Efficient solution of the non-linear Reynolds equation for compressible fluid using the finite element method

Abstract

An efficient finite element scheme for solving the non-linear Reynolds equation for compressible fluid coupled to compliant structures is presented. The method is general and fast and can be used in the analysis of airfoil bearings with simplified or complex foil structure models. To illustrate the computational performance, it is applied to the analysis of a compliant foil bearing modelled using the simple elastic foundation model. The model is derived and perturbed using complex notation. Top foil sagging effect is added to the bump foil compliance in terms of a close-form periodic function. For a foil bearing utilized in an industrial turbo compressor, the influence of boundary conditions and sagging on the pressure profile, shaft equilibrium position and dynamic coefficients is numerically simulated. The proposed scheme is faster, leading to the conclusion that it is suitable, not only for steady-state analysis, but also for non-linear time domain analysis of rotors supported by airfoil bearings.

Appendices

Appendix 1: Top foil deflection

To include the ’sagging’ effect of the top foil, as illustrated in Fig. 8a, into the mathematical model of the foil bearing, a periodic expression for the top foil flexibility, dependant on the angle $\theta$, is sought. If the top foil is assumed to have unit width, the uniform pressure P becomes a uniformly distributed load along x (Fig. 8b). The top foil is assumed in pure bending and the bump foil deflection is kept at zero. Requiring the infinitesimal element of the top foil, Fig. 8b, to be in static equilibrium one obtain:

Fig. 8

Schematics and nomenclature of a the foil structure (bump foil and top foil) together with an infinitesimal element of the deformed top foil between two consecutive bump tops

$$\begin{aligned} M + {\text{d}}M - M + \frac{1}{2}P{\text{d}}\tilde{\theta }^{2} + (T + {\text{d}}T){\text{d}}\tilde{\theta } = 0\quad \Rightarrow {\mkern 1mu} \frac{{{\text{d}}M}}{{d\tilde{\theta }}} = T \hfill \\ - N + N + {\text{d}}N = 0\quad \Rightarrow {\text{d}}N = 0 \hfill \\ T + {\text{d}}T - T - P{\text{d}}\tilde{\theta } = 0\quad \Rightarrow {\mkern 1mu} {\mkern 1mu} \frac{{{\text{d}}T}}{{{\text{d}}\tilde{\theta }}} = P \hfill \\ \end{aligned}$$

(31)

rewriting and differentiating (31) yields

$$\begin{aligned} \frac{{\rm d}M}{{\rm d}\tilde{\theta }} = T \,\,\Rightarrow \,\, \frac{{\rm d}^2M}{{\rm d}\tilde{\theta }^2} = \frac{{\rm d}T}{{\rm d}\tilde{\theta }} = P. \end{aligned}$$

(32)

If pure bending in one direction is assumed, then Kirchhoff–Love plate theory for isotropic plates, describes the relation between the bending moment M and the curvature $\frac{{\rm d}^2w_t}{{\rm d}\tilde{\theta }^2}$ as

$$\begin{aligned} M = D_t \frac{{\rm d}^2 w_t}{{\rm d}\tilde{\theta }^2},\,\,\, D_t = \frac{E t_t^3}{12(1-\nu ^2)} \end{aligned}$$

(33)

where $D_{\rm t}$ is the flexural rigidity. Integrating (32) twice and inserting (33) leads to

$$\begin{aligned} D_{\rm t} \frac{{\rm d}^2 w_t}{{\rm d}\tilde{\theta }^2} = \frac{P^2}{2}\tilde{\theta }^2 + c_1\tilde{\theta }+ c_2 \end{aligned}$$

(34)

which upon double integration yields

$$\begin{aligned} w_{\rm t}(\tilde{\theta }) = \frac{1}{D_{\rm t}} \left( \frac{P}{24}\tilde{\theta }^4 + \frac{c_1}{6}\tilde{\theta }^3 + \frac{c_2}{2}\tilde{\theta }^2 + c_3\tilde{\theta } + c_4\right) . \end{aligned}$$

(35)

