscoelastic and curvature corrections to the classical Reynolds lubrication approximation. Comparison of our approximation with the classical Reynolds approximation suggests that viscoelasticity can have a significant influence on the lubrication characteristics, at least for certain values of the film thickness and of the eccentricity ratios of the journal bearing.     

使用不等距多重网格技术求解线接触热弹流润滑问题

使用不等距多重网格法求解了线接触热弹流润滑问题 .结果表明 :与等距多重网格法相比 ,新解法虽不能提高数值求解的稳定性 ,却能得到更为精确的压力、膜厚及温度分布 ;计算结果表明中载下第二压力峰是光滑的     

滚子摩擦副弹流润滑特性及其应用──Ⅰ 有限长直母线滚子的弹流数值解

运用改进的数值方法求得了与实验结果相符的等温有限长直母线滚子的弹流数值解，与无限长线接触弹流结果相比较，揭示了润滑状态下滚子摩擦副的边缘效应和端泄对油膜分布的影响，在低速和重载时滚子的两端都几乎不能形成全膜润滑，因而有必要通过修形消除滚子端部的边缘效应．     

Reynolds方程在纹理表面动压润滑计算中的有效性评价

从理论上阐述了纹理表面动压润滑计算中决定Reynolds方程有效性的两个关键因素为油膜厚度与纹理特征长度的比值h/L和缩减的雷诺数re;只有当h/L和re同时趋近于零时Reynolds方程才能够获得准确的结果,并由此在h/L-Re平面上标注了Reynolds方程的适用范围.继而以二维矩形沟槽为实例,采用数值方法计算了h/L和re对Reynolds方程误差的影响规律;分析了Reynolds方程在不同条件下的失效机制;分析了矩形沟槽纹理表面Reynolds方程有效性的评价标准:当缩减的雷诺数re小于0.20,并且h/L小于0.015时能够保证Reynolds方程的误差在10%以下.     

ON THE NUMERICAL SOLUTION OF QUASILINEAR WAVE EQUATION WITH STRONG DISSIPATIVE TERM

The numerical solution for a type of quasilinear wave equation is studied. The three-level difference scheme for quasi-linear waver equation with strong dissipative term is constructed and the convergence is proved. The error of the difference solution is estimated. The theoretical results are controlled on a numerical example.     

New numerical method for volterra integral equation of the second kind in piezoelastic dynamic problems

The elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly simplifies the solving procedure for such elastodynamic problems. Meanwhile, it becomes very important to find a way to solve the second kind Volterra integral equation effectively and quickly. By using an interpolation function to approximate the unknown function, two new recursive formulae were derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently. It is also very stable for long time calculating.     

步态条件下人工膝关节线接触弹流润滑分析

参照关节模拟试验机的运动和力学参数,利用多重网格技术对人工膝关节摩擦副进行了1个步态周期仿人体环境线接触弹流润滑仿真,关节支承表面的弹性变形按半无限体计算.同时,观察了几种参数对流体压力分布和膜厚形状的影响.结果表明:在1个步态周期内,中心压力与载荷变化基本相同,且等效曲率半径的变化会引起中心压力的跳跃.站立相时在卷吸和挤压膜效应的共同作用下,中心膜厚呈逐步减小并伴随着波动;摆动相时,载荷固定,膜厚的变化主要与卷吸速度有关.减小胫骨平台曲率半径有助于提高滑液膜厚度;延长步态周期的时间,会使滑液膜厚减小.     

摆动工况下有限长线接触弹流的实验研究

针对对数型凸度滚子,在自制的滚子接触光弹流试验装置上开展了摆动工况弹流的实验研究.在2种不同摆动频率下,测量了弹流润滑油膜厚度和形状.结果表明:在一定载荷范围内,对数凸型滚子弹流润滑状态下的油膜厚度沿轴向呈均匀分布.摆动工况下润滑油膜的变化受到楔形效应和挤压效应的共同作用;低频时卷吸速度变化较小,以挤压效应为主;高频时卷吸速度变化较大,楔形效应增强.在卷吸速度为零时,油膜被封入接触区内,高频时形成的凹陷更为明显.     

