Nguyen Van Liem Zhang Jianrun Jiao Renqiang Huang Dacheng

(1School of Mechanical Engineering, Southeast University, Nanjing 211189, China)(2School of Mechanical and Electrical Engineering, Hubei Polytechnic University, Huangshi 435003, China)(3Faculty of Automotive and Power Machinery Engineering, Thai Nguyen University of Technology, Thai Nguyen 23000, Vietnam)

Abstract：A new method combining the slider-crank mechanism dynamic (SCM) and crankpin bearing (CB) lubrication models is proposed to analyze the effects of CB dimensions and engine speed on the lubrication efficiency and friction power loss (LE-FPL) of an engine. The dynamic and lubrication equations are then solved on the basis of the combined model via an algorithm developed in MATLAB. To enhance the reliability of the research results, the experimental data of combustion gas pressure is applied for simulation. The load bearing capacity (or oil film pressure), friction force, friction coefficient, and eccentricity ratio of the CB are selected as objective functions to evaluate the LE-FPL. The effects of engine speed, bearing width, and bearing radius on the LE-FPL are then evaluated. Results show that reductions in engine speed, bearing width, or bearing radius can decrease the FPL but reduce the LE of the engine and vice versa. In particular, the LE-FPL can effectively be improved by slightly reducing the bearing width and bearing radius or maintaining engine speed at 2 000 r/min.

Key words：slider-crank mechanism; crankpin bearing; lubrication performance; friction loss

Given greater demands for fuel economy, energy conservation, and emission reduction in oil-fueled automobiles, technologies that can reduce friction power losses (FPLs) to improve the power of internal combustion engines (ICE) is a major concern for designers. The main causes of engine power loss include the friction between the piston skirt and the ring against the cylinder bore during motion and the friction between moving surface pairs of joints in the slider-crank mechanism (SCM)[1-3].

Studies on the influence of various SCM design parameters, the inertial mass of the piston, and the eccentricity between the crankshaft and cylinder centers on the horizontal impact force of the piston in the cylinder bore have been carried out over the last few decades[4-6]. Friction and noise vibration between the piston skirt and ring around the cylinder bore could significantly be reduced by optimizing the structure of the SCM. Optimization studies have also investigated the characteristics of lubrication factors, such as the friction forceFand friction coefficientμ, between the moving surfaces of a piston and cylinder[1-2,6-7]. Researchers have found that FPLs are significantly reduced when a stable oil film exists on the lubricating surfaces.

The lubrication model of crankpin bearings (CBs) under an external load acting on the shaft moving at high speed was previously studied to illustrate the stability of the lubricating oil film thicknesshon the friction surface pairs of the engine[8-10]according to the evaluation indices of load bearing capacity (LBC) andF. Results showed that the stability ofhgreatly depends on the pressure generated by the oil film. The effects of several factors, such as the temperature[11], radial clearance[7, 12-13], and load and speed of the shaft, on the oil film pressure have also been discussed[9-11,14]. Previous studies mainly assessed the lubrication efficiency (LE) of the CB under the condition of a static load on a shaft moving at high speed. In actual applications, the shear stress of the oil film generated under various speeds and the dimensions of the CB could affect the resistance of the oil film. Moreover, in the working cycle of an ICE, the dynamic load acting on the CB changes very quickly under various engine speedsωin terms of direction and intensity. Therefore, the effects ofωand CB dimensions should be taken into account during analyses of the LE-FPL of an engine. However, investigations on the LE-FPL of engines consideringωand CB dimensions together have rarely been reported.

The current study establishes a new numerical method by coupling the SCM dynamic and CB lubrication models to investigate their effect on the LE-FPL of an engine. An algorithm program based on the hybrid model was developed in MATLAB, and numerical simulations were performed to solve the hydrodynamic equations of the SCM and CB. The LBC,F,μ, and eccentricity ratioεbetween the shaft and bearing were then selected as objective functions. Finally, the influence ofω, bearing radiusrb, and bearing widthBon the LE-FPL was clarified.

1 SCM Dynamic and CB Lubrication Models

1.1 SCM dynamic model

Assuming that the center of a crankshaft coincides with the cylinder center, the SCM dynamic model can be established as shown in Fig.1. Here,LandRare the connecting rod length and rotation radius of the crankshaft, respectively;Pis the combustion gas pressure acting on the piston peak;NandFcare the piston forces impacting on the cylinder wall and connecting rod, respectively;Fis the total force of the piston;Fic,Fc1, andFc2are the respective centrifugal inertial, radial, and tangent forces of the large-rod-end mass of the connecting rod impacting on the CB;Fipis the inertial force of the small-rod-end and piston;W0is the impacting force on the CB; andφandωare the respective angle and angular velocity of the crankshaft.

