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Abstract
The medium-frequency error on the surface of ultraprecision flycutting has an important effect on the performance of the optical crystal. In this paper, firstly, the characteristic phenomenon of “knife-like grain” in the medium frequency surface of the square and circular optical crystal machined by ultraprecision fly-cutting is revealed. Besides, the error traceability is realized and the results show that the periodic low-frequency fluctuation of 0.3 Hz between the tool tip and the workpiece is the cause of the medium frequency error of “knife-like grain”. Secondly, through the frequency domain waterfall diagram of vibration signal and the analysis of spindle speed signal, it is proved that the surface shape characteristic is caused by the fluctuation of spindle speed during the cutting process. Then, the variation rule of the cutting amount caused by the fluctuation of spindle speed is explored theoretically and experimentally, and the formation mechanism of medium frequency error in flycutting is revealed. Finally, in order to reduce the medium frequency error of “knife-like grain”, the control parameters of the aerostatic spindle are reasonably optimized based on the analysis of the mechanical and electrical coupling control performance of the spindle, so that the RMS values in the medium frequency band of the workpiece are greatly improved after machining.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Potassium Dihydrogen Phosphate (KDP) crystal is an important optical material for the realization of Potassium switch and frequency conversion in large laser devices. Inertial Confinement Fusion (ICF) system requires the wavefront mass of KDP crystal and other optical elements to be strictly controlled in the full frequency band especially in the middle frequency band. Medium-frequency bands of optical elements are easy to cause beam quality deterioration, central bright spot darkening and beam broadening, which is also an important reason for nonlinear self-focusing of strong laser [1–3]. Large laser devices require a large number of high-precision large-aperture KDP crystals, and extreme manufacturing requirements are put forward for the surface quality, surface precision and surface defects of KDP crystals. KDP crystals are characterized by low hardness, great brittleness, strong anisotropy and easy cracking, etc. It is extremely difficult to process KDP crystals with high precision and quality [4–5]. Currently, ultraprecision fly-cutting machine tool is mainly used for KDP crystal cutting [6], it can make the surface damage of optical elements, shape error and surface roughness control in small error range, and can ensure accurate orientation axis and crystals’ surface and side effect, thus to improve the optical element laser damage threshold [7]. In view of its advantages such as high machining efficiency and easy control, diamond fly-cutting technology is widely used in ultraprecision machining of KDP crystal and pure copper planar parts [8–9].
The medium-frequency error of optical elements is generally expressed by power spectral density (PSD), which is divided into PSD1 and PSD2 frequency bands. The period of PSD1 is generally 2.5 mm∼33 mm (wavelength), and the period of PSD2 is generally 0.125 mm∼2.5 mm (wavelength), which is mainly caused by the relative vibration between the tool and workpiece, and the relative vibration is mainly affected by the vibration of the foundation, the fluctuation of the air source pressure, the fluctuation of the oil pressure, the fluctuation of the spindle speed and the cutting force. Different influencing factors lead to larger medium and low frequency vibration by stimulating the modal vibration of the machine tool system or components, which have a greater influence on the surface waviness error of the optical crystal. Li adopted an improved bi-dimensional empirical mode decomposition method to perform multi-scale adaptive decomposition on the surface shape of ultraprecision fly-cutting, and the spatial frequency identification of the relative vibration between the tool and the workpiece was realized [10]. Lee established a single-degree-of freedom dynamic equation to analyze the vibration between the tool and the workpiece, and explained the cause of low frequency errors in the circular direction of ultra-precision turning surface [11]. Kim [12] identified theoretically and experimentally the joint stiffnesses of the bearings and guideways of the ultraprecision machine by comparing the predicted and measured loop stiffnesses. Besides, the Finite Element Analysis was utilized to obtain the optimum design and reveal the relationship between the heat generated and cutting accuracy [13]. Suet [14] proposed effective methods to machining high precision smooth surface on brittle materials with low tool wear and high machining efficiency, and the formation process of brittle materials chip was also obtained by simulation. Cheng [15] revealed the relationship between tool inclination angle and machined surface quality of workpiece, and found that the surface roughness could be reduced by increasing the tool inclination angle.
