A calculation method for static characteristics of gas hydrostatic bearings based on finite difference method

Abstract

The present invention relates to a method for calculating the static characteristics of a gas hydrostatic bearing based on a finite difference method. S1. Define the structural dimensions of the bearing unit, give the gas source pressure and gas film thickness, and divide the grid to determine the coordinates of each node and the gas film height; S2. Give the throttle outlet pressure P d S3, calculate the distribution of pressure in the gas film; S4, calculate the mass flow rate of gas inflow and the mass flow rate of gas outflow, and calculate the relative error of mass flow rate; S5, determine whether the relative error meets the given accuracy, if so, enter step S6, if not, use the binary method to revise P d The value of , returns to step S3; S6, calculates the static characteristics of the gas static pressure bearing according to the result of the gas film pressure. The static characteristics of the gas static pressure bearing are calculated more accurately, and a calculation method for angular stiffness is given, which can further improve the bearing capacity and angular stiffness in the structural design of the gas static pressure bearing and increase the scope of application of the gas static pressure bearing.

Description

Technical Field

[0001] The invention relates to the design and production of a gas static pressure bearing, and in particular to a calculation method for the static characteristics of a gas static pressure bearing based on a finite difference method. Background Art

[0002] Gas hydrostatic bearings are provided with high-pressure gas from the outside, and a gas film with a certain pressure is formed in the bearing gap to support the supported parts. They have the advantages of low friction heat generation, high precision, and long life, and realize the improvement of the precision of the motion mechanism from micron level to nanometer level. They are widely used in precision and ultra-precision processing equipment, aerospace equipment, precision measurement equipment and other fields. Most of the existing gas hydrostatic bearings are in the small hole throttling mode, and the small hole throttling is prone to air hammer vibration due to the existence of air cavity, which affects the working stability; and the bearing capacity and stiffness are low. In addition, the compressibility of gas makes it easy to become unstable, and unreasonable design will lead to serious self-excited vibration and air hammer phenomenon, which limits its application in heavy equipment, high-precision processing and other occasions. Improving the static and dynamic characteristics of gas hydrostatic bearings is a current research hotspot. Relevant researchers still need to continuously study new structural forms, new materials, and new processing methods to develop gas hydrostatic bearings with higher bearing capacity and stiffness and simpler processing to expand their application fields.

[0003] When the air-floating bearing unit is subjected to an asymmetric force, the working surface will tilt, the thickness distribution of the air film will no longer be uniform, and the pressure distribution of the air film will also change. A moment M will be generated in the air film to resist the change of the air film. The angular stiffness of the air-floating bearing unit reflects its ability to resist tilting. In the case of low angular stiffness, the air-floating bearing unit is prone to tilting, so it is necessary to study its angular stiffness to improve its anti-tilting ability.

[0004] Therefore, a method for calculating the static characteristics of gas hydrostatic bearings is needed that can be more accurate. Summary of the invention

[0005] The technical problem to be solved by the present invention is to provide a method for calculating the static characteristics of a gas hydrostatic bearing based on a finite difference method, which accurately calculates the static characteristics of a gas hydrostatic bearing and facilitates the study and improvement of its static characteristics.

[0006] The technical solution adopted by the present invention to solve the technical problem is: a calculation method for the static characteristics of a gas hydrostatic bearing based on a finite difference method, the calculation method is:

[0007] S1. Define the structural dimensions of the load-bearing unit in the gas hydrostatic bearing, give the gas source pressure and gas film thickness parameters, and divide the grid to determine the coordinates of each node and the gas film height;

[0008] S2, given throttle outlet pressure P d The initial value of

[0009] S3. According to P d The value of is used to calculate the pressure distribution inside the gas membrane;

[0010] S4, by P d The mass flow rate of gas inflow is calculated by the value of, the mass flow rate of gas outflow is calculated by the gas film pressure distribution, and the relative error of the mass flow rates of the two is calculated;

[0011] S5, determine whether the relative error of mass flow meets the given accuracy, if yes, proceed to step S6, if not, use the binary method to revise P d The value of , returns to step S3;

[0012] S6. Calculate the bearing capacity, vertical stiffness, angular stiffness and gas consumption of the gas hydrostatic bearing based on the results of the gas film pressure.

