Surface quality and cylindricity of ultrasonic elliptical vibration–assisted centerless grinding of micro-rod YAG single crystals

Micro-rod YAG single crystals are the preferred laser crystals for laser gain mediums with a high power. However, brittle fracture and crack damages are easy to occur in the grinding process of micro-rod brittle crystals due to their large length-to-diameter ratio, high brittleness, and high hardness. In this work, the modal, frequency, and harmonic response of the transducer under ultrasonic elliptical vibration are analyzed by using finite element simulation. Then, the mechanical structure of ultrasonic elliptical vibration system was designed and optimized based on the ultrasonic elliptical vibration theory and finite element simulation. To verify the reliability of the transducer, ultrasonic vibration experiments were carried out to measure the resonance frequency, amplitude, and impedance characteristics of the transducer. The vibration synthesis experiments under different phase differences and voltages were performed to verify the rationality of the structural design of the ultrasonic elliptical vibration system. An experimental platform of ultrasonic elliptical vibration–assisted centerless grinding (UEVCG) was developed, and UEVCG tests of micro-rod YAG crystals were performed. The influences of the voltage, phase difference, and pallet inclined angle on surface roughness, peak-to-valley value, and cylindricity of the micro-rod YAG crystals were systematically analyzed. The ultrasonic elliptical vibration parameters were optimized based on the range analysis results of the orthogonal test. The results indicated that ultrasonic elliptical vibration effectively improved the surface quality and cylindricity of the micro-rod YAG crystals compared with traditional grinding. This work will not only enhance the understanding of the ultrasonic elliptical vibration principle, but also provide a technical support for precision and high-efficiency machining of micro-rod brittle materials.