Many scholars have done great work in recent years to reduce the medium-frequency errors. Rui [16] adopted transfer matrix method to establish the multi-rigid-fexible-body dynamics model of ultraprecision fly-cutting machine tool, and he concluded that the stiffness of aerostatic bearing spindle had great influence on surface. Gao [17] successfully reduced the micro waviness of the machined surface by adopting the combination of the computer-controlled optical surface technique and principles of micro-dissolution water polishing. Li [18] described mid-spatial frequency removal on an aluminum mirror, deformed to a saddle-like freeform shape, using power spectral density as a diagnostic. Chen [19] found that the PSD of actual surface waviness and the rotation frequency of spindle was highly correlated by establishing an evaluation. Chen [20] came up with the effective control methods to reduce micro waviness and built the relationship between the surface waviness and the dynamic characteristics of flycutting machine tool. Li [21] established 5-DOF dynamic model to study the dynamics of the aerostaic bearing spindle with considering the translation motions and tilting motions for fly-cutting machine tool. Chen [22] analyzed the spatial frequency domain errors of ultraprecision machine tool and revealed the effect of each error on the machined surface. He found that the fluctuation of oil pressure had great influence on spatial frequency-based specifications. Lu [23] investigated the external aerodynamic forces on ultraprecision fly-cutting machine tool, which was caused by the intermittent and periodical sweep between the workpiece and tool holder. Chen [24] introduced the dynamic accuracy design for fly-cutting machine tool which was based on the error allocation of frequency domain. An [25] established the prediction model of machining surface topography, and he concluded that the dynamic characteristics of the air spindle were the main cause of the surface strips. Li [26] explained the phenomenon of arc shape in the medium-frequency waviness error by establishing the dynamic model of tool tip position of ultraprecision fly-cutting machine tool, and the surface shape characteristics caused by dynamic vibration were also analyzed. Wei and Li [27] analyzed the surface characteristic caused by dynamic vibration in ultraprecision machining. A phased, self-regulated mode of forced vibration caused by intermittent cutting force was then proposed, and the forming of the arc shape feature was also explained.
However, the existing researches have not explained the phenomenon of “knife-like grain” appearing in the surface shape of the optical crystal. In this paper, the cutting parameters affecting the spatial frequency are firstly analyzed, and the surface shape error is found to be caused by the fluctuation of spindle speed through the vibration waterfall diagram. The influence mechanism of the surface shape error caused by the fluctuation of spindle speed is further analyzed. Secondly, the main reason for the PSD2 medium frequency error caused by flycutting is revealed. Besides, the causes of spindle speed fluctuation and the solutions are also obtained. Finally, the control parameters of spindle speed are optimized to improve the precision of machining optical crystal.
2. Tracing analysis of the surface shape error of the “knife-like grain”
2.1 Analysis of the “knife-like grain” phenomenon
In the process of cutting test, as shown in Fig. 1, the vibration signal is obtained by using a one-way acceleration sensor (333B50, PCB Piezotronics Inc., USA). The machining surface of workpiece is tested by using 4D technology dynamic laser interferometer. After machining surface filtering, it is found that periodic patterns appear along the feeding direction of the optical crystal. The patterns are similar to the cutting grain of the diamond tool, but their spatial frequency is much lower than that of cutting grain. Such “knife-like grain” phenomenon exists in both square and circular optical crystals, as shown in Fig. 2. Along the direction of feed, “knife-like grain” contour curve is extracted, and the spectrum analysis is carried on. It can be seen that under different cutting parameters, the spatial spectrum frequency mainly concentrated in the 1.2 mm−1, 1.5 mm−1 and 2.0 mm−1, these frequencies are in PSD2 range, as shown in Fig. 3. But the space frequency of the cutting trajectory of the diamond tool belongs to the range of roughness, so the micro ripple is obviously not the cutting grain of diamond tool.
Fig. 3. Space frequencies of “knife-like grain” at different feed speeds. (a) contour curve at feed rate of 15 mm/min; (b) spatial frequency corresponding to contour curve is 1.2 mm−1; (c) contour curve at feed rate of 12 mm/min; (d) spatial frequency corresponding to contour curve is 1.5 mm−1; (e) contour curve at feed rate of 9 mm/min; (f) spatial frequency corresponding to contour curve is 2.0 mm−1.
Table 1 shows the spatial frequency of “knife-like grain” under different cutting parameters. It can be seen that under the same feed speed, the spatial frequency is the same when the spindle speed is 260 rpm, 280 rpm and 300 rpm, respectively. The values of the spatial frequencies have nothing to do with the spindle speed and cutting depth, but only with the feed rate. The larger the feed rate, the smaller the spatial frequency.
Table 1. Spatial frequencies of “knife-like grain” under different cutting parameters
The vibration frequency can be calculated by the following equation:
where, ${\textrm{v}_\textrm{t}}$ means feed rate, ${f_k}$ denotes the spatial frequency of the contour curve extracted along the feed direction.When the feed rate and corresponding spatial frequency values in Table 1 are substituted into Eq. (1), ft=0.3 Hz can be obtained. That is, there is a vibration phenomenon between the cutter and the optical crystal with a vibration frequency of 0.3 Hz and a period of 3.3 seconds. Based on the surface prediction model simulation algorithm [27], the simulated surface shape of “knife-like grain” under this vibration frequency can be obtained, as shown in Fig. 4. The fringe distribution of simulated surface shape is consistent with that of experimental surface shape.
