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Two-dimensional lubrication analysis and designoptimization of a Scotch Yoke engine linear bearing

X Wang1, A Subic1,and H Watson2

1School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Melbourne, Australia2Department of Mechanical and Manufacturing Engineering, The University of Melbourne, Melbourne, Australia

The manuscript was received on 12 December 2005 and was accepted after revision for publication on 19 June 2006.

DOI: 10.1243/0954406JMES260

Abstract: Recent study has shown that the application of a Scotch Yoke crank mechanism to areciprocating internal combustion engine reduces the engines size and weight and generatessinusoidal piston motion that allows for complete balance of the engine.

This paper describes detailed investigation of the performance of a linear bearing assembly,which is one of the key components of the Scotch Yoke mechanism. The investigation starts bysolving Reynolds equation for the Scotch Yoke linear bearing. The two-dimensional lubricantflow is numerically simulated and the calculated results are compared with experimental resultsfrom a linear bearing test rig. The performance characteristics and a design sensitivity analysisof the bearing are presented. Dynamic testing and analysis of an instrumented linear bearing ona test rig are used to validate the numerical simulation model. The oil supply and lubricationmechanism in the linear bearing are analysed and described in detail. This work aims to providenew insights into Scotch Yoke linear bearing design. In addition, strategies for optimization ofthe linear bearing are discussed.

Keywords: hydrodynamic lubrication, linear bearing, Scotch Yoke engine, engine friction,vehicle power system, internal combustion engine

1 INTRODUCTION

Patents on the Scotch Yoke crank mechanismdate backto the last century [14]. This type of mechanism hasbeen used primarily in reciprocating machinery,including water pumps and internal combustionengines. Recent research on the Scotch Yoke enginehas mainly focused on the reduction of engine sizeand weight and on improvement of aspects of theengine performance including drivability and enginebalance, as reported in references [13]. However, lim-itedpublished information exists in relation to thelubri-cation mechanism of the Scotch Yoke linear bearing.

In a Scotch Yoke engine, pairs of opposed pistonsare rigidly connected by very short connecting rods

and four C plates, which link the connecting rods,as shown in Fig. 1. The linear bearing surfaces arethe opposed parallel guide surfaces built on theconnecting rods transverse to the axes of the pistons.The bearing block contains the linear bearing and

journal bearing shells, which run on a crankpin ofthe crankshaft. The lubrication performance of thelinear bearing determines the effectiveness of theScotch Yoke mechanism and influences the perform-ance of a Scotch Yoke engine in areas such as fuelconsumption, output power, and reliability.

This paper presents the results of experimentalinvestigation, theoretical analysis, and numericalsimulation of the Scotch Yoke linear bearing lubrica-tion mechanism. In particular, the following threeaspects are investigated in detail.

1. Solving the Reynolds equation for the linear bear-ing using a two-dimensional lubricant flow modeland theoretical analysis of the behaviour of the

Corresponding author: School of Mechanical and Manufactur-

ing Engineering, RMIT University, Building 251, Level 3, 264Plenty Road, PO Box 71, Bundoora, Victoria 3083, Australia.

email: [email protected]

1575

JMES260 IMechE 2006 Proc. IMechE Vol. 220 Part C: J. Mechanical Engineering Science

Citation: Wang, X, Subic, A and Watson, H 2006, 'Two-dimensional lubrication analysis and design

optimization of a Scotch Yoke engine linear bearing', Institution of Mechanical Engineers.

Proceedings. Part C: Journal of Mechanical Engineering Science, vol. 220, no. 10, pp. 1575-1587.

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bearing oil-film, considering both sliding andsqueezing actions of the film.

2. Establishing a sliding velocity model of the linearbearing on a test rig and verifying lubricationmodelling effectiveness of the test rig by compar-ing the linear bearing reaction forces in the test rig

and in the CMC Scotch Yoke engine.3. Analysing and discussing the simulation results

and their applications.

The main purpose of this research was to solvethe Reynolds equation for the oil-film pressure inthe linear bearing as a two-dimensional lubricantflow problem and to compare analytical results

with experimental ones. The theoretical model isapplied to illustrate and explain the lubricationprocess in the linear bearing and to evaluate thelubrication performance of the linear bearing with

respect to different geometrical design parameters,oil supply pressure, oil supply timing, and enginespeeds.

