Approaches enhancing the process accuracy of fluid jet polishing for making ultra-precision optical components

Accepted Manuscript Approaches Enhancing the Process Accuracy of Fluid Jet Polishing for Making Ultra-Precision Optical Components

Zongfu Guo, Tan Jin, Guizhi Xie, Ange Lu, Meina Qu PII:

S0141-6359(17)30577-9

DOI:

10.1016/j.precisioneng.2018.08.021

Reference:

PRE 6832

To appear in:

Precision Engineering

Received Date:

05 October 2017

Accepted Date:

31 August 2018

Please cite this article as: Zongfu Guo, Tan Jin, Guizhi Xie, Ange Lu, Meina Qu, Approaches Enhancing the Process Accuracy of Fluid Jet Polishing for Making Ultra-Precision Optical Components, Precision Engineering (2018), doi: 10.1016/j.precisioneng.2018.08.021

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ACCEPTED MANUSCRIPT

Approaches Enhancing the Process Accuracy of Fluid Jet Polishing for Making Ultra-Precision Optical Components ZONGFU GUO, 1 TAN JIN, 2, * GUIZHI XIE, 2 ANGE LU, 2 AND MEINA QU2 1School

of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou 310018, China of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China *Corresponding author: [email protected] 2College

ABSTRACT:

Recent research has shown that the computer-controlled fluid jet polishing (FJP) process can generate nanometerlevel form accuracy on free surface optical components. The key issues affecting the machining accuracy of the FJP process, including the deviation of the removal function, dynamic response of the moving tables and the change of material removal rates at a small-scale dwell time, have been investigated. Approaches using signal filtering and alignment methods to reduce the measurement errors of the removal functions in FJP are presented. An enhancing algorithm is proposed to improve the rotational symmetry of the removal function, with correction factors being included in the quality assessment to adjust the change in the quality of the removal function with different dwell times. The dynamic responses and moving accuracy of the worktable were measured, and their effects on the FJP polishing process were analyzed. Fluid jet polishing experiments using small-scale dwell time (less than 5 s) under intermittent motion and continuous motion modes were also performed. The experimental results show that the fluctuation of enhanced removal functions is less than that of the original functions. The residual information on the surface, related to the path distance, is barely noticeable when the distance is less than 0.4 mm, when a nozzle with a diameter of 1 mm diameter is used. The material removal experiments with small-scale dwell times indicate that the removal rate is stable when the worktable is in continuous motion. Keywords: Optical design and fabrication; Optical fabrication; fluid jet Polishing (FJP).

1. INTRODUCTION Optical components with complex shapes and ultra-precision are widely used in the metrological and observatory devices used in astronomy, aerospace and biomedical applications [1-3]. A prohibitively long time is required to manufacture these parts using conventional process methods, in which the form accuracy of the components relies heavily on the level of the operator’s skill [4]. The computer control optical surfacing (CCOS) technique was proposed in the 1970s, in which surface error correction capability is achieved using a polishing or lapping tool head, usually under constant contact pressure, running along the component surface according to precalculated “dwell time” program [5-7]. The dwell time program or dwell time map is typically computed according to the removal function, i.e., the size and shape of the local material removed by the tool under certain contact pressure and contact time, and the measured component surface error map. The fundamental requirements to realize deterministic surface form correction include the accurate description of the removal functions, and the precise execution of the

worktable movements to incorporate the dwell time map accordingly. The removal function of a polishing tool is usually acquired using the fixed-point polishing method, i.e., by making a dent or pot by the tool at a fixed position under the same machining parameters as in the actual polishing process. However, the removal function directly acquired often does not accurately reflect the removal features of the polishing tool, mainly owing to the measurement error of the removal profiles. Another important issue is that, the dwell time scale in fixed-point polishing is much larger than that in the actual form correction process, therefore, the applicability of the removal function determined from the fixed-point tests in an actual polishing process must be carefully investigated. Elmar Pitschke et al. developed a mathematical method to analyze the removal function used in a computer-controlled polishing process. The method is able to determine asymmetries in an influence function. The application of this method yields a value to judge the quality of an influence function. The resulting algorithm renders a symmetrical influence function and filters noisy data. Practical polishing tests with magnetorheological finishing have been performed to verify this new

