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Abstract
The mold core fabrication of a freeform surface on die steel by ultra-precision machining can make the optical elements of freeform be mass-produced by plastic injection and glass mold pressing. However, because steel is a typical difficult-to-cut material, the technical limitations of existing machining methods hardly meet the current requirements of design. In this paper, a novel machining method, one-dimension ultrasonic-assisted slow tool servo (UASTS) turning, is proposed to manufacture the freeform surface with high-precision and large-steepness on the die steel. Aiming at the characteristics of UASTS turning, the tool trajectory is generated by analyzing the compensation of tool radius and confirming the position of ultrasonic displacement. The 2D surface model contours of residual tool marks are established for predicting the 3D surface topography based on considering the effects of kinematics, material elastic recovery and plastic side flow. In the experiments, the large-amplitude bidirectional sinusoidal wave grid (BSWG) surface is successfully fabricated by UASTS technology on the material, Polmax steel, for which the value Rt of surface roughness is less than 70nm and the value Ra of surface roughness can achieve to 3.298nm. The results show that the freeform surface with high-precision and large-steepness can be machined by UASTS turning technology on mold steel.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The optical element containing freeform surface can improve the properties of optical systems and simplify the system structure, which is diffusely applied in illumination, aerospace, optical communication, new energy and other fields [1]. With the rapid development of precision plastic injection technology and precision glass mold pressing technology in optical field, the optical elements of freeform surfaces can be mass-produced [2]. As large-scale application of mold core with steel, the surface machining quality plays an extremely important role in replication manufacturing [3].
For the fabrication of freeform surface in conventional diamond turning (CDT), there were two main turning techniques, namely slow tool servo (STS) and fast tool servo (FTS) [4–7]. According to the comparison between the two machining methods, Davis et al. [8] made the relevant conclusion that STS possessed a huge advantage in processing freeform surfaces with large azimuth and large curvature. In the machining process of freeform surface, the strategies of tool trajectory and the prediction of surface topography were important research content, which decided the fabrication accuracy and the time cost. Yin et al. [9] analyzed the tool path error for manufacturing off-axis aspheric surface by STS turning. Ji et al. [7] processed sinusoidal grid wave surfaces on aspheric substrates and analyzed the influence of different tool parameters for tool path formation. Zhu et al. [10] put forward a new machining method, that the FTS device is perpendicular to the Z axis of machine tool to overcome drawbacks for the requirements of bandwidth and resolution. Furthermore, He et al. [11] established the accurate prediction model of 3D surface morphology in the CDT through considering the effects of kinematics, material elastic recovery, plastic side flow and material defects. Liang et al. [12] proposed a theoretical model to predict surface topography according to importantly analyze the relative vibration between tool and workpiece.
However, because diamond tools cannot directly cut ferrous materials, the above machining methods hardly realized the high-precision manufacture of mold steel [13]. The affinity between the carbon element and the iron element caused tool rapid wear even at very small cutting distances. The wear was mainly caused by chemical reactions, including diffusion effects, oxidation, graphitization and carbide-formation [14]. The ultrasonic-assisted diamond turning (UADT) could effectively solve this problem because of its characteristic of intermittent cutting [15].
Ultrasonic-assisted machining was firstly used in the 1960s for the macro-scale applications of nonferrous metals on the traditional lathe [16]. As the progress of ultra-precision machining technology, it was about 1970s that the UADT technology was further developed to machine difficult-to-cut materials on the precision lathe, especially ferrous materials [17]. Compared with CDT, the UADT presented more manufacturing advantages. Xiao et al. [18] found that the chatter phenomenon could be effectively suppressed by UADT. Song et al. [19] summarized that the decisive factor of diamond tool wear was the contact time of tool-workpiece during cutting ferrous materials. Zhou et al. [20] found that the transverse pushing force and the longitudinal cutting force could be significantly reduced by UADT. Gan et al. [21] notarized that the surface quality of brittle materials by UADT processing was better than CDT processing. For the fabrication of steel materials in the UADT, Gaidys et al. [15] used the UTS2 ultrasonic auxiliary equipment to cut aspheric steel molds for which the surface roughness Ra could achieve 2.63nm. For the fabrication of optical die steel, it was usually coated with a nickel layer. Due to the application of UADT technology, the optical surface with high-precision could be directly cut on the mold steel so that the production cost was decreased and the mold life was increased. Therefore, the potential and benefit of machining complex surface or freeform surface on die steel with UADT was exceptionally high.