Since the distributed load $P$ is assumed uniform and the deflection of the bump foil is kept zero, the boundary conditions for a section of the top foil between two consecutive bump tops over the length S, as depicted in Fig. 8a, are

$$\begin{aligned} w_{\rm t}(0)=w_{\rm t}^\prime (0)=w_{\rm t}(S)=w_{\rm t}^\prime (S)=0. \end{aligned}$$

(36)

Applying these boundary conditions leads to the integration constants $c_1 = -PS/2$, $c_2=PS^3/12$ and $c_3=c_4=0$ which by insertion in (35) leads to the foil deflection function

$$\begin{aligned} w_{\rm t}(\tilde{\theta }) = (p-p_{\rm a})K_{\rm t} \end{aligned}$$

(37)

where $(p-p_{\rm a}) = P$ and the top foil flexibility per unit width is

$$\begin{aligned} K_{\rm t}(\tilde{\theta }) = \frac{(1-\nu ^2)}{2E t_t^3} \left( \tilde{\theta }^4 - 2S\tilde{\theta }^3 + S^2\tilde{\theta }^2\right) ,\,\,\,\tilde{\theta }\in [0:S]. \end{aligned}$$

(38)

The top foil flexibility $K_{\rm t}(\tilde{\theta })$ is defined over a section of the length S, i.e. in a closed interval between two bump tops. To develop an expression for the top foil flexibility over several bump tops, i.e. a periodic expression (38) is expanded into a Fourier series as

$$\begin{aligned} K_{\rm t}(\tilde{\theta }) = \frac{(1-\nu ^2)}{2E t_{\rm t}^3} \left( \frac{a_0}{2} + a_1 \cos \left( \frac{2\pi \tilde{\theta }}{S} \right) + \ldots \right) \end{aligned}$$

(39)

where

$$\begin{aligned} a_0 = \frac{S^4}{15},\,\,\, a_1 = -3\left( \frac{S}{\pi }\right) ^4, \,\,\, \ldots \end{aligned}$$

(40)

It can be shown, that the first two terms of (39) approximate the top foil flexibility with sufficient accuracy and thereby the top foil flexibility per unit width can be written as:

$$\begin{aligned} K_{\rm t}(\tilde{\theta }) \approx \frac{S^4(1-\nu ^2)}{E t_{\rm t}^3} \left( \frac{1}{60} - \frac{3}{2\pi ^4} \cos \left( \frac{2 \pi \tilde{\theta }}{S}\right) \right) \end{aligned}$$

(41)

Appendix 2: Iterative solution based on Nl method

The pressure p is found iteratively by trying to satisfy the non-linear equilibrium condition [16] which can be written in residual form as:

$$\begin{aligned} R(p) = R_{\rm ext}(p) - R_{\rm int}(p). \end{aligned}$$

(42)

If $p_i$ is an approximate solution to the exact solution p, then a first-order Taylor expansion gives an equilibrium equation for the next Nl step as

$$\begin{aligned} R(p_{i+1}) \approx R(p_i) + \frac{{\rm d} R_{\rm int}(p_i)}{{\rm d}p} \Delta p_i = 0. \end{aligned}$$

(43)

If we now define the tangent as

$$\begin{aligned} K_{\rm t} \equiv \frac{{\rm d} R_{\rm int}(p_i)}{{\rm d}p} \end{aligned}$$

(44)

then the equilibrium equation (43) can be written as

$$\begin{aligned} K_t \Delta p_i = -R(p_i) \end{aligned}$$

(45)

or inserting (42)

$$\begin{aligned} K_{\rm t} \Delta p_i = -R_{\rm ext}(p_i) + R_{\rm int}(p_i). \end{aligned}$$

(46)

When the equilibrium equation (46) has been solved the pressures are updated from

$$\begin{aligned} p_{i+1} = p_i + \Delta p_i. \end{aligned}$$

(47)

The tangent is then updated with the new pressure $p_i = p_{i+1}$ and the procedure is repeated. We repeat until the norm of the residual is sufficiently small. Even though the Nl method, as shown above, was derived for a scalar problem, it is directly applicable to vector problems as well.

Appendix 3: Solution algorithm

Based on the iterative Nl method outlined in Appendix 2, a pseudo algorithm is given as: