Subsections

• Formulation
• Performance

16 Journal Bearing

Given the eccentricity $\epsilon$ of the journal bearing, find the pressure distribution in the lubricant separating the shaft from the bearing.

Formulation

The journal bearing problem [10] requires determining the pressure between two circular cylinders of length $L$ and radii $R$ and $R + c$. The separation between the cylinders is $\varepsilon c$, where $\varepsilon$ is the eccentricity. The pressure $v$ minimizes the quadratic $q:K \mapsto \mbox{${\mathbb{R}}$}$,

$$q(v)= \int_{\cal D} \left\{ \frac{1}{2} w_q(x) \Vert\nabla v(x)\Vert^2 - w_l(x) v(x) \right\} dx ,$$

over the convex set $K$, where ${\cal D}=(0,2 \pi) \times (0,2 b)$,

$$K=\{v \in H_0^1 ({\cal D}): v \ge 0 \} ,$$

$H_0^1 ({{\cal D}})$ is the space of functions with gradients in $L^2 ( {\cal D} )$ that vanish on the boundary of $\cal {D}$, and the functions $w_q:{\cal D} \mapsto \mbox{${\mathbb{R}}$}$ and $w_l:{\cal D} \mapsto \mbox{${\mathbb{R}}$}$ are defined by

$$w_q (\xi_1,\xi_2)=(1+\varepsilon \cos \xi_1)^3, \qquad \ w_l(\xi_1,\xi_2) = \varepsilon \sin \xi_1 ,$$

with $\varepsilon \in (0,1)$ the eccentricity of the bearing.

A finite element approximation to the journal bearing problem is obtained by triangulating ${{\cal D}}$ and minimizing $q$ over the space of piecewise linear functions with values $v_{i,j}$ at the vertices of the triangulation. We follow [3] by using a triangulation with, respectively, $n_x$ and $n_y$ internal grid points in the coordinate directions. Data for this problem appears in Table 16.1.

Table 16.1: Journal bearing problem data

Variables	$n_x n_y$
Constraints	0
Bounds	$n_x n_y$
Linear equality constraints	0
Linear inequality constraints	0
Nonlinear equality constraints	0
Nonlinear inequality constraints	0
Nonzeros in $\nabla ^2 f(x)$	$5n_x n_y -2(n_x+n_y)$
Nonzeros in $c'(x)$	0

Performance

We provide results with the AMPL formulation in Table 16.2 with $b=10$ and $\epsilon = 0.1$. For these results we fix $n_x = 50$ and vary $n_y$. The starting guess is the function $\max\{\sin (x),0\}$ evaluated at the grid nodes. Figure 16.1 shows the pressure distribution for the journal bearing problem.

Table 16.2: Performance on pressure in journal bearing problem

Solver	$n_y = 25$	$n_y = 50$	$n_y = 75$	$n_y = 100$
LANCELOT	3.02 s	7.19 s	11.77 s	17.89 s
$f$	-1.54015e-01	-1.54824e-01	-1.54984e-01	-1.55042e-01
$c$ violation	0.00000e+00	0.00000e+00	0.00000e+00	0.00000e+00
iterations	12	11	10	10
LOQO	3.36 s	5.71 s	9.56 s	13.33 s
$f$	-1.54015e-01	-1.54824e-01	-1.54984e-01	-1.55042e-01
$c$ violation	2.2e-16	3.1e-16	3.7e-16	4.2e-16
iterations	26	19	20	21
MINOS	173.65 s	964.59 s	2850.41 s	$\ddagger$
$f$	-1.54015e-01	-1.54824e-01	-1.54984e-01	$\ddagger$
$c$ violation	0.0e+00	0.0e+00	0.0e+00	$\ddagger$
iterations	1	1	1	$\ddagger$
SNOPT	722.68 s	$\ddagger$	$\ddagger$	$\ddagger$
$f$	-1.54015e-01	$\ddagger$	$\ddagger$	$\ddagger$
$c$ violation	0.0e+00	$\ddagger$	$\ddagger$	$\ddagger$
iterations	197	$\ddagger$	$\ddagger$	$\ddagger$
$\dagger$ Errors or warnings. $\ddagger$ Timed out.

Figure 16.1: Journal bearing problem with $\epsilon = 0.1$