A method for predicting motion error of a gas hydrostatic guide rail considering dynamic fluid-structure interaction

Abstract

A method for predicting the motion error of a gas static pressure guide considering dynamic fluid-structure interaction is disclosed, relating to the field of mechanical design and manufacturing technology. This method addresses the problem that existing technologies cannot be applied to real-time calculation of dynamic irregular boundary film flow fields considering manufacturing errors, thus hindering the prediction of the five-degree-of-freedom quasi-static motion error of a gas static pressure guide considering dynamic fluid-structure interaction. This application achieves real-time calculation of irregular boundary film flow fields considering manufacturing errors, thereby enabling the prediction of the five-degree-of-freedom quasi-static motion error of a gas static pressure guide considering dynamic fluid-structure interaction. This application overcomes the problems of complex modeling processes and poor computational convergence in traditional computational fluid dynamics for irregular film thicknesses.

Description

Technical Field

[0001] This invention relates to the field of mechanical design and manufacturing technology, specifically to a method for predicting motion errors of gas hydrostatic guideways that considers dynamic fluid-structure interaction. Background Technology

[0002] Gas static pressure guides, with their superior properties such as low friction and high precision, are widely used in key fields such as ultra-precision machine tools and measuring equipment. Significant fluid-structure interaction (FSI) behavior exists between the solid field and the film fluid field of the structural components of a gas static pressure guide. This FSI behavior refers to the deformation and displacement of the structural components under the load of the film fluid field due to the interaction between the film fluid field and the structural field of the gas static pressure guide system. The changes in the structural field, in turn, cause changes in the boundary conditions of the film fluid field, ultimately reaching a dynamic equilibrium at the FSI interface. This FSI behavior leads to significant changes in the boundary conditions and pressure distribution of the film fluid field, which ultimately affect the key performance characteristics of the gas static pressure guide, such as its motion accuracy. The motion error of the gas static pressure guide mainly originates from the manufacturing errors of the structural components. These manufacturing errors, along with the motion errors of the sliding plate assembly, cause continuous changes in the boundary conditions of the film fluid field during the sliding plate assembly's movement, resulting in dynamic changes in the aforementioned FSI behavior and introducing a dynamic FSI effect during motion. Due to limitations such as poor computational convergence and low computational efficiency, the current mainstream two-way transient fluid-structure interaction calculation method based on dynamic mesh technology and computational fluid dynamics three-dimensional air film flow field model cannot be applied to the real-time calculation of dynamic irregular boundary air film flow field considering processing errors. This leads to the problem of low accuracy in predicting the five-degree-of-freedom quasi-static motion error of gas hydrostatic guide rails that consider dynamic fluid-structure interaction behavior. Summary of the Invention

[0003] The purpose of this invention is to address the problem that existing technologies are not applicable to the real-time calculation of dynamic irregular boundary air film flow fields that take into account processing errors, thus leading to the inability to predict the five-degree-of-freedom quasi-static motion error of gas hydrostatic guides that consider dynamic fluid-structure interaction. Therefore, this invention proposes a method for predicting the motion error of gas hydrostatic guides that considers dynamic fluid-structure interaction.

[0004] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0005] A method for predicting motion errors of a gas hydrostatic guide rail considering dynamic fluid-structure interaction includes the following steps:

[0006] Step 1: Obtain the machining errors of the structural components of the gas static pressure guide rail system and the parameters of the gas static pressure guide rail system. Based on the parameters of the gas static pressure guide rail system, obtain the gas film thickness change Δh caused by the ideal fluid-structure interaction effect through bidirectional transient fluid-structure interaction calculation. FSI-ideal ;

[0007] Step 2: Calculate the ideal gas film thickness h0 and Δh in the parameters of the gas static pressure guide rail system. FSI-ideal Import the two-dimensional air film flow field finite element model to obtain the initial air film force. Then, import the initial air film force into the finite element model of the skateboard assembly to obtain the initial deformation parameters of the air film and initialize the motion position i of the skateboard assembly.

[0008] Step 3: Based on the machining error of the structural components of the gas static pressure guide rail system, obtain the change in air film thickness Δh caused by the machining error of the guide rail component at the motion position i of the slide plate component. man-err The thickness of the air film at this time is expressed as:

[0009] h ij =h0+Δh FSI-ideal +Δh man-err ;

[0010] Step 4: Obtain the motion error of the skateboard component corresponding to the current air film thickness, and calculate the change in air film thickness Δh caused by the motion error of the skateboard component. mot-err Then, a straightness error array E is constructed. stp1 and deflection angle error array E stp2 Set the current fluid-structure interaction cycle number c = 0, and store the motion error of the slide plate assembly corresponding to the current air film thickness into E. stp1 (0) and E stp2 (0), the thickness of the air film at this time is expressed as:

[0011] h ij =h0+Δh FSI-ideal +Δh man-err +Δh mot-err ;

[0012] Step 5: Import the current air film thickness into the two-dimensional air film flow field finite element model to obtain the actual air film force;

[0013] Step 6: Import the actual air film force into the finite element model of the slide plate component to obtain the actual deformation parameters of the air film;

[0014] Step 7: Subtract the actual deformation parameters of the air film from the initial deformation parameters of the air film to obtain the change in air film thickness Δh caused by the deformation component of the dynamic fluid-structure interaction effect during the motion. FSI-dyn The dynamic fluid-structure interaction cycle number c = c + 1 is updated, and the film thickness is expressed as:

[0015] h ij =h0+Δh FSI-ideal +Δh man-err +Δh mot-err +Δh FSI-dyn ;

[0016] Step 8: Obtain the motion error of the skateboard component corresponding to the current air film thickness, and store this motion error in E. stp1 (c) and E stp2 (c) Based on the motion error of the skateboard component corresponding to the current air film thickness, the change in air film thickness Δh caused by the motion error of the skateboard component is obtained. mot-err The thickness of the air film at this time is expressed as:

[0017] h ij =h0+Δh FSI-ideal +Δh man-err +Δh mot-err '+Δh FSI-dyn ;

[0018] Step 9: Based on the motion error of the slide plate assembly when the number of dynamic fluid-structure interaction cycles is c and the motion error of the slide plate assembly when the number of cycles is c-1, determine whether the dynamic fluid-structure interaction process has reached a steady state. If a steady state has been reached, proceed directly to step 10. If a steady state has not been reached, use the air film thickness at this time in step 8 as the current air film thickness in step 5, repeat steps 5 to 8 until a steady state is reached, and then proceed to step 10.

[0019] Step 10: Record the motion error of the skateboard component corresponding to the motion position i, i.e., E stp1 (c) and E stp2 (c) and store it in E total (i);

[0020] Step 11: Let the motion position of the skateboard component be i = i + 1. Repeat steps 3 to 10 until the motion position reaches an extreme value. Derive all recorded motion errors, i.e., the total motion error E. total .

