Aerostatic bearing rotation error modeling method based on bidirectional fluid-solid coupling

Abstract

The invention discloses an aerostatic bearing rotation error modeling method based on bidirectional fluid-solid coupling, belongs to the technical field of ultra-precision mechanical design and performance simulation, and aims to solve the problems of insufficient model applicability and low calculation efficiency in the existing aerostatic spindle analysis technology. The method comprises the following steps: establishing a five-degree-of-freedom mechanical model of the gas static pressure main shaft; constructing a thrust gas film initial model and a neck gas film initial model containing comprehensive manufacturing errors, and recording unbalanced torque; calculating the total unbalanced torque of the system, carrying out iterative circulation until the total torque, the axial force and the radial force of the system reach convergence conditions, traversing all angle positions of the main shaft rotating 360 degrees for a circle, integrating main shaft balance attitude coordinates obtained by calculation under all angles, generating a mass center trajectory diagram and an axis space inclination trajectory diagram of the main shaft, and calculating the balance attitude of the main shaft. And an accurate rotation error model is obtained.

Description

Technical Field

[0001] This invention relates to the field of ultra-precision mechanical design and performance simulation technology, and in particular to a high-precision modeling method for the rotational error of a gas hydrostatic spindle that considers manufacturing errors and the two-way fluid-structure interaction (FSI) effect. Background Technology

[0002] Gas hydrostatic spindles, with their advantages of high precision, low friction, and low heat generation, have become core components of high-end equipment such as ultra-precision machine tools and measuring instruments. Their performance hinges on the gas film formed between the spindle and the bearings; the stability of this gas film directly determines the spindle's final rotational accuracy.

[0003] In practical applications, two key factors significantly affect the film gas characteristics and spindle performance: First, the fluid-structure interaction effect. A significant fluid-structure interaction effect exists between the solid field of the structural components and the film gas fluid field of the gas hydrostatic spindle. The pressure generated by the film gas flow field causes elastic deformation of the bearings and spindle structure. This deformation, in turn, alters the geometry and boundary conditions of the film gas, thus affecting the pressure distribution and forming an interactive coupling process. Second, manufacturing and assembly errors. Due to limitations in machining accuracy, errors inevitably exist in the geometry (such as flatness and cylindricity) and assembly position (such as coaxiality) of the bearings and spindle parts. These structural machining errors affect the actual film gas shape and cause the spindle rotation trajectory to deviate from the ideal state.

[0004] Currently, mainstream computational fluid dynamics (CFD) software suffers from insufficient model applicability and low computational efficiency when performing mesh generation and fluid-structure interaction analysis of irregular film flow fields. Summary of the Invention

[0005] To address the limitations of existing gas hydrostatic spindle analysis techniques, such as insufficient model applicability and low computational efficiency, this invention provides a method for modeling the rotational error of gas hydrostatic bearings based on two-way fluid-structure interaction (FSI). This method, by incorporating manufacturing error data and employing a two-way FSI and 5-DOF static equilibrium iterative solution strategy, can accurately simulate the attitude changes of a real spindle during rotation, thereby achieving precise prediction of rotational errors.

[0006] The present invention discloses a method for modeling the rotational error of a gas hydrostatic bearing based on bidirectional fluid-structure interaction. This method includes the following steps:

[0007] S1. Establish a 5-DOF mechanical model of the gas hydrostatic principal shaft. The 5 degrees of freedom include axial translation Y, two radial translations X and Z, and two radial tilt rotational degrees of freedom. , This model is used to describe the equilibrium state of the spindle under the action of air film force, and the displacement of the center of mass and the rotation tilt distance of the spindle axis under equilibrium state are defined as rotation error.