ACCEPTED MANUSCRIPT technique. The improvement of the peak-valley (PV) value of the surfaces polished with the rendered symmetrical influence function was observed to be on average 14% greater than the PV value improvement of surfaces polished with the unmodified influence function [8, 9]. To fulfil the requirements of precision and ultra-precision surface processing quality, many new polishing techniques have been developed including chemical mechanical polishing (CMP), stress lap polishing (SLP), magnetorheological finishing (MRF), ion beam finishing (IBF), bonnet polishing, and fluid jet polishing (FJP) [10-15]. In the FJP process, the pre-mixed abrasive-liquid polishing media, driven by a high pressure pump, strikes the workpiece surface through a jet nozzle to perform material removal function. Compared to other types of polishing methods, FJP has several rather unique advantages. It is suitable for the machining of complex shapes and the wear of the nozzle is slow, allowing long-time consistent polishing of the components [14, 16, 17]. Compared with IBF and MRF devices, the FJP system does not require sophisticated functional supports, such as a vacuum environment or magnetic field assistance, and thus, the overall equipment cost for an FJP system is significantly lower than that for IBF and MRF devices. The process accuracy of the ion beam polishing method can reach the sub-nanometer level (PV), while that of other types of polishing techniques can reach a level of approximately λ/20 (PV) (λ = 632.8 nm). For the machining of optical components with ultra-precision, magnetorheological finishing (MRF) or bonnet polishing (BP) are often used as pre-polishing methods, whereas ion beam finishing (IBF) is used as the final surface form correction process. The IBF process removes the workpiece surface material at molecular or even atomic level, with an extremely low material removal rate and very high expense for the equipment. Recent research shows that there is a rather high possibility that the surface form correction ability of FJP can reach levels of as low as 1/50λ or even 1/100λ [18]. It is possible to use FJP to accomplish a high percentage of the surface form correction performed by the IBF, significantly reducing the workload and process cost, and increasing the process efficiency for the manufacture of those ultra-precision parts, as show in Fig. 1. According to previous research, the smaller the width of the removal functions, the easier it is to correct the small-scale surface error [19].

Fig. 1. Research objective of precision fluid jet polishing (FJP) technology To explore the process capacity of fluid jet polishing (FJP), researchers have conducted investigations focusing on different aspects of the process. Oliver W. Fähnle et al. from the Netherlands proposed and verified the feasibility of using the abrasive fluid jet machining technique as a precision polishing method, for fixed point local shape modification and polishing optical surfaces with complex shapes [14]. Silvia M. Booij et al. investigated the possibility of using FJP to remove surface material in the ductile regime from both theoretical and experimental aspects [20]. They proposed a method to shape surfaces with FJP by adjusting the influence function (the shape of the footprint of the nozzle) instead of changing the dwell time of the nozzle on the workpiece surface [21]. Osamu Horiuchi et al. studied the effects of jet pressure and alumina abrasive grit size on the removal depth and surface roughness. Their findings show that the removal depth and surface roughness increased with the jet pressure, while the removal depth decreased which reduction in the jet tilt angle. Circular motion machining was introduced and a preferable profile with an axis-symmetric V-shape was realized.