[0021] Furthermore, the steps for constructing the two-dimensional air film flow field finite element model are as follows:

[0022] The six air films in the skateboard assembly are evenly divided into m×n regions, and a quasi-static equilibrium equation for the skateboard assembly is established in the five-degree-of-freedom motion error direction. The six air films are upper air film 1, upper air film 2, side air film 1, side air film 2, lower air film 1, and lower air film 2. The quasi-static equilibrium equation for the skateboard assembly is shown in equation (1):

[0023]

[0024] Among them, F U1-ij F U2-ij F represents the local air film force of upper air film 1 and upper air film 2 in their respective ij regions. L1-ij F L2-ijF represents the local air film force of side air film 1 and side air film 2 in their respective ij regions. I1-ij F I2-ij Let x be the local air film force of lower air film 1 and lower air film 2 in their respective ij regions. U1-ij y U1-ij Let x be the coordinates of region ij of the upper air film 1 along the X and Y directions. U2-ij y U2-ij Let x be the coordinates of region ij of the upper air film 2 along the X and Y directions. I1-ij y I1-ij Let x be the coordinates of region ij of the lower air film 1 along the X and Y directions. I2-ij y I2-ij Let x be the coordinates of region ij of the lower air film 2 along the X and Y directions. L1-ij x L2-ij Let W be the coordinates of the ij region of side air film 1 and side air film 2 along the X direction. load Let i = 1, 2...m, j = 1, 2...n, and j be the weight of the skateboard assembly and the external load.

[0025] The steady-state compressibility equation for the film flow field in any region is then expressed in dimensionless form, as follows:

[0026]

[0027] in, The dimensionless thickness of the regional air film. For dimensionless coordinates of the region, For dimensionless coordinates of the region, For the dimensionless pressure of the region, δ is the mass flow coefficient of the orifice throttling. i The symbol is Kronecker, set to 1 in the computational region where an orifice exists, and 0 otherwise. x Let Λ be the number of dimensionless bearings in the X direction. y The number of dimensionless bearings in the Y direction;

[0028] Define the square function of pressure As shown in equation (3), and expressed as a square function The variable is constructed according to equation (1), as shown in equation (4):

[0029]

[0030]

[0031] Where Ω represents the computational region of the air film;

[0032] Solving the square function of pressure using triangular finite element method The interpolation function f divides the air film into 2(q*k) triangular elements, where q is an integer multiple of n and k is an integer multiple of m. The triangular finite element interpolation function is expressed as:

[0033] f = N eT f e (5)

[0034] f e =[f i f j f m ] T (6)

[0035] N e =[N i N j N m ] T (7)

[0036] Among them, f e N is the square of the pressure at each node of the triangular element e. e Let be the shape function corresponding to the triangular element e, and let i, j, and m be the three nodes of any triangular finite element element.

[0037] The pressure square function is replaced by an approximate interpolation function f. Substituting into equation (4) and applying it to the finite element method of the entire computational domain Ω, we obtain equation (8);

[0038]

[0039] Take Φ(f) with respect to f i Find the extreme values ​​of equation (8) by taking the partial derivatives of i, i = 1, 2, ..., n. The partial derivatives are shown in equation (9):

[0040]

[0041] c e =[c i c j c m ] T (10)

[0042] b e =[b i b j b m ] T (11)

[0043] Among them, c e b are the coefficients of the triangular unit shape function. e The coefficients of the triangular unit shape function are given, k1 is a constant coefficient, and μ is a constant coefficient. r is the proportionality coefficient of the area of ​​the throttling orifice r within the corresponding element. Δ is the mass flow rate out of the throttling orifice r. i Let i be the element body with node i, and c and b be the triangular element shape function coefficients at the node;

[0044] Setting equation (9) to 0, we obtain the pressure function relationship at node i where Φ is an extremum. We then extend this relationship to the entire computational domain Ω and rewrite it in matrix form, as shown in equation (12):

[0045] KF=T(12)

[0046] Where K is an n×n square matrix, F is an n×1 column matrix of the squared nodal pressure function to be determined, and T is an n×1 column matrix of the right-hand side of the equation, with three types of elements: 0, ... as well as

[0047] in, and For the coefficients of the triangular unit function in different directions, Δ k Let the area be the area of ​​the k-th triangular unit;

[0048] K is represented as:

[0049]

[0050] Where, Δ i For a triangular element with node i, Δ j Let e ​​be a triangular element with node j, and e be an element with both node i and node j. e For Δ i and Δ j The total area of ​​the intersection;

[0051] The pressure at each node is obtained according to equation (12), let Repeat formulas (4)-(13) for iteration. The iteration uses the node pressure containing the throttling orifice as the convergence criterion. When the relative error between two adjacent iterations is less than the threshold, it is considered to have reached convergence. The gas film force of each region is calculated by integrating the node pressure.

[0052] The flow state in the orifice is simplified to compressible flow through an ideal throttle. When air flows through the orifice, the air pressure immediately changes from the supply pressure P. s Reduced to the pressure P at the end of the throttle orifice d Ideal theoretical mass flow rate of a small-orifice throttle From equation (14), we get:

[0053]

[0054] Where A is the cross-sectional area of ​​the throttling orifice, ψ sFor flow rate function;

[0055] Introducing the flow coefficient C d To correct the mass flow rate, as shown in equation (15):

[0056]

[0057] in, The corrected ideal theoretical mass flow rate for the orifice throttle.

[0058] This completes the construction of the two-dimensional air film flow field finite element model.

[0059] Furthermore, the threshold is 10. -6 .

[0060] Furthermore, the flow coefficient C d =0.8.

[0061] Furthermore, the flow coefficient C d Represented as:

[0062]

[0063] Furthermore, the flow function ψ s Represented as:

[0064]

[0065] Where, β k is a coefficient, and k is the gas constant.

[0066] Furthermore, the specific steps for obtaining the motion error of the skateboard assembly corresponding to the current air film thickness are as follows:

[0067] Step 1: Obtain the current air film thickness and initialize the motion error of the skateboard assembly. Calculate the air film force in each region using a two-dimensional air film flow field finite element model.

[0068] Step 2: For each of the five degrees of freedom, calculate the resultant force of the skateboard assembly in each direction. When the absolute value of the ratio of the resultant force in a certain direction to the reference value in that direction is not greater than a threshold, it is determined that the direction has reached equilibrium. If it is greater than the threshold, the motion error value e in that direction is set. i Adjust to e i +Δe i Δe i Adjust the step resolution, Δe, to the set error. i The sign of the resultant force is related to the direction of the resultant force. The current air film thickness in step 1 is updated using the motion error value in this direction. Then the resultant force in this direction is recalculated. This process is repeated multiple times until the direction reaches a state of equilibrium, and the motion error in this direction is obtained.