[0008] S2. Based on the machining error data of the structural components of the thrust bearing and the spindle, and the bearing design parameters, construct an initial model of the thrust bearing air film containing comprehensive manufacturing errors; calculate the pressure distribution and structural deformation of this air film under the bidirectional fluid-structure interaction effect using the finite element method, and iteratively solve the problem by adjusting the axial displacement of the spindle until the axial force is balanced, recording the axial displacement at this point. and unbalanced torque around the X and Z axes , ;

[0009] S3. Based on the machining error data of the structural components of the radial bearing and the spindle, and the bearing design parameters, construct an initial model of the radial bearing air film containing comprehensive manufacturing errors; calculate the pressure distribution and structural deformation of this air film under the bidirectional fluid-structure interaction effect using the finite element method, and iteratively solve the problem by adjusting the radial displacement of the spindle until the radial force is balanced, recording the radial displacement at this point. and unbalanced torque around the X and Z axes , ;

[0010] S4, Calculate the total unbalanced torque of the system. Its value is the vector synthesis result of the torque obtained in steps S2 and S3; determine Is it less than the preset torque convergence threshold? ;

[0011] S5. If step S4 determines otherwise, adjust the tilt angle of the spindle. and / or This process is repeated to change the air film morphology; then, with a new attitude angle, steps S2 and S3 are executed again to recalculate the axial force, radial force, and new unbalanced torque, and step S4 is performed again. This iterative cycle continues until the total torque of the system meets the requirements. ;

[0012] S6. If step S4 is correct, then further determine whether the axial force and radial force are both less than the preset force convergence threshold. If not, return to steps S2 to S5; if yes, record the 5-DOF coordinates of the spindle when it reaches equilibrium. In the above steps, 'i' represents the variable corresponding to the i-th change in the spindle's rotation angle around the Y-axis.

[0013] S7. The (i+1)th time, the spindle rotation angle around the Y-axis is changed to... The working angle of the spindle rotation around the Y-axis compared to the previous one Step Update based on the current rotation angle The thrust and journal bearing air film model; using the updated air film model as the new initial condition, repeat steps S2 to S6 to solve the spindle balance attitude under the new angle;

[0014] S8. Traverse all angular positions of the spindle in a 360° rotation, integrate the spindle balance attitude coordinates calculated under all angles, and generate the spindle centroid trajectory diagram and axis spatial tilt trajectory diagram, thus obtaining the accurate rotation error model.

[0015] Preferably, the specific process for calculating the bidirectional fluid-structure interaction effect of the gas film containing manufacturing errors in steps S2 and S3 is as follows:

[0016] S21. Perform finite element mesh generation based on the current air film geometry and set the flow field boundary conditions;

[0017] S22. Solve the Reynolds equation for the film flow field using the finite element method to obtain the pressure distribution matrix of the discrete nodes;

[0018] S23. Apply the air film pressure distribution as a surface load to the solid structure finite element model of the main shaft and bearing;

[0019] S24. Solve the governing equations of solid mechanics to obtain the deformation field of the solid structure under load;

[0020] S25. Update the thickness and shape of the air film according to the deformation of the structural field;

[0021] S26. Using the updated air film shape as the new computational domain, repeat steps S21 to S25 until the changes in air film pressure distribution and solid deformation are less than the set tolerance, thus achieving fluid-structure interaction equilibrium.

[0022] Preferably, the machining error data of the thrust bearing and spindle structural components in step S2 mainly includes the flatness error of the working surfaces of the thrust plate and bushing, as well as the assembly perpendicularity and parallelism errors of each component.

[0023] Preferably, the machining error data of the structural components of the radial bearing and the spindle in step S3 mainly includes the roundness and cylindricity errors of the spindle and bushing, as well as the assembly coaxiality error of each component.

[0024] Preferably, the equilibrium condition of the 5-DOF mechanical model in step S1 is: the resultant force of the main shaft in the axial direction and the two radial directions is zero, and the resultant torque about the two radial axes is zero.

[0025] Preferably, the torque convergence threshold .

[0026] Preferably, the force convergence threshold .

