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The water-lubricated bearings have been paid attention for their advantages to reduce the power loss and temperature rise and increase load capacity at high speed. To fully study the complete dynamic coefficients of two water-lubricated, hydrostatic journal bearings used to support a rigid rotor, a four-degree-of-freedom model considering the translational and tilting motion is presented. The effects of tilting ratio, rotary speed, and eccentricity ratio on the static and dynamic performances of the bearings are investigated. The bulk turbulent Reynolds equation is adopted. The finite difference method and a linear perturbation method are used to calculate the zeroth- and first-order pressure fields to obtain the static and dynamic coefficients. The results suggest that when the tilting ratio is smaller than 0.4 or the eccentricity ratio is smaller than 0.1, the static and dynamic characteristics are relatively insensitive to the tilting and eccentricity ratios; however, for larger tilting or eccentricity ratios, the tilting and eccentric effects should be fully considered. Meanwhile, the rotary speed significantly affects the performance of the hydrostatic, water-lubricated bearings.

Hydrostatic journal bearings are applied widely in spindle-bearing systems owning to their favorable performance characteristics. However, with the requirement of higher machining speed, the limitations of the conventional oil film bearings are apparently due to their remarkable power loss as well as the temperature rise. Therefore, the water-lubricated bearings were developed and have been studied to fulfill the targets of lower power loss, lower temperature rise, and heavier load capacity at high speed.

Many studies related to water-lubricated bearings have been reported in the literatures in the past few years. Liu et al. compared the oil-lubricated and water-lubricated hybrid sliding bearings, and the results show that the latter benefits more from improved processing precision and efficiency [

In actual practice, the bearings and the journals may not be properly aligned as a result of improper assembly or noncentral loading. As a consequence, not only should the translational motion of the rotor be studied, but also the tilting motion of the rotor should be investigated. As a result, the complete stiffness and damping coefficients of a journal bearing, which is important to the vibration of a rotor, should be taken into consideration in four degrees of freedom, including the translation in

Unlike the conventional oil film bearings, water-lubricated bearings utilized in spindle are different in working conditions and characteristics. However, for hydrostatic water-lubricated journal bearings, we are not aware of any previous investigations to study their dynamic characteristics considering the translational and titling motions. Therefore, in this work, we aim to fully study the complete dynamic coefficients for two water-lubricated, hydrostatic journal bearings used to support a rigid rotor. The dynamic characteristics will be categorized into four groups: coefficients of force to displacement, coefficients of force to angle, coefficients of moment to angle, and coefficients of moment to displacement. In the present study, in order to fully study the variations of the complete static and dynamic characteristics of the proposed water-lubricated bearings, the influences of the tilting ratio, rotary speed, and eccentricity ratio on the bearings have been studied.

Figure

The arrangement and coordinate system of the bearings [

The rigid rotor-bearing model.

For an isoviscous, incompressible fluid, the Reynolds equation governing the turbulent bulk flow in nondimensional form is given as

With the increase of rotary speed or film depth, the flow is likely to become turbulent from laminar state. The turbulent coefficients

At the bearing exit plane, the pressure takes a constant value equal to the ambient pressure.

The dimensionless continuity equation at the recess is defined by the global balance between the flow through the orifice restrictor and the recess outflow into the film lands:

The journal center rotates about its steady-state position

The dimensionless perturbed film expression considering the tilting angles is [

The dimensionless perturbed pressure expression is

Substitution of the perturbed equations (

The perturbed quality into the orifice diameter can be obtained by Taylor expansion:

Substitution of the perturbed equations (

The finite difference method (FDM) and a successive over-relaxation (SOR) scheme are implemented to solve (

The quality, frictional power loss, and pump power are calculated by integration of the pressure field on the bearing surfaces:

San Andres et al. carried out a systematic research on the water-lubricated hydrostatic journal bearings both theoretically and experimentally [

The dynamic coefficients of the journal bearings can be calculated by integrating the perturbed pressure across the fluid film. There are 16 stiffness and damping coefficients, and the dynamic coefficients can be grouped into four categories: coefficients of force to displacement, coefficients of force to angle, coefficients of moment to angle, and coefficients of moment to displacement:

The finite difference method (FDM) and a successive over-relaxation (SOR) scheme are implemented to solve (

The fluid film was discretized by rectangular grid with unequal intervals, which are 0.033, 0.088, and 0.1154 for the film land, recess, and return groove, respectively, in the circumferential direction and 0.02 and 0.04 for the film land and recess area in the axial direction. The total number of the grid is 31 × 102. It takes about 35 seconds for the static performance to achieve convergence. A few validation tests were made with a coarser grid of 31 × 86 and finer grid of 37 × 158 with different intervals, and in no case did the predicted results of the static characteristics differ by more than 0.1 percent from those obtained by the initial grid.

