1 Introduction

The connecting-rod or main bearing is one of main frictional pairs in engine. The operating condition of connecting-rod or main bearing affects directly the operational economy, reliability, durability and lifetime of engine. The capability and rationality of design are one of most essential factors to determine the performance of connecting-rod or main bearing. By the unremitting efforts of researchers and the progress of correlation technique, current design capability of engine connecting-rod and main bearing is improved continually, the factors considered in the design calculation of bearing are perfected increasingly, and the performance of bearing is predicted more accurately. Many researches on the lubrication of connecting-rod or main bearing have been done. For example, Liu analyzed the transient tribodynamic performance of crankshaft-main bearing system in engine starting up [1]. Zammit investigated the lubrication of crankshaft bearing during engine warm-up [2]. Inui, Jia and Galera et al. [3,4,5] studied the factors affecting the lubrication of crankshaft bearing. Gu et al. [6] researched the performance of textured crankshaft bearing during an engine cycle. Mohammadpour, Mahdi, and Liu et al. [7,8,9] analyzed the effect of cylinder deactivation on the tribo-dynamic-acoustic and thermo-hydrodynamic characteristics of crankshaft bearing. Francisco, Lavie and Zhang et al. [10,11,12] studied the optimization of crankshaft bearing lubrication. Ozasa et al. [13] proposed a simplified estimation method of engine bearing lubrication. Shahmohamadi et al. [14] analyzed the mixed thermo-hydrodynamic lubrication of engine connecting rod bearing. Refs. [15,16,17,18] study the thermo-elastohydrodynamic lubrication of engine crankshaft bearing. Tabrizi et al. [19] compared three kinds of simulation models of connecting rod bearing. Zhang et al. [20] researched the mixed lubrication performance of high power-density engine main bearing. Toshihiro et al. [21] analyzed the lubrication of gasoline engine connecting rod bearing. Bi and Chen et al. [22, 23] studied the lubrication of turbocharged engine main bearing. Choi et al. [24] analyzed the dynamically bearing elastohydrodynamic lubrication considering multi-flexible-body dynamics. Wei et al. [25,26,27] investigated the thermo-elasto-hydrodynamic mixed lubrication of marine engine main bearing. Shao et al. [28] analyzed the lubrication performance of engine main bearing coupling cylinder block and crankshaft-connecting rod system. Zhao et al. [29] analyzed the lubrication performance of marine engine main bearing in typical operating conditions. Yang et al. [30] analyzed the lubrication of gasoline engine main bearing considering transient heat transfer. However, only the lubrication performance of connecting-rod and main bearing in rated engine operating condition was generally analyzed in current design and research. In actual use, the engine (especially used in vehicle) does not always operate in rated operating condition and its operating condition changes constantly. Therefore, it is necessary to study the lubrication performance of engine connecting-rod and main bearing in different operating conditions, which is helpful to perfect the lubrication analysis theory of engine bearing and can provide more comprehensive reference base for the design of connecting-rod and main bearing.

In this paper, a four-stroke four-cylinder engine is taken as the studying object, the loads and lubrication characteristics of its connecting-rod bearing and all main bearings in different engine operating conditions are calculated, and the effects of engine operating condition on the lubrication performances of connecting-rod and main bearings are researched. The parameters of the engine and its bearings are shown in Table 1.

Table 1 Parameters of engine and bearings
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2 Calculation of the Loads of Bearing in Different Engine Operating Conditions

The load of connecting-rod bearing is calculated by the dynamic calculation method [31], which is obtained by analyzing the applied forces on the engine crank-rod mechanism based on the engine cylinder pressure indicator diagram (some shown in Figure 1) measured in engine bench test and configuration parameters. The loads of all main bearings are calculated by the finite element method of whole crankshaft beam-element [31]. The beam-element finite element model of whole crankshaft is shown in Figure 2.

Figure 1
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Indicator diagram

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Figure 2
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Beam-element finite element model of whole crankshaft

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The loads of connecting-rod bearing and No. 2 main bearing under full engine load at 1200 r/min and 3200 r/min are shown in Figures 3 and 4, respectively. There are great differences of the change rule and numerical value of bearing load at corresponding moment in an engine operating cycle in different engine operating conditions.

Figure 3
figure 3

Load of connecting-rod bearing

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Figure 4
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Load of No. 2 main bearing

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The maximum loads of connecting-rod bearing and all main bearings in an engine operating cycle under full engine load at 1200, 1600, 1800, 2000, 2200, 2400, 2800 and 3200 r/min are shown in Table 2. The maximum load of connecting-rod bearing in an engine operating cycle reduces with the increase of engine speed under same engine load.

