An Optimization Method for the Main Influence Factors on the Performance of Aerostatic Spindle

)is paper proposes a novel optimization method to improve the performance of aerostatic spindle. )e optimization process is based on the main factors affecting the load and stiffness of aerostatic spindle, which are determined by cross-correlation analysis. )e maximum values of load capacity and stiffness are obtained from the relationship between the optimization function and the main influence factors. After optimization, the corresponding natural frequency has been improved, which indicates that the optimization method proposed in this paper is more efficient and reliable in the field of precision machining. )e results will enforce the industrial application of aerostatic spindle system.


Introduction
e aerostatic spindle is widely used in precision and ultraprecision machining by the advantages of low friction and high accuracy [1].When the aerostatic bearing is working, the air supply is provided by an external air compressor.
en, the gas enters the air cavity through the orifice to form a layer of gas film which can support the workpiece to machine and achieve frictionless lubrication.
e bearing capacity of the aerostatic spindle mainly comes from the gas film between the aerostatic bearing and the spindle.For ultraprecision machining, the stability of aerostatic spindle should be guaranteed.In fact, the performance of the aerostatic spindle is completely different due to some factors, such as structural parameters of air cavity, orifice diameter, gas film thickness, and air supply pressure.And the performance of the aerostatic spindle is a key factor affecting the quality of the machine [2].Besides, it will be also directly reflected to the surface of the workpiece and affect the quality of the workpiece.erefore, it is important to study the performance of the aerostatic spindle to improve the precision of ultraprecision machining.
ere are two major aspects for the optimization of aerostatic spindle: one focuses on the static characteristics analysis of aerostatic spindle to optimize basic parameters [3].Aguirre et al. [4] proposed an active compensation strategy to achieve nanometer position control, which can improve the performance of the thrust bearing.Chang and Jeng [5] researched the performance of aerostatic bearing by optimizing the bearing diameter, orifice diameter, orifice structure, gas supply pressure, and so on.Li et al. [6] proposed an optimization model by the maximum Mach number to improve dynamic stability.However, this is only one aspect of the optimization of aerostatic spindle.
Another aspect of research is that the final performance is determined by optimization methods.Shie and Shih [7] obtained a numerical solution of the pressure distribution between the surface of the aerostatic bearing and the worktable using the finite difference method.Li et al. [8,9] proposed a coupling optimization method to optimize the spindle structure by magnetic circuit calculation and magnetic field calculation.Wang and Chang [10] used Pareto ranking method and genetic algorithm to optimize the stiffness and air mass flow of porous air bearing.Besides, they used the hypercube-diving method to research the multiobjective optimization of porous air bearing [11].ese optimization methods have different objective function for the different structure of gas cavity and the aerostatic bearing with orifice has a high load capacity.e performance of gas film in microscale is different from the macroscale, and the corresponding optimization in microscale of structure of gas film is seldom.
In this paper, an optimization method is proposed to optimize the factors affecting the performance of the aerostatic spindle.ese factors include gas film thickness, diameter of orifice, and the diameter of stomatic distribution circle.e relationship between load capacity, stiffness, and the main influence factors determined by correlation analysis is deduced.e results are verified from static and dynamic characteristics.And the corresponding simulation and experiment of the bearing capacity and stiffness of the aerostatic spindle are carried out to achieve enhanced performance in widely application.

Structure of Aerostatic Spindle.
e circular flat pad aerostatic bearing with eight orifice-type restrictor is shown in Figure 1.According to the structure characteristics and actual working situation of the aerostatic bearing, the main design parameters contain bearing clearance h, orifice diameter d 3 , the diameter of orifice distribution circle d 2 , the rotate speed of the spindle, and so on.e bearing clearance h, orifice diameter d 3 , and the diameter of orifice distribution circle d 2 are considered as design variables because they have more influence on the aerostatic bearing performance, and the ranges are 5-20 μm for h, 2 mm-5 mm for d 3 , and 75-85 mm for d 2 , respectively.e air supply pressure p s is given between 0.3 MPa and 0.6 MPa.

