Theoretical and experimental research on 3D ultrasonic vibration–assisted turning driven by a single actuator

In this paper, the peak heights of several turning methods were analyzed theoretically. Based on the results of theoretical analysis, a 3D-UVAT device driven by a single actuator was developed. To validate this design, the theory and experiment of 3D-UVAT have been undertaken. The FEA results show that design is safe, and can achieve obvious displacements in X-axis, Y-axis, and Z-axis. For comparison, CT, UVAT, and UEVT experiments were also carried out. The experimental results show that with the help of ultrasonic vibration, the surface grooves and defects are significantly reduced. This phenomenon is more obvious on machined surface obtained by 3D-UVAT. Three-dimensional surface topography shows that the roughness value Sa obtained by 3D-UVAT is smaller than CT and UVAT. Under some cutting conditions, the roughness value Sa of machined surface obtained by 3D-UVAT is smaller than UEVT. Taguchi experiments show that vc has the most significant impact on the Ra, followed by the fr, then the A. Lastly, the ap has the least impact on the Ra. Thus, the results of theory and experiment proved that the 3D ultrasonic vibration–assisted turning driven by a single actuator has a great potential in improving the quality of machined surface.


Introduction
In recent decades, with the development of manufacturing technology and materials science, the difficult-to-cut materials with superior performance high hardness, high wear, and temperature resistance are widely used [1]. The traditional machining methods are sometimes difficult to satisfy the needs of today's advanced materials. The ultrasonic vibration-assisted turning technology is a machining method that combines traditional turning and ultrasonic vibration, which can reduce cutting force [2][3][4], decrease cutting heat [5][6][7], and improve surface quality [8][9][10] of the workpiece. The ultrasonic vibration turning includes line ultrasonic vibration turning, two-dimensional elliptical ultrasonic vibration turning, and 3D ultrasonic vibration turning, which refers to the vibration pattern during the turning process. The current researches had been verified that many advantages can be achieved by 3D ultrasonic vibration-assisted turning compared with conventional turning [11], line ultrasonic vibration-assisted turning [12], and two-dimensional ultrasonic vibration-assisted turning [13].
Wang et al. [14] developed a 3D ultrasonic elliptical vibration device which combined one-dimensional stepped horn and two-dimensional stepped horn, according to the vibration superposition principle. Different tool nose trajectories can be obtained by adjusting the vibration parameters during the turning process. Building on the research of Lin et al. [15], Lu et al. [16] described a 3D elliptical vibration-assisted cutting device, which was driven by four parallel piezoelectric ceramic plates. During the cutting process, four piezoelectric stacks can generate high-frequency vibration after receiving the control signal. The superposition of the four high-frequency vibration signals will cause the tool nose to generate a high-frequency elliptical orbit. As for the development of non-resonant 3D elliptical vibration-assisted turning apparatus, some scholars presented the structure based on flexure hinge driven by piezoelectric stacks [17,18]. In these devices, the flexure hinges were utilized to increase the output displacement of turning tool. Cui et al. [19] proposed a method of 3D ultrasonic vibration-assisted cutting device, driven by three parallel piezoelectric ceramics. The vibration signal is generated by three parallel piezoelectric ceramics and transmitted to diaphragm platform. Driven by the diaphragm platform, the tool generated a 3D vibration trajectory. Lu et al. [20] developed a resonant T-shaped hinge 3D ultrasonic vibration turning structure, which is driven by three groups of piezoelectric ceramics. The trajectory can be adapted depending on demand. Kurniawan et al. and Hosseinabadi et al. [21,22] developed a 3D ultrasonic elliptical vibration device composed of a longitudinal sandwich piezoelectric ceramic and two bending piezoelectric ceramics. During turning process, three groups of piezoelectric ceramics have similar resonance frequencies to realize the three-dimensional curved surface vibration trajectory of turning tool.
