CN112632827A - Method for calculating static characteristics of aerostatic bearing based on finite difference method - Google Patents
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Abstract
The invention relates to a method for calculating the static characteristics of a gas hydrostatic bearing based on a finite difference method, S1, defining the structural size of a bearing unit, giving the pressure of a gas source and the thickness of a gas film, and dividing grids to determine the coordinates of each node and the height of the gas film; s2 given orifice outlet pressure PdAn initial value of (d); s3, calculating the distribution of the pressure in the air film; s4, calculating the mass flow rate of the gas inflow and the mass flow rate of the gas outflow, and calculating the relative error of the mass flow rates; s5, judging whether the relative error meets the given precision, if so, entering the step S6, otherwise, revising P again by adopting dichotomydReturns to step S3; and S6, calculating the static characteristic of the aerostatic bearing according to the result of the air film pressure. The static characteristic of the aerostatic bearing is calculated more accurately, and an angle steel is providedThe calculation method of the degree can further improve the bearing capacity and the angular rigidity of the aerostatic bearing structure design, and increase the application range of the aerostatic bearing.
Description
Technical Field
The invention relates to design and production of aerostatic bearings, in particular to a method for calculating static characteristics of aerostatic bearings based on a finite difference method.
Background
The aerostatic bearing provides high-pressure gas from the outside, and a gas film with certain pressure is formed in a bearing gap to support a supported piece, so that the aerostatic bearing has the advantages of small friction heating, high precision, long service life and the like, realizes the promotion of the precision of a motion mechanism from micron-scale to nanometer-scale, and is widely applied to the fields of precision and ultra-precision machining equipment, aerospace instruments, precision measurement equipment and the like. The existing gas hydrostatic bearing adopts a small hole throttling mode mostly, and the small hole throttling easily generates a gas hammer vibration phenomenon due to the existence of a gas cavity, so that the working stability is influenced; and the bearing capacity and rigidity are low. In addition, the gas is easy to destabilize due to the compressibility of the gas, and serious self-excited vibration and air hammering phenomena can be caused due to improper design, so that the application of the gas in heavy equipment, high-precision machining and other occasions is limited. The improvement of the static and dynamic characteristics of the aerostatic bearing is a research hotspot at present, and relevant researchers still need to continuously research new structural forms, new materials and new processing methods to develop aerostatic bearings with higher bearing capacity and rigidity and simpler processing to expand the application fields thereof.
When the air-float bearing unit is acted by asymmetric force, the working surface can be inclined, the thickness distribution of the air film is not uniform any more, the pressure distribution of the air film can be changed, a moment M is generated in the air film to resist the change of the air film, and the angular rigidity of the air-float bearing unit reflects the inclination resistance of the air-float bearing unit. In the case of low angular stiffness, the air bearing unit is prone to tilt, so it is necessary to study its angular stiffness to improve the tilt resistance.
Therefore, a method for calculating the static characteristics of the aerostatic bearing is needed, which can be more accurate.
Disclosure of Invention
The invention aims to solve the technical problem of providing 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 the gas hydrostatic bearing, and is convenient for researching and improving the static characteristics of the gas hydrostatic bearing.
The technical scheme adopted by the invention for solving the technical problems is as follows: a method for calculating the static characteristics of aerostatic bearing based on finite difference method includes such steps as calculating the static characteristics of aerostatic bearing,
s1, defining the structural size of a bearing unit in the aerostatic bearing, giving air source pressure and air film thickness parameters, and dividing grids to determine the coordinates of each node and the air film height;
s2 given orifice outlet pressure PdAn initial value of (d);
s3, according to PdCalculating the distribution of the pressure in the gas ji film;
s4, from PdCalculating the mass flow rate of gas inflow, calculating the mass flow rate of gas outflow according to the gas film pressure distribution, and calculating the relative error of the mass flow rates of the gas inflow and the gas outflow;
s5, judging whether the relative error of the mass flow meets the given precision, if so, entering the step S6, and if not, revising P again by adopting a dichotomydReturns to step S3;
and S6, calculating the bearing capacity, the vertical rigidity, the angular rigidity and the air consumption of the aerostatic bearing according to the result of the air film pressure.
More specifically, the structural dimensions of the bearing unit in the aerostatic bearing in step S1 include the dimensions of the throttle hole, the air chamber, the throttle groove and the working surface of the bearing unit.
More specifically, MATALB software is adopted to write a program and the parameters are input into the program for simulation.
More specifically, the gas consumption is obtained by subtracting the mass flow rate of the gas flowing in from the mass flow rate of the gas flowing out.
More specifically, the calculation formula of the bearing capacity W is as follows:
wherein P (i, j) is the pressure with coordinates of (i, j), PaFor 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.
More specifically, the calculation formula of the vertical stiffness K is as follows:
wherein, Δ W is the change of the bearing capacity and Δ h is the change of the height of the gas film.
More specifically, the calculation formula of the angular stiffness is as follows:
wherein, KβIn order to obtain angular rigidity, Δ β is the angular change of the rotation of the working surface of the aerostatic bearing unit, and Δ M is the change of the torque generated by the gas film when the working surface of the aerostatic bearing unit rotates by Δ β.