A local‐analytic‐based discretization procedure for the numerical solution of incompressible flows

An Erratum has been published for this article in International Journal for Numerical Methods in Fluids 2005, 49(8): 933. We present a local‐analytic‐based discretization procedure for the numerical solution of viscous fluid flows governed by the incompressible Navier–Stokes equations. The general procedure consists of building local interpolants obtained from local analytic solutions of the linear multi‐dimensional advection–diffusion equation, prototypical of the linearized momentum equations. In view of the local analytic behaviour, the resulting computational stencil and coefficient values are functions of the local flow conditions. The velocity–pressure coupling is achieved by a discrete projection method. Numerical examples in the form of well‐established verification and validation benchmarks are presented to demonstrate the capabilities of the formulation. The discretization procedure is implemented alongside the ability to treat embedded and non‐matching grids with relative motion. Of interest are flows at high Reynolds number, ??(10⁵)–??(10⁷), for which the formulation is found to be robust. Applications include flow past a circular cylinder undergoing vortex‐induced vibrations (VIV) at high Reynolds number. Copyright © 2005 John Wiley & Sons, Ltd.     

考虑摩擦动力学特性的渐开线齿轮润滑分析

以渐开线齿轮为研究对象,综合考虑齿面摩擦、油膜刚度和阻尼的影响,将摩擦动力学特性和齿面形貌考虑到有限长线接触弹流润滑理论(EHL)中,采用多重网格数值求解法获得了渐开线齿轮在不同啮合位置处的动载荷分布,压力与膜厚分布,并对比分析了干摩擦与润滑条件下压力分布的不同特征以及动载荷对油膜分布的影响.计算结果表明:低速时,动载荷接近稳态分布,在单双齿交替点有明显的高频冲击,随着转速的增加,动载荷变化趋于平缓;与干摩擦相比,润滑油膜可以减小最大压力峰值以及入口区和出口区的压力,但对粗糙峰凹谷处压力有所增加.沿啮合线方向,压力在节点位置附近达到最大值,膜厚受动载荷的影响较小,近似呈线性增加趋势.     

一类非线性奇异积分方程及其数值方法研究

探讨了一类非线性奇异积分方程的数理性质以及在颗粒雷诺数Rep＜1时此类方程解的存在条件，然后详细研究了该方程的数值计算方法并构造称之为P（EC）^k多步法的差分格式，分析了该格式的收敛性和代数精度，得到时域离散步长的约束关系。运用该格式计算了静止流场和均匀振荡流中球形小颗粒的非恒定运动，将计算结果与其解析解及有关实验数据的比较表明，上述数值方法具有良好的计算精度。     

A least-squares procedure for the solution of transport problems

In this paper a least-squares formulation associated with a conjugate gradient algorithm is proposed for the solution of transport problems. In this procedure the advection–diffusion equation is first discretized in time using an implicit scheme. At each time step the resulting partial differential equation is replaced by an optimal control problem. This minimization problem involves the minimization of a functional defined via a state equation. This functional is chosen in order to force the numerical solution of the advection–diffusion equation to be equal to the hyperbolic advective part of this equation. The effectiveness of the method is shown through a one-dimensional example involving advective and diffusive transport. No oscillation and high accuracy have been obtained for the entire range of Peclet numbers with a Courant number well in excess of unity.     

小波插值方法自适应数值求解时间进化微分方程

应用小波自相关函数的插值性质,得到任意给定函数的插值小波表达式,然后对其直接求导,可以得到函数导数的表达式。导数运算不再应用差分算法,扩展了小波方法在数值求解微分方程中的应用。由于小波基函数的有限支撑特点,小波方法可以有效地处理微分方程中解的局部突变问题。通过设定小波系数阀值,实现了求解过程的自适应。本文给出了两个算例,结果表明了算法的自适应特点及其向二维空间问题推广的有效性。     

Weak coupling of a Reynolds model and a Stokes model for hydrodynamic lubrication

The Reynolds model is a reduced Stokes model, valid for narrow lubrication regions. In order to be able to handle locally non‐narrow regions such as pits or grooves, often displaying rapid geometrical variations, there is a need to be able to transit to the more accurate Stokes model. A fundamental problem is how to couple the two models in a numerical simulation, preferably allowing for different meshes in the different domains. In this paper, we present a weak coupling method for Reynolds and Stokes models for lubrication computations, including the possibility of cavitation in the different regions. The paper concludes with a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.