Introduction
Micro-rod YAG single crystals are the preferred laser crystals for laser gain mediums with a high power due to their high gain, low threshold, and stable physical and chemical properties [1][2][3]. In addition to developing advanced growth technology of laser crystals with low defects, hard and brittle crystal elements must achieve their satisfactory surface integrity through precision machining technologies, such as grinding and polishing [4][5][6]. At present, the plastic damage mechanism of the laser crystals was revealed by the nanoindentation and nanoscratch tests combined with the crosssectional TEM observation [7][8][9], which was dominated by polycrystalline nanocrystalline, dislocations, stacking faults, and lattice distortions. In addition, the ultra-smooth surface of planar laser crystal elements can be achieved by polishing and ductile grinding technologies [10][11][12]. However, microrod brittle crystals have large length-to-diameter ratio, high brittleness, and high hardness; therefore, brittle fracture and crack damages are easy to occur in the grinding process [13][14][15], which seriously affect the service accuracy and life of the solid-state lasers. At present, centerless grinding is the most effective machining technology for micro-rod work materials [16][17][18][19][20], which can avoid the concentrated stress caused by the workpiece clamping and improve the roundness and surface quality of the workpiece. Barrenetxea et al. [16] performed centerless grinding tests of rod workpieces based on an active damping system, and the results demonstrated that centerless grinding technology effectively suppressed the chatter of the machine tool and improved the accuracy and productivity of the workpiece. Xu and Wu [18,19] performed centerless grinding tests of rod stainless steels on a surface grinder, and a precision rod workpiece with a roundness of 0.9 μm was achieved by using the optimized process parameters. Hashimoto et al. [20] reviewed the development history of the centerless grinding technology, and they pointed that centerless grinding had a broad application prospect in the industrial production and highprecision manufacturing of micro-rod elements.
To improve the surface integrity of the hard and brittle components, many efforts have been made to reduce the brittle damages generated in the grinding process of hard and brittle materials, such as combining ultrasonic vibration into the traditional grinding [21][22][23][24][25], using intelligent algorithms to optimize grinding parameters [26,27], and developing innovative grinding fluid [28,29]. Many scholars demonstrated that ultrasonic vibration effectively improved the ground surface quality, decreased the subsurface damage, and decreased the wheel wear [21][22][23][24][25]. Li et al. [21] performed ultrasonic vibration-assisted grinding tests of deep holes of zirconia ceramics, and the results indicated that compared with traditional grinding, ultrasonic vibration-assisted grinding decreased the edge-chipping size and improved the surface integrity of the holes. Wang et al. [22] analyzed the power spectrum density during ultrasonic vibration-assisted grinding of sapphire single crystals, and they found that power spectrum density could well characterize the wheel wear and ultrasonic vibration effectively decreased the brittle fractures and wheel wear. Li et al. [24] studied the damage mechanism and force modeling during the ultrasonic vibration-assisted grinding of SiC ceramics, and the results demonstrated that compared with traditional grinding, ultrasonic vibration-assisted grinding decreased the subsurface damage depth, surface roughness value, and grinding force. Kumar et al. [25] summarized the advances in the machining of brittle optical materials, and they believed that ultrasonic vibration could reduce the brittle damages and improve the surface quality during the machining of brittle optical materials. Moriwaki and Shamoto [30] found that a high-frequency voltage with a phase difference excited two transducers to generate elliptical vibration, and proposed ultrasonic elliptical vibration-assisted machining technology based on the ultrasonic vibration technology. Compared with the traditional grinding technology, the ultrasonic elliptical vibration-assisted centerless grinding (UEVCG) technology can obtain a larger range of elliptical vibration and a larger amplitude amplification, which has been widely used in the high-efficiency and precision machining of micro-rod difficult-to-machine materials [31][32][33][34][35][36][37]. Wu et al. [31,32] fabricated the micro-scale cylindrical components of tungsten carbide using the UEVCG technology, and a cylindrical component of 60 μm in diameter and 15 mm in length was achieved, which demonstrated that UEVCG had significant advantages in fabricating micro-rod elements with a large aspect ratio. Xu and Wu [33,34] conducted UEVCG tests of K-grade cemented carbide to fabricated micro-rod components, and a micro-rod component with an aspect ratio of 310:1 and a diameter of 42 μm was successfully achieved, which indicated that UEVCG was an effective method to manufacture micro-rod components with a large aspect ratio. Fan et al. [35,36] performed UEVCG tests of rod carbon steels, and the results demonstrated that the UEVCG technology effectively improved the cylindricity and surface quality of the work material. Nevertheless, there is hardly report on ultrasonic vibration-assisted centerless grinding of rod laser crystals, which hinders the precision and industrial production of rod laser crystals.
In this work, the modal, frequency, and harmonic response of the transducer under the ultrasonic elliptical vibration were analyzed by using finite element simulation. Then, the mechanical structure of ultrasonic elliptical vibration system was designed and optimized based on the ultrasonic elliptical vibration theory and finite element simulation. To verify the reliability of the transducer, ultrasonic vibration experiments were carried out to measure the resonance frequency, amplitude, and impedance characteristics of the transducer. The vibration synthesis experiments under different phase differences and voltages were performed to verify the rationality of the structural design of the ultrasonic elliptical vibration system. An experimental platform of ultrasonic elliptical vibration-assisted centerless grinding was developed, and UEVCG tests of micro-rod YAG single crystals were performed. The influences of voltage, phase difference, and pallet inclined angle on surface roughness, peak-to-valley (PV) value, and cylindricity of the micro-rod YAG single crystals were systematically analyzed, based on which the processing parameters were optimized. The results will not only enhance the understanding of the ultrasonic elliptical vibration principle, but also provide a technical support for precision and high-efficiency machining of micro-rod brittle solids.

Structural design of ultrasonic transducer
The structure of the ultrasonic transducer consists of front cover, back cover, piezoelectric ceramics, and fastening bolt. The function of the front cover was to transmit the energy generated by the piezoelectric ceramics and impedance conversion, which was made of duralumin. The back cover was made of 45# steel, which was used to decrease the energy dissipation. PZT-8 piezoelectric ceramics were used in the design of the ultrasonic transducer, which had low mechanical loss, high piezoelectric constant, and high electromechanical conversion coefficient. The detailed parameters of covers and PZT-8 piezoelectric ceramics were given in Tables 1 and 2, respectively. The longitudinal propagation speed of the sound wave in PZT-8 piezoelectric ceramics is 3560 m/s, so the lengths of the ultrasonic transducer, front cover, back cover, and piezoelectric ceramics are calculated as 92.1 mm, 46 mm, 14 mm, and 20 mm, respectively. The number of piezoelectric ceramic sheets is generally set as an even number; therefore, four piezoelectric ceramic sheets with a thickness of 5 mm are used in the ultrasonic transducer. As shown in Fig. 1, the thickness of the flange ring is set as 1 mm to reduce the influence of the flange ring on the vibration of the ultrasonic transducer.