More specifically, the air film pressure distribution in the inclined state is obtained by a finite difference method, and the torque M is obtained,
wherein, P (i, j) is the pressure with coordinates of point (i, j), β is the inclined angle of the working surface of the aerostatic bearing unit, x (j) is the x-axis coordinate of point j, Δ x is the difference value of the x-axis coordinates of two adjacent points, and Δ y is the difference value of the y-axis coordinates of two adjacent points.
More specifically, the correction formula of the gas film height is as follows:
h (i, j) is the height of the air film at the point (i, j), H is the initial air film thickness, x (j) is the x-axis coordinate of the point j, beta is the inclination angle of the bearing surface, delta is the air cavity depth, HgThe depth of the throttling groove.
The invention has the beneficial effects that: the calculation method enables the static characteristics of the aerostatic bearing to be calculated more accurately, and particularly provides a calculation method of angular stiffness, so that the bearing capacity and the angular stiffness can be further improved on the structural design of the aerostatic bearing, and the application range of the aerostatic bearing is enlarged.
Drawings
FIG. 1 is a flow chart of the present invention;
Detailed Description
In order to make the implementation objects, technical solutions and advantages of the present invention clearer, the technical solutions in the embodiments of the present invention will be described in more detail below with reference to the accompanying drawings in the embodiments of the present invention. In the drawings, the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The described embodiments are only some, but not all embodiments of the invention. The embodiments described below with reference to the drawings are illustrative and intended to be illustrative of the invention and are not to be construed as limiting the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
In the description of the present invention, it is to be understood that the terms "center", "longitudinal", "lateral", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc., indicate orientations or positional relationships based on those shown in the drawings, and are used merely for convenience in describing the present invention and for simplifying the description, but do not indicate or imply that the device or element being referred to must have a particular orientation, be constructed and operated in a particular orientation, and thus, should not be construed as limiting the scope of the present invention.
The present invention will be described in detail below with reference to the accompanying drawings.
As shown in fig. 1, a method for calculating the static characteristics of the aerostatic bearing based on the finite difference method is that,
s1, defining the structural size of a bearing unit in the aerostatic bearing, wherein the structural size of the bearing unit comprises the sizes of a throttling hole, an air cavity, a throttling groove and a bearing unit working surface, drawing the position and the shape of the throttling groove in CAD, giving air source pressure and air film thickness parameters, and dividing grids to determine the coordinates of each node and the air film height;
s2 given orifice outlet pressure PdAn initial value of (d);
s3, according to PdCalculating the distribution of the pressure in the air film;
s4, from PdCalculating the mass flow rate of gas inflow, calculating the mass flow rate of gas outflow from the gas film pressure distribution, and calculating the relative error of the mass flow rates of the gas inflow and the gas outflow;
s5, judging whether the relative error of the mass flow meets the given precision, if so, entering the step S6, and if not, revising P again by adopting a dichotomydReturns to step S3 until a given accuracy is satisfied;
and S6, calculating the bearing capacity, the vertical rigidity, the angular rigidity and the air consumption of the aerostatic bearing according to the result of the air film pressure.
Key points such as the sizes of the throttling holes, the air cavities, the throttling grooves and the working faces of the bearing units, the air source pressure, the air film thickness, the coordinates of all nodes of the divided grids and the heights of the air films at all the nodes are input into MATALB software, corresponding application programs are compiled through the MATALB software to simulate the aerostatic bearings, and related results close to actual use can be obtained.
The static characteristics of the calculation required in step S6 are directly obtained by the application operation, and the calculation formula corresponding to the operation is described in detail below.
And (3) calculating the bearing capacity W of the aerostatic bearing, wherein the calculation formula is as follows:
wherein P (i, j) is the pressure with coordinates of (i, j), PaFor 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.
And (3) calculating the vertical rigidity of the aerostatic bearing, wherein the calculation formula is as follows:
wherein, Δ W is the change of the bearing capacity and Δ h is the change of the height of the gas film.
And (3) calculating the gas consumption Q of the aerostatic bearing, wherein the calculation formula is as follows:
Q＝Qinflow into-QOutflow of the liquid；
The gas consumption is determined by subtracting the mass flow of the gas flowing in from the mass flow of the gas flowing out.
For the present invention, the main objective is the calculation of the angular stiffness, and the correction for the gas film thickness.
When this tilted state is stabilized, the thickness of the gas film at the x position becomes:
h(x)＝h-x tan(β)；
wherein h is the initial gas film thickness.
When the support surface and the working surface of the air-float bearing unit are relatively inclined, the acting force direction of the air film is still perpendicular to the working surface of the bearing unit, and the included angle between the acting force direction and the support surface is the complementary angle of beta, so that the acting force generates torque to the support surface along the component parallel to the surface direction of the bearing unit to resist the inclination of the support surface.
