A nano-positioning compensation device for a fly-cutting machine tool and a method for suppressing low-frequency shape errors

Abstract

The present invention discloses a nano-positioning compensation device for a fly-cutting machine tool and a method for suppressing low-frequency shape errors, which relates to the field of ultra-precision machining technology. The key points of its technical solution are as follows: A three-degree-of-freedom nano-positioning compensation device is arranged on a horizontal fly-cutting machine tool, and the workpiece to be machined is fixed to the three-degree-of-freedom nano-positioning compensation device; According to the static and dynamic parameters of the air-bearing spindle, a precise prediction model of the swing trajectory of the axis inclination angle of the air-bearing spindle under the action of interrupted cutting force is established; The expansion and contraction amounts of the displacement driving support points required for compensating the reverse attitude of the swing trajectory of the axis inclination angle predicted by the precise prediction model are converted into displacement control commands, and the displacement control commands are input into the three-degree-of-freedom nano-positioning compensation device to keep the center line of the workpiece to be machined parallel to the axis of the air-bearing spindle for axis swing error compensation. The present invention effectively solves the problem of low-frequency "saddle-like" shape errors existing on the machined surface of existing machine tools, and realizes the effective elimination of low-frequency "saddle-like" shape errors.

Description

Technical Field

[0001] The present invention relates to the technical field of ultra-precision machining, and more specifically, it relates to a nano-positioning compensation device for a fly-cutting machine tool and a method for suppressing low-frequency shape errors. Background Art

[0002] Due to excellent optical properties such as a high laser damage threshold and good light transmittance, potassium dihydrogen phosphate (KDP) crystals are widely used in major national scientific projects such as high-power laser devices and are important optical materials for realizing optoelectronic switches and frequency doubling conversion. The performance of KDP crystal optical elements is closely related to the accuracy of their surface topography. For example, laser inertial confinement fusion has put forward different extremely high manufacturing accuracy requirements for the low-frequency surface shape error, intermediate-frequency ripple error, and high-frequency roughness error of KDP crystal optical elements.

[0003] However, due to characteristics such as softness, brittleness, easy deliquescence, and anisotropy, KDP crystals are recognized as one of the most difficult optical materials to process. Traditional grinding and polishing processes are difficult to meet the accuracy requirements of KDP crystal optical elements. Ultra-precision fly-cutting is currently the most effective and widely used processing method for KDP crystal optical elements, but the cutting surface usually has low-frequency "saddle-like" shape errors. For high-power laser devices, such low-frequency geometric shape errors are likely to cause the deflection or divergence of laser beams, making the beams exceed the receiving range of the target pellets, and even causing abnormal focusing of the laser and damaging the optical elements. Research shows that the swing error of the spindle inclination angle during the processing process is the main reason for the low-frequency "saddle-like" shape error on the fly-cut surface, as Figure 1 shown. For such low-frequency errors, existing suppression methods usually involve optimizing the design of the machine tool structure and replacing the spindle, wasting a lot of time and cost, and on existing fly-cutting machine tools, such low-frequency errors are usually difficult to effectively eliminate.

[0004] Therefore, how to research and design a nano-positioning compensation device for a fly-cutting machine tool and a method for suppressing low-frequency shape errors is an urgent problem we need to solve currently. Summary of the Invention

[0005] To solve the deficiencies in the prior art, the purpose of the present invention is to provide a nano-positioning compensation device for a fly-cutting machine tool and a method for suppressing low-frequency shape errors, which can reduce the influence of the swing of the spindle inclination angle on the processed surface topography by real-time positioning, effectively suppress the low-frequency "saddle-like" shape error on the fly-cut surface, and thus improve the machining accuracy of the horizontal fly-cutting machine tool.

[0006] The above technical objectives of the present invention are achieved through the following technical solutions:

[0007] In the first aspect, a nano-positioning compensation device for a fly-cutting machine tool is provided, which includes a base and a support part. The support part is provided with an installation platform for fixing a workpiece to be machined. The support part is provided with three flexible hinges distributed in a triangular shape around the installation platform, and at least one displacement driving member and at least one displacement detecting member are correspondingly arranged between each flexible hinge and the base.

[0008] Furthermore, the support part is in the shape of a triangular hollow frame, and the flexible hinges are distributed in the middle of the corresponding side frames of the support part.

[0009] Furthermore, the displacement driving member is located between the opposite surfaces of the flexible hinge and the base, and the displacement detecting member is located outside the support part. Both the displacement driving member and the displacement detecting member are arranged perpendicular to the surface of the base.