Some researchers believed that micro/nano structures could be generated on the workpiece surface by controlling the tool vibration direction and amplitude. Liu et al. [22] proposed a machining method that the tool vibration was along Z axis direction to generate micro-textured end face improving surface property by one-dimensional UADT. Guo et al. [23] studied the surface generation mechanics of micro-structure by the elliptical vibration texturing process. On this basis, the employ of ultrasonic technique to fabricate the freeform surface on steel mold was further developed. Zhou et al. [24,25] firstly proposed the application of double-frequency elliptical vibration cutting to fabricate freeform surface on difficult-to-cut material, where the principle of machining method was investigated in details. However, this processing method was similar to the traditional FTS turning, which still had bandwidth limitations in cutting workpiece with large curvature. Yuan et al. [26,27] designed the new machining method, for which a non-resonant compliant vibrator with low vibration frequency was used to cut freeform surface and a resonant ultrasonic vibrator was applied to dispose the workpiece surface quality. Although this method could improve the limitation of workpiece shape, the surface quality still needed to be further improved due to the average surface roughness obtained with tens or even hundreds nanometers.
Therefore, in this paper, a novel machining method, one-dimension ultrasonic-assisted slow tool servo (UASTS) turning technology, is proposed to fabricate optical freeform surface on die steel to overcome previous machining drawbacks. The UASTS turning technology based on STS can provide larger machining stroke to cut complex surface or freeform surface compared with the double-frequency elliptical vibration cutting. And the optical surface quality with high-precision can be obtained by one-dimension UADT with the larger vibration frequency. Furthermore, the rest of this paper is organized as follows. The tool trajectory of UASTS technology is generated in Section 2. The prediction model of 2D surface profile is established by considering effects of kinematic, material elastic recovery and plastic side flow in Section 3. The generation of tool path and surface topography for large-amplitude bidirectional sinusoidal wave grid (BSWG) surface is analyzed in Section 4. The experiment and discussion for large-amplitude BSWG surface is displayed in Section 5. Finally, the conclusion is put in Section 6.
2. Generation of tool trajectory
2.1 UASTS turning technology
The turning technology of the UASTS is mainly constituted by two parts, namely the STS part and the UADT part. The whole layout of the UASTS machine tools in details is displayed in Fig. 1. The X axis is linear movement, on which the rotary C axis is also installed. The diamond tool is mounted on ultrasonic-assisted system. The ultrasonic system, which is composed of a longitudinal actuated transducer and a bending sonotrode, makes the tool perform continuous vibration in the method of simple harmonic motion along the cutting direction [15]. The system is installed on the Z axis, where its motion direction is perpendicular to the X axis. The UASTS turning technology changes the spindle rotation pattern from speed control to position control, and adopts the linkage of C, X and Z axes to form a tool trajectory in polar coordinates and cylindrical coordinates. In the UASTS, the STS part is used to cut the surface shape and the UADT part is used to improve surface quality. Therefore, the high-precision fabrication of complex surface or freeform surfaces for difficult-to-cut materials can be realized by the UASTS technology.
2.2 UASTS tool path
In the machining process of the UASTS for cutting freeform surface or complex optical surface, the arc center point of tool is regarded as the tool position and the points trace is seen as the tool motion path. In the process, the macroscopic tool path is formed by the Archimedes spiral and the microscopic tool path is formed by the oscillatory movement. The tool in X direction moves from the outer edge to the center of workpiece. Moreover, the tool path of STS needs to be established firstly, because the generation of UASTS tool path is based on the traditional STS turning trajectory. Due to the special correlation among the three axes, the tool path is expressed mainly through cylindrical-coordinate system $({\rho ^{(t)}},{\theta ^{(t)}},{z^{(t)}})$. In order to facilitate the subsequent compensation calculation, the cylindrical-coordinate expression of tool path could be converted into rectangular-coordinate. It can be expressed as:
In order to avert the overcutting phenomenon on the workpiece surface, the edges of tool circular arc could be tangent to the workpiece contour machined all the way after the tool radius compensation (TRC) is implemented. The specific method of TRC is displayed in Fig. 2. In the schematic diagram, the original tool position $({\rho ^{(t)}},{z^{(t)}})$ at the red dot is changed to the compensation position $(\rho _r^{(t)},z_r^{(t)})$ along the Z and X directions, where the diamond tool has only one point on the workpiece surface. After the arc center point of tool is changed, the slope angle between the original and compensatory position can be calculated with the slope variation of freeform surfaces at every cutting points along the radial direction. Therefore, the TRC can be expressed as followed:
For the UADT, it is the main difference compared with CDT that the continuous vibration of tool is parallel to cutting direction in whole machining course and the workpiece moves with concentric rotation of spindle displayed in Fig. 1. As shown in Fig. 3(a), the diamond tool makes a simple harmonic motion relative to the conventional tool position, whose motion trajectory is approximate to the sinusoidal curve. Furthermore, the cutting variation of tool vibration during one cycle is simulated in details by finite element method (FEM) as shown in Fig. 3(b). In this simulation, the Johnson-Cook constitutive model is adopted to describe the variations of material physical properties during machining process, where the vibration parameters of tool are set as frequency of 103 KHz and amplitude of 1μm.
