## Stresses in a Piston Ring - and Friction Losses

Stresses in Piston Rings

This section provides a simplified way of calculating the stresses in a piston ring. A example (from a Nissan 2 l engine) is analysed, see below, and the results compared with a finite element analysis (FEA).

The material is assumed to be cast iron, E = 100GPa

The main assumption in the hand calculation is that the ratio of the depth of the beam to its radius of curvature is small and the error in assuming that the beam is initially straight will only be small.

Ring width = 0.00115m

Ring depth = 0.0031m

Free ring radius approx. = 45mm

Free ring gap approx. = 10mm

Angle subtended at centre of ring by gap = 10/45 = 0.222radians

Installing the ring, almost closing the gap (there is still a small gap when the ring is installed, between 0.2 and 1mm) causes a rotation of one end of the ring relative to the other of 0.22 rad.

Starting with the bending equation: EId2y/dx2 = M

Then integrating: EIdy/dx = Mx + C

The simple analysis, above shows that if it assumed that a uniform bending moment is applied to the ring 0.222 Nm is required to close the gap, with this moment applied, the maximum stress in the ring is 120 MPa.

An FEA model of the ring was produced, one end restrained and one end loaded with a moment of 0.222 Nm. This gives essentially a uniform moment throughout the ring however the deflection of the free end does not result in the gap being 'neatly almost closed' as would occur on actual installation.

The computed von Mises stress was 123 Mpa on the ring bore and 117 MPa on the outer periphery of the ring.

The minimum principal stress, on the ring bore was -123 MPa and the maximum principal stress, on the outer periphery was 117 Mpa.

These computed results are in close agreement with the hand calculations and the stresses are well below the strength of grade 30 cast iron - approximately 200 MPa.

Further FEA indicates that to almost close the gap (by 9.2 mm of the 10 mm gap in the 'x' direction requires a radial inward pressure on the ring of just under 75000 N/m2 which corresponds to a total inward radial force of about 21.6N. However there is still a shift of the free end in the inward radial (-'y') direction.
Moreover this loading does not give a reasonably uniform stress on either the bore or the OD - von Mises stresses vary betwen 110 and 190 MPa, so this is also not a satisfactory loading.

Rotating the free ring end by 0.2 radians also does not work, closing the gap by only 5 mm and moving the free end in by 4.5mm.

This example shows that modeling even an apparently simple configuration and carrying out FEA can be problematic.

 Very Approximate Calculation of Friction Losses From Piston Ring - Cylinder Walls

1. Calculation involving only the residual stress of the installed piston ring

Assuming that the engine is 2 litre capacity and 80mm bore, the stroke is determined from:

Swept vol. = stroke x 3.14159 (bore)2/4 giving the stroke as close to 100mm.

Assume there are 3 compression piston rings for each piston (+1 oil control ring which is a different design).

The nominal contact area per cylinder is 3 x ring width x outer periphery (80mm bore):

3 x 0.00115 x 3.14159 x 0.08 m2 = 0.867E-6 m2 If the normal pressure is 75000N/m2 the total radial (or normal) force per cylinder is 0.867E-6 x 75000 N = 65N

Assume the coefficient of friction between the piston rings and the cylinder wall is an average of 0.2

Then the friction force opposing the motion of each piston will be about 0.2 x 65N = 13N

Assuming the engine crank is rotating at 4000rpm, this equals 4000/60 = 66.7 revs per s.

In 1 second each piston travels 0.1 x 2 x 66m = 13.2m

In 1 second the work done against friction by each piston is 13.2 x 13 = 171.6Nm

Total for four pistons in engine is 686Nm/s = 686 watts.

2. Effect of Gas Pressure Behind Piston Rings

In reality the design of pistons and rings directs gas pressure arising from combustion on to the top and to behind the piston rings, pushing them against the cylinder with much higher pressure, improving their sealing, but causing greater friction losses.

If it is assumed that the brake mean effective pressure in the cylinders is about 0.9MPa, then this pressure can be assumed to be the mean pressure pushing the rings against the cylinder walls.
This gives rise to a force between each piston ring and cylinder of:

0.00115 x 3.14159 x 0.08 x 900000 N = 260N Assume coefficient of friction = 0.2, then friction force per ring is: 260 x 0.2 = 52N,

For a four cylinder engine there are 12 piston rings, so total friction force is 624N.

This is acting only during the power stroke. Work done per second by this friction is given by: 624 x 0.1 x 0.5 x 66 = 2060 watts.

3. Effect Side Force on Piston due to Connecting Rod Orientation
The figure below shows schematically the forces acting on the piston during the combustion stroke:

If it is assumed that the 'average' angle of inclination of the connecting rod to the line joining the centre of the crank to the piston pin ('a' in diagram) is about 20o, then resolving vertically gives:

p x piston area = cf x cos(a) + f .... (equation i) and resolving horizontally gives wf = cf x sin(a) ...... (equation ii) The Coefficient of friction between cylinder wall and piston, mu = f/wf

substituting for wf in (ii)

f/mu = cf x sin(a) .... so cf = f/(mu x sin(a)) Substituting back into (i): p x piston area = f x cos(a)/(mu x sin(a)) + f Rearranging: p x piston area = f x (1 + 1/(mu x tan(a)))

f = p x piston area/(1 + 1/(mu x tan(a)))

Assume coefficient of friction is 0.2, tan 20o = 0.364, substituting values:

f = 900000 x 3.14159 x 0.04 x 0.04/(1 + (1/(0.2 x 0.364))) = 307N

For 4 cylinders, work done per second is given by: 4 x 307 x 0.1 x 0.5 x 66 = 4052 watts.

Very Approximate Total:
Adding these 3 components together: 686 + 2060 + 4052 = 6800 watts

NB: This is a very rough approximation, the coefficient of friction varies with the engine speed and with the position of the piston during the stroke, maintaining an oil film near the ends of the stroke where the piston is moving more slowly is more difficult. The force between the cylinder wall and the piston also varies with the connecting rod orientation. Because of the finite radii of the piston pin (or gudgeon pin) and the big end bearing, the friction in these will also effect the normal force between the cylinder wall and the piston, and consequently the friction force between the cylinder and the piston.
These calculations serve only as a very rough order of magnitude approximation.