[0010] Furthermore, the displacement driving member is a piezoelectric ceramic.

[0011] Furthermore, the displacement detecting member is a capacitive sensor.

[0012] Furthermore, the installation platform is a vacuum chuck with holes.

[0013] In the second aspect, a method for suppressing low-frequency shape error of a nano-positioning compensation device for a fly-cutting machine tool as described in any item of the first aspect is provided, including the following steps:

[0014] Arrange a three-degree-of-freedom nano-positioning compensation device on a horizontal fly-cutting machine tool, and fix the workpiece to be machined to the three-degree-of-freedom nano-positioning compensation device;

[0015] Establish an accurate prediction model for the swing trajectory of the axis inclination of the aerostatic spindle under the action of intermittent cutting force according to the static and dynamic parameters of the aerostatic spindle;

[0016] Convert the telescopic amounts of the displacement driving support points required for compensating the anti-attitude of the swing trajectory of the axis inclination predicted by the accurate prediction model into displacement control commands, and input the displacement control commands into the three-degree-of-freedom nano-positioning compensation device to keep the center line of the workpiece to be machined parallel to the axis of the aerostatic spindle for axis swing error compensation.

[0017] Furthermore, the accurate prediction model is specifically:

[0018] Calculate the information of the gas lubrication Reynolds equation of the air bearing based on the air static bearing lubrication theory;

[0019] Calculate the air film thickness distribution information at each point in the bearing based on the bearing being slightly disturbed at the steady-state position;

[0020] Use the Taylor expansion method to convert the air film pressure into a function of the steady-state pressure and the disturbance amounts of the journal, and obtain the air film pressure distribution information;

[0021] Fuse and calculate the air film pressure distribution information, air film thickness distribution information and gas lubrication Reynolds equation information, and calculate the steady-state Reynolds equation information and perturbation equation information based on the linear perturbation method;

[0022] Based on the flow balance algorithm, use the finite difference method and the differential quadrature method to solve the steady-state Reynolds equation information and perturbation equation information to obtain the steady-state and dynamic air film pressure distribution information of the bearing;

[0023] Integrate the steady-state and dynamic air film pressure distribution information along the entire air film area to obtain the steady-state bearing capacity, dynamic stiffness and damping of the bearing;

[0024] Equivalent the air film in the air bearing to a spring-damper to support the rotor according to the steady-state bearing capacity, dynamic stiffness and damping of the bearing. Based on the intermittent cutting force and combined with the differential equation of the spindle rotation motion, establish a dynamic model of the bearing-rotor system;

[0025] Use the hammer impact method modal test to identify the model parameters, and solve based on the Runge-Kutta method for the coupled vibration analysis of the air bearing-spindle system to predict the spindle tilt swing response.

[0026] Furthermore, the process of generating the displacement control command is specifically as follows:

[0027] Construct an axis swing trajectory prediction algorithm according to the optimized cutting parameters;

[0028] Measure the spindle tilt angle and feed position in real time, and combine the size of the crystal to be processed and the clamping space points to obtain the axis swing trajectory through the axis swing trajectory prediction algorithm;

[0029] According to the axis swing trajectory and the spatial geometric relationship between the spindle tilt angle and the support points of the three displacement driving parts, calculate the telescopic amounts of the support points of each displacement driving part required for the anti-attitude compensation of the compensation device;

[0030] Convert the telescopic amounts of the support points into displacement commands of the corresponding displacement driving parts.

[0031] Furthermore, the cutting parameters include spindle speed, feed speed, and cutting depth.

[0032] Compared with the prior art, the present invention has the following beneficial effects:

[0033] 1. The three-degree-of-freedom nano-positioning and compensation device designed by the present invention consists of three groups of circular flexible hinges and three groups of stacked piezoelectric ceramics of the same model, which are symmetrically distributed in an equilateral triangle to form a three-dimensional virtual quick tool device. The structure layout is reasonable and has high stiffness. The number of degrees of freedom is the same as the number of driving elements, without redundancy and without decoupling, and can more accurately realize the attitude adjustment function with nanometer-level accuracy;

