Ultra-precision grinding of light-weighted SiC aspheric mirror

Based on the approximate 3D model of light-weighted SiC aspheric mirror, the deformation and stress at different positions along the radius caused by the grinding force on the mirror surface was analysed. Then the influence of processing parameters on grinding force was experimentally studied, and the prediction model of grinding force was established. Finally, the grinding experiment of light-weighted SiC aspheric mirror was carried out. The PV value of the aspheric surface form error was 4.15μm, and the Ra of surface roughness was about 28nm. The experimental results verified the feasibility of ultra-precision forming of lightweight SiC mirror by controlling grinding force based on process parameters.


Introduction
Compared with traditional spherical optical elements, aspheric optics are wildly used in the large and complex optical systems due to their extremely advantages such as eliminating the adverse effects of spherical aberration, comet aberration, reducing the loss of light energy in the beam transfer process and improving the precision of focusing precision when the beam is focused [1].There are many advanced materials for the fabrication of modem optics.Silicon carbide (SiC) is one of the best candidate materials for large scale space optical system mirrors owing to its high specific stiffness, low coefficient of thermal expansion, good thermal conductivity and so on [2].However, due to hardness and brittleness, it is difficult to obtain high-precision surface of large scale SiC aspheric mirrors using traditional optical manufacturing technology [3].At present, in order to shorten the overall manufacturing time of the SiC aspheric mirror, the mainstream manufacturing method of the world is directly processing the desired aspheric surface on the high rigidity and ultra-precision grinder by using the consolidated abrasive diamond wheel [4].Liming Xu investigated a novel methodology called equivalent-sphere swing grinding for the grinding of large revolving aspheric and spherical surfaces by cup wheel, which proposed to improve the machining efficiency and reduce the complexity of machine tools.A parabola surface with diameter of 420mm was machined by this methodology, and the average value of the form error was less than 0.056mm [5].Ping Li employed an Infeed Grinding (IG) mode on the Schneider Surfacing Center SCG 600 with a rotary table and a cup wheel to process a Φ300mm diameter plano mirror, whose form error of Pt was about 2μm [6].Jianfeng Liu grinded a Φ90mm diameter aspheric mirror with cup wheel on five-axis computer numerical control machine, and the PV value of profile error was about 4.77μm [7].Guangpeng Yan investigated a novel approach of three-linear-axis ultra-precision grinding with ball wheel path generation, tool interference checking and profile compensation for fabricating aspheric surfaces.And the experimental results demonstrated that the profile error could be reduced to 0.1μm or lower on a 5mm aperture concave aspheric tungsten carbide mould [8].
In the ultra-precision grinding process of SiC materials, the diamond grinding wheel is easy to wear, which will increase the grinding force sharply and deteriorate machining accuracy and surface roughness of component [9].For the thin-walled space mirror with a large number of light-weighted holes on the back, the increase of grinding force will easily lead to micro-cracks on the mirror body.In the subsequent using process, the micro-cracks will expand easily under the load, resulting in the mirror breakage, and ultimately affecting the performance of the entire optical system [10].In order to ultra-precision form large scale light-weighted SiC aspheric mirror, based on the approximate 3D model of the mirror, the deformation and stress at different positions along the radius caused by the grinding force on the mirror surface was analyzed using finite element method (FEM) in this article.Then the influence of processing parameters on the grinding force of processing SiC material was experimentally studied, and the grinding force prediction model was established.Finally, the ultraprecision grinding experiment of large scale light-weighted SiC mirror was verified, and the ultraprecision forming of aspheric mirror was realized.

Modelling and analysis of aspheric mirror
Aspheric surface is defined as the curved surface rotated by the normal line passing through the vertex [11].The general equation of aspheric surface is shown in Formula (1).In this formula, c=-1/R 0 , and R 0 is the vertex radius of aspheric surface.k is the cone coefficient.When k<-1, the surface is a hyperboloid.When k=-1, the surface is a paraboloid.When -1<k<0, the surface is a semi-ellipsoid with the major axis of the ellipse as the axis of symmetry.When k=0, it is a sphere.When k>0, the surface is a semi-ellipsoid with the minor axis of ellipse as the symmetry axis.α i is the high-order aspheric coefficient.The aspheric parameters of SiC mirror are shown in Table 1.The light-weighted holes on the back of the mirror are hexagonal honeycomb structure, as shown in Figure 1.