Fig. 3. The principle and simulation of UADT process: (a) principle of vibration cutting, and (b) simulation of cutting process in one cycle with FEM.
This process is mainly divided into two phases, namely the tool-workpiece cutting stagein red curve and the separation stage ${t_s}$ in blue curve. In the to ${t_c}$ ol-workpiece cutting stage between ${a_i}$ and ${b_i}$, the vibration-up-velocity of tool is faster than the rotational speed of workpiece to cut material until the vibration-up-velocity of tool gradually reduces to zero and the vibration-down-velocity of tool begins rising. In the tool-workpiece separation stage between ${b_i}$ and ${a_{i + 1}}$, the tool begins to withdraw and separate from the workpiece when the tool vibration-down-velocity is higher than the rotational speed of workpiece. After the tool retreats to the peakedness point in minimum velocity, the tool movement turns to the cutting direction again to cut material. The course of contact and separation between tool and workpiece periodically happen until the end of the machining process. The tool vibration positionand velocity ${V_{Y{Y^{(t)}}}}$ relative to the conventional tool position is expressed as:
Based on the position confirmation of tool trajectory points for STS with TRC, the UASTS position points are generated after the UADT is added to the cutting process as shown in Fig. 4. Due to the variation of linear vibration on the cutting direction, the points of original STS are moved from $P$ to $P^{\prime}$ along Y direction to form the points of UASTS. The trajectory points of the UASTS have a linear displacement $\varDelta Y{Y^{(t)}}$ compared with the original trajectory points of the STS. These trajectory points generated by linear displacement can be confirmed by the compensation calculation of polar radius and polar angle. The polar radius length of the UASTS is slightly longer than the STS one. The polar angle exist differences at the different trajectory points which may be more or less than the original polar angle, because of the characteristics of the oscillating cutting in the UASTS. Therefore, the polar radius and polar angle after vibration compensation can be expressed as:
The tool trajectory points of the UASTS in actual machining can be expressed as:
3. Model of surface generation
Based on the generation of the UASTS tool path, the fundamental model of 2D surface profile should be firstly analyzed in order to establish the prediction model of 3D surface topography. On the machined workpiece surface, the 2D surface profile is formed by the residual tool mark based on the turning characteristics of the UASTS technology. It is mainly influenced by three factors, namely the effects of kinematic, material elastic recovery and plastic side flow. The process is displayed in Figs. 5(a)–5(d). Among them, the red line represents the diamond tool profile, the green line represents the residual tool mark in the ideal condition and the purple line represents the residual tool mark with different influence factors. Furthermore, the residual tool mark can be approximately calculated by introducing a quadratic function model in one period, which guarantees the continuity and smoothness in Fig. 5(e). In this model, the boundary condition of both top points are consisted by kinematic and plastic side flow, and the boundary condition of bottom point is obtained by material elastic recovery. Therefore, the final expression of tool mark profile is displayed as followed:
where ${h_f},\,{h_e}$ and ${h_p}$ are the value of kinematic, material elastic recovery and plastic side flow respectively, and $g$ is the points of 2D surface profile.Fig. 5. The residual height of tool mark for different influence factors in UASTS: (a) kinematic, (b) kinematic and material elastic recovery, (c) kinematic and material plastic side flow, (d) kinematic, material elastic recovery and plastic side flow, and (e) calculation model of 2D surface profile.
