The effect of ultrasonic amplitude on the performance of ultrasonic vibration-assisted EDM micro-hole machining

A study on ultrasonic vibration-assisted electrical discharge machining (UVEDM) micro-hole, which mainly includes three key designs, was conducted to improve the machining performance of the electric discharge machining (EDM) micro-holes on TC4 titanium alloy. First, a mathematical model for the velocity of the flow field in the interelectrode gap was established. Consequently, the effect of the presence or absence of ultrasonic vibration on the flow velocity was analyzed. Furthermore, based on COMSOL Multiphysics, the two distributions of the flow field and the debris in the interelectrode gap were simulated, and the influence of ultrasonic amplitude on them was analyzed. Finally, the study of UVEDM micro-hole machining with different ultrasonic amplitudes was conducted. The machining performance of the UVEDM micro-hole was found to increase with the increment of the ultrasonic amplitude and the optimal value of 5.22 μm, while the performance decreases when the ultrasonic amplitude goes beyond the optimal value. Compared with EDM micro-hole, the MMR is increased by 2.4 times, and the electrode loss and hole taper are reduced by 56% and 24%, respectively.


Introduction
With excellent corrosion resistance and high specific strength, titanium alloy Ti-6Al-4 V (TC4) is widely used in aerospace, automotive, and medical device applications. However, due to its low thermal conductivity and high toughness, problems (e.g., high cutting forces and severe tool wear) are noted when conventional drilling is used for machining micro-holes in TC4 [1,2]. Electrical discharge machining (EDM), which is a noncontact machining process using pulse discharge, can remove material by the heat generated between the tool electrode and workpiece electrode regardless of material strength and hardness [3,4]. Hence, EDM is more suitable for machining micro-holes because no macroscopic mechanical force appears during the process [5]. Nevertheless, the EDM of micro-holes still faces the problems of accumulating debris at the micro-hole bottom, which will produce abnormal discharge like arcing and short circuits, leading to poor shape accuracy and low material removal rate (MRR) [6].
To enhance debris removal efficiency during EDM of micro-holes, applying ultrasonic vibrations to the tool electrode, the workpiece, or the dielectric has continuously attracted widespread attention. During ultrasonic vibrationassisted electrical discharge machining (UVEDM) of microholes, several other approaches (e.g., using single-notch electrode [7], helical electrode [8], and planetarily moved electrode [9]) have been taken to improve the machining performance (e.g., efficiency, tool wear, and hole taper). For example, Zhao et al. [7] adopted a single-notch electrode to increase the capacity of debris removal and developed a mathematical model of the dielectric pressure when ultrasonic vibration was applied to the electrode. In addition, they pointed out that higher MRR and smaller electrode wear rate could be achieved with the increase of ