Xin Q. Diesel Engine System Design

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670 Diesel engine system design

of surface roughness have different effects. The transverse roughness (i.e.,

at right angles to the sliding direction) should enhance the load support and

the oil lm thickness, while the longitudinal roughness would reduce them

(Spikes, 1997). Surface topography has an important inuence on the load

carrying capability of the oil lm. On the other hand, the oil lm pressure in

the elastohydrodynamic lubrication has an impact on the surface roughness

to cause it to deform and affect the load supported by the asperity contacts

in the mixed lubrication. These interactions become more important as the

thickness-to-roughness ratio reduces. The widely used models related to

surface topography for the mixed lubrication regime include the following:

(1) the Patir and Cheng (1979) average ow model, which accounts for the

inuence of surface roughness on the lubricant ow and on the load carrying

capacity of the lubricating oil lm; and (2) the Greenwood and Tripp (1971)

asperity contact model, which accounts for the load carrying capacity of the

asperity contacts in the mixed and boundary lubrication.

Surface topography also changes after engine running-in (break-in) and

gradually changes during the engine life due to wear. This affects the friction

power of all the sliding surfaces, especially the power cylinder. During

break-in, some of the asperities are worn off and the sliding surfaces become

smoother. This reduces the metal-to-metal contact and the boundary friction.

As the break-in progresses, the engine friction reduces rapidly for an initial

period, then stabilizes or changes very slowly.

The importance of surface topography on engine friction and wear was

emphasized by Taylor (1998). The surface topography of cylinder liner, piston

skirt or piston rings and the effect on engine friction and wear are discussed

by Zhu et al. (1992), Pawlus (1996, 1997), and Rao et al. (1999).

10.2.5 Lubricant properties and impact on engine friction

Lubricant oil reduces friction and wear between the components, serves as

a coolant to remove the frictional heat, and also removes impurities/debris.

Lubricant oil properties have a direct inuence on engine friction and wear.

Lubricant oil additives include friction modiers, viscosity index improvers

(VI), anti-wear agents, detergents, anti-rust agents, oxidation inhibitors, and

antifoam agents. Some important effects of the lubricant on engine friction

include oil starvation or partially ooded (related to oil feed designs),

cavitation, thermal effects, and the changes in viscosity with temperature,

pressure and shear rate.

The engine oil viscosity classication is given in SAE J300 (2004). Common

monogrades of viscosity include the SAE 5, 10, 30, 40, 45, and 50. The higher

the grade, the higher the viscosity. The multigrade oils introduced since the

1950s use viscosity index improvers (the additives of temperature-sensitive

polymers) to stabilize the viscosity at high temperatures so that the viscosity

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671Friction and lubrication in diesel engine system design

does not reduce as much as the monograde oils over the engine operating

temperature range. Common multigrade oils include the SAE 10W-30, 10W-

40, 10W-50, 15W-40, 15W-50, and 20W-50. The 10W-30 means that the oil

has the viscosity of the SAE 10 when it is cold (at 15∞C) and the viscosity of

the SAE 30 when it is hot (at 90∞C). Multigrade oil may reduce the engine

start friction at very cold temperatures without having the problems of an

unacceptably low viscosity and metal contact at high temperatures. Monaghan

(1988) indicated that the multigrade oil may achieve a reduction in viscosity

equivalent to about one grade at intermediate temperatures, hence reducing

the viscous friction by about 20%. Multigrade oils contain the polymetric

additives with relatively high molecular weight, and exhibit both temporary

and permanent shear thinning as the non-Newtonian ow behavior, which

affects their viscosity. Newtonian uid obeys a linear relationship between the

shear stress and the shear rate, and its viscosity is independent of the shear

rate. Under high pressures, the lubricant exhibits non-Newtonian behavior

and the viscosity decreases as the shear rate increases. Moreover, lubricant

oil viscosity increases with the level of soot and dispersant in the oil (George

et al., 2007). Other fundamental information on engine oils can be found in

SAE J357 (2006), J2227 (2006), J1423 (2003), and J183 (2006).

Lubricant viscosity is a strong function of temperature (e.g., Vogel’s

equation) and pressure (e.g., Barus’ equation). The viscosity decreases with

increasing temperature, and increases with increasing pressure (important

for the elastohydrodynamic conditions). The non-Newtonian and thermal

effects can be important in oil lm thickness calculations, especially for

rough surfaces and in the elastohydrodynamic lubrication. Under the non-

Newtonian condition the shear thinning and viscoelasticity effects (i.e., the

inuence of shear rate and pressure on viscosity) need to be considered.