[0069] Among them, e y The directional reference value is the total film force F of the side film 1. L1 e z The directional reference value is the total film force F of the upper film 1. U1 e α The directional reference value is the resultant torque M of the upper air film 1 in that direction. U1-α e β The directional reference value is half of the resultant torque M of the upper air film 1 in that direction. U1-0.5-β e γ The directional reference value is the resultant moment M of the half-side air film 1 in that direction. L1-0.5-γ ;

[0070] Step 3: Calculate the five-degree-of-freedom motion error sequentially through the above process. After the five-degree-of-freedom motion error is calculated, verify the balance state in the five directions. If the balance in any direction is broken, update the current air film thickness based on the current five-degree-of-freedom motion error and repeat the above calculation process until the balance state is satisfied in all five directions, and obtain the final five-degree-of-freedom motion error.

[0071] Furthermore, the threshold is 0.1%.

[0072] Furthermore, the resultant force is expressed as:

[0073] F y =∑F L1-ij -∑F L2-ij

[0074] F z =∑F U1-ij +∑F U2-ij -∑F I1-ij -∑F I2-ij -W load

[0075] M α =∑F U1-ij y U1-ij +∑F U2-ij y U2-ij -∑F I1-ij y I1-ij -∑F I2-ij y I2-ij

[0076] M β =-∑F U1-ij x U1-ij -∑F U2-ij x U2-ij +∑F I1-ij x I1-ij +∑F I2-ij x I2-ij

[0077] M γ =∑F L1-ij x L1-ij -∑F L2-ij x L2-ij .

[0078] Furthermore, when the air film is the upper air film 1, the change in air film thickness Δh caused by the motion error of the skateboard assembly... mot-err For e z +e α y U1-ij -e β x U1-ij ;

[0079] When the air film is the upper air film 2, the change in air film thickness Δh caused by the motion error of the skateboard assembly mot-err For e z +e α y U2-ij -e β x U2-ij ;

[0080] When the air film is side air film 1, the change in air film thickness Δh caused by the motion error of the slide plate assembly mot-err for -e y -e γ x L1-ij ;

[0081] When the air film is side air film 2, the change in air film thickness Δh caused by the motion error of the slide plate assembly mot-err For e y +e γ x L2-ij ;

[0082] When the air film is the lower air film 1, the change in air film thickness Δh caused by the motion error of the skateboard assembly mot-err for -e z -e α y I1-ij +e β x I1-ij ;

[0083] When the air film is the lower air film 2, the change in air film thickness Δh caused by the motion error of the skateboard assembly mot-err for -e z -e α y I2-ij +e β x I2-ij .

[0084] The beneficial effects of this invention are:

[0085] This application enables real-time calculation of irregular boundary film flow fields considering manufacturing errors, and further enables prediction of the five-degree-of-freedom quasi-static motion error of a gas hydrostatic guide considering dynamic fluid-structure interaction behavior. This application overcomes the problems of complex modeling process and poor computational convergence in traditional computational fluid dynamics for irregular film thickness. Attached Figure Description

[0086] Figure 1 This is a schematic diagram of the five-degree-of-freedom motion error of a gas static pressure guide rail.

[0087] Figure 2 This is a schematic diagram showing the division of the air-supported membrane area;

[0088] Figure 3 This is a schematic diagram of a two-dimensional film flow finite element model.

[0089] Figure 4 This is a flowchart for calculating motion errors without considering dynamic fluid-structure interaction.

[0090] Figure 5 Flowchart for calculating the overall motion error considering dynamic fluid-structure interaction;

[0091] Figure 6 The straightness error of the 38CrMoAl skateboard assembly;

[0092] Figure 7 The deviation angle error of the 38CrMoAl skateboard assembly;

[0093] Figure 8 The straightness error of the 7075 skateboard assembly;

[0094] Figure 9 The error is the deflection angle of the 7075 skateboard assembly. Detailed Implementation

[0095] It should be noted that, where there is no conflict, the various embodiments disclosed in this application can be combined with each other.

[0096] Specific implementation method one: Refer to Figure 1 This embodiment specifically describes a method for predicting motion errors of a gas hydrostatic guide rail that considers dynamic fluid-structure interaction, comprising the following steps:

[0097] Step 1: Obtain the machining errors of the structural components of the gas static pressure guide rail system and the parameters of the gas static pressure guide rail system. Based on the parameters of the gas static pressure guide rail system, obtain the gas film thickness change Δh caused by the ideal fluid-structure interaction effect through bidirectional transient fluid-structure interaction calculation. FSI-ideal ;

[0098] Step 2: Calculate the ideal gas film thickness h0 and Δh in the parameters of the gas static pressure guide rail system. FSI-idealImport the two-dimensional air film flow field finite element model to obtain the initial air film force. Then, import the initial air film force into the finite element model of the skateboard assembly to obtain the initial deformation parameters of the air film and initialize the motion position i of the skateboard assembly.

[0099] Step 3: Based on the machining error of the structural components of the gas static pressure guide rail system, obtain the change in air film thickness Δh caused by the machining error of the guide rail component at the motion position i of the slide plate component. man-err The thickness of the air film at this time is expressed as:

[0100] h ij =h0+Δh FSI-ideal +Δh man-err ;

[0101] Step 4: Obtain the motion error of the skateboard component corresponding to the current air film thickness, and calculate the change in air film thickness Δh caused by the motion error of the skateboard component. mot-err Then, a straightness error array E is constructed. stp1 and deflection angle error array E stp2 Set the current fluid-structure interaction cycle number c = 0, and store the motion error of the slide plate assembly corresponding to the current air film thickness into E. stp1 (0) and E stp2 (0), the thickness of the air film at this time is expressed as:

[0102] h ij =h0+Δh FSI-ideal +Δh man-err +Δh mot-err ;

[0103] Step 5: Import the current air film thickness into the two-dimensional air film flow field finite element model to obtain the actual air film force;

[0104] Step 6: Import the actual air film force into the finite element model of the slide plate component to obtain the actual deformation parameters of the air film;

[0105] Step 7: Subtract the actual deformation parameters of the air film from the initial deformation parameters of the air film to obtain the change in air film thickness Δh caused by the deformation component of the dynamic fluid-structure interaction effect during the motion. FSI-dyn The dynamic fluid-structure interaction cycle number c = c + 1 is updated, and the film thickness is expressed as:

[0106] h ij =h0+Δh FSI-ideal +Δh man-err +Δh mot-err +Δh FSI-dyn ;

[0107] Step 8: Obtain the motion error of the skateboard component corresponding to the current air film thickness, and store this motion error in E. stp1(c) and E stp2 (c) Based on the motion error of the skateboard component corresponding to the current air film thickness, the change in air film thickness Δh caused by the motion error of the skateboard component is obtained. mot-err The thickness of the air film at this time is expressed as:

[0108] h ij =h0+Δh FSI-ideal +Δh man-err +Δh mot-err '+Δh FSI-dyn ;

[0109] Step 9: Based on the motion error of the slide plate assembly when the number of dynamic fluid-structure interaction cycles is c and the motion error of the slide plate assembly when the number of cycles is c-1, determine whether the dynamic fluid-structure interaction process has reached a steady state. If a steady state has been reached, proceed directly to step 10. If a steady state has not been reached, use the air film thickness at this time in step 8 as the current air film thickness in step 5, repeat steps 5 to 8 until a steady state is reached, and then proceed to step 10.

