Xin Q. Diesel Engine System Design

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730 Diesel engine system design

∑ High rotational speed of the follower results in a large reduction in the

instantaneous friction coefcient. Follower rotation may result in an

asymmetric distribution of the ‘instantaneous friction coefcient vs.

cam angle’.

The above characteristics can be explained by the behavior of the instantaneous

cam–follower friction force, which determines the follower rotational speed

and the closeness toward rolling friction. The friction force can be regarded

as the product of a friction coefcient and the cam load. The friction

coefcient is low at the cam anks due to large oil lm thickness, and

gradually increases as it approaches the cam nose region where the boundary

lubrication dominates and the oil lm is very thin. At higher engine speeds,

the friction coefcient tends to decrease across the cam surface due to a

stronger hydrodynamic lubrication effect. Note that the measured friction

coefcient at the cam–follower interface may not be symmetric with respect

to the cam nose, as discovered by Willermet and Pieprzak (1989), indicating

the complex nature of the lubrication regimes involved.

The cam load (force) reects the characteristics of the valve spring load

at low engine speeds with negligible effect of dynamic inertia loading. At

high engine speeds, the dynamic inertia effect of the valvetrain mass may

become dominant. The dynamic inertia effect increases the cam force at the

cam anks and decreases the cam force at the cam nose by offsetting the

spring force. It induces several vibration peaks as well. Note that the valve

spring force gradually peaks at the cam nose without vibration in a basically

symmetrical fashion with respect to the cam nose. The cam force at high

engine speeds looks more uniform across the entire cam event cycle. Due to

the rocker shaft bearing friction, the cam force is not exactly symmetric at

low engine speeds. At high speeds, the vibratory valvetrain dynamic force

can easily destroy any symmetric pattern of the cam force.

The cam–follower friction force is the net complex dynamic effect of the

above factors, and it may exhibit a high symmetric peak at the cam nose at

low speeds, and become more asymmetric and uniform at high speeds. The

follower rotation is driven by the friction force at the cam–follower interface,

hence exhibits similar behavior to the dynamic friction force. In general,

the greater the friction force, the faster the follower rotation. Therefore, the

follower rotation is directly dependent on the specic valvetrain dynamics

characteristics at different engine speeds and the cam–follower design

geometry. A theoretical follower rotation model may predict the rotation very

accurately at low engine speeds because the valvetrain force is relatively

simple (i.e., static spring force dominated) and the friction force is easier to

predict. However, at high engine speeds, the errors in valvetrain dynamics

simulation can cause large discrepancies in the prediction of the friction

force and the follower’s rotational speed.

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731Friction and lubrication in diesel engine system design

10.7.6 Cam–follower friction analysis for the roller

follower

Roller followers have been widely used for both pushrod and overhead-

cam valvetrains to signicantly reduce friction. Other reasons for using the

roller follower to replace the at-faced follower include the following: (1)

the design difculty of controlling the ash temperature on the cam with

the at-faced follower; (2) the higher allowable cam stress limit of rolling

contact than sliding contact; and (3) a more aggressive cam acceleration

prole enabled by using a negative radius of curvature. Cam roller follower

design was elaborated by Korte et al. (2000).

Staron and Willermet (1983) conducted a lubrication analysis for a 1.6-liter

engine valvetrain to investigate the design effects on each major component.

They found that using a roller follower could reduce the valvetrain friction by

approximately 50%. Lee et al. (1994) conducted experimental work to nd

that the roller follower friction (including the roller pin and the cam–roller

interface) accounted for 11.8% of the total valvetrain friction at 750 rpm,

12.8% at 1100 rpm, and 13.3% at 1500 rpm, compared with the 70% given

by a at-faced follower researched by Paranjpe and Gecim (1992). There are

three important issues related to the friction in roller follower valvetrains:

(1) roller pin bearing friction/wear; (2) the elastohydrodynamic lubrication

rolling friction on the cam; and (3) roller slip (sliding or skidding) and its

associated friction and wear. These issues are inter-related.

