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Abstract
Various micro-structure surface texturing methods have been used to produce optical functional surface in the grinding, such as the textured grinding wheel, wheel path control and off-spindle-axis grinding. However, those grinding technologies are inherently challenged to employ in large-scale surface grinding due to the extremely high requirement for wheel cutting profile dressing. In this study, a novel phase shift modulation based on wheel oscillation motion was proposed to generate the micro-structure array in ultra-precision grinding. The phase shift effect involved in the surface micro-structure generation is investigated, in which the role of the second phase shift on superimposed mode and micro-waviness forms is discussed. A theoretical model based on the tool superimposed oscillation is established to study the micro-structure texture generation mechanism by considering the second phase shift. The influence of modulation frequency in the case of phase shift and out of phase shift on the surface texture generation both for the striation pattern and micro-structure is compared to clarify the transition between the continuous grooves and the discrete micro-structure array. The study indicates that the phase shift modulation represents a novel paradigm for fabricating micro-structure array with considerable capability and high efficiency in ultra-precision grinding.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Micro-structure array, such as micro-Fresnel array [1], micro-pyramid [2], and micro-lenses [3] are playing an increasingly essential role in optical applications due to flexible design and compact construction [4]. Currently, the single-point diamond turning with fast tool servo technology [5], electrical discharge machining (EDM) [6] and high-speed micro-milling [6] have been primarily used to fabricate a large variety of complex micro-structural arrays with high form accuracy and fine surface finish. However, those methods are restricted to easy-to-machine materials, such as brass [7], aluminum [8], copper alloys [9,10], and optical plastics [11]. In fact, most of the advanced optical material share hard and brittle features, which make them difficult to shape by using conventional turning and milling due to serious tool wear and extremely small allowed depth of cut [12–14]. Shibata et al. [15] investigated the plastic regime removal mechanism of single-crystal in ultra-precision turning. A better surface quality can be achieved along all the cutting directions when the depth of cut of 100 nm is adopted. Ultra-precision grinding acts as a powerful machining process in cutting difficult-to-machine materials, such as glass, silicon carbon, tungsten carbide, and infrared materials, regarded as a promising machining technology in shaping complex optical components [16–19]. Li et al. [20,21] systematically studied the surface integrity evolution mechanism in grinding of single-crystal gallium nitride and gallium gadolinium garnet in both theoretical and experimental aspects, and a high-quality super-smooth surface without brittle fracture and crack can be achieved by selecting suitable grinding parameters. However, the size of the cutting tool for grinding (grinding wheel) is relatively larger than that of diamond lathe and micro-milling in general, which demonstrated that there are more challenges to fabricating micro-structures array by using grinding operation. At present, some simple optic micro-structure surfaces have been fabricated by grinding process [22,23], in which the wheel cutting profile acted as a “copy tool” is exploited to leave a series of crisscrossed grooves to generate micro-structure arrays. Yamamoto et al. [24] adopted a synchronous movement between the grinding wheel and workpiece, in which the speed and rotational direction of the grinding wheel were the same as the workpiece to keep a constant distance between the cutting point and the center of the workpiece rotation. A microstructural unit can be obtained when the workpiece rotates by half a circle and generates a micro-lens array by changing the micro-lens position on the workpiece surface. Zhang et al. [25] used a V-shape grinding wheel to fabricate a micro-groove array on a freeform surface in a 3-axial coordinated control micro-grinding. Chen and Lin [26] developed a new type of thin diamond grinding by electroplating co-deposition technology to obtain micro-crisscrossing grooves on the optical glass by using the in situ grinding method. Aurich et al. [27] made an electroplated diamond micro-grinding pen combined with a high-speed and precision air wheel spindle to generate a micro-groove structure on carbide and carbide ceramic materials. Guo and Zhao [28] investigated the influence of dressing variables on the wheel profile evolution and generated a series of micro-structured array surfaces by using the diamond wheel with various shapes. Combined with the advantages of micro-grinding and forming grinding, the micro-structured surface can be achieved by using the designed grinding wheel with regular texture grooves [29–31]. Su et al. [32] shaped a textured diamond grinding wheel by using micro-abrasive water jet operation to machine highly accurate cylindrical microlens array on RB-SiC mold. However, the textured grinding wheel method can only be used to generate grooves, which are parallel to the grinding feed direction, and the microstructure distribution characteristics are highly dependent on the surface structure of the grinding wheel [33]. Different from the fast tool servo machining utilizing the high frequency tool vibration to generate the micro-structure array surface, the vibration assisted grinding mainly have been exploited to reduce the size of chips to achieve ductile mode material removal in the machining of hard and brittle materials, in which the direction of vibration is parallel to the workpiece surface [34]. In ultra-precision grinding, the vibration due to wheel unbalance is inevitably induced and causes small regular mark formation on the ground surface, in which the period of vibration corresponds to the rotational frequency of the grinding wheel [35,36]. In our previous studies, the modelling for spiral waviness pattern generation was established and found that the primary phase shift played a key role in waviness pattern evolution, which directly determined the spiral waviness geometry [37,38]. However, the second phase shift acted a dominant role in spiral waviness formation when the primary phase shift was equal to π [39]. In this case, the maximum difference for wheel vibration displacement between the neighboring tool paths and it shares the same vibration phase for every another tool pass interval. It is very beneficial for fabricating discrete micro-structure surfaces. However, it only uncovers the waviness pattern generation mechanism both for the primary phase shift and the second phase shift, and there is a lack of a theoretical model to relate the tool vibration conditions to the discrete micro-structure unit generation.
