Longitudinal-torsional composite ultrasonic vibration polishing method for inhibiting intermediate frequency error of hard and brittle free-form surface

Abstract

The invention relates to a polishing method for a hard and brittle free-form surface, in particular to a longitudinal-torsional composite ultrasonic vibration polishing method for inhibiting intermediate-frequency errors of the hard and brittle free-form surface, and solves the technical problem that an existing ultrasonic vibration polishing method cannot inhibit the intermediate-frequency errors of polishing of the free-form surface of a hard and brittle material. According to the method, a mechanical polishing method is adopted, longitudinal-torsional composite ultrasonic vibration is added to the polishing tool, so that the rotating motion and the feeding motion of the polishing tool are combined with the longitudinal-torsional composite ultrasonic vibration to form a space complex coupling polishing track, and the purpose that the polishing tool conducts polishing in the material removing direction according to the space complex coupling polishing track is achieved; by changing the amplitude and frequency of longitudinal-torsional composite ultrasonic vibration, the spatial attitude of a longitudinal-torsional composite ultrasonic vibration polishing time-varying removal function is changed in real time, so that the removal function of each polishing point is different under the action of a spatial complex coupling polishing track; therefore, the suppression of the intermediate-frequency error is realized.

Description

Technical Field

[0001] The present invention relates to a polishing method for hard and brittle free-form surfaces, and particularly to a longitudinal-torsional composite ultrasonic vibration polishing method for suppressing medium-frequency errors of hard and brittle free-form surfaces. Background Art

[0002] Due to the unique advantages of improving the imaging quality, light weight, and integration degree of the system, high-precision free-form optical elements have become a new trend in the optical application field. The manufacturing accuracy requirements for hard and brittle free-form surfaces in optical systems with high resolution, large numerical aperture, and ultra-small distortion have been extended to the full frequency domain, that is, while achieving nanometer-level accuracy of the low-frequency surface shape error, the medium and high-frequency errors must also reach sub-nanometer-level accuracy. Most of the efficient shaping methods that adapt to the multi-degree-of-freedom changes of hard and brittle free-form surfaces are sub-aperture polishing methods. As a principle by-product of the sub-aperture polishing method, medium and high-frequency errors can only be suppressed by adjusting the shape of the removal function, reducing the regularity and spacing of the polishing path, reducing the abrasive grain size, and increasing the elastic-plastic removal ratio of hard and brittle materials. The complex polishing path of free-form surfaces requires high dynamic characteristics of the machine tool, while the miniaturization of tool size and process parameters will severely restrict the polishing efficiency, and the commonly used methods for suppressing medium and high-frequency errors in shaping and polishing are also limited.

[0003] Aiming at problems such as weakening of the surface spatial texture of hard and brittle materials, difficulty in suppressing medium-frequency errors, and improving processing efficiency, using ultrasonic vibration to polish free-form surfaces will be one of the important future development directions and research hotspots. At present, the ultrasonic vibration polishing method has shown significant comprehensive advantages in the polishing of flat surfaces: improving the machinability of hard and brittle materials, increasing the number of effective abrasive grains and the uniformity of movement trajectories, extending the tool life, improving the polishing efficiency, reducing the spatial frequency of the polishing surface texture, reducing the surface roughness and the depth of subsurface damage, and has been maturely applied to flat optical elements. However, there is little research on ultrasonic vibration polishing of hard and brittle free-form surfaces, and the ultrasonic vibration polishing method for flat surfaces cannot be applied to free-form surfaces. Summary of the Invention

[0004] The purpose of the present invention is to solve the technical problem that the existing ultrasonic vibration polishing method cannot suppress the medium-frequency errors in the polishing of hard and brittle material free-form surfaces, and to provide a longitudinal-torsional composite ultrasonic vibration polishing method for suppressing medium-frequency errors of hard and brittle free-form surfaces.

