Elsevier

Tribology International

Volume 46, Issue 1, February 2012, Pages 30-40
Tribology International

Novel approach to solve the dynamical porous journal bearing problem

https://doi.org/10.1016/j.triboint.2011.03.017Get rights and content

Abstract

Analytical pressure solutions are a simple and robust way to model plain journal bearings in rotordynamics, but cannot be extended to porous journal bearings. Using the method of weighted residuals—viz. Galerkin's method with global shape functions—a novel and fast approach to solve the dynamical porous journal bearing problem is proposed. This approach allows for the influence of rough surfaces on the hydrodynamic pressure implemented through flow-factors, journal misalignment and calculation of stiffness and damping coefficients for finite bearings. The proposed method will be verified using analytical expressions and new results will be shown for porous journal bearings including the influence of rough surfaces.

Highlights

► A novel and fast approach to solve the dynamical journal bearing problem. ► Surface roughness with flow-factors for porous journal bearings. ► Stiffness and damping coefficients for porous bearings including influence of surface roughness.

Introduction

Porous journal bearings find a wide range of applications in present-day machinery due to their favorable self-lubricating properties and high load capacity in combination with low production costs. The bush of a porous journal bearing is typically made of porous metals such as sintered bronze or sintered iron giving it the ability to transmit fluid. This property enables the bush to stimulate lubrication and act as a lubricant reservoir.

Since the first theoretical study on porous journal bearings [1] much development has taken place in this field leading to a wide range of models. Many of these models account for the influence of the porous bush, being described by d’Arcy's equation [1], [2], [3], [4], [5], [6] or the influence of the slip conditions between fluid film and the porous bush [6], [7], [8], [9], [10].

Only few studies address the influence of rough surfaces in porous bearings [11], [12], [13], [14]. All of these studies utilized a stochastic Reynolds equation which can be derived using stochastic process theory [15], [16]. This method is, however, restricted to either transverse or longitudinal roughness. To overcome this shortcoming a modified form of the Reynolds equation was introduced by [17], [18], which relies on the so-called flow factors accounting for the influence of the surface structure on the fluid flow. It can be summarized that the modeling of porous journal bearings gives rise to various extensions to the classical Reynolds equation and thus demands for advanced solution strategies.

The Reynolds equation can be solved analytically, however, these analytical solutions cannot be extended to porous journal bearings nor are they valid for the full Reynolds equation [19]. A solution for the full Reynolds equation for porous journal bearings has to be generated using numerical discretization schemes [10], [20], [21], [22], which are time-consuming and are therefore not practical in use for high performance applications like multibody system simulations. In such cases, the common approach is to calculate high-dimensional look-up tables. Practical applications show that retrieving specific values from these tables using numerical interpolation often turns out to be inaccurate and not very robust due to issues related with high-dimensional interpolation.

The goal of this paper is to offer an approach without using complex, time-consuming numerical discretization methods or look-up tables and in doing so preserving the advantages of analytical solutions, namely speed and robustness. Since for the generalized Reynolds equation—accounting for effects like porous bushes, rough surfaces and misalignment for instance—analytical pressure functions are not available, a low-dimensional semi-analytical approximation is constructed using Galerkin's method [23], [5], [24]. It is shown that using appropriate global basis functions and taking advantage of orthogonality conditions eventually yields sparse, band-structured linear equations, which can be solved very efficiently.

The proposed approach is verified by comparing its results for the plain journal bearing with the analytical short and long bearing solution. Subsequently, results for the porous journal bearing problem are shown for several loading conditions. Eventually, the flexibility of the proposed method is demonstrated by solving the hydrodynamic porous journal problem under the influence of rough surfaces.

Section snippets

Governing equations

The Reynolds equation governs the pressure profile in the fluid film of a porous journal bearing. This elliptic differential equation has to be solved to obtain the pressure profile. We adopt the dimensionless formulation from [19] for the axial coordinate, film thickness, pressure and time respectively:z=z/L,h=H/c,p=(c/Ri)2p/6ηΩandτ=Ωt,where L is the bearing length, H is fluid film thickness, c is the radial clearance, Ri is the inner bearing radius, η is the fluid viscosity and Ω is the

Analytical solutions for special cases

For the plain journal bearing the third term on the left hand side of Eq. (2) vanishes and the equation simplifies to the classical standard form. Ref. [19] showed that the journal bearing problem can be solved analytically for the long and short bearing simplifications. For a short journal bearing (L/D<1/2) the term p/θ can be ignored. Therefore, the first term on the left hand side of Eq. (2) vanishes and when taking the boundary conditions from Eq. (4) the pressure function becomesp(θ,z)

Approximate solution by Galerkin's method

In order to obtain a low-dimensional approximation Galerkin's method [29] with global basis functions is used to approximate the solution of the problem. Since for constant κ Eq. (3) is a Laplace equation, it can be solved by separation of variables [5]. By assuming the solution can be written as a product of a circumferential, an axial and a radial partp^(θ,z,r)Θ(θ)Z(z)ρ(r)the problem is separated into three differential equations2Θθ2+β2Θ=0,2Zz2+α2/κ2Z=0,r22ρr2+rρr(α2r2β2)

Results and discussion

In order to verify the proposed method, the classical journal bearing problem is solved and the results are compared to known analytical solutions as well as full numerical solutions. After this verification, the method is applied to a porous journal bearing and finally its flexibility is demonstrated by applying it to the generalized formulation of the Reynolds equation to account for the influence of rough surfaces.

Conclusion

Most models for porous journal bearing are based on advanced numerical discretization schemes. Although these methods can be very accurate, practical applications often are numerically very costly, time-consuming and therefore not practical in use for many applications like multibody simulations for instance.

To solve the Reynolds equation Galerkin's method is used to obtain a low-dimensional semi-analytical approximation of the pressure function. Advantages of this approach are that only a

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