The previous work in micro-structure array fabrication in grinding has mainly focused on wheel cutting edge preparation by the dressing operation. The wheel acted as a copy tool, and the tool edge structure was transformed onto the workpiece surface. However, the micro-structure unit geometry heavily depends on wheel dressing operation. Our previous study shows that the wheel oscillation combined with phase shift control is capable of fabricating regular micro-structure on the ground surface. However, there is a lack of a theoretical model to relate the wheel oscillation and micro-structure formation, which is essential to control micro-structure unit geometry. To achieve the machining micro-structure array by ultra-precision grinding, a superimposed low-frequency oscillation has been introduced, and a novel phase shift modulation control has been proposed in ultra-precision grinding. The phase shift modulation grinding that superimposed a low oscillation frequency onto the grinding wheel offers an alternative method for machining micro-structure arrays. No additional equipment is required, and its striation pattern and the micro-structure unit shape can be more easier controlled just by adjusting the tool oscillation frequency in comparing other methods for fabricating the micro-structure array. The second phase shift makes it possible to fabricate discrete micro-structure arrays in the grinding. The second phase modulation is determined by changing the workpiece speed to keep the fractional part of the ratio (0.5) of the tool oscillation frequency to the rotation frequency of the workpiece spindle. Based on that, a theoretical model for the micro-structure array formation by considering the interaction of wheel geometry and the second phase shift effect, in which interference of cutting profile for the grinding wheel both in circumferential and feed direction has been taken into account. It provides a flexible and alternative solution to generating complex micro-structure arrays in ultra-precision grinding.
2. Phase shift modulation in ultra-precision grinding
For conventional ultra-precision grinding, the high-speed rotating grinding wheel is superimposed on the linear and rotating motion of the workpiece, which forms a spiral cutting path on the ground surface. The grooves left by the grinding wheel are also a spiral mark from the outer of the workpiece to the rotational center of the workpiece, and the profile height for the groove is generally the same (ignoring tool wear) on all whole workpiece surface due to the consistent depth of cut for the wheel. To fabricate the micro-structure arrays, the motion of tool low-frequency oscillation is added to the normal direction of the workpiece surface, in which the depth of cut follows a sine wave track, as shown in Fig. 1. However, the uniform waviness was generated if the same phase occurred in the same angular displacement. In that case, there are a series of sine waviness profile formed on the ground surface rather than the micro-structure arrays.
Fig. 1. Schematic of the micro-structure array generation with a phase shift modulation in ultra-precision grinding
To realize the different depths of cut for the grinding wheel, the phase shift modulation is introduced for superimposed low-frequency oscillation motion in the adjacent cutting passes. With the development of phase shift, there are two types of parodical structures on the surface generation, one is waviness structure along the feed direction, and another is micro-structure texture along the circumferential direction. Therefore, this novel method can not only generate the micro-structure array but also control the undulating radial striation pattern of the structured arrays in the ultra-precision grinding.