[0005] To achieve the above purpose, the technical solution adopted by the present invention is as follows:

[0006] A longitudinal-torsional composite ultrasonic vibration polishing method for suppressing medium-frequency errors of hard and brittle free-form surfaces, which is characterized in that it includes the following steps:

[0007] Step 1: Perform surface shape detection on the hard and brittle free-form surface to be processed. Determine the polishing removal amount of each polishing point on the hard and brittle free-form surface to be processed according to the surface shape detection results, as well as the amplitude and frequency range of the longitudinal-torsional composite ultrasonic vibration applied to the polishing tool; the longitudinal-torsional composite ultrasonic vibration includes spindle longitudinal ultrasonic vibration and torsional ultrasonic vibration;

[0008] Step 2: Select the feed motion path of the polishing tool;

[0009] Step 3: Determine the time-varying removal function of longitudinal-torsional vibration polishing at different amplitudes and frequencies of the longitudinal-torsional composite ultrasonic vibration; at the same time, superimpose the longitudinal-torsional composite ultrasonic vibration on the feed motion path of the polishing tool to obtain a spatially complex coupled polishing trajectory;

[0010] Step 4: Calculate the dwell time of the polishing tool at each polishing point on the hard and brittle free-form surface to be processed according to the time-varying removal function of longitudinal-torsional vibration polishing and the polishing removal amount of each polishing point on the hard and brittle free-form surface to be processed;

[0011] Step 5: Adopt a mechanical polishing method. During the polishing process, change the amplitude and frequency of the longitudinal-torsional composite ultrasonic vibration, so that the polishing tool polishes the hard and brittle free-form surface to be processed according to the spatially complex coupled polishing trajectory and the dwell time at each polishing point on the hard and brittle free-form surface to be processed;

[0012] Step 6: Perform surface shape detection on the polished hard and brittle free-form surface to be processed, and judge whether the polishing target is achieved. If the polishing target is achieved, the polishing of the hard and brittle free-form surface to be processed is completed; otherwise, return to Step 5 and continue polishing.

[0013] Further, in Step 3, the time-varying removal function of longitudinal-torsional vibration polishing is:

[0014] TIF(x,y,t) = TIF(x,y) × f(w,A)

[0015] where TIF(x,y,t) is the time-varying removal function of longitudinal-torsional vibration polishing, (x,y) is the two-dimensional plane coordinates of the polishing point on the hard and brittle free-form surface to be processed, and t is the polishing time;

[0016] TIF(x,y) is the removal function per unit time;

[0017] f(w,A) is a function that varies with the frequency and amplitude of the longitudinal-torsional composite ultrasonic vibration, w is the frequency of the longitudinal-torsional composite ultrasonic vibration, and A is the amplitude of the longitudinal-torsional composite ultrasonic vibration.

[0018] Further, in Step 4, the dwell time of the polishing tool on the hard and brittle free-form surface to be processed is calculated by the following formula:

[0019]

[0020] Among them, h(x, y) is the polishing removal amount of the polishing point on the free-form surface of the hard and brittle material to be processed, j = 1, 2, …, m, where m is the number of scanning rows in the spatially complex coupled polishing trajectory, and i = 1, 2, …, n, where n is the number of polishing points in each row of the spatially complex coupled polishing trajectory; T(x, y) is the dwell time of the polishing tool at the polishing point on the free-form surface of the hard and brittle material to be processed.

[0021] Further, in step 3, the time-varying removal function of longitudinal-torsional vibration polishing is determined through dotting tests.

[0022] Further, step 1 is specifically as follows:

[0023] 1.1. Use LuphoScan or an interferometer to perform surface shape detection on the free-form surface of the hard and brittle material to be processed, and extract the intermediate-frequency error frequency distribution data through FFT analysis or PSD analysis;

[0024] 1.2. Set the polishing removal amount of each polishing point on the free-form surface of the hard and brittle material to be processed according to the surface shape detection result, and set the amplitude and frequency range of the longitudinal-torsional composite ultrasonic vibration applied to the polishing tool according to the intermediate-frequency error frequency distribution data.

[0025] Further, in step 2, the feed motion path of the polishing tool is a grating path, and its scanning line spacing is the scanning spacing of the grating path.

[0026] Further, in step 5, the frequency and amplitude of the longitudinal-torsional composite ultrasonic vibration vary randomly within the amplitude and frequency range of the longitudinal-torsional composite ultrasonic vibration set in step 1.

[0027] Further, in step 2, the polishing tool is a polishing disc, a polishing wheel or an airbag tool, and its loading device is a machine tool or a six-degree-of-freedom robot.