For the residual height caused by kinematic in the UASTS, it is similar as that of the CDT because of the ultrasonic vibration in the Y-axis direction, which is mainly generated by feed rate, tool radius and curvature radius. Furthermore, the two calculations including convex and concave surfaces should be considered on account of the continuous variation of curvature radius for workpiece surface profile in the machining process of freeform surface as shown in Fig. 6. Therefore, the curvature radius of the designed space curve need be firstly calculated as:
In the concave surface, the residual height ${h_f}$ can be expressed as:
In the convex surface, the residual height ${h_f}$ can be expressed as:
For the influence of material elastic recovery on the residual height, it stems from the inherent attribute of vickers hardness $H$ and elasticity modulus. It is written by [28]:
where ${k_1}$ is the coefficients which is set as 43 in this work and ${r_n}$ is tool cutting edge radius.For the influence of plastic side flow on the residual heigh $E$t, it is greatly different between CDT and UASTS. At the same position of the machined surface, the tool in the UASTS performs two or more reciprocating cuttings rather than single turning in Y-axis direction, which depends on the frequency and the amplitude of tool. The pile-up process is displayed in Fig. 7(a). The diamond tool relative to the residual height of workpiece is huge. Due to the tool cuts repeatedly many times, the residual height caused by plastic side flow is accumulated again and again until tool no longer passes here. Therefore, the calculation model aiming at the plastic side flow of the UASTS needs to be further established.
Fig. 7. The pile-up process of plastic side flow in UASTS: (a) accumulation height after tool reciprocating cutting, and (b) simulation results of positive cutting force and lateral thrust force by FEM.
The rheological coefficient $\kappa$ represents the flow ability of material as followed by [29,30]:
where $\theta$ is the semi-apical angle of the tool cutting edge and ${\sigma _y}$ is the flow stress.The semi-apical angle could be expressed as a function of the cutting depth ${a_p}$ and the tool nose radius $r$ as following:
In order to obtain the flow yield stress in the shear zone, the friction coefficient $\mu$ firstly needs to be calculated as:
The flow stress can be expressed by:
where ${k_2}$ is an empirical constant. In this work, it is set to be 4.1.In the UASTS, the accumulation height caused by plastic side flow is determined by tool cut times at same workpiece position. Therefore, it can be calculated by:
where ${b_i}$ is the integrated coefficients,$m$ is cutting times.4. Generation of tool path and surface topography for large-amplitude BSWG surface
In this paper, the large-amplitude bidirectional sinusoidal wave grid (BSWG) surface is selected as representative freeform surface to analyze the generation of tool path and surface topography in UASTS turning technology. For the manufacture of BSWG surfaces in the past, the sinusoidal amplitudes are usually set as no more than 10 microns [4,7,24–26,31]. To demonstrate that the UASTS technique can process freeform surfaces with large-steepness and ensure machining accuracy, the large-amplitude BSWG surfaces with the amplitude of 25 micron are designed and manufactured. The relative machining parameters in the simulation and experiments with the UASTS are list in Table 1, including the parameters of ultrasonic vibration, finish turning, workpiece, surface and tool. The expression of large-amplitude BSWG surfaces is followed by:
Table 1. Machining parameters in the simulation and experiments
According to the machining parameters in Table 1, the macroscopic tool trajectory for large-amplitude BSWG surface is generated including the ${A_y}$STS and the UASTS in Fig. 8(a). It is found that the microscopic tool path of the UASTS technology is reciprocating and periodic forward movement caused by tool vibration in Y direction, which is different from the STS turning strategies in continuous cutting manner as shown in Fig. 8(b).
Fig. 8. The comparison of tool paths between the STS and the UASTS with $A$=1μm=100 KHz: (a) macroscopic tool path, and (b) microscopic tool trajectory.
In Fig. 8, the tool path is displayed in the ultrasonic parameters with the frequency of 100 KHz and the amplitude of 1 μm. The influence of different vibration frequencies for the tool trajectory is shown in Figs. 9(a)–9(e). These frequencies are set as 20 KHz, 40 KHz, 60 KHz, 80 KHz and 100 KHz with the vibration amplitude of 1μm, which are also the development process of ultrasonic device in recent decades. Furthermore, the tool moving velocity becomes faster in Y axis direction with the augment of vibration frequency so that the cutting times increase significantly at the same position of the workpiece. The workpiece surface in this position with the tool frequency of 100 KHz is cut four times or even more, while the same position with the tool frequency of 20 KHz is cut twice or even only once. In the same time, the more times the surface is cut, the better results the machining quality is obtained. Therefore, in the actual machining process, the equipment of faster vibration frequency should be selected to enhance surface quality. Furthermore, the wear of diamond tool can be effectively restrained due to the contact time between tool and workpiece become shorter with the larger vibration frequency.
