Open-loop control of elliptical shaft center orbit

Abstract

In order to realize the machining of elliptical holes, the article researches control simulation of shaft center movement orbit using Eulerian method. A new hydrostatic bearing and a hydraulic control system are proposed, and quantitative relation between the expected shaft center coordinates and spool displacement of electro-hydraulic servo valve is studied. With the Eulerian method, open-loop control orbit of shaft center is simulated. And the simulation results indicate that open-loop control method not only controls shaft center movement to form ellipse orbits but also has strong robustness against dynamic disturbing force or instantaneous static disturbing force. This article lays theoretical basis for machining of elliptical holes.

Introduction

As mechanical equipment power and rotating speed increase, the load strength the hole in mechanical equipment has to stand becomes increasingly high. The traditional shape of a hole is a circle, but the inner side of the circular hole has severe stress concentration, which reduces the hole’s working life. Comparing with circular holes, non-circular holes have stronger fatigue strength. Elliptical hole, such as pin hole of piston,^1,2 is one of the non-circular holes in common use. There are many traditional machining methods for the elliptical hole. The German KS corporation adopts mechanical coping apparatus to machine elliptical holes.^3,4 The experiment indicates that the machining precision of elliptical hole with that method is at micron level. Zhang and Hu ⁵ use electromagnetic actuation device for boring elliptical holes. The boring feeding is controlled by electricity in the actuation device. Gosiewski et al. ⁶ also present the machining method of non-circular holes based on the electromagnetic actuation and conduct the fine boring experiments of elliptical pin holes. Wilson et al. ⁷ and Beach ⁸ design the fast tool system for machining piston hole, and the tool system adjusts tool feed rate with piezoelectric actuator. George ⁹ also proposes a type of fast tool system based on piezoceramic material, and deformable boring bar is adopted in fast tool system. By controlling coil current to change the length of giant magnetostrictive material, Zhang et al. ¹⁰ use the bending deflection of boring bar to realize the boring of elliptical hole. Moller ¹¹ put forward a new kind of magnetic-suspension spindle system, which can be applied to boring non-circular holes with any shape. Later, Kim et al. ¹² also use a magnetic bearing to control the displacement of the rotor for boring the non-circular hole and carry out some experiments using the boring machine developed by themselves. The cutting error is within 2 mm at the rotational speed of 1500 r min⁻¹.

Now, there exist many boring methods for elliptical holes. But it can be found that these methods have some disadvantages, such as complex mechanical composition, relatively small installation space for boring rod and tough accuracy control. So it is necessary to quest a much better boring method. A new machining method for elliptical holes is presented in the literature, ¹³ and the conclusion is obtained: when the shaft center revolution orbit is an ellipse, the motion synthesis between shaft center revolution orbit and tool rotation orbit can form the elliptical tool nose orbit. And thus, the machining of elliptical hole can be realized. So, as long as the shape of shaft center orbit is an expected ellipse, the elliptical tool nose orbit can be obtained. This article aims at controlling shaft center movement orbit. In this article, open-loop control simulation of shaft center orbit is carried out, which proves the feasibility of hydraulic control scheme and lays theoretical basis for the application of new machining method. ¹³

At present, the main simulation methods of shaft center orbit are Eulerian method, ¹⁴ Hahn method,^15,16 Holland method ¹⁷ and mobility method. ¹⁸ Eulerian method is a dynamics simulation method, but the other three ones are static methods which ignore the accelerated speed of shaft. Comparing with those three methods, the solution of Eulerian method is much better in consistency with the actual situation and the simulation precision of shaft orbit is much higher. Based on those advantages, many experts use the Eulerian method^19,20 to simulate the shaft center orbit.

In this article, a new type of hydrostatic bearing is proposed, and the corresponding hydraulic control scheme is presented as well. The quantitative relation between the shaft center coordinates and bearing load capacity is obtained. On the basis of previous work,^21,22 using flow continuity equation, one can get the analytical expressions of recess pressure and bearing load capacity. Then, the quantitative relation between load capacity and spool displacement of electro-hydraulic servo valve can be obtained. This article studies open-loop control simulation of shaft center movement orbit with Eulerian method. The simulation results manifest that open-loop control method can control shaft center movement to form ellipses with different ovalities and has strong robustness against dynamic disturbing force or instantaneous static disturbing force. This article provides theoretical foundation for practical application of new movement forming method.

Analysis

New hydrostatic bearing and corresponding hydraulic control scheme

Because the machining of elliptical holes is based on hydrostatic bearing control, the hydrostatic bearing is an important part of hydraulic control system. If bearing is a common four-recess hydrostatic bearing with circumferential return chute, hydraulic control system has disadvantages of control delay and low control sensitivity because of big recess area and big flow. In order to solve these problems, this research designs a new hydrostatic bearing with new recess structure, and the structure details are presented in Figure 1. New hydrostatic bearing has four big pockets, and inside each big pocket (it is named as load pocket), there is a small pocket (it is named as control pocket). Because the small pocket is inside the big pocket, this new structure bearing is named as pocket-nest hydrostatic bearing. There is isolation surface between control pocket and load pocket. Two kinds of pockets have their own oil supply holes separately, and the supply pressure of control pocket is higher than that of load pocket. One of load pocket’s design purposes is to offset the shaft gravity, and the other purpose is to increase system stiffness. Comparing with the pocket of common hydrostatic bearing, the control pocket has smaller pocket area and smaller flow, so the design of control pocket can improve the system control sensitivity and control speed.

Figure 1.

(a) Structure of pocket-nest hydrostatic bearing and (b) A-A section view.

Based on the previous work, ¹³ this article uses open-loop control method to control shaft center movement orbit and make it an ellipse, which synthesizes with rotation of the tool nose around shaft center. Thus, the machining of elliptical hole can be achieved. New hydraulic control system is designed based on the structure of pocket-nest hydrostatic bearing, and its details are presented in Figure 2. Hydraulic system can control the shaft center movement orbit by changing flow and pressure of control pocket. In Figure 2, four restrictors are all capillary restrictors, and four servo valves are all electro-hydraulic servo valves. Because the new bearing has two kinds of pockets (i.e. load pocket and control pocket), hydraulic system has two kinds of oil supply circuits ((1) lubricant flows into control pocket through electro-hydraulic servo valve and (2) lubricant flows into load pocket through capillary restrictor). To ensure the pressure of control pocket to be higher than that of load pocket, the supply pressure $p_{s1}$ of first oil circuit is bigger than supply pressure $p_{s2}$ of second oil circuit.

Figure 2.

Hydraulic control system.

The principle to realize the expected elliptical shaft center movement orbit using this hydraulic control system is as follows: in original state, servo valves 1 and 3 have the same spool displacement. When spool displacement of servo valve 1 becomes bigger and that of servo valve 3 becomes smaller, the pressure of control pocket 1 becomes bigger and that of control pocket 3 becomes smaller. Then, pressure difference between control pockets 1 and 3 generates thrust to push the shaft and the shaft center moves in the left direction. Conversely, the pressure difference makes the shaft center move in the right direction. In a similar way, the active control of servo valves 2 and 4 can control y-coordinate of shaft center. So, by active control of four servo valves, pressure difference can push the shaft and make the shaft center movement orbit an expected ellipse.