2 REYNOLDS EQUATION FOR A SCOTCHYOKE LINEAR BEARING

The Scotch Yoke engine linear bearing block movingat a general crank angle position is shown in Fig. 1. Inorder to investigate the lubrication mechanism of the

Scotch Yoke linear bearing, the relevant Reynoldsequation for the linear bearing must be establishedand solved. The assumptions made in establishingthe Reynolds equation are the following.

1. The bearing is completely filled with incompressi-ble oil.

2. There are no spatial viscosity variations.3. The linear bearing inclination angle is small

because of the small clearance between thelinear bearing block and the connecting-rodsurfaces.

The simplified Reynolds equation for the linear bear-ing takes the form of a two-dimensional lubricant

Fig. 1 Force resolution of the single piston-con-rod assembly in a CMC Scotch Yoke engine

where pistons, connecting rods, and the linear bearing are presented [2]

1576 X Wang, A Subic, and H Watson

Proc. IMechE Vol. 220 Part C: J. Mechanical Engineering Science JMES260 IMechE 2006

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flow, as in reference [5]

1

6m

@

@x h3

@p

@x

@

@z h3

@p

@z

2V

@h

@xU (1)

whereVis the squeezing speed of the linear bearingblock with respect to the connecting rod and Uis thesliding speed of the linear bearing block with respectto the connecting rod at a crank angle position asillustrated in Fig. 1; oil viscositym is evenly distribu-ted and independent ofxand z; andp is the oil-filmpressure. The oil-film thickness h is assumed con-stant in the z-direction and is a function ofxalone.It can be written as

h(x) h2h1

L

xh1 (2)

where h1 and h2 are the oil-film thicknesses at thetrailing and leading edges. L is the length of thelinear bearing and x 0 is at the thinner oil-filmend (h h1) of the bearing.

On the right-hand side (RHS) of equation (1), thereare two terms: the first term representing squeezinghydrodynamic action of the linear bearing and thesecond term representing sliding hydrodynamicaction of the linear bearing.

If only the squeezing hydrodynamic actionof thelinear bearing is considered in equation (1), assum-

ing that the sliding speedU 0 and that there is con-stant pressure distribution along the length of thelinear bearing, i.e. (@p=@x) 0, then equation (1)becomes

@

@z h3

@p

@z

12mV (3)

For the boundary conditions at z (Bw=2),(dp=dz)0 and at z 0 orz Bw,p 0, integrating equation(3) twice and considering equation (2) for h(x) gives

p(x,z) 6mVz((h2 h1=L)xh1)

3zBw (4)

The bearing load capacity is, therefore, given by

F L

Bw0

6mV

h3 z2 zBwdz

mVLB3wh3

(5)

It is seen from equations (4) and (5) that under thesqueezing effect, as the squeezing velocity and theoil viscosity increase and as the oil-film thickness

decreases, the oil-film pressure and the bearing loadcapacity increase. The load capacity is proportionalto the cube of the bearing width and is proportional

to the bearing length. The oil-film pressure and thebearing load capacity are inversely proportional tothe cube of the oil-film thickness.

Similarly, if only the sliding hydrodynamic action

is considered in equation (1), assuming that thesqueezing speed V 0 and ignoring the pressurevariations across the bearing width, i.e. @p=@z 0,equation (1) becomes

1

6m

@

@x h3

@p

@x

@h

@xU (6)

By integrating equation (6) twice and consideringequation (2) as well as the boundary conditions, at

x 0, h h1, p 0, and at x L, h h2, p 0, thefollowing equation is obtained

p(x) 6mLU

h2h1

1

((h2h1=L)xh1)

1

((h2 h1=L)xh1)2

h1h2h1h2

1

h1h2

(7)

It is seen from equation (7) that under the slidingeffect, as the sliding velocity, the bearing length,and the oil viscosity increase, the oil-film pressureand the bearing load capacity increase. The oil-filmpressure and the bearing load capacity are inverselyproportional to the oil-film thickness and the incli-nation angle.

In the Scotch Yoke engine, both the squeezing andsliding effects of the linear bearing must be con-sidered and, therefore, equation (1) will have to besolved for a two-dimensional lubricant flow asfollows.