Fig. 9. Microscopic tool path in different frequency and amplitude: (a)$A$=1μm=20 KHz, (b)$A$=1μm$f$=40 KHz, (c)$A$=1μm$f$=60 KHz, (d)=1μm$f$=80 KHz,(e)=1μm=100 KHz, (f)$A$=0.1μm$f$=100 KHz, (g)$A$=0.5μm$f$=100 KHz, (h)$A$=1.5μm$f$=100 KHz, and (i)$A$=2μm$f$=100 KHz.
The variation of tool vibration amplitude is another main ultrasonic parameter that affects the tool trajectory of the UASTS during machining process as shown in Figs. 9(e)–9(i). The vibration amplitudes are respectively set as 0.1μm, 0.5μm, 1.0μm, 1.5μm and 2.0μm with the frequency of 100 KHz. It can be found that the larger vibration amplitude is set, the more obvious tool displacement distance in path is generated when the vibration frequency and the spindle speed remain unchanged. The larger vibration amplitude increases the tool cutting length in a cycle so that the same workpiece position is machined multiple times. However, excessive amplitudes may lead to cutting errors during machining, where the process of tool retracting and recutting could interfere with the machined workpiece surface. Too small vibration amplitude could lead to a reduction in the number of cutting times at same position. In this situation, the contact time is also increased between the tool and workpiece. The possibility of tool wear is enhanced so that the effect of ultrasonic-assisted machining is weakened. Therefore, the reasonable vibration amplitude of tool should be carefully selected before manufacturing.
After the tool trajectory is determined, based on the previous establishment of 2D surface contour model in section 3, the 3D morphology of the large-amplitude BSWG surface is predicted with the frequency of 100 KHz and the amplitude of 1μm in Fig. 10. The prediction of the whole surface morphology is displayed in Fig. 10(a) with using parameters in Table 1. In the prediction of local surface morphology in Fig. 10(b), the residual tool marks caused by kinematics, material elastic recovery and plastic side flow can be clearly observed.
Fig. 10. The 3D surface topography prediction of large-amplitude BSWG surface: (a) the whole topography, and (b) the local topography.
5. Experiment verification
To validate the feasibility and capacity of the UASTS turning technology, the large-amplitude BSWG surface is processed. By measuring the surface quality of workpiece machined, it is to prove that the freeform surface with large-steepness and high-precision can be machined on die steel.
5.1 Experimental setup
In this experiment, the ultrasonic-assisted system provided by Son-X Company is installed on the Nanoform 700 single point diamond lathe. The relevant vibration parameters are set as the amplitude of 1.0 µm and the frequency of 103.79 KHz in this ultrasonic-assisted system. It is displayed in Figs. 11(a)–11(c). After pre-processing, the roundness and parallelism of the workpiece are ensured by traditional precision lathes and electro discharge cutting machine. The pretreated circular workpiece with a radius of 5mm is the material of Polmax steel provided by Switzerland ASSAB Company, which is mounted in a rotating fixture. Furthermore, in order to avoid the phenomenon of tool interference in the cutting process for large-steepness workpiece surface, the critical parameters of nature diamond tool is selected as radius of 1.035mm, clearance angle of 12.5° and warp angle of 120°. The other details of specific parameters are listed in Table 1.
Fig. 11. Fabrication equipment: (a) single point diamond lathe, (b) ultrasonic-assisted turning device, and (c) ultrasonic control system.
5.2 Results and discussion
After the preparation of the experimental device, cutting simulation need be used to confirm the tool trajectory before fabrication of the large-amplitude BSWG surface with the UASTS. In the previous section, the UASTS technique is verified by theoretical modeling, and the 3D surface topography is predicted by numerical simulation which is shown in Fig. 12(a). In Fig. 12(b), the parameters of workpiece, tool and compensation calculation are edited by commercial software DIFFSYS. The cutting path, including rough, semi-finish and finish is gained in prediction simulation. Under the stable operation of ultrasonic equipment and the tool setting of precise laser, the surface profile of workpiece is manufactured after roughed machining for 5 times and semi-finished machining for 2 times. Furthermore, a large number of cutting points are produced in the finishing process when the method with constant-angle cutting is selected. The slower spindle speed is required to ensure machining accuracy at every cutting point, because the vertical stroke is 50 microns between the top of the convex surface and the bottom of the concave surface for the large-amplitude BSWG surface. Finally, the workpiece surface is successfully fabricated as shown in Figs. 12(c) and 12(d), where the finishing machining parameters is the cutting depth of 2μm, the tool feed rate of 2μm/rev and the spindle speed of 75 RPM. In Fig. 12(c), the residual tool marks of surface morphology of the machined workpiece are consistent with that predicted in numerical simulation. Furthermore, it can be clearly observed that the workpiece surface shows the degree of specular gloss, and there is no obvious pitting or rainbow spots. The shape and size of each lenslet cell are consistent with the design results.
