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Fig 728: The relationship of stiffness with air pressure [25]
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Fig 729: A Toshiba work spindle [26]
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axial stiffness of 1,140 N/ m [27] The Precitech high-speed air-bearing spindle has a speed range of 10 10,000 rpm with a load capacity of 18 kg (40 lb) and an axial stiffness of 31 N/ m (175,000 lb/in) [28] Figure 732 shows the schematic diagram of a work spindle and a grinding spindle in a Moore ultra-precision machine An air turbine drives the grinding spindle shown in
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Fig 730: A Moore aerostatic work spindle for ultra-precision machines [27]
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Fig 731: A Precitech Nanoform 200 work spindle [28] Fig 732: Front view of machine work [29]
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Figure 733 It has a maximum speed of 70,000 rpm [29] Due to the high stiffness and rigidity required, the grinding spindle is also supported by air bearings Passive air bearings are used in spindles for woodworking The spindle shown in Figure 734 basically consists of a hollow shaft (1), journal bearings (2 and 10), thrust bearing (4) and an 8 kW asynchronous motor The spindle is capable of 36,000 rpm The runout at 0 rpm is less than 1 m in both the axial and radial directions, whereas the stiffness is 100 N/ m in the axial direction and around 40 N/ m in the radial direction [4]
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738 Mathematical Approximation of Aerostatic Bearings
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As in hydrostatic bearings, the pressure and load capacity of the aerostatic bearing can be roughly
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Fig 733: A Moore air bearing grinding spindle attachment [29]
approximated by a one-dimensional flow (Figure 735) Further design details will be discussed based on an example in the next section It may be beneficial to compare the pressure profile for the two
Fig 734: A spindle with passive air bearings for woodworking [4]
types of bearings In the half station, feeding aerostatic bearing where only one row of jet is used at the middle of the bearing, the pressure profile is of a triangular shape However, if two rows of jets are used either at the one-quarter or the one-eighth feeding station, the pressure profiles will overlap resembling that of the hydrostatic bearing (Figure 736)
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Fig 735: The load and pressure relationship based on a one-dimensional flow
739 Theory of Aerostatic Lubrication
Most of the mathematical analysis behind aerostatic lubrication is based on fluid mechanics This information is of little practical importance for designers For this reason, only the basic theory is illustrated in order to maintain the simplicity of the design
Gas flow lines
Pressure distributions (a) Axial flow model (b) Real bearing
Fig 736: The theoretical and actual pressure distribution [3]
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The flow in the clearance of gas bearings is usually laminar with pressure losses mainly due to the viscous shear in the air film The study of this effect is assumed as flow between parallel plates However, it is different for rectangular slots and circular plates which are mainly applied for journal and thrust bearings, respectively [3] A number of assumptions listed next are made to simplify the analysis and to apply the Navier Stokes equation: Inertia forces due to acceleration can be neglected compared with frictional forces due to viscous shearing Laminar flow conditions exist at all points in the gas film Pressure is constant over any section normal to the direction of flow There is no slip at the boundaries between the fluid and the plates The complete analysis begins with the basic Navier Stokes equation which is given as
1 P 2u = 2 x y
where u is the velocity of the gas at any point, P the pressure, the viscosity of the gas, g the gravitational field and h is the thickness of the gas film By applying boundary conditions as in Figure 737 and the principle of the conservation of mass, expressions representing the pressure and mass flow can be obtained Detailed steps can be obtained from Design of Aerostatic Bearings by JW Powell [3] and any fluid mechanics book that deals with the fundamentals
Fig 737: Munson et al s illustration of flow between two stationary parallel plates [30]
The load capacity and the stiffness of an aerostatic journal bearing depend upon the design value of the gauge pressure ratio, Kgo, which varies with the eccentricity ratio as shown in Figure 76 The optimum value of Kgo for various eccentricity ratios as recommended by Shires is shown in
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