[0027] The beneficial effects of this invention are as follows: This invention solves the problem that existing methods for calculating the performance of gas hydrostatic bearings cannot consider the impact of manufacturing errors and solid structure deformation under load on rotational accuracy. This invention is the first to deeply integrate manufacturing errors and the two-way fluid-structure interaction effect into the model, greatly improving the consistency between the predicted results and the actual physical working state. A solution model for the gas film pressure distribution under manufacturing errors is established using the FEM (Finite Element Method), avoiding the meshing difficulties encountered by CFD when dealing with irregular error gas films. The calculation process is simpler, more efficient, and the values ​​are more stable. The gas film thickness and shape deformation under the interaction between the fluid field and the solid structure of the gas hydrostatic spindle are solved using the two-way fluid-structure interaction method. Then, by changing the axial and radial thickness of the gas film and the spindle tilt angle, axial and radial force balance and total torque balance are obtained. This achieves the solution of the rotational error of the gas hydrostatic spindle considering manufacturing errors and the two-way fluid-structure interaction effect. This invention is based on the principles of gas lubrication, solid structure statics, 5-DOF equilibrium equations, and numerical calculation methods. It boasts advantages such as high accuracy, high computational efficiency, and a model that closely matches actual working conditions. It can accurately solve for the rotational error of a gas hydrostatic spindle. The final output includes not only the radial error trajectory but also the complete axial error trajectory and the axis rotation tilt trajectory, providing unprecedented detailed methodological support for comprehensively evaluating and optimizing spindle rotational accuracy. Attached Figure Description

[0028] Figure 1 This is a schematic diagram of the six degrees of freedom motion of the gas static pressure principal axis;

[0029] Figure 2 This is a schematic diagram of the gas film under the influence of manufacturing errors and fluid-structure interaction effects of a gas static pressure spindle;

[0030] Figure 3 This is a flowchart of a gas hydrostatic bearing rotation error modeling method based on bidirectional fluid-structure interaction as described in this invention;

[0031] Figure 4 It is a rotational error trajectory diagram of a gas static pressure spindle with bidirectional fluid-structure interaction. Detailed Implementation

[0032] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0033] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0034] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but this is not intended to limit the scope of the invention.

[0035] Specific Implementation Method 1: The following is combined with... Figures 1 to 4 This embodiment describes a method for modeling the rotational error of a gas hydrostatic bearing based on bidirectional fluid-structure interaction. The method includes the following steps:

[0036] S1. Establish a 5-DOF mechanical model of the gas hydrostatic principal shaft. The 5 degrees of freedom include axial translation Y, two radial translations X and Z, and two radial tilt rotational degrees of freedom. , This model is used to describe the equilibrium state of the spindle under the action of air film force, and the displacement of the center of mass and the rotation tilt distance of the spindle axis under equilibrium state are defined as rotation error.

[0037] S2. Based on the machining error data of the structural components of the thrust bearing and the spindle, and the bearing design parameters, construct an initial model of the thrust bearing air film containing comprehensive manufacturing errors; calculate the pressure distribution and structural deformation of this air film under the bidirectional fluid-structure interaction effect using the finite element method (FEM), and iteratively solve the problem by adjusting the axial displacement of the spindle until the axial force is balanced, recording the axial displacement at this point. and unbalanced torque around the X and Z axes , ;

[0038] S3. Based on the machining error data of the structural components of the radial bearing and the spindle, and the bearing design parameters, construct an initial model of the radial bearing air film containing comprehensive manufacturing errors; calculate the pressure distribution and structural deformation of this air film under the bidirectional fluid-structure interaction effect using FEM, and iteratively solve the problem by adjusting the radial displacement of the spindle until the radial force is balanced, recording the radial displacement at this point. and unbalanced torque around the X and Z axes , ;

[0039] S4, Calculate the total unbalanced torque of the system. Its value is the vector synthesis result of the torque obtained in steps S2 and S3; determine Is it less than the preset torque convergence threshold? ;

[0040] S5. If step S4 determines otherwise, adjust the tilt angle of the spindle. and / or This process is repeated to change the air film morphology; then, with a new attitude angle, steps S2 and S3 are executed again to recalculate the axial force, radial force, and new unbalanced torque, and step S4 is performed again. This iterative cycle continues until the total torque of the system meets the requirements. ;

[0041] S6. If step S4 is correct, then further determine whether the axial force and radial force are both less than the preset force convergence threshold. If not, return to steps S2 to S5; if yes, record the 5-DOF coordinates of the spindle when it reaches equilibrium. In the above steps, 'i' represents the variable corresponding to the i-th change in the spindle's rotation angle around the Y-axis.