3. Micro-structure arrays generated in the low-frequency oscillation grinding
Figure 2 is the schematic representation of the model workflow based on the proposed phase shift modulation method. In comparison with the conventional grinding operation, the wheel-work engagement is constantly changing throughout the whole machining process, The surface texture is generated by the interference of the cutting profile in the feed direction (nose radius) and wheel profile in the cutting direction. Combined with the shape and relative oscillatory motion of the grinding wheel, the formation of waviness on both radial and tangential direction were calculated respectively. Considering the interference of neighbouring tool path of grinding wheel engagement due to the superimposed oscillation motion, the micro-structure array texture can be established. Based on the model, the relationship between the structured texture and oscillation parameters (including modulation amplitude and modulation frequency) can be determined. In this way, the oscillatory tool path can be calculated once the micro-structure element is designed.
Fig. 2. Schematic representation of micro-structure array generation under superimposed tool oscillation
3.1 Geometric and kinematic modelling of Micro-structure arrays generation in the feed direction of the grinding wheel engagement
Phase describes the initial phase of the sinusoidal modulation path for each revolution of the workpiece, and the phase shift represents the phase difference in the wheel oscillation locus in successive workpiece spindle revolutions. The phase shift played a critical role in determining micro-structure array generation as discussed above. The phase shift is defined as the ratio of tool oscillation frequency to workpiece spindle speed in radians, which can be calculated as
where ${f_m}$ is the modulation frequency, ${V_2}$ is workpiece spindle speed, and $INT\;()$ denotes integral function.During wheel oscillation motion, the phase shift (-π≤δ≤π) reflects the phase difference between the neighbouring oscillation waveforms at the angular position, which is related to the fractional part of the ratio of the frequency of wheel oscillation to workpiece spindle speed. The integral part of the ratio is equal to the number of waviness striations If the phase shift is slight. The second phase shift presents the difference between oscillation waveforms at every two feed intervals. The second phase shift plays a significant role in waviness structure generation if the phase shift is substantial (especially for δ=π or δ=-π). In that case, the number of waviness striations is double the integral part of the ratio of tool oscillation frequency to workpiece rotation frequency.
The non-integer part (phase shift) of the ratio of the modulation frequency to the workpiece spindle speed determines the relative position of the grinding wheel center between two adjacent tool loci, therefore, the intersection points between two neighboring cutting profiles of the grinding wheel are different and different surface textures are formed. Especially, the second phase shift contributes the maximum difference for the wheel engagement depth between the adjacent modulation tool paths, which is in favor of the finer microstructure unit formation.
To determine the tool interference geometry in 3D space, it is necessary to describe the wheel geometry. The grinding wheel has a small nose radius ($r$) and a large grinding wheel radius ($R$), which can be approximately regarded as a three-dimensional ellipsoid, as shown in Fig. 3. The equation for modelling of contour grinding wheel is given as:
From the cross section for grinding wheel as shown in Fig. 4, it is known $a = c = R + {r_s}$ and the ellipsoid can be described as
Fig. 4. Schematic representation of grinding wheel geometry and surface generation under superimposed tool oscillation (a) grinding wheel geometry in Z-Y plane, (b) grinding wheel geometry in Z-X plane, (c) wheel profile interference under different phase shifts
However, the parameter $b$ in Eq. (3) is unknown. In order to calculate b, the equation for the wheel cross section is described as
The instantaneous curvature radius of the 2D elliptic equation can be calculated as
From the Eq. (5), the first and second order derivatives can be calculated as
The instantaneous curvature radius can be described as
In the center of the contact area between the grinding wheel and workpiece ($y = 0$), the instantaneous curvature radius of the elliptic is equal to the nose radius of the grinding wheel. Therefore, Eq. (5) can be rewritten as
The minor axis of the ellipse $b$ can be taken as
Finally, the 3D ellipsoid can be expressed as
Due to $z \le 0$
In the Y-Z plane, the radius for each cross section can be calculated as
When the phase shift = 0, the intersection $P({x_{i,i + 1}},{y_{i,i + 1}},{z_{i,i + 1}})$ of the adjacent cutting profiles of grinding wheels in Y-Z plane can be calculated as
When the phase shift≠0, the intersection $P({x_{i,i + 1}},{y_{i,i + 1}},{z_{i,i + 1}})$ of the adjacent cutting profiles of grinding wheels in Y-Z plane can be calculated as in Y-Z plane
During the ultra-precision grinding, each cutting profile of the wheel moves on a defined 3D trajectory with an oscillation movement. In order to establish the model of tool cutting edge path over the whole workpiece surface, three coordinate systems are selected, one is used for the grinding wheel Os-XsYsZs, one for the workpiece Ow-XwYwZw and one for the fixed global coordinate system Oc-XcYcZc, as shown in Fig. 5. Since the surface topography generation resulted from the superimposed movement between the wheel and workpiece, a coordinate transformation between the tool and workpiece is required for the representation of the rotational grinding wheel and its movement relative to the workpiece surface in a fixed coordinate system instantly.