To sum up, the current 3D ultrasonic vibrationassisted turning mechanism faces three vital problems. First, a large number of compliant linkages and complicated structures were designed to isolate input motions, which may lead to slow response speed because of heavy moving inertia and over-sized dimensions. Second, most 3D ultrasonic vibration-assisted turning mechanisms required two or more ultrasonic actuators. Third, the working conditions of horns are special, that is, the same working frequency. To solve the problems mentioned above, a novel 3D ultrasonic vibration-assisted turning system was proposed. The objective of this paper is to characterize a new 3D-UVAT (Three-dimensional ultrasonic vibration-assisted turning) driven by a single actuator. Simulation and comparison experiment are also submitted to validate the satisfaction of this turning system.

Theoretical analysis
The schematic diagrams of the four turning methods are indicated in Fig. 1, which are CT (Common turning), UVAT (Ultrasonic vibration-assisted turning), UEVT (Ultrasonic ellipse-assisted turning) and 3D-UVAT. In order to compare several turning methods, some theoretical analysis was carried out.
Turning process is actually under the squeezing of turning tool, the surface material of workpiece undergoes deformation, and is separated from the workpiece. The topography of machined surface is related to various parameters, such as turning tool nose radius, feed rate, and so on. Figure 2a is a schematic diagram of the maximum residual height of machined surface obtained by CT. (1) In the above equation, H 1 refers to the maximum residual height of workpiece surface obtained by 3D-UVAT (μm); r refers to the turning tool nose (μm). The H 1 further calculated as: The ultrasonic vibration-assisted turning is different from common turning. Ultrasonic vibration-assisted turning has changed the working mechanism of turning process and has a positive effect on improving surface quality of workpiece. Figure 2b is the schematic diagram of the maximum residual height of machined surface by 3D-UVAT.
where H 2 refers to the maximum residual height of workpiece surface obtained by 3D-UVAT (μm). The H 2 further calculated as: In order to compare H 1 and H 2 , the relationship between them can be expressed by the following formula: According to Eq. (7), within a certain range, the difference between H 1 and H 2 is always greater than 0. In other words, the maximum residual height of machined surface obtained by CT is greater than 3D-UVAT. The reduction of surface residual height is conducive to reduce surface roughness. Figure 3 is a schematic diagram of turning width of UVAT and 3D-UVAT. In Fig. 3a, regardless of turning tool wear and other factors, turning width of UVAT can be calculated by the following formula.
where a e1 stands for turning width of UVAT (μm); r represents turning tool nose radius (mm); S means advancing length of turning tool nose in one vibration cycle (μm); a p is cutting depth (mm); F r indicates feed (mm/r); n stands for speed of the lathe spindle (r/min); f is vibration frequency (Hz). The turning width of 3D-UVAT is shown in Fig. 3b.
(11) a e2 = S + 2l 2  where a e2 stands for turning width (μm); R is the crosssectional radius of turning belt (mm). When the feed is the same, the larger the turning width, the finer the bulge between two adjacent turning belts. According the above equation, when the feed rate, cutting depth, and ultrasonic vibration frequency are the same, the width of bulge distributed on machined surface obtained by 3D-UVAT is finer. The reduction of bulge width contributes to the reduction of surface roughness. Most UEVT is equivalent to orthogonal turning, but 3D-UVAT is closer to oblique cutting. Compared with orthogonal cutting, oblique cutting has many advantages. According to Fig. 1, in the process of UEVT, the curl form of chip is closer to the up-curling. However, in the process of 3D-UVAT, the curl form of chips shows obvious lateralcurling. In this case, chips are not only easier to be taken away, but also can prevent the machined surface from being (12) scratched. On the other hand, the lateral outflow of chips helps to reduce the friction resistance between chips and turning tool. The reduction of friction resistance is helpful to prolong the life of turning tool, reduce the generation of built-up edge, and improve the surface quality of machined workpiece. Figure 4 is the three-dimensional schematic diagram of machined surface obtained by UEVT and 3D-UVAT. During UEVT process, the turning belt is perpendicular to feed direction. This turning method has many similarities with CT. However, in 3D-UVAT process, there is an acute angle between the turning belt and feed direction. Figure 5 is the cross-sectional view of two machined surfaces.
