This paper proposes a novel optimization method to improve the performance of aerostatic spindle. The optimization process is based on the main factors affecting the load and stiffness of aerostatic spindle, which are determined by crosscorrelation analysis. The 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. The results will enforce the industrial application of aerostatic spindle system.
The aerostatic spindle is widely used in precision and ultraprecision machining by the advantages of low friction and high accuracy [
There are two major aspects for the optimization of aerostatic spindle: one focuses on the static characteristics analysis of aerostatic spindle to optimize basic parameters [
Another aspect of research is that the final performance is determined by optimization methods. Shie and Shih [
In this paper, an optimization method is proposed to optimize the factors affecting the performance of the aerostatic spindle. These factors include gas film thickness, diameter of orifice, and the diameter of stomatic distribution circle. The relationship between load capacity, stiffness, and the main influence factors determined by correlation analysis is deduced. The 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.
The circular flat pad aerostatic bearing with eight orificetype restrictor is shown in Figure
Structure of bearing.
Microfactors will lead to decrease of the gas film density and increase of the gas compressibility. Therefore, the traditional Reynolds equation is no longer applicable to describe the microscale flow. The flow factor
Equation (
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
Performance of aerostatic spindle in axial direction. (a) Gas pressure distribution. (b) Relation between bearing stiffness and the gas film clearance.
Then, use the disturbance method to establish unsteady Reynolds equation. The 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 firstorder 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.
The 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
According to the pressure distributions, the load capacity
At the same time, the stiffness of bearing can be obtained by calculating the increment ratio of the bearing capacity and bearing clearance.
The relationship between stiffness and bearing clearance is shown in Figure
Theoretical calculations show that the factors affecting performance of aerostatic spindle include the bearing clearance
In order to analyze the influence of the single factor more intuitively on the final performance of aerostatic bearing, the crosscorrelation 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. This is also known as a sliding dot product or sliding innerproduct. To characterize the correlation between two random variables
After analysis the above, the influence extent of single factor on the stiffness of the aerostatic bearing is obtained, as shown in Figure
Correlation coefficient of the stiffness and influence factors.
In order to find the relationship between the performance of aerostatic spindle and the influence factors. An array
According to the assumption of uniform radial flow of gas, the motion equation in cylindrical coordinates can be simplified as follows:
By integrating equation (
The expression of bearing capacity can be obtained by substituting equations (
Then, we can get
And then, we define the gauge pressure ratio by using Powell’s method.
So, we write the bearing capacity as follows:
Then, we introduce the slit factor
According to the equation (
Using equation (
Import of goal function and constrains.
Optimization results of capacity.
Use equation (
Import of goal function and constrains.
Optimization results of stiffness.
The bearing capacity and stiffness 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 stiffness. The optimization results are shown in Table
Optimal results of single objective function.
Optimization values  Optimization parameters  



 

8.301  3.213  82.531 

12.342  2.912  82.125 
It can be seen from Table
The optimal values of bearing capacity and stiffness calculated above are brought into equation (
The fitness function is shown in formula (
Optimization process.
The result of optimization is shown in Figure
Optimization results of multiple objective.
The comparison of optimization parameters obtained by multiobjective function
Comparison of optimization parameters and original parameters.




 

Original parameters  12  3.5  82.5  1052  120 
Optimization parameters  10.521  3.501  82.214  1136  131 
Comparison of bearing capacity.
Comparison of bearing stiffness.
The aerostatic spindle system and the static stiffness measurement system of aerostatic bearing are, respectively, established by the optimization parameters of the bearing, as shown in Figure
Measurement system of aerostatic bearing static stiffness.
Air compressor provides gas to the aerostatic spindle system, and the supply pressure is 0.5 MPa
Comparison of bearing stiffness.
By optimizing the parameters, the static performance of the aerostatic spindle is improved; however, the dynamic performance of the spindle can be affected. This is because the corresponding frequency varies with the change of bearing capacity and stiffness. Here, the COMBIN14 springdamper unit is used in modal analysis of the spindle model, in which the thrust bearings can be seen as the longitudinal springdamper. The node of each orifice in the gas cavity is only stretched or compressed and not affected by bending and torsion. The radial bearings can be seen as the longitudinal and torsional springdamper, and the node of each orifice in the gas cavity has the rotary element with three degrees of freedom in addition to the tensioncompression in the normal direction. The model of aerostatic spindle is shown in Figure
Model of aerostatic spindle.
And the natural frequencies of the first six orders before and after optimization are calculated and shown in Table
First 6order frequency of spindle in two cases.
Orders  1  2  3  4  5  6 

Original parameters (Hz)  494  496  1439  4857  4876  5591 
Optimization (Hz)  516  518  1503  5075  5094  5842 
The dynamic characteristics of the aerostatic spindle are important in actual work, and the performance in axial direction is the key factor affecting the machining accuracy of the machine tool. In order to better verify the optimization effect of aerostatic bearing, the axial dynamic response is analyzed. The motion equation which has natural frequency and damping coefficient is proposed.
And the response curve is shown in Figure
Dynamic response of aerostatic spindle.
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 affecting the bearing capacity and stiffness of aerostatic bearings are determined by crosscorrelation analysis, and the corresponding actual experimental platform of the aerostatic spindle system is established for analysis. The following major conclusions are obtained:
A new optimization method for the performance of aerostatic spindle considering the main influence factors was proposed and proved experimentally. Taking the load capacity and stiffness of the spindle as the optimize function in the multiobjective function. The relationship between the optimize function and the main influence factors is proposed. Then, the maximum values are obtained by genetic algorithm.
The influence factors affecting the performance of the aerostatic spindle were evaluated, whose influence degree were obtained by crosscorrelation analysis. And the correlation coefficients of more than 0.7 were determined, which are imported into the optimization process as the independent variables.
The dynamic characteristics after optimization were analyzed, which show that the optimization method proposed in the paper provided effective parameters for the aerostatic spindle. The 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. The optimization of aerostatic bearing is proceeded combining the theory with microscale characteristics. The performance of aerostatic bearing is improved.
The data used to support the findings of this study are included within the article.
The authors declare no conflicts of interest in preparing this article.
This research was funded by the National Natural Science Foundation of China (grant nos. 51875005 and 51475010), Beijing Nova Program (Z161100004916156), and Natural Science Foundation of Beijing Municipality (grant no. 3142005).