Flow Field Analysis in Microscale.
Microfactors will lead to decrease of the gas film density and increase of the gas compressibility.erefore, the traditional Reynolds equation is no longer applicable to describe the microscale flow.
e flow factor Q embodied microfactor whose equation is PH is introduced into the Reynolds equation, where C 1 and C 2 can be obtained from the literature [12].So, the Reynolds equation after introducing the flow factor Q is as follows: where p is the film pressure, h is the bearing film thickness, μ is the dynamic viscosity of the lubricating gas, U is the circumferential speed of the bearing surface, and U � ωR 0 .ω is the angular velocity of spindle rotation, R 0 is the bearing radius, and t is the time.Equation ( 1) is converted into the polar coordinate for the convenience of calculation. ( Equation ( 2) is dimensionless for the convenience of calculation.e characteristic pressure of the film is p 0 and the characteristic thickness of film is bearing radius clearance h 0 .Let p � p 0 P, p 0 � ((6μR Two key steps for solving Reynolds equation by finite difference method are (1) dividing the solution domain into nodal grids and (2) transforming partial differential Reynolds equation into difference equation.Finally, the pressure distribution of p a can be obtained by substituting boundary conditions, as shown in Figure 2.
en, use the disturbance method to establish unsteady Reynolds equation.e basic idea is to assume that when the input disturbance is a sinusoidal or cosine function, the output is the superposition of sinusoidal and cosine functions when the object of study is a first-order linear system.So, the integral of the pressure disturbance term in phase with the input disturbance can be defined as dynamic stiffness, and the integral of the pressure disturbance term with phase lag can be defined as dynamic damping coefficient. h where p a is the film pressure in the steady state, h 0 is steadystate film thickness, ω 1 is the perturbation frequency, p b is the pressure perturbation term in phase with the input perturbation, and p c is the pressure perturbation term with phase lag.Substituting equation (4) in equation ( 1), let p A � p a • p b , p B � p a • p c , the disturbance quantities in quadratic and above are removed.e perturbed Reynolds equation is as follows: e dynamic performance is solved by finite difference method.First, the perturbation Reynolds equation is discretized and then meshed.Finally, the boundary conditions of steady state and perturbation are introduced into the equation to solve the problem.By substituting the steady pressure p a and the steady film thickness h 0 obtained from the steady-state solution into the perturbation equation, the dynamic pressure and the dynamic film gap can be obtained.p b and p c can be obtained by solving equation (5).

Performance Analysis of Aerostatic Spindle.
According to the pressure distributions, the load capacity W is calculated by integrating the pressure along the lower surface of bearing.

2
Shock and Vibration At the same time, the sti ness of bearing can be obtained by calculating the increment ratio of the bearing capacity and bearing clearance.
e relationship between sti ness and bearing clearance is shown in Figure 2(b).
e bearing capacity and sti ness of the aerostatic spindle are a ected by many factors, including structural and working parameters.Next, we will evaluate the e ect of the main factors on the spindle performance.

In uence Factors on the Performance of Aerostatic Spindle.
eoretical calculations show that the factors a ecting performance of aerostatic spindle include the bearing clearance h, ori ce diameter d 3 and the diameter of ori ce distribution circle d 2 , the depth of the air cavity d and the pressure supply p s .However, the in uence extent of these factors is di erent.For example, some factors improve the performance of the aerostatic bearing, and others reduce the performance.
erefore, the rst step is to assess the inuence extent of these factors.

Analysis of the Impact Extent of In uence Factors.
In order to analyze the in uence of the single factor more intuitively on the nal performance of aerostatic bearing, the cross-correlation analysis method is proposed, which uses similarity of two series as a function of the lag of one relative to the other to evaluate correlation extent of the single parameter on the performance of aerostatic spindle. is is also known as a sliding dot product or sliding inner-product.To characterize the correlation between two random variables x and y 1,2 , which produce yield data sets x x 1 , . . ., x n and y 1,2 y 1 , . . ., y n , the correlation coe cient is usually used to express the degree of linear correlation between the two variables [13].Shock and Vibration 3 where x is the in uence factors including the bearing clearance h, ori ce diameter d 3 , the diameter of stomatic distribution circle d 2 , the depth of the air cavity d, and the pressure supply p s .e ranges are 5 μm-20 μm for h, 2 mm-5 mm for d 3 , 75 mm-85 mm for d 2 , 0.75 mm-1.0 mm for d, and 0.3 MPa-0.6 MPa for p s .i is the signal number of correlation analysis and y 1 and y 2 are the load capacity and sti ness of aerostatic spindle, respectively.
After analysis the above, the in uence extent of single factor on the sti ness of the aerostatic bearing is obtained, as shown in Figure 3. From this, we can get that the correlation coe cients of the in uence factors h, d 3 , and d 2 are all more than 0.7; however, the correlation values of in uence factor d and p s are both less than 0.6, which indicates the main in uence factors are the bearing clearance h, ori ce diameter d 3 , and the diameter of stomatic distribution circle d 2 .erefore, next, we will optimize the three parameters.