Structure simulation and vibration analysis of ultrasonic transducer
The 3D model of the ultrasonic transducer was established in SolidWorks, and then, the mode analysis was performed in the workbench. The materials of the front cover, back cover, and piezoelectric ceramics were duralumin, 45# steel, and PZT-8 piezoelectric ceramics, respectively. In this model, the number of the mesh and node are 97,596 and 197,952, respectively. The mesh quality is 0.86, which indicates that the quality of the mesh division is appropriate. The elastic, piezoelectric constant, and dielectric constant matrixes of PZT-8 piezoelectric ceramics are given in Eqs. (1)-(3). (1) where ε 0 is the permittivity of vacuum, which is equal to 8.8542 × 10 −12 F/m. The transducer will output the maximum displacement and efficiency when the voltage frequency is equal to the resonance frequency. The flange ring is set as a fixed constraint. The results of the modal analysis of the ultrasonic transducer are shown in Fig. 2, which indicates that only longitudinal vibration with a frequency of 29,968 Hz occurs in the results of the 10 th mode of the ultrasonic transducer. The vibration frequency of the simulated results is close to the design index of 28 kHz, and the error is approximately 7.03%.

Harmonic response simulation of ultrasonic transducer
Harmonic response analysis is usually used to determine the steady-state response of a linear structure subjected to a sinusoidal load [38,39]. The relationship between the amplitude and frequency under different voltages was analyzed in this work. The range of the analysis frequency was chosen as 29.5-30.5 kHz according to the modal analysis results. The voltage of the piezoelectric ceramic was selected as 100 V. The simulated results of the amplitude and phase are shown in Fig. 3. As shown in Fig. 4, the displacement reaches to a maximum value when the frequency is 29,968 Hz. The output displacement is the same as the phase of the input voltage, and the maximum elongation length of the transducer is 6 μm.

Vibration analysis of ultrasonic transducer
The ultrasonic transducer is shown in Fig. 5a, where the piezoelectric ceramic is clamped between the front and back covers by the fastening bolt. The flange ring is between the front cover and piezoelectric ceramics. The fastening bolt is used to connect with other components. As shown in Fig. 5b, an impedance analyzer (Keysight, E4990A) is used to analyze the impedance characteristics to obtain the resonant frequency. The impedance curves of the two ultrasonic transducers are shown in Fig. 5c and d, respectively. The resonant frequencies are 28,145 Hz and 27,810 Hz, respectively. The anti-resonant frequencies are 29,830 Hz and 29,617 Hz, respectively. When the ultrasonic transducer is in the resonant state, the internal impedance is the smallest, and the phase is equal to 0. The equivalent impedance of the two ultrasonic transducers in resonant state is 105.2 Ω and 86.4 Ω, respectively. When the ultrasonic transducer is in antiresonant state, the internal impedance is the largest, and the phase is equal to π/2. The equivalent impedance of the two ultrasonic transducers in anti-resonant state is 6777.1 Ω and 3650.8 Ω, respectively.
The vibration amplitude under different driving voltages can be determined by measuring the vibration displacement of the resonant frequency. The displacement amplitude for the end face of the front cover is measured by a laser displacement sensor (Keyence, LK-H020) whose repetition accuracy and sampling period are 0.2 μm and 2.5 μs, respectively. The vibration measurement system is shown in Fig. 6, and the voltage parameters and measurement results are given in Table 3.
When the input voltage is 200 peak-to-peak voltage (V p-p ), the vibration displacement curve and its partial enlarged view are shown in Fig. 7a Fig. 7c and d, which indicate that the vibration amplitude is approximately proportional to the voltage.