3 NUMERICAL SOLUTION OF THE REYNOLDSEQUATION FOR A TWO-DIMENSIONALLUBRICANT FLOW

To analyse the two-dimensional lubricant flow, afinite-difference method is employed to obtain anumerical solution of the Reynolds equation. Thefinite difference mesh is formed for the bearing, andeach inner node, where the pressure is not known,is assigned a node number. As shown in Fig. 2, thebearing surface is divided into 105 elements, i.e. thenumber of nodes is 8 16 128. The node (i,j) rep-resents the ith count in the x-direction and the jthcount in the z-direction where the coordinate is

(xi,j, zi,j). The generalized subscripts are introducedto form the partial derivatives ofp andh in equation(1) at the node (i, j), using central differentiation as

Optimization of a Scotch Yoke engine 1577


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shown in Fig. 2. The following relationships arederived

@

@x h3

@p

@x

h3i0:5,j(pi1,j pi,j=Dx)

h3i0:5,j(pi,j pi1,j=Dx)

Dx (8)

@

@z h3

@p

@z

h3i,j0:5(pi,j1pi,j=Dz)h3i,j0:5(pi,j pi,j1=Dz)

Dz (9)

@h

@x

hi0:5,j hi0:5,j

Dx (10)

where Dx L/15 and Dz Bw/7.Substituting equations (8) to (10) into equation (1)

gives

(h3i0:5,j h3i0:5,j h

3i,j0:5h

3i,j0:5)pi,j

h3i0:5,jpi1,j h3i0:5,jpi1,j h

3i,j0:5pi,j1

h3ij0:5pi,j1 12mDx2Dz2V

6mDxDz2(hi0:5,jhi0:5,j)U (11)

wherei 1,. . ., 16 and j 1,. . ., 8.Pressure levels at these nodes are unknown except

at the boundary points, i.e. the four edges of the bear-ing surface where (i 1,. . ., 16,j 1), (i 1,. . ., 16,

j 8), (i 1, j 2,. . ., 7) and (i 16, j 2,. . ., 7),

and where the gauge pressure is zero. At the fourpoints representing the oil supply orifices at thecentral grid elements (i 8, j 4), (i 8, j 5),

(i 9, j 4), and (i 9, j 5), the pressure isassumed to be equal to the engine oil supplypressure p0. Hence, the 84 linear homogeneousequations can be developed from equation (11) atthe 84 unknown pressure nodes. The equations canbe written in the matrix form as

H{p} {R} (12)

where elements of [H] represent the coefficients ofthe unknown pressures on the left-hand side ofequation (11). fpg is the unknown pressure matrixand fRg is the matrix of constants defined on theRHS of equation (11).

If oil-film thickness variation and the instan-taneous relative velocities U and V are knownfor a particular crank angle position, the instan-taneous pressure distribution over the linear bearingsurface can be found by solving equation (12).From the resulting pressure distribution, the

key bearing characteristics can be calculated,including the load carrying capacity and the frictioncoefficient.

3.1 Load carrying capacity

The load carrying capacity of the linear bearing isobtained by integrating the linear bearing pressuredistribution over the bearing surface and can beexpressed as

F Xi7,j15

i1,j1

(pi,j pi1,jpi,j1pi1,j1)dxdz

4

(13)

Fig. 2 Resolution of the linear bearing force for hydrodynamic lubrication at one instantaneouscrank angle position and node numbering for uniform finite difference meshes



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3.2 Friction coefficient

The friction coefficient of the linear bearing can becalculated using the following expression [6]

fc Xi7,j15

i1,j1

(mU=hi,j)(hi,j=4dx)(pi1,j

pi1,j1 pi,j pi,j1)dxdz

F (h2h1)=L (14)

To obtain specified oil-film thickness andvelocities U and V, a linear bearing test rig wasdesigned and built. The oil-film thickness, tempera-ture, and pressure were measured at several nodesas described in the following section.

4 DYNAMIC ANALYSIS AND LUBRICATIONMEASUREMENT USING LINEAR BEARINGTEST RIG

In order to determine the critical lubrication para-meters of the linear bearing, a special-purposelinear bearing test rig was designed and built, asshown in Fig. 3.

A piezo-electric pressure transducer, four oil-filmthickness sensors, and two K-type thermocouples(located as shown in Fig. 3) were used to measureoil pressure, oil-film thickness, and oil temperaturebetween the linear bearing surfaces.