[0013] More specifically, the structural dimensions of the bearing unit in the gas hydrostatic bearing in step S1 include the dimensions of the throttle hole, the air cavity, the throttle groove and the working surface of the bearing unit.

[0014] More specifically, the program is written using MATALB software and the above parameters are input into the program for simulation.

[0015] More specifically, the gas consumption is obtained by subtracting the outflow mass flow rate from the inflow mass flow rate of the gas.

[0016] More specifically, the calculation formula of the bearing capacity W is as follows:

[0017]

[0018] Where P(i,j) is the pressure at the point with coordinates (i,j), P a is the atmospheric pressure, x(j+1) is the x-axis coordinate of point (j+1), x(j) is the x-axis coordinate of point j, y(i+1) is the y-axis coordinate of (i+1), and y(i) is the y-axis coordinate of point i.

[0019] More specifically, the calculation formula of the vertical stiffness K is as follows:

[0020]

[0021] Among them, ΔW is the change of bearing capacity, and Δh is the change of air film height.

[0022] More specifically, the calculation formula of the angular stiffness is as follows:

[0023]

[0024] Among them, Kβ is the angular stiffness, Δβ is the angular change of the working surface of the gas static pressure bearing unit, and ΔM is the change of the torque generated by the gas film when the working surface of the gas static pressure bearing unit rotates Δβ.

[0025] More specifically, the air film pressure distribution in the tilted state can be obtained by the finite difference method, so that the torque M can be obtained as follows:

[0026]

[0027] Wherein, P(i,j) is the pressure at the point with coordinates (i,j), β is the inclination angle of the working surface of the gas static pressure bearing unit, x(j) is the x-axis coordinate of point j, Δx is the difference between the x-axis coordinates of two adjacent points, and Δy is the difference between the y-axis coordinates of two adjacent points.

[0028] More specifically, the correction formula for the air film height is as follows:

[0029]

[0030] Where H(i,j) is the height of the air film at point (i,j), h is the initial air film thickness, x(j) is the x-axis coordinate of point j, β is the inclination angle of the bearing surface, δ is the depth of the air cavity, and h g Depth of throttle groove.

[0031] The beneficial effect of the present invention is that the static characteristics of the gas hydrostatic bearing can be calculated more accurately through the above-mentioned calculation method, especially a calculation method of angular stiffness is provided, which can further improve the bearing capacity and angular stiffness in the structural design of the gas hydrostatic bearing and increase the application range of the gas hydrostatic bearing. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 It is a flow chart of the present invention; DETAILED DESCRIPTION

[0033] In order to make the purpose, technical scheme and advantages of the implementation of the present invention clearer, the technical scheme in the embodiment of the present invention will be described in more detail below in conjunction with the drawings in the embodiment of the present invention. In the drawings, the same or similar reference numerals throughout represent the same or similar elements or elements with the same or similar functions. The described embodiments are part of the embodiments of the present invention, not all of the embodiments. The embodiments described below with reference to the drawings are exemplary and are intended to be used to explain the present invention, and should not be construed as limitations on the present invention. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0034] In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside" and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be understood as limiting the scope of protection of the present invention.

[0035] The present invention is described in detail below in conjunction with the accompanying drawings.