In order to realize the control of shaft center movement orbit, this research should at first get quantitative relation between the shaft center coordinates and bearing load capacity. The quantitative relation between bearing load capacity and spool displacement should also be solved. Then, the quantitative relation between the shaft center coordinates and spool displacement is obtained.

Quantitative relation between shaft center coordinates and bearing load capacity

Suppose the parameter equations of the shaft center’s expected ellipse orbit are

$$\{\begin{array}{c}{x}_J=a\cos {\omega }_{0}t\hfill \\ \multicolumn{1}{c}{y_J=b\sin {\omega }_{0}t}\end{array}$$

(1)

where $(x_J,y_J)$ are the shaft center coordinates, a is the semi-major axis length of the expected ellipse, b is the semi-minor axis length of the expected ellipse, ${\omega }_0$ is the shaft center angular velocity, and t is the time.

Taking the first-order derivative of equation (1) gives

$$\{\begin{array}{c}{\stackrel{·}{x}}_J=-a{\omega }_0\sin {\omega }_{0}t\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{y}}_J=b{\omega }_0\cos {\omega }_{0}t}\end{array}$$

(2)

where $({\stackrel{·}{x}}_J,{\stackrel{·}{y}}_J)$ are the shaft center velocities.

Taking the second-order derivative of equation (1) gives

$$\{\begin{array}{c}{\stackrel{··}{x}}_J=-a{\omega }_0^2\cos {\omega }_{0}t\hfill \\ \multicolumn{1}{c}{{\stackrel{··}{y}}_J=-b{\omega }_0^2\sin {\omega }_{0}t}\end{array}$$

(3)

where $({\stackrel{··}{x}}_J,{\stackrel{··}{y}}_J)$ are the shaft center accelerated velocities.

When the shaft moves inside the bearing, the force analysis of the shaft is presented in Figure 3. From Figure 3, the kinetic equations of the shaft can be obtained as

$$\{\begin{array}{c}m{\stackrel{··}{x}}_J=F_x+q_x\hfill \\ \multicolumn{1}{c}{m{\stackrel{··}{y}}_J=F_y-\mathrm{mg}+q_y}\end{array}$$

(4)

where m is the shaft mass, g is the gravitational acceleration, $(F_x,F_y)$ are the components of bearing load capacity, and $(q_x,q_y)$ are the components of external force.

Figure 3.

Force analysis of shaft.

Combining equations (3) and (4), one can obtain the bearing load capacity’s expressions

$$\{\begin{array}{c}{F}_x=-\mathrm{ma}{\omega }_0^2\cos {\omega }_{0}t-q_x\hfill \\ \multicolumn{1}{c}{F_y=-\mathrm{mb}{\omega }_0^2\sin {\omega }_{0}t+\mathrm{mg}-q_y}\end{array}$$

(5)

And the dimensionless expressions of equation (4) are

$$\{\begin{array}{c}{\stackrel{··}{X}}_J=\alpha {\overline{F}}_x+{\overline{q}}_x\hfill \\ \multicolumn{1}{c}{{\stackrel{··}{Y}}_J=\alpha {\overline{F}}_y-\beta +{\overline{q}}_y}\end{array}$$

(6)

where $({\stackrel{··}{X}}_J,{\stackrel{··}{Y}}_J)=({\stackrel{··}{x}}_J,{\stackrel{··}{y}}_J)/({\omega }_0^{2}c)$ , $\alpha =(p_{s1}\mathrm{rL})/(2m{\omega }_0^{2}c)$ , $({\overline{F}}_x,{\overline{F}}_y)=(2(F_x,F_y))/(p_{s1}\mathrm{rL})$ , $\beta =g/({\omega }_0^{2}c)$ , $({\overline{q}}_x,{\overline{q}}_y)=(q_x,q_y)/(m{\omega }_0^{2}c)$ , c is the bearing clearance, $p_{s1}$ is the supply pressure of control pocket circuit, r is the journal radius, and L is the bearing length.

Quantitative relation between bearing load capacity and spool displacement

Based on the previous work,^21,22 when the hydrostatic bearing has small bearing land, hydrodynamic effect on bearing land is very weak and can be neglected, and oil film force on bearing land is obtained using linear valuating method, and then analytical expressions of recess pressure can be solved using flow continuity equation. Moreover, from the literature, ²² the analytical expression of recess pressure has not only a wide application range but also a high computational accuracy. Because bearing land and isolation surface of new hydrostatic bearing in this article are both narrow, their hydrodynamic effect is weak and can be nearly ignored. Then, static performance parameters (such as recess pressure, flow of control pocket, and load pocket) can be solved without considering hydrodynamic effect. Analytical expressions of recess pressure and bearing load capacity are obtained using flow continuity equation as well as Gauss–Legendre integral formula, and the deduction details can be seen as follows.

The pocket-nest hydrostatic bearing has eight pockets; so there are eight flow continuity equations, as

$$\{\begin{array}{c}{Q}_{\mathrm{kin}}^{(i)}=Q_{\mathrm{kout}}^{(i)}\hfill \\ \multicolumn{1}{c}{Q_{\mathrm{kout}}^{(i)}+Q_{\mathrm{jin}}^{(i)}=Q_{\mathrm{jout}}^{(i)}}\end{array}(i=1,2,3,4)$$

(7)

where $Q_{\mathrm{kin}}^{(i)}$ is the flow of lubricant flowing into the ith control pocket, $Q_{\mathrm{kout}}^{(i)}$ is the flow of lubricant flowing out from the ith control pocket, $Q_{\mathrm{jin}}^{(i)}$ is the flow of lubricant flowing into the ith load pocket through capillary restrictor, and $Q_{\mathrm{jout}}^{(i)}$ is the flow of lubricant flowing out from the ith load pocket.

Using non-dimensional expression $\overline{Q}=Q((12\eta )/(p_{s1}{c}^3))$ , one can obtain the dimensionless expressions of equation (7), which are

$$\{\begin{array}{c}{\overline{Q}}_{\mathrm{kin}}^{(i)}={\overline{Q}}_{\mathrm{kout}}^{(i)}\hfill \\ \multicolumn{1}{c}{{\overline{Q}}_{\mathrm{kout}}^{(i)}+{\overline{Q}}_{\mathrm{jin}}^{(i)}={\overline{Q}}_{\mathrm{jout}}^{(i)}}\end{array}(i=1,2,3,4)$$

(8)

where η is the dynamic viscosity of lubricant.

Flow of control pocket

In Figure 4, the flow of lubricant flowing out of ith control pocket consists of two parts, that is, circumferential flow ( $Q_{k\theta 1}^{(i)}$ , $Q_{k\theta 2}^{(i)}$ ) and axial flow ( $Q_{\mathrm{kz}1}^{(i)}$ , $Q_{\mathrm{kz}2}^{(i)}$ )

$$Q_{\mathrm{kout}}^{(i)}=Q_{k\theta 2}^{(i)}-Q_{k\theta 1}^{(i)}+Q_{\mathrm{kz}1}^{(i)}+Q_{\mathrm{kz}2}^{(i)}$$

(9)

Figure 4.