Structural design and analysis of ultrasonic elliptical vibration transducer
Suppose that a particle vibrates in two mutually perpendicular directions at the same frequency and with a certain phase difference, and the displacement equation of this particle in the X and Y directions is shown in Eq. (4).
The coordinate equation can be obtained by eliminating the parameter t, as shown in Eq. (5).
The inclination angle of the major axis of the ellipse is shown in Eq. (6).
The major axis and minor axis of the ellipse can be calculated by Eq. (7).
When the vibration frequencies of two simple harmonic vibrations perpendicular to each other are the same, a stable . 6 a Overfall vibration measurement system, b drive system, c data acquisition and control systems, and d laser displacement sensor elliptical trajectory can be synthesized. In addition, the synthesized elliptical trajectory can be adjusted by changing the amplitude ratio and phase difference. For ultrasonic vibration transducer, the elliptical trajectory can be adjusted by changing the voltage and phase difference of the piezoelectric ceramics. The diagrammatic sketch of the overall structure of two transducers is shown in Fig. 8a, and the vibration joint is connected with two transducers. Figure 8b shows the modal analysis result of the transducer when the vibration frequency is 27,911 Hz. The voltage amplitude and frequency between the piezoelectric ceramic sheets are 100 V and 27,911 Hz, respectively. The initial phases of the transverse and longitudinal transducers are 0° and 45°, respectively. The amplitudes and displacements of two transducers are shown in Fig. 8c-f, which indicate that the elliptical vibration is well realized.
Ultrasonic elliptical vibration transducer is shown in Fig. 9a, which shows that the elliptical vibration is generated at the end face of the vibrating joint. It is driven by two high-frequency alternating voltages, as shown in Fig. 9b. A digital signal generator (AFG1022) generates sinusoidal voltage with a phase difference, and a two-channel power amplifier (ATA-2042) amplifies the voltage signal and connects it to two transducers. Then, elliptical vibration is generated on the end face of the vibration joint.
The displacement trajectory can be obtained by the endface displacement of the transverse and longitudinal transducers during the vibration process. When the voltage is 160 V p-p , the synthetic trajectories under different phase differences are shown in Fig. 10. Because the resonant frequencies of the two transducers are different and the longitudinal wave attenuates during the vibration process, therefore, there is a small error in the synthesized elliptical trajectory. When the phase difference is 15°, the trajectory vibrates at a major axis of the ellipse, whose major axis and minor axis are 14.6 μm and 5.4 μm, respectively. With the increase of the phase difference, the major axis becomes shorter and the minor axis becomes longer. When the phase difference is 90°, the synthesized trajectory is approximately a circle.
When the phase difference is 45°, the synthetic trajectories under different voltages are shown in Fig. 11. Both major and short axes increase as the voltage increases; therefore, the area of the ellipse becomes larger. When the voltage is low, the shape of the ellipse is irregular. With the increase of voltage, the shape of the ellipse becomes regular and the vibration amplitude of the transducer increases. Therefore, high voltage should be selected in the grinding test.

Experiment
The schematic diagram of ultrasonic elliptical vibration-assisted centerless grinding of micro-rod YAG single crystals is shown in Fig. 12a, where n g , V f , R w , and φ are the wheel rotational speed, feed speed, workpiece radius, and pallet inclined angle, respectively. The ultrasonic elliptical vibration transducer can produce high-frequency vibration at the end face, which drives the rotation of the workpiece. The position of the transducer and plate is fixed. The grinding wheel moves downward in the radial direction during the grinding process. As shown in Fig. 12b, ultrasonic elliptical vibration-assisted centerless grinding tests of microrod YAG single crystals were performed on an ultrasonic elliptical centerless grinding device. The guide wheel was replaced by the elliptical vibration device, so the rotation of the workpiece changes periodically. The grinding platform was built on a precision grinder with a feed resolution of 1 μm, and worktable is an electromagnetic chuck with a movement range of 300 mm × 500 mm. The lifting range and rotation speed of the spindle is 300 mm and 2800 r/ min, respectively. A sliding table with a groove is placed under the plate, whose lifting range is 10 mm. The inclined plane and vibration device are fixed by the electromagnetic chuck. The ultrasonic elliptical vibration device is driven by the signal generator when the voltage is amplified by the power amplifier. Metal-bonded diamond wheel was used in grinding experiment, whose diameter, width, and abrasive size were 200 mm, 15 mm, and 40 μm, respectively. The positional relationship between the work material and UEVCG experimental device is shown in Fig. 12c. The YAG workpiece before grinding is shown in Fig. 12d, whose diameter, length, and surface roughness were 1.2 mm, 12 mm, and 0.91 μm in Ra, respectively. The micro-rod YAG crystals were ground to 1.0 mm by ultrasonic elliptical vibration-assisted centerless grinding. The grinding depth is 1 μm. The influences of the abrasive size, grinding speed and feed speed, and grinding depth on the grinding process of laser crystals have been reported in other papers [11][12][13]. Therefore, this paper focused on the influences of the voltage, phase difference, and pallet inclined angle. As shown in Table 4, the orthogonal test was designed to analyze the ultrasonic vibration parameters on the grinding quality. Before the grinding tests, the wheels were shaped and dressed by electrolytic dressing method. Copper electrode, discharge voltage of 100 V, pulse width of 20 μs, and pulse gap of 50 μs were used during the truing and dressing process. After the grinding, the surface roughness of the work material was measured by a profilometer (Form Talysurf PGI, UK), and the PV value and cylindricity were measured by a laser confocal microscope (OLS3000, Japan).