2.2 Traceability of vibration frequency
The low vibration frequency (0.3 Hz) may be caused by a variety of reasons, including pressure fluctuations of air source, oil source, and so on, resulting in periodic fluctuations in the position of aerostatic bearing or hydraulic guide rail. During the process of machining square workpiece, the vibration signal is measured at the same time. The vibration acceleration response signal of acquisition signal waterfall diagrams is amplified and analyzed. As shown in Fig. 5(a), by randomly selecting a period of time (26.8 seconds) in the time domain and calculating the number of peak values of the fluctuation curve (8), the period of the fluctuation curve (3.35 seconds) can be calculated. The same analysis method is used for the circular workpiece. As shown in Fig. 5(b), by randomly selecting a period of time (6.69 seconds) in the time domain and calculating the number of peak values of the fluctuation curve (2), the period of the fluctuation curve (3.35 seconds) can be calculated. Therefore, after the amplification analysis of the waterfall diagram of the vibration acceleration response signal, it is found that when the square or circular pieces are processed, the curves with periodic changes appear on the vibration waterfall diagram. The time period is about 3.3 seconds, which is consistent with the frequency 0.3 Hz.
Fig. 5. The periodic variation curves in the response signal waterfall diagram. (a) vibration local magnification waterfall view of the square optical crystal; (b) vibration local magnification waterfall view of the round optical crystal.
In order to study the physical meanings of the wave curve, the frequency interval between the curves is studied. As shown in Fig. 6, it can be seen that the frequency interval between the curves is consistent with the spindle speed. 300 rpm corresponds to the 5 Hz interval, 280 rpm corresponds to the 4.67 Hz interval, and 260 rpm corresponds to the 4.33 Hz interval. Therefore, the curve represents the harmonics of the spindle speed, that is, the spindle speed fluctuates with a period of 3.3 s.
Fig. 6. Local magnification of vibration waterfall when machining round optical crystal: the interval frequency of the periodic change curve. (a) 300 rpm, 5 Hz; (b) 280 rpm, 4.67 Hz; (c) 260 rpm, 4.33 Hz.
3 Formation mechanism and control method of spindle speed fluctuation
3.1 Influence of spindle speed fluctuation on cutting depth
The cutter disc diameter of the fly-cutting machine tool is 650 mm and the cutter head is made of gray cast iron. Two diamond tools are symmetrically mounted on the edge of the cutter head, one for cutting the workpiece and the other configured to maintain dynamic balance. Besides, the corner radius of the diamond tool tip is 5 mm. The effects of different spindle speeds on cutting depth of ultraprecision fly-cutting machine tool is studied theoretically and experimentally. Firstly, the finite element model of the cutter head and the tool rest are established in Ansys software. The top position of the cutter head is fixed, and the rotation speed of 200-400 rpm is given respectively. The simulation results in Fig. 7 show that with the increase of spindle speed, due to the partial eccentricity of the tool rest relative to the horizontal rotating surface of vibration, a bending moment will be generated under the action of centrifugal force, making the edge bend upward. With the increase of rotating speed, the bending moment generated by centrifugal force also increases, and the bending deformation will also become larger. At the position of the diamond tool tip, it shows a vertical displacement, which reduces the cutting depth.
Fig. 7. Simulation results of centrifugal force by using Ansys software. (a) spindle speed is 200 rpm; (b) spindle speed is 280 rpm; (c) spindle speed is 340 rpm; (d) spindle speed is 400 rpm.
Then, a cutting test in ultreprecision fly-cutting machine tool is designed as shown in Fig. 8. A short distance is cut on the pure copper, the size of which is 50 mm×50 mm, when spindle speed is 400 rpm. Then stop feeding, reduce the spindle speed to a near-zero value (50 rpm), and increase the spindle speed to 375 rpm. After reaching the speed, continue feeding for a distance, repeat the process, each time reduce the speed of 25 rpm, so as to process a stepped surface on the pure copper. Figure 9 shows the test results of the pure copper processed by the method on the white light interferometer. It can be clearly seen that the higher the spindle speed difference, the wider the groove width, which also indicates the deeper the groove cutting depth.
Fig. 9. Test results of white light interferometer when cutting square workpiece. (a) Numbered path; (b) spindle speed 400-375 rpm; (c) spindle speed 350-325 rpm; (d) spindle speed 300-275 rpm; (e) spindle speed 250-225 rpm; (f) spindle speed 200-175 rpm.