Schematic diagram of boundary flow of pocket.

The flow of lubricant flowing through circumferential boundaries ( ${\Gamma }_{k\theta 1}^{(i)}$ and ${\Gamma }_{k\theta 2}^{(i)}$ ) of control pocket at per unit time is

$$q_{k\theta 1}^{(i)}=(\frac{{\mathrm{Uh}}_{k\theta 1}^{(i)}}{2}-\frac{h_{k\theta 1}^{(i)3}}{12\eta }\frac{1}{R}\frac{\partial p}{\partial \theta })\Delta z$$

(10)

$$q_{k\theta 2}^{(i)}=(\frac{{\mathrm{Uh}}_{k\theta 2}^{(i)}}{2}-\frac{h_{k\theta 2}^{(i)3}}{12\eta }\frac{1}{R}\frac{\partial p}{\partial \theta })\Delta z$$

(11)

where $({\Gamma }_{k\theta 1}^{(i)},{\Gamma }_{k\theta 2}^{(i)})(i=1,2,3,4)$ are the circumferential boundaries of ith control pocket, $(h_{k\theta 1}^{(i)},h_{k\theta 2}^{(i)})$ are the oil film thickness on circumferential boundaries $({\Gamma }_{k\theta 1}^{(i)},{\Gamma }_{k\theta 2}^{(i)})$ , p is the oil film pressure, and $\partial p/\partial \theta$ is the pressure gradient of isolation surface. Here, the hydrodynamic effect is ignored, and pressure distribution of circumferential isolation surface decreases linearly (for $q_{k\theta 1}^{(i)}$ , $\partial p/\partial \theta =(p_k^{(i)}-p_j^{(i)})/{\gamma }_k$ ; for $q_{k\theta 2}^{(i)}$ , $\partial p/\partial \theta =-(p_k^{(i)}-p_j^{(i)})/{\gamma }_k$ ). ${\gamma }_k$ is the isolation surface wrap angle in the circumferential direction, $p_k^{(i)}$ is the pressure of ith control pocket, $p_j^{(i)}$ is the pressure of ith load pocket, and R is the bearing radius. The equation $U={\omega }_{J}r$ exists, U is the speed of shaft external boundary, and ${\omega }_J$ is the journal rotational speed.

Substituting the equations of pressure gradient into equations (10) and (11) gives the flow of lubricant flowing through circumferential boundaries ( ${\Gamma }_{k\theta 1}^{(i)}$ and ${\Gamma }_{k\theta 2}^{(i)}$ ) of the ith control pocket, and the flow is expressed as

$$Q_{k\theta 1}^{(i)}=(\frac{{\mathrm{Uh}}_{k\theta 1}^{(i)}}{2}-\frac{h_{k\theta 1}^{(i)3}}{12\eta }·\frac{p_k^{(i)}-p_j^{(i)}}{R{\gamma }_k})B_k$$

(12)

$$Q_{k\theta 2}^{(i)}=(\frac{{\mathrm{Uh}}_{k\theta 2}^{(i)}}{2}+\frac{h_{k\theta 2}^{(i)3}}{12\eta }·\frac{p_k^{(i)}-p_j^{(i)}}{R{\gamma }_k})B_k$$

(13)

According to the non-dimensional equation $\overline{Q}=Q((12\eta )/(p_{s1}{c}^3))$ , the dimensionless forms of equations (12) and (13) can be given as

$${\overline{Q}}_{k\theta 1}^{(i)}=\frac{L}{R}\frac{{\overline{B}}_k}{{\gamma }_k}[\Lambda {\gamma }_k{\overline{h}}_{k\theta 1}^{(i)}-{\overline{h}}_{k\theta 1}^{(i)3}({\overline{p}}_k^{(i)}-{\overline{p}}_j^{(i)})]$$

(14)

$${\overline{Q}}_{k\theta 2}^{(i)}=\frac{L}{R}\frac{{\overline{B}}_k}{{\gamma }_k}[\Lambda {\gamma }_k{\overline{h}}_{k\theta 2}^{(i)}+{\overline{h}}_{k\theta 2}^{(i)3}({\overline{p}}_k^{(i)}-{\overline{p}}_j^{(i)})]$$

(15)

where ${\overline{Q}}_{k\theta 1}^{(i)}=12\eta Q_{k\theta 1}^{(i)}/(c^3{p}_{s1})$ and ${\overline{Q}}_{k\theta 2}^{(i)}=12\eta Q_{k\theta 2}^{(i)}/(c^3{p}_{s1})$ are the non-dimensional flow, $\Lambda =6\eta {\omega }_J{R}^2/(p_{s1}{c}^2)$ is the bearing speed parameter, and ${\overline{p}}_k^{(i)}=p_k^{(i)}/p_{s1}$ and ${\overline{p}}_j^{(i)}=p_j^{(i)}/p_{s1}$ are the non-dimensional pressure of control pocket and load pocket, respectively. In Figure 5, $B_k$ is the axial length of control pocket, ${\overline{B}}_k=B_k/L$ is the non-dimensional axial length of control pocket, and L is the bearing length.

Figure 5.

Schematic diagram of boundary of pocket.

Flow of lubricant flowing through axis boundaries of control pocket at per unit time is

$${\mathrm{dq}}_{\mathrm{kz}1}^{(i)}=\frac{h^3}{12\eta }\frac{\partial p}{\partial z}\mathrm{Rd}\theta$$

(16)

where $\partial p/\partial z$ is the pressure gradient of axis isolation surface. When the hydrodynamic effect is ignored, pressure distribution ( $\partial p/\partial z=(p_k^{(i)}-p_j^{(i)})/b_k$ ) of axis isolation surface decreases linearly, and $b_k$ is the isolation surface width in the axial direction.

According to geometrical relationship, ²¹ there exists equation $h=c+x_J\sin \theta -y_J\cos \theta$ . Taking the equation $h=c+x_J\sin \theta -y_J\cos \theta$ into equation (16), one can obtain the non-dimensional flow of lubricant flowing through axis boundaries ( ${\Gamma }_{\mathrm{kz}1}^{(i)}$ and ${\Gamma }_{\mathrm{kz}2}^{(i)}$ )

$${\overline{Q}}_{kz1}^{(i)}={\overline{Q}}_{kz2}^{(i)}=\frac{R}{L}\frac{{\overline{p}}_k^{(i)}-{\overline{p}}_j^{(i)}}{{\overline{b}}_k}{\displaystyle \underset{{\theta }_{k1}^{(i)}}{\overset{{\theta }_{k2}^{(i)}}{\int }}{(1+X_J\sin \theta -Y_J\cos \theta )}^{3}d\theta }$$

(17)

where ${\overline{Q}}_{\mathrm{kz}1}^{(i)}=12\eta Q_{\mathrm{kz}1}^{(i)}/(c^3{p}_{s1})$ , ${\overline{Q}}_{\mathrm{kz}2}^{(i)}=12\eta Q_{\mathrm{kz}2}^{(i)}/(c^3{p}_{s1})$ , ${\overline{b}}_k=b_k/L$ is the non-dimensional isolation surface width in the axial direction, and $(X_J,Y_J)$ are the non-dimensional shaft center coordinates. ${\theta }_{k1}^{(i)}$ and ${\theta }_{k2}^{(i)}$ ( ${\theta }_{k1}^{(i)}<{\theta }_{k2}^{(i)},i=1,2,3,4$ ) are the circumferential boundaries’ coordinates of ith control pocket.