Relationship between Performance and In uence Factors.
In order to nd the relationship between the performance of aerostatic spindle and the in uence factors.An array X with these design variables which include the bearing clearance h, ori ce diameter d 3 , and the diameter of . e industrial applications require high bearing capacity and sti ness, but their values vary with design variables.erefore, the multiobjective optimization containing the bearing capacity and sti ness is proposed, and the multiobjective optimization functions F(x) including the maximum bearing capacity function f 1 (x) and the maximum sti ness function f 2 (x) are established.
where N 1 and N 2 represent weighting factors which re ect the speci c gravity of bearing capacity and sti ness in the total objective function, respectively.And, they are calculated by the equation: where f i (X * ) represents the optimal solution of single constraint problem for the i-th subobjective as the objective function.

Function Determination of the Performance of Aerostatic
Spindle.According to the assumption of uniform radial ow of gas, the motion equation in cylindrical coordinates can be simpli ed as follows: By integrating equation (11) with z twice and combining the equivalent mass ow equation and the gas state equation, we can get the expression of pressure p as follows: e expression of bearing capacity can be obtained by substituting equations ( 12) and ( 13) in equation ( 6): en, we can get f 1 (x).
And then, we de ne the gauge pressure ratio by using Powell's method.
According to the equation ( 7), we can get the objective function f 2 (x) of the sti ness as follows: where x 1 represents the gas lm thickness h, x 2 is the ori ce diameter   e optimal value of the sti ness is max f 2 (x) 135.4.

Optimization Process
e bearing capacity and sti ness of aerostatic spindle are optimized by using genetic algorithm toolbox in simulation system software that combines the optimization theory and genetic algorithm.First, the performance of aerostatic bearing is optimized by the single objective function of the capacity and the sti ness.e optimization results are shown in Table 1.
It can be seen from Table 1 that the contradiction is present in the optimal result of the sti ness and load capacity function, respectively.us multiobjective optimization function is used to optimize the performance of the aerostatic spindle in order to get ideal consistent results.

Optimization of Multiobjective Function.
e optimal values of bearing capacity and sti ness calculated above are brought into equation (10) to obtain the weighting factors N 1 and N 2 .Substituting formulas (15) and ( 19) in ( 9), the multiobjective function including bearing capacity, sti ness, and in uence parameters of aerostatic bearing is determined.
e tness function is shown in formula (12).e multiobjective function and constrains are imported to the genetic algorithm toolbox in the simulation.
e optimization process of the solution is shown as Figure 8.
e result of optimization is shown in Figure 9.When the design variable X [10.521 3.501 82.214], that is, bearing clearance is 10.521 μm, ori ce diameter is 3.501 mm, and the diameter of stomatic distribution circle is 82.214 mm, the optimal result of multiobjective function F(X) is max F(X) 1.8682.

Comparison of Static Performance by Optimization
Parameters.
e comparison of optimization parameters obtained by multiobjective function F(X) and original parameters are shown in Table 2. Using optimization parameters and original parameters, the bearing capacity and sti ness of the di erent bearing clearance are simulated.e bearing performance before and after optimization is shown in Figures 10 and 11.Compared with original parameters, the performance of bearing with optimized parameters is improved.
e bearing capacity and the sti ness are increased 7.98% and 9.17%, respectively.e aerostatic spindle system and the static sti ness measurement system of aerostatic bearing are, respectively, established by the optimization parameters of the bearing, as   shown in Figure 12.An inductive micrometer is used to measure the variation of the displacement of the spindle table, and it is necessary to perform zero adjustment rst.e speci c operation is to determine the height of the induction microprobe by adjusting the height of the measuring gantry.e contact between the probe and the table is within ±2 μm to ensure the accuracy of the test.
Air compressor provides gas to the aerostatic spindle system, and the supply pressure is 0.5 MPa.When the mass block as load is applied to worktable of aerostatic spindle, the workbench will have vertical displacement change.e change is sensed by the probe of the inductive micrometer and transmitted to the inductive micrometer which shows the change in display.e weight of mass block is 50 N, and the method of stepwise loading is used to perform multiple measurements and average.According to loaded load of the aerostatic bearing and the displacement change detected by the inductance micrometer, the static sti ness value is calculated by the sti ness calculation formula.e values of sti ness test are shown in Figure 13 which shows that the sti ness after optimization is more close to the values of sti ness test.So, the correctness and the feasibility of optimization method in this paper are veri ed by sti ness test.