From their experimental results, it was clarified that the nanometerabrasion machining was applicable to corrective figuring of brittle materials [22]. Fang H et al. investigated the dependence of material removal and surface roughness on the characteristics of abrasive particles and the incidence angle [23]. They also presented an effective method, a discrete convolution algorithm, to compute the dwell function for controlling the figuring process. This method avoids the deconvolution operation, which usually fails to converge [24]. Their experimental results demonstrate that the classical edge effect can be neglected in FJP [25]. Zhaoze Li et al. analyzed the material removal mechanism of FJP technology, based on the fluid impact dynamics theory. Combined with the computational fluid dynamics simulation and process experiments, influence functions at different impingement angles were obtained. The influence function was then optimized to obtain an ideal Gaussian shape by rotating the oblique nozzle [26]. T. Matsumura et al. conducted FJP experiments for the crack-free machining of micro grooves and fluid polishing of micro channels along with a CFD analysis. With the jet pressure reaching up to 35 MPa, the stagnation under the jet and the horizontal flow on the machining area were controlled to generate crack-free surfaces by the mask shape [27]. Anthony Beaucamp et al. proposed a computational fluid dynamics model, simulating dynamically the interface between fluid and air. The particles trajectories were computed for studying the removal mechanism. A feedback control loop was established to improve the pressure stability and the form deviation was decreased under ductile removal [28]. Tan Wang et al. used an eccentric rotational device to get the Gaussian-like removal character in magnetorheological jet polishing. In addition, an algorithm using so called tending genes was proposed to estimate the optimal eccentric distances. The experiment results proved that the novel setup was suitable for high precision polishing of complex surfaces [29]. W. Peng et al. investigated in detail the material removal mode affected by the particle size, the particle trajectories with different sizes in the FJP process were numerically simulated, and polishing experiments were conducted to verify the feasibility of the theoretical calculations [30]. Zhong-Chen Cao et al. built a ductile-mode erosion model to predict the volume removed by a single particle and a modified Gaussian function was used to describe the spatial distribution of abrasive particles in fluid jet polishing. The material removal characteristics in FJP were predicted by combining the erosion model and experimental results [31]. A theoretical model was built for predicting and characterizing the material removal characteristics and surface generation in FJP based on computational fluid dynamic modelling. The results show that the theoretical model satisfactorily predicts the surface generation under different polishing conditions and enables a better understanding of the polishing process in FJP [32]. Lv L et al. studied the laser damage resistances of fused silica optics by the FJP process [33]. Several important issues for the successful conduction of the FJP process are addressed in this paper. To reduce the measurement errors of the removal functions, methods for eliminating the effects of original workpiece surface form error, positioning error, jet machining residual features and signal noise, are suggested. An enhancing algorithm is proposed to improve the rotational symmetry of the removal functions, with correction factors added in the quality assessment to adjust the change in the quality of the removal function corresponding to different dwell time. As the FJP process control relies crucially on the accurate execution of the dwell time map, which is realized by the worktable stops and movements, the effects of dynamic responses and moving accuracy of the worktable on the FJP process were investigated in detail. To identify the appropriate process parameters and working mode to achieve good process stability, the effects of dwell time and path distance on the variation of RF were experimentally studied through fixed-point flow jet tests. The process performance under intermittent

ACCEPTED MANUSCRIPT and continuous motion modes (using small-scale dwell time < 5 s) was compared, and a suitable working mode was suggested.

2. PROCESSING AND ENHANCEMENT OF REMOVAL FUNCTION 2.1. Processing method The fix-point jet polishing method, which produces regular shaped dents by the jet flow at fixed positions on the workpiece surface, is typically used to obtain the removal function in FJP. In our tests, a ZYGO GPI laser interferometer (λ = 632.8 nm) was used to measure and compare the surface shape before and after jet machining to obtain the removal profile under certain flow jet machining time. To obtain the disticnt removal profile and reduce the influence of the interferometer’s measurement accuracy (surface measurement accuracy PV < 1/20 λ @ ø = 100mm) the flow jet machining time is usually chosen to be more than 60s. To accurately measure the removal function, the original surface form error of the flat mirror workpiece used in the tests has to be removed, performing a surface form difference algorithm using the surface form measurement data obtained before and after the fix-point test. To reduce the measurement errors, positioning errors (induced by repeated measurements) and signal noise, the measurement results are processed using the flow chart shown below to obtain an ideal removal function. Map 1

Feature points

Low pass filter 1

{F(z(x,y)),0, ss <≥ kk

(1)

z (x,y) = real(IF(s))

(2)

s= '

Z(x, y) represents the raw data of the 3D surface form measured, F is the Fourier transformation, k is the cutoff bandwidth, IF is the inverse Fourier transformation, and z’(x, y) is the 3D surface form data after filtering. By adjusting the cutoff bandwidth, k, the jet machining features (machined dents) can be removed, which helps in the quick alignment of the surface images measured before and after the fixed-point tests. The figure below shows the original and filtered surface images.

Map 2

Low pass filter 1

Alignment Δ(m, n, ø, δ)

Fig. 3. Comparison of fixed-point jet surface before and after low - pass filtering

Difference Low pass filter 2 Removal

Enhancement function

Idea RF

Fig. 2. Flow chart of measurement data processing procedure In Fig. 2, Map 1 and Map 2 are the profile data measured before and after the fixed-point jet flow test. Low pass filter 1 is used to filter out the highfrequency information and jet machining residual features on the surface of the workpiece (see Fig. 3), so that the following surface alignment procedure can be performed more easily, sometimes by simply using obvious geometrical features on the non-machined area. Because the distribution of surface features (dents machined) on the workpiece surface is known (5 mm distance between the flow jet machining points), as shown in Fig. 3, low pass filter 1 uses an ideal low pass filter shown in Equation 1 and 2.