Function $f(\theta )=(1+X_J\sin \theta -Y_J\cos \theta {)}^3$ and variable ${\Delta }_k^i={\int }_{{\theta }_{k1}^{(i)}}^{{\theta }_{k2}^{(i)}}f(\theta )d\theta$ are defined. With mathematical approach, the variable ${\Delta }_k^i$ is written as

$${\Delta }_k^i=\frac{{\theta }_{k2}^{(i)}-{\theta }_{k1}^{(i)}}{2}\underset{-1}{\overset{1}{\int }}f(\frac{{\theta }_{k2}^{(i)}-{\theta }_{k1}^{(i)}}{2}t+\frac{{\theta }_{k2}^{(i)}+{\theta }_{k1}^{(i)}}{2})\mathrm{dt}$$

(18)

The three-point Gauss–Legendre integral formula ²² can be written as

$$\underset{-1}{\overset{1}{\int }}f(t)\mathrm{dt}\approx \frac{5}{9}f(-\frac{\sqrt{15}}{5})+\frac{8}{9}f(0)+\frac{5}{9}f\left(\frac{\sqrt{15}}{5}\right)$$

Using three-point Gauss–Legendre integral formula to solve variable ${\Delta }_k^i$ yields

$$\begin{array}{c}\multicolumn{1}{c}{{\Delta }_k^i=\frac{{\theta }_{k2}^{(i)}-{\theta }_{k1}^{(i)}}{2}\underset{-1}{\overset{1}{\int }}f(\frac{{\theta }_{k2}^{(i)}-{\theta }_{k1}^{(i)}}{2}t+\frac{{\theta }_{k2}^{(i)}+{\theta }_{k1}^{(i)}}{2})\mathrm{dt}\approx \frac{{\theta }_{k2}^{(i)}-{\theta }_{k1}^{(i)}}{2}}\\ \multicolumn{1}{c}{[\frac{5}{9}f(-\frac{\sqrt{15}}{5}\frac{{\theta }_{k2}^{(i)}-{\theta }_{k1}^{(i)}}{2}+\frac{{\theta }_{k2}^{(i)}+{\theta }_{k1}^{(i)}}{2})+\frac{8}{9}f\left(\frac{{\theta }_{k2}^{(i)}+{\theta }_{k1}^{(i)}}{2}\right)}\\ \multicolumn{1}{c}{+\frac{5}{9}f(\frac{\sqrt{15}}{5}\frac{{\theta }_{k2}^{(i)}-{\theta }_{k1}^{(i)}}{2}+\frac{{\theta }_{k2}^{(i)}+{\theta }_{k1}^{(i)}}{2})]}\end{array}$$

(19)

Equation (19) can be used to approximately solve the value of definite integral, so equation (17) is changed into

$${\overline{Q}}_{\mathrm{kz}1}^{(i)}={\overline{Q}}_{\mathrm{kz}2}^{(i)}=\frac{R}{L}\frac{{\overline{p}}_k^{(i)}-{\overline{p}}_j^{(i)}}{{\overline{b}}_k}{\Delta }_k^i$$

(20)

Substituting equations (14), (15), and (20) into equation (9) gives

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{Q}}_{\mathrm{kout}}^{(i)}=\frac{L}{R}\frac{{\overline{B}}_k}{{\gamma }_k}}\\ \multicolumn{1}{c}{[\Lambda {\gamma }_k({\overline{h}}_{k\theta 2}^{(i)}-{\overline{h}}_{k\theta 1}^{(i)})+({\overline{h}}_{k\theta 2}^{(i)3}+{\overline{h}}_{k\theta 1}^{(i)3})({\overline{p}}_k^{(i)}-{\overline{p}}_j^{(i)})]}\\ \multicolumn{1}{c}{+2\frac{R}{L}\frac{{\overline{p}}_k^{(i)}-{\overline{p}}_j^{(i)}}{{\overline{b}}_k}{\Delta }_k^i}\end{array}$$

(21)

Flow of lubricant flowing into control pocket through electro-hydraulic servo valve is

$${\overline{Q}}_{\mathrm{kin}}^{(i)}=C_s^{(i)}\sqrt{1-{\overline{p}}_k^{(i)}}$$

(22)

where $C_s^{(i)}=(12\sqrt{2}\eta C_d{x}_v^{(i)}\pi d_v)/(\sqrt{\rho p_{s1}}{c}^3)$ exists. Here, $C_d$ is the discharge coefficient of servo valve, $x_v^{(i)}$ is the spool displacement of ith electro-hydraulic servo valve, and $d_v$ is the valve core diameter.

Flow of load pocket

In a similar way, based on Figure 4, the flow equation of lubricant flowing out of ith load pocket is written as

$${\overline{Q}}_{\mathrm{jout}}^{(i)}={\overline{Q}}_{j\theta 2}^{(i)}-{\overline{Q}}_{j\theta 1}^{(i)}+{\overline{Q}}_{\mathrm{jz}1}^{(i)}+{\overline{Q}}_{\mathrm{jz}2}^{(i)}$$

(23)

where the flow equations for lubricant flowing through circumferential boundaries $({\Gamma }_{j\theta 1}^{(i)},{\Gamma }_{j\theta 2}^{(i)})$ are

$${\overline{Q}}_{j\theta 1}^{(i)}=\frac{L}{R}\frac{{\overline{B}}_j}{{\gamma }_j}(\Lambda {\gamma }_j{\overline{h}}_{j\theta 1}^{(i)}-{\overline{h}}_{j\theta 1}^3{\overline{p}}_j^{(i)})$$

(24)

$${\overline{Q}}_{j\theta 2}^{(i)}=\frac{L}{R}\frac{{\overline{B}}_j}{{\gamma }_j}(\Lambda {\gamma }_j{\overline{h}}_{j\theta 2}^{(i)}+{\overline{h}}_{j\theta 2}^3{\overline{p}}_j^{(i)})$$

(25)

The flow of lubricant flowing through axial boundaries $({\Gamma }_{\mathrm{jz}1}^{(i)},{\Gamma }_{\mathrm{jz}2}^{(i)})$ is

$${\overline{Q}}_{\mathrm{jz}1}^{(i)}={\overline{Q}}_{\mathrm{jz}2}^{(i)}=\frac{R}{L}\frac{{\overline{p}}_j^{(i)}}{{\overline{b}}_j}\underset{{\theta }_{j1}^{(i)}}{\overset{{\theta }_{j2}^{(i)}}{\int }}{(1+X\sin \theta -Y\cos \theta )}^{3}d\theta$$

(26)

where ( ${\theta }_{j1}^{(i)}$ , ${\theta }_{j2}^{(i)}$ ) are the coordinates of circumferential boundaries of ith load pocket ( ${\theta }_{j1}^{(i)}<{\theta }_{j2}^{(i)},i=1,2,3,4$ ), and ( ${\overline{h}}_{j\theta 1}^{(i)}$ , ${\overline{h}}_{j\theta 2}^{(i)}$ ) are the oil film thickness on circumferential boundaries ( ${\theta }_{j1}^{(i)}$ and ${\theta }_{j2}^{(i)}$ ). ${\gamma }_j$ is the bearing land wrap angle in the circumferential direction, $b_j$ is the bearing land width in the axial direction, ${\overline{b}}_j=b_j/L$ , and $B_j$ is the axial length of load pocket, ${\overline{B}}_j=B_j/L$ .