[0028] Compared with the prior art, the beneficial technical effects of the present invention are as follows:

[0029] 1. The longitudinal-torsional composite ultrasonic vibration polishing method for suppressing intermediate-frequency errors of a free-form surface of a hard and brittle material provided by the present invention adopts a mechanical polishing method, and attaches longitudinal-torsional composite ultrasonic vibration to the polishing tool. By combining the rotational motion and feed motion of the polishing tool itself with the longitudinal-torsional composite ultrasonic vibration, the spatial attitude of the time-varying removal function of longitudinal-torsional vibration polishing is changed in real time, thereby realizing the suppression of intermediate-frequency errors;

[0030] 2. The longitudinal-torsional composite ultrasonic vibration polishing method for suppressing the medium-frequency error of hard and brittle free-form surfaces provided by the present invention adopts the method of adding longitudinal-torsional composite ultrasonic vibration to apply two-dimensional vibration to the tangent plane of the hard and brittle free-form surface, which can adapt to different curvature positions of the hard and brittle free-form surface, achieve point-to-point fitting, effectively reduce the polishing force and surface roughness of the hard and brittle free-form surface, and enhance the plastic shear effect in polishing. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 It is a schematic diagram of the principle of the mechanical polishing method for existing hard and brittle free-form surfaces;

[0032] Figure 2 It is a schematic diagram of the principle of the method of the embodiment of the present invention;

[0033] Figure 3 It is a simulation schematic diagram of the traditional mechanical polishing method. The scanning line spacing of the feed motion path of the polishing tool is 2 mm. Among them, (a) is the three-dimensional topography map of the polished hard and brittle free-form surface to be processed, and (b) is the two-dimensional cross-sectional view of the polished hard and brittle free-form surface to be processed;

[0034] Figure 4 It is a simulation schematic diagram of the method of the embodiment of the present invention. The scanning line spacing of the feed motion path of the polishing tool is 2 mm. Among them, (a) is the three-dimensional topography map of the polished hard and brittle free-form surface to be processed, and (b) is the two-dimensional cross-sectional view of the polished hard and brittle free-form surface to be processed;

[0035] Figure 5 For Figure 3 The frequency-domain characteristic peak map obtained by FFT analysis;

[0036] Figure 6 For Figure 4 The frequency-domain characteristic peak map obtained by FFT analysis. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0037] The following describes in detail a longitudinal-torsional composite ultrasonic vibration polishing method for suppressing the medium-frequency error of hard and brittle free-form surfaces proposed by the present invention in conjunction with the drawings and specific embodiments. Those skilled in the art should understand that these embodiments are only used to explain the technical principle of the present invention, and the purpose is not to limit the protection scope of the present invention.

[0038] A longitudinal-torsional composite ultrasonic vibration polishing method for suppressing the medium-frequency error of hard and brittle free-form surfaces, characterized by comprising the following steps:

[0039] Step 1: Perform surface shape detection on the hard and brittle freeform surface to be processed. According to the surface shape detection results, determine the polishing removal amount of each polishing point on the hard and brittle freeform surface to be processed, as well as the amplitude and frequency range of the longitudinal-torsional composite ultrasonic vibration applied to the polishing tool. Among them, the longitudinal-torsional composite ultrasonic vibration includes the spindle longitudinal ultrasonic vibration and the torsional ultrasonic vibration. Step 1 is specifically as follows:

[0040] 1.1. Use LuphoScan or an interferometer to perform surface shape detection on the hard and brittle freeform surface to be processed, and extract the intermediate frequency error frequency distribution data through FFT (Fast Fourier Transform) analysis or PSD (Power Spectral Density) analysis;

[0041] 1.2. Set the polishing removal amount of each polishing point on the hard and brittle freeform surface to be processed according to the surface shape detection results, and set the amplitude and frequency range of the longitudinal-torsional composite ultrasonic vibration applied to the polishing tool according to the intermediate frequency error frequency distribution data.

[0042] Step 2: Select the feed motion path of the polishing tool. In this embodiment, the feed motion path of the polishing tool is a raster path, and its scanning line spacing is the scanning spacing of the raster path.

[0043] Step 3: According to the amplitude and frequency range of the longitudinal-torsional composite ultrasonic vibration, determine the time-varying removal function of longitudinal-torsional vibration polishing at different amplitudes and frequencies through a dotting test; at the same time, superimpose the longitudinal-torsional composite ultrasonic vibration on the feed motion path of the polishing tool to obtain a spatially complex coupled polishing trajectory. Among them, the spatially complex coupled polishing trajectory is composed of the rotational motion and feed motion of the polishing tool itself and the superimposed longitudinal-torsional composite ultrasonic vibration. The time-varying removal function of longitudinal-torsional vibration polishing is:

[0044] TIF(x,y,t) = TIF(x,y) × f(w,A)

[0045] Wherein, TIF(x,y,t) is the time-varying removal function of longitudinal-torsional vibration polishing, (x,y) is the two-dimensional plane coordinates of the polishing point on the hard and brittle freeform surface to be processed, and t is the polishing time;

[0046] TIF(x,y) is the removal function per unit time, obtained through a dotting test;

[0047] f(w,A) is a function that varies with the frequency and amplitude of the longitudinal-torsional composite ultrasonic vibration, w is the frequency of the longitudinal-torsional composite ultrasonic vibration, and A is the amplitude of the longitudinal-torsional composite ultrasonic vibration.