Figure 3 shows the resolution of motion and forcesof the linear bearing in this rig. The displacementof the centre of gravity of the con-rod assembly in

Fig. 3 Force and motion resolution for the linear bearing test rig and the cross-section through

connecting rod, showing transducer locations and installation details [7]



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the x- and y-directions at one instantaneous crankangle position vtis expressed as

yc l sin u

xc ll cos u

(15)

wherelis the support arm length, vthe angular vel-ocity of the crankshaft,tthe time, and uthe angle ofthe sway arm with respect to the vertical axis xat thecrank angle position vt. The crankshaft rotates in theclockwise direction and a zero crank angle is definedby the horizontaly-axis.

The displacement of the centre of gravity of thelinear bearing block in the x- and y-directions canbe written as

yb rsin vtp

2

xb rrcos vtp

2

(16)

whereris the crank throw radius and ycis assumedto be equal to yb, which leads to

sin ur

l

sin vtp

2 (17)

cos u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

r

l

2cos2 vt

r (18)

u rv

l

sinvt

cos u(19)

u u2 sin u

cos u

rv2

l

cosvt

cos u(20)

The sliding velocity of the linear bearing block withrespect to the connecting-rod surface can be writtenas

U rvsin vt p2

lusin u (21)

The inertia forces of the connecting-rod assemblyin thex- andy-directions can be written as

Fix Mcl( usin u u2cos u) (22)

and

FiyMcl( ucos u u2sin u) (23)

whereMcis the mass of the connecting-rod assem-bly. When the support arm length l becomes verylarge, as shown in Fig. 3, the linear bearing reaction

force iny-directionFbcan be written as

Fb Fiy Mcrv2 cosvt (24)

The inertia force of the connecting-rod assembly inthex-directionFixtends to zero.

Figure 1 shows the Scotch Yoke connecting rodand the linear bearing in a sectional view where theforces acting on the connecting rod and the bearingblock are given. The linear bearing reaction force ina firing Scotch Yoke engine can be written as

Fb P1(vt)P2(vt)pD2

4 Mcrv

2 cosvt (25)

where P1(vt) and P2(vt) are the combustion press-

ures in the two opposite cylinders acting on thetwo piston crowns and D is the cylinder bore. Thecrankpin reaction forceTin both the linear bearingtest rig and the firing Scotch Yoke engine can be

written as

T ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

F2b M2b r

2v4 2FbMbrv2 cosvtq

(26)

u1 tan1 Mbrv

2 sinvt

FbMbrv2 cosvt (27)

whereMbis the mass of the linear bearing block and

u1 is the crankpin reaction force vector angle withrespect to the y-axis. The linear bearing reactionforce and the crankpin reaction force in the CMC422 Scotch Yoke engine can be calculated by usingequations (25) to (27). The linear bearing reactionforce is plotted in Fig. 4.

The solid line curve in Fig. 4 shows the linearbearing reaction at a low speed of 1500 r/min, andthe dashed line curve in Fig. 4 shows the bearingreaction force at a high speed of 5500 r/min. It canbe seen from Fig. 4 that the combustion pressureeffects near 3608 and 5408 crank angle positions arelarge at a low engine speed of 1500 r/min. The inertiaforce effects at 1808 and 7208 crank angle positionsare large at a high engine speed of 5500 r/min, asconfirmed in reference [4]. The dotted square wavecurves in Fig. 4 show the oil supply timing of thepresent Scotch Yoke engine linear bearing. Figure 4indicates that at a low speed of 1500 r/min and at3608 and 5408 crank angles corresponding to thetop dead centre (TDC) positions of the combustionstrokes, the linear bearing reaction force is a maxi-mum. At those points, the square wave value isunity, indicating that the oil supply ports in the bear-ing plates open, and that lubricating oil is supplied

from the crankpin ports to the oil passages of thebearing block through the leading and trailing portsof the journal bearing shell, as shown in Figs 1 and 4.



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This indicates that at a low engine speed, the engineoil supply pressure has an important effect on thelinear bearing load capacity. At a high engine speedand at 1808 and 7208 crank angles corresponding tothe bottom dead centre and TDC of the inductionstroke, the linear bearing reaction force is maximum.

At those points, the square wave values are not unity,indicating that at the crank angles, the oil supply tothe linear bearings is isolated from the engine oilsupply, and that the linear bearing load capacitydepends upon the squeezing hydrodynamic actionof the residual oil between the linear bearingsurfaces.

