CN105972081B - Aerostatic bearing performance optimization method under a kind of minute yardstick - Google Patents

Aerostatic bearing performance optimization method under a kind of minute yardstick Download PDF

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CN105972081B
CN105972081B CN201610426093.3A CN201610426093A CN105972081B CN 105972081 B CN105972081 B CN 105972081B CN 201610426093 A CN201610426093 A CN 201610426093A CN 105972081 B CN105972081 B CN 105972081B
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CN105972081A (en
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陈东菊
高雪
董丽华
范晋伟
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Beijing University of Technology
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Beijing University of Technology
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16C—SHAFTS; FLEXIBLE SHAFTS; ELEMENTS OR CRANKSHAFT MECHANISMS; ROTARY BODIES OTHER THAN GEARING ELEMENTS; BEARINGS
    • F16C32/00—Bearings not otherwise provided for
    • F16C32/06—Bearings not otherwise provided for with moving member supported by a fluid cushion formed, at least to a large extent, otherwise than by movement of the shaft, e.g. hydrostatic air-cushion bearings
    • F16C32/0603—Bearings not otherwise provided for with moving member supported by a fluid cushion formed, at least to a large extent, otherwise than by movement of the shaft, e.g. hydrostatic air-cushion bearings supported by a gas cushion, e.g. an air cushion

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  • Engineering & Computer Science (AREA)
  • General Engineering & Computer Science (AREA)
  • Mechanical Engineering (AREA)
  • Magnetic Bearings And Hydrostatic Bearings (AREA)
  • Other Investigation Or Analysis Of Materials By Electrical Means (AREA)

Abstract

The invention discloses one kind to be applied to aerostatic bearing performance optimization method under minute yardstick, belongs to hydrokinetics calculation field.This method combination aerostatic bearing operation principle, characteristic coefficient Q is quoted to embody the rarified flow phenomenon of Bearing inner gas；According to bearing arrangement characteristic, it is determined that influenceing the parametric variable of bearing performance；According to bearing concepts, the scope of parametric variable is determined；The carrying force function and stiffness function expression formula of bearing are determined, is laid the first stone for the multiple objective function of next step；Multiple objective function is determined, draws bearing capacity and rigidity value after optimization.The present invention determines multiple objective function during bearing optimization according to the bearing airfilm pressure drawn under minute yardstick, carry the combination of force function and stiffness function, the lifting of bearing capacity and rigidity is realized, there is certain directive significance for aerostatic bearing performance study.

