Department of Mathematics


Distortion Inside a Piston Bore*

Proposer/Liaison: Bruce Geist

Found inside of an engine block are piston bores and corresponding piston rings. The piston ring may be viewed as a C-shaped, structural elastic beam that makes contact with the walls of the piston bore. The piston ring is guided into a grove around the piston as the piston / piston-ring assembly is inserted into a cylinder of the engine block. The C-shaped ring is forced closed, so that the C-shape becomes roughly a circle surrounding the piston. Because the ring wants to revert to its “free” shape, it exerts a pressure on the cylinder walls. Ideally, the piston ring forms a “light-tight” seal with the surrounding cylinder wall and piston, meaning that no gaps are present through which combustion gasses can escape. The ring seals a piston in its combustion chamber, so that pressure from exploding gasoline can be converted to mechanical energy.

Unfortunately, forces exerted on an engine block produce distortions of the cylinder in which the piston and piston ring reciprocate. Ideally, a piston ring is able to accommodate itself to the distorted shape. However, this is not always the case. In practice, three types of distortions are observed in opposition to the desirable and standard circular piston ring shape. These distortions may be classified by their shapes: oval, three-leafed clover, and four-leafed cloverleaf distortion patterns corresponding to terms in a certain Fourier series.

Suppose the piston ring is to have constant radius r, and in reality, it has radius r + Dr(q), where q is a polar angle. Reasonable amounts of distortion Dr(q) can be accommodated by the ring. However when a threshold level of distortion is reached, the piston ring fails to make a tight seal. The level of distortion at which the ring no longer makes a good seal can be estimated, and bounds on the types of distortion can be derived. A standard method for determining these bounds involves decomposing Dr(q) into a Fourier Series. Individual terms in the series give rise to the different distortion types mentioned above. A common method for assessing the acceptability of distortion involves setting bounds for select coefficients of the Fourier series expansion of Dr(q). That is, usually about the first five Fourier coefficients in a series expansion are computed. If the higher order coefficients from this group of five are underneath certain bounds, the distortion pattern is deemed acceptable. In reality, the various distortion patterns corresponding to these Fourier components can have an additive effect. This additive affect is not well accounted for with the current technique. Coefficient bounds can be derived based on a threshold value of overall curvature. Hence, it may be that that a piston ring’s overall curvature could be computed and assessed directly. This then, is one project idea: determine acceptable limits on overall curvature that will not disturb a “light-tight” seal of the ring in its piston bore and write software to compute a rings curvature based on measured data (a sequence of points in R2) that defines the ring shape.

Secondly, there are two competing ways to derive Fourier bounds referred to in the previous paragraph. See [1] and [2] for one approach, and [5] and [6] for an alternate approach. Which way is the best way? There is no clear understanding of this issue. A second project idea is to analyze the papers cited above and put forward a rational for advocating one or the other methodology based on your analysis. If this is not possible, simply clearly describing what the underlying assumptions for each method would be a useful document for us.

Many project deliverables are possible. Example deliverables could include:

  • A paper or technical report that documents analytical work that helps flesh out formulas for bounds that do not rely on Fourier series expansions.
  • A paper or technical report that addresses the question of which of the competing techniques for calculating Fourier bounds is best.
  • A computer program that calculates the extent that a piston ring’s distortion deviates from its nominal curvature (nominal curvature would be something close to 1/r, where r is the radius of the piston cylinder). Such a deviation could be compared to the bound determined by analytical work suggested above.


1) Shizuo Abe and Makoto Suzuki. Analysis of cylinder bore distortion during engine operation. Society of Automotive Engineers, 950541:9--14, 1995.

2) V. V. Dunaevsky. Analysis of distortions of cylinders and conformability of piston rings. Tribology Transactions, 33(1):33--40, 990.

3) Carl Englisch. KolbenRinge. Springer-Verlag, Viena, 1958.

4. Stephen H. Hill. Piston ring designs for reduced friction. Society of Automotive Engineers, 841022:1--20, 1984.

5) Klaus Loenne and R.~Ziemb. The goetze cylinder distortion measurement system and the possibilities of reducing cylinder distortions. Society of Automotive Engineers, 880142:25--33, 1988.

6) Reinhard Mueller. Zur grage des formuellungsvermoegens von kolbenringen in von der kreisform abweichenden bohrungen gleicher umfanglaenge. MTZ, 31:79--82, 1970.

7) Eric W. Schneider. Effect of cylinder bore out-of-roundness on piston ring rotation and engine oil consumption. Society of Automotive Engineers, 930796:139--160, 1993.

8) Eduardo Tomanik. Piston ring conformability in a distorted bore. Society of Automotive Engineers, 960356:169--180, 1996.

*Summary prepared by Ryan Mellot in collaboration with Bruce Geist, Applied Mathematician, DaimlerChrysler Corp.

Top of Page


For any additional questions regarding the program curriculum and/or the extension deadline for the application to the MSIM program, contact us at


Department of Mathematics
Michigan State University
619 Red Cedar Road
C212 Wells Hall
East Lansing, MI 48824

Phone: (517) 353-0844
Fax: (517) 432-1562

College of Natural Science