[0110] Step 10: Record the motion error of the skateboard component corresponding to the motion position i, i.e., E stp1 (c) and E stp2 (c) and store it in E total (i);

[0111] Step 11: Let the motion position of the skateboard component be i = i + 1. Repeat steps 3 to 10 until the motion position reaches an extreme value. Derive all recorded motion errors, i.e., the total motion error E. total .

[0112] This application regionalizes the dynamic air film during the motion process, solves the local air film force in each region using a local air film force calculation method based on the finite element method and flow coefficient model, establishes the quasi-static equilibrium equation of the slide plate component, and performs data transfer and interactive calculation with the finite element model of the structural component on the basis of achieving static equilibrium. Through multiple iterations, the final steady-state equilibrium is reached, realizing the quasi-static motion error calculation of the gas static pressure guide rail system considering dynamic fluid-structure interaction behavior.

[0113] This application realizes the calculation of irregular dynamic air film flow field considering manufacturing errors, overcoming the problems of complex modeling process and poor convergence of traditional computational fluid dynamics for irregular air film thickness. It lays the foundation for the calculation of actual air film force in regionalized air film areas considering manufacturing errors of structural components. Simultaneously, the air film flow field is regionalized, and the air film thickness of each region is decomposed into components influenced by five factors: initial air film thickness, ideal fluid-structure interaction, manufacturing errors, motion errors, and dynamic fluid-structure interaction. The motion error is solved using the quasi-static equilibrium equation of the sliding plate assembly, and then iteratively calculated with the static model of the sliding plate assembly based on the finite element method to solve for the dynamic fluid-structure interaction components, achieving accurate prediction of the quasi-static motion error of the gas hydrostatic guide rail. Taking two sets of sliding plate assemblies as examples, the five-degree-of-freedom motion error is calculated using the calculation method proposed in this application, and the results are as follows. Figures 6-9 As shown, the two skateboard assemblies have the same cross-sectional dimensions and share a single guide rail assembly. The only differences between the two sets of skateboard assemblies are their materials (38CrMoAl stainless steel and 7075 aluminum alloy, respectively, corresponding to different structural rigidities) and skateboard lengths. It can be observed that, except for the torsional yaw angle error e... γ Except for those that are almost unaffected by dynamic fluid-structure interaction (FSI), all other motion errors are influenced by FSI behavior. For 38CrMoAl stainless steel slide plates, FSI results in straightness errors e in both directions. y e z The error increased significantly, with the peak error points increasing by approximately 18.3% and 10.6% respectively, and the pitch angle error e β The influence of dynamic fluid-structure interaction is moderate, with its maximum difference accounting for 12.3% of its maximum amplitude; the roll angle error e α The effect of dynamic fluid-structure interaction is extremely significant, with the maximum difference accounting for 57.6% of its maximum amplitude. Furthermore, the 7075 aluminum alloy slide plate, which has a weaker structural rigidity, experiences a straightness error e in both directions due to dynamic fluid-structure interaction. y e z The peak values ​​increased by 41.4% and 24.3% respectively, and the pitch angle error e β The amplitude at the maximum deflection angle increases by approximately 20.2%, while the roll angle error e α The amplitude at the maximum deflection angle increases by approximately 81.0%.

[0114] The motion error of the gas hydrostatic guide slide assembly includes six degrees of freedom, such as... Figure 1 As shown, the motion error e along the X direction x This is commonly referred to as positioning error. Positioning error is mainly determined by the feedback unit and motion control algorithm, while the motion errors in the other five directions (including two translational errors e) y e z and three rotational errors e αe β e γ The equilibrium state of the skateboard assembly at different positions is determined by the air film bearing capacity provided by the six air films inside the skateboard assembly and its external load. This application mainly calculates the five-degree-of-freedom quasi-static motion error caused by the difference in the static equilibrium position of the skateboard assembly. The processing errors of the guide rail assembly and the skateboard assembly, as well as the motion error of the skateboard assembly itself, all lead to continuous changes in the air film boundary conditions during the motion, resulting in significant changes in the air film force. Therefore, the process of solving the five-degree-of-freedom quasi-static motion error can be equivalent to solving the quasi-static equilibrium equation of the skateboard assembly at different positions.

[0115] Each of the six air films is uniformly divided into m×n regions (the finer the division, the higher the calculation accuracy, but the lower the calculation efficiency), such as... Figure 2 As shown, the quasi-static equilibrium equation of the skateboard assembly established in the direction of five degrees of freedom motion error is shown in equation (1).

[0116]

[0117] in,

[0118] F U1-ij ,F U2-ij —The local air film forces of upper air film 1 and upper air film 2 in their respective ij regions;

[0119] F L1-ij ,F L2-ij —The local air film forces of side air film 1 and side air film 2 in their respective ij regions;

[0120] F I1-ij ,F I2-ij —The local air film forces of lower air film 1 and lower air film 2 in their respective ij regions;

[0121] x U1-ij ,y U1-ij —Coordinates of region ij of upper air film 1 along the X and Y directions;

[0122] x U2-ij ,y U2-ij —Coordinates of region ij of upper air film 2 along the X and Y directions;

[0123] x I1-ij ,y I1-ij —Coordinates of region ij of lower air film 1 along the X and Y directions;

[0124] x I2-ij ,y I2-ij —Coordinates of region ij of lower air film 2 along the X and Y directions;

[0125] xL1-ij ,x L2-ij —The coordinates of the ij region of each of the side air films 1 and 2 along the X direction;

[0126] W load —The weight of the skateboard assembly and external load.

[0127] Considering manufacturing errors, the shape of the air film along the thickness direction will be irregular, and the shape and thickness of the air film will continuously change as the sliding plate assembly moves. Therefore, this application adopts a two-dimensional air film flow field model based on the finite element method and the flow coefficient model to calculate the regionalized air film force considering manufacturing errors. The air film thickness of different regions can be directly loaded onto the corresponding nodes by assignment, avoiding the limitation of needing to re-mesh at different positions in the air film flow field calculation. The steady-state compressible equation of the air film flow field in any region is given in dimensionless form, as shown in equation (2).