Fig. 5. Schematic of micro-structure array generation in coordinate systems under superimposed tool oscillation (a) superimposed oscillation in grinding operation, (b) engagement geometry of the grinding wheel, (c) micro-structure arrays generation
Firstly, the coordinate transformation between the wheel and workpiece is applied, in which the tool coordinate system (Os-XsYsZs) is transformed to the workpiece coordinate system (Ow-XwYwZw). There are a rotational movement and a translational movement, which are used to transform into the same frame. According to the geometric relationship shown in Fig. 5, the instant relative motion of the point (Pi) on the wheel can be described as
To transformed the workpiece coordinate system (Ow-XwYwZw) into the global coordinate system (Oc-XcYcZc), rotation of workpiece coordinate about the axis Z is applied and the operator of the rotation matrix can be expressed as
The point on the grinding wheel with respect to the workpiece surface is given as
3.2 Modelling of micro-structure arrays generation in the peripheral direction of grinding wheel engagement
The phase shift modulation grinding involves the superposition of grinding wheel oscillation control. The resulting modulation of the wheel oscillation alters the relative cutting position between neighboring cutting path, in which the successive cutting paths can occur unsynchronized phase shift at the same angular position, which leads to a relative variation of the depth of cut and generates different micro-structure arrays, as shown in Fig. 6. The phase difference depends on the fractional part of the ratio of wheel oscillation frequency to the rotational frequency of the workpiece. For the in-phase (phase = 0) condition, the distribution of peaks and troughs of the wheel oscillation coincident tool cutting path as the grinding wheel moves towards the rotating center of the workpiece radically, which results in continuous micro-waviness striation on the ground surface, as shown in Fig. 6(a). For the out-of-phase, especially in the second phase (phase=π), however, it shares the same phase for every two intervals of tool feed rate spacing have in the machining cycles. In that case, the maximum difference for the depth of cut in the adjacent cutting paths in the grinding, which has the best potential for discrete micro-structures array, as shown in Fig. 6(b). In addition, a slight change for the second phase shift (increased by 0.1π) can result a remarkable variation of micro-structures array arrangement, as shown in Fig. 6(c).
Fig. 6. Schematic representation of tool path formation and micro-structure array generation under superimposed tool oscillation (a) phase shift = 0, (b) phase shift =π, (c) phase shift =1.1π
A sinusoidal modulation is superimposed in the normal direction to the workpiece surface, in which the instantaneous depth of cut for the grinding wheel varies with time. The phase difference between adjacent two cutting paths contributes to different intersections of cutting tool profiles at the same angular position, which generate a series of regular intersections for a given phase. Especially for the large phase (second phase shift), the maximum difference in depth of cut between two neighbouring cutting paths is conducive to producing some discrete and regular microstructure units as the machining cycles. Therefore, the generation of the micro-structured array is derived from phase shift modulation.
In the peripheral direction of grinding wheel engagement, the arc length for grinding wheel traveling is gradually decreasing as the tool moves the feed direction continuously within a given time, which results from the variable relative linear velocity of the workpiece, as shown in Fig. 7. It means the size of the micro-structure unit along the circumferential direction of the workpiece is decreased as the machining cycles.
Fig. 7. Schematic of varying arc length the wheel travels under different phase shift modulations, (a) phase shift = 0, (b) phase shift=π, (c) phase shift = 1.1π
In the case of the wheel oscillation, the depth of cut of the grinding wheel changes periodically and the relative contact point for each cross-section of the wheel cutting profile changes along the circumferential direction of the workpiece with modulation cycles. The phase shift (in phase) is various at each cutting cross section between neighbouring cutting paths, resulting in different depth of cut for the grinding wheel at the same angular position (adjacent cutting paths). Therefore, the presence of phase shift can result in different tool profile interference conditions and various intersection points of the wheel cutting profile. as shown in Fig. 8. It is not always that the lowest point of cutting profile of grinding wheel contributes to the final surface generation.