Fig. 3 shows that all the jet machining features are removed after the low-pass filtering. The translation error can be evaluated by comparing the mean square errors under different overlapping situations. The measured surface images can also be rotated, resized and translated to match according to the obvious features such as the lowest and highest point on the non-machined area.

𝑆=

∑𝑁 (𝑍𝑖 ‒ 𝑍)2 𝑖=1 𝑁

(3)

where N is the number of data points in the overlapping area and 𝒁 is the average value of the measured height data of the data points. After the alignment treatment, the translation error (m, n), rotating error ø, and magnifying and reducing error δ, introduced during repeated measurements can be effectively reduced. The aligned surface map data mainly consists of jet removal features, measurement noise and residual measurement errors. Low pass filter 2 (Fig. 2) filters out the measurement noise while maintaining the integrity of the jet machining features. The cut-off frequency of low pass filter 2 can be determined according to the distribution distance of the jet flow machining points. Fig. 4 shows the surface images, after filtering by low pass filter 2, with clear jet flow removal features. The filtered surface features are smoother than the un-filtered ones. Fig. 5 shows that the high frequency errors are filtered out, the noise signals do not include obvious

ACCEPTED MANUSCRIPT machining information, and the amplitude of the noise signals is approximately ±5 nm.

Fig. 6. Shapes of the removal functions before (a) and after (b) enhancement

Fig. 4. Surface image maps before and after filtering the measurement noise

Fig. 7 shows the surface height difference (in the z direction) between the original [Fig. 6(a)] and the enhanced removal function [Fig. 6(b)]. Equation 8 is used to assess the quality of the removal function [7]. To avoid the change in the symmetry of the removal functions with flow jet time, a correction factor δ is proposed to correct the removal function quality η, and the quality assessment method is accordingly adjusted (equation 9).

Fig. 5. Distribution of filtered measurement noise The individual dents machined out in the fixed point flow jet tests shown in Fig. 4 do not have ideal axisymmetric shapes. To improve the axial symmetry of the removal functions, a rotary enhancing algorithm is proposed in the following section, further, correction factors are added in the quality evaluation algorithm to study the change in the quality of removal functions with jet flow time.

Fig. 7. Surface height difference between the original and enhanced removal function '

∆Z = Z ‒ Z

2.2. Enhancing algorithm and assessment of quality of removal functions This paper proposes an integration method related to the individual concentric circular centers of the machined out dents, to average the surface height value of the features to obtain ideal symmetrical rotary shapes: 1 𝑁

g(r) = 𝑁∑𝑖 = 1𝑓(𝑟)

(4)

where f (r) represents an element whose distance to the central point of the dent is r. When g(r) is smaller than a pre-set fluctuation range, then g(r) is set to 0. Fig. 6 shows the 3D shapes of the original and enhanced removal function machined out using the process parameters listed in Table 1. The enhanced removal function shows an intact axial symmetric shape. It must be highlighted that the rotary enhancing algorithm does not change the volume of the material removed. Table 1 The experimental parameters used for getting the removal function Concentration：

20 g/L

impingement angle：

90°

Pressure： Nozzle diameter:

1 MPa

5 mm

1 mm

Jet distance： Material:

Time：

60 s

Grain：

CeO2

BK7

∆𝑍 =

∑𝑵 ∆𝒁 𝒊=𝟏 𝑵

𝑇

δ=𝑇 𝜂=

0

1 𝑁 2 ∑ (∆𝑍𝑖 ‒ ∆𝑍) 𝑁 𝑖=1

𝜂

σ=𝛿

(5) (6) (7) (8) (9)

where Z’ is the modified removal function matrix obtained using the enhancement algorithm, Z is the original removal function matrix measured, N is the number of elements in the matrix, T0 is the unit time constant and T is the fix-point flow jet time.