Comparison of equation (24) with equation (25)indicates that the linear bearing test rig with verylong support arms is able to model the lubrication

of the linear bearing in the firing Scotch Yokeengine at no load, where the combustion pressuresP1(vt) and P2(vt) are relatively small. The first term(P12 P2)pD

2/4 can be ignored in comparison withthe second term in equation (25), which makesequation (25) identical to equation (24). The lubrica-tion mechanism of the linear bearing in the test rigtends to be same as that in the firing CMC Scotch

Yoke engine at zero load, which is the most criticalcondition for the linear bearing, as illustrated inreference [7].

By the substitution of equation (21) into equation(11) or (12), the coefficient array fRg on the RHS of

equation (11) can be determined, except that thebearing squeezing speed V is unknown. In order todetermine the bearing squeezing speed V, i t i s

Fig. 4 Linear bearing reaction force for the CMC 422 Scotch Yoke engine at 1500 and 5500 r/min

and a sectional view of the linear bearing and crankpin configuration



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assumed that the linear bearing reaction force actingupon the bearing block is equal to the bearing loadcapacity at any crank angle position. By assigningany initial value to the bearing squeezing speed V,

equation (11) can be solved at a specified crankangle position and the pressure distribution overthe bearing surface can be found; and by integratingthe pressure distribution over the bearing surface,the bearing load capacity can be determined.Iteration can be performed until the bearing loadcapacity is equal to the linear bearing reactionforce at this crank angle position. The bearingsqueezing speed V and the pressure distributionafter iteration can be used to calculate the linearbearing performance characteristics and arecompared with the experimental results as follows.

5 TWO-DIMENSIONAL LUBRICATION FLOWSIMULATION RESULTS

A MATLAB program is used to solve equation (11)and to conduct the required numerical iteration.Table 1 lists the input parameters for this program,

which include the measured parameters from thetest rig, namely, the oil supply pressures at the inletport of the linear bearing and the oil-film thicknessesat the leading and trailing edges at differentrotational speeds and crank angles.

From the numerical simulation, the linear bearingoil-film pressure distributions have been obtained.Figures 5 and 6 show the linear bearing pressure

distributions at the crank angle position of 2608 forthe rotational speeds of 1000 and 1500 r/min.Figure 7 shows the bearing oil-film pressure distri-bution at the crank angle position of 2308 for therotational speed of 1500 r/min.

From Figs 5 and 6, it can be seen that the volumeunder the oil pressure distribution profile increasesas the rotational speed increases, indicating that thelinear bearing load capacity increases as the rotationalspeed increases. It can be seen here that the oil-filmpressure distributions are symmetrical about thecentre plane of the bearing width and asymmetricalabout the centre plane of the bearing length. The inte-gration of the oil-film pressure distribution over thebearing surface gives the bearing load capacity.

As the rotational speed increases, the bearing load

capacity increases, enabling the maintenance ofnormal bearing lubrication with the increased bear-ing reaction force. As the rotational speed increases,the connecting-rod assembly reciprocating inertiaforce and the bearing reaction force increase.

At low rotational speeds, the oil-film pressure dis-tributions only have a single peak, whereas at high

Table 1 Input parameters for the simulation program

Parameter names Symbols and numbers Unit

Crank throw r 0.0375 mConnecting rod-C-plates reciprocating mass Mc 2.37 kg Mass of the linear bearing Mb 0.5465 kg The linear bearing test rig sway arm length l 0.445 mThe linear bearing length L 0.082 mThe linear bearing width Bw 0.021 mMesh number in the bearing length direction nx 15Mesh number in the bearing width direction nz 7

2608crank angle positionCrank rotational speed (r/min) 500 1000 1500Measured oil supply pressure (kPa) 300 400 500Measured oil-film thickness at the trailing edge (mm) 27.58 33.17 18.57Measured oil-film thickness at the leading edge (mm) 46.9 29.83 24.04

2308crank angle positionCrank rotational speed (r/min) 500 1500

Measured oil supply pressure (kPa) 300 500Measured oil-film thickness at the trailing edge (mm) 21.94 25.17Measured oil-film thickness at the leading edge (mm) 45.62 18.52

Fig. 5 Oil-film pressure distribution of the linear

bearing at 1000 r/min 2608crank angle position



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speeds, the oil-film pressure distributions have twopeaks, as shown in Fig. 7. As the rotational speedincreases, the maximum oil-film pressure increases.The single peak distribution means that the oiltends to come out of the supply ports and tospread outwards. The twin peak distribution meansthat some oil in-between the bearing surfaces tendsto flow back to or be sucked back to the supplyports. The phenomenon of the bearing oil flowinginto or out of the supply ports at different crankangle positions characterizes the linear bearing

lubrication at high speeds. The number of thepeaks also depends on the crank angle position.