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Table 1.Parameters of SiC aspheric mirror.According to the aspheric parameters, the closest spherical radius of the aspheric surface is calculated to be 5244.2mm,and the asphericity is 32.6μm.Relative to the mirror thickness of 7mm ± 0.2mm, the influence of asphericity on the surface shape can be ignored.Therefore, the approximate 3D model of light-weighted SiC aspheric mirror was established by replacing the aspheric surface with a sphere whose radius was 5244.2mm.The model error was less than 0.5%.Then the deformation and stress of the component under the grinding force during the processing was simulated by the finite element software.In order to improve the analysis efficiency, the mounting hole on the back of the element was simplified.SiC material parameters were set as shown in Table 2.During processing, the component was fixed on the grinder worktable, so the back of the component was fixed when simulation.The mesh division was shown in Figure 2, with 1060754 elements and 1723769 nods.According to the element structure as shown in Figure 1, the lightweight holes are rotationally symmetric.Because of the weakest stiffness at the center of the light-weighted hole of the element, in the grinding process of aspheric surface with arc diamond wheel, the action point of grinding force could be simplified to a circle region with diameter of 1mm.The force acting points of simulation analysis were from point a to point g as shown in Figure 1, which were at the center of the hexagon.The simulated deformation and stress by FEM under 100N grinding force of approximate element at different weakest positions was shown in Figure 3.The maximum deformation was about 0.64μm at point d as shown in Figure 4, whose influence on the processing accuracy could be ignored.Points b, c, d, f and g were located in the center of the hexagonal hole with the rib thickness of 3.04mm, whose rigidity were the weakest.So the deformations of those points were larger than other area.The maximum stress was about 63MPa at point g, which was far less than the yield strength and fracture limit of SiC material, as shown in Figure 5.This indicated that the mirror would not break under 100N grinding force.

Experimental design
Figure 6 shows the principle of parallel grinding of aspheric surface using arc diamond wheel on a three-axis linkage ultra-precision grinder.According to the aspheric parameters and the size of the grinding wheel, the coordinates of the wheel in the grinding process are calculated.And then the coordinates are dispersed according to the parallel grating machining path to control the arc diamond grinding wheel moving along the surface of element.At the meantime, the material of element is removed rapidly by the grinding wheel with high rotation speed and the high-precision optical surface is obtained.There are full of large-scale light-weighted holes on the back of element, which results in weak rigidity.It is easy to deform or even crack under the action of grinding force during the processing.Therefore, the grinding force test experiment was carried out on the ultra-precision grinder by processing a SiC work piece with size of 200mm × 200mm using diamond wheel.The grinding force was measured by the dynamometer, as shown in Figure 7.The parameters of diamond grinding wheel were shown in Table 3, and the processing parameters were shown in Table 4.

Result and discussions
Figure 8 shown the tangential grinding force F x , axial grinding force F y and normal grinding force F z of grinding SiC work piece test at the processing parameters of grinding wheel speed of 1500r/min, each grinding depth of 10μm, feed speed 5000mm/min, step distance 2mm.The sampling frequency of grinding force signals was 50Hz, and all the signals were smoothed by moving mean method with window size of 10.The tangential grinding force was about 10N, the axial grinding force was almost 0, and the normal grinding force was about 50N, which was far greater than the tangential grinding force and the axial grinding force.In the processing, the normal grinding force was the main factor leading to deformation and fragmentation.So the normal grinding force under different processing parameters was mainly investigated.Figure 9 shown the normal grinding force when machining SiC work piece under different wheel speeds at grinding depth of 10μm, feed speed of 5000mm/min and step distance of 2mm.When other process parameters were unchanged, with the increase of grinding wheel speed, the SiC material to be removed per revolution decreased, and then the normal grinding force decreased.Figure 10 shown the normal grinding force when machining SiC work piece under different grinding depths at wheel speed of 1500r/min, feed speed of 5000mm/min and step distance of 2mm.When other process parameters were unchanged, with the increase of grinding depth, the SiC material to be removed per unit time increased exponentially, and the normal grinding force increased sharply.Figure 11 shown the normal grinding force when machining SiC work piece under different feed speeds at wheel speed of 1500r/min, grinding depth of 10μm and step distance of 2mm.When other process parameters were unchanged, the normal grinding force increased with the increase of feed speed.Figure 12 shown the normal grinding force when machining SiC components under different step distances at wheel speed of 1500 r/min, grinding depth of 10μm and feed speed of 5000mm/min.When other process parameters unchanged, the normal grinding force increased sharply with the increase of machining step distance.It's necessary to accurately define the direct relationship between various physical quantities to establish the theoretical model of grinding force, and there is a large error due to the influence of actual working conditions.For the empirical model, if all important variables are fully considered, the model usually has high prediction accuracy for grinding force [12].The empirical grinding force model of SiC ultra-precision grinding was established by multiple linear regression analysis method, as shown in Formula (2), where α, β, γ, δ were coefficients.After taking logarithmic on both sides of Formula (2) and linear transformation, the regression coefficients were solved by the least square method, and the grinding force prediction model was shown in Formula (3).According to the prediction model, the grinding force calculated under the above process parameters was shown in Figure 9 ~ Figure 12.The relative deviation from the measured value was not more than 8%, which indicated that the grinding force prediction model had high accuracy.(3)

