A comprehensive analysis of factors affecting the accuracy of the precision hydrostatic spindle with mid-thrust bearing layout

Abstract

This paper investigates the influence of rotor coaxiality error on spindle accuracy under different operating conditions in the precision hydrostatic spindle with a mid-thrust bearing structure. Firstly, a linearized model of rotor dynamics equation with five degrees-of-freedom is established based on Newton’s law of motion and angular momentum principle. Then, by solving the Reynolds equation and the flow continuity equation using the mathematical perturbation method, the steady and transient pressure distribution functions of the hydrostatic bearings are obtained; subsequently, the stiffness and damping coefficient matrices of the spindle are determined. Furthermore, the linearized stiffness and damping coefficients are substituted into the rotor dynamics equation to obtain the instantaneous acceleration of the rotor, and the motion trajectory of the rotor is solved iteratively by the Euler method. Finally, the paper studies the influences of different rotor coaxiality, rotating speed, cutting load, oil supply pressure, and oil film clearance on dynamic characteristics, amplitude amplification factor, and radial runout of the spindle by computer simulation. The simulation results show that the spindle runout is more sensitive to rotor coaxiality error and oil film clearance.

Appendix

1.1 Nomenclature

A _r :

Recess area

C ₀ :

Design oil film thickness of the journal and thrust bearing

C_L, C_R, C_T:

Damping matrices of left journal bearing, right journal bearing, and thrust bearing

${c}_{ij},{\overline{c}}_{ij}$ :

Damping coefficients and their dimensionless form (i, j = x, y, z, θ_x, θ_y)

d _c :

Diameter of orifice restrictor

D :

Diameter of journal bearing

e_e,${\overline{e}}_e$:

Are the rotor coaxiality error and its dimensionless form

F_x, F_y, F_z, M_x, M_y:

Cutting force and cutting moment

g :

The acceleration of gravity

$h,\overline{h}$ :

Oil film thickness and its dimensionless form

${\overline{h}}_j$, ${\overline{h}}_t$:

The oil film thicknesses of the journal and thrust bearings

H _f :

Friction power (H_f = η(πDL − 0.75A_r)/C₀)

H _p :

Pumping power (H_p = p_sq)

I_x, I_y:

The mass moments of inertia of the spindle rotor

K_L, K_R, K_T:

Stiffness matrices of left journal bearing, right journal bearing, and thrust bearing

K _p :

Concentric power ratio (K_p = H_f/H_p)

${k}_{ij},{\overline{k}}_{ij}$ :

Stiffness coefficients and their dimensionless form (i, j = x, y, z, θ_x, θ_y)

L, R, T:

Left journal bearing, right journal bearing, and thrust bearing

L _j :

Bearing axial length

L _a :

The width of axial land

m:

The mass of rotor

${M}_x^f$ and ${M}_y^f$:

The transient moments of oil film along x/y axis

$p,\overline{p}$ :

Oil film pressure and its dimensionless form

p _r :

Absolute recess pressure

p _s :

Oil supply pressure

${Q}_{in},{\overline{Q}}_{in}$ :

Inflow rate of recess and its dimensionless form

${Q}_{out},{\overline{Q}}_{out}$ :

Outflow rate of recess and its dimensionless form

${r}_i,\overline{R}$ :

Radius of thrust bearing and its dimensionless form

${\overline{r}}_i$ :

The dimensionless oil film thickness of thrust bearing

R₁,R₄:

Inner and outer radius of thrust bearing

R₂,R₃:

Inner and outer radius of recess in the thrust bearing

S _h :

Velocity coefficients of journal bearing

S _t :

Velocity coefficients of thrust bearing

$t,\overline{t}$ :

Times, $\overline{t}=\omega t$

U :

Velocity on the surface of journal bearing

x, y, z, θ_x, θ_y:

Cartesian coordinates

$\overline{z}$ :

Dimensionless coordinate of axial distance

${z}_i,{\overline{Z}}_i$ :

The axial distance from the rotation center and its dimensionless form

α :

Flow coefficient of orifice restrictor

β :

Concentric pressure ratio (β = p_r/p_s)

λ :

Frequency ratio

χ :

Damping coefficient

δ :

Dimensionless orifice restrictor coefficient

ω :

Rotary speed of spindle

ω _n :

Natural frequency

ρ :

Density of oil

η :

Viscosity of oil

θ :

Dimensionless circumferential direction

∆p:

Pressure difference of orifice restrictor

∆t:

The simulation time step

$\Delta x,\Delta y,\Delta z,{\theta}_x,{\theta}_y,\Delta \overline{x},\Delta \overline{y},\Delta \overline{z}$ :

The displacement of spindle rotor and their dimensionless form

ξ :

x, y, z, θ_x, θ_y

ζ :

$0,x,y,{\theta}_x{\theta}_y,\dot{x},\dot{y},\dot{z},{\dot{\theta}}_x,{\dot{\theta}}_y$

ς :

L, R