[0048] Step 4: According to the time-varying removal function of longitudinal-torsional vibration polishing and the polishing removal amount of each polishing point on the hard and brittle freeform surface to be processed, respectively calculate the dwell time of the polishing tool at each polishing point on the hard and brittle freeform surface to be processed through the following formula:

[0049]

[0050] Among them, h(x, y) is the polishing removal amount of the polishing point on the free-form surface of the hard and brittle material to be processed, j = 1, 2, …, m, where m is the number of scanning lines in the spatially complex coupled polishing trajectory, and i = 1, 2, …, n, where n is the number of polishing points in each line of the spatially complex coupled polishing trajectory; T(x, y) is the dwell time of the polishing tool at the polishing point on the free-form surface of the hard and brittle material to be processed.

[0051] Step 5: Adopt the mechanical polishing method. During the polishing process, change the amplitude and frequency of the longitudinal-torsional composite ultrasonic vibration, so that the polishing tool polishes the free-form surface of the hard and brittle material to be processed according to the spatially complex coupled polishing trajectory and the dwell time at each polishing point on the free-form surface of the hard and brittle material to be processed. Among them, the frequency and amplitude of the longitudinal-torsional composite ultrasonic vibration vary randomly within the amplitude and frequency range of the longitudinal-torsional composite ultrasonic vibration set in Step 1. The length of the dwell time determines the polishing speed of the polishing tool. The longer the dwell time, the slower the polishing speed.

[0052] Step 6: Perform surface shape detection on the polished free-form surface of the hard and brittle material to be processed, and judge whether the polishing target is achieved. If the polishing target is achieved, the polishing of the free-form surface of the hard and brittle material to be processed is completed; otherwise, return to Step 5 and continue polishing.

[0053] In this embodiment, the mechanical polishing method is adopted, and the longitudinal-torsional composite ultrasonic vibration is added to the polishing tool, so that the rotational motion and the feeding motion of the polishing tool itself are combined with the longitudinal-torsional composite ultrasonic vibration to form a spatially complex coupled polishing trajectory, realizing that the polishing tool polishes along the material removal direction with the spatially complex coupled polishing trajectory, thereby realizing the longitudinal-torsional composite ultrasonic vibration polishing of the free-form surface of the hard and brittle material. By changing the amplitude and frequency of the longitudinal-torsional composite ultrasonic vibration, the spatial attitude of the removal function is changed in real time, so that the removal functions at each polishing point are different under the action of the spatially complex coupled polishing trajectory, thereby realizing the suppression of the intermediate frequency error.

[0054] Specifically, in this embodiment, a spatially complex coupled polishing trajectory with a time-varying removal function is formed through the form of longitudinal-torsional composite ultrasonic vibration + rotational motion + feeding motion. As the amplitude and frequency of the longitudinal-torsional composite ultrasonic vibration change, the spatial shape of the removal function in the spatially complex coupled polishing trajectory changes at all times, and the intermediate frequency error can be suppressed during the time-varying scanning process of the removal function. When the spatial shape of the removal function changes at all times, the removal method is no longer the convolution principle of the traditional mechanical polishing method, so the introduction of the intermediate frequency error, especially the intermediate frequency ripple error, can be reduced from the root cause.

[0055] The present invention does not limit the type of polishing tool. The polishing tool can be a polishing disc, a polishing wheel or an airbag tool, and its carrying device can be a machine tool or a six-degree-of-freedom robot. Longitudinal-torsional composite ultrasonic vibration can be added. Especially when polishing a hard and brittle free-form surface, the polishing tool can adapt to different curvature positions of the hard and brittle free-form surface, achieve point-to-point fitting, and thus perform polishing processing. On the other hand, the longitudinal-torsional composite ultrasonic vibration is a two-dimensional vibration applied to the tangent plane of the hard and brittle free-form surface. Compared with the one-dimensional vibration applied longitudinally, it can effectively reduce the polishing force and surface roughness and enhance the plastic shear effect during the polishing process.