Description

Aerostatic bearing performance optimization method under a kind of minute yardstick
Technical field
The present invention relates to one kind to be applied to aerostatic bearing performance optimization method under minute yardstick, belongs to hydrodynamics meter Calculation field.
Background technology
Accurate and ultra-precision machine tool is in accurate and Ultra-precision Turning in occupation of very important status, static air pressure static pressure Support member of the bearing as lathe, the influence for lathe precision at work is very important, just because of static pressure axle The secure support held could cause main shaft in the course of the work held stationary, efficiently, precisely.Therefore, hydrostatic bearing is super in precision Influence in Precision Machining is very crucial, and the raising of the optimization of bearing performance for machine finish has very important meaning Justice.
Optimization design is the new technology currently developed rapidly, and it is related to subject than wide, comprising Structural Dynamics, Design method, optimized algorithm, computer technology and various branchs of mechanics etc..The final purpose of optimization is according to system performance It is required that design one can bear maximum load, operational shock is small and has the product structure of superperformance.Aerostatic bearing The lifting of performance influences very big, it is necessary to be optimized to bearing parameter for the machining accuracy of lathe, and then realizes bearing performance Lifting.Air-film thickness is smaller inside aerostatic bearing, generally lies in micron dimension, therefore flowing of the air in bearing Belong to the research category of minute yardstick, and then the analysis that bearing performance is different under macro-scale, this will necessarily make traditional Simulation analysis result and actual result produce certain error, it is necessary to minute yardstick stream is combined in the research of air film flowing law The research method of body flowing, in combination with optimum theory, certain lifting is carried out to minute yardstick lower bearing performance.
The content of the invention
For above-mentioned the problem of technically existing at present, the invention provides static pressure axle under the influence of one kind consideration rarified flow Performance optimization method is held, this method considers the influence that a kind of characteristic factor in rarified flow is brought, and this method, which has, to be calculated The advantages that simplicity, controllability, realize the lifting of hydrostatic bearing performance.
Aerostatic bearing performance optimization method under a kind of minute yardstick, including following steps：
S1 is introduced under minute yardstick a kind of characteristic in rarified flow, establishes aerostatic bearing pressure distribution side under minute yardstick Journey, such as formula (1)
In formula, n be axis system rotating speed, Q be rarified flow in characteristic coefficient, p be bearing in gas pressure value, η For the dynamic viscosity of air；H is bearing clearance；R is along bearing radial direction polar coordinates；θ is along bearing circumferential direction coordinate, t For the time.
S2 carries out nondimensionalization processing to aerostatic bearing pressure distribution equation (1), and it is air to take bearing reference pressure Pressure p0, bearing axial direction reference length is bearing clearance h0, bearing radial direction reference length is throttling pore size distribution radius of circle r0, make p= p0P, P are gas pressure value in dimensionless bearing, whereinH=Hh0, H be dimensionless bearing clearance value, r=Rr0, R To be nondimensional along bearing radial direction polar value.Reynolds equation after nondimensionalization is：
S3 by equation (2) linearization process, obtains following lienarized equation using finite difference calculus：
Wherein, Δ r be along bearing radial direction Gridding length, Δ θ be along bearing circumferential direction Gridding length, (i, J) it is position coordinates in bearing, ri,jFor length of (i, the j) place along bearing radial direction, Pi,jFor in (i, j) place dimensionless bearing Gas pressure value；Using over-relaxation iterative method combination MATLAB software platform numerical solution lienarized equations (3), thin effect is produced Gas pressure in lower bearing is answered to be distributed.