[0034] 2. Based on the variation law of the inclination angle swing of the air-bearing spindle of the fly-cutting machine tool during the machining process, and combined with the angle signal measured in real time by the angle encoder of the spindle of the fly-cutting machine tool, by controlling the attitude of the KDP crystal adsorbed on the three-degree-of-freedom nano-positioning device, the workpiece center line of the KDP crystal is always parallel to the spindle axis during the machining process, effectively suppressing the low-frequency "saddle-like" shape error of the fly-cutting surface; the present invention effectively solves the problem of the low-frequency "saddle-like" shape error existing on the machining surface of the existing ultra-precision fly-cutting machine tool. Under the condition that there is no particularly stringent requirement for the rotational accuracy of the air-bearing spindle, through the three-degree-of-freedom nano-positioning device without decoupling, according to the swing law of the spindle axis, the effective elimination of the low-frequency "saddle-like" shape error is realized, ensuring the precision use requirements of the KDP crystal for fly-cutting in high-power laser devices.

[0035] 3. By analyzing the dynamic characteristics of the aerostatic bearing under the action of the interrupted cutting force, the dynamic stiffness and dynamic damping coefficient of the air-bearing under the action of the interrupted cutting force are obtained, a dynamic model of the bearing-rotor system is established, and an accurate prediction model of the inclination angle swing of the horizontal fly-cutter cutting spindle is obtained. Brief Description of the Drawings

[0036] The drawings described herein are used to provide a further understanding of the embodiments of the present invention, form a part of this application, and do not limit the embodiments of the present invention. In the drawings:

[0037] Figure 1 is a schematic diagram of the low-frequency "saddle-like" shape error of the fly-cutting surface caused by the inclination angle swing of the spindle in the embodiment of the present invention;

[0038] Figure 2 is a schematic diagram of the structure of the three-degree-of-freedom nano-positioning compensation device in the embodiment of the present invention;

[0039] Figure 3 is a schematic diagram of the integrated installation of the three-degree-of-freedom nano-positioning compensation device and the fly-cutting machine tool in the embodiment of the present invention;

[0040] Figure 4 is a flowchart of the prediction modeling of the axis swing trajectory of the air-bearing spindle of the fly-cutting machine tool in the embodiment of the present invention;

[0041] Figure 5 is a schematic diagram of the compensation of the low-frequency "saddle-like" shape error of the fly-cutting surface in the embodiment of the present invention.

[0042] Marks in the drawings and corresponding component names:

[0043] 101. Base; 102. Support part; 103. Flexible hinge; 104. Installation platform; 105. Displacement detection part; 106. Displacement driving part; 107. Three-degree-of-freedom nano-positioning and compensation device; 201. Bed body; 202. Feeding assembly; 203. Aerostatic spindle; 204. Positioning assembly; 205. Vibration isolation assembly; 206. Support assembly. Detailed implementation manner

[0044] To make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below in conjunction with embodiments and the accompanying drawings. The illustrative embodiments and descriptions of the present invention are only used to explain the present invention and are not intended to limit the present invention.

[0045] It should be noted that when a component is referred to as "fixed to" or "arranged on" another component, it can be directly on the other component or indirectly on the other component. When a component is referred to as "connected to" another component, it can be directly or indirectly connected to the other component.

[0046] It should be understood that the orientation or positional relationship indicated by terms such as "length", "width", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. is based on the orientation or positional relationship shown in the drawings, and is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation to the present invention.

[0047] In addition, the terms "first" and "second" are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the quantity of the indicated technical features. Thus, the features defined with "first" and "second" may explicitly or implicitly include one or more of such features. In the description of the present invention, "a plurality" means two or more, unless otherwise specifically defined.

[0048] Embodiment 1: A nano-positioning and compensation device for a fly-cutting machine tool, as Figure 2 shown, includes a base 101 and a support part 102. The support part 102 is provided with an installation platform 104 for fixing a workpiece to be processed. The support part 102 is provided with three flexible hinges 103 distributed in a triangular shape around the installation platform 104. A displacement driving part 106 and a displacement detection part 105 are correspondingly provided between each flexible hinge 103 and the base 101. In this embodiment, the displacement driving part 106 is a piezoelectric ceramic, the displacement detection part 105 is a capacitive sensor, and the installation platform 104 is a vacuum chuck with holes. The vacuum chuck clamps the KDP crystal by means of vacuum adsorption.

[0049] The support part 102 is a regular triangular hollow frame, and the flexible hinges 103 are distributed in the middle of the corresponding side frames of the support part 102.

[0050] The displacement driving member 106 is located between the opposite surfaces of the flexible hinge 103 and the base 101, and the displacement detecting member 105 is located outside the support part 102. Both the displacement driving member 106 and the displacement detecting member 105 are arranged perpendicular to the surface of the base 101.