Both the roller pin friction torque and the roller inertia torque are usually

quite high in magnitude, and the latter is responsible for roller slip. The roller

pin bearing is believed to operate primarily in the hydrodynamic lubrication

regime. Like in other journal bearings, the oil feed hole position needs to be

carefully designed so that it is located in the unloaded region and does not

interfere with the hydrodynamic lubrication. Colechin et al. (1993) presented

a kinematic and lubrication analysis for a roller follower valvetrain and its

roller pin bearing. Lee and Patterson (1995) measured roller pin friction and

found that the roller pin operated in the hydrodynamic lubrication in the

valve opening period before the cam nose (with a friction coefcient in the

order of 0.001–0.002) and operated in mixed lubrication during the valve

closing period after the cam nose. They found at the same Sommerfeld duty

parameter the friction coefcient at the valve opening side was lower than

that at the valve closing side. They suspected that the reasons could be the

change of angle between the oil supply hole and the loading position, or

cavitation. This again suggests the importance of oil feed hole. The roller pin

friction is related to the instantaneous rotational speed of the roller, which

is affected by roller slip.

The roller friction coefcient at the cam–follower interface is in the

order of 0.003–0.006, while a at-faced follower has a friction coefcient

of 0.11–0.14 (Lee and Patterson, 1995). Lee and Patterson (1995) conducted

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732 Diesel engine system design

a more detailed mixed lubrication analysis for the cam–roller contact by

considering the surface roughness effect on the elastohydrodynamic  lm

thickness. The cam–roller friction force is the sum of the rolling friction

and the boundary asperity contact friction. When the roller rolls without

slippage, the instantaneous velocity of the roller is equal to that of the cam,

which is equal to the oil entraining velocity. The rolling friction can be

calculated by using an elastohydrodynamic line contact model (Goksem,

1978) as follows:

ollin

gCAMgC

fr,,fr

ollin,,ollin

gC,,gC

(

)

(

318

(

318

(

658

..658..

..0..

0126

012660126

10.49

where C is a pressure viscosity coef cient, and the de nition of G

lub

, U

lub

and r

ROC

are the same as in equation 10.43.

When roller slip occurs, the additional shear friction caused by the sliding

can be calculated by using the viscous shear principle as follows:

lip

fs,,fs

lip,,lip

10.50

where v

slip

is the relative velocity due to the slip, h

is the oil  lm thickness,

and A

is the contact area.

Roller slip is basically inevitable and it causes much higher friction than

the rolling friction. Predicting roller slip is the most important topic in roller

friction analysis. A pure rolling contact means the relative speed between the

cam and the roller at the contact point is zero (i.e., equal ‘surface’ velocities).

For convenience, a slip or sliding ratio is usually simply de ned as a relative

difference between the ‘rotational’ speeds of the roller and the cam. It should

be noted that an equal rotational speed of the cam and the roller does not

mean pure rolling. The complex kinematic relationship (e.g., radii, offset)

of the cam–roller schematic determines a variable relationship between the

cam speed and the roller speed at each contact point within a cam cycle in

order to maintain a pure rolling. Any deviation in the cam and roller speeds

from that prescribed kinematic relationship, for example caused by a force

change on the roller, causes roller slip to occur.

There have been very few published studies in the area of roller slip

since investigations started about 30 years ago. Among the published work,

Duffy (1993), Lee et al. (1994), Lee and Patterson (1995) and Ji and Taylor

(1998) are probably the most important. Experimental evidence revealed that

roller slip occurred at the cam base circle and peaked at the two symmetric

locations on the cam  anks (with respect to the cam nose) corresponding to

the two large peaks of oil  lm thickness. There was almost no slip near the

cam nose. The slip tended to increase with engine speed. In fact, the slip

ratio could be either positive or negative (meaning that the roller can spin

either faster or slower). The slip ratio could reach as high as 10%. Previous

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733Friction and lubrication in diesel engine system design

studies showed that roller slip became less when larger surface roughness

at the cam–roller interface or lower oil viscosity was used.

Analogous to the vehicle wheel slip on a slippery road surface where

the vehicle tractive force from the engine is greater than the road adhesion

limit, the roller slip occurs when the ‘tractive force’ mismatches the total

friction force at the roller follower contact point. What is different in the

case of roller slip is that the ‘tractive force’ is the roller pin bearing friction

force. The pin bearing friction force essentially comes from the rotation of

the roller, which in turn originally comes from the camshaft driving torque.