The shear rate impact (e.g., the Cross equation) is especially important

for multigrade oils. In the highly dynamically loaded lubricated contacts,

the effects of temperature thinning, pressure thickening (the viscoelastic

effect) and shear thinning can all be important. The lubricant behavior in

the elastohydrodynamic lubrication contacts is reviewed by Jacobson (1996).

Coy (1997) and Taylor (1997) provide the details on the models of lubricant

rheology, friction and wear based on the Stribek-type diagrams. Note that

these models are suitable for diesel engine system design as the level-2

models (to be detailed later).

As seen from the earlier discussion of the Stribeck diagram, the friction

coefcient in the hydrodynamic lubrication regime decreases as the lubricant

viscosity decreases. However, in the boundary lubrication regime (e.g., for

some cam–follower contact and piston oil rings), lower viscosity results

in higher friction. Therefore, the bearings and the piston skirt usually give

lower friction when the lubricant viscosity is reduced, while the valvetrain

gives increased friction. The effectiveness of using the low viscosity oil to

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672 Diesel engine system design

reduce engine friction depends largely on how much the boundary lubrication

accounts for the total engine friction. Note that the valvetrain friction can be

reduced by using the oil containing a friction modier additive. Since the

engine has both hydrodynamic and mixed/boundary lubrication components,

an optimum viscosity that provides a minimum engine friction exists at a

given engine speed and load.

The fuel economy sensitivity to lubricants is different for heavy-duty

and light-duty engines. Taylor (2000) indicated that because the proportion

of the valvetrain friction losses is much higher in light-duty engines than

in heavy-duty engines, the latter can be regarded more ‘hydrodynamically’

overall in the lubrication regime. Because the valvetrain is the main boundary

lubricated component in an engine, the friction modier additives can appear

to be more effective for light-duty than heavy-duty engines. Taylor (2000)

claimed that the fuel consumption savings for heavy-duty diesel engines under

mixed driving conditions (low, medium, and high loads) by using carefully

formulated engine and transmission lubricants (with reduced viscosity and

optimized additive packages) can reach the order of 5%. Another investigation

into the lubricant effects on light-duty and heavy-duty vehicle fuel economy

was conducted by Kapadia et al. (2007).

Reducing viscosity moves the lubrication regime from hydrodynamic

toward boundary. This increases the risk of wear and scufng due to oil

lm thickness reduction. Therefore, there is a balance between reducing

friction and maintaining acceptable durability. In fact, the Association des

Constructeurs Européens d’Automobiles (ACEA) has mandated that heavy-duty

diesel engine lubricants should have a HTHSV (high-temperature high-shear

viscosity) of at least 0.0035 Pa.s (Taylor, 2000). Lubricant specications also

affect emissions and aftertreatment performance (McGeehan et al., 2007).

10.3 Overall engine friction characteristics

10.3.1 Engine friction characteristics in different

lubrication regimes

Engine friction has been measured using various methods, including the

indicator diagram method, the motoring and teardown method, the pressurized

motoring method (i.e., simulating the ring cylinder pressure effect with

high intake manifold pressure without actually ring), the Morse test or

electronic cylinder disablement method, the Willan’s Line method, the hot

shutdown method, the free Deceleration Curves method, the instantaneous

IMEP method (Uras and Patterson, 1983), and the instantaneous friction

torque method (P–w method, Rezeka and Henein, 1984). The advantages and

disadvantages of each experimental method are reviewed by Wakuri et al.

(1995) and Richardson (2000). Because the diesel engine and the gasoline

engine, either heavy-duty or light-duty, may share similar characteristics of

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673Friction and lubrication in diesel engine system design

engine friction within a common range, the references in this chapter also

include the gasoline engine.

Different engine components operate in different lubrication regimes at

different operating conditions due to their various load and speed characteristics.

Their friction losses respond in different ways to the changes in design

parameters and operating conditions. The understanding and modeling of

engine friction characteristics should be built on each individual component.