where h e is peak height of machined surfaces obtained by UEVT (μm); s e represents the width of turning belt produced by UEVT in a cutting cycle (mm); f w represents the feed of turning tool in Y-axis direction in a vibration period (mm), which can be expressed by (14)  In the above formula, h t is peak height of machined surfaces obtained by 3D-UVAT (μm); s t represents the turning belt width produced by 3D-UVAT in a turning cycle (μm). According to Eq. (19), compared with UEVT, 3D-UEVT can obtain a smaller peak-valley height under the same turning conditions.
According to the calculation principle of surface roughness, the reduction of peak-valley height and bulge width helps to reduce roughness of machined surface. From the above analysis, it can be concluded that compared with CT, UVAT and UEVT, 3D-UVAT is more friendly in improving the machined surface quality.

Analysis of cutting force
In Fig. 6, there is the oblique turning schematic of 3D-UVAT. The cutting force is composed of tangential F t , thrust F h , and lateral F l . F t is along the turning speed direction, F h is along the turning depth direction, and F l is along the feed direction. It is known that cutting force component consists of rake face, flank face, and cutting edge. Experimental observations have shown that a single oblique cutting edge, cutting force components of F t and F h are nearly independent of inclination angle [23], so F t and F h can be expressed as: where F tc and F hc are cutting force that consists of rake face; F te and F he are cutting force that consists of cutting edge; F x is cutting force that consists of flank face; K tc , K hc , K te , K he are cutting force coefficients; A s is elemental chip area; L is cutting edge contact length [24,25]. where β n is normal friction angle; φ n is normal shear angle; α n is normal rake angle; τ s .is shear stress, b is cutting width; h is cutting depth; A is cross section area of cut; r ε is tool nose radius; k r is approach angle; f is feed; ϕ is inclination angle.
where β is friction angle; r t is chip thickness ratio, which is given by cut-thickness divided by chip thickness. The directions and magnitudes of F t and F h are known, the third component F l can be found with respect to the fact that resultant cutting force must lie in the plane normal to tool cutting face in chip flow direction. Combining with chip flow angle η, F t , and F h , the third component that satisfies this condition can be obtained by [24].
According to Fig. 1, in the 3D-UVAT process, the turning tool path is divided into two parts, the main turning and the no-load turning. During no-load turning, there is only friction between the turning tool cutting edge and workpiece. This frictional force is negligible compared to the cutting forces during main turning process.
During 3D-UVAT, the cutting force, normal rake angle, shear angle, friction angle, and chip flow angle all change with the cutting path. Among the parameters mentioned above, the normal rake angle and friction angle are the main factors affecting the cutting force. Actually, in one vibration cycle, the contact time between turning tool and chip is very short, and the direction of turning speed changes from time (26) to time. For simplicity, without considering the contribution of friction angle, only the effect of normal rake angle on the cutting force is discussed. Figure 7 is the schematic diagram of normal rake angle during 3D-UVAT process. Therefore, the normal rake angle in 3D-UVAT can be expressed as: In the above equation, θ is the turning angle of turning tool in one vibration period. Based on the above theoretical analysis, the cutting force of 3D-UVAT can be obtained by numerical calculation in Matlab software.

Design of 3D-UVAT system
The 3D-UVAT system driven by a single actuator is composed of several elements: the ultrasonic vibration actuators, the vibration-carrying rod with spiral grooves, holder, and several plates, as indicated in Fig. 8.
The role of ultrasonic vibration actuator is to convert the high-frequency electrical signal into mechanical vibration signals and transmit it. It includes two parts: a transducer and a horn. The vibration-carrying rod with spiral grooves is used to transmit the ultrasonic vibration signals from ultrasonic vibration actuator to turning tool holder. The connecting plate is used to connect the horn and vibration-carrying rod with spiral grooves. Another function of the connecting plate is to transmit vibration signals to vibration-carrying rod. The horn is fixed on the support plate.