In fact, for this case with two friction forces involved (one at the roller pin,

the other at the cam–roller contact), it is immaterial to dene which one is

the ‘tractive force’ and which one is the ‘friction or resistance force’. The

pin bearing force tries to slow down the roller, while the cam–roller contact

friction force tries to speed up the roller. The roller’s angular moment is equal

to the product of the mass moment of inertia and the angular acceleration of

the roller. The moment balance of the roller is given by equating the roller’s

angular moment to the sum of the torques coming from the cam–roller friction

and the roller pin bearing friction.

As mentioned earlier, it is assumed that at a given constant camshaft

rotational speed and at any moment within the cam event cycle, a unique

value of the required roller rotational speed corresponding to pure rolling

can be computed based on the kinematic relationship of the mechanism. That

speed is denoted as the required rolling speed of the roller. The moment

balance of the roller must satisfy the required rolling speed at each moment

if pure rolling is to be maintained. Any deviation from that balance caused

by a force change or camshaft speed change results in a violation of the pure

rolling condition, i.e., roller slip will occur. The roller pin friction torque is

governed by the hydrodynamic lubrication theory and affected by roller mass,

pin size, and rotational speed. The cam–roller friction torque is governed by

the elastohydrodynamic/mixed lubrication theory and affected by valvetrain

dynamic loading, camshaft speed and surface roughness. Therefore, it is

impossible to always maintain the zero slip (i.e., pure rolling) condition

within the cam cycle. The angular speed of the roller calculated from its

moment balance equation cannot always satisfy the pure rolling condition

that is demanded by the kinematic relationship. This is the root cause of

roller slip. Note that the roller does not always slip when the friction force

is smaller than the tractive force.

From the above analysis, it can be observed that an increase in the roller

pin bearing friction force, due to any design change or operating reason, tends

to decelerate the roller to slip lagging behind the cam (i.e., slip-behind), and

vice versa. An increase in the cam–roller friction force tends to accelerate

the roller to slip further ahead of the cam (i.e., slip-ahead), and vice versa.

Any design or operation changes affecting those two friction forces affect

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734 Diesel engine system design

the roller slip. Engine speed affects roller slip in a complicated way through

its impact on the valvetrain dynamic load, the elastohydrodynamic friction at

the cam–roller contact and the hydrodynamic friction at the roller pin. The

experimental phenomena observed from the literature can be well explained

by this analysis. For example, Lee and Patterson (1995) found that slip-

down usually occurred at the cam ank areas where the oil lm thickness

is high and the friction force is low. Slip-down also occurs on the cam base

circle where the cam–roller friction is very low or zero. The slip effect was

especially prominent at high speeds where the roller pin bearing friction force

was high while the cam–roller friction became low. Duffy (1993) found in

his experiment that low oil viscosity reduced roller slip due to an increase in

the cam–roller friction force caused by the reduced elastohydrodynamic lm

thickness and the more severe asperity contacts. He also indicated that a slip

index, dened as the angular integral of the product of the slip percentage

and the follower lift over one camshaft revolution, was roughly proportional

to the frictional power loss. This conrms that roller slip is very important

for the friction of the roller–follower valvetain.

Roller follower slip and friction control presents an interesting and

challenging task for modern engines. Coupled dynamic and lubrication

modeling plays a vital role in the system optimization of roller follower

kinematics and valvetrain dynamics in order to minimize roller slip, friction,

and wear. Another important topic of roller motion is the skew (tilting) of the

roller which affects cam stress and friction. The dynamic skew motion of

roller followers was investigated by Ito (2003, 2004, 2006) and Ito and Yang

(2002) based on a theoretical method developed by Ahmadi et al. (1983).

10.7.7 Friction analysis of valvetrain bearings and guides

Although the friction at the cam–follower interface is usually dominant

in valvetrain friction (especially for the at-faced followers), the friction

from the rocker arm bearing, the camshaft bearings, and the follower/valve

guides also plays an important role. Paranjpe and Gecim (1992) presented a

simulation for a at-faced-follower pushrod valvetrain at 2000 rpm engine

speed, and showed that the cam–follower friction accounted for 70% of the

total valvetrain friction, the rocker pivot 16%, the follower (lifter) guide 11%,

and others 3%. Teodorescu et al. (2002) reported in a at-faced-follower

measurement that the rocker shaft bearing friction contributed 10% of the

total valvetrain friction. In fact, the friction power at the cam–follower

interface can be less important or lower than the total friction power of all

other components, especially at high speeds. Experimental evidence on this

fact was presented by Dowson et al. (1986).

