1. Introduction
In the fields of high-power laser systems and inertial confinement fusion, potassium dihydrogen phosphate (KH2PO4, KDP) crystal is widely used as the key material for nonlinear frequency-doubled components [
1,
2]. Machined KDP surface topographies can impact the optical performance of KDP components [
3,
4]. Therefore, mechanism analysis for the forming of machined surface topography is useful for improving machining quality and enhancing the optical performance of KDP components. The forming of machined surface topography is directly related to the machining process. All factors in the machining process, including the various vibrations of the cutter, workpiece, spindle, guide way, and the processing parameters such as feed rate, cutting speed, cutting depth, lubrication, material parameters, and the geometric parameters of the cutting tool can act on the machined surface topography [
5,
6].
Many experimental studies have been carried out that show that the conditions of the machining process can directly impact the topographies and performances of machined surfaces. Frequency or waviness features in the machined surfaces are the main aspects affected by the various machining process factors [
7]. The cutting parameters under dry conditions and a minimum quantity of lubrication were compared to analyze their influence on surface roughness and topographies [
8,
9]. Lubrication conditions between tool and material, the ambient temperature, and the cutting parameters of the machining process could affect the surface roughness of machined material [
10]. A good level of lubrication and cooling in the process of machining could obviously improve the surface quality of the machined surface [
11]. The cutting parameters, such as feed rates, cutting speeds, and cutting depth were shown to have a direct influence on machined surface roughness, two-dimensional (2D) profile, and surface topography [
12]. Topographies of the machined surface could severely impact the optical performance of the KDP components. Scratches in machined KDP surfaces could impact the laser-damage threshold of KDP components [
13]. The main frequency features in KDP crystal surfaces machined using the fly cutting method could reduce optical field performance [
14]. The frequency features of different wavelengths machined using the micro-milling method had different influences on the optical performance of KDP crystals [
15]. To obtain the effects of surface topography on the performance of machined KDP crystals, the important prerequisite is to analyze the forming mechanism of the machined surface topography.
Some researchers have built theoretical models to predict machined surface topographies. The response surface method is used to verify the relation between the cutting parameters and surface roughness in the turning process [
16,
17]. The surface topography machined by abrasive belt grinding was numerically calculated based on the Johnson transformation system and a filter impulse function [
18]. A least squares support vector machine-based algorithm was developed to predict surface roughness in machined surfaces [
19]. A combination of a finite element model and cellular automata model was used to predict microstructure in the machined surface based on the material model [
20]. The integration methods of fast Fourier, discrete wavelet, and discrete shearlet transforms were used to predict surface roughness on machined surfaces in [
21]. Machined surface errors and behavior of materials have been simulated and calculated using a dynamic fixture-workpiece system based on finite element analysis [
22,
23]. The regression method and an artificial neural network were used to predict three-dimensional (3D) surface topographies based on cutting parameters and cutter vibration [
24]. The 3D form errors of machined surfaces were evaluated and analyzed using the integration of the unified Jacobian–Torsor model and skin model shapes [
25]. Recently, machine learning has been an efficient method for predicting and evaluating the relationship between machined surface characteristics and the machining process. Machine learning based on cutting force variation modeling was used to improve machined surface shape prediction [
26]. Machined surface roughness and 2D profiles of wire electrical discharge machining were also predicted by machine learning algorithms [
27].
In the above predictions of machined surface topographies, the study subjects of most studies were the 2D profiles and their roughness. Compared with 3D surface topographies, 2D profiles cannot completely reflect the topographic information of the machined surfaces. For the above studies, the machining process factors that were used to analyze the machined surface topographies were mainly cutting parameters, material parameters, lubrication conditions, and vibration of the cutter. However, from the systematic perspective, other factors such as environment vibration, runout of the spindle, tool wear, the geometric parameters of the cutting tool, the output noise of the fast tool servo, etc., could also impact the topographies of machined surfaces. The purpose of this investigation was to comprehensively reveal the relationship between the machining process and the 3D surface topographies of machined KDP crystals. To do this, this study provides a dynamic response model, which includes almost all the factors of the machining system to predict the 3D surface topographies of machined KDP crystals. With this prediction model, the entire process of machining KDP is simulated to optimize machining parameters and achieve the optimum surface topography to enhance the optical performance of the machined KDP crystals.
3. Results and Discussion
In order to reduce the computation, the radius of the workpiece in the turning simulations was 3.6 mm. The entire machining process of the workpiece was simulated and its 3D surface topographies are shown in
Figure 8. It can be observed that the obvious waviness textures around the center of the workpiece are distributed in the machined surfaces. The maximum height of the entire simulated surface for the input parameters ap = 9 μm, n = 1300 r/min, and f = 12 μm/r was about 60 nm and higher than the other surface. To clearly compare the results of the experiment and simulation, local 3D topographies with an area of 360 μm × 360 μm were extracted from the entire simulation surface.