Table 2 Maximum loads of connecting-rod bearing and all main bearings at full load and different speed
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The maximum value of the maximum load of connecting-rod bearing in an engine operating cycle appears at low engine speed (1200 r/min), and the numerical value of the maximum load of connecting-rod bearing in an engine operating cycle at high engine speed (3200 r/min, rated engine operating condition) is small. The main reason is that the load of connecting-rod bearing is decided by the actual condition of the movement and applied force of engine crank-rod mechanism. The maximum load of connecting-rod bearing in an engine operating cycle appears near to the combustion top dead center. Based on the applied force analysis of engine crank-rod mechanism, the load of connecting-rod bearing is the resultant force of the forces resulted from the cylinder pressure, the reciprocating inertial force and the rotational inertial force. The reciprocating inertial force and the rotational inertial force will be increased with the increase of engine speed, but the acting directions of the reciprocating inertial force and the rotational inertial force at the combustion top dead center are opposite to the direction of force resulted from the cylinder pressure, so the higher the engine speed the more the reduction of the maximum load of connecting-rod bearing. The change situation of the maximum load of one main bearing in an engine operating cycle against the engine speed is different from that of another main bearing under same engine load. The loads of No. 1 and No. 5 main bearing are affected mainly by the force on single crank, and the maximum loads of No. 1 and No. 5 main bearing happen at the lower engine speed (1200 r/min). The loads of No. 2, No. 3 and No. 4 main bearing are decided by the forces of adjacent two cranks, and the maximum loads of No. 2, No. 3 and No. 4 main bearing happen at the higher engine speed.

The maximum loads of connecting-rod bearing and all main bearings in an engine operating cycle at 2200 r/min under 20%, 40%, 60%, 80% and full engine load are shown in Table 3. Under the same engine speed, the maximum loads of connecting-rod bearing and all main bearings in an engine operating cycle increase basically with the increase of engine load, and the maximum loads of connecting-rod bearing and all main bearings in an engine operating cycle take place at full engine load.

Table 3 Maximum loads of connecting-rod bearing and all main bearings at 2200 r/min and different load percentage
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3 Lubrication Analysis of Connecting-Rod and Main Bearing in Different Engine Operating Condition

3.1 Method and Formulation

The journal axis orbits of all bearings are calculated by the dynamic method in the lubrication analyses of all connecting-rod or main bearings [32].

3.1.1 Reynolds’ Equation [33]

$$\frac{\partial }{\partial \theta }\left( {h^{3} \frac{\partial p}{\partial \theta }} \right) + R_\text{b}^{2} \frac{\partial }{\partial y}\left( {h^{3} \frac{\partial p}{\partial y}} \right) = 6\eta R_\text{b} \left( {u\frac{\partial h}{\partial \theta } + 2R_\text{b} \frac{\partial h}{\partial t}} \right),$$
(1)

where p is the oil film pressure, h is the oil film thickness, η is dynamic viscosity of lubricating oil, u = uj + ub, uj is the velocity of journal surface and uj = Rjωj, Rj is the journal radius, ωj is the angular velocity of journal, ub is the velocity of bearing surface and ub = Rbωb, Rb is the bearing radius, ωb is the angular velocity of bearing.

Reynolds’ equation is solved by the finite difference method.

3.1.2 Oil Film Thickness [34]

$$h = c + e\cos (\theta - \psi ) + \delta ,$$
(2)

where c is the radial clearance of bearing, e is the journal eccentric distance of bearing, ψ is the attitude angle of bearing, δ is the change of oil film thickness caused by elastic deformation of bush surface of bearing under oil film pressure, and the elastic deformation of bush surface of bearing under oil film pressure is calculated by the compliance matrix method.

3.1.3 Load Equilibrium Equation

If the effect of the oil film inertia is not considered, the motion of the journal axes of bearing conforms to the Newton second law, that is,

$$\varvec{P} + \varvec{F} = m_{\text{j}} \frac{{{\text{d}}\varvec{v}}}{{{\text{d}}t}},$$
(3)

where P is the load vector of bearing, F is the resultant oil film force vector of bearing, v is the velocity vector of journal axes.