3. MACHINE TOOL DYNAMIC MEASUREMENT AND ANALYSIS

RESPONSE

In the deterministic polishing (surface form correction) process, the dwell time map (time distribution matrix), calculated using the removal function and measured surface form errors, can be used in two modes to perform workpiece surface machining, i.e., intermittent and continuous worktable movements. Intermittent movement uses a constant worktable movement speed between two dwell points, and stopping at each individual dwell point for a certain time; in contrast, in continuous movement, the worktable moves continuously in variable speeds, which are calculated using the path distances divided by the dwell time of the dwell points. In the FJP process, the dwell time is usually less than 10 seconds, and the path distance is a fraction of the jet diameter (millimeter level), which requires a rather rapid dynamic

ACCEPTED MANUSCRIPT response for the worktable. Fig. 8 shows a simulated removal pit shape obtained by using the removal function shown in Fig. 6(b), with a simulation path distance of 0.2 mm and a dwell time of 0.1 s at each point. It shows that the overlapped error using a small-scale dwell time 0.1 s, results in a removal depth difference of up to 10 nm, which would have a significant effect on the surface form accuracy of ultra-precision components.

Fig. 10. Measurement procedure of the worktable movements 3.2. Influence of dwell time When using FJP to carry out ultra-precision polishing, the calculated dwell time in the dwell time map ranges from zero to several seconds. The change in the movement speed and displacement of the worktable under different dwell time is measured. The results are shown in Figs. 11 and 12.

Fig. 8. Simulated removal pit profile with path spacing 0.2 mm and dwell time 0.1 s 3.1. Dynamic response measurement In the FJP process, the nozzle and workpiece are mounted on the spindle and worktable of a CNC machine tool (control system is GSK 208D). As mentioned in the previous sections, to perform well-controlled surface form correction of the workpiece, the movement accuracy and dynamic response of the worktable need to be measured. A laser interferometer Renishaw XL80 (linear measurement accuracy ± 0.5 ppm, readings can be taken at up to 50 kHz, with a maximum linear measurement speed of 4 m/s and a linear resolution of 1 nm) is used to measure the worktable movements and evaluate its dynamic response, as shown in Fig. 9. Fig. 10 shows the measurement procedure. The worktable moves for seven path distances (d) and then moves back, after each path distance (d) it stops for a dwell time Δt.

Fig. 11. Change in worktable speed under different dwell time

Fig. 12. Displacement of worktable under different dwell time

Fig. 9. Layout of measurement system using laser interferometer

The measured results show that, the worktable cannot reach the pre-set movement speed when using a path distance of 0.2 mm, most of the time it is in the acceleration and deceleration phase. When the dwell time is less than 0.1 s, a ‘lost step’ phenomenon occurs, and a smaller dwell time corresponds to a higher number of steps lost. A lost step indicates that, at a dwell point the pre-set dwell time is not fully reached. When the pre-set dwell time is 0.1 s, although there are no losing steps, the dwell time (Δt) at each dwell point is not equal to the pre-set time. When the dwell time is 0.05 s, there exists a positioning error (Fig. 12), which may be caused by a higher movement speed due to step losing. The issues pertaining to worktable movement and dynamic response mentioned above could lead to significant differences between the actual material removal and predicted removal. 3.3. Influence of movement speed Fig. 13 shows the variation of worktable speed under different pre-sett speeds.

ACCEPTED MANUSCRIPT

Fig. 13. Variation of worktable speed under different pre-sett speeds

Fig. 15. Positioning accuracy of the worktable

The worktable reaches the pre-set speed when the path distance d is 0.2 mm and the pre-set speed is less than 530 mm/min. A higher time is required to move between the dwell points when the pre-set speed < 530 mm/min. When the pre-set speed is more than 530 mm/min, the movement between the dwell points is either decelerating or accelerating; therefore, the actual movement speed is the same without the occurrence of step loss.

Fig. 15 shows that the worktable can reach the set position according to the given instruction, but the dwell times at the start and final dwell points are not the same. The dwell time at the start point is much smaller than the set value; it remains reasonably constant at the middle points, and becomes larger than the set value at the ending point. The above analysis shows that the positioning accuracy of the worktable satisfies the requirement of the polishing process; however, when the moving direction changes or the end point of the movement is reached, there is a difference between the actual dwell time and programme set time. The selected worktable speed must be less than a certain threshold value, and the dwell time also needs to be larger than a certain threshold value. These issues must be considered when selecting the FJP parameters, i.e., setting an appropriate dwell time scale and movement speed, so that the programmed dwell time map can be accurately executed and workpiece surface form correction can be achieved as predicted.