(a)(b)

The piston’s motion can be determined as

zp=L+R-Rcosφ-Lcosθ

(1)

The kinematic relation of SCM can be determined by

(2)

whereλ=R/Lis the ratio between the rotation radius of the crankshaft and the length of the connecting rod.

Substituting Eq.(2) into Eq.(1), the piston’s motion and acceleration can be rewritten as

(3)

(4)

mscandmbcare assumed to be the lumped masses of the small- and large-rod-ends of the connecting rod atosandob, respectively. Thus, theFipof the small-rod-end and piston is written as

Fip=-(msc+mp)Rω2(cosφ+λcos2φ)

(5)

wherempis the piston mass.

Under the impact ofPon the piston peak, the dynamic forces impacting the cylinder wall and connecting rod can be determined by

(6)

The impact ofFc, which has two components of tangential and radial forces, on the CB is determined by

(7)

The large-rod-end mass of the connecting rod is rotated withω, and itsFiccan be defined as

Fic=-mbcRω2

(8)

The bearing of the connecting rod provides a rotating motion and transmits loads between the large-rod-end and crankpin. Thus, the dynamic loadW0of the connecting rod impacting on the CB is determined by

(9)

Changes inW0in both direction and intensity could be applied to analyze the LE-FPL of the CB.

1.2 CB lubrication model

1.2.1 Lubrication model in the computational region

In the working cycle of the engine, the crankpin is impacted byW0and rotated with an angular velocityωinside the bearing of the connecting rod, as shown in Fig.1. Therefore, the LE-FPL of the CB could be evaluated by modeling the crankpin under the impact ofW0and rotated atωinside the bearing as shown in Fig.2.

In Fig.2,Bandrbare the bearing width and radius, respectively;rsis the shaft radius andrs

h=rb-rs+ecosφ=c(1+εcosφ)

(10)

Fig.2 The CB hydrodynamic model

1.2.2 Application of the Reynolds equations

Assume that the bearing surface is fixed in thex- andy-directions and that the shaft surface only moves with a constant velocityu0in thex-direction, as seen in Fig.2. The boundary conditions of the fluid velocity on the CB surface in thex- andy-directions are determined by

(11)

The lubricant density and viscosity are also assumed to be constant, and the influence of the inertia of the lubricant flow on the working process is negligible. Therefore, the pressure distribution of the lubrication film can be described as[11, 15]

(12)

Taking the second integral of Eq.(12) with respect tozand considering the initial conditions in Eq.(11), the velocity fluid equations are written as

(13)

whereηis the dynamic viscosity of the oil film.

The volumetric flows ofqxandqyin thex- andy-directions are calculated by integrating Eq.(13) over thez-direction and written as

The equation of the fluid continuity condition is given by

(15)

(16)

In Eq.(10), the oil film thicknesshis constantly changing according to the rotation angleφ. Thus, the substitution of Eq.(14) into (16) and mathematical transformation lead to a new form of Eq.(16)[16-17],

(17)

The dimensionless form of the Reynolds equation in Eq.(17) can be expressed as

(18)

whereβ=B/rb,H=h/c,φ=x/rb,Y=y/B,P=p/p0,T=t/t0,Π=6ηωB2/(c2p0),Λ=12ηB2/(c2p0t0), andp0=101.325 kPa.

The boundary conditions must be determined to obtain the pressure distribution of the oil film over the computational domain of Eq.(17). In this study, we assume thathexists over the CB surfaces and that the inlet and outlet lubricants are at the maximum position ofh. The ambient pressure around the CB and the inlet and outlet pressures are equal to the atmospheric pressurep0. The computational regionΓof the CB is plotted in Fig.3. Here,iandfare the respective boundary lines of the initial and final pressures at the maximum position ofhcorresponding toφ= 0° and 360°, andrandlare the respective boundary lines of the right and left pressures of the bearing aty=0 andB.