[0128]

[0129] in,

[0130] —Dimensionless thickness of the regional air film;

[0131] —Dimensionless coordinates of the region;

[0132] —Dimensionless coordinates of the region;

[0133] —Dimensionless regional pressure;

[0134] —The mass flow rate coefficient of orifice throttling;

[0135] δ i — Kronecker symbol, set to 1 in the computational region where an orifice exists, otherwise 0;

[0136] Λ x —Number of dimensionless bearings in the X direction;

[0137] Λ y —Number of dimensionless bearings in the Y direction.

[0138] Define the square function of pressure As shown in equation (3), and expressed as a square function The variable Φ can be constructed according to equation (1) as shown in equation (4).

[0139]

[0140]

[0141] Where Ω represents the computational domain of the air film, it has been mathematically proven that, under certain boundary conditions, the extremum of equation (4) can only be determined by a certain pressure square function. Therefore, the function is the solution to equation (1). In this application, triangular finite element methods are used to solve the square function of pressure. The interpolation function f divides the air film into 2(q*k) triangular elements, where q is an integer multiple of n and k is an integer multiple of m. i, j, and m are the three nodes of any triangular finite element element. The triangular finite element interpolation function can be defined by equations (5), (6), and (7).

[0142] f = N eT f e (5)

[0143] f e =[f i f j f m ] T (6)

[0144] N e =[N i N j N m ] T (7)

[0145] in,

[0146] f e —The square of the pressure at each node of the triangular element e;

[0147] N e —The shape function corresponding to the triangular unit e.

[0148] The pressure square function is replaced by an approximate interpolation function f. Substituting into equation (4) and applying it to the finite element method across the entire computational domain, we obtain equation (8).

[0149]

[0150] To find the extreme value of equation (8), we take Φ(f) with respect to f i The partial derivatives of (i = 1, 2, ..., n) involve f i The partial derivative of this partial derivative is such that it only has a non-zero value for elements containing node i. Therefore, the computational domain of this partial derivative is reduced to Δ. i Δ i The partial derivatives obtained for the unit cell with node i are shown in equation (9).

[0151]

[0152] in,

[0153] c e — Coefficients of triangular unit shape functions;

[0154] b e — Coefficients of triangular unit shape functions;

[0155] k1 — constant coefficient;

[0156] μ r —The proportionality coefficient of the area of ​​the throttling orifice r within the corresponding unit;

[0157] —The mass flow rate of the flow from the orifice r.

[0158] By setting equation (9) to 0, we can obtain the pressure function relationship at node i that makes Φ an extreme value. We can then extend it to the entire computational domain Ω and rewrite it in matrix form, as shown in equation (10).

[0159] KF=T(10)

[0160] in,

[0161] K is an n×n dimensional square matrix, which can be calculated by equation (11);

[0162] F — the n×1 dimensional column matrix of the squared nodal pressure function to be determined;

[0163] T — An n×1 dimensional column matrix of the right-hand side of the equation, containing three types of elements: 0, ... as well as

[0164]

[0165] in, and For the coefficients of the triangular unit function in different directions, c (k) and b (k) The difference between c and b lies in the coefficients of the triangular unit functions in different directions, Δ k Let be the area of ​​the k-th triangular unit.

[0166] After the above process is completed, the pressure of each node can be obtained, and the node pressure can be set. Repeat Equation 4-11 iteratively, using the nodal pressure containing the orifice as the convergence criterion for multiple iterations. The iteration continues until the relative error between two adjacent iterations is less than 10. -6 When convergence is considered achieved, the film force in each computational region can be obtained through nodal pressure integration. The finite element model of the film force of the gas static pressure guide rail is as follows: Figure 3As shown. The flow state of a throttle orifice is generally simplified to compressible flow through an ideal throttle. When air flows through the throttle orifice, the air pressure immediately changes from the supply pressure P. s Reduced to P d Ideal theoretical mass flow rate of a small-orifice throttle It can be calculated using equation (12).

[0167]

[0168] in,

[0169] A—Cross-sectional area of ​​the throttling orifice;

[0170] ψ s — The flow function can be calculated using equation (13).

[0171]

[0172] in,

[0173] P d —Pressure at the end of the throttle orifice;

[0174] β k --coefficient;

[0175] k — gas constant.

[0176] Considering the actual mass flow rate of the orifice throttle Compared with theoretical mass flow rate The difference between them introduces the flow coefficient C. d To correct the mass flow rate, as shown in equation (14).

[0177]

[0178] In hydrostatic guide rail modeling, the value of the flow coefficient has a significant impact on the calculation accuracy. In many studies, C is generally chosen. d =0.8. However, the academic community generally believes that C d The orifice throttle is highly sensitive to geometric parameters (orifice diameter, film thickness, etc.) and flow parameters (Reynolds number, etc.). Machining errors in the guide rail assembly and fluid-structure interaction can alter the film thickness, thereby changing the flow parameters and ultimately the flow coefficient. To accurately simulate the throttling effect of the orifice throttle, these phenomena and their impact on the flow coefficient should be considered in the finite element modeling. d Is the pressure ratio P d / P s The function is shown in Equation (15), which is used to correct the mass flow rate of the gas film.

[0179]

[0180] Manufacturing errors in the gas static pressure guide system result in varying film thicknesses in different regions. By obtaining the actual film thickness in each region and inputting it into the aforementioned finite element model of the film flow field, the local film force in each region can be solved by integrating the film pressure. For a gas static pressure guide system with predetermined geometric parameters and operating conditions, parameters such as the gas supply pressure are known. Therefore, when calculating the film pressure distribution of any film in the gas static pressure guide system using the aforementioned two-dimensional finite element model of the film flow field, the only input variable is the actual film thickness at each node. The actual film thickness h in any region... ij It consists of five factors and influences, as shown in equation (16), where h is... ij Converted to a dimensionless quantity, this is the dimensionless true gas film thickness of the corresponding unit in equation (2). Among the five parameters, h0 and Δh FSI-ideal Regardless of the position of the skateboard components, the value of Δh remains constant throughout the entire motion process, and the values ​​for each region are independent of the position of the skateboard components. FSI-ideal It can be obtained through conventional two-way transient fluid-structure interaction model calculation.