The camshaft bearing load can be calculated with a similar approach as

for the crankshaft main bearing in the multi-cylinder engine. For a four-

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735Friction and lubrication in diesel engine system design

stroke engine, the camshaft rotates at a half speed of the crankshaft. At very

low engine speeds, the hydrodynamic lubrication oil lm sometimes cannot

be effectively developed in the camshaft bearing and it results in mixed

lubrication with surface asperity contacts. Without knowing the details of

the bearing minimum oil lm thickness, its friction power calculation can be

simplied by assuming hydrodynamic lubrication and following the method

introduced in Section 10.6. For valvetrains equipped with mechanical lash

adjusters, the camshaft load may have certain zero-load period (depending

on the number of cylinders in the engine) which corresponds to the valve

closed period. For valvetrains equipped with hydraulic lash adjusters, the

camshaft load is always greater than zero because of the static load from the

hydraulic pressure. During the unloaded period, the viscous shear friction

force should be accounted for with a 2p lm assumption without cavitation.

During the statically or dynamically loaded period, the friction from both

the viscous shear and the hydrodynamic pressure term should be accounted

for with appropriate cavitation assumption. Staron and Willermet (1983)

and Crane and Meyer (1990) used a slightly different model of the camshaft

bearing to calculate the journal eccentricity, compared to the equations in

Section 10.6. Staron and Willermet (1983) found that using needle bearings

at the camshaft journal bearings did not reduce the friction at high speeds

but could reduce the friction at very low speeds.

Staron and Willermet (1983) found that using needle bearings at the rocker

arm fulcrum reduced the friction by about 10%. The rocker arm bearing

friction occurs only when the valves move in each individual cylinder. The

rocker bearing friction can be signicant if a shaft bearing is used rather than

a roller or ball bearing. The rocker shaft does not rotate constantly like the

camshaft. The rocker arm is stationary when the follower is on the cam base

circle, squeezing out the oil lm and leading to the boundary lubrication in

the rocker bearing. When the valve opens and closes, the rocker arm rocks

slightly back and forth at a low velocity with heavy loading acting only on

a small region of the bearing (similar to the piston pin bearing). The rocker

bearing friction can be observed from the pushrod or cam load trace vs. cam

angle. The load instantly drops immediately after the moment of maximum

valve lift from a higher load at the valve opening side to a low load at the

valve closing side. Note that the valve moving direction reverses at the

maximum valve lift. This instant drop is caused by the change of direction

of the friction forces. Teodorescu et al. (2002) measured the instantaneous

rocker shaft bearing friction torque within an engine cycle. They observed

that the rocker friction force varied instantaneously to follow the pattern of

the pushrod force (the normal loading), indicating a boundary lubrication

characteristic. They measured the friction coefcient of the rocker shaft as a

constant (around 0.1–0.2), not affected by engine speed. They conrmed that

the friction in the rocker bearing never reached hydrodynamic lubrication.

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736 Diesel engine system design

The rocker friction can be calculated by multiplying the bearing load by an

equivalent friction coefcient (e.g., a rolling friction coefcient for a rolling

contact). Note that the direct acting OHC does not have a rocker arm.

The friction losses of the valvetrain oscillating components (e.g., the

valves and the followers in the guides) can be determined by assuming the

clearance between the oscillating parts and the guides is constant at the

concentric value by ignoring the side loading, the tilting effect, and the

cavitation for simplicity. The clearance space can be assumed fully lled by

the lubricant oil. The friction force of these components is the hydrodynamic

viscous shear force, which is proportional to the oil dynamic viscosity, the

sliding velocity, and the contact area, and inversely proportional to the guide

clearance. A more sophisticated model may assume boundary lubrication at the

ends of the oscillating strokes with zero sliding velocity under side loading.

Paranjpe and Gecim (1992) and Wang (2007) provided the calculations of

the side loading. Teodorescu et al. (2002) presented the measurement data

of the instantaneous valve stem friction force within an engine cycle. The

data showed this force was very small and only accounted for 1.5–2% of the

force acting on the top of the valve stem and contributes to 2% of the total

valvetrain friction loss. Their data showed this friction force did not increase

with engine speed as the usual hydrodynamic friction behavior would exhibit.