S4 determines to influence the major parameter of bearing performance, that is, the determination of design variable in optimizing, this method is by bearing clearance H, bore dia d is supplied3, air vent pitch circle diameter d2As design variable.
The determination of S5 multiple objective functions
The mode that carrying force function is combined with stiffness function is taken as multi-goal optimizing function, this multiple-objection optimization letter Number includes：Object function k under bearing capacity is optimal1(x) the object function k under, rigidity is optimal2(x).Total object function be by What both were formed in the form of linear weighted function, i.e.,：
F (x)=N1k1(x)+N2k2(x) (4)
Wherein N1And N2Weighted factor is represented, embodies proportion of the bearing capacity and stiffness in catalogue scalar functions respectively.
Wherein ki(X*) expression is designated as the optimal solution of single restricted problem of object function with i-th of subhead.
S6 carries the determination of force function
In formula, W is bearing capacity.
Therefore, subhead scalar functions k1(x) it is：
In formula, psFor supply gas pressure value, piFor the Bearing inner gas pressure value p being calculated in S3I, j。
The determination of S7 stiffness functions
In formula, K is rigidity.
Therefore, subhead scalar functions k2(x) it is：
In formula, paFor atmospheric pressure.d1For bearing outside diameter, d2For air vent pitch circle diameter, d3Supply bore dia, d4Bearing Internal diameter.
S8 constraintss determine
For the characteristic for the aerostatic bearing studied, the scope of design variable is determined.
S9 writes program solution
Optimization object function and constraints are respectively written as M files, are updated in MATLAB GAtoolboxes, solution obtains Bearing parameter and bearing performance after must optimizing.
Compared with prior art, the present invention has advantages below：
The present invention draws gas rarified flow under the minute yardstick that will ignore in traditional design and introduced, and realizes Bearing inner gas Specificity analysis yardstick reaches minute yardstick research category, and is determined that bearing optimizes according to the bearing airfilm pressure drawn under minute yardstick When multiple objective function, that is, carry the combination of force function and stiffness function, realize the lifting of bearing capacity and rigidity, for Aerostatic bearing performance study has certain directive significance.Using the bearing performance that emulation mode of the present invention is drawn and tradition Emulation mode is compared, and wherein bearing capacity improves 7.98%, and rigidity improves 9.17%, and this is for the excellent of aerostatic bearing Changing design has certain realistic meaning.
Brief description of the drawings
Fig. 1 is the flow chart of method involved in the present invention.
Fig. 2 is aerostatic bearing structural representation.
In figure：d1For bearing outside diameter, d2For air vent pitch circle diameter, d3To supply bore dia, d4For bearing bore diameter
Embodiment
The method of the invention is realized by MATLAB software programs.
The flow chart of the method for the invention is as shown in figure 1, specifically include following steps：
Step 1, with reference to aerostatic bearing operation principle, characteristic coefficient Q is quoted to embody the thin of Bearing inner gas Effect phenomenon, establish aerostatic bearing internal gas pressure distribution equation under minute yardstick.
Step 2, using finite difference method pressure distribution equation, minute yardstick lower bearing pressure values are obtained.
Step 3, according to bearing arrangement characteristic, it is determined that influenceing the parametric variable of bearing performance.
Step 4, according to bearing concepts, the constraints during scope, i.e. bearing optimization of parametric variable is determined.
Step 5, the pressure values drawn according to step 2, the carrying force function and stiffness function expression formula of bearing is determined, is The multiple objective function of next step lays the first stone.
Step 6, multiple objective function is determined, while object function and constraints are written as M files.
Step 7, the M files write are imported into MATLAB GAs Toolboxes, Optimization Solution, draws the bearing after optimization Bearing capacity and rigidity value.
Table 1 gives the contrast of the front and rear bearing parameter of optimization and performance number, it can be seen that the bearing capacity after optimization 7.98% is improved, rigidity improves 9.17%.
Table 1 optimizes front and rear parameter comparison