[0042] S7. The (i+1)th time, the spindle rotation angle around the Y-axis is changed to... The working angle of the spindle rotation around the Y-axis compared to the previous one Step Update based on the current rotation angle The thrust and journal bearing air film model; using the updated air film model as the new initial condition, repeat steps S2 to S6 to solve the spindle balance attitude under the new angle;

[0043] S8. Traverse all angular positions of the spindle in a 360° rotation, integrate the spindle balance attitude coordinates calculated under all angles, and generate the spindle centroid trajectory diagram and axis spatial tilt trajectory diagram, thus obtaining the accurate rotation error trajectory.

[0044] The core of the method of this invention lies in solving the precise balance posture of the main shaft at each rotation angle by using an iterative architecture of an "outer loop" containing an "inner loop".

[0045] External circulation (step S7): Controls the working angle of the spindle rotation around its axis. It traverses from 0° to 360° to simulate the working condition of a full rotation of the spindle.

[0046] Inner loop (steps S2 to S6): at each fixed rotation angle The static equilibrium of the principal axis with 5 degrees of freedom is solved iteratively.

[0047] Force balance iteration (S2, S3): First, solve the axial force balance of the thrust bearing and the radial force balance of the journal bearing respectively, and record the unbalanced torque that cannot be eliminated.

[0048] Torque balancing and system coordination (S4, S5): Considering the overall torque, determine if the total system torque is balanced. If unbalanced, adjust the global tilt attitude of the spindle. , This process is repeated until the system's torque balance is achieved. Then, force balance is verified and calculated until the global force and torque are simultaneously balanced.

[0049] Result Record (S6): Record the angle. The 5-DOF coordinates of the principal axis after convergence.

[0050] Finally, result synthesis (S8): Integrating the principal axis attitude at all angles, the centroid trajectory and axis rotation tilt trajectory characterizing the rotation error are plotted. See [link to relevant documentation]. Figure 4 As shown.

[0051] See Figure 1 It is a theoretical model with 6 degrees of freedom for the main axis. The 6 degrees of freedom include axial translation (Y) and rotation about the axis, two radial translations (X and Z), and two radial tilt rotations. , The model established in this invention has 5 degrees of freedom, namely, five degrees of freedom other than the working degree of freedom of rotation around the axis. The equilibrium condition of the 5-degree-of-freedom mechanical model in step S1 is: the resultant force of the main shaft in the axial direction and the two radial directions is zero, and the resultant torque about the two radial axes is zero.

[0052] In step S2, the machining error data of the thrust bearing and the spindle structural components, along with the bearing design parameters, serve as known inputs and form the starting point and foundation of the entire high-precision, high-fidelity rotational error modeling method of this invention. The innovation of this invention lies not in changing these parameters, but in creating a method capable of absorbing these parameters and accurately calculating their ultimate impact on spindle accuracy.

[0053] The machining error data of the structural components of the thrust bearing and the main shaft mainly include the flatness error of the working surface of the thrust plate and the bushing, as well as the assembly perpendicularity and parallelism errors of each component.

[0054] Similarly, in step S3, the machining error data of the structural components of the radial bearing and the spindle includes the roundness and cylindricity errors of the spindle and bushing, as well as the assembly coaxiality errors of each component.

[0055] The data for S2 and S3 are derived from actual measurements or simulations. The inclusion of these error parameters reflects the true bearing performance, which is limited by manufacturing capabilities.