The vibration-carrying rod with spiral grooves is an important part to realize 3D curved trajectory of turning tool. It composes of connecting plate and twists barrels. The realization of 3D curved surface trajectory of turning tool is excited by a set of ultrasonic vibration signals. In the The schematic diagram of normal rake angle during 3D-UVAT process: A and C are the highest point positions of turning tool, B is the lowest point of turning tool The International Journal of Advanced Manufacturing Technology (2022) 121:1173-1190 displacement output structure, considering the deformation characteristics of spring, two twist barrels were selected. When the twist barrel is stretched and compressed, expansion and torsion deformation will occur, as showing in Fig. 8b. Divided vibration-carrying rod into upper twist barrel and lower twist barrel, there are ten right-hand spiral grooves arranged on the sidewall of the upper twist barrel. The force acting on the twist barrel is transmitted and decomposed along tangent direction of the twist barrel and perpendicular to the tangent surface of the spiral grooves. The structure of lower twist barrel is the same as that of upper twist barrel, but the spiral grooves rotate in the opposite direction. There is a holder designed on the upper twist barrel to install turning tool.
When the connecting plate is subjected to periodic displacement in the form of sinusoidal, twist barrels will be tensed and compressed following the movement of connecting plate. When twist barrels are moving, the turning tool will move periodically following twist barrels in the direction of Z-axis. During telescopic deformation of twist barrels, the turning tool also rotates with twist barrels in the planar of O-XY. Therefore, displacement output in three directions can be achieved by this device.
In the established coordinate system, turning tool movement trajectory is expressed by mathematical equations. The structure of upper twist barrel shows in Fig. 8c. It can be seen from the above figure that several spiral grooves divide the whole twist barrel into several rectangular cross-section springs. l refers to the effective length of spring (m), d 2 is the outer diameter of spring (m), d 1 stands for the inner diameter of spring (m), h refers to the width of the spring (m), b is the thickness of spring (m). The deformation of twist barrels divides into torsional deformation in the planar of O-XY and telescopic deformation in the direction of the Z-axis. The stiffness coefficient of spring group formed by several rectangular cross-section springs calculated as: where, G refers to the modulus of rigidity (MPa); d = (d 1+ d 2 )/2 is the middle diameter of spring (m); n stands for the spring number of active coils. In actual calculations, i only involves the first three. Considering that Eq. (31) is very complicated, the following simplified formula is used to replace the above formula in general engineering applications [26].
The cross section of twist barrel is shown in Fig. 8d. The vibration-carrying rod can be divided into several open thin-walled twist barrels, and the number of which is equal to the number of spiral grooves. The ultrasonic signal that transmits to the spiral grooves decomposes into two components, the component force in the tangential The movement of turning tool is excited by the ultrasonic vibration signal applied on vibration-carrying rod with spiral grooves, the relationship between vertical displacement of turning tool and working time can be calculated as: In Eq. (37), U is the displacement of horn in the vertical direction which can be expressed as: In Eq. (38), A is the vibration amplitude (μm), f refers to the frequency of ultrasonic vibration (Hz), and φ 0 is the initial phase angle of ultrasonic signal.
When the ultrasonic vibration signal is transmitted to the spiral grooves, the force is decomposed into a component force F x (t) in the tangential direction of spiral grooves and a component force F y (t) in the direction of perpendicular to spiral grooves, F x (t) is given in Eq. (40): In the above equation, F(t) is calculated by Hooke's law, namely F = Δl ⋅ k.