Recall that the duty parameter or the lubrication parameter in the Stribeck-

type diagrams consists of lubricant viscosity, loading, and sliding velocity to

characterize the lubrication. The proportionality between the hydrodynamic

and the mixed/boundary lubrication regimes depends on many factors such

as engine speed, load, lubricant viscosity, state of break-in, etc.

The piston skirt has a periodic sliding velocity between the TDC and the

BDC, and carries the side thrust load generated from the cylinder gas pressure

and the piston-assembly inertia force (Munro and Parker, 1975). The piston

skirt predominantly operates in the hydrodynamic or elastohydrodynamic

lubrication regime, especially in the middle of each stroke where the piston

speed is higher. In the expansion stroke and the late compression stroke when

the gas loading is high, some pistons may operate in mixed lubrication due to

surface asperity contact. Under dynamic conditions when sudden increases in

load or decreases in relative speed occur, a ‘squeeze lm’ effect is developed

to maintain a certain oil lm thickness to separate the surfaces within a

short moment. Near the TDC and the BDC where the piston sliding velocity

approaches zero, there is a tendency to transition to the mixed lubrication

regime although the large ‘squeeze lm’ effect of the entire piston skirt tries

to overcome such a tendency. Similar ‘squeeze lm’ effects may exist for the

piston rings. Note that the piston side thrust can reach zero at some crank

angle locations within one engine cycle (Fig. 10.2).

The piston compression rings (the top and second rings) have the same

periodic sliding velocity as the skirt, and their loads basically include the ring

tension and the gas pressure acting on the back of the rings. The ring tension

is usually dened as the diametrical compression force required to compress

the free end of the ring to a specied clearance gap. The compression rings

operate primarily in the hydrodynamic or elastohydrodynamic lubrication

regime during the mid-stroke and in the mixed lubrication regime near

the TDC and the BDC, especially in the mixed or even the boundary

lubrication at the ring TDC where the cylinder gas pressure loading acting

on the rings is high and the lubricant viscosity is low due to the hot wall

temperature. Under that condition, the lubricating oil lm breaks down and

surface asperities come into contact, causing higher rubbing friction and

wear. The second compression ring is commonly designed to have a tapered

face (i.e., smaller ring diameter at the top than the bottom) to help control

oil through a scraping action in the mixed or boundary lubrication on the

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674 Diesel engine system design

down stroke. Note that the piston ring load in the diametrical direction is

always greater than zero.

Among the piston rings, the oil control ring contributes much higher

friction than the compression rings during the most of the strokes except

around the ring TDC where the friction force of the compression rings may

be higher due to the high cylinder pressure gas loading. The oil ring usually

has two very thin rails and very high tension. Each rail of the oil control ring

is handled as a separate single ring in the analysis. The rails usually are too

thin to generate a hydrodynamic lubricating oil lm when the tension is high

and the speed is low. In that case, they operate primarily in the boundary

lubrication regime with strong surface asperity contact between the ring face

and the cylinder bore.

The instantaneous friction force acting on the piston and the rings may

experience a sudden increase at the TDC or a sudden decrease at the BDC

when the piston reverses direction due to the abrupt change between the

dynamic and the static friction coefcients. This step change in the piston

friction force is impulsive in nature and contains a broad range of frequencies

which can excite the crankshaft resonant frequencies and produce a knocking

noise at low speeds. Such a piston ‘stick slip’ noise was discovered in gasoline

engines (Beardmore, 1982; Werner, 1987). Low-friction coating materials

and friction modiers in the lubricant were found to be effective to depress

the abrupt changes between the static and the dynamic friction coefcients

in order to reduce the ‘stick slip’ noise.

The engine bearings (the crankshaft main bearings, the connecting rod

bearings at the big end and the small end, the camshaft bearings, and the

rocker shaft bearings) carry different types of loading, and operate at different

sliding speeds. For example, the crankshaft main bearing has a basically

constant sliding velocity at a given engine speed, and it operates mainly in the

hydrodynamic or elastohydrodynamic lubrication regime. At low speeds the

heavily loaded engine bearings may operate in the mixed lubrication regime.

The crankshaft main bearing seals operate in the boundary lubrication regime

due to the direct contact between the seal lips and the crankshaft surface.