[0056] The following further illustrates the technical principle and beneficial effects of the present invention through a comparative analysis of the traditional mechanical polishing method and the method of this embodiment.

[0057] In the existing traditional mechanical polishing method, the removal amount of the workpiece material is in the form of convolution:

[0058] R(x,y) = TIF 1 (x,y) * T 1 (x,y)

[0059] Among them, R(x,y) is the polishing removal amount in traditional mechanical polishing, TIF 1 (x,y) is the removal function per unit time in traditional mechanical polishing, and T 1 (x,y) is the dwell time of the polishing tool at the polishing point in traditional mechanical polishing.

[0060] As Figure 1 shown, the removal function of the traditional mechanical polishing method is a space-invariant function. E’ is the one-dimensional profile of the removal function TIF 1 (x,y), and F’ is the surface texture after traditional mechanical polishing. It can be seen that under a fixed grating scanning pitch, traditional mechanical polishing is prone to generate medium-frequency errors with regular waviness.

[0061] As Figure 2 shown, in this embodiment, the time-varying removal function of longitudinal-torsional vibration polishing is a space-varying function. f 1 (x), …, f n (x) are the one-dimensional profiles of the time-varying removal function TIF(x,y,t) of longitudinal-torsional vibration polishing at different times, and h(x) is the surface texture after longitudinal-torsional composite ultrasonic vibration polishing. It can be seen that this embodiment uses the strategy of longitudinal-torsional composite ultrasonic vibration and randomly changing its frequency and amplitude. After the superposition of the time-varying removal function of longitudinal-torsional vibration polishing, the surface of the hard and brittle free-form surface to be processed becomes a texture morphology with an irregular shape and very small peak-to-valley values, and the effective suppression and reduction of medium-frequency errors can be achieved.

[0062] As Figure 3 、 Figure 4As shown, they are respectively the simulation diagrams of the traditional mechanical polishing method and the method of this embodiment when the scanning line spacing in the feed motion path is 2 mm. Figure 4 In Figure 4 , the amplitude of the longitudinal-torsional composite ultrasonic vibration is 1 μm and the frequency is 22 KHz. It can be seen that after longitudinal-torsional vibration polishing, both the PV (peak-to-valley value difference) and RMS (root mean square) values of the surface decrease, and the surface accuracy is also significantly improved.

[0063] As Figure 5 、 Figure 6 shown, FFT analysis is respectively performed on Figure 3 Figure 3 、 Figure 4 to extract their frequency-domain characteristics and obtain characteristic peaks. When the traditional mechanical polishing method is used, a relatively high intermediate-frequency ripple error will be introduced on the surface to be machined. When the longitudinal-torsional composite ultrasonic vibration polishing method of this embodiment is used, the characteristic peak decreases by 50% compared with the characteristic peak of the traditional mechanical polishing method, and the periodic intermediate-frequency waviness error generated by the original fixed grating scanning pitch is significantly suppressed.

[0064] In summary, based on traditional mechanical polishing, the present invention combines longitudinal-torsional composite ultrasonic vibration, can effectively eliminate the intermediate-frequency ripple error and improve the error in the full frequency band, can overcome the defect of generating intermediate-frequency error in the existing mechanical polishing method, and has good practicability.

Claims

1. A longitudinal-torsion composite ultrasonic vibration polishing method for suppressing the mid-frequency error of a hard and brittle free-form surface, characterized in that: The following steps are involved: Step 1, performing a surface shape detection on the hard and brittle free-form surface to be processed, and determining the polishing removal amount of each polishing point of the hard and brittle free-form surface to be processed, and the amplitude and frequency range of the longitudinal-torsional composite ultrasonic vibration attached to the polishing tool according to the surface shape detection result; the longitudinal-torsional composite ultrasonic vibration includes the longitudinal ultrasonic vibration of the main shaft and the torsional ultrasonic vibration; Step 2, select the feed motion path of the polishing tool; Step 3: According to the amplitude and frequency range of the longitudinal-torsional composite ultrasonic vibration, determine the time-varying removal function of the longitudinal-torsional composite ultrasonic vibration polishing at different amplitudes and frequencies; at the same time, superimpose the longitudinal-torsional composite ultrasonic vibration on the feed motion path of the polishing tool to obtain a spatially complex coupled polishing trajectory; Step 4, according to the time-varying removal function of longitudinal-torsional vibration polishing and the polishing removal amount of each polishing point of the hard and brittle free-form surface to be processed, respectively calculate the residence time of the polishing tool at each polishing point of the hard and brittle free-form surface to be processed; Step 5, using a mechanical polishing method, changing the amplitude and frequency of the longitudinal-torsional composite ultrasonic vibration during the polishing process, so that the polishing tool polishes the hard and brittle free-form surface to be processed according to the spatial complex coupling polishing trajectory and the residence time at each polishing point of the hard and brittle free-form surface to be processed; Step 6: Perform a surface shape inspection on the hard and brittle free-form surface to be processed after polishing to determine whether the polishing target is reached. If the polishing target is reached, the polishing of the hard and brittle free-form surface to be processed is completed. Otherwise, return to step 5 to continue polishing.