[0051] As Figure 3 shown, the horizontal ultra-precision fly-cutting machine tool includes a support assembly 206, four vibration isolation assemblies 205, and a machine body 201 arranged in sequence from bottom to top. The upper surface of the machine body 201 is provided with a feed assembly 202 and a positioning assembly 204. An air-floating spindle 203 for installing a fly cutter head is arranged on the feed assembly 202. The three-degree-of-freedom nano-positioning and compensation device 107 is installed on the positioning assembly 204. By controlling the linear movement of the air-floating spindle 203, the feeding of the diamond tool on the fly cutter head is realized. By controlling the three piezoelectric ceramics according to different input commands, the attitude adjustment of the KDP crystal is realized, so that the workpiece center line of the KDP crystal is always parallel to the axis of the air-floating spindle 203 during the machining process. In this way, the influence of the inclination angle swing of the axis of the air-floating spindle 203 on the fly-cutting surface error during the interrupted cutting process is reduced, and the low-frequency "saddle-like" shape error generated by the swing of the axis of the air-floating spindle 203 is suppressed. Therefore, it is possible to effectively suppress the low-frequency "saddle-like" shape error more flexibly without replacing the air-floating spindle 203.

[0052] It should be noted that the three-degree-of-freedom nano-positioning and compensation device 107 is installed on the Z-axis positioning assembly 204 of the horizontal ultra-precision fly-cutting machine tool, and the air-floating spindle 203 and the fly cutter head with a diamond tool are installed on the X-axis.

[0053] Embodiment 2: A method for suppressing the low-frequency shape error of a fly-cutting machine tool nano-positioning and compensation device, as Figure 5 shown, includes the following steps:

[0054] S1: Arrange a three-degree-of-freedom nano-positioning and compensation device on a horizontal fly-cutting machine tool, and fix the workpiece to be machined to the three-degree-of-freedom nano-positioning and compensation device; the three-degree-of-freedom nano-positioning and compensation device can adopt the nano-positioning and compensation device described in Embodiment 1;

[0055] S2: Establish an accurate prediction model for the inclination angle swing trajectory of the axis of the air-floating spindle under the action of the interrupted cutting force according to the static and dynamic parameters of the air-floating spindle;

[0056] S3: The expansion and contraction of each displacement drive support point required to compensate the anti-posture by the axis inclination swing trajectory predicted by the precise prediction model is converted into a displacement control instruction, and the displacement control instruction is input into the three-degree-of-freedom nano-attitude adjustment compensation device to keep the center line of the workpiece and the axis of the air-floating spindle parallel to each other for real-time response compensation, thereby obtaining a compensated fly-cut surface and suppressing the low-frequency "saddle-like" shape error of the machining surface.

[0057] In step S2, if Figure 4 As shown in the figure, the precise prediction model is as follows:

[0058] S21: Based on the theory of air hydrostatic bearing lubrication, the Navier-Stokes equation, the law of conservation of mass and the gas state equation are combined to calculate the gas lubrication Reynolds equation information of the air bearing;

[0059] S22: calculating the air film thickness distribution information of each point in the bearing based on the small disturbance effect on the bearing at the steady-state position;

[0060] S23: The Taylor expansion method is used to convert the air film pressure into a function of the steady-state pressure and the disturbance quantities of the journal to obtain the air film pressure distribution information;

[0061] S24: integrating and calculating the air film pressure distribution information, the air film thickness distribution information and the gas lubrication Reynolds equation information, and obtaining the steady-state Reynolds equation information and the perturbation equation information based on the linear perturbation method;

[0062] S25: Based on the flow balance algorithm, the finite difference method and differential quadrature method are used to solve the steady-state Reynolds equation information and the perturbation equation information to obtain the steady-state and dynamic air film pressure distribution information of the bearing;

[0063] S26: Integrate the steady-state and dynamic air film pressure distribution information along the entire air film area to obtain the steady-state bearing capacity, dynamic stiffness and damping of the bearing;

[0064] S27: According to the steady-state bearing capacity, dynamic stiffness and damping of the bearing, the air film in the air bearing is equivalent to the spring damping supporting rotor. Based on the intermittent cutting force and combined with the differential equation of spindle rotation motion, the dynamic model of the bearing-rotor system is established;

[0065] S28: The model parameters are identified by the hammer method modal test, and the coupled vibration analysis of the air bearing-spindle system is carried out based on the Runge-Kutta method to predict the spindle inclination swing response.