[0181] h ij =h0+Δh FSI-ideal +Δh man-err +Δh mot-err +Δh FSI-dyn (16)

[0182] in,

[0183] h0—The ideal thickness of the air film, determined by design parameters;

[0184] Δh FSI -ideal——The change in air film thickness caused by the ideal fluid-structure interaction effect obtained from the calculation of air film design parameters;

[0185] Δh man -err——Changes in gas film thickness caused by machining errors in the structural components of the gas static pressure guide rail system;

[0186] Δh mot-err —Changes in air film thickness caused by motion errors of skateboard components;

[0187] Δh FSI -dyn——The change in air film thickness caused by the deformation component of the dynamic fluid-structure interaction effect during motion.

[0188] The last three terms of air film thickness Δh man-err Δh mot-err and Δh FSI-dyn This will continuously change as the position of the skateboard components moves, with the third term Δh... man-errIt can be obtained by measuring the machining errors of the actual guide rail assembly and slide plate assembly, while the fourth item Δh mot-err The error e is caused by the five degrees of freedom motion of the skateboard components during movement. y e z e α e β e γ The change in air film thickness at region ij caused by the motion error of the skateboard assembly can be calculated using equation (17). The five-degree-of-freedom motion error of the skateboard assembly can be obtained through iterative calculation of the quasi-static equilibrium equations at different motion positions. The flowchart for calculating the motion error at any motion position is shown below. Figure 4 As shown. First, the current air film thickness of each region is read and the motion error is initialized. The local air film force of each region is calculated using the two-dimensional air film flow field finite element model described above. For any degree of freedom, the resultant force of the slide plate assembly in that direction can be calculated using equation (1). When the absolute value of the ratio of the resultant force in that direction to the reference value in that direction is less than 0.1%, it is determined that the direction has reached equilibrium. If the value is greater than 0.1%, the motion error value e in that direction needs to be adjusted. i to e i +Δe i Δe i The sign of the resultant force is related to the direction of the net force. After adjustment, the net force in that direction is recalculated, and this process is repeated multiple times until equilibrium is reached in that direction. The five-degree-of-freedom motion error is calculated sequentially through the above process, where e... y The directional reference value is the total film force F of the side film 1. L1 e z The directional reference value is the total film force F of the upper film 1. U1 e α The directional reference value is the resultant torque M of the upper air film 1 in that direction. U1-α e β The directional reference value is half of the resultant torque M of the upper air film 1 in that direction. U1-0.5-β e γ The directional reference value is the resultant moment M of the half-side air film 1 in that direction. L1-0.5-γ It is important to note that there is coupling between the five degrees of freedom. Motion errors in subsequent calculations may disrupt the equilibrium state of the preceding motion errors. Therefore, after the five-degree-of-freedom motion error calculation is completed, the equilibrium state in the five directions needs to be verified. If the equilibrium is disrupted, the current air film thickness needs to be updated based on the current motion error, and the above calculation process needs to be repeated multiple times until the equilibrium state in each direction is reached, thus deriving the five-degree-of-freedom motion error.

[0189]

[0190] The last term of the air film thickness, Δh FSI-dynThe change in air film thickness during motion is caused by the dynamic fluid-structure interaction effect resulting from the variations in the third and fourth parameters. It is worth noting that the dynamic fluid-structure interaction effect includes both displacement and deformation components, with the displacement component already included in the fourth parameter Δh. mot-err Therefore, the fifth parameter Δh FSI-dyn It only includes the deformation component of the dynamic fluid-structure interaction effect, that is, the dynamic deformation of the skateboard component structure caused by the dynamic changes of the air film force during motion. Because the computational resources required for dynamic fluid-structure interaction during motion are too enormous, it is almost impossible to calculate it using existing two-way transient fluid-structure interaction models. However, due to the fifth term Δh... FSI-dyn This calculation only includes the deformation component of the dynamic fluid-structure interaction effect, therefore this parameter can be obtained through the computationally efficient static finite element method. First, a static finite element model of the skateboard assembly is established. Using the previous two-dimensional air film flow finite element model, the air film force distribution corresponding to the air film thickness (including only the first two parameters) is calculated. This air film force distribution is then loaded onto the air film surface of the skateboard assembly finite element model, and its initial deformation parameters are obtained through calculation. Next, the air film force distribution corresponding to the air film thickness considering the first four parameters is calculated using the air film flow finite element model, and this is loaded into the skateboard assembly finite element model to obtain its actual deformation parameters. The difference between the actual deformation parameters and the initial deformation parameters in each region is the air film thickness change Δh caused by the dynamic deformation of the skateboard assembly. FSI-dyn However, it is important to note that Δh FSI-dyn This itself will also cause changes in the local air film force in various regions, therefore, it is necessary to consider this in the motion error calculation process. Figure 3 The process involves multiple iterations between the process flow and the static finite element model of the skateboard component until the motion error between two adjacent iterations stabilizes.

[0191] Taking into account the above five parameters, the comprehensive calculation process for the five-degree-of-freedom motion error of the gas hydrostatic guide rail system considering the dynamic fluid-structure interaction effect is as follows: Figure 5 As shown, the main steps for iterative calculation of motion error are as follows:

[0192] (a) Import the system parameters of the gas static pressure guide rail system (including but not limited to air supply pressure, slide plate assembly structural dimensions, and air film geometry) and the machining error Δh of the slide plate assembly and guide rail assembly obtained by measurement. man-err The ideal air film thickness in each region (including the first two parameters of equation (16)) is obtained by bidirectional transient fluid-structure interaction calculation.

[0193] (b) Import the ideal air film thickness into the two-dimensional air film flow field finite element model to calculate the ideal air film force in each region, import the ideal air film force in each region into the finite element model of the skateboard assembly to calculate the initial deformation parameters of each air film, and initialize the motion position of the skateboard assembly.

[0194] (c) Read the current position machining error of the skateboard component and initialize the fourth term Δh. mot-err (Reset to zero), update the air film thickness of each region (including the first three parameters of formula (16));

[0195] (d) By Figure 4 The process calculates the motion error of the skateboard assembly at that location (without considering dynamic fluid-structure interaction effects) and stores it in E. stp1 (0) and E stp2 (0); Initialize the number of dynamic fluid-structure interaction cycles c, and initialize the fifth term Δh. FSI-dyn Update the air film thickness of each region (including the first four parameters of formula (16));

[0196] (e) Import the film thickness of each region into the two-dimensional film flow field finite element model to calculate the actual film force in each region. Import the actual film force of each region into the finite element model of the slide plate assembly to calculate the actual deformation parameters of each film. Subtract the actual deformation parameters from the actual deformation parameters to obtain the fifth term Δh. FSI-dyn The number of dynamic fluid-structure interaction cycles c is updated to c+1, and the air film thickness of each region is updated (including the first five parameters of equation (16));

[0197] (f) By Figure 4 The process calculates the motion error of the skateboard assembly at that location (considering dynamic fluid-structure interaction effects) and stores it in E. stp1 (c) and E stp2 (c) Determine whether the dynamic fluid-structure interaction process has reached a steady state by the motion error between the cth and c-1th iterations. If it has not reached a steady state, return to (e) for iteration until the dynamic fluid-structure interaction reaches a steady state.