The present numerical solution has been correlated and validated with the experimental results available in the literature. A five-recess, turbulent-flow, water-lubricated hydrostatic bearing operating at a high rotational speed is tested by San Andres et al. [

Operating condition of the test bearing [

Orifice diameter (mm) | 2.49 | Supply temperature (°C) | 55 |

Length (mm) | 76.2 | Supply pressure (MPa) | 4 |

Film thickness (um) | 127 | Rotary speed (rpm) | 24600 |

Geometry of the test water-lubricated hydrostatic journal bearing [

Comparison of static performance of experimental results [

Comparison of load capacity

Comparison of mass flow rate

The geometric parameters for the bearings have been presented in Figure

Figure

In order to obtain a better physical insight into the effect of misalignment, the static characteristics have been presented with eccentricity ratio equal to zero. Figures

The effect of tilting angle on the static performance of the water-lubricated bearing.

The effect of tilting angle on the quality

The effect of tilting angle on the power loss

The effect of tilting angle on the temperature rise

Figures

The effect of tilting ratio on the stiffness.

The stiffness of force to displacement versus tilting ratio

The stiffness of force to angle versus tilting ratio

The stiffness of moment to angle versus tilting ratio

The stiffness of moment to displacement versus tilting ratio

The effect of tilting ratio on the damping coefficients.

The damping of force to displacement versus tilting ratio

The damping of force to angle versus tilting ratio

The damping of moment to angle versus tilting ratio

The damping of moment to displacement versus tilting ratio

According to the results, the dynamic coefficients of force to displacement and moment to angle for the front bearing are approximately equal to those for the rear bearing due to the fact that the tilting center almost coincides with the bearing span center. However, the coupled dynamic coefficients of force to angle and moment to displacement for the two bearings have close proximity magnitudes but in the opposite direction due to the fact that the two bearings are arranged at both sides of the mass center which is the origin of the tilting motions. It is observed that, with the increase of the tilting ratio, the stiffness and damping coefficients of the bearings keep almost independent of the tilting ratio when it is not larger than 0.4; however, when the tilting ratio continues to increase, the effect on dynamic coefficients is significant, with a maximum variation rate of 16.7% for the stiffness and 49.2% for the damping coefficients. Furthermore, the larger the tilting ratio is, the greater the differences among the coefficients are. This can be ascribed to the variations of the film thickness induced by the tilting angles and bearing span, which can be calculated according to (

Figure

The effect of rotary speed on the static performance of the water-lubricated bearing.

The effect of rotary speed on the quality

The effect of rotary speed on the power loss

The effect of rotary speed on the temperature rise

Figures

The effect of rotary speed on the stiffness coefficients.

The stiffness of force to displacement versys rotary speed

The stiffness of force to angle versus rotary speed

The stiffness of moment to angle versus rotary speed

The stiffness of moment to displacement versus rotary speed

The effect of rotary speed on the damping coefficients.

The damping of force to displacement versus rotary speed

The damping of force to angle versus rotary speed

The damping of moment to angle versus rotary speed

The damping of moment to displacement versus rotary speed

The influence of eccentricity ratio in aligned condition on static performances of each bearing is as shown in Figure

The effect of eccentricity ratio on the static performance of the water-lubricated bearing.

The effect of rotary speed on the quality

The effect of rotary speed on the power loss

The effect of rotary speed on the temperature rise

Figures

The effect of eccentricity ratio on the stiffness coefficients.

The stiffness of force to displacement versus eccentricity ratio

The stiffness of force to angle versus eccentricity ratio

The stiffness of moment to angle versus eccentricity ratio

The stiffness of moment to displacement versus eccentricity ratio

The effect of eccentricity ratio on the damping coefficients.

The damping of force to displacement versus eccentricity ratio

The damping of force to angle versus eccentricity ratio

The damping of moment to angle versus eccentricity ratio

The damping of moment to displacement versus eccentricity ratio

This paper investigated the complete dynamic coefficients for two hydrostatic, water-lubricated journal bearings used to support a rigid rotor considering the translational and tilting motion. The bulk turbulent flow model and FDM method is used to numerically predict the performance of the bearings. The results show that the proposed water-lubricated hydrostatic bearing has the potential to fulfill the target of lower power loss, temperature rise, and larger load capacity at high speed. On the basis of the results presented, the following conclusions can be drawn.

For a small tilting ratio (<0.4), the influence of tilting ratio on the static and dynamic characteristics of a water-lubricated hydrostatic journal bearing is relatively small; however, when the tilting ratio continues to increase, the power loss and temperature rise increase gradually while the quality decreases, and the effect of tilting ratio on the dynamic coefficients should be taken into consideration.

The quality of the bearings is relatively insensitive to the rotary speed; however, the power loss and temperature rise increase sharply with the rotary speed in an aligned condition. The direct stiffness coefficients vary significantly with the rotary speed due to the hydrodynamic effect while the damping coefficients are almost constant.

For a relatively smaller eccentric condition (≤0.1), the static and dynamic characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricity ratio. However, for a larger eccentric condition, the dynamic characteristics increase or decrease significantly with the eccentricity ratio.

Design film thickness

Orifice diameter

Film thickness

Distance between the mass center and front journal bearing center

Distance between the mass center and rear journal bearing center

Distance between the mass center and the left edge of front journal bearing

Distance between the mass center and the right edge of rear journal bearing

Radius of the bearing

Diameter of a journal bearing

The bearing force

The length of a journal bearing

Supply pressure

The external load

Rotary speed

Flow coefficient

Density

Viscosity

Orifice design coefficient.

The authors declare that there is no conflict of interests regarding the publication of this paper.