[0066] In this embodiment, the calculation of dynamic parameters of the air bearing includes the calculation of static and dynamic parameters of the radial air bearing and the calculation of static and dynamic parameters of the thrust air bearing.

[0067] A. Calculation of static and dynamic parameters of radial air bearings

[0068] The Reynolds equation is calculated using the finite difference method, and the dynamic characteristic parameters can be obtained. To simplify the calculation of the journal bearing, cylindrical coordinates are used. The gas lubrication Reynolds equation for a radial air bearing is as follows:

[0069]

[0070] where R is the journal radius; H is the film thickness; p is the film pressure; μ is the gas viscosity coefficient; Ω is the journal angular velocity.

[0071] Higher-order terms above the second order of perturbation are ignored:

[0072]

[0073] The following dimensionless parameters are substituted:

[0074] τ = γΩt,

[0075] where is the dimensionless film pressure; p a is the ambient pressure; C is the average radius clearance of the journal; γ is the whirl ratio; τ is the dimensionless time; Λ is the bearing number; c0 is the average radius clearance of the bearing, and t is the time.

[0076] Substitute the above dimensionless parameters into Equation (2):

[0077]

[0078] Ignoring the motion along the z-axis, the dimensionless film pressure and film thickness caused by the micro-perturbation can be expressed as:

[0079]

[0080]

[0081] where is the dimensionless film thickness at the steady state of the bearing; Φ m0 is the steady-state attitude angle of the bearing; ε0 is the steady-state eccentricity of the bearing; and are the tilt angles about the x-direction and y-direction, respectively.

[0082] The displacements and tilt angles generated by small perturbations are small. Substitute Equations (3) and (4) into Equation (1), and only retain the zero-order and first-order terms. The finite difference method is used to calculate the change in pressure. By integrating the film pressure, the dynamic characteristic coefficients of the radial air bearing can be determined:

[0083]

[0084] In the formula, and are the steady-state bearing capacities of the radial bearing in the x and y directions, with the unit of N; M x0 and M y0 are the steady-state bearing capacity moments of the radial bearing in the x and y directions, with the unit of Nm; P0 is the steady-state air film pressure distribution, with the unit of N / m 2 .

[0085]

[0086]

[0087] In the formula, K JB is the dynamic stiffness of the radial bearing; c JB is the dynamic stiffness of the radial bearing; is the dynamic pressure in each direction of the radial bearing.

[0088] B. Calculation of Static and Dynamic Parameters of Thrust Air Bearing

[0089] The Reynolds equation of the thrust air static bearing in the β-γ coordinate system is as follows:

[0090]

[0091] Without considering the radial translation motion, the air film thickness and air film pressure caused by micro-perturbations are:

[0092]

[0093]

[0094] Among them, H0 is the air film thickness at the steady state of the bearing.

[0095] Substitute the following dimensionless parameters into Equation (9), Equation (10) and Equation (11):

[0096]

[0097] The dimensionless form of the Reynolds equation of the thrust air static bearing and the expressions of the dimensionless form of the air film pressure and thickness can be obtained:

[0098]

[0099]

[0100]

[0101] In the formula, It is the displacement of the z-axis movement caused by micro-perturbations and the tilting angles around the x-axis and y-axis.

[0102] Substitute Eqs. (13) and (14) into Eq. (12), and only retain the zero-order terms and first-order terms. Use the finite difference method to solve the steady-state Reynolds equation and perturbation equation of the thrust bearing, and the steady-state and dynamic air film pressures of the thrust bearing can be obtained. By integrating the pressure in the air film along the entire air film area, the dynamic characteristic coefficients of the thrust air bearing can be obtained:

[0103]

[0104] In the formula, is the steady-state bearing capacity of the thrust bearing in the z direction, with the unit of N; M T x0 , M T y0 are the steady-state bearing moments of the thrust bearing in the x and y directions respectively, with the unit of Nm; P0 is the steady-state air film pressure distribution, N / m 2 ;

[0105]

[0106]

[0107] In the formula, K TB is the dynamic stiffness of the thrust bearing; c TB is the dynamic stiffness of the thrust bearing; is the dynamic pressure in each direction of the thrust bearing.