Using three-point Gauss–Legendre integral formula to approximately solve the value of definite integral in equation (26) yields

$${\Delta }_j^i={\displaystyle \underset{{\theta }_{j1}^{(i)}}{\overset{{\theta }_{j2}^{(i)}}{\int }}{(1+X\sin \theta -Y\cos \theta )}^{3}d\theta \approx \frac{{\theta }_{j2}^{(i)}-{\theta }_{j1}^{(i)}}{2}\left[\frac{5}{9}f\left(-\frac{\sqrt{15}}{5}\frac{{\theta }_{j2}^{(i)}-{\theta }_{j1}^{(i)}}{2}+\frac{{\theta }_{j2}^{(i)}+{\theta }_{j1}^{(i)}}{2}\right)+\frac{8}{9}f\left(\frac{{\theta }_{j2}^{(i)}+{\theta }_{j1}^{(i)}}{2}\right)+\frac{5}{9}f\left(\frac{\sqrt{15}}{5}\frac{{\theta }_{j2}^{(i)}-{\theta }_{j1}^{(i)}}{2}+\frac{{\theta }_{j2}^{(i)}+{\theta }_{j1}^{(i)}}{2}\right)\right]}$$

(27)

Substituting equations (24)–(27) into equation (23) yields

$${\overline{Q}}_{\mathrm{jout}}^{(i)}=\frac{L}{R}\frac{{\overline{B}}_j}{{\gamma }_j}[\Lambda {\gamma }_j({\overline{h}}_{j\theta 2}^{(i)}-{\overline{h}}_{j\theta 1}^{(i)})+({\overline{h}}_{j\theta 2}^{(i)3}+{\overline{h}}_{j\theta 1}^{(i)3}){\overline{p}}_j^{(i)}]+2\frac{R}{L}\frac{{\overline{p}}_j^{(i)}}{{\overline{b}}_j}{\Delta }_j^i$$

(28)

Non-dimensional flow of capillary restrictor is

$${\overline{Q}}_{\mathrm{jin}}^{(i)}=\frac{(\pi d_c^4(p_{s2}-p_j^{(i)}))/(128\eta l_c)}{(p_{s1}{c}^3)/(12\eta )}=\frac{3\pi d_c^4}{32l_c{c}^3}({\overline{p}}_{s2}-{\overline{p}}_j^{(i)})$$

(29)

In the equation above, $d_c$ is the diameter of capillary restrictor, $l_c$ is the length of capillary restrictor, $p_{s2}$ is the supply pressure of load pocket circuit, and there is ${\overline{p}}_{s2}=p_{s2}/p_{s1}$ .

Analytical expressions of recess pressure and load capacity

Substituting equations (21), (28), and (29) into equation (8) can produce flow continuity equation of the ith load pocket, as

$$\begin{array}{l}\frac{L}{R}\frac{{\overline{B}}_k}{{\gamma }_k}\left[\Lambda {\gamma }_k({\overline{h}}_{k\theta 2}^{(i)}-{\overline{h}}_{k\theta 1}^{(i)})+({\overline{h}}_{k\theta 2}^{(i)3}+{\overline{h}}_{k\theta 1}^{(i)3})({\overline{p}}_k^{(i)}-{\overline{p}}_j^{(i)})\right]+2\frac{R}{L}\frac{{\overline{p}}_k^{(i)}-{\overline{p}}_j^{(i)}}{{\overline{b}}_k}{\Delta }_k^i\\ +\frac{3\pi d_c^4}{32l_c{c}^3}({\overline{p}}_{s2}-{\overline{p}}_j^{(i)})=\frac{L}{R}\frac{{\overline{B}}_j}{{\gamma }_j}\left[\Lambda {\gamma }_j({\overline{h}}_{j\theta 2}^{(i)}-{\overline{h}}_{j\theta 1}^{(i)})+({\overline{h}}_{j\theta 2}^{(i)3}+{\overline{h}}_{j\theta 1}^{(i)3}){\overline{p}}_j^{(i)}\right]+2\frac{R}{L}\frac{{\overline{p}}_j^{(i)}}{{\overline{b}}_j}{\Delta }_j^i\end{array}$$

(30)

Suppose $K_1^{(i)}=(L/R)({\overline{B}}_k/{\gamma }_k)\Lambda {\gamma }_k({\overline{h}}_{k\theta 2}^{(i)}-{\overline{h}}_{k\theta 1}^{(i)})$ , $K_2^{(i)}=(L/R)({\overline{B}}_k/{\gamma }_k){\overline{h}}_{k\theta 2}^{(i)3}+{\overline{h}}_{k\theta 1}^{(i)3}+2(R/L)({\Delta }_k^i/{\overline{b}}_k)$ , $J_1^{(i)}=(L/R)({\overline{B}}_j/{\gamma }_j)\Lambda {\gamma }_j({\overline{h}}_{j\theta 2}^{(i)}-{\overline{h}}_{j\theta 1}^{(i)})$ , $J_2^{(i)}=(L/R)({\overline{B}}_j/{\gamma }_j)({\overline{h}}_{j\theta 2}^{(i)3}+{\overline{h}}_{j\theta 1}^{(i)3})+2(R/L)({\Delta }_j^i/{\overline{b}}_j)$ , and $J_3^{(i)}=(3\pi d_c^4)/(32l_c{c}^3)$ exist, equation (30) can be simplified as

$$K_1^{(i)}+K_2^{(i)}({\overline{p}}_k^{(i)}-{\overline{p}}_j^{(i)})+J_3^{(i)}({\overline{p}}_{s2}-{\overline{p}}_j^{(i)})=J_1^{(\mathrm{i})}+J_2^{(i)}{\overline{p}}_j^{(i)}$$

(31)

The expression of control pocket pressure is obtained by solving equation (31), that is

$${\overline{p}}_k^{(i)}=\frac{(J_2^{(i)}+J_3^{(i)}+K_2^{(i)})}{K_2^{(i)}}{\overline{p}}_j^{(i)}+\frac{J_1^{(\mathrm{i})}-K_1^{(i)}-J_3^{(i)}{\overline{p}}_{s2}}{K_2^{(i)}}$$

(32)

Substituting equations (21) and (22) into equation (8), one can obtain the flow continuity equation of control pocket

$$K_1^{(i)}+K_2^{(i)}({\overline{p}}_k^{(i)}-{\overline{p}}_j^{(i)})=C_s^{(i)}\sqrt{1-{\overline{p}}_k^{(i)}}$$