Thus, F x (t) is further expressed as: In Eq. (42), θ is the rise angle of the spiral grooves. Combining Eqs. (42) and (36), the relationship of torsion angle and time is as follows: The above equation can be calculated: The displacement of the turning tool nose along the X-axis and Y-axis can be calculated by the following: where R L refers to the distance between turning tool nose and central axis of vibration-carrying rod (m). Equations (39), (45) and (46) combined with the feed speed of turning tool can be used to describe the 3D surface movement trajectory of turning tool nose.
Combining the size of vibration-carrying rod, the feed rate, the amplitude and frequency of ultrasonic vibration signal, the trajectory of turning tool nose that can be gotten by the MATLAB software is shown in Fig. 9.
It can be seen from Fig. 9 that the trajectory of turning tool nose is a semi-elliptical 3D space surface. Theoretical calculations can prove that applying a set of ultrasonic vibration signals to this vibration-carrying rod with spiral grooves can realize the 3D space surface movement of tool nose by a single actuator.

Simulation of vibration-carrying rod with spiral grooves
During the process of 3D ultrasonic vibration-assisted turning, the movement trajectory and turning quality are related to the vibration-carrying rod with spiral grooves. To further refine the geometrical parameters and verify the performance of vibration-carrying rod, finite element analysis (FEA) was performed. Fine structure is not only the vibration-carrying rod with spiral grooves that can sustain stress but also can produce satisfactory displacement. In the simulation, 0 displacements were applied to the fixed platform. Input displacement is assumed 20 μm provided by the piezoelectric actuator. In Fig. 10 are the FEA results of vibration-carrying rod with spiral grooves. It can be observed that the maximum deformation is about 14 µm, which occurs at the nose of turning tool. The maximum stress is 15 MPa, which occurs at the spiral grooves, far less than the permissible stress range of the material 65 Mn (around 800 MPa). Thus, the design of 3D-UVAT system is perfectly safe.
Extract the output displacement when the input displacement is 10 μm for the analysis of the 3D-UVAT system. Take a node on the turning tool nose, obtain the maximum displacement in the direction of three coordinate axes (X, Y, Z), which are shown in Fig. 11.
It can be seen from Fig. 11 that there are obvious displacements in three directions. The maximum displacement in the X-axis direction is about 0.51 µm, in the Y-axis direction is about 0.75 µm, and in the Z-axis direction is about 2.6 µm.

Experimental and setups
After the finite element simulation, the structural parameters of 3D-UVAT system were optimized and determined. The Kath-Matic laser vibrometer (KV-HB4525S) was used to measure the actual amplitude of the turning tool, as shown in Fig. 12. Table 1 is the actual output displacement of turning tool nose.
To further investigate the performance of the 3D-UVAT device, a series of experiments were conducted to verify its performance. The material of vibration-carrying rod with spiral grooves is 65 Mn, and other parts are 45 steel. In the experimental setup, the working frequency of ultrasonic horn is 20.031 kHz. The material of workpiece is 304 stainless steel, whose length is 60 mm and diameter is 35 mm. The raw workpiece surface roughness is 3.2 μm. The experimental setup is illustrated in Fig. 13.
The available research conclusions, appropriate cutting speed (v c ), feed rate (f r ), depth of cut (a p ) and amplitude of ultrasonic (A) are helpful to improve the quality of machined surface [28][29][30]. In the UEVT process, the amplitude ratio has an important influence on the turning surface. When the amplitude ratio is 0.9, a better machined surface can be obtained [31,32]. For comparison, several groups of single-factor experiments and Taguchi experimental were carried out. To eliminate the negative effects of experimental errors on the results, each group of experiments was repeated three times. The experimental conditions are summarized in Table 2.