3.1.4 End Leakage Flow-Rate of Bearing

The lubricating oil flow-rate Q1 from the front-end plane of bearing and the lubricating oil flow-rate Q2 from the rear-end plane of bearing are given by

$$Q_{1} = - \int_{0}^{2\pi } {\frac{{h^{3} }}{12\eta } \cdot \left. {\frac{\partial p}{\partial y}} \right|_{y = 0} \cdot R_\text{b} \text{d}\theta } ,$$
$$Q_{2} = - \int_{0}^{2\pi } {\frac{{h^{3} }}{12\eta } \cdot \left. {\frac{\partial p}{\partial y}} \right|_{y = L} \cdot R_\text{b} \text{d}\theta } .$$

The total end leakage flow-rate Q of the lubricating oil is then given by

$$Q = \left| {Q_{1} } \right| + \left| {Q_{2} } \right|.$$
(4)

3.1.5 Frictional Coefficient and Average Frictional Power Loss of Bearing

The frictional force on the journal surface Fj can be calculated from

$$F_\text{j} = \int_{0}^{L} {\int_{0}^{2\pi } {\left( {\frac{h}{2}\frac{\partial p}{{R_\text{j} \partial \theta }} + \frac{u\eta }{h}} \right)R_\text{j} \text{d}\theta \text{d}y} } .$$

The frictional coefficient of the journal surface μj is then given by

$$\mu_\text{j} = \frac{{F_\text{j} }}{F}.$$
(5)

The average frictional power loss of bearing in an engine operating cycle is then given by

$$w_\text{m} = \sum\limits_{i = 1}^{720} {(F_\text{j} )_{i} u} /720.$$
(6)

3.2 Results and Discussion

The journal axes orbits, maximum oil film pressures, minimum oil film thicknesses, end leakage flow-rates and frictional coefficients of connecting-rod bearing and No. 2 main bearing in an engine operating cycle under full engine load at 1200 r/min and 3200 r/min are shown in Figures 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. In different engine operating conditions, there are major differences of the lubrication performance of bearings, and there are obvious differences in the changes and numerical values of the journal axes orbits, maximum oil film pressures, minimum oil film thicknesses, end leakage flow-rates and frictional coefficients of bearings in an engine operating cycle.

Figure 5
figure 5

Journal axis orbit of connecting-rod bearing

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Figure 6
figure 6

Journal axis orbit of No. 2 main bearing

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Figure 7
figure 7

Maximum oil film pressure of connecting-rod bearing

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Figure 8
figure 8

Maximum oil film pressure of No. 2 main bearing

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Figure 9
figure 9

Minimum oil film thickness of connecting-rod bearing

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Figure 10
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Minimum oil film thickness of No. 2 main bearing

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Figure 11
figure 11

End leakage flow-rate of connecting-rod bearing

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Figure 12
figure 12

End leakage flow-rate of No. 2 main bearing

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Figure 13
figure 13

Frictional coefficient of connecting-rod bearing

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Figure 14
figure 14

Frictional coefficient of No. 2 main bearing

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The maximum oil film pressures, minimum oil film thicknesses and average frictional power losses of connecting-rod bearing and all main bearings in an engine operating cycle under full engine load and at 1200, 1600, 1800, 2000, 2200, 2400, 2800 and 3200 r/min respectively are shown in Tables 4, 5, 6.

Table 4 Maximum film pressures of connecting-rod bearing and main bearings at full load and different speed
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Table 5 Minimum film thicknesses of connecting-rod bearing and main bearings at full load and different speed
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Table 6 Average friction power losses of connecting-rod bearing and main bearings at full load and different speed
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When the engine load is same, the maximum oil film pressure of connecting-rod bearing in an engine operating cycle is decreased generally with the increase of engine speed, and the maximum oil film pressure (372.52 MPa) of connecting-rod bearing at 1200 r/min is as 4.02 times as that (92.74 MPa) at 3200 r/min, which shows that the maximum oil film pressure of connecting-rod bearing at lower engine speed is larger remarkably than the one at higher engine speed under the same engine load. The main reason why the maximum oil film pressure of connecting-rod bearing at lower engine speed is larger than the one at higher engine speed under the same engine load is that, when the engine load is same, the maximum load of connecting-rod is decreased basically with the increase of engine speed, and the maximum load of connecting-rod at 1200 r/min is larger remarkably than the one at 3200 r/min.

When the engine load is same, the changes of the maximum oil film pressures of all main bearings in an engine operating cycle is different one another with the change of engine speed. The maximum oil film pressures of No. 1 and No. 5 main bearings do not have obvious change at different engine speed, the maximum oil film pressures of No. 2 and No. 4 main bearings are larger at lower engine speed, and the maximum oil film pressure of No. 3 main bearings is larger at higher engine speed.

When the engine load is same, the minimum oil film thicknesses of connecting-rod bearing and all main bearings in an engine operating cycle do not have the same change one another with the change of engine speed. The minimum oil film thicknesses of connecting-rod bearing and all main bearings in an engine operating cycle at higher engine speed (3200 r/min) are generally smaller in addition to the individual circumstances.

