A new method calculating load balance of sliding bearing by using neural network PID algorithm

2.1. Oil film pressure solution

The dynamic pressure radial sliding bearing is simplified, and the oil film thickness calculation is shown in Fig. 1.

In Fig. 1, $O_b$ is the center of the bearing, $O_j(x,y)$ donates the center of the journal. $R$ represents the bearing radius, $r$ is the journal radius, $e$ donates the eccentricity, $\varphi$ is the position angle, and $h$ is the oil film thickness at point A, $\theta$ donates shaft rotation angle. According to the geometric relationship, the expression of the oil film thickness is $h=C+x\mathrm{c}\mathrm{o}\mathrm{s}\theta +y\mathrm{s}\mathrm{i}\mathrm{n}\theta$, where $C=R-r$ donates the radius gap, the eccentricity is $\epsilon =e/C.$ The relationship between axial coordinate and the eccentricty position and angle is:

Under certain assumptions, unsteady Reynolds equations [5] can be obtained in an incompressible (invariant), laminar state (without considering turbulence effects) and viscosity does not change with pressure:

where $p$ donates oil film pressure, $z$ donates width of bearing, ${\omega }_0$ represents journal angular velocity, $\eta$ is oil viscosity. By using the finite difference method and substituting the boundary conditions to solve the Reynolds equation, the pressure distribution of the bearing oil film can be obtained.

Fig. 1Oil film thickness diagram of dynamic pressure sliding bearing

2.2. Bearing oil film joint force solution

Through the known oil film pressure, the resultant oil film force can be integrated, and the oil film joint force is the embodiment of bearing bearing capacity. The oil film pressure is integrated in the x and y directions to obtain the oil film force $F_x$ and $F_y$:

The discretization of the oil film force is solved, and the double integral is converted into the cumulative summation for calculation. The component in the x direction is $P(i,j)\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_i\right)$, and the component in the $y$ direction is $P(i,j)\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_i\right)$. Therefore, the oil film force component discretization summation formula is in the form:

where $\lambda$ is a coefficient whose value is 1/4 of the vertices of the summation region, 1/2 for the boundary, and 1 for the internal node. The oil film resultant force is: $F=\sqrt{{\stackrel{-}{F}}_x^2+{\stackrel{-}{F}}_y^2}$ and the load angle is $\mathrm{t}\mathrm{a}\mathrm{n}\phi ={\stackrel{-}{F}}_y/{\stackrel{-}{F}}_x$.

3.1. Calculation process

Under the condition of load and other parameters, a set of pressure solutions that reach the convergence condition is calculated by the Reynolds equation iteration, and then the bearing oil film force is solved by integral, and whether the load balance condition is satisfied is judged. If the load balancing condition is not met, the bearing eccentricity needs to be modified until the resulting oil film force satisfies the load balancing condition. The calculation process is shown in Fig. 2.

The load balancing condition is $\mathrm{\Delta }\mathrm{W}=|W-F|\le {\delta }_w$, where $\mathrm{\Delta }W$ is the load error, $W$ donates the external load, and ${\delta }_w$ is the load balance tolerance. Under certain premise, the size of the oil film force is related to the eccentricity. Therefore, when solving the oil film pressure, it is necessary to find the eccentricity, so that the oil film combined force is equal to the bearing load. The difficulty of the above solution process is how to modify the eccentricity according to the load error, in order that the calculated oil film force can quickly and accurately approach the external load, which requires the algorithm to be simple and efficient, and has strong adaptability.

Fig. 2Flow chart of load balance calculation

3.2. Step-down method

The step-down method is a variable-step load balancing algorithm. The adjustment step size $\mathrm{\Delta }\epsilon$ of the fixed eccentricity is not easy to converge, so the general calculation changes the size of the step $\mathrm{\Delta }\epsilon$ according to the number of iterations. Primary selection eccentricity ${\epsilon }_0$, then adjust the eccentricity according to the load error $\epsilon$:

After $k$-round calculation, the load error is continuously reduced. At this time, the error value is further reduced by the binary law $\mathrm{\Delta }{\epsilon }_k=\mathrm{\Delta }{\epsilon }_{k-1}/2$ until the convergence accuracy is satisfied. This method is simple, but the problem is how to choose the initial eccentricity ${\epsilon }_0$ reasonably and how to determine $k$ and start the binary calculation. This method has certain limitations for variable load calculation.

3.3. Single neural network PID method

Neural network PID method has been widely used in control systems [6-8]. The typical single neural network PID control structure is shown in Fig. 3. The characteristic of the algorithm is that the adaptive control function is realized by adjusting the weight coefficients, and the weight coefficient is adjusted according to the supervised Hebb learning rule [9, 10].

Fig. 3Single neural network PID control structure diagram

In order to realize the constraint control of output error and control incremental weighting, with the idea of optimal control, a quadratic performance index is introduced in the adjustment of the weighting coefficient, and the weighting coefficient is adjusted by minimizing the output error and the control incremental weighted square sum. Set performance indicators $E\left(k\right)$ as:

where $M$, $N$ are the weighting coefficients of the output error and the control increment, respectively, and $y_d\left(k\right)$, $y\left(k\right)$ represent the reference input and controller output at time $k$.

The neuron input $u\left(k\right)$ and weight coefficients $w\left(k\right)$ are expressed as:

where ${\eta }_P$, ${\eta }_I$, ${\eta }_D$ are the learning rates for adjusting the weighting coefficients of the control parameters $P$, $I$, and $D$ respectively, and $b_0$ is the first values of the output response, and there is:

The neural network PID algorithm is applied to the sliding bearing load balance solving problem. The reference input $y_d$ in the algorithm corresponds to the bearing load $W$, the system output $y$ corresponds to the bearing oil film force $F$, the control quantity $u$ corresponds to the eccentricity $\epsilon$, and the system control object corresponds to the bearing elastohydrodynamic lubrication. model.