[0198] (g) Record the final motion error E at the current position. total (i) The motion position i changes to i+1, return to step (c) to calculate the motion error, iterate multiple times until the motion position reaches an extreme value, and derive the full-stroke motion error E. total .

[0199] It should be noted that the specific embodiments are merely explanations and illustrations of the technical solution of the present invention and should not be used to limit the scope of protection. Any modifications made in accordance with the claims and specification of the present invention that are only partial should still fall within the protection scope of the present invention.

Claims

1. A method for predicting motion errors of a gas hydrostatic guide rail considering dynamic fluid-structure interaction, characterized in that... Includes the following steps: Step 1: Obtain the machining errors of the structural components of the gas static pressure guide rail system and the parameters of the gas static pressure guide rail system. Based on the parameters of the gas static pressure guide rail system, obtain the change in gas film thickness caused by the ideal fluid-structure interaction effect through bidirectional transient fluid-structure interaction calculation. ; Step 2: Calculate the ideal gas film thickness in the parameters of the gas static pressure guide rail system. and Import the two-dimensional air film flow field finite element model to obtain the initial air film force. Then, import the initial air film force into the finite element model of the skateboard assembly to obtain the initial deformation parameters of the air film and initialize the motion position i of the skateboard assembly. Step 3: Based on the machining error of the structural components of the gas static pressure guide rail system, obtain the change in air film thickness caused by the machining error of the guide rail components at the motion position i of the slide block component. The thickness of the air film at this time is expressed as: ； Step 4: Obtain the motion error of the skateboard component corresponding to the current air film thickness, and calculate the change in air film thickness Δh caused by the motion error of the skateboard component. mot-err Then construct the straightness error array E stp1 and deflection angle error array E stp2 Set the current fluid-structure interaction cycle number c=0, and store the motion error of the slide plate assembly corresponding to the current air film thickness into E. stp1 (0) and E stp2 (0), the thickness of the air film at this time is expressed as: ； Step 5: Import the current air film thickness into the two-dimensional air film flow field finite element model to obtain the actual air film force; Step 6: Import the actual air film force into the finite element model of the slide plate component to obtain the actual deformation parameters of the air film; Step 7: Subtract the actual deformation parameters of the air film from the initial deformation parameters of the air film to obtain the change in air film thickness caused by the deformation component of the dynamic fluid-structure interaction effect during the motion. The dynamic fluid-structure interaction cycle number c = c + 1 is updated, and the film thickness is expressed as: ； Step 8: Obtain the motion error of the skateboard component corresponding to the current air film thickness, and store this motion error in E. stp1 (c) and E stp2 (c) Based on the motion error of the skateboard component corresponding to the current air film thickness, the change in air film thickness caused by the motion error of the skateboard component is obtained. The thickness of the air film at this time is expressed as: ； Step 9: Based on the motion error of the slide plate assembly when the number of dynamic fluid-structure interaction cycles is c and the motion error of the slide plate assembly when the number of cycles is c-1, determine whether the dynamic fluid-structure interaction process has reached a steady state. If a steady state has been reached, proceed directly to step 10. If a steady state has not been reached, use the air film thickness at this time in step 8 as the current air film thickness in step 5, repeat steps 5 to 8 until a steady state is reached, and then proceed to step 10. Step 10: Record the motion error of the skateboard component corresponding to the motion position i, i.e., E stp1 (c) and E stp2 (c) and store it in E total (i); Step 11: Let the motion position of the skateboard component be i = i + 1. Repeat steps 3 to 10 until the motion position reaches an extreme value. Derive all recorded motion errors, i.e., the total motion error E. total .

2. The method for predicting motion error of a gas hydrostatic guide rail considering dynamic fluid-structure interaction as described in claim 1, characterized in that... The steps for constructing the two-dimensional air film flow field finite element model are as follows: The six air films in the skateboard assembly are evenly divided into m×n regions, and a quasi-static equilibrium equation for the skateboard assembly is established in the five-degree-of-freedom motion error direction. The six air films are upper air film 1, upper air film 2, side air film 1, side air film 2, lower air film 1, and lower air film 2. The quasi-static equilibrium equation for the skateboard assembly is shown in equation (1): (1) Among them, F U1-ij F U2-ij F represents the local air film force of upper air film 1 and upper air film 2 in their respective ij regions. L1-ij F L2-ij F represents the local air film force of side air film 1 and side air film 2 in their respective ij regions. I1-ij F I2-ij Let x be the local air film force of lower air film 1 and lower air film 2 in their respective ij regions. U1-ij y U1-ij Let x be the coordinates of region ij of the upper air film 1 along the X and Y directions. U2-ij y U2-ij Let x be the coordinates of region ij of the upper air film 2 along the X and Y directions. I1-ij y I1-ij Let x be the coordinates of region ij of the lower air film 1 along the X and Y directions. I2-ij y I2-ij Let x be the coordinates of region ij of the lower air film 2 along the X and Y directions. L1-ij x L2-ij Let W be the coordinates of the ij region of side air film 1 and side air film 2 along the X direction. load Let i = 1, 2...m, j = 1, 2...n, and j be the weight of the skateboard assembly and the external load. The steady-state compressibility equation for the film flow field in any region is then expressed in dimensionless form, as follows: (2) in, The dimensionless thickness of the regional air film. For dimensionless coordinates of the region, For dimensionless coordinates of the region, For the dimensionless pressure of the region, The mass flow rate coefficient for orifice throttling is... The symbol is Kronecker, set to 1 in the computational region where an orifice exists, and 0 otherwise. Let X be the number of dimensionless bearings in the X direction. The number of dimensionless bearings in the Y direction; Define the square function of pressure As shown in equation (3), and expressed as a square function As a variable, construct the constructor Φ according to equation (1), as shown in equation (4): (3) (4) Where Ω represents the computational region of the air film; Solving the square function of pressure using triangular finite element method The interpolation function f divides the air film into 2(q*k) triangular elements, where q is an integer multiple of n and k is an integer multiple of m. The triangular finite element interpolation function is expressed as: (5) (6) (7) Among them, f e N is the square of the pressure at each node of the triangular element e. e Let be the shape function corresponding to the triangular element e, and let i, j, and m be the three nodes of any triangular finite element element. The pressure square function is replaced by an approximate interpolation function f. Substituting into equation (4) and applying it to the finite element method of the entire computational domain Ω, we obtain equation (8); (8) Take Φ(f) with respect to f i Find the extreme values ​​of equation (8) by taking the partial derivatives of i, i = 1, 2, ..., n. The partial derivatives are shown in equation (9): (9) (10) (11) Among them, c e b are the coefficients of the triangular unit shape function. e The coefficients of the triangular unit shape function are given, k1 is a constant coefficient, and μ is a constant coefficient. r is the proportionality coefficient of the area of ​​the throttling orifice r within the corresponding element. Δ is the mass flow rate out of the throttling orifice r. i Let i be the unit cell with node i. and These are the shape function coefficients of the triangular element at the nodes; Setting equation (9) to 0, we obtain the pressure function relationship at node i where Φ is an extremum. We then extend this relationship to the entire computational domain Ω and rewrite it in matrix form, as shown in equation (12): (12) Where K is an n×n square matrix, F is an n×1 column matrix of the squared nodal pressure function to be determined, and T is an n×1 column matrix of the right-hand side of the equation, with three types of elements: 0, ... ,as well as , in, and For the coefficients of the triangular unit functions in different directions, Let the area be the area of ​​the k-th triangular unit; K is represented as: (13) in, For a triangular element with node i, For a triangular element with node j, For an element that simultaneously has node i and node j, for and The total area of ​​the intersection; According to formula (12), the pressure of each node is obtained. Let the node pressure = f̅. Repeat formula (4)-(13) for iteration. The iteration takes the node pressure containing the throttle hole as the convergence criterion. When the relative error between two adjacent iterations is less than the threshold, it is considered to have reached convergence. The gas film force of each region is calculated by integrating the node pressure. The flow state in the orifice is simplified to compressible flow through an ideal throttle. When air flows through the orifice, the air pressure immediately changes from the supply pressure P. s Reduced to the pressure P at the end of the throttle orifice d Ideal theoretical mass flow rate of a small-orifice throttle From equation (14), we get: (14) Where A is the cross-sectional area of ​​the throttling orifice, ψ s For flow rate function; Introducing the flow coefficient C d To correct the mass flow rate, as shown in equation (15): (15) in, The corrected ideal theoretical mass flow rate for the orifice throttle. This completes the construction of the two-dimensional air film flow field finite element model.