Grinding experiment of large scale light-weighted SiC aspheric mirror
Grinding experiment of large scale light-weighted SiC aspheric mirror was carried out on the ultraprecision grinder using a diamond wheel with parameters as shown in Table 3 and under the processing parameters as shown in Table 5.According to the grinding force model in Formula (3), the normal grinding force in the material removal grinding process was predicted to be about 136N, and that in the smooth grinding process was predicted to be about 11N.By FEM simulation, the maximal deformation caused by the maximal grinding force at the center of the lightweight hole was about 0.9μm, whose influence on processing accuracy could be ignored.And the maximal stress was about 79MPa, which was more less than the yield strength and fracture limit of SiC material.In the material removal grinding process, the purpose of adopting larger process parameters was to improve the material removal efficiency.And in the smooth grinding process, the purpose of adopting smaller process parameters was to improve the surface quality of component.The grinding process of SiC aspheric mirror was shown in Figure 13.The processed component was shown in Figure 14.After fine processing, the aspheric form error was measured by on-machine measurement system with chromatic confocal distance sensor whose accuracy was 0.06μm.As shown in Figure 15, the PV value of form error was 4.15μm.The surface roughness was measured by white light interferometer as shown in Figure 16, and the Ra was about 28nm.The roughness values at the up-grinding area and the downgrinding area were similar, because the difference between the two relative grinding velocities at those two areas were less than 0.5%.

Conclusions
During the grinding process of large scale light-weighted SiC mirror, it's easy to cause large deformation or even fracture under the action of grinding force because of the weak rigidity of lightweight hole.In order to form light-weighted SiC aspheric mirror efficiently and precisely, this paper first established the approximate 3D model of the component whose error was less than 5%, and simulated the surface deformation of the component under the grinding force of 100N.Then the grinding force under different processing parameters was experimentally measured, and the grinding force prediction model was established.According to the grinding force prediction model, the processing optimal parameters were determined.And the grinding force was predicted by the model.With theoretical simulation analysis, under this process parameter, the component deformation caused by grinding force would not affect the final machining accuracy and would not lead to fragmentation.Finally, the ultra-precision grinding experiment of large scale light-weighted SiC mirror was carried out, and the PV value of aspheric form error was 4.15μm.The Ra of surface roughness was about 28nm, which formed the large scale light-weighted SiC aspheric mirror precisely.

1 Figure 1 .
Figure 1.Light-weighted structure on the back of the mirror.

Figure 2 .
Figure 2. FEM mesh division (Left is the front surface and right is the back).

Figure 3 .
Figure 3. Simulated deformation and stress at different positions.

Figure 4 .
Figure 4. Deformation distribution at point d.Figure5.Stress distribution at point g.

Figure 5 .
Figure 4. Deformation distribution at point d.Figure5.Stress distribution at point g.

Figure 7 .
Figure 7. Schematic diagram of grinding experimental system.

Figure 9 .
Figure 9. Grinding force under different wheel speeds.

Figure 11 .
Figure 11.Grinding force under different feed speeds.

Figure 12 .
Figure 12.Grinding force under different step distances.

Table 2 .
Material property of SiC.

Table 4 .
Processing parameters of grinding force test.

Table 5 .
Processing parameters of grinding SiC aspheric element.Figure 7 13.Grinding process of element.Figure14.SiC element after grinding.