Fig. 12. The large-amplitude BSWG surface: (a) numerical simulation with MATLAB, (b) prediction simulation with DIFFSYS, (c) tool marks of surface, and (d) experimental result.
To further analyze the surface quality of workpiece machined, the five areas which are displayed in Fig. 12(b) are selected to measure the surface roughness and topography by the Bruker X3 White Light Interferometer. The testing equipment and measuring results are shown in Figs. 13(a)–13(f). It can be found that the value Ra of average surface roughness of the measured area, whether convex or concave, is between 3.298nm and 4.341nm within the area of 60μm${\times}$45μm for the large-amplitude BSWG surface. To be different from the asymmetrical variation of surface quality after traditional STS turning, there is no significant deviation in the machining results at the different position of the workpiece. This phenomenon is mainly caused by the fact that, although the cutting movement of tool is based on the trajectory points programmed, the high frequency vibration enables the tool multiple reciprocating cutting between adjacent path points. In other words, the reciprocating cutting process of UASTS technology can be regarded as more similar to the traditional non-servo turning method relative to STS technology when the freeform surface or complex surface is fabricated. The faster the ultrasonic vibration frequency is applied to the tool, the closer the surface machining quality of traditional non-servo turning is obtained. The problem is effectively solved, that the uneven surface quality of the workpiece is generated due to the uneven distribution of the cutting points between the center and the edge caused by the cutting method of equal-angle or equal-arc. Therefore, the measurement results of these five areas prove that the uniform surface quality on the overall workpiece can be obtained by UASTS turning technology. Furthermore, the value Rt of surface roughness is less than 70nm within these measurement areas in the Figs. 13(b)–13(f). According to observing the three-dimensional surface morphology, the distinct mutation is not generated on the machined workpiece surface. It is indicated that even if a larger tool radius is selected in the cutting environment with high frequency, the chatter vibration between workpiece and tool can be effectively restrained by using the UASTS technique. And the steady cutting status can be achieved when the die steel of hard-brittle materials are machined. Finally, it is proved that the surface quality with high-precision can be obtained by adopting UASTS turning technology for the fabrication of freeform surface with large-steepness on die steel.
Fig. 13. Measurement results of surface roughness, (a) Bruker X3 White Light Interferometer, and (b)-(f) measurement results of area A-E.
6. Conclusion
In this paper, the novel machining method, one-dimensional ultrasonic-assisted slow tool servo (UASTS) turning, is proposed to cut the freeform surface with large-sharpness and high-precision on the die steel through theoretical analysis and experimental verification. The main conclusions are summarized as follows:
- (1) The tool path of UASTS is based on the traditional STS trajectory. In order to prevent the overcut phenomenon, the compensation strategy of tool radius is proposed to make tool tangent to the surface contours of workpiece during cutting. And then, the position points of UASTS turning are confirmed by compensation of pole radius and pole angle. Finally, the tool trajectory of UASTS is established in Cylindrical-coordinate system and Cartesian-coordinates system.
- (2) In order to predict the 3D surface topography of workpiece, the 2D surface contours model of residual tool marks are established by considering the effects of kinematics, material elastic recovery and plastic side flow aiming at the characteristics of UASTS turning.
- (3) The variation influence of different ultrasonic frequency and amplitude for tool trajectory are predicted by the numerical simulation, when the large-amplitude BSWG surface is selected as representative freeform optical surface. It is proved that UASTS turning technology can be realized by theoretical analysis.
- (4) The BSWG surface with the amplitude of 25μm is successfully machined by using UASTS technology on the optical mold material of Polmax steel. After measurement, the value Rt of surface roughness is less than 70nm and the value Ra of surface roughness can achieve to 3.298nm. The experimental results verify that the UASTS turning technology can process freeform surfaces with large-steepness and high-precision on die steel.
Funding
Key Science and Technology Program of Jilin Province (20180201030GX); National Natural Science Foundation of China (61905024).
Acknowledgments
Authors thank Prof. Xue for his academic guidance and Dr. Li for his help in the experiments.
Disclosures
The authors declare that there are no conflicts of interest related to this paper.
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