3.4. Influence of path distance

4. FJP MACHINING EXPERIMENTS AND ANALYSIS

Fig. 14. Displacement of moving platform under different path distance When the path distance is increased, the step loss becomes more pronounced. At the same path distance, the step loss becomes more severe as the dwell time decreases. The accumulated time error increases as the step loss becomes more severe. The step lengths for the horizontal line sections in Fig. 14 are set equal to the dwell time Δt, while the dwell time at the start point is much lower than the set value, and the time at the final point is larger than that when no step loss occurs. 3.5. Positioning accuracy measurement

Three types of experiments are designed and carried out on a computer numerical control (CNC) FJP machining platform developed in our Lab (Fig. 16). 1) Fixed-point fluid jet experiments: analyzing the features of the obtained removal function and enhancing it by using the enhancement method, in addition, the quality of the removal function was evaluated according to the quality assessment algorithm. 2) Fluid jet machining experiments using different path distances: analyzing the relationship between the path distance and quality of the processing surface. 3) Small-scale dwell time removal experiments: analyzing the effects of change in material removal rate at small-scale dwell time.

ACCEPTED MANUSCRIPT equivalent jet time is 30 s/mm. Table 3 lists the process parameters for workpiece 1. Small-scale dwell time (Δt < 5 s) jet machining experiments were performed on workpiece 2. Two types of experiments were conducted, namely, continuous movement FJP machining tests and intermittent movement machining tests, using small-scale dwell time. To maintain constant test conditions (nozzle distance and original workpiece surface status), the experiments were performed on the same workpiece. Experiment areas 1 to 5 corresponded to the continuous mode and the machining parameters are listed in Table 4, areas 6 to 10 corresponded to the intermittent mode and their machining parameters are listed in Table 5.

Fig. 16. a) Schematic of computer numerical control (CNC) FJP machining system b) Hydraulic system of FJP c) FJP tool head d) Removal function contour from system stability experiment e) Removal function’s depth and width distribution Before the experiments, the systems stability was tested by machining a series of removal spots on the workpiece surface. Table 2 lists the experimental parameters. The system stability test results [Fig. 16(d) and (e)] show that the fluctuation of the removal depth is within ±10 nm and the width is within ±0.5 mm. Table 2 The experimental parameters used for test the systems stability 20 g/L 1 MPa Concentration： Pressure： Jet distance： Nozzle diameter: Impingement angle： Grain size:

5 mm

Time

60 s

1 mm 90°

Material: Grain:

BK7 CeO2

≈ 1 um

4.1. Experiments arrangement The experiments were performed on the flat surfaces of two BK7 workpieces with a diameter of 50 mm. The jet nozzle outlet diameter used was 1 mm, with a distance of 5 mm to the workpiece surface and a jet pressure of 1 MPa; the polishing fluid was a CeO2 suspension liquid with a concentration of 20 g/L. Fig. 17 shows the experimental arrangement of the machining areas. On workpiece 1, area 1 is the fixed-point machining zone, with different jet machining times of 60, 90 and 120 s used at adjacent points and executed in a loop, thereby making a total of 35 points. Area 2 is the jet path distance experiment zone; the flow jet path distance is 0.2, 0.4 and 0.6 mm, and the dwell time is 6, 12 and 18 s, respectively, from top to bottom, as show in Fig. 17. The movement speed is 300 mm/min, the worktable reciprocates three times, and the

Fig. 17. Arrangement of experimental areas on the specimen surfaces Table 3 The experimental parameters used for fixed-point jet and path distance experiment area Dwell Path Speed Passes time distance (mm/min) times (s) (mm) 60 5 1000 1 90 5 1000 1 1 120 5 1000 1 6 0.2 300 3 2 12 0.4 300 3 18 0.6 300 3

4.2. Fixed-point fluid jet experiment The figure below shows the distribution of the measured removal depths under different jet dwell times, and the corresponding depths calculated using enhanced removal functions,.

ACCEPTED MANUSCRIPT workpiece surface and the jet polished surface. Fig. 20 also shows that the depth over width ratio increases with the dwell time. Fig. 21 shows the distribution of the values of quality of the removal functions assessed using equation 8. The quality of the removal function was corrected using equation 9. The unit time T0 is set as 60 s.