(33)

Solving equation (33) yields the other expression of control pocket pressure, as

$${\overline{p}}_k^{(i)}=\frac{-[2(K_1^{(i)}-K_2^{(i)}{\overline{p}}_j^{(i)})K_2^{(i)}+C_s^{{(i)}^2}]+\sqrt{C_s^{{(i)}^4}+4(K_1^{(i)}-K_2^{(i)}{\overline{p}}_j^{(i)})K_2^{(i)}{C}_s^{{(i)}^4}+4{(K_1^{(i)})}^2{C}_s^{{(i)}^2}}}{2K_2^{{(i)}^2}}$$

(34)

Combining equations (32) and (34) can give

$$\begin{array}{c}\multicolumn{1}{c}{{4K}_2^2(J_2^{(i)}+J_3^{(i)}{)}^2{\overline{p}}_j^2+4K_2^{(i)}(J_2^{(i)}+J_3^{(i)})}\\ \multicolumn{1}{c}{{{[C}_s^{(i)}}^2+2K_2^{(i)}(J_1^{(\mathrm{i})}-J_3^{(i)}{\overline{p}}_{s2})]{\overline{p}}_j^{(i)}}\\ \multicolumn{1}{c}{+{4K}_2^2{C}_s^2{\overline{p}}_j^{(i)}+{{{[C}_s^{(i)}}^2+2K_2^{(i)}(J_1^{(\mathrm{i})}-J_3^{(i)}{\overline{p}}_{s2})]}^2}\\ \multicolumn{1}{c}{{-C}_s^4-4K_1^{(i)}{K}_2^{(i)}{C}_s^2-4(K_2^{(i)}{)}^2{C}_s^2=0}\end{array}$$

(35)

Suppose

$$\left\{\begin{array}{c}{a}^{(i)}=4K_2^{{(i)}^2}(J_2^{(i)}+J_3^{(i)}{)}^2\hfill \\ \multicolumn{1}{c}{b^{(i)}=4K_2^{(i)}(J_2^{(i)}+J_3^{(i)})[C_s^{{(i)}^2}+2K_2^{(i)}(J_1^{(i)}+J_3^{(i)}{\overline{p}}_{s2})]+4K_2^{{(i)}^2}{C}_s^{{(i)}^2}}\\ \multicolumn{1}{c}{c^{(i)}=[C_s^{{(i)}^2}+2K_2^{(i)}(J_1^{(i)}+J_3^{(i)}{\overline{p}}_{s2}){]}^2-C_s^{{(i)}^4}-4K_1^{(i)}{K}_2^{(i)}{C}_s^{{(i)}^2}-4(K_2^{(i)}{)}^2{C}_s^{{(i)}^2}}\end{array}\right\}$$

exists, equation (35) can be simplified as

$$a^{(i)}{\overline{p}}_j^{{(i)}^2}+b^{(i)}{\overline{p}}_j^{(i)}+c^{(i)}=0$$

(36)

From equation (36), analytical expression of load pocket pressure can be obtained as

$${\overline{p}}_j^{(i)}=\frac{-b^{(i)}+\sqrt{{(b^{(i)})}^2-4a^{(i)}{c}^{(i)}}}{2a^{(i)}}$$

(37)

Summarize pressure expressions of load pocket and control pocket as

$$\{\begin{array}{c}{\overline{p}}_j^{(i)}=\frac{-b^{(i)}+\sqrt{{(b^{(i)})}^2-4a^{(i)}{c}^{(i)}}}{2a^{(i)}}\hfill \\ \multicolumn{1}{c}{{\overline{p}}_k^{(i)}=\frac{(J_2^{(i)}+J_3^{(i)}+K_2^{(i)})}{K_2^{(i)}}{\overline{p}}_j^{(i)}+\frac{J_1^{(\mathrm{i})}-K_1^{(i)}-J_3^{(i)}{\overline{p}}_{s2}}{K_2^{(i)}}}\end{array}$$

(38)

For the computational convenience, only through considering pressure of load pocket and control pocket (without considering the pressure of bearing land and isolation surface), can this research get components of load capacity in the directions of x and y, and those components are expressed as

$$\{\begin{array}{c}{F}_x=\sum _1^4[p_j^{(i)}R{B}_j(\cos {\theta }_{j1}^{(i)}-\cos {\theta }_{j2}^{(i)})]\hfill \\ \multicolumn{1}{c}{+\sum _1^4[(p_k^{(i)}-p_j^{(i)})R{B}_k(\cos {\theta }_{k1}^{(i)}-\cos {\theta }_{k2}^{(i)})]}\\ \multicolumn{1}{c}{F_y=\sum _1^4[p_j^{(i)}R{B}_j(\sin {\theta }_{j1}^{(i)}-\sin {\theta }_{j2}^{(i)})]}\\ \multicolumn{1}{c}{+\sum _1^4[(p_k^{(i)}-p_j^{(i)})R{B}_k(\sin {\theta }_{k1}^{(i)}-\sin {\theta }_{k2}^{(i)})]}\end{array}$$

(39)

If the value of ${\omega }_{0}t$ is given, the value of shaft center coordinates and the needed bearing load capacity $(F_x,F_y)$ can be obtained by equations (1) and (5). In equation (39), there are two expressions and four unknown parameters (i.e. four spool displacement). So in order to obtain their individual unique solutions of four spool displacements, another two equations should be added. The two added equations are

$$x_v^{(1)}+x_v^{(3)}=x_v$$

(40)

$$x_v^{(2)}+x_v^{(4)}=x_v$$

(41)

where $x_v$ is defined as a constant quantity. Therefore, when spool displacement $x_v^{(1)}$ of servo valve 1 increases, spool displacement $x_v^{(3)}$ of servo valve 3 will decrease, so that the pressure of control pocket 1 increases, and that of control pocket 3 decreases. Thus, combining equations (39)–(41) can obtain the unique solution. Using analytical expressions (38) and (39) to solve recess pressure and load capacity can significantly simplify computing process and reduce computing time.

Eulerian method

Combining equations (5), (39), (40), and (41) yields quantitative relation between the expected shaft center orbit coordinates and spool displacement of electro-hydraulic servo valve. Using obtained spool displacement parameters and Eulerian method, this research can simulate shaft center’s open-loop control orbits.