Fig. 8. Schematic diagram model of the wheel track and interference under superimposed tool oscillation
The relative trajectory of the grinding wheel along the circumferential direction of the workpiece is an Archimedes’ spiral in the X-Y plane, in which the arc length the wheel traveled can be described with a polar equation as
where ${R_w}$ is the radius of workpieceThe arc length the wheel travelled can be calculated as
According to the geometrical relationship, as shown in Fig. 7, the intersection points of each neighboring wheel cutting profile along the circumferential direction can be calculated as
For each circle of the grinding wheel cutting profile, solving the intersection points (Xi, Zi), (Xi + 1, Zi + 1)…(Xi + n, Zi + n) with all other circles along the wheel track in one cycle and the lowest point of the cutting is the valid intersection, the rest is the invalid intersection.
4. Experimental results and discussion
The tests for the high-precision micro-structure array fabrication were examined on a 4-axis Ultra-precision grinding machine (Nanotech 450UPL V2), as shown in Fig. 9. Table 1 summarizes the experimental conditions for fabricating the micro-structure array. Before the experiment, all samples were ground and then polished to a surface roughness of 20 nm. To reduce the impact of grinding wheel wear on micro-structure arrays generation, the dressing operation is conducted before each test and dressing conditions are shown in Table 1. To investigate the effect of phase shift on the evolution of surface micro-structure arrays, hold the rotational speed of the workpiece and feed speed constant at 1mm/min and 20 rpm respectively in the grinding and adjust the oscillational frequency (140-152 cycles per minute) to change the phase shift (0, π, 1.1π, 1.2π respectively). The phase shift can be adjusted by changing the tool oscillation frequency, in which the phase shift is determined by the fraction part of the ratio of wheel oscillation frequency to the workpiece rotational frequency. Therefore, tool modulation frequency can be obtained by multiplying the ratio and workpiece speed. Take the second phase shift for example, if the desired integer and fractional part of the ratio are 7 and 0.5 respectively, the oscillation frequency is equal to 7.5 × 20 = 150 cycles/min. The ground surfaces were measured using a 3D optical surface profiler (NewView 9000) to obtain the representation of the micro-structure array.
Table 1. Conditions for phase shift modulation grinding
4.1. Role of the second phase shift in micro-structure arrays generation
The phase shift resulted in different tool-work engagement, especially for the phase shift=π, in which the oscillation phase is the same with another every other tool path at the same angular displacement. In this case, the width of micro-structure unit is double to feed rate (2S), and the shape of the structed texture is approximately rhombus-shaped. The number of micro-structure array’s striation is double the ratio of tool oscillation frequency to the rotational speed of workpiece spindle and it is called the second phase shift, which is different from primary phase in tool vibration (the number of vibration marks is equal to the ratio). As shown in Fig. 10, the second phase shift resulted in different surface textures.
Fig. 10. SEM images for the micro-structure surface with different second phase shifts in high frequency oscillation with workpiece speed 1500 rpm, feed speed 10 mm/min, depth of cut 10 μm, (a) wheel speed 39150 rpm, (b) wheel speed 39750 rpm
It is observed that the surface microstructure texture appears as a regular rhombus-shaped array surface and the radial striation presents a linear pattern texture on the ground surface, as shown in Fig. 10(a). In addition, the surface texture presents a rhomboid structure with a curved striation mode for the micro-structure array arrangement when the phase shift is 1.1π, as shown in Fig. 10(b). Therefore, it indicated that the phase shift has a considerable influence on the formation of the surface micro-structure texture.
Figure 11 shows the surface topography when the phase shift = 0, it is found that the number of radial striation is equal to the ratio between the workpiece speed and modulation (140/20 = 7). The striation pattern represents a linear mode, which diverges outward from the workpiece center. It is observed that there are only cutting grooves generated by the cutting profile of the grinding wheel, in which the grooves developed along a spiral locus. It is impossible to generate the micro-structured array due to the same depth of cut for the grinding wheel at the same angular position but just a series of waviness structure formed on the ground surface (no superposition phenomenon).
