Mathematical modelling of Piston rings and Piston motion

The piston-ring-liner system is an important source of oil consumption in modern diesel engines. One of the important factors affecting oil consumption from the piston-ring-liner system is the distortion of the cylinder bore. To the naked eye, a cylinder bore may look perfectly round but when examined more closely at a µm level, it can be highly distorted. Combustion pressure and temperature gradients during engine operation can lead to further distortions. In fact, bore distortion in internal combustion engines is normal and difficult to prevent.

Figure 1 illustrates the result of measurements of a distorted cylinder bore.

Bore deformity affects the conformability or ability of piston rings to maintain contact with the cylinder liner. On a macro scale, bore deformity can be described by a Fourier series, Figure 2 [Ma 1997][Andersson 2002]:

(1)R(φ) = ∑ (Ai cos(φ) + Bi sin(φ))

where:

R(φ) - radial coordinate,

φ - angular coordinate,

Ai, Bi - amplitude constants,

i - order, with the summation from i=0 to i=n,

n - highest order distortion to be considered.

[schematic]

Figure 2. Coordinate System and Fourier Series Bore Distortion Orders

Bore distortion is caused by a number of factors including machining tolerance (zero and 1st order distortions), forces generated from tightening head bolts (distortion order = number of head bolts), variations in liner cooling resulting in differences in thermal expansion, distortions caused by gas pressure—especially with thinner cylinder walls. Clamping of the liner between the cylinder head and the engine block can also cause outward deformation of the liner. The maximum distortion usually occurs near the upper edge of cylinder [Andersson 2002].

One way to reduce bore distortion is to carry out the final machining of the bore with a deck plate that can reproduce some of the distortions that occur from the cylinder head bolts. In one case, this was estimated to reduce in oil consumption by about 20% [Yamada 2003].

Other ways to reduce bore distortion include cylinder liner material selection and engine design details that reduce distorting forces. For example, in 2012, Federal Mogul announced a dual-material cylinder liner technology (Hybrid Liner) to reduce bore distortion in highly-loaded gasoline engines with an aluminum engine block. Conventionally, an aluminum engine block with a cast-in liner would not maintain the contact necessary to keep cylinder liner distortions to a minimum. However, the bonding between the sleeve and block surfaces can be improved by spray coating the outside of a cast iron sleeve with an AlSi12 alloy having a melting point below that of the aluminium engine block. This can not only reduce bore distortion and oil consumption but considerably improve heat transfer through the liner 

Conformability is defined as the ability of a piston ring to conform to a deformed cylinder bore. A long process of calculation is required to derive the mathematically exact solution from theory. In practice, however, a simplified equation may be used derived from the theory of the closed (uncut) ring and the ring with constant radial pressure.

The conformability opposite the ring gap (see also Fig. 5) in a cylinder with an "i"th order radial deformation ui under which the ring is still light tight at a contact pressure p = 0 is calculated as:

ui = radial deformation of the cylinder by the "i"th order from its nominal radius

k = piston ring parameter (equation 10)

Since with increasing i the conformability decreases by approximately the 4th power, it follows that high order cylinder distortions are particularly critical for the functioning of piston rings.

It should be noted that the simplified theory only indicates the conformability opposite the ring gap and not the local conformability around the ring periphery. 

In the case of self-conforming piston rings the conformability decreases progressively from the region opposite the ring gap towards the gap. (Fig. 5)

Conformability is improved by the gas pressure pz acting behind the piston ring:

Spring backed rings have a very uniform conformability around the whole periphery.

Accordingly, for a spring loaded ring

The ring is pressed against the cylinder wall under a contact pressure p which is governed by the dimensions and total free gap of the ring and by the modulus of elasticity of the material used. The total free gap is defined as the distance, measured along the neutral axis, between the ends of a piston ring in its uncompressed state (Fig. 2). 

A ring can be given a constant or a variable contact pressure [2, 3], the latter being a function of the angle (φ) (Fig. 3).

Measurement of the contact pressure is extremely difficult. Therefore, in practice it is calculated from the tangential force. This is the force which, when applied tangentially to the ends of the ring, is sufficient to compress the ring to the specified closed gap. By comparing the bending moment of the tangential force against that of the constant contact pressure, the following relationships are established (Fig. 4):

A criterion for piston ring conformability to distorted cylinders is described. The criterion allows one to quantify magnitudes and order of bore distortions regarding the piston ring conformability. By comparing magnitudes of distortions which are critical relative to conformability with the magnitudes of actual distortions, determined through Fourier analysis of bore profile, the quality of piston ring/cylinder interface (expressed in terms of conformability/non-conformability of the piston rings) can be evaluated. The criterion was developed based upon the understanding that curvatures of the bore profile and of the piston ring running band are equal at full conformability of the ring to the cylinder bore. A description of the bore profile by random function was instrumental in development of the criterion and yielded opportunity for the presentation of the piston ring conformability problem in a statistical aspect that is important for quality control operations. The critical-to-conform-ability values of bore distortions were obtained with the aid of a general theory of piston rings.

Conformability of a piston ring is a measure of the ring ability to conform to the ever-present cylinder bore deviations from a perfect circle by elastic deformation of the ring. Conformability is usually defined as the limit of bore distortion at which the clearance between the piston ring and bore can be kept at zero by elastic deformation of the piston ring. Retention of good conformability is important for ring/bore sealing and reduction of oil passing and blow-by.

The most known technique for conformability analysis was developed almost sixty years ago and considers only two-dimensional deformation of the ring in its plane by the force of piston ring elasticity. It is known, however, that the piston rings with asymmetrical cross-section, a prevalent type of the modern compression piston rings, undergo a complicated three-dimensional installation displacements (twist) which affect many functions of the rings. Piston rings with a symmetrical cross-section will also produce displacement normal to the ring plane if for example they have taper face and thus are loaded in the plane off the centroidal axis of the ring. Hence a need exists for improvement of the conformability analysis to account for torsional distortions of the rings.

This paper, which is based on the work of the authors demonstrated in the preceding SAE conferences and papers, illustrates development of a model of conformability of the piston rings with arbitrary cross-section that, unlike the conventional model, takes into account three-dimensional torsional distortion of the rings. As a measure of verification of the analysis torsional distortions of a piston ring calculated with the formulae developed are compared favorably with the values of ring twist measured experimentally.

reference : Federal Mogul