2. The longitudinal-torsion composite ultrasonic vibration polishing method for suppressing the mid-frequency error of hard and brittle free-form surfaces according to claim 1 is characterized in that: In step 3, the time-varying removal function of the longitudinal-torsional vibration polishing is: TIF(x,y,t)=TIF(x,y)×f(w,A) Among them, TIF(x, y, t) is the time-varying removal function of longitudinal-torsional vibration polishing, (x, y) is the two-dimensional plane coordinates of the polishing point on the hard and brittle free-form surface to be machined, and t is the polishing time; TIF(x,y) is the removal function per unit time; f(w,A) is a function that changes with the frequency and amplitude of the longitudinal-torsional composite ultrasonic vibration, w is the frequency of the longitudinal-torsional composite ultrasonic vibration, and A is the amplitude of the longitudinal-torsional composite ultrasonic vibration.

3. The longitudinal-torsion composite ultrasonic vibration polishing method for suppressing the mid-frequency error of hard and brittle free-form surfaces according to claim 2 is characterized in that: In step 4, the residence time of the polishing tool on the hard and brittle free-form surface to be processed is calculated by the following formula: Wherein, h(x,y) is the polishing removal amount of the polishing point of the hard and brittle free-form surface to be processed, j=1,2,…,m, m is the number of scanning lines in the spatial complex coupled polishing trajectory, i=1,2,…,n, n is the number of polishing points in each line in the spatial complex coupled polishing trajectory; T(x,y) is the residence time of the polishing tool at the polishing point of the hard and brittle free-form surface to be processed.

4. The longitudinal-torsion composite ultrasonic vibration polishing method for suppressing the mid-frequency error of a hard and brittle free-form surface according to any one of claims 1 to 3, characterized in that: In step 3, the time-varying removal function of the longitudinal-torsional vibration polishing is determined by a dot-marking test.

5. The longitudinal-torsion composite ultrasonic vibration polishing method for suppressing the mid-frequency error of hard and brittle free-form surfaces according to claim 4 is characterized in that: Step 1 is as follows: 1.

1. Use LuphoScan or interferometer to detect the hard and brittle free-form surface to be processed, and extract the frequency distribution data of the intermediate frequency error through FFT analysis or PSD analysis; 1.

2. The polishing removal amount of each polishing point of the hard and brittle free-form surface to be processed is set according to the surface shape detection results, and the amplitude and frequency range of the longitudinal-torsional composite ultrasonic vibration attached to the polishing tool are set according to the intermediate frequency error frequency distribution data.

6. The longitudinal-torsion composite ultrasonic vibration polishing method for suppressing the mid-frequency error of hard and brittle free-form surfaces according to claim 5 is characterized in that: In step 2, the feeding motion path of the polishing tool is a grating path, and the polishing line spacing is the scanning spacing of the grating path.

7. The longitudinal-torsion composite ultrasonic vibration polishing method for suppressing the mid-frequency error of hard and brittle free-form surfaces according to claim 6 is characterized in that: In step 5, the frequency and amplitude of the longitudinal-torsional composite ultrasonic vibration are randomly changed within the amplitude and frequency range of the longitudinal-torsional composite ultrasonic vibration set in step 1.

8. The longitudinal-torsion composite ultrasonic vibration polishing method for suppressing the mid-frequency error of hard and brittle free-form surfaces according to claim 7 is characterized in that: In step 2, the polishing tool is a polishing disc, a polishing wheel or an airbag tool, and its supporting equipment is a machine tool or a six-degree-of-freedom robot.

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