[0056] The calculation of the two-way fluid-structure interaction effect of the air film containing manufacturing errors described in steps S2 and S3 is a multi-step, iterative, and precise simulation process. Its goal is to determine the equilibrium state that the air film and solid structure will ultimately reach under realistic conditions with manufacturing errors. The specific process is as follows:

[0057] S21. Perform finite element mesh generation based on the current air film geometry and set the flow field boundary conditions;

[0058] S22. Solve the Reynolds equation for the film gas flow field using FEM to obtain the pressure distribution matrix of discrete nodes; obtain the gas pressure value at each point in the entire film gas region, i.e., the pressure distribution contour map. Due to manufacturing errors causing uneven film gas thickness, the pressure distribution here is naturally irregular.

[0059] S23. Apply the gas film pressure distribution as a surface load to the solid structure finite element model of the main shaft and bearing; apply the gas film pressure calculated in the previous step as an external load to the solid structure model of the thrust plate and main shaft. This structural model is also meshed and material properties (such as elastic modulus) are defined.

[0060] S24. Solve the governing equations of solid mechanics to obtain the deformation field of the solid structure under load, and calculate the deformation (δ) of the thrust plate and main shaft under film pressure. For example, the thrust plate near the outer diameter region will be slightly bent.

[0061] S25. Update the thickness and shape of the air film based on the structural deformation field; specifically, the program automatically updates the geometry of the air film based on the calculated structural deformation (δ). The original air film thickness (h) with errors is now increased by the deformation caused by the force, forming a new air film shape of "error + deformation".

[0062] S26. Using the updated film gas shape as the new computational domain, repeat steps S21 to S25 until the changes in film gas pressure distribution and solid deformation are less than the set tolerance, achieving fluid-structure interaction equilibrium. This step is a judgment process; the acquired new shape is compared with the film gas shape used in the previous cycle. If the shape change is significant, it indicates that the system is not in equilibrium, so return to step one and recalculate the flow field pressure using this new shape. If the shape change is negligible (less than the set convergence threshold), it indicates that the film gas pressure and solid deformation have adapted to each other and reached dynamic equilibrium. The cycle stops.

[0063] Traditional methods either ignore manufacturing errors (assuming an ideal shape) or ignore bidirectional coupling (assuming a rigid structure). The method of this invention, however, considers both and, through the rigorous iterative process described above, enables calculation results to closely approximate the actual working state of the bearing in the real world, thus achieving accurate prediction of "rotational error." This precisely demonstrates the advanced nature and practicality of this invention.

[0064] Step S2 temporarily only concerns the axial displacement Y, fixing the radial position (X, Z) and tilt angle of the spindle. , The axial force balance of the thrust bearing is sought solely by adjusting the axial displacement (Y). In step S3, only the X and Z radial directions are considered for the time being. The axial position (Y) and tilt angle of the spindle are fixed. , The radial force balance of the thrust bearing is sought solely by adjusting the radial displacement (X, Z). Both steps address the balance only in their respective directions and record any unresolved "side effects" (i.e., unbalanced torque). The spindle is a rigid body, and its orientation (including position and angle) must simultaneously satisfy all bearing constraints. Changes in one direction will inevitably affect the other. Step S2 addresses the unbalanced torque generated by the thrust bearing. , This can cause the spindle to tilt, which alters the film shape of the radial bearing. Conversely, the stiffness and force of the radial bearing also affect the spindle's tilt attitude. Therefore, the results of these two steps need to be fused in steps S4 and S5.

[0065] The fusion process is completed through iterative loops: steps S2 and S3 respectively output the "unbalanced torque" that they cannot eliminate on their own to step S4. After receiving these two torques, step S4 performs a synthesis judgment. If the total torque does not meet the convergence condition, it indicates that the overall attitude of the current spindle (especially the tilt angle) is not satisfactory. , This is incorrect. Adjust the spindle's attitude (e.g., by adjusting (...) , (Change the tilt direction and angle of the spindle), then return to steps S2 and S3, and perform independent calculations again with this new orientation. This cycle continues until, in a certain spindle orientation, the axial force calculated in step S2, the radial force calculated in step S3, and the total torque calculated in step S4 simultaneously satisfy the convergence condition. At this point, we have found the true 5-DOF equilibrium state of the spindle at that rotation angle.

