Analysis of Diesel Engine Main Bearing Lubrication Under Single-cylinder Misfiring Situation

2.1 Lubrication model

In the form of average Reynolds equation with oil filing ratio [20]:

$-\frac{\partial}{\partial \mathrm{x}}\left(\cdot \bar{\theta} \cdot \varphi_{x} \cdot \frac{h^{3}}{12 \eta} \cdot \frac{\partial \overline{\mathrm{p}}}{\partial x}\right)-\frac{\partial}{\partial \mathrm{y}}\left(\bar{\theta} \cdot \varphi_{y} \cdot \frac{h^{3}}{12 \eta \cdot} \frac{\partial \overline{\mathrm{p}}}{\partial \mathrm{y}}\right)+  \partial\left(\bar{\theta} \cdot\left(\bar{h}_{T}+\sigma \cdot \varphi_{s}\right) \cdot \frac{u_{1}+u_{2}}{2}\right)\frac{\partial x}{\partial \mathrm{t}}+\frac{\partial\left(\bar{\theta} \cdot \bar{h}_{T}\right)}{\partial x}=0$       (1)

Where: $\bar{\theta}$ was referred to as the oil filling rate, that was, the ratio of the oil film thickness of shaft journal and bearing shell and the clearance height of shaft journal and bearing shell in the bearing friction pair; p was oil film pressure; η was the dynamic viscosity of lubricating oil; u1 was the circumferential velocity of shaft journal; u2 was the circumferential velocity of bearing shell; t was the time; h was the oil film thickness; x and y represented the peripheral direction of bearing; z was the axial direction of the bearing; $\bar{h}_{T}$ was the average value of actual oil film thickness; $\varphi_{S}$ represented the shear flow factor and; .. was the standard deviation of the comprehensive contact surface roughness.

To solve, the following boundary conditions were adopted:

(1)Axial boundary conditions:  

$\bar{p}=p_{a}\left(y=\pm \frac{B}{2}\right)$ 

(2) Circumferential boundary conditions:

(i) Oil film starting side: as $\theta=0$ ,

$\left\{\begin{array}{l}p=p_{n}\left(|x| \leq \frac{1}{2}\right) \\ p=p_{n} \frac{B-2|x|}{B-1}\left(\frac{1}{2}<|x| \leq \frac{B}{2}\right)\end{array}\right.$

(ii) Oil film ending side: as $\theta=\frac{y}{r}, \frac{\partial p}{\partial \theta}=0$ and $p=p_{a}$

(3) Oil supplying area: $\bar{p}=p_{\text {in }}$

(4) Cavitation boundary conditions:  

$\left\{\begin{array}{l}\bar{p}>0(\bar{\theta}=1) \\ \bar{p}=0(0<\bar{\theta}<1)\end{array}\right.$

2.2 Contact model

In this paper, the asperity contact model proposed by Greenwood and Tripp was used. In consideration of elastic deformation of the contact body, the contact force was expressed as:

$p_{a}=k \sqrt{\frac{\sigma_{s}}{\beta}} E^{*} F_{5 / 2}\left(h_{S}\right)$      (2)

In which:

$E^{*}=\frac{1}{\left(\frac{1-v_{1}^{2}}{E_{1}}+\frac{1-v_{2}^{2}}{E_{2}}\right)}$

$F_{5 / 2}=\left\{\begin{array}{ll}4.408610^{-5}\left(4-H_{s}\right)^{6.804}, H_{s}<4 \\ 0     , H_{s} \geq 4\end{array}\right.$

$k=\frac{16 \sqrt{2} \pi}{15}\left(\sigma_{s} \bar{\beta} \eta_{2}\right)^{2}$

$\sigma_{s}=\sqrt{\frac{1}{\eta_{s}} \sum_{q=1}^{\eta_{s}} \delta_{s}^{2}}$ ;

In this formula, $F_{5 / 2}\left(H_{s}\right)$  was the function of oil film thickness, that was, to judge whether the bearing was in the state of elastohydrodynamic contact. $E^{*}$ was the overall elasticity modulus. $E_{1}$ and $E_{2}$ were the elasticity modulus of bearings and shaft journal materials, respectively. $v_{1}$ and $v_{2}$ were the Poisson’s ratios of bearings and shaft journal materials, respectively. $k$ was the elastic contact factor and related to the roughness of the materials. Generally, it was difficult to determine and usually in the range $0.0003 \leq k \leq 0.003$ .$\sigma_{s}$ was the comprehensive roughness.

2.3 Flexible multi-body system dynamics model

The motion of the flexible body meets Lagrange equation:

$\left\{\begin{array}{c}\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{\xi}}\right)-\frac{\partial L}{\partial \xi}+\frac{\partial \Gamma}{\partial \dot{\xi}}+\left[\frac{\partial \Psi}{\partial \xi}\right]^{T} \lambda-Q=0 \\ \Psi=0\end{array}\right.$      (3)

In which, $L$ represented the Lagrange equation. Here, $L=T-W$ , where $T$ was kinetic energy and $W$ was potential energy. $\boldsymbol{\Psi}$ represented the constraint equation. $\xi$ represented generalized coordinates. $\lambda$ represented the Lagrange multiplier of the constraint equation. $\boldsymbol{Q}$ was the force of the generalized coordinate system $\xi$  $\Gamma$ represented the energy loss function.