The equations of Eulerian method are

$${\stackrel{·}{X}}_J(t+\Delta t)={\stackrel{·}{X}}_J(t)+{\stackrel{··}{X}}_J(t){\omega }_0\Delta t$$

(42)

$$X_J(t+\Delta t)=X_J(t)+{\stackrel{·}{X}}_J(t+\Delta t){\omega }_0\Delta t$$

(43)

$${\stackrel{·}{Y}}_J(t+\Delta t)={\stackrel{·}{Y}}_J(t)+{\stackrel{··}{Y}}_J(t){\omega }_0\Delta t$$

(44)

$$Y_J(t+\Delta t)=Y_J(t)+{\stackrel{·}{Y}}_J(t+\Delta t){\omega }_0\Delta t$$

(45)

where $(X_J(t),Y_J(t)),({\stackrel{·}{X}}_J(t),{\stackrel{·}{Y}}_J(t)),({\stackrel{··}{X}}_J(t),{\stackrel{··}{Y}}_J(t))$ are the shaft center coordinates, velocities, and accelerated velocities, respectively, at the moment of t. $(X_J(t+\Delta t),Y_J(t+\Delta t)),({\stackrel{.}{X}}_J(t+\Delta t),{\stackrel{.}{Y}}_J(t+\Delta t)),({\stackrel{··}{X}}_J(t+\Delta t),{\stackrel{··}{Y}}_J(t+\Delta t))$ are the shaft center coordinates, velocities, and accelerated velocities, respectively, at the moment of t + Δt. $\Delta t$ is the time interval for simulation.

The computing steps of Eulerian method are as follows:

1. Setting the value of external loads $q_x$ and $q_y$ (for computational convenience, they are supposed as 0).

2. Setting the value of the initial shaft center location coordinates $(X_{J0},Y_{J0})$ , velocities $({\stackrel{·}{X}}_{J0},{\stackrel{·}{Y}}_{J0})$ , and spool displacement $x_{v0}^{(i)}(i=1,2,3,4)$ of electro-hydraulic servo valves. Equation (38) is solved to get control pocket pressure and load pocket pressure, and bearing load capacity ( ${\overline{F}}_x$ , ${\overline{F}}_y$ ) can be obtained by solving equation (39).

3. Substituting the value of ${\overline{F}}_x$ and ${\overline{F}}_y$ into equation (6), initial accelerated velocities $({\stackrel{··}{X}}_{J0},{\stackrel{··}{Y}}_{J0})$ can be obtained when t = 0 exists.

4. Valuing $\Delta t$ , and using equations (42)–(45) to calculate the next location coordinates $(X_{J1},Y_{J1})$ and velocities $({\stackrel{·}{X}}_{J1},{\stackrel{·}{Y}}_{J1})$ of shaft center with Eulerian method.

5. Using the next shaft center location coordinates $(X_{J1},Y_{J1})$ , velocities $({\stackrel{·}{X}}_{J1},{\stackrel{·}{Y}}_{J1})$ , and the next servo valve’s spool displacement $x_{v1}^{(i)}(i=1,2,3,4)$ , and repeating steps 2–4 until it comes to simulation time t.

Results

First, an example is used to demonstrate the change law of system control parameter (i.e. spool displacement of four electro-hydraulic servo valves) when shaft center movement orbit is an ellipse.

Suppose non-dimensional equations of the expected elliptical orbit of shaft center as

$$\{\begin{array}{c}{X}_J=A\cos {\omega }_{0}t\hfill \\ \multicolumn{1}{c}{Y_J=B\sin {\omega }_{0}t}\end{array}$$

(46)

where $(X_J,Y_J)=(x_J,y_J)/c$ are the non-dimensional shaft center coordinates, and (A, B) are the non-dimensional length of the expected ellipse’s semi-major axis and semi-minor axis, respectively.

Structure parameters of pocket-nest hydrostatic bearing are shown in Table 1. The capillary restrictor diameter $d_c$ is 0.85 mm, and its length $l_c$ is 60 mm. The servo valve core’s diameter $d_v$ is 4 mm, and the spool displacement constant $x_v$ is 0.12 mm. The shaft rotation speed is 1500 r min⁻¹, and the angular speed ${\omega }_0$ in equation (46) is $50\pi \mathrm{rad}{\mathrm{s}}^{-1}$ . The supply pressure $p_{s1}$ of control pocket circuit is 3.5 N mm⁻², and the supply pressure $p_{s2}$ of load pocket circuit is 1 N mm⁻². The lubricant density ρ is 870 kg m⁻³, and the dynamic viscosity η is 0.0087 N s m⁻². Substituting the equation ${\omega }_0=50\pi \mathrm{rad}{\mathrm{s}}^{-1}$ into equation (46) yields the expected shaft center orbit equation. Equation (5) is adopted to solve bearing load capacity, and equations (5) and (39)–(41) are combined together to get change laws of spool displacement $x_v^{(i)}(i=1,2,3,4)$ of electro-hydraulic servo valves. The change laws are shown in Figures 6 and 7. From the two figures, the change laws of four spool displacement are approximate to sine or cosine. According to equation (40), the sum of spool displacement of servo valves 1 and 3 is constant, so is that for servo valves 2 and 4.

Table 1.

Geometric structure parameters and operating parameters of pocket-nest hydrostatic bearing.

Parameters	Value
Journal diameter (DJ)	60.00 mm
Bearing diameter (Db)	60.06 mm
Bearing length (L)	60 mm
Radius clearance (c)	0.03 mm
Wrap angle of load pocket ( ${\alpha }_j$ )	66°
Wrap angle of control pocket ( ${\alpha }_k$ )	20°
Axial length of load pocket (Bj)	48 mm
Axial length of control pocket (Bk)	17.5 mm
Bearing land wrap angle in circumferential direction ( ${\gamma }_j$ )	7°
Bearing land width in axial direction (bj)	6 mm
Isolation surface wrap angle in circumferential direction ( ${\gamma }_k$ )	5°
Isolation surface width in axial direction (bk)	2.5 mm
Recess shape	Rectangular

Figure 6.

Change law of spool displacement of servo valves 1 and 3.

Figure 7.

Change law of spool displacement of servo valves 2 and 4.

In Figures 8–16, (X, Y) and (ΔX, ΔY) are the non-dimensional coordinates, that is, the units of the axes in these figures are non-dimensional. The equations $(X,Y)=(x,y)/c$ and $(\Delta X,\Delta Y)=(\Delta x,\Delta y)/c$ exist, and c is the radial clearance of bearing. The unit of c is millimeter, so are the coordinates (x, y) and (Δx, Δy). According to Eulerian method steps, open-loop control simulations of shaft center movement orbit are taken and shaft center orbits (seen in Figures 8–10) with different ovalities are obtained. In the three figures, the initial non-dimensional location coordinates of shaft center are (0, 0), initial velocities are (0, 0), and non-dimensional semi-minor axes of small ellipses formed by shaft center movement orbits are all 0.1. In Figures 8–10, their non-dimensional semi-major axes are 0.2, 0.3, and 0.4, respectively. From the three figures, open-loop control can control shaft movement orbit and make shaft center form elliptical orbits with different ovalities, which enlarges application range of movement forming method ¹³ and provides the theoretical support for application of movement forming method. Even if the shaft center’s initial location coordinates are not on the expected ellipse (in the three figures, the initial location coordinates are on the origin point), the open-loop control method can make the shaft center movement orbit quickly approach the expected ellipse, forming the expected elliptical shaft center orbit.

Figure 8.

(a) Open-loop control simulations of shaft center movement orbit (A = 0.2, B = 0.1), (b) X-coordinate of shaft center, and (c) Y-coordinate of shaft center.

Figure 9.

(a) Open-loop control simulations of shaft center movement orbit (A = 0.3, B = 0.1), (b) X-coordinate of shaft center, and (c) Y-coordinate of shaft center.