Single-factor experiment design analysis
A comparison between the CT, UVAT, UEVT and 3D-UVAT methods was implemented. In Fig. 14   micrograph of four turning methods, which were magnified two hundred times. As can be viewed, there are obvious irregular scratches and notches on the machined surface obtained by CT. The reason for this phenomenon may be extrusion, bonding and wear between the turning tool and the workpiece. However, machined surface quality obtained by the other turning methods (UVAT, UEVT, and 3D-UVAT) has been significantly improved. In other words, ultrasonic vibration-assisted turning contributes to reducing machined surface defects. The reason for the improvement of machined surface quality is that compared with common turning, ultrasonic vibration-assisted turning changes the turning mechanism. The contact between tool nose and workpiece has changed from continuous to intermittent. The reduction of contact time helps to reduce the cutting force and cutting heat, thereby reducing the friction between turning tool and workpiece, and weakening the bonding between turning tool and workpiece surface.
Although the machined surfaces obtained by several ultrasonic vibration-assisted turning methods are better than that of common turning, there are also obvious differences among several ultrasonic vibration-assisted turning methods. According to Fig. 14, compared with UVAT and UEVT, the material shedding and surface furrow of machined workpiece obtained by 3D-UVAT are significantly reduced. The reduction of these two aspects helps to reduce the roughness of machined surface. The reason for this phenomenon may be that during 3D-UVAT process, the turning tool trajectory is a 3D space curve that periodically changes in the directions of the X, Y, and Z coordinate axes. This special motion trajectory of turning tool will leave regular concaves on workpiece surface. These regular concaves help reduce the friction resistance of workpiece surface [33].
The three-dimensional surface topographies measured by the 3D surface profiler are shown in Fig. 15. In the figure, there is a selection of several surface topography pictures from the experimental results. Figure 15a is the threedimensional topography of machined surface obtained by CT when the v c = 10 m/min, a p = 0.05 mm, f r = 0.08 mm/r. Deep furrows are distributed on the machined surface in Fig. 15a. The height of peak-valley is about 15.1 μm, which is greater than that obtained by the other three turning methods. Figure 15b is the three-dimensional topography of machined surface obtained by UVAT when the v c = 20 m/min, a p = 0.05 mm, f r = 0.08 mm/r, A = 5.7 μm. It is observed that the peak-valley of machined surface obtained by UVAT is about 13.0 μm. There are still furrows on the machined surface, but the width of furrows  is narrower than that of CT. Figure 15c is the threedimensional topography of machined surface obtained by UEVT when the v c = 25 m/min, a p = 0.05 mm, f r = 0.08 mm/r, A = 3.9 μm. Figure 15c shows that the machined surface obtained by UEVT is better than the above two turning methods. The advantages are mainly reflected in fewer furrows, shallower furrow depth and narrower furrow width. Figure 15d is the three-dimensional topography of machined surface obtained by 3D-UVAT when the v c = 25 m/min, a p = 0.1 mm, f r = 0.08 mm/r, A = 4.8 μm. Under this turning condition, the machined surface peak-valley height is 5.7 μm, which is less than the other three turning methods. According to Fig. 15d, we can observe that the workpiece surface obtained by 3D-UVAT not only has no furrow, but also has uniform peak height distribution. In sum, among several turning methods, the 3D-UVAT has obvious advantages in improving the machined surface quality. This conclusion is consistent with the above theoretical analysis of surface residual peak height. The surface morphology of machined workpiece is threedimensional. Therefore, it is more reasonable to use threedimensional evaluation parameters to evaluate the machined surface. There are many three-dimensional evaluation parameters, of which the S a (Arithmetic mean deviation of 3D surface topography) is the most common [34]. Figure 16 shows the S a of machined surface obtained by CT, UVAT, UEVT, and 3D-UVAT.  According to Fig. 16, compared with CT, the UVAT, UEVT, and 3D-UVAT can significantly reduce the roughness of machined surface. In Fig. 16a, the graph shows the relationship between S a and A. With the increase of ultrasonic amplitude, the S a decreases at first and then increases during the 3D-UVAT process. When the A is greater than 4 μm, the surface roughness obtained by 3D-UVAT is lower than other turning methods. However, A is 3 μm, the surface roughness of 3D-UVAT is close to CT. In other words, within a certain range, the quality of machined surface increases with the increase of A. But beyond the proper amplitude range, the quality of machined surface will decrease. Therefore, during the ultrasonic vibration-assisted turning process, the appropriate amplitude contributes to improving the finished surface quality.