Finally, the differential equation of motion of a flexible body might be written as:

$M \ddot{\xi}+\dot{M} \dot{\xi}-\frac{1}{2}\left[\frac{\partial M}{\partial \xi} \dot{\xi}\right]^{T} \dot{\xi}+K \xi+f_{g}+D \dot{\xi}+\left[\frac{\partial \Psi}{\partial \xi}\right]^{T} \lambda=Q$        (4)

In the formula, $\xi$ represented generalized coordinates. $\boldsymbol{M}$ was the mass matrix. $\frac{\partial \boldsymbol{M}}{\partial \boldsymbol{\xi}}$ was the partial derivative of mass matrix to generalized coordinates. $K$ represented the stiffness matrix. $f_{g}$ was the inertial force. $D$ was the damping matrix. $M$ was the modal number.

2.4 Single-cylinder pressure as misfires

With regard to the lubricating properties of main bearings with single-cylinder misfires, related research work has previously been carried out. Typically, a misfired cylinder was processed as follows: it is assumed that the pressure of the misfired cylinder was opened by the indicator valve of the misfired cylinder to maintain atmospheric pressure. Pressure of the misfired cylinder was handled as follows: there was no fuel burning in the misfire cylinder, but the gas distribution mechanism and crank link mechanism were in normal working states. That is, in the cylinder, the pressure varied with crank angle and the gas still went through the processes of inhaling, compression, expansion and exhaling. Since there was no fuel burning, the mean effective pressure of the misfired cylinder had a negative value; work was consumed rather than created by the misfired cylinder due to mechanical losses.

4.1 Loading conditions

Now, an inline six cylinder was taken as a research object to analyze the lubrication of main bearings through coupling calculation of dynamics and tribology. Diesel engine parameters are shown in Table 1. Data from indicator diagram with 2,000 r/min rated speed was selected as the initial load boundary conditions, as shown in Fig 1. 0°CA of crank angle means Cylinder I was located on the compression Top Dead Center (TDC). Single-cylinder pressure during misfires was computed by conditions and parameters of the diesel engine in 1.4, as shown in Fig. 1.

Table 1. Parameters of diesel engine

1.png

Figure 1. Cylinder Pressure Diagram Under Single-cylinder Misfiring Situation

It can be seen from Fig 1 that as the single cylinder misfired, the pressure in the combustion chamber of the misfired cylinder experienced compression-expansion process under the motion of gas distribution mechanism and crank link mechanism. However, since there was no fuel combustion process, the maximum pressure of the combustion chamber was lower than under normal ignition. During normal ignition, the maximum pressure was shown after compression top dead center due to fuel combustion. But, the highest pressure point was just on the compression top dead center because rising pressure of the misfired cylinder was only led by gas compression.

4.2 Dynamics and lubrication calculation

A complete engine body and crankshaft was selected as a system for solution, for the purpose of more accurately describing the stress and motion of the crankshaft. Firstly, the following operation should be done: three-dimensional geometric model, mesh generation, and finite element model. Finite element models of the crankshaft and engine body were made as shown in Fig 2. Then, modal was reduced by modal reduction technology and the multi-body dynamic and tribology coupled solution model was established based on substructure condensation, as shown in Fig 3. X axis was the rotation axis of the crankshaft and direction of free end towards flywheel was set to be positive. Z axis was parallel to the center line of the cylinder. Dynamic coupling units of crankshaft main journal, rod journal, flywheel, and shock absorber were established respectively to constrain the degrees of freedom of coupling nodes to six directions. The internal surface of the bearing shell of the engine body was used to constrain the translational degrees of freedom in the Y and Z directions. Primary and secondary thrust surfaces of the cylinder sleeve were used to constrain translational degrees of freedom to the Y and Z directions. Thrust bearing was used to constrain translational degrees of freedom to the X direction. Bottom of firepower surface of cylinder cover was used to constrain translational degrees of freedom to the Z direction.

The parameters for solving oil film lubrication were as shown in Table 2.

Table 2. Lubrication calculation parameters

4.3 Results analysis

Table 2 showed the maximum bear force for each main bearing. Figure 4 showed the graph of bear forces for each main bearing under rated conditions and of Cylinder I misfired. From the table and the figure, we concluded that:

(1)The maximum bear force decreased when one cylinder misfired;

(2)As for rated conditions, the bear force of the first and second main bearing had certain difference compared with first cylinder misfire conditions; the difference percentage between rated conditions and cylinder 1 misfire conditions of bearing 1 and bearing 2 were 27.22 % and 13.82 % respectively.

(3)Single cylinder misfire had a greater influence on the loads and axis orbits of the two adjacent main bearings, but less effect on the loads and axis orbits of other main bearings.