The air film pressure distribution in the inclined state can be obtained by a finite difference method, and the torque M is further obtained as follows:
wherein, P (i, j) is the pressure with coordinates of point (i, j), β is the inclined angle of the working surface of the aerostatic bearing unit, x (j) is the x-axis coordinate of point j, Δ x is the difference value of the x-axis coordinates of two adjacent points, and Δ y is the difference value of the y-axis coordinates of two adjacent points.
Along with the change of the inclination angle beta, the anti-inclination torque correspondingly changes, and the ratio of the two changes is the anti-inclination angle rigidity. Similar to the stiffness, the angular stiffness can be calculated by the following formula, where Δ β is the variation of the rotation angle of the working surface of the air bearing unit, and Δ M is the variation of the torque generated by the air film when the working surface of the air bearing unit rotates by Δ β.
The calculation process of the finite difference form of the Reynolds equation and the air film pressure distribution deduced in the foregoing is general, and can also be used for calculating the air film pressure distribution of the air floatation bearing unit which deflects, so that the torque and the angular stiffness of the air floatation bearing unit can still be obtained through simulation after an MATLAB programming program. However, the calculation of the gas film thickness distribution of the nodes in the area needs to be modified, and the finite difference form of the gas film thickness distribution is as follows:
h (i, j) is the height of the air film at the point (i, j), H is the initial air film thickness, x (j) is the x-axis coordinate of the point j, beta is the inclination angle of the bearing surface, delta is the air cavity depth, HgThe depth of the throttling groove.
In conclusion, by the aid of the calculation method, the static characteristics of the aerostatic bearing can be well simulated and calculated, particularly, the calculation of the angular stiffness is more accurate, meanwhile, the bearing capacity and the angular stiffness can be further improved on the aerostatic bearing structure design, and the application range of the aerostatic bearing is widened
It is to be emphasized that: the above embodiments are only preferred embodiments of the present invention, and are not intended to limit the present invention in any way, and all simple modifications, equivalent changes and modifications made to the above embodiments according to the technical spirit of the present invention are within the scope of the technical solution of the present invention.
Claims (9)
1. A method for calculating the static characteristics of a gas hydrostatic bearing based on a finite difference method is characterized in that the method comprises the following steps,
s1, defining the structural size of a bearing unit in the aerostatic bearing, giving air source pressure and air film thickness parameters, and dividing grids to determine the coordinates of each node and the air film height;
s2 given orifice outlet pressure PdAn initial value of (d);
s3, according to PdCalculating the distribution of the pressure in the air film;
s4, from PdCalculating the mass flow rate of gas inflow, calculating the mass flow rate of gas outflow according to the gas film pressure distribution, and calculating the relative error of the mass flow rates of the gas inflow and the gas outflow;
s5, judging whether the relative error of the mass flow meets the given precision, if so, entering the step S6, and if not, revising P again by adopting a dichotomydReturns to step S3;
and S6, calculating the bearing capacity, the vertical rigidity, the angular rigidity and the air consumption of the aerostatic bearing according to the result of the air film pressure.
2. The method for calculating the static characteristics of the aerostatic bearing according to claim 1, wherein the structural dimensions of the load-bearing unit in the aerostatic bearing of step S1 include the dimensions of the throttle hole, the air cavity, the throttle slot and the working surface of the load-bearing unit.
3. The method of claim 1, wherein the finite difference method is used to calculate the static behavior of the aerostatic bearing, wherein the MATALB software is used to write a program and the parameters are input into the program for simulation.
4. The method of claim 1, wherein the gas consumption is determined by subtracting the mass flow of the gas flowing in from the mass flow of the gas flowing out.
5. The method for calculating the static characteristics of the aerostatic bearing according to claim 1, characterized in that the load capacity W is calculated as follows:
wherein P (i, j) is the pressure with coordinates of (i, j), PaFor 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.
6. The method for calculating the static characteristics of the aerostatic bearing based on finite difference method according to claim 1, characterized in that the vertical stiffness K is calculated as follows:
wherein, Δ W is the change of the bearing capacity and Δ h is the change of the height of the gas film.
7. The method of claim 1, wherein the angular stiffness is calculated by the following formula:
wherein, KβIn order to obtain angular rigidity, Δ β is the angular change of the rotation of the working surface of the aerostatic bearing unit, and Δ M is the change of the torque generated by the gas film when the working surface of the aerostatic bearing unit rotates by Δ β.
8. The method of claim 7, wherein the torque M is obtained by obtaining the gas film pressure distribution in the inclined state by the finite difference method,
wherein, P (i, j) is the pressure with coordinates of point (i, j), β is the inclined angle of the working surface of the aerostatic bearing unit, x (j) is the x-axis coordinate of point j, Δ x is the difference value of the x-axis coordinates of two adjacent points, and Δ y is the difference value of the y-axis coordinates of two adjacent points.
9. The finite difference method-based calculation method for the static characteristics of aerostatic bearings according to claim 1, characterized in that the formula for correcting the gas film height is as follows:
h (i, j) is the height of the air film at the point (i, j), H is the initial air film thickness, x (j) is the x-axis coordinate of the point j, beta is the inclination angle of the bearing surface, delta is the air cavity depth, HgThe depth of the throttling groove.
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