The method of Christopherson for solving free boundary problems for infinite journal bearings by means of finite differences

Abstract:

A method for solving free boundary problems for journal bearings by means of finite differences has been proposed by Christopherson. We analyse Christopherson’s method in detail for the case of an infinite journal bearing where the free boundary problem is as follows: Given $T > 0$ and $h(t)$ find $\tau \in (0,T]$ and $p(t)$ such that (i) $[{h^3}p’]’ = h’$ for $t \in (0,\tau )$, (ii) $p(0) = 0$, (iii) $p(t) = 0$ for $t \in [\tau ,T]$, and (iv) $p’(\tau - 0) = 0$. First, it is shown that the discrete approximation is accurate to $O({[\Delta t]^2})$ where $\Delta t$ is the step size. Next, it is shown that the discrete problem is equivalent to a quadratic programming problem. Then, the iterative method for computing the discrete approximation is analysed. Finally, some numerical results are given.