At low rotational speeds and for a single peak oil-film pressure distribution, the sliding effect is greaterthan the squeezing effect; at high speeds and in atwin-peak oil-film pressure distribution, the squeez-ing effect is greater than the sliding effect. This isbecause the sliding speed increases linearly withthe rotational speed, as per equation (21), whereas

the squeezing speed increases linearly with theinertia force of the con-rod assembly, which is pro-portional to the square of the rotational speed, asper equation (24). The squeezing speed increases

much faster than the sliding speed at high rotationalspeeds. When the bearing oil supply opens throughthe crankpin channel, the oil supply pressure has alarger influence on the bearing load capacity atlow-rotational speeds than at high-rotationalspeeds. At high rotational speeds, the large squeezingeffect produces enough load capacity and maintainshydrodynamic lubrication of the bearing.

When the bearing oil supply is cut off from thecrankpin oil channel, the residual oil between thelinear bearing surfaces is squeezed and the oil-filmbecomes thinner, which provides sufficient loadcapacity. This is because the oil pressure from the

squeezing effect is inversely proportional to thecube of the oil-film thickness, whereas the oilpressure from the sliding effect is inversely pro-portional to the oil-film thickness, as per equations(4) and (7).

Table 2 lists the measured and predicted oil-filmpressures for the test rig at different conditions usingthe MATLAB program. From the calculated oil-filmpressure distributions, the pressures at two specificpoints are obtained (Table 2). One of the points was55 mm away from the leading edge and in the centreplane of the bearing width, and the other point was

also in the centre plane of the bearing width and46.8 mm away from the leading edge. Figure 3 showsthe pressure sensors inserted through the con-rod,

which measured oil-film pressures at the two specificpoints. The predicted results coincided with themeasured results except for the case of 1500 r/min,at the position of 2308 crank angle, as shown, inFig. 7. According to the measured data in this case inTable 2, the oil supply port pressure at the centrehole of the linear bearing surface was 500 kPa, andthe measured oil-film pressure at the point 46.8 mmaway from the leading edge was 386 kPa.

If the oil pressure distribution had a single peak atthe centre of the bearing surface as indicated by themeasured pressure data, the bearing load capacityobtained by integrating the oil pressure distributionover the bearing surface would be less than the pro-duct of the pressure peak value and the linear bearingsurface area, which equals 861 N. The linear bearingreaction force in this case was 1409 N according toequation (24), which means that the bearing loadcapacity was smaller than the bearing side force. Ifthat had occurred, hydrodynamic lubrication couldnot be maintained, the oil-film could break down,and there could be contacts between metal surfaces

in the bearing. In practice, the linear bearing surfacedid not have any signs of scratches or damage,suggesting that the oil-film pressure distribution







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was not a single peak pressure distribution. Themeasured pressure in this case must be incorrect.This might be caused by the effect of measurementnoise on the measured signals. Only the twin peakoil pressure distribution as in Fig. 7 would make it

possible for the linear bearing load capacity to equalthe linear bearing reaction force.

6 LINEAR BEARING PERFORMANCECHARACTERISTICS AND GEOMETRYDESIGN OPTIMIZATION

From the two-dimensional lubricant flow modelling,some of the linear bearing performance character-istics can be obtained. Figures 8 and 9 predict therelationships of bearing load capacity and frictioncoefficient with the bearing length and width.

These results were obtained at the rotational speedof 1000 r/min, at 2608crank angle under the assump-tion that the bearing oil-film thickness, oil-filmshape, and the oil-film squeezing speed were stableand reproducible. The bearing performance par-ameters were calculated from equations (13) and (14).

