## 4 Conical Liners  
The selection of the tapering angle for the conical liner was based on reversing the angle resulted from the cylindrical liner. Figure 2 shows the resulting bore deformation for the circular liner on the thrust, antithrust, front, and rear sides. The figure shows the deformation in a strongly exaggerated representation to highlight the differences between the upper and lower part deformations. It is noticeable that this difference varies from one side to another. The upper part is always more deformed than the lower part. The difference of the deformation is about 5 \(\upmu\)m on the thrust and antithrust sides 7 \(\upmu\)m on the front and rear sides. For the setup of an optimized liner, a conical shaped liner was assumed for the cold state. It was formed with 5-\(\upmu\)m increments of the lower radius (see Figure 3). This design is compatible with the generally known shapes for noncylindrical liners used by some manufacturers [18; 19]. For the sake of simplicity, the design proposed in this work was based on one fixed operation point. In later applications, an optimal inclination design should be determined on the basis of a statistical analysis for different operation points of the engine.  
Figure 1: One and half cylinder model used in this study with the different sides of the cylinder.  
Figure 3 also shows the cross section in the top plane of the conical liner. Two designs were investigated here. Commonly, the cross section has a circular form. In a preceding work [14], it was shown that an elliptical liner cross section in the cold state can reduce the roundness error in the fired state by more than 75% for the terminal cylinder in four-cylinder gasoline engine due to the inhomogeneous thermal conditions. It was found that the deformation of a circular liner leads to a nearly elliptical shape in the fired state, while an initially elliptic liner with opposite ellipticity in the cold state deforms to an almost circular shape in the fired state. This does not affect the piston shape, since the amount of ellipticity is small and can be absorbed by the piston rings in the cold state. Furthermore, it can be assumed that the piston itself keeps its circular shape in the fired state due to its geometry.  
Figure 3: Dimensions and cross sections for conical liners (Dimensions in millimeter).  
Figure 2: Simulation of the bore deformation for fired conditions for the originally straight circular liner for the four sides of the terminal cylinder.  
The deformation of both the conical circular and the conical elliptical liners in the cold state were simulated at the same operational conditions (4000 rpm at full load). The results for the bore deformation for circular and elliptical conical liners are presented in Figures 4 and 5. For comparison, the figures also show the deformations of the initially straight wall liners, being either circular (Figure 4) or elliptical (Figure 5). The comparison between the straight and conical liners shows that the conical liners deformed to a better straight and parallel shape in the fired state. This led to friction reduction due to both increasing the oil film thickness between piston ring and cylinder liner and eliminating the secondary movement [12; 15; 18]. It is worth noting that although the values and trends for circular and elliptical liners look the same, the final shape of deformation was totally different, as explained previously.  
Figure 4: Bore deformation for straight (**top**) and conical circular liners (**bottom**) for the hot operation case.  
Figure 5: Bore deformation for straight (**top**) and conical elliptical liners (**bottom**) for the hot operation case.