[0066] The method of the present invention will be illustrated below with a specific example.

[0067] In the example, the torque convergence threshold Force convergence threshold .

[0068] Input preparation: Input bearing geometric parameters, air supply pressure, etc. according to the design drawings; define error functions such as flatness of the thrust plate and bushing working surfaces, and cylindricity of the working surfaces of the spindle and bushing according to the measured or assumed error values.

[0069] Outer loop initialization: Set the initial spindle rotation angle when i=0. =0°, angle step size =1°.

[0070] Solving the inner loop:

[0071] a) Construct the current perspective The initial air film model (including errors) is shown below.

[0072] b) Perform a two-way fluid-structure interaction analysis to calculate the film force of the thrust bearing and the neck bearing respectively.

[0073] c) Determine if the axial and radial forces are balanced. If not, adjust Y or X, Z, and re-perform the fluid-structure interaction calculation.

[0074] d) After force balance, calculate the total torque of the system. If unbalanced, adjust accordingly. and / or Return to b) and recalculate.

[0075] e) Once both force and torque have converged, record the current attitude.

[0076] Angle step: .like If the angle is less than 360°, then i = i + 1 to execute the inner loop; otherwise, proceed to the next step.

[0077] Final processing: Draw all rotation working angles Obtain the centroid trajectory and draw ( , The trajectory of the inclined axis is obtained.

[0078] In gas hydrostatic bearings, the primary influence of thrust bearings is on axial errors, such as the flatness error of the thrust plate's working surface. These errors lead to uneven thickness of the planar gas film at different radial positions. The gas film stiffness and deformation of the thrust bearing directly affect the axial movement and tilting of the spindle.

[0079] The primary impact of radial bearings is on radial errors, such as the roundness and cylindricity of the bushing, and the coaxiality error of the assembly with the spindle. These errors lead to uneven thickness of the cylindrical air film on the circumference. The air film stiffness and deformation of the radial bearing directly affect the radial runout and tilting oscillation of the spindle.

[0080] In this invention, models (S2 and S3) are established separately, and their force balance and torque balance are solved separately. Finally, they are coupled and integrated into the 5-DOF model of the main shaft. This invention solves the core problem of how to accurately predict the performance (rotation error) of gas hydrostatic bearings during manufacturing and operation (i.e., when errors and deformations exist).