From Fig. 8, it can be seen that when the linear bear-ing length is kept constant and the bearing widthincreases, the bearing surface area, the bearing loadcapacity, and the friction coefficient increases. Forexample, doubling the width from 20 to 40 mm

increases the load capacity by a factor of four. Thisresult is different from the conclusion from equation(5), which indicates that the load capacity is pro-portional to the cube of the bearing width. This isbecause equation (5) is a one-dimensional solution ofthe Reynolds equation, which assumes that the slidingspeedU 0 and that there is constant pressure distri-bution along the length of the linear bearing. The one-dimensional solution overestimates the effect of thebearing width on the load capacity. In other words,the pressure variation along the bearing length andthe sliding speed may play a role in reducing theeffect of the bearing width on the load capacity.

Corresponding to this 2 increase in the bearingwidth is an increase in friction coefficient by about2 as well. It can be seen from Fig. 9 that when thelinear bearing width is kept constant and the bearinglength increases, the bearing load capacity and thefriction coefficient increase. In general, it can be

Table 2 Measured and predicted oil-film pressures for the linear bearing test rig at different

conditions

Oil film pressure at the position of 55 mm from the bearing leading edge at 260 8crank angle positionCrank rotational speed (r/min) 500 1000 1500

Measured oil-film pressure (kPa) 158 218 456Calculated oil-film pressure (kPa) 118.7 218.8 422.5

Oil-film pressure at the position of 46.8 mm from the bearing leading edge at 2308crank angle positionCrank rotational speed (r/min) 500 1500Measured oil-film pressure (kPa) 168 386Calculated oil-film pressure (kPa) 194.6 1275.5

Fig. 8 Relationships of the load capacity and the friction coefficient with the bearing width for the

linear bearing at 1000 r/min 2608crank angle



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concluded that these parameters are much moresensitive to the changes in width than to the changesin length.

The preceding geometry sensitivity analysis indi-cates that the dimensional changes of the bearingshould be made to achieve sufficient load capacity

with the minimum bearing surface area or the mini-

mum friction coefficient. As the width change isrestricted by the connecting-rod design, only bearinglength change is possible. If the bearing lengthreduces from 0.082 to 0.068 m and the squeezingspeed was not changed, the instantaneous oil-filmthickness would have to be reduced to maintain thebearing load capacity at a specific rotational speed.

Figure 10 shows the bearing load capacities at1000 and 1500 r/min, which are equal to the appliedreaction forces on the bearing block at the tworotational speeds. As the bearing length increases,the bearing load capacity increases. The frictioncoefficients at the two speeds decrease as the bear-ing length decreases. In a new design version of the

CMC 422 Scotch Yoke engine, the linear bearinglength was reduced from 0.082 to 0.07 m; it wasshown to work well as the engine friction wasreduced. It should be noted that the reduction ofthe linear bearing length was also restricted by thedimensions of the crank pin and the bearingconnecting bolts.

Fig. 9 Relationships of the load capacity and the friction coefficient with the bearing length for the

linear bearing at 1000 r/min 2608crank angle

Fig. 10 Bearing load capacities and the bearing friction coefficient versus the bearing length at

1000 and 1500 r/min where the bearing load capacities were enough



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7 CONCLUSIONS

A numerical model of a two-dimensional lubricationflow has been developed and used to simulate lubri-

cation of the linear bearing of a Scotch Yoke engine.The results of the simulation were compared withexperimental results obtained using a custom-builtbearing test rig. The following conclusions havebeen drawn.

1. Dynamic analyses for the linear bearing test rigand the linear bearing assembly of the Scotch

Yoke engine showed that the test rig satisfactorilysimulated the lubrication mechanism for theScotch Yoke engine at a no-load condition.

2. The bearing load capacity associated with the

squeezing effect of the lubricant film increaseswith the squeezing speed and the bearing width.The bearing oil-film pressure associated with thesliding effect of the film increases with the slidingspeed and the bearing length.

3. The bearing oil-film pressure and the loadcapacity are inversely proportional to the cubeof the oil-film thickness for the squeezing effectand are inversely proportional to the oil-filmthickness for the sliding effect. The oil-film thick-ness depends on the bearing clearance.

4. The bearing oil supply timing also has an import-ant effect on the linear bearing lubrication. Whenthe linear bearing oil supply ports are open andconnected to the engine oil supply, the oilsupply pressure will contribute to the bearingoil-film pressure and the load capacity. Whenthe bearing oil supply ports are cut off awayfrom the engine oil supply, the residual oilbetween the bearing surfaces will be squeezedand the oil-film thickness will be reduced, provid-ing the required bearing load capacity. The oil-film pressure distribution may have either asingle or twin peaks.