Fig. 11. Ground surface topography generated in low frequency oscillation of the grinding wheel with phase shift = 0, (a) surface striation pattern, (b) surface micro-structure
4.2 Experimental verification of the micro-structure arrays generation
The simulated micro-structure array for different second phase shifts control is shown in Fig. 12. The optical photographs of the ground surface are presented in Fig. 13, both for in and out of phase shift modulation, it is observed that both the structured texture array and the radial striation pattern are accurately predicted. The resulting surface texture evolves in wheel oscillation motion with the phase shift control are quite different from those generated out of phase shift in grinding, in which the continuous grooves left by the tool are disrupted into the discrete micro-structure array. In Fig. 11(a) and Fig. 12(a), it clearly demonstrates that the waviness structures are centered at the workpiece rotation angle, and the number of radial striations is equivalent to the ratio of modulation frequency to workpiece spindle speed, therefore there is no discrete micro-structure generation when the phase shift is 0. For the phase shift modulation. However, a distinct difference between the surface waves of striation pattern and surface micro-structure is formed, and the number of waviness stripe is double to the ratio and discontinuous micro-structure array generated on the machined surface under the presence of the second phase shift, in which the oscillation phase has a slight difference every two intervals for tool path, especially the phase is the same every another wheel path interval. In this case, the number of structured surface striation is double as large as the integral part of the speed ratio. However, the huge differences for the oscillation phase between the adjacent tool path at the same angular position, which lead to the district difference for the depth of cut and discrete micro-structure element formed on the ground surface.
Fig. 12. Simulated micro-structure array (a) phase shift = 0, (b) phase shift=π, (b) phase shift = 1.1π, (c) phase shift = 1.2π
Fig. 13. Micro-structure array surface generation with second phase shift control (a), (b) phase shift=π
For the phase shift=π, the striation pattern centers align as shown in Fig. 12(b), which creates a serial of approximated rhombus-shaped micro-structure texture with a rotationally symmetric pattern around the workpiece center. The size of radial micro-structure along the circumference direction presents a linear decrease from the outer to the center for variable phase shifts, which resulted from the continuously decreased relative linear speed under the constant angular speed for the workpiece, as shown in Fig. 12. Furthermore, the width of the micro-structure along the radial direction holds constant at about 100 μm for all the phase shifts, which is equal to the twice of feed rate (2Vf/V2) in presence of second phase shift. It demonstrates that the dimension of the micro-structure array can be controlled by adjusting the workpiece speed and feed speed, in which the maximum width of microstructure unit is double to feed rate.
There is a substantial change in the striation pattern when the phase shift is modified from π to 1.1π despite a small adjustment, in which the micro-structure array pattern changed from the linear mode to a sharp curve pattern along an anti-clockwise direction from the outer diameter to the rotation center of the workpiece as shown in Fig. 13(a) and Fig. 14(a). when the phase shift increases up to1.2π, it is found the degree of curvature also increases, as shown in Fig. 15(a). The results indicate the phase shift modulation is capable of fabricating the micro-structure array in a flexible way, which provides an alternative method to both control micro-structure locations and the radial striation pattern.
Fig. 14. Micro-structure array surface generation with second phase shift control (a), (b) phase shift = 1.1π
Fig. 15. Micro-structure array surface generation with second phase shift control (a), (b) phase shift = 1.2π
It shows that phase shift modulation grinding is capable of achieving small-sized micro-structure array fabrication. The results demonstrate that the proposed novel grinding process is both controllable and flexible, which has the potential to generate a micro-size and a wide range of micro-structure array by modifying the phase shift. It provides a new texturing method to achieve complex and discrete micro-structure array.
4. Conclusions
In this study, a new method for fabrication of the high-precision micro-structure array was proposed by using phase shift modulation of grinding wheel oscillation and a theoretical model of micro-structure generation based on the superimposed tool path was developed. This research was shown that phase shift modulation is a controllable and flexible texturing technology in ultra-precision grinding for fabricating micro-structure array. The main conclusions from this study are as follows:
- (1) The phase shift plays a vital role in fabricating micro-structure array, in which only continuous grooves are left by the grinding wheel when the oscillation frequency is out of phase shift, and there is no discrete micro-structure generation.
- (2) The presence of the second phase shift imposed a considerable translation for the surface texture and striation pattern, which is prone to generate discontinuous micro-structure array in the grinding. The number of radial striations is double to the ratio of the tool osculation frequency and workpiece spindle speed. For the modulation frequency out of phase shift, the number of the waviness pattern is equivalent to the ratio.
- (3) The striation pattern is sensitive to the change of the second phase shift, in which a slight change for the phase shift imposed a significant influence on degree of striation curvature.
- (4) The approximated rhombus-shaped micro-structure array is generated when the modulation frequency is in the second phase shift, and the width of the unit structure is equal to the double feed rate. It provides a flexible method to achieve a large number of micro-structured arrays by simply adjusting the feed rate and the phase shift.
Funding
National Natural Science Foundation of China (52105481, 52005397); Natural Science Foundation of Zhejiang Province (LQ21E050010); Key Research and Development Program of Shaanxi Province (2020DLGY04-02).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results represented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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