[0036] like Figure 1 A calculation method for the static characteristics of a gas hydrostatic bearing based on the finite difference method is shown. The calculation method is:

[0037] S1. Define the structural dimensions of the load-bearing unit in the gas hydrostatic bearing. The structural dimensions of the load-bearing unit include the dimensions of the throttle hole, the air cavity, the throttle groove, and the working surface of the load-bearing unit. Draw the position and shape of the throttle groove in CAD, give the gas source pressure and gas film thickness parameters, and divide the grid to determine the coordinates of each node and the gas film height;

[0038] S2, given throttle outlet pressure P d The initial value of

[0039] S3. According to P d The value of is used to calculate the pressure distribution inside the air film;

[0040] S4, by P d The mass flow rate of gas inflow is calculated by the value of, the mass flow rate of gas outflow is calculated by the gas film pressure distribution, and the relative error of the mass flow rates of the two is calculated;

[0041] S5, determine whether the relative error of mass flow meets the given accuracy, if yes, proceed to step S6, if not, use the binary method to revise P d The value of , returns to step S3 until the given accuracy is met;

[0042] S6. Calculate the bearing capacity, vertical stiffness, angular stiffness and gas consumption of the gas hydrostatic bearing based on the results of the gas film pressure.

[0043] The key points such as the dimensions of the throttle hole, air cavity, throttle groove and the working surface of the load-bearing unit, the gas source pressure, the gas film thickness, the coordinates of each node of the grid and the height of the gas film at each node are input into the MATALB software. The corresponding application is written by MATALB software to simulate the gas hydrostatic bearing, and relevant results close to actual use can be obtained.

[0044] The static characteristics required to be calculated in the above step S6 are directly obtained through application program operations, and the corresponding calculation formula of the operation is described in detail below.

[0045] The calculation formula for the load-bearing capacity W of the gas hydrostatic bearing is as follows:

[0046]

[0047] Where P(i,j) is the pressure at the point with coordinates (i,j), P a is the atmospheric pressure, x(j+1) is the x-axis coordinate of point (j+1), x(j) is the x-axis coordinate of point j, y(i+1) is the y-axis coordinate of (i+1), and y(i) is the y-axis coordinate of point i.

[0048] The calculation formula for the vertical stiffness of the gas hydrostatic bearing is as follows:

[0049]

[0050] Among them, ΔW is the change of bearing capacity, and Δh is the change of air film height.

[0051] The calculation formula for the gas consumption Q of the gas hydrostatic bearing is as follows:

[0052] Q=Q 流入 -Q 流出 ;

[0053] The gas consumption is calculated by subtracting the gas outflow mass flow from the gas inflow mass flow.

[0054] For the present invention, the main purpose is the calculation of the angular stiffness and the correction of the air film thickness.

[0055] When the tilt state is stable, the thickness of the air film at position x becomes:

[0056] h(x) = hx tan(β);

[0057] Where h is the initial air film thickness.

[0058] When the supporting surface and the working surface of the air-floating bearing unit are relatively tilted, the direction of the force of the air film is still perpendicular to the working surface of the bearing unit, and the angle between the air film and the supporting surface is the complementary angle β. Therefore, the component of the force parallel to the surface of the bearing unit generates a torque on the supporting surface to resist the tilt of the supporting surface.

[0059] The air film pressure distribution under the tilted state can be obtained by the finite difference method, and then the torque M can be obtained as:

[0060]

[0061] Wherein, P(i,j) is the pressure at the point with coordinates (i,j), β is the inclination angle of the working surface of the gas static pressure bearing unit, x(j) is the x-axis coordinate of point j, Δx is the difference between the x-axis coordinates of two adjacent points, and Δy is the difference between the y-axis coordinates of two adjacent points.

[0062] As the tilt angle β changes, the anti-tilt torque also changes accordingly, and the ratio of the two changes is the anti-tilt angular stiffness. Similar to stiffness, angular stiffness can be calculated by the following formula, where Δβ is the change in the rotation angle of the working surface of the air-floating bearing unit, and ΔM is the change in the torque generated by the air film when the working surface of the air-floating bearing unit rotates Δβ.