3. The method for predicting motion error of a gas hydrostatic guide rail considering dynamic fluid-structure interaction as described in claim 2, characterized in that... The threshold is 10 -6 .

4. The method for predicting motion error of a gas hydrostatic guide rail considering dynamic fluid-structure interaction according to claim 2, characterized in that... The flow coefficient =0.

8.

5. The method for predicting motion error of a gas hydrostatic guide rail considering dynamic fluid-structure interaction according to claim 2, characterized in that... The flow coefficient Represented as: 。 6. The method for predicting motion error of a gas hydrostatic guide rail considering dynamic fluid-structure interaction according to claim 2, characterized in that... The flow function ψ s Represented as: Where, β k is a coefficient, and k is the gas constant.

7. The method for predicting motion error of a gas hydrostatic guide rail considering dynamic fluid-structure interaction according to claim 1, characterized in that... The specific steps for obtaining the motion error of the skateboard component corresponding to the current air film thickness are as follows: Step 1: Obtain the current air film thickness and initialize the motion error of the skateboard assembly. Calculate the air film force in each region using a two-dimensional air film flow field finite element model. Step 2: For each of the five degrees of freedom, calculate the resultant force of the skateboard assembly in each direction. When the absolute value of the ratio of the resultant force in a certain direction to the reference value in that direction is not greater than a threshold, it is determined that the direction has reached equilibrium. If it is greater than the threshold, the motion error value e in that direction is set. i Adjust to e i +Δe i Δe i Adjust the step resolution, Δe, to the set error. i The sign of the resultant force is related to the direction of the resultant force. The current air film thickness in step 1 is updated using the motion error value in this direction. Then the resultant force in this direction is recalculated. This process is repeated multiple times until the direction reaches a state of equilibrium, and the motion error in this direction is obtained. Among them, e y The directional reference value is the total film force F of the side film 1. L1 e z The directional reference value is the total film force F of the upper film 1. U1 e α The directional reference value is the resultant torque M of the upper air film 1 in that direction. U1-α e β The directional reference value is half of the resultant torque M of the upper air film 1 in that direction. U1-0.5-β e γ The directional reference value is the resultant moment M of the half-side air film 1 in that direction. L1-0.5-γ ; Step 3: Calculate the five-degree-of-freedom motion error sequentially through the above process. After the five-degree-of-freedom motion error is calculated, verify the balance state in the five directions. If the balance in any direction is broken, update the current air film thickness based on the current five-degree-of-freedom motion error and repeat the above calculation process until the balance state is satisfied in all five directions, and obtain the final five-degree-of-freedom motion error.

8. The method for predicting motion error of a gas hydrostatic guide rail considering dynamic fluid-structure interaction according to claim 7, characterized in that... The threshold is 0.1%.

9. A method for predicting motion error of a gas hydrostatic guide rail considering dynamic fluid-structure interaction as described in claim 7, characterized in that... The resultant force is expressed as: ， Among them, F U1-ij F U2-ij F represents the local air film force of upper air film 1 and upper air film 2 in their respective ij regions. L1-ij F L2-ij F represents the local air film force of side air film 1 and side air film 2 in their respective ij regions. I1-ij F I2-ij Let x be the local air film force of lower air film 1 and lower air film 2 in their respective ij regions. U1-ij y U1-ij Let x be the coordinates of region ij of the upper air film 1 along the X and Y directions. U2-ij y U2-ij Let x be the coordinates of region ij of the upper air film 2 along the X and Y directions. I1-ij y I1-ij Let x be the coordinates of region ij of the lower air film 1 along the X and Y directions. I2-ij y I2-ij Let x be the coordinates of region ij of the lower air film 2 along the X and Y directions. L1-ij x L2-ij Let W be the coordinates of the ij region of side air film 1 and side air film 2 along the X direction. load Let i = 1, 2...m and j = 1, 2...n be the weight of the skateboard assembly and the external load.

10. The method for predicting motion error of a gas hydrostatic guide rail considering dynamic fluid-structure interaction according to claim 7, characterized in that, When the air film is the upper air film 1, the change in air film thickness Δh caused by the motion error of the skateboard assembly mot-err for ; When the air film is the upper air film 2, the change in air film thickness Δh caused by the motion error of the skateboard assembly mot-err for ; When the air film is side air film 1, the change in air film thickness Δh caused by the motion error of the slide plate assembly mot-err for ; When the air film is side air film 2, the change in air film thickness Δh caused by the motion error of the slide plate assembly mot-err for ; When the air film is the lower air film 1, the change in air film thickness Δh caused by the motion error of the skateboard assembly mot-err for ; When the air film is the lower air film 2, the change in air film thickness Δh caused by the motion error of the skateboard assembly mot-err for .

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