The instantaneous resultant force of the connecting rod small-end bearing

can reach zero. The instantaneous resultant force of the connecting rod big-

end bearing is usually greater than zero. The instantaneous resultant force

of the crankshaft main bearing is greater than zero most of the time during

an engine cycle but occasionally it can reach zero. The camshaft resultant

force does not reach zero within an engine cycle.

The main friction interface in the valvetrain is between the cam and the

follower (tappet). The cam loading consists of the valve spring force and

the valvetrain dynamic inertia load. The contact can be sliding (for a at-

faced follower) or rolling (for a roller follower). The cam–follower contact

is mainly in the severe mixed lubrication or boundary lubrication regime

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675Friction and lubrication in diesel engine system design

(including elastohydrodynamic) due to the very high load acting on a small

contact area with a relatively slow speed.

The frictional losses in the oil pump consist of bearing friction, internal

uid friction/restriction and intermeshing friction (Baba and Hoshi, 1986).

The pump friction torque is proportional to the third power of the gear’s

outside diameter and the pump speed which is related to the drive ratio

(Kovach et al., 1982).

Note that in the hydrodynamic lubrication regime for all the above engine

components, elastohydrodynamic lubrication may occur in some portion of

the engine cycle, because of the elastic deformation of the contact surfaces

due to very high unit load, and because of the inuence of the lubricating

oil lm pressure on viscosity. Also note that mixed or boundary lubrication

happens for almost all the components during engine start-up and shut-down

because of the very low engine speed, oil starvation, and less effective oil

lm. Moreover, there is a difference between friction force and friction power.

Although poor lubrication at the ends of the stroke results in a large friction

force, wear and even scufng between the piston rings and the cylinder bore,

it has little effect on the friction power because the piston sliding velocity

is very low there. In contrast, the hydrodynamic drag at the mid-stroke,

although lower in terms of friction force, is the major contributor to the

friction power due to the high piston velocity.

10.3.2 Engine friction characteristics at different speeds

and loads

In boundary lubrication, friction force is independent of speed. In hydrodynamic

lubrication, the viscous friction force increases with increasing speed. The

uid pumping torque usually increases with speed squared. Moreover, some

component loading changes with engine speed, for example the valvetrain.

Basically, the friction torque or the FMEP of all the components increases

with engine speed, except for the valvetrain friction torque which decreases

as the engine speed increases. The reason is that the boundary lubrication

dominates in the valvetrain and the valvetrain load decreases with increasing

engine speed. As the engine speed decreases, the proportion of the boundary

lubrication friction increases.

The trend of overall engine friction at different loads is more complex,

primarily because of the complex trend of the power cylinder friction. When

the diesel engine load or fueling rate changes (with an extreme of comparing

motoring with full-load ring), the following three aspects change in the

power cylinder: (1) gas pressure loading; (2) metal/wall temperature; and

(3) the clearances in the power cylinder. Higher gas loading increases the

piston assembly friction during the expansion stroke and the late compression

stroke in the boundary, mixed and even hydrodynamic lubrication regimes,

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676 Diesel engine system design

especially near the ring TDC. The temperature of the oil on the cylinder liner

is controlled mainly by the coolant temperature. At higher load, the coolant

temperature is higher. The higher metal/wall temperature reduces the lubricant

viscosity and the hydrodynamic viscous friction force, particularly in the mid-

stroke where the gas pressure effect is small. The smaller clearances due to

components’ thermal expansion tend to increase the friction force. Another

effect to consider is that the high friction force (evidenced by the wear at

the top ring reversal) does not translate to high friction power near the TDC

because the piston speed is low there. During the mid-stroke of an engine

cycle the high temperature effect plays a major role due to the low cylinder

pressure. This affects the friction power signicantly since the piston velocity

is high. The net result depends on which factor plays a dominant role.

It is generally believed that the bearing forces increase with engine load.

Accessory friction losses also increase with load. Most authors believe that

the overall engine friction torque increases with load, based on experimental

ndings. Accurate modeling of the engine friction at various loads is important

because the fuel economy benet of friction reduction is most prominent at

the low-load conditions, as encountered in many real-world usage conditions,

where the proportion of the friction is high compared to the brake power.