[0063]

[0064] The finite difference form of the Reynolds equation and the calculation process of the air film pressure distribution derived above are general and can also be used to calculate the air film pressure distribution of the deflected air-floating bearing unit. Therefore, the torque and angular stiffness of the air-floating bearing unit can still be obtained by simulating the program written in MATLAB. However, the air film thickness distribution of the nodes in the calculation area needs to be modified. The finite difference form of the air film thickness distribution is:

[0065]

[0066] Where H(i,j) is the height of the air film at point (i,j), h is the initial air film thickness, x(j) is the x-axis coordinate of point j, β is the inclination angle of the bearing surface, δ is the depth of the air cavity, and h g Depth of throttle groove.

[0067] In summary, the above calculation method can be used to simulate the static characteristics of the gas hydrostatic bearing, especially the calculation of the angular stiffness is more accurate. At the same time, this method can further improve the bearing capacity and angular stiffness of the gas hydrostatic bearing structure design and increase the scope of application of the gas hydrostatic bearing.

[0068] It should be emphasized that the above are only preferred embodiments of the present invention and do not limit the present invention in any form. Any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention are still within the scope of the technical solution of the present invention.

Claims

1. A method for calculating the static characteristics of a gas hydrostatic bearing based on a finite difference method, characterized in that: The calculation method is, S1. Define the structural dimensions of the load-bearing unit in the gas hydrostatic bearing, give the gas source pressure and gas film thickness parameters, and divide the grid to determine the coordinates of each node and the gas film height; S2, given throttle outlet pressure P d The initial value of S3. According to P d The value of is used to calculate the pressure distribution inside the air film; S4, by P d The mass flow rate of gas inflow is calculated by the value of, the mass flow rate of gas outflow is calculated by the gas film pressure distribution, and the relative error of the mass flow rates of the two is calculated; S5, determine whether the relative error of mass flow meets the given accuracy, if yes, proceed to step S6, if not, use the binary method to revise P d The value of , returns to step S3; S6. Calculate the bearing capacity, vertical stiffness, angular stiffness and gas consumption of the gas hydrostatic bearing according to the result of the gas film pressure; The structural dimensions of the bearing unit in the gas static pressure bearing in step S1 include the dimensions of the throttle hole, the air cavity, the throttle groove and the working surface of the bearing unit; The calculation formula of the bearing capacity W is as follows: Where P(i,j) is the pressure at the point with coordinates (i,j), P a is the atmospheric pressure, x(j+1) is the x-axis coordinate of point (j+1), x(j) is the x-axis coordinate of point j, y(i+1) is the y-axis coordinate of (i+1), and y(i) is the y-axis coordinate of point i; The calculation formula of the vertical stiffness K is as follows: Among them, ΔW is the change of bearing capacity, Δh is the change of air film height; The calculation formula of the angular stiffness is as follows: Among them, K β is the angular stiffness, Δβ is the angular change of the working surface of the gas static pressure bearing unit, and ΔM is the change of the torque generated by the gas film when the working surface of the gas static pressure bearing unit rotates Δβ; The finite difference method can be used to obtain the air film pressure distribution under the tilt state, and thus the torque M can be obtained as follows: Wherein, P(i,j) is the pressure at the point with coordinates (i,j), β is the inclination angle of the working surface of the gas static pressure bearing unit, x(j) is the x-axis coordinate of point j, Δx is the difference between the x-axis coordinates of two adjacent points, and Δy is the difference between the y-axis coordinates of two adjacent points; The correction formula of the air film height is as follows: Where H(i,j) is the height of the air film at point (i,j), h is the initial air film thickness, x(j) is the x-axis coordinate of point j, β is the inclination angle of the bearing surface, δ is the depth of the air cavity, and h g Depth of throttle groove.

2. The method for calculating the static characteristics of a gas hydrostatic bearing based on the finite difference method according to claim 1 is characterized in that: Use MATALB software to write a program and input the above parameters into the program for simulation.

3. The method for calculating the static characteristics of a gas hydrostatic bearing based on the finite difference method according to claim 1, characterized in that: The gas consumption is obtained by subtracting the outflow mass flow rate from the inflow mass flow rate of the gas.

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