Claims (2)

  1. A kind of 1. aerostatic bearing performance optimization method under minute yardstick, it is characterised in that：This method includes following steps：
    S1 is introduced under minute yardstick a kind of characteristic in rarified flow, establishes aerostatic bearing pressure distribution equation under minute yardstick, such as Formula (1)
    <mrow> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <mi>r</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <msup> <mi>ph</mi> <mn>3</mn> </msup> <mi>Q</mi> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>p</mi> </mrow> <mrow> <mo>&amp;part;</mo> <mi>r</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mo>&amp;part;</mo> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <msup> <mi>ph</mi> <mn>3</mn> </msup> <mi>Q</mi> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>p</mi> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>=</mo> <mn>6</mn> <mi>&amp;eta;</mi> <mi>n</mi> <mfrac> <mrow> <mo>&amp;part;</mo> <mrow> <mo>(</mo> <mi>p</mi> <mi>h</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> <mo>+</mo> <mn>12</mn> <mi>&amp;eta;</mi> <mfrac> <mrow> <mo>&amp;part;</mo> <mrow> <mo>(</mo> <mi>p</mi> <mi>h</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>t</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    In formula, n is the rotating speed of axis system, and Q is the characteristic coefficient in rarified flow, and p is gas pressure value in bearing, and η is sky The dynamic viscosity of gas；H is bearing clearance；R is along bearing radial direction polar coordinates；θ is along bearing circumferential direction coordinate, when t is Between；
    S2 carries out nondimensionalization processing to aerostatic bearing pressure distribution equation (1), and it is atmospheric pressure to take bearing reference pressure p0, bearing axial direction reference length is bearing clearance h0, bearing radial direction reference length is throttling pore size distribution radius of circle r0, make p=p0P, P is gas pressure value in dimensionless bearing, whereinH=Hh0, H be dimensionless bearing clearance value, r=Rr0, R is nothing Dimension along bearing radial direction polar value；Reynolds equation after nondimensionalization is：
    <mrow> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <mi>R</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <msup> <mi>PH</mi> <mn>3</mn> </msup> <mi>Q</mi> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>P</mi> </mrow> <mrow> <mo>&amp;part;</mo> <mi>R</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mo>&amp;part;</mo> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <msup> <mi>PH</mi> <mn>3</mn> </msup> <mi>Q</mi> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>P</mi> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>6</mn> <mi>n</mi> <mo>&amp;part;</mo> <mrow> <mo>(</mo> <mi>P</mi> <mi>H</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>12</mn> <mo>&amp;part;</mo> <mrow> <mo>(</mo> <mi>P</mi> <mi>H</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>t</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    S3 by equation (2) linearization process, obtains following lienarized equation using finite difference calculus：
    <mrow> <msup> <mi>h</mi> <mn>3</mn> </msup> <mi>Q</mi> <mfrac> <mrow> <msubsup> <mi>P</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mrow> <msup> <mrow> <mo>(</mo> <mi>&amp;Delta;</mi> <mi>r</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>+</mo> <msup> <mi>h</mi> <mn>3</mn> </msup> <mi>Q</mi> <mfrac> <mrow> <msub> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msub> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mi>&amp;Delta;</mi> <mi>r</mi> </mrow> </mfrac> <mo>+</mo> <msup> <mi>h</mi> <mn>3</mn> </msup> <mi>Q</mi> <mfrac> <mrow> <msub> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mn>2</mn> <msub> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <msup> <mi>P</mi> <mn>2</mn> </msup> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mi>&amp;Delta;</mi> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>=</mo> <mn>12</mn> <mi>n</mi> <mi>h</mi> <mfrac> <mrow> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi>&amp;Delta;</mi> <mi>&amp;theta;</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, Δ r is the Gridding length along bearing radial direction, and Δ θ is the Gridding length along bearing circumferential direction, and (i, j) is Position coordinates in bearing, ri,jFor length of (i, the j) place along bearing radial direction, Pi,jFor gas in (i, j) place dimensionless bearing Pressure value；Using over-relaxation iterative method combination MATLAB software platform numerical solution lienarized equations (3), produce under rarified flow Gas pressure is distributed in bearing；
    S4 determine influence bearing performance major parameter, that is, optimize in design variable determination, this method by bearing clearance h, supply Hole diameter d3, air vent pitch circle diameter d2As design variable；
    The determination of S5 multiple objective functions
    The mode that carrying force function is combined with stiffness function is taken as multi-goal optimizing function, this multi-goal optimizing function bag Include：Object function k under bearing capacity is optimal1(x) the object function k under, rigidity is optimal2(x)；Total object function is by both Formed in the form of linear weighted function, i.e.,：
    F (x)=N1k1(x)+N2k2(x) (4)
    Wherein N1And N2Weighted factor is represented, embodies proportion of the bearing capacity and stiffness in catalogue scalar functions respectively；
    <mrow> <msub> <mi>N</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>X</mi> <mo>*</mo> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    Wherein ki(X*) expression is designated as the optimal solution of single restricted problem of object function with i-th of subhead；
    S6 carries the determination of force function