Figure 10.

(a) Open-loop control simulations of shaft center movement orbit (A = 0.4, B = 0.1), (b) X-coordinate of shaft center, and (c) Y-coordinate of shaft center.

Figure 11.

(a) Open-loop control simulations of shaft center movement orbit (q_x = 500 N), (b) X-coordinate of shaft center, and (c) Y-coordinate of shaft center.

Figure 12.

(a) Open-loop control simulations of shaft center movement orbit (q_y = −500 N), (b) X-coordinate of shaft center, and (c) Y-coordinate of shaft center.

Figure 13.

(a) Open-loop control simulations of shaft center movement orbit (q_x = 500 N, q_y = −500 N), (b) X-coordinate of shaft center, and (c) Y-coordinate of shaft center.

Figure 14.

(a) Open-loop control simulations of shaft center movement orbit ( $q_x=q_0\cos \omega t$ , $q_y=q_0\sin \omega t$ , q₀ = 200 N), (b) X-coordinate of shaft center, and (c) Y-coordinate of shaft center.

Figure 15.

(a) Tracking errors of shaft center coordinates in X-direction under different ovalities and (b) tracking errors of shaft center coordinates in Y-direction under different ovalities.

Figure 16.

(a) Tracking errors of shaft center coordinates in X-direction under different disturbing forces and (b) tracking errors of shaft center coordinates in Y-direction under different disturbing forces.

The shaft is often disturbed by instantaneous external force in moving process, so this article makes open-loop control simulations under that situation. Figures 11–13 are the open-loop control orbits of shaft center under the condition of instantaneous external disturbing forces. In the three figures, the non-dimensional semi-major axis length A of ellipse formed by shaft center movement orbit is 0.4 and semi-minor axis length B is 0.3. The non-dimensional initial coordinates of shaft center are (0, −0.3), and the initial velocities are (0, 0). Figure 11 is the open-loop control movement orbit of shaft center when the shaft is disturbed by an instantaneous external force (q_x = 500 N) in the X-axis direction. It can be seen that shaft center has small vibration in the X-axis direction, but shaft center will quickly approach the expected ellipse, which indicates the good anti-interference of open-loop control method. Figure 12 is the open-loop control movement orbit of shaft center when the shaft is disturbed by an instantaneous external force (q_y = −500 N) in the Y-axis direction. It can also be seen that shaft center has small vibration in the Y-axis direction, but shaft center will quickly approach the expected ellipse. Figure 13 is the open-loop control movement orbit of shaft center when the shaft suffers from two instantaneous external forces (q_x = 500 N, q_y = −500 N) in the direction of X- and Y-axes. It can be seen that shaft center has small vibration both in the direction of X- and Y-axes, but shaft center will quickly approach the expected ellipse. Figures 11–13 indicate that open-loop control method has good controlling effect and robustness for instantaneous external disturbing force.

When the shaft rotates inside the bearing, it may be disturbed by dynamical force whose value and direction vary with time. In this article, the sine dynamic load and cosine dynamic load are chosen as examples for dynamical disturbance force. Figure 14 is the open-loop control simulations of shaft center movement orbit when the shaft is disturbed by dynamical disturbance forces ( $q_x=q_0\cos \omega t$ , $q_y=q_0\sin \omega t$ , q₀ = 200 N). In the simulation process, the value and direction of resultant quantity ( $\overrightarrow{q}={\overrightarrow{q}}_x+{\overrightarrow{q}}_y$ ) vary with time. From Figure 14, it can be seen that the control orbit of shaft center deviates from the expected orbit, but the degree of deviation is little. The control orbit is approximately an ellipse, and this indicates that the anti-interference of open-loop control for dynamical disturbance force is also good.

Figure 15 indicates the tracking errors of the shaft center coordinates under different ovalities. Figure 15(a) and (b) shows the tracking errors in the X- and Y-directions, respectively. It can be seen from Figure 15 that the tracking errors of initial stage are big, because the initial point of shaft center is on the original point instead of the expected ellipse (it can be seen in Figures 8–10). But after the adjustment for short time, the tracking errors in the X- and Y-directions become small and the tracking precisions become high. As can be seen in Figure 15(a), the variation range of tracking errors in the X-direction becomes wider as the value of A increases, that is, the variation range becomes wider when the length of the expected ellipse’s semi-major axis increases. When A = 0.4, the range of the tracking errors in the X-direction is widest, the biggest tracking error in steady state is 0.006421, and the percentage error is 1.61%. When A = 0.2, the range of the tracking errors in the X-direction is narrowest, the biggest tracking error in steady state is 0.00264, and the corresponding percentage error is 1.32%. It can be seen from Figure 15(b) that the tracking error in the Y-direction is also small.

Figure 16 shows the tracking errors of the shaft center coordinates under different disturbing forces. In the figure, the blue curve indicates the tracking errors under the condition of instantaneous external disturbing forces (q_x = 500 N, q_y = −500 N) and the red curve indicates the tracking errors under the condition of dynamical forces ( $q_x=q_0\cos \omega t$ , $q_y=q_0\sin \omega t$ , q₀ = 200 N). From Figure 16(a), the blue curve has a big tracking error (the value is 0.04066, and the percentage error is 10.17%) at the moment of t = 0.025 s. But after the adjustment for short time, the tracking errors of shaft center will become small and stable. When the shaft center is disturbed by dynamical force, the tracking errors (i.e. the red curve in Figure 16(a)) have the periodicity and seem like a sine curve. The biggest tracking error of red curve in Figure 16(a) is 0.01732 and the percentage error is 4.33%. The curves in Figure 16(b) are like those in Figure 16(a), and to avoid repetition, this article does not discuss them. From Figures 15 and 16, the tracking errors in steady state are always small under different ovalities and different disturbing forces.

Conclusion

On the basis of previous work,^13,21,22 this article simulates open-loop control movement orbits of shaft center using Eulerian method. With open-loop control method, this research can control shaft center movement orbit and make it an expected ellipse. The above work lays theoretical foundation for machining elliptical holes. Several conclusions can be gained by summarizing the above work:

1. A new hydrostatic bearing (i.e. pocket-nest hydrostatic bearing) is proposed. Comparing with the pocket of common hydrostatic bearing, control pocket of pocket-nest hydrostatic bearing has smaller area and flow. So control sensitivity of hydraulic control system can be enhanced with this new hydrostatic bearing.

2. When the hydrodynamic effect of bearing land is ignored, oil film pressure of bearing land can be linearizingly valued. Analytical expressions of pocket pressure and bearing load capacity are deduced. The method of using analytical expressions to solve pocket pressure and bearing load capacity is very simple.

3. Open-loop control method can control shaft movement and make the shaft center movement orbits’ expected ellipses with different ovalities. Even if the initial coordinates of shaft center are not on the expected ellipse, open-loop control method can make the shaft center close to the expected ellipse.

4. Open-loop control method is of high anti-interference. The simulation results indicate that when the shaft center is disturbed by instantaneous external force or dynamic load during the moving process, open-loop control method has strong robustness against those disturbance forces and make the shaft center quickly approach the expected ellipse.