Fig. 18. Depth distribution of fixed-point jet removal under different dwell time Fig. 21. Distribution of the quality of removal function The above figure shows that as dwell time increases, the removal function quality deteriorates (a higher value indicates lower rotary symmetry). Under the same dwell time, the quality of the removal functions gradually deteriorates. With increase in the dwell time and the adjustment value δ, the removal functions quality improves, which means that the asymmetry errors are homogenized. 4.3. Path distance experiments The graph below shows contours of three-dimensional removal under different path distances; the process parameters are listed in Table 3. Fig. 19. Width distribution of fixed point jet removal under different dwell time

Fig. 22. Three-dimensional contours of removal pits machined using different path distances Fig. 20. Variation of the removal depth and width with dwell time Fig. 18 shows that the depths obtained using the enhanced removal function are approximately 20 nm less than those by the original function. Under a larger dwell time, the fluctuation of the removal depth becomes more significant. The removal depth increases at the beginning of the tests and gets stabilized, perhaps because the polishing media needs time to mix well to reach stable machining performance. Fig. 19 shows that the width of the removal function also increases with the dwell time. Fig. 20 shows that the enhanced removal function leads to not only smaller removal depth values than the original removal function, but also less fluctuations. The removal depth increases with the time; however, the variation trend of the enhanced removal depth follows a straight line not passing through the zero depth point. Theoretically, when the flow jet time is zero, the removal depth should also be zero. This may be caused by the difference in roughness of the

ACCEPTED MANUSCRIPT Fig. 23. Cross sectional profiles of the removal pits Fig. 23 shows that the maximum removal depths under different path distances are the same. Under a path distance of 0.6 mm, the surface removal profile shows obvious periodical information corresponding to the path distance; the removal depth fluctuation is approximately 20 nm. When the path distance is smaller than 0.6 mm, there is no clear periodical information. Fig. 24 shows the cross-sectional profiles measured using a Talay profilometer. When the path distance is larger than 0.6 mm, the residual information of the path distance becomes obvious, and the effect on the surface roughness is also distinct.

Fig. 25. Three-dimensional images of removal slots under continuous movement speeds

Fig. 24. Surface roughness under different path distance 4.4. Removing experiments with small-scale dwell time 1. Continuous movement experiments In the continuous movement tests, the fluid jet moves back and forth along a 10 mm long straight line under a certain speed. The total jet machining time along this length is determined by the movement speed and the number of reciprocation movements. After several reciprocating movements, sufficient removal depth is achieved for carrying out measurements. Table 4 lists the experimental parameters used. To compare the change in removal rates at different movement speed, the number of reciprocations is adjusted so that the total dwell time on the 0.2 mm step length under different parameter settings is equally set to 12 s.

The blue curve in Fig. 25 is the measured removal profile, which is also shown in Fig. 26. To compare the removal difference under long and small-scale dwell times, a fixed-point jet removal experiment, with a dwell time of 60 s, was performed before each continuous movement test. The change in the removal depth is shown as the red line in Fig. 26. The enhanced removal functions are evaluated, and then the average depths and widths are determined. Using the enhanced removal functions consistent with the solved average depths and widths, the removal contours under different speeds are predicted, Fig. 26 (predicted value 1, yellow line). According to the measured removal functions, the average depths and widths are directly determined, and the removal contours are predicted (prediction value 2, dark blue line).

Table 4 Continuous movement experiments parameters Area 1 2 3 4 5

Speed（ mm/min） 60 24 12 6 3

Repeat times

Equivalent time

Fig. 26. Removal profiles under different speeds and predictions

60 24 12 6 3

0.2s/0.2mm*60 0.5s/0.2mm*24 1s/0.2mm*12 2s/0.2mm*6 4s/0.2mm*3

It can be seen from the figure that the actual removal depths at different movement speeds are basically the same, but the depths and widths of the actual removal contours are slightly larger than the predicted values.

Fig. 25 shows the measured three-dimensional images of the slots machined (areas 1 to 5 on workpiece 2 in Fig. 17), viewed upward from the workpiece bottom. It is seen that under a slow movement speed of 3 mm/min, an obvious singularity point appears at the end of the slot.

2. Intermittent movement experiments In the intermittent movement tests (areas 6 to 10, workpiece 2, Fig. 17), the nozzle jet moves back and forth along a 12 mm long straight line and stops every 3 mm for a certain time period. The experimental parameters used are listed in Table 5. In each jet machining area, two experiments, with dwell times of Δt and 2Δt, were conducted. The removal profile under a certain dwell time Δt is later obtained by performing a difference operation over the two measured removal profiles, i.e., taking off the Δt profile (left) from the 2Δt (right) profile. This approach is designed to eliminate the material removal error introduced by the worktable movement and the influence of the initial workpiece surface quality on the material removal. Fig. 27 shows the removal contours obtained using the difference operation.