A typical argument on the engine friction trend vs. load is whether the

motoring friction torque is lower than the friction torque in ring. This topic is

important because the motoring test has been widely used as a convenient way

to estimate the engine friction at ring. As mentioned above, the mechanisms

of the friction losses in motoring and ring are very different due to the

changes in cylinder pressure loading, wall temperatures and clearances, even

if the overall engine friction is similar. The pressurized motoring tests with

high intake manifold pressures without ring at Cummins (Richardson, 2000)

conrmed that an increase in the peak cylinder pressure increases the piston-

assembly friction force. Although controversial experimental ndings have

been published regarding the friction power difference between motoring and

ring, it appears that most people believe the friction force in a ring engine

is greater than that in a motored engine around the ring TDC under the

combined inuence of higher cylinder pressure and temperature. Richardson

(2000) reviewed this topic and concluded that the piston ring pack friction

power in hot motoring was lower than that in ring. In general, the cold

motoring friction power was close to the hot ring power. He concluded that

the overall average power cylinder friction of a ring engine was 0–20%

more than the friction of a motored engine.

10.3.3 Design measures for engine friction reduction

Effective design measures to reduce engine friction should be planned for

each individual component according to their different lubrication regimes.

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677Friction and lubrication in diesel engine system design

The general guidelines for friction reduction are as follows:

∑ Durability and reliability take precedence over friction reduction and

fuel economy. Any friction reduction should not jeopardize durability.

∑ Always minimize the load on the frictional interface. This is difcult to

achieve in many situations because of the nature of the high load (e.g.,

caused by the demand of high power rating and the corresponding high

cylinder pressure) and the compromises involved (e.g., using a shorter

stroke to reduce the piston side thrust may cause an unfavorable stroke-

to-bore ratio for the combustion performance). However, in certain

scenarios the design optimization to reduce the load is achievable. For

example, reducing the reciprocating mass of the piston assembly may

reduce mechanical noise and even friction, although the magnitude of

friction reduction depends on the particular application (e.g., engine speed).

Offsetting the piston pin may change the side thrust and the moments

acting on the piston so that less violent piston secondary motions can

be achieved to reduce the friction loss. Another example is to reduce

the valve spring force through valvetrain optimization.

∑ For boundary lubrication (e.g., cam–follower interface, oil control ring),

reducing the normal load acting on the contact is effective. However,

this needs to be done carefully. For example, excessively decreasing the

tension of the oil control ring will increase oil consumption, especially

when the cylinder bore distortion is large. Other design measures need to

be taken to ensure the ring–bore conformability for adequate oil control

(e.g., by reducing the ring stiffness through a reduction in the width or

the radial wall thickness of the oil ring).

∑ Reduce friction coefcient by replacing sliding friction with rolling

friction, especially in situations where high loads must be carried at

relatively low sliding velocities (e.g., the follower and the rocker arm

pivot in the valvetrain). One approach is to use rolling-element or needle

bearings instead of plain bearings. A roller design may increase the

contact stress and require a better material to withstand the stress.

∑ For hydrodynamic lubrication (e.g., the piston skirt, the bearings, the

compression rings), reducing the lubricated area or the lubricant viscosity

is the most effective way to reduce the shear friction force. Examples

include using shorter piston skirt, smaller bearing width and diameter,

and fewer and thinner piston rings. However, this needs to be done

carefully without decreasing the oil lm thickness or increasing the

bearing temperature to an unacceptable level. Decreasing the number of

compression rings tends to increase blow-by. As shown in the Stribeck

diagram, there is a minimum achievable friction coefcient that is

located at the verge before the transitioning from the hydrodynamic to

the mixed lubrication regime. Reducing the load carrying area increases

the unit load and decreases the oil lm thickness. This reduction in oil

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678 Diesel engine system design

lm thickness may offset the benet of reduced area, as Kovach et al.

(1982) showed in their piston skirt experiment. Moreover, if the design

change causes the oil lm thickness to fall into the mixed lubrication

regime, the coefcient of friction starts to increase again. For example,

if the piston skirt length reduction causes a drastic decrease in the

side-thrust carrying capacity so that the skirt-to-bore asperity contact

happens, then there is no friction reduction. In fact, the friction may

even increase due to wear or scufng. Moreover, although reducing the

bearing size (especially the diameter) may reduce the hydrodynamic

lubrication friction, journal misalignment and edge loading should be

avoided in order to prevent structural failures or scufng.

∑ Using multigrade oils with high viscosity indexes can reduce friction.

When lower viscosity oil is used to reduce friction, care needs to be

taken on the improved oil formulation/additives and the wear resistant

materials to prevent degradation in wear.