Instead, their valve–guide friction force stayed constant across the speed

range and exhibited a mixed lubrication characteristic. The measurement data

of the follower–bore friction force by Teodorescu et al. (2003) in a diesel

pushrod valvetrain exhibited hydrodynamic lubrication characteristics (i.e.,

proportional to valve velocity). The friction force was in the same order of

magnitude as the at-faced cam–follower friction force.

Valve stem seals are used in modern diesel engines to drastically reduce the

amount of oil passing the valve stem–guide area and the oil ow transported

into the combustion chamber in order to minimize lube oil consumption and

particulate emissions (Marlin et al., 1994). The valve stem seals operate in

the boundary lubrication regime (Netzer and Maus, 1998).

10.8 Engine friction models for system design

10.8.1 Level-1 engine friction model

The level-1 engine friction model is empirical or semi-empirical, non-

instantaneous, and expressed by cycle-average FMEP. It has two types:

(1) a simpler ‘lumped overall’ model to lump the friction of all different

components together; and (2) a more complex ‘breakdown overall’ model

to handle each component separately.

The foundation of the level-1 overall engine friction model can be traced

back to about half a century ago. The effect of cylinder pressure (or engine

load) on engine friction was explored by Gish et al. (1958). The effect of

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737Friction and lubrication in diesel engine system design

engine compression ratio on mechanical friction and thermal ef ciency was

studied in their work on a gasoline engine by using the indicator diagram

method. They discovered a second-order correlation between the engine

mechanical FMEP and the peak cylinder pressure regardless of whether the

cylinder pressure was produced by the change in engine compression ratio

or other measures. They concluded that the main reason for the mechanical

friction to increase rapidly with the cylinder gas pressure was the high

pressures acting behind the top compression ring.

The effect of peak cylinder pressure on engine friction is especially

important due to the concern of possible decrease in mechanical ef ciency

in modern diesel engines, which usually run at very high boost and cylinder

pressure levels. Peak cylinder pressure and the shape of the cylinder pressure

trace are affected by intake manifold boost pressure, engine compression

ratio, intake valve closing timing, and fuel injection timing.

The effect of design variables on engine friction was  rst systematically

studied by Bishop (1964) at Ford Motor Company. He tested a wide range

of different water-cooled four-stroke engines and derived certain empirical

formulae for the following FMEP:

∑ piston FMEP as a function of cylinder bore, stroke, engine compression

ratio, intake pressure, mean piston speed, and piston skirt length

∑ bearings FMEP as a function of cylinder bore, stroke, bearing diameter

and length, and engine speed

∑ valvetrain FMEP as a function of cylinder bore, stroke, number of valves,

valve diameter, and linearly decreasing with engine speed

∑ accessory FMEP, which is proportional to engine displacement and

15.15

where N

is engine speed.

Chen and Flynn (1965) at International Harvester proposed a popular formula

of engine mechanical friction that has been widely used since. The engine

mechanical friction they analyzed included rubbing friction and accessory

loss. It was based on single cylinder diesel engine test data in the form of

FMEP

= C

+ C

max

+ C

10.51

where v

is the mean piston speed, and p

max

is the peak cylinder pressure.

They also indicated that the engine oil temperature had a signi cant impact

on engine friction.

Millington and Hartles (1968) analyzed frictional losses in direction

injection diesel engines with motoring breakdown test, and proposed a

correlation on FMEP, which included both engine mechanical friction and

pumping loss, as follows:

FMEPFM

EmpEm

= (

–

)

Em12Em

10.52

where W is the engine compression ratio, N

is the engine speed, and the term

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738 Diesel engine system design

is primarily for the pumping loss. They studied many design effects

on engine friction, including piston rings, piston weight, skirt area, engine

compression ratio, oil viscosity, bearing diameter, and accessory power.

Winterbone and Tennant (1981) proposed the following formula

FMEP

= C

+ C

max

+ C

10.53

by assuming the effects of the independent variables, speed and load, may

be linearly superimposed. They also analyzed diesel engine transient friction

and concluded that the appreciably increased friction during transients

compared to the steady-state levels at the same speed and load was caused

by the crankshaft de ection during deceleration and acceleration and the

associated changes in bearing friction.

Kouremenos et al. (2001) proposed a formula in the following form based

on engine experimental data and the simulation of instantaneous friction

force:

FMEP

= C

IMEP

+ C

max

10.54

Kouremenos et al. (2001) and Rakopoulos et al. (2002) believed that

the in uence of peak cylinder pressure is rather small (imperceptible or

negligible).