Results and discussions
The optical image of the surface of the ground micro-rod YAG crystal is shown in Fig. 13a. The measured size and number of the scanning points are 640 μm × 640 μm and 1024 × 1024, respectively. The 3D morphology of the ground surface and its cross-sectional profile are shown in Fig. 13b and c, respectively. According to the points of the cross-sectional profile, the radius and coordinates of the circle center were fitted by using the least squares method, as shown in Fig. 13d. It can be found that the radius and coordinates are approximately 446.3 μm, 333.8 μm, and − 382.3 μm, respectively. For each workpiece, 20 groups of fitted radius and coordinates can be obtained. The minimum value of the circumscribed circle diameter containing 20 circle centers was taken as the straightness error. The roundness of each fitted circle can be expressed by the radius difference between   Tables 5 and 6, respectively. Table 6 shows that the voltage has the greatest influence on the surface roughness and cylindricity, and the phase difference has the smallest influence on the surface and cylindricity. The phase difference and voltage have the greatest and smallest influences on the PV value, respectively. The (8) f Cylindricity = f Straightness(max) + 1 2 f Roundness(max) Fig. 13 a The scanning area of YAG cylinder. b The 3D topography. c The cross section of grinding. d The fitting of cross section optimized grinding parameters are given in Table 7 based on the range analysis results. Under the same grinding parameters, the surface roughness and 3D surface morphology of micro-rod YAG crystals measured in traditional centerless grinding and ultrasonic elliptical vibration-assisted centerless grinding are shown in Fig. 14a and b, respectively. The surface roughness and PV value measured in traditional centerless grinding were 0.91 μm and 9.29 μm, respectively, and they were 0.56 μm and 5.48 μm measured in UEVCG. The results indicated that ultrasonic elliptical vibration improved the surface quality of the micro-rod YAG crystals compared with traditional grinding. In addition, the 3D surface morphologies also showed that the cylindricity in UEVCG was better than that in traditional grinding. This is because ultrasonic elliptical vibration increases the cutting arc length of the single abrasive and enlarges the interaction area between the abrasives and workpiece [40][41][42], which results in the decrease of maximum undeformed chip thickness, grinding force, and wheel wear and in the increase of the brittle-to-ductile transition depth [43][44][45]. In addition, there is no regulating wheel the in UEVCG technology of this work, which can avoid the Fig. 14 Surface roughness and 3D surface morphology measured in a traditional centerless grinding and b ultrasonic elliptical vibration-assisted centerless grinding roundness errors caused by the regulating wheel [31][32][33][34]. Therefore, compared with the traditional grinding, UEVCG effectively reduced the surface roughness and cylindricity error of the ground workpiece.

Conclusions
• The modal, frequency, and harmonic response of the transducer under ultrasonic elliptical vibration are analyzed by using finite element simulation. Then, the mechanical structure of ultrasonic elliptical vibration system was designed and optimized based on the ultrasonic elliptical vibration theory and finite element simulation. • To verify the reliability of the transducer, ultrasonic vibration experiments were carried out to measure the resonance frequency, amplitude, and impedance characteristics of the transducer. The vibration synthesis experiments under different phase differences and different voltages were performed to verify the rationality of the structural design of the ultrasonic elliptical vibration system. The results indicated that the elliptical vibration trajectory was well synthesized by adjusting the phase difference and voltage. • An experimental platform of ultrasonic elliptical vibration-assisted centerless grinding was developed, and UEVCG tests of micro-rod YAG crystals were performed. The influences of voltage, phase difference, and pallet inclined angle on surface roughness, PV value, and cylindricity of the micro-rod YAG crystals were systematically analyzed, based on which the processing parameters were optimized. The surface roughness and PV value measured in traditional centerless grinding were 0.91 μm and 9.29 μm, respectively, and they were 0.56 μm and 5.48 μm measured in UEVCG. The results indicated that ultrasonic elliptical vibration improved the surface quality and cylindricity of the micro-rod YAG crystals compared with traditional grinding.
Author contribution Chen Li contributed in the ideal and paper writing, Xin Wang contributed in the grinding experiment and paper writing, and Yuxiu Hu, Feihu Zhang, Yanquan Geng, and Guijian Xiao contributed in the paper writing and proofreading.