∑ For hydrodynamic lubrication, reducing the clearance at the frictional

interface may reduce friction, especially at low speeds. However, this

may cause mechanical noise problems, cylinder liner cavitation due to

worse piston slap and a larger oil ow requirement that result in a higher

oil pump power. Moreover, changing the clearance usually affects the

operating oil viscosity.

∑ For hydrodynamic lubrication, in order to increase the overall oil lm

thickness on the lubricated contact (i.e., for the purpose of reducing the

viscous friction force) or promote a better lubricating oil lm pressure

distribution (e.g., through the lubrication ‘wedge’ effect or the ‘squeeze

lm’ effect), component prole design is an important measure for friction

reduction and tribological motion control. Examples include the piston

skirt’s barrel prole and the piston ring’s off-centered barrel prole.

∑ For mixed or boundary lubrication, if the load cannot be reduced, in

order to reduce the coefcient of friction, design measures to change the

lubrication regime to hydrodynamic should be considered. This can be

done by changing the component clearance or prole or by decreasing the

surface roughness to increase the lm thickness-to-roughness ratio.

∑ For boundary lubrication, although the lubricant viscosity is not important

for the friction characteristics, friction-modier additives in the lubricant

(e.g., the extreme pressure additive) may establish a crucial thin reaction

lm in the boundary lubrication, and this may reduce the boundary

friction coefcient hence reduce friction. When the lubricant viscosity is

reduced, the frictions from the piston assembly and the bearings can be

reduced, but the valvetrain friction may increase because the valvetrain

(e.g., cam–follower) operates mainly in the boundary or mixed lubrication

regime. Friction-modier oils may reduce the boundary friction in the

engine, if the oil is supplied effectively into the lubricated interface.

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679Friction and lubrication in diesel engine system design

∑ Using special face materials or coatings on the piston rings and the piston

skirt can reduce friction. Changing the bearing material may substantially

reduce friction in the mixed or boundary lubrication regime.

A comprehensive list of detailed design methods for power cylinder friction

reduction are provided by Richardson (2000).

10.3.4 System design effects on engine friction

An engine system design engineer often faces the challenges of selecting

appropriate system-level designs and operating parameters with a minimum

penalty on engine friction. The system design parameters usually include the

following: engine displacement, the number of cylinders, cylinder centreline

distance, cylinder bore, engine stroke, connecting rod length, the ratio of

connecting rod length to crank radius, valve size, valvetrain conguration

parameters, crankshaft offset, piston compression height, piston pin offset,

piston assembly weight, engine weight, engine compression ratio, cold-start

friction torque, peak cylinder pressure, and engine speed range. An accurate

modeling of the impact of these parameters on engine friction has a direct

inuence on system design results.

Engine testing results indicated that the engine friction reduces as the

cylinder size or the number of cylinders increases (Monaghan, 1988).

Therefore, for a given engine displacement, the choice of the number of

cylinders may have a complex impact on engine friction. Ciulli (1993)

supported the previous published experimental work (e.g., Bishop, 1964) by

stating that fewer larger cylinders result in lower friction. In general, a shorter

engine stroke (Millington and Hartles, 1968), a lower compression ratio,

or lower peak cylinder pressure can reduce overall engine friction. Bishop

(1964) showed that the stroke-to-bore ratio has only a very small effect on

the tested engine friction. Patton et al. (1989) used a friction model to nd

that the total engine friction decreases with decreasing stroke-to-bore ratio.

Millington and Hartles (1968) showed that the ratio of connecting rod length

to crank radius has little effect on the piston friction. Larger bore distortion

usually increases friction (Monaghan, 1988). Offsetting the crankshaft may

signicantly affect the piston side thrust and the sliding speed of the piston,

hence reducing the piston skirt friction force.

Engine friction fundamentals are systematically introduced by Heywood

(1988), Ferguson and Kirkpatrick (2001), and Taylor (1993a). The mechanisms

of engine mechanical friction for each major component are introduced by

Comfort (2003). Overviews on engine friction are provided by Rosenberg

(1982), Parker and Adams (1982), Kovach et al. (1982), Furuhama (1987),

Monaghan (1988), Parker (1990), Wakuri et al. (1995), Ciulli (1992, 1993),

and Richardson (2000).

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