[0081] While the invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A method for modeling the rotational error of a gas hydrostatic bearing based on two-way fluid-structure interaction, characterized in that, The method includes the following steps: S1. Establish a 5-DOF mechanical model of the gas hydrostatic principal shaft. The 5 degrees of freedom include axial translation Y, two radial translations X and Z, and two radial tilt rotational degrees of freedom. , This model is used to describe the equilibrium state of the spindle under the action of air film force, and the displacement of the center of mass and the rotation tilt distance of the spindle axis under equilibrium state are defined as rotation error. S2. Based on the machining error data of the structural components of the thrust bearing and the spindle, and the bearing design parameters, construct an initial model of the thrust bearing air film containing comprehensive manufacturing errors; calculate the pressure distribution and structural deformation of this air film under the bidirectional fluid-structure interaction effect using the finite element method, and iteratively solve the problem by adjusting the axial displacement of the spindle until the axial force is balanced, recording the axial displacement at this point. and unbalanced torque around the X and Z axes , ; S3. Based on the machining error data of the structural components of the radial bearing and the spindle, and the bearing design parameters, construct an initial model of the radial bearing air film containing comprehensive manufacturing errors; calculate the pressure distribution and structural deformation of this air film under the bidirectional fluid-structure interaction effect using the finite element method, and iteratively solve the problem by adjusting the radial displacement of the spindle until the radial force is balanced, recording the radial displacement at this point. and unbalanced torque around the X and Z axes , ; S4, Calculate the total unbalanced torque of the system. Its value is the vector synthesis result of the torque obtained in steps S2 and S3; determine Is it less than the preset torque convergence threshold? ; S5. If step S4 determines otherwise, adjust the tilt angle of the spindle. and / or This process is repeated to change the air film morphology; then, with a new attitude angle, steps S2 and S3 are executed again to recalculate the axial force, radial force, and new unbalanced torque, and step S4 is performed again. This iterative cycle continues until the total torque of the system meets the requirements. ; S6. If step S4 is correct, then further determine whether the axial force and radial force are both less than the preset force convergence threshold. If not, return to steps S2 to S5; if yes, record the 5-DOF coordinates of the spindle when it reaches equilibrium. In the above steps, 'i' represents the variable corresponding to the i-th change in the spindle's rotation angle around the Y-axis. S7. The (i+1)th time, the working angle of the spindle rotation around the Y-axis is changed to... The working angle of the spindle rotation around the Y-axis compared to the previous one Step Update based on the current rotation angle The thrust and nadir bearing air film model; using the updated air film model as the new initial condition, repeat steps S2 to S6 to solve the spindle balance attitude under the new angle; S8. Traverse all angular positions of the spindle in a 360° rotation, integrate the spindle balance attitude coordinates calculated under all angles, and generate the spindle centroid trajectory diagram and axis spatial tilt trajectory diagram, thus obtaining the accurate rotation error trajectory.

2. The method for modeling the rotational error of a gas hydrostatic bearing based on bidirectional fluid-structure interaction as described in claim 1, characterized in that, The specific process for calculating the bidirectional fluid-structure interaction effect of the gas film containing manufacturing errors in steps S2 and S3 is as follows: S21. Perform finite element mesh generation based on the current air film geometry and set the flow field boundary conditions; S22. Solve the Reynolds equation for the film flow field using the finite element method to obtain the pressure distribution matrix of the discrete nodes; S23. Apply the air film pressure distribution as a surface load to the solid structure finite element model of the main shaft and bearing; S24. Solve the governing equations of solid mechanics to obtain the deformation field of the solid structure under load; S25. Update the thickness and shape of the air film according to the deformation of the structural field; S26. Using the updated air film shape as the new computational domain, repeat steps S21 to S25 until the changes in air film pressure distribution and solid deformation are less than the set tolerance, thus achieving fluid-structure interaction equilibrium.

3. The method for modeling the rotational error of a gas hydrostatic bearing based on bidirectional fluid-structure interaction as described in claim 1, characterized in that, The machining error data of the thrust bearing and spindle structural components mentioned in step S2 mainly include the flatness error of the working surface of the thrust plate and bushing, as well as the assembly perpendicularity and parallelism errors of each component.

4. The method for modeling the rotational error of a gas hydrostatic bearing based on bidirectional fluid-structure interaction as described in claim 1, characterized in that, The machining error data of the structural components of the radial bearing and spindle mentioned in step S3 mainly include the roundness and cylindricity errors of the spindle and bushing, as well as the assembly coaxiality error of each component.

5. The method for modeling the rotational error of a gas hydrostatic bearing based on bidirectional fluid-structure interaction as described in claim 1, characterized in that, The equilibrium condition of the 5-DOF mechanical model in step S1 is: the resultant force of the main shaft in the axial direction and the two radial directions is zero, and the resultant torque about the two radial axes is zero.

6. The method for modeling the rotational error of a gas hydrostatic bearing based on bidirectional fluid-structure interaction as described in claim 1, characterized in that, Torque convergence threshold .

7. The method for modeling the rotational error of a gas hydrostatic bearing based on bidirectional fluid-structure interaction as described in claim 1, characterized in that, Force convergence threshold .

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