When the engine load is same, the average frictional power losses of connecting-rod bearing and all main bearings increase with the increase of engine speed, and the average frictional power losses of connecting-rod bearing and all main bearings are the largest at higher engine speed (3200 r/min).

The maximum oil film pressures, minimum oil film thicknesses and average frictional power losses of connecting-rod bearing and all main bearings in an engine operating cycle at 2200 r/min under 20%, 40%, 60%, 80% and full engine load are shown in Tables 7, 8, 9.

Table 7 Maximum film pressures of connecting-rod bearing and main bearings at 2200 r/min and different load percentage
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Table 8 Minimum film thicknesses of connecting-rod bearing and main bearings at 2200 r/min and different load percentage
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Table 9 Average friction power losses of connecting-rod bearing and main bearings at 2200 r/min and different load percentage
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When the engine speed is same, the maximum oil film pressures of connecting-rod bearing and all main bearings in an engine operating cycle are increased generally with the increase of engine load, but the maximum value of the maximum oil film pressures of No. 3 main bearing is appeared at the smaller engine load (40%).

When the engine speed is same, the minimum oil film thickness of connecting-rod bearing is decreased with the increase of engine load, and the maximum value appears at the full engine load (100%). The minimum oil film thicknesses of all main bearings do not have the same change with the change of engine load, the minimum values of the minimum oil film thicknesses of Nos. 1, 4, 5 main bearings all appear at the full engine load (100%), and the minimum values of the minimum oil film thicknesses of Nos. 2, 3 main bearings appear at the smaller engine load.

When the engine speed is same, the average frictional power losses of connecting-rod bearing and all main bearings in an engine operating cycle do not change obviously with the change of engine load. The average frictional power losses of connecting-rod bearing and all main bearings in an engine operating cycle are generally increased slightly with the increase of engine load in addition to the individual engine loads.

In addition, the corresponding comparisons between the lubrication performance of connecting-rod bearing and the one of main bearings at same engine operating condition (shown in Tables 4, 5, 6, 7, 8, 9) show that, the maximum oil film pressure of connecting-rod bearing in an engine operating cycle is larger than those of all main bearings, the minimum oil film thickness of connecting-rod bearing in an engine operating cycle is smaller than those of all main bearings, and the average frictional power losses of connecting-rod bearing in an engine operating cycle is smaller than those of all main bearings. Also, there is the corresponding difference between the lubrication performances (the maximum oil film pressure, minimum oil film thickness and average frictional power loss in an engine operating cycle) of all main bearings one another at same engine operating condition, and some difference is larger.

4 Conclusions

  1. (1)

    In different engine operating conditions, there are great differences of the loads of connecting-rod and main bearings. When the engine load is same, the maximum load of connecting-rod bearing appears at the lower engine speed, the maximum loads of some main bearings appear at the lower engine speed but the others appear at the higher engine speed. When the engine speed is same, the maximum loads of connecting-rod bearing and all main bearings appear at the full engine load.

  2. (2)

    In different engine operating conditions, there are obvious difference of the lubrication performance of connecting-rod and main bearing.

    When the engine load is same, the maximum oil film pressure of connecting-rod bearing at lower engine speed is larger obviously than the one at higher engine speed; the change of the maximum oil film pressure of each main bearing with the engine speed is not the same; the smaller values of the minimum oil film thicknesses of connecting-rod bearing and all main bearings appear generally at higher engine speed; the maximum average frictional power losses of connecting-rod bearing and all main bearings happen at higher engine speed.

    When the engine speed is same, the maximum oil film pressures of connecting-rod bearing and main bearings increase generally with the increase of engine load; the minimum oil film thickness of connecting-rod bearing happens at the full engine load, the minimum oil film thicknesses of some main bearings happen at the full engine load but some other happen at the lower engine load; the average frictional power losses of connecting-rod bearing and all main bearings do not have evident change with the change of engine load.

  3. (3)

    Under the same engine operating condition, the maximum oil film pressure of connecting-rod bearing is larger than the one of main bearing; the minimum oil film thickness of connecting-rod bearing is smaller than the one of main bearing; the average frictional power loss of connecting-rod bearing is smaller than the one of main bearing.

  4. (4)

    There are the larger difference between the lubrication performances of main bearings one another under same engine operating condition.

Therefore, it is not comprehensive and reasonable that only the lubrication performance of a certain connecting-rod bearing or main bearing at rated engine operating condition is analyzed. It is necessary to analyze simultaneously the lubrication performance of connecting-rod bearing and all main bearings in different engine operating conditions in the design of engine.