In Fig. 16b, there is a graph of the relationship between S a and a p . We can get that the roughness of machined (47) surface obtained by UEVT and 3D-UVAT increases with the value of a p . However, during CT and UVAT process, the effect of a p on surface roughness is shallow. When the a p is 0.06-0.15 mm, 3D-UVAT has more obvious advantages than UEVT in terms of finished surface quality. However, if the a p is greater than 0.18 mm, the quality of 3D-UVAT will drop sharply. Figure 16c shows that, compared with other turning methods, 3D-UVAT can obtain better surface quality during the turning process. Through in-depth study, the curve can be found that as f r increases, the surface quality decreases. The reason for this phenomenon may be that as the f r increases, the turning area in unit time increases, resulting in an increase in turning deformation and plastic side flow [35].
In Fig. 16d, compared with CT and UVAT, the 3D-UVAT method can achieve better surface quality, within the range of experimental conditions. The curves of UEVT and 3D-UVAT show that the greater the v c , the smaller the S a . When v c is lower than 18 m/min, 3D-UVAT can obtain better quality compared with UEVT. Out of the blue, if v c exceeds 20 m/min, compared with UEVT, 3D-UVAT can obtain a better machined surface.

Taguchi experimental design analysis
The cutting parameters regarding their impacts on the response variables R a are presented in Tables 3. It can be observed from Table 4 that v c has the most significant impact on the R a , followed by f r , then the A. Lastly, a p has the least impact on the R a . The delta rankings in terms of the percentage contribution of each of the cutting parameters are given in Table 5, wherein v c was found to have 47.49%, followed by f r with 38.46%, then A having 2.34%. And finally, a p contributed 2.34%. This is similar to the published results in Usman et al. [36].
The mean effect plots presented in Fig. 17 highlighted the surface roughness variation trend under each of the cutting variables' influence. As can be observed from Fig. 17, the R a downtrends with the increase in the v c . Whereas the rise in both f r and the A tends to increase the R a , while a p has a negligible influence on R a .

Conclusion
In this paper, based on the theoretical analysis of several turning methods, a novel 3D-UVAT driven by a single actuator was developed. Simulation and experimental results prove the feasibility of the 3D-UVAT device. The conclusions of the above work are summarized as follows: 1. The machined surfaces obtained by four turning methods are compared from peak-valley height and bulge width.
According to the equation, the peak-valley height of machined surface obtained by 3D-UVAT is smaller than CT and UEVT. Compared with UVAT, the surface bulge width obtained by 3D-UVAT is finer. It shows that the 3D-UVAT method is feasible. 2. The finite element analysis results show that, within the range of simulation parameters, the turning tool nose produces significant displacement in three coordinate axis directions. Under this premise, the maximum stress    1 3 is less than permissible stress. The simulation results prove that this 3D-UVAT device is positive. 3. Compared with CT, UVAT and UEVT, the peak-valley height on machined surface obtained by 3D-UVAT is the smallest. This is consistent with the results of theoretical analysis in this paper. Thus, this 3D-UVAT device has greater potential in improving the quality of machined surface. 4. Taguchi experiments show that v c has the most significant impact on R a , followed by f r , then A. Lastly, a p has the least impact on R a of finished surface obtained by 3D-UVAT.
Author contribution Shiyu Wei: methodology, experiment, validation, editing, and writing-original draft; Ping Zou: resources and supervision; Jingwei Duan and Rui Fang: investigation. All authors read and approved the final manuscript.
Funding This project is supported by the National Natural Science Foundation of China (Grant No. 51875097).
Availability of data and materials All data used in the manuscript are available as submitted.

Declarations
Ethics approval and consent to participate Not applicable.

Competing interests
The authors declare no competing interests.