Fig.3 CB computational region and meshing diagram

The boundary conditions of the oil film pressure can be written as

(19)

In general, the oil film in the small gap of the CB can usually be divided into the liquid and cavitation zones when the oil film is ruptured[11]. Because negative pressure due to gas dissolution and cavitation effects exists, Eq.(17) above cannot be used to calculate the cavitation region[18-19]. Thus, instead of ignoring negative pressures in the cavitation zone, only pressures below the saturation pressurepsare varied to achieve the effective pressurepe[15]. The oil film pressure in the cavitation zone can then be expressed as

(20)

1.2.3 Lubrication forces of mixed hydrodynamics

Under the impact ofW0on the CB, the LBC of the oil film pressure generated inΓcan be determined by[16]

(21)

where load bearing capacityWequals the dynamic loadW0on the CB.

Fgenerated from the interfacial shear stressτacting on the shaft inΓcan be calculated by[17]

(22)

WhenWandFare obtained, theμof the CB can be given by[15]

(23)

In this study, theW,F,μ, andεbetween the shaft and bearing are selected as indices to evaluate the LE-FPL of the engine.

2 Algorithm

An algorithm written in MATLAB was used to estimate the objective functions and solve the equations of the system models. The simulation process includes three main steps as follows.

Step1The initial parameters that must be set include the dimensions of the SCM and CB, the combustion pressure on the piston peak, the zero matrices of the oil film pressure and shear stress (a×b), and the zero matrices of the force vectors (1×b). TheW0and coefficients of Eq.(18) are then calculated, and the initial pressure and shear stress matrices are also determined. Herein, the number of grid nodes isa=b=120.

Step2When the pressure and shear stress have been determined, the initial LE-FPL can be obtained. In theory,WequalsW0. However, in actual conditions,Wis not in equilibrium withW0; thus, the computational algorithm cannot give the stopping condition for the computational process. This issue is addressed by setting a stopping condition ofW0-W0≤ζwith a very small value ofζ. Reaching the stop condition means that the difference betweenWbandW0is very small. In this case, the distribution of the oil film pressure and shear stress inΓandεbetween the journal and bearing are considered acceptable, and the computational algorithm is stopped. The computational algorithm is continuously performed until the stopping condition is satisfied.

Step3The time required to complete a cycle of the engine corresponding to a crankshaft rotation angle of 720° ist0. Parametert0is divided into 120 equal parts, andtrefers to each time node corresponding to a crankshaft rotational angle oft′=6°. The oil film pressure and shear stress are calculated at each timet, and the computational algorithm is continued for the next loop att=t+t′ untilt≥t0. The complete computational algorithm and objective functions of the LE-FPL are finally obtained. A schematic of the calculation process is provided in Fig.4.

Fig.4 Flowchart of the algorithm of the computational model

3 Simulation Results and Analysis

The necessary parameters for simulation listed in Tab.1 and the combustion gas pressure acting on the piston peak, which is obtained from the experimental data in Ref.[14], as shown in Fig.5(a), were applied to calculateW0under variousωand analyze the LE-FPL. TheW0results are plotted in Fig.5(b).

As shown in Fig.5(b), the impact load is constantly varied under differentωand rotation angles. As the crankshaft rotational angle moves from 360° to 410°, corresponding to the combustion stroke of the engine, the maximum and minimum values ofW0are 2 000 and 6 000 r/min, respectively. However, outside the rotational angles of 360°- 410°, the maximumW0is 6 000 r/min. This finding may be attributed to the effect of theFicof the large-rod-end of the connecting rod andωin Eq.(8). Thus, the LE-FPL of the CB is affected by the operating parameters of the CB. The effect ofω,rb, andBof the CB are investigated in the next subsections.

Tab.1 Parameters of the SCM and CB

(a)

(b)

3.1 Effect of engine speed

The relative speed between the shaft and bearing surfaces isu0=ωrb; thus,ωof 1 000, 2 000, 4 000, and 6 000 r/min with the correspondingW0are applied to study their effects on the LE-FPL of the CB, as shown in Fig.5(b).

The simulation results of the oil film pressure and shear stress distributions on the CB surfaces under differentW0at 2 000 r/min are shown in Fig.6. Thepvalues are mainly distributed over the range of 90°-180° withB, and the maximumpat 158° is 88.8 MPa. This finding is identical to the results presented in Refs.[1,14-15]. The shear stress is also uniformly distributed alongB.

(a)

(b)

However, in the circumferential direction, the shear stress remarkably varies and peaks at 178°.

The oil film pressure and shear stress distributions at a bearing width ofB/2 with variousωare illustrated in Fig.7. Fig.7(a) shows that the oil film pressure strongly depends onW0andω. The maximum and minimum pressures corresponding toW0are also obtained at 2 000 and 6 000 r/min, and the results are shown in Fig.5(b). The oil film shear stress mainly depends onω, and the shear stress increases with increasingωand vice versa.