Figure 10 shows the test results of the machined square workpiece at variable speeds. It can be seen that the lowest position of each groove is basically on a straight line because the lowest rotation speed (50 rpm) is the same. With the decrease of the rotating speed, the position of the tool tip becomes lower. The test results show that the cutting depth corresponding to each rotating speed in the Table 2, and the different spindle speeds will cause significant changes in cutting depth. Therefore, when the spindle speed parameter fluctuates during the cutting process, micro ripples will be formed on the cutting surface. This is the main reason for the medium frequency error of machined surface in ultraprecion fly-cutting machine tool.
Fig. 10. Test result of the square workpiece on profilometer with variable speeds. (a) change curve of groove depth of workpiece; (b) local curve of groove depth change of workpiece.
Table 2. Comparison of the relationship between spindle speed and cutting depth
3.2 Electrical control system model of spindle speed
The spindle speed control of fly-cutting machine tool adopts the control strategy of the driver itself. The driver is the ASDA-AB series driver of DELTA. Its speed control framework is shown in Fig. 11, and its corresponding spindle speed electrical servo control block diagram is shown in Fig. 12.
According to Fig. 12, the transfer function of spindle speed control system can be obtained as follows:
The cutter head is made of cast iron and the moment of inertia is large, the natural frequency and damping ratio of the servo system are small, so it is easy to cause system vibration. Figure 13 shows the fluctuation of spindle speeds and the counter curve along the feed direction under the current control parameters, and it can be seen that the spindle speed has obvious underdamp fluctuation. Due to the influence of rotation speed fluctuation on the cutting depth, a similar curve also appears along the feed direction of the straight spool on the machined surface.
Fig. 13. Spindle speed fluctuation curve and counter curve of the machining surface. (a) spindle speed fluctuation curve; (b) the counter curve of the machining surface.
3.3 Optimization results of electrical control system parameters
The Eq. (4) of natural frequency and damping ratio that increasing the integral parameter of the system can improve the natural frequency of the system, but at the same time the damping ratio of the system reduces. By increasing the proportional control coefficient, the natural frequency and damping ratio of the system can be improved. Therefore, to reduce the rotation speed fluctuation of the spindle, the proportional parameters can be increased. Due to the limitation of the motor itself, the proportional control parameters cannot be increased indefinitely. Figure 14 shows the vibration signal waterfall diagram of the cutting process after the proportional control parameter is increased by about four times. It can be seen from the Fig. 14 that before the control parameter optimization, the frequency doubling order curve of the spindle speed has obvious oscillation. However, after the control parameters optimization, the vibration of the frequency doubling order curve has obvious inhibition, indicating that the spindle speed fluctuation has been well controlled.
Fig. 14. Waterfall diagrams of vibration signal before and after control parameters optimization. (a) before control parameters optimization; (b) after control parameters optimization.
Figure 15 shows the machined surface in PSD2 band of workpiece processed by fly-cutting machine tool before and after optimization of control parameters. Before the control parameters optimization, the RMS value is 15 nm. After optimizing the control parameters of the spindle speed, the RMS value is reduced to 5 nm, and the surface shape in the PSD2 band is greatly improved.
Fig. 15. Machining surface in PSD2 frequency band before and after control parameters optimization. (a) before control parameters optimization; (b) after control parameters optimization.
4 Conclusions
In this paper, the causes of the medium-frequency error are analyzed theoretically and experimentally in detail for the “knife-like grain” feature in both square and circular optical crystal processed by ultra-precision flycutting. Besides, by analyzing the electromechanical coupling control model, the medium-frequency error of “knife-like grain” is suppressed, and the surface quality and performance of ultra-precision flycutting are improved greatly. The main conclusions are summarized as follows :
- (1) The periodic low-frequency fluctuation of the spindle speed with a frequency of 0.3 Hz is the cause of the medium-frequency error of “knife-like grain”.
- (2) The finite element simulation and test results show that the difference of spindle speed will cause significant changes in the amount of cutting tool on the tip back. That is, the higher the spindle speed is, the smaller the amount of cutting tool on the tip back will be.
- (3) By optimizing the control parameters of the aerostatic spindle, the vibration signal of the cutting process shows that the oscillation of the frequency order curve of the spindle speed is obviously suppressed, which indicates that the fluctuation of the spindle speed is well controlled.
- (4) The cutting experiments based on the optimized spindle control parameters shows that the surface shape of the medium-frequency PSD2 band of the optical crystal is improved greatly.
Funding
National Natural Science Foundation of China (52005460); China Postdoctoral Science Foundation (2021M693906); Science Challenge Project (JCKY2016212A506-0105).
Disclosures
The authors declared that they have no conflicts of interest to this work.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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