    <mrow> <mtable> <mtr> <mtd> <mrow> <mi>W</mi> <mo>=</mo> <msup> <msub> <mi>&amp;pi;d</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mi>hp</mi> <mi>i</mi> </msub> <mo>&amp;lsqb;</mo> <mn>1</mn> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <msub> <mi>p</mi> <mi>s</mi> </msub> <msub> <mi>p</mi> <mi>i</mi> </msub> </mfrac> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mn>2</mn> <mi>ln</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>d</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>4</mn> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </msup> <mo>&amp;rsqb;</mo> <mo>+</mo> <msup> <msub> <mi>&amp;pi;d</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mi>hp</mi> <mi>i</mi> </msub> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mfrac> <msub> <mi>p</mi> <mi>s</mi> </msub> <msub> <mi>p</mi> <mi>i</mi> </msub> </mfrac> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mn>2</mn> <mi>ln</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>d</mi> <mn>1</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <msup> <msub> <mi>d</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <mo>-</mo> <msup> <msub> <mi>d</mi> <mn>4</mn> </msub> <mn>2</mn> </msup> <mo>)</mo> </mrow> <msub> <mi>p</mi> <mi>s</mi> </msub> <mo>+</mo> <msup> <msub> <mi>&amp;pi;d</mi> <mn>3</mn> </msub> <mn>2</mn> </msup> <msub> <mi>p</mi> <mi>i</mi> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    In formula, W is bearing capacity；
    Therefore, subhead scalar functions k1(x) it is：
    <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <msub> <mi>&amp;pi;d</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>&amp;lsqb;</mo> <mn>1</mn> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <msub> <mi>p</mi> <mi>s</mi> </msub> <msub> <mi>p</mi> <mi>i</mi> </msub> </mfrac> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mn>2</mn> <mi>ln</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>x</mi> <mn>3</mn> </msub> <msub> <mi>d</mi> <mn>4</mn> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </msup> <mo>&amp;rsqb;</mo> <mo>+</mo> <msup> <msub> <mi>&amp;pi;d</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mfrac> <msub> <mi>p</mi> <mi>s</mi> </msub> <msub> <mi>p</mi> <mi>i</mi> </msub> </mfrac> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mn>2</mn> <mi>ln</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>d</mi> <mn>1</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>&amp;rsqb;</mo> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <msup> <msub> <mi>d</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <mo>-</mo> <msup> <msub> <mi>d</mi> <mn>4</mn> </msub> <mn>2</mn> </msup> <mo>)</mo> </mrow> <msub> <mi>p</mi> <mi>s</mi> </msub> <mo>+</mo> <msup> <msub> <mi>&amp;pi;x</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mi>p</mi> <mi>i</mi> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
    In formula, psFor supply gas pressure value；piFor the Bearing inner gas pressure value p being calculated in S3I, j；
    The determination of S7 stiffness functions
    <mrow> <mi>K</mi> <mo>=</mo> <mfrac> <mrow> <mi>d</mi> <mi>W</mi> </mrow> <mrow> <mi>d</mi> <mi>h</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <mn>0.98</mn> </mrow> <mi>h</mi> </mfrac> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mi>s</mi> </msub> <mo>-</mo> <msub> <mi>p</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mfrac> <mi>&amp;pi;</mi> <mn>2</mn> </mfrac> <msub> <mi>d</mi> <mn>3</mn> </msub> <mo>&amp;lsqb;</mo> <mfrac> <mrow> <msup> <msub> <mi>d</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>-</mo> <msup> <msub> <mi>d</mi> <mn>4</mn> </msub> <mn>2</mn> </msup> </mrow> <mrow> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>d</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>4</mn> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <msub> <mi>d</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <mo>-</mo> <msup> <msub> <mi>d</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> <mrow> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>d</mi> <mn>1</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&amp;rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
    In formula, K is rigidity；
    Therefore, subhead scalar functions k2(x) it is：
    <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <mn>0.98</mn> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </mfrac> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mi>s</mi> </msub> <mo>-</mo> <msub> <mi>p</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mfrac> <mi>&amp;pi;</mi> <mn>2</mn> </mfrac> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>&amp;lsqb;</mo> <mfrac> <mrow> <msup> <msub> <mi>x</mi> <mn>3</mn> </msub> <mn>2</mn> </msup> <mo>-</mo> <msup> <msub> <mi>d</mi> <mn>4</mn> </msub> <mn>2</mn> </msup> </mrow> <mrow> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>x</mi> <mn>3</mn> </msub> <msub> <mi>d</mi> <mn>4</mn> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <msub> <mi>d</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <mo>-</mo> <msup> <msub> <mi>x</mi> <mn>3</mn> </msub> <mn>2</mn> </msup> </mrow> <mrow> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>d</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&amp;rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
    In formula, paFor atmospheric pressure；d1For bearing outside diameter, d2For air vent pitch circle diameter, d3Supply bore dia, d4In bearing Footpath；
    S8 constraintss determine
    For the characteristic for the aerostatic bearing studied, the scope of design variable is determined；
    S9 writes program solution
    Optimization object function and constraints are respectively written as M files, are updated in MATLAB GAtoolboxes, it is excellent to solve acquisition Bearing parameter and bearing performance after change.
  2. 2. aerostatic bearing performance optimization method under a kind of minute yardstick according to claim 1, it is characterised in that：We Method is realized by MATLAB software programs, specifically includes following steps：
    Step 1, with reference to aerostatic bearing operation principle, characteristic coefficient Q is quoted to embody the rarified flow of Bearing inner gas Phenomenon, establish aerostatic bearing internal gas pressure distribution equation under minute yardstick；
    Step 2, using finite difference method pressure distribution equation, minute yardstick lower bearing pressure values are obtained；
    Step 3, according to bearing arrangement characteristic, it is determined that influenceing the parametric variable of bearing performance；
    Step 4, according to bearing concepts, the constraints during scope, i.e. bearing optimization of parametric variable is determined；
    Step 5, the pressure values drawn according to step 2, the carrying force function and stiffness function expression formula of bearing is determined, is next The multiple objective function of step lays the first stone；
    Step 6, multiple objective function is determined, while object function and constraints are written as M files；
    Step 7, the M files write are imported into MATLAB GAs Toolboxes, Optimization Solution, draws the loading ability of bearing after optimization Power and rigidity value.
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