[0108] In step S3, the process of generating the displacement control instruction is specifically as follows:

[0109] S31: Construct an axis swing trajectory prediction algorithm according to the optimized cutting parameters; the cutting parameters include but are not limited to the spindle speed, feed rate, and cutting depth;

[0110] S32: Measure the spindle tilting angle and feed position in real time, and combine the size of the crystal to be processed and the clamping space points, and obtain the axis swing trajectory through the axis swing trajectory prediction algorithm;

[0111] S33: Calculate the expansion and contraction amounts of the support points of each displacement drive required for the anti-attitude compensation of the compensation device according to the axis swing trajectory and the spatial geometric relationship between the spindle tilting angle and the support points of the three displacement drives;

[0112] S34: Convert the expansion and contraction amounts of the support points into displacement instructions of the corresponding displacement drives.

[0113] The specific embodiments described above further elaborate on the purpose, technical solutions, and beneficial effects of the present invention. It should be understood that the above description is only for the specific embodiments of the present invention and is not used to limit the protection scope of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A method for suppressing low-frequency shape errors of a nano-positioning compensation device for a fly-cutting machine tool, characterized in that, The nano-positioning and compensation device of the fly-cutting machine tool includes a base (101) and a support part (102). The support part (102) is provided with an installation platform (104) for fixing the workpiece to be machined. The support part (102) is provided with three flexible hinges (103) distributed in a triangle along the periphery of the installation platform (104). At least one displacement driving part (106) and at least one displacement detecting part (105) are correspondingly arranged between each flexible hinge (103) and the base (101). The method includes the following steps: Arrange a three-degree-of-freedom nano-positioning and compensation device on a horizontal fly-cutting machine tool, and fix the workpiece to be machined to the three-degree-of-freedom nano-positioning and compensation device; Establish an accurate prediction model of the swing trajectory of the axis inclination angle of the air-bearing spindle under the action of intermittent cutting force according to the static and dynamic parameters of the air-bearing spindle; Convert the expansion and contraction amounts of the displacement driving support points required for compensating the reverse attitude of the swing trajectory of the axis inclination angle predicted by the accurate prediction model into displacement control instructions, and input the displacement control instructions into the three-degree-of-freedom nano-positioning and compensation device to keep the center line of the workpiece to be machined parallel to the axis of the air-bearing spindle for axis swing error compensation; The specific form of the accurate prediction model is as follows: Calculate the information of the gas lubrication Reynolds equation of the air bearing based on the air static pressure bearing lubrication theory; Calculate the air film thickness distribution information at each point in the bearing based on the bearing being subjected to a small disturbance at the steady-state position; Use the Taylor expansion method to convert the air film pressure into a function of the steady-state pressure and the perturbation amounts of the journal, and obtain the air film pressure distribution information; Fuse and calculate the air film pressure distribution information, the air film thickness distribution information and the gas lubrication Reynolds equation information, and calculate the steady-state Reynolds equation information and the perturbation equation information based on the linear perturbation method; Based on the flow balance algorithm, use the finite difference method and the differential quadrature method to solve the steady-state Reynolds equation information and the perturbation equation information, and obtain the steady-state and dynamic air film pressure distribution information of the bearing; Integrate the steady-state and dynamic air film pressure distribution information along the entire air film area to obtain the steady-state bearing capacity, dynamic stiffness and damping of the bearing; Equivalent the air film in the air-bearing to a spring-damping supported rotor according to the steady-state bearing capacity, dynamic stiffness and damping of the bearing. Based on the action of intermittent cutting force and combined with the differential equation of the spindle rotation motion, establish a dynamic model of the bearing-rotor system; Use the hammering method modal test to identify the model parameters, and perform the coupled vibration analysis of the air-bearing-spindle system based on the Runge-Kutta method to predict the spindle inclination angle swing response.

2. The low-frequency shape error suppression method of a nano-positioning compensation device for a fly-cutting machine tool according to claim 1, characterized in that The specific process of generating the displacement control instructions is as follows: Construct an axis swing trajectory prediction algorithm according to the optimized cutting parameters; Measure the spindle inclination angle and the feed position in real time, and combine the size of the crystal to be machined and the clamping space points to obtain the axis swing trajectory through the axis swing trajectory prediction algorithm; Calculate the expansion and contraction amounts of the displacement driving support points required for the reverse attitude compensation of the compensation device according to the axis swing trajectory and the spatial geometric relationship between the spindle inclination angle and the support points of the three displacement driving parts; Convert the expansion and contraction amounts of the support points into displacement instructions of the corresponding displacement driving parts.

3. The low-frequency shape error suppression method of a nano-positioning compensation device for a fly-cutting machine tool according to claim 2, characterized in that The cutting parameters include spindle speed, feed speed, and cutting depth.

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