An and Stodolsky (1995) simulated the effect of engine assembly mass

on engine friction and vehicle fuel economy. They proposed a formula in

the following form:

FMEPFM

= (

CC +CC

NC NC

NC+ NC

)

+ (

NC +NC

CC +CC

CC CC

EP2EP

EE34EE

–

EE34EE

–

mmNm

)

10.55

where m

is the piston assembly mass, m

is the valvetrain mass. The

terms in the two parentheses represent the FMEP from the piston assembly

and the valvetrain, respectively. The terms

and

represent

the contributions from the dynamic inertia loading of the piston assembly

and the valvetrain, respectively. Martin (1985) commented in his bearing

friction review that in general reducing mass of relevant moving parts would

result in friction reduction. An and Stodolsky (1995) commented that at

high engine speed (e.g., 4000 rpm) the

term can become dominant.

Previous simulations conducted by Goenka and Meernik (1992), Keribar et

al. (1993), Livanos and Kyrtatos (2006) as well as the experimental work by

Uras and Patterson (1987) reported that the piston skirt and bearing friction

forces were insensitive to load change. In fact, the effect of the inertia terms

can be complex or small for two reasons. First, unlike the piston ring that

is circumferentially and uniformly loaded to produce friction on one side

(i.e., the ring face side), the piston skirt and bearings have different friction

forces on both the loaded side and the unloaded side. When the load level

(including the cylinder gas load and inertia load) changes, the load pushes

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739Friction and lubrication in diesel engine system design

the skirt or the journal to move within the clearance space eccentrically. This

causes the lubricating oil  lm thinner on one side and thicker on the other

side. Although the oil  lm cavitates on the unloaded side, the changes in the

viscous shear friction forces on the two sides may offset each other to a great

extent as long as the lubrication stays within the hydrodynamic regime. As a

result, the total viscous friction force of the skirt or the bearing may possibly

stay unchanged. Second, as the load increases, the friction coef cient in the

hydrodynamic lubrication regime actually decreases.

Lee et al. (1999) at Ford Motor Company summarized their experience

of using the overall engine friction models for engine system concept

assessment. Their evaluations, including the critical geometry and weight of

the powertrain components, were conducted based on the traditional model

established by Bishop (1964) at Ford.

Ciulli (1993) reviewed the differences between various overall engine

friction models and applied the models to a four-cylinder four-stroke diesel

DI engine. He observed that most formulae are not directly comparable

because they were obtained from different types of engines and different

operating conditions.

To summarize the above  ndings, the level-1 lumped overall engine friction

model without differentiating the contributions from each engine component

can be written in the following form, as a function of certain basic engine

design and operating parameters:

FMEPFM

xmpExm

maxmma

CC=+CC

pC++pC

vC++vC

NC +NC

mN mN

NC NC

Em,Em

CC=+CC

xm23xm

pE23pE

xmpExm23xmpExm

xm23xm

pE23pE

xmpExm

xmpExm23xmpExm

xmpExm

xm23xm

vC++vC

xm23xm

pE23pE

xmpExm

xmpExm23xmpExm

xmpExm

EE5EE

222

10.56

In equation 10.56, the constant term C

represents the boundary lubrication

friction which is independent of engine speed. The peak cylinder pressure

term C

max

represents the mixed and boundary lubrication friction of the

piston compression rings caused by cylinder gas pressure loading (also

related to engine compression ratio). The linear term C

represents the

hydrodynamic lubrication friction of the sliding components (e.g., the piston

skirt and the piston rings) which is proportional to the mean piston speed.

The linear term C

represents the hydrodynamic lubrication friction of

the rotating components (e.g., the bearings) which is proportional to the

crankshaft speed. The second-order term

Em,Em

represents a correction

term related to the dynamic inertia force of the engine assembly mass, and

is usually very small for well designed diesel engines. The second-order

term

represents the  uid pumping in the accessory work losses. Note

that there is no

term in equation 10.56 for engine pumping loss since

the pumping loss is not a part of the engine friction.

The level-1 overall friction model can be further expanded to include design

geometries such as cylinder bore and stroke (like the formulae reviewed by

Ciulli, 1993), or broken down to each component (e.g., the piston, the rings,

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