ACCEPTED MANUSCRIPT Table 5 Experiment parameters of intermittent movements using small-scale dwell time Dwell Path Speed Number Rightarea time distance (mm/mi of Left (s) (mm) n) repeats Left 0.2 3 300 300 0.2s×3 6 00 Right 0.4 3 300 300 Left 0.5 3 300 120 0.5s×1 7 20 Right 1 3 300 120 Left 1 3 300 60 8 1s×60 Right 2 3 300 60 Left 2 3 300 30 9 2s×30 Right 4 3 300 30 Left 4 3 300 15 1 4s×15 0 Right 8 3 300 15

Fig. 27. Removal profiles under small-scale dwell time Fig. 27 shows that the removal depths at the end points of the moving line are significantly smaller than those at the middle ones, as the total dwell time at both the ends is smaller than that at the middle points. In each jet machining area, the removal depth values (less than 150 nm) at the middle points are rather close, which means that the FJP system used for the present work is reasonably stable. As shown in Figs. 18 and 19, when the removal depth reaches up to approximately 160 nm, the width of the removal function is less than 6 mm, i.e., the radius of influence is less than the distance between the dwell points (3 mm), which means that the adjacent dwell points do not influence the material removal at a certain dwell point. Fig. 28 shows the change in the removal depths under different dwell times. It is seen that the material removal rates under small-scale dwell times are significantly larger than those under a long dwell time, which is typically used in fixed point tests (60 s, baseline in Fig. 28). Further, the removal rate changes nonlinearly following the increase in dwell time.

Fig. 28. Change of removal depth under different dwell time The investigation of material removal under small-scale dwell times indicates that the material removal rate is rather stable in the continuous movement mode, and it has a high level of fluctuation in the intermittent movement mode, although it is larger than that obtained in the long dwell time fixed point tests for both modes.

5. CONCLUSIONS This study investigated the key issues affecting the machining accuracy of the FJP process, including the deviation of the removal function, dynamic response of the moving tables and the change in material removal rates at small-scale dwell times. The dynamic responses and moving accuracy of the worktable was measured, and their effects on the FJP polishing process were analysed. The measurement results show that when the moving direction changes or the end point of the movement is reached, a difference between the actual dwell time and programme set time appears. The worktable speed must be selected to be less than a certain threshold value (< 530 mm/min), and the dwell time must be larger than a certain threshold value (≥ 0.1 s). The experimental results show that the fluctuation of enhanced removal functions is less than that of the original ones, and the depth over width ratio increases with the dwell time. The residual information on the surface, related with the path distance, is hardly noticeable when the distance is less than 0.4 mm for the 1-mm-diameter nozzle used. In continuous movement, the predicted material removal depths and widths under small-scale dell time are smaller than the measured values, whereas the removal profile is rather stable at different speeds. In intermittent movement, the material removal depths and widths are significantly larger than those under a long dwell time, and the removal profile changes nonlinearly following the increase in dwell time. In this study, it was noted that the depth of the removal function at a small-scale dwell time has a nonlinear relationship with that obtained at the large-scale dwell time, and the depth over width ratios of the removal function under these two conditions are different. Funding Information. National Key Technology Support Program "Large-scale optical components high speed and high efficiency ultraprecision machining, testing technology and equipment" (Project No.: 2012BAZ03623); The Hunan Province graduate student research and innovation projects “Development of Modular Ultra - precision Jet Polishing System and Control Software” (Project No.: 521293339)

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ACCEPTED MANUSCRIPT Highlights ●

Key issues affecting the machining accuracy of the FJP process, including the deviation of the removal function, dynamic response of the moving tables and the change of material removal rates at small-scale dwell time, have been investigated.

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Approaches using signal filtering and alignment methods to reduce the measurement errors of the removal functions in FJP are presented.

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An enhancing algorithm is proposed to improve the rotational symmetry of the removal function, with correction factors added in the quality assessment adjusting the change of removal function quality with different dwell time.

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It has been found that the depth of the removal function at the small scale dwell time has a nonlinear relationship with that obtained at the large scale dwell time, whilst the depth over width ratios of the removal function under these two conditions are also different.