5. As the bearing width or length increases, its load

capacity and friction coefficient increase. Thelatter are much more sensitive to change of thebearing width than to change of the bearinglength.

6. The optimization strategy for the Scotch Yokeengine will be the minimization of the bearing sur-face area and the bearing friction coefficient whilemaintaining sufficient bearing load capacity.

ACKNOWLEDGEMENTS

This authors would like to express their thanksto Richard Gabler for providing experimental results,Dr Hans G. Rosenkranz for his advice and support.

REFERENCES

1 Aitken, A., Daniel, M., and Fountain, G. The CMC ScotchYoke engine a family of engines for automotive use. SAEpaper no. 901532, 1990.

2 Close, H., Szydlowski, W., and Downton, C. Perfectengine balance via collins Scotch Yoke technology. SAEpaper no. 941039, 1994.

3 Close, H.The compact collins Scotch yoke engine moreprogress. SAE paper no. 930314, 1993.

4 Daniel, M. Interconnecting rotary and reciprocatingmotion. International Patent Classification: F01B 9/02,(WO 93/04269), 1993.

5 Fuller, D. D. Theory and practice of lubrication for engin-eers, 1956 (Wiley, New York).

6 Bagci, C.and Singh, A. P. Hydrodynamic lubrication offinite slider bearing: effect of one dimensional filmshape, and their computer aided optimum design.

J. Lubr. Technol. Trans. ASME, 1983, 105, 138.7 Wang, X. Dynamic analysis of the Scotch Yoke com-

ponents. CMC Research Report 94/2, The University ofMelbourne, 1994.

APPENDIX

Notation

Bw the width of the linear bearingblock

D the cylinder bore

Dx, Dy, Dz length increments in thedirectionsx,y, andz

F the load capacity of the linearbearing

fc the friction coefficient of the linearbearing

Fb the linear bearing side reactionforce

Fx,Fz the forces in thex- andz-directions

Fix,Fiy the inertia forces of the con-rodassembly in thex- and

y-directionsFa1,Fa2 the side forces on the top and

bottom pistonshi, j oil-film thickness at (xi, j,zi, j)hi20.5, j oil-film thickness at

(xi, j2Dx/2,zi, j)hi0.5, j oil-film thickness at

(xi, j Dx/2,zi, j)hi, j20.5 oil-film thickness at

(xi, j,zi, j2Dz/2)hi, j0.5 oil-film thickness at

(xi, j,zi, jDz/2)h(x),h oil-film thickness at the

position xh1,h2 oil-film thicknesses at the trailing

and leading edges



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[H] 84 84 oil-film thicknesscoefficient matrix

l the length of the sway arm of thelinear bearing test rig

L the length of the linear bearingblock

Mb the mass of the linear bearingblock

Mc the mass of the connecting-rodassembly

nx,nz the mesh numbers in thex- andz-directions

p oil pressurefpg 841 oil pressure matrixpi, j oil pressure at (xi, j,zi, j)pi1, j oil pressure at (xi, jDx,zi, j)

pi21, j oil pressure at (xi, j2

Dx,zi, j)pi, j21 oil pressure at (xi, j,zi, j2Dz)pi, j1 oil pressure at (xi, j,zi, j Dz)p1(vt),p2(vt) the combustion pressures in the

top and bottom cylindersr the crank throw radiusfRg 841 matrix of constantst timeT the crankpin reaction forceu,v,w oil flow velocities in the directions

x,y, andz

U,V the velocities of the linear bearingblock in the directions xand y

x,y,z oil-film element coordinatesxb,yb the centre of gravity

displacements of the linearbearing block in the directionsxandy

xc,yc the centre of gravitydisplacements of the connecting-rod assembly in the directionsxandy

u the angle of the sway arm of thelinear bearing test rig

u the angular velocity of the swayarm of the linear bearing test rig

u the angular acceleration of thesway arm of the linear bearing testrig

u1 the crankpin reaction force vectorangle with respect to the y-axis

m dynamic viscositytx, tz the shear stresses producing

forces in the directions xand z@ partial differentiationvt crank anglev crankshaft rotational speed



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