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

Author:
Colin W. Cryer

Journal:
Math. Comp. **25** (1971), 435-443

MSC:
Primary 65N05

DOI:
https://doi.org/10.1090/S0025-5718-1971-0298961-7

MathSciNet review:
0298961

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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 and find and such that (i) for , (ii) , (iii) for , and (iv) .

First, it is shown that the discrete approximation is accurate to where 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.

**[1]**Garrett Birkhoff and Donald F. Hays,*Free boundaries in partial lubrication*, J. Math. and Phys.**42**(1963), 126–138. MR**0153199****[2]**A. Cameron & W. L. Wood, ``The full journal bearing,''*Inst. Mech. Engrs. J. Proc.*, v. 161, 1949, pp. 59-64.**[3]**Derman G. Christopherson,*A new mathematical method for the solution of film lubrication problems*, Inst. Mech. Engrs. J. Proc.**146**(1941), 126–135. MR**0006295****[4]**C. W. Cryer,*The Method of Christopherson for Solving Free Boundary Problems for Infinite Journal Bearings by Means of Finite Differences*, Technical Report #72, Computer Sciences Dept., University of Wisconsin, Madison, Wisconsin, 1969.**[5]**Colin W. Cryer,*The solution of a quadratic programming problem using systematic overrelaxation*, SIAM J. Control**9**(1971), 385–392. MR**0298922****[6]**A. A. Gnanadoss & M. R. Osborne, ``The numerical solution of Reynolds' equation for a journal bearing,''*Quart. J. Mech. Appl. Math.*, v. 17, 1964, pp. 241-246.**[7]**O. Pinkus & B. Sternlicht,*Theory of Hydrodynamic Lubrication*, McGraw-Hill, New York, 1961.**[8]**A. A. Raimondi & J. Boyd, ``A solution for the finite journal bearing and its application to analysis and design. III,*Trans. Amer. Soc. Lubrication Engrs.*, v. 1, 1958, pp. 194-209.**[9]**Richard S. Varga,*Matrix iterative analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0158502**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0298961-7

Keywords:
Christopherson's method,
free boundary problems,
finite differences,
journal bearings,
quadratic programming,
lubrication theory

Article copyright:
© Copyright 1971
American Mathematical Society