Comparison of Dynamic Response by Optimization
Parameters.By optimizing the parameters, the static performance of the aerostatic spindle is improved; however, the dynamic performance of the spindle can be a ected.is is because the corresponding frequency varies with the change of bearing capacity and sti ness.Here, the COMBIN14 spring-damper unit is used in modal analysis of the spindle model, in which the thrust bearings can be seen as the longitudinal spring-damper.e node of each ori ce in the gas cavity is only stretched or compressed and not a ected by bending and torsion.e radial bearings can be seen as the longitudinal and torsional spring-damper, and the node of each ori ce in the gas cavity has the rotary element with three degrees of freedom in addition to the tensioncompression in the normal direction.e model of aerostatic spindle is shown in Figure 14.
And the natural frequencies of the rst six orders before and after optimization are calculated and shown in Table 3, in which we can get that the corresponding frequency has been increased 5% after optimization.e dynamic characteristics of the aerostatic spindle are important in actual work, and the performance in axial direction is the key factor a ecting the machining accuracy

Shock and Vibration
of the machine tool.In order to better verify the optimization e ect of aerostatic bearing, the axial dynamic response is analyzed.e motion equation which has natural frequency and damping coe cient is proposed.
where ω n is the natural frequency, m is the mass of the spindle, which is 3.7 kg, and C is the damping coe cient.e solution of above equation is as shown.
And the response curve is shown in Figure 15 which indicates that the response speed after optimization has been improved, but it still in the same order of magnitude with the values before optimization.

Conclusions
In this paper, the performance of aerostatic spindle has been improved by a novel optimization approach, and the results of before and after optimization are compared in static and dynamic aspect.In the optimization process, the main factors a ecting the bearing capacity and sti ness of aerostatic bearings are determined by cross-correlation analysis, and the corresponding actual experimental platform of the aerostatic spindle system is established for analysis.e following major conclusions are obtained: (1) A new optimization method for the performance of aerostatic spindle considering the main in uence factors was proposed and proved experimentally.
Taking the load capacity and sti ness of the spindle as the optimize function in the multiobjective function.e relationship between the optimize function and the main in uence factors is proposed.
en, the maximum values are obtained by genetic algorithm.
(2) e in uence factors a ecting the performance of the aerostatic spindle were evaluated, whose in uence degree were obtained by cross-correlation analysis.And the correlation coe cients of more than 0.7 were determined, which are imported into the optimization process as the independent variables.(3) e dynamic characteristics after optimization were analyzed, which show that the optimization method proposed in the paper provided e ective parameters for the aerostatic spindle.e corresponding natural frequency and the response speed have been increased when the response amplitude is in the same order magnitude with the value before optimization.e optimization of aerostatic bearing is proceeded combining the theory with microscale characteristics.
e performance of aerostatic bearing is improved.
e o f t h e c ir c u m f e r e n t ia l d ir e c t io n N o d e o f th e ra d ia l d ir ec ti o bearing clearance h (μm) (b)

Figure 2 :
Figure 2: Performance of aerostatic spindle in axial direction.(a) Gas pressure distribution.(b) Relation between bearing sti ness and the gas lm clearance.

Figure 3 :
Figure 3: Correlation coe cient of the sti ness and in uence factors.

d 3 ,
and x 3 is the diameter of distribution circle d 2 .d 1 represents the external diameter of bearing, d 4 represents the inner diameter of bearing, and p s represents the air supply pressure.p 0 represents the atmospheric pressure and the discharge coe cient C D 0.8.

Figure 9 :
Figure 9: Optimization results of multiple objective.

Table 2 :
Comparison of optimization parameters and original parameters.

Table 1 :
Optimal results of single objective function.

Table 3 :
First 6-order frequency of spindle in two cases.