(a)

(b)

The CB’s load bearing capacityWandFare calculated from the results in Fig.7 and plotted in Figs.8(a) and (b). TheWshown in Fig.8(a) is similar to theW0in Fig.5(b) under differentωbecause the oil film pressure varies to satisfy the condition thatWis equivalent toW0. Fig.8(b) reveals thatFincreases sharply with increasingω. Moreover,Fpeaks at a highωof 6 000 r/min, which means that theFacting on the CB is greatest at this point. This result may be due to the increase in relative velocity between the shaft and the bearing surface, which increases the shear stress and friction resistance of the oil film.

(a)

(b)

(c)

(d)

Fig.8(c) presents theμof the CB. The values ofμchange with the variation ofWandFand decrease with increasingω. Comprehensive comparative analysis shows that, in the combustion stroke of the engine (i.e., a crankpin angle of 360°-410°), the maximumW0affects not only the CB’s LE but also its engine power. In particular, whenωis maintained at 2 000 r/min,W0peaks, butFandμremain relatively small. Thus, this is also a reason that most engines should work at this speed[1, 6].

The effect ofεbetween the journal and bearing is also investigated to analyze the stability of the CB under variousω, as shown in Fig.8(d). The results indicate that increasingωcan reduceεand vice versa. Moreover,εpeaks at a lowωof 1 000 r/min, and its minimum value occurs at a highωof 6 000 r/min. Therefore,εcould also affect the LE and stability of the CB.εshows less variation whenωis 6 000 r/min than at other speeds, which means that the lubrication capacity and the rotation of the journal are fairly stable at this point.

3.2 Effect of bearing radius

Considering the analysis above, the maximumW0atωof 2 000 r/min is selected to investigate the effect ofrbandBon the LE-FPL of the engine.

The simulation results of oil film pressure and shear stress distributions at a bearing width ofB/2 andrb=15, 20, 25, 30, 35 mm are plotted in Figs.9(a) and (b). The results show that the oil film pressure decreases, whereas the shear stress increases asrbgradually increases from 15 to 35 mm.

(a)

(b)

Figs.10(a) and (b), respectively, reflect changes inWandFunder differentrb. Changes inrbappear to have less effect onWin comparison with that onW0under the condition of equivalentWandW0, as illustrated in Fig.10(a). This result may be attributed to an increase inΓdue to the increase inrb, which leads to a decrease in oil film pressure, as shown in Fig.9(a), to stabilizeWand vice versa. This finding also verifies the accuracy and robustness of the mathematical model and algorithm developed in this study.

(a)

(b)

(c)

(d)

Fig.10(b) reveals thatFtends to increase with increasingrb. This finding may be attributed to increases in the relative speed of the oil film (u0=ωrb) andΓaffecting the oil film shear stress and friction resistance. BecauseWdoes not vary with differentrb,μincreases because of the increase inFof the CB, as shown in Fig.10(c). Moreover, as seen in Fig.10(d), the influence ofrbonεis not significant, and the gap between the shaft and bearing of the CB shows an obvious tendency to decrease with increasingrb.

3.3 Effect of bearing width

Fig.11 demonstrates the influence ofB=10, 15, 20, 25, 30 mm on the LE-FPL under a constantW0at 2 000 r/min. Fig.11(a) shows that theWgenerated from the oil film pressure does not change significantly in comparison withW0asBis gradually increased from 10 to 30 mm, likely because the oil film pressure changes to stabilizeW.Fandμalso tend to increase, as shown in Figs.

(a)

(b)

(c)

(d)

11(b) and (c), which directly causes greater FPL. However,Bvariations have little effect on theεbetween the journal and bearing[6, 16-17], as indicated in Fig.11(d).

4 Conclusions

1) Increasingωnot only effectively reduces theεbetween the journal and bearing but also increases the minimumh, thereby directly improving the CB LE. However, increases inωalso increaseF, thereby increasing the FPL of the engine. The simulation results show that the engine’s LE-FPL can be improved better whenωis maintained at 2 000 r/min.

2) Increasing therborBof the CB can help decreaseε, thereby improving the LE. In addition, the dimensional change of the CB has a great influence onFandμ, which, in turn, increase the FPL of the engine.

3) Improving LE while reducing FPL is a challenging issue. Therefore, the present study not only contributes to the existing body of knowledge on the LE-FPL of automotive engines but also provides an important reference for optimal SCM design parameters to improve this engine property further.