The 3D simulation surfaces in
Figure 8 can reflect the surface roughness and geometric error of machined surfaces and are similar to the surface texture of machined KDP components. However, the weakness of the entire simulation surface is that it cannot clearly reflect the details of 3D topography. The extraction positions of the local 3D topographies are pointed out in
Figure 8, and the local 3D simulation and experiment results are presented in
Figure 9 to compare the details of the 3D surface topographies. Comparing the two results, it can be seen that there are seven and six clear peaks in the simulation, shown in
Figure 9a,b respectively. The comparison of the two results reveals that the distributions of peaks and wavelengths in the simulation surfaces are accordant with those in the experiment surfaces. The amplitude of local 3D topographies for the input parameters ap = 9μm, n = 1300 r/min, and f = 12 μm/r was about 10 nm higher than that with cutting parameters of ap = 6 μm, n = 1400 r/min, and f = 14 μm/r; this is consistent with the experiment results and accords with the cutting theory. It can be found that the distribution directions of waviness in the results are different. This is induced by differences in sampling position in the experiment and simulation surfaces. However, in general the entire simulation surface maintains the directional features of the machined surface.
To further analyze the effects of the dynamic response model on the details of the machined KDP surface topography, the 2D simulation surface profiles were extracted from the simulated local 3D topographies; the extracted positions are shown in
Figure 9 and the extraction results of the 2D profiles are shown in
Figure 10. Comparing the results, it can be seen that there are five and four recognizable peaks in the simulation and experiment, as shown in
Figure 10a,b, respectively. The peak number of the simulation profiles is equal to that of the experiment profiles. From
Figure 10, it can be inferred that the average interval distance of peaks in the simulation profile with the parameters of ap = 9 μm, n = 1300 r/min, and f = 12 μm/r was 55 μm, and that of the other simulation profile was 66 μm. The average interval distances of peaks in the simulation profiles were consistent with the experiment results. The amplitude of the 2D simulation profile with input parameters of ap = 9 μm, n = 1300 r/min, and f = 12 μm/r was 60 nm and about 10 nm higher than that with cutting parameters of ap = 6 μm, n = 1400 r/min, and f = 14 μm/r. The amplitudes of the 2D simulation profile are very comparable to those of the experiment profiles. The distinct difference between the experiment and simulation profiles in
Figure 10 is that many micro-waves are overlapped with the main frequencies in the experiment profiles and the simulation profiles are smoother. The reasons for the forming of micro-waviness in the experiment profiles is that brittle fractures in the KDP occurred during machining as well as instrument noise in measurement.
There are various vibrations, such as the runout of the spindle, environment vibration, the noise of the fast tool servo, etc., in the machining process. These vibrations can be reflected in the machined surfaces. To analyze the effect of processing frequency factors on the frequency features of machined surfaces, the PSD values of the simulation and experiment profiles in
Figure 10 were calculated and are shown in
Figure 11. From the results of the PSD analysis, the main frequency of all profiles was 0.0138 μm
−1. In general, the main frequency of the machined surface was mainly influenced by the cutting parameters. Because the feed rates of 12 μm/r and 14 μm/r are very close, the main frequencies of all profiles were unchanged. Through many attempts to change the input parameters of various vibrations, it was found that the axial runout of the spindle and the output noise of the fast tool servo can respectively impact the low and high frequencies of the simulation surfaces. For the profiles with cutting parameters of ap = 9 μm, n = 1300 r/min, and f = 12 μm/r, the low frequencies of experiment and simulation profiles were 0.0027 μm
−1, and this shows that the axial runout of the spindle defined in the dynamic response model was nearly the same as that in the experiment. The high frequencies of the experiment and simulation profiles were 0.0387 μm
−1 and 0.0332 μm
−1, respectively, and this shows that the output noise of the fast tool servo in the simulation was slightly different from that in the experiment. For the profiles with cutting parameters of ap = 6 μm, n = 1400 r/min, and f = 14 μm/r, the low and high frequencies of the experiment and simulation profiles were 0.0055 μm
−1, 0.0249 μm
−1, 0.0027 μm
−1, and 0.0276 μm
−1; the differences of the low and high frequencies were about 0.003 μm
−1 between the experiment and simulation profiles. The difference also suggests that the axial runout of the spindle and output noise of the fast tool servo are not constants and changed with the variation of cutting parameters.
In this study, the runout of the spindle and the output noise of the fast tool servo were input to the simulation model as constants, and this created slight errors in the prediction of frequency features in the machined surfaces. Further research should be undertaken to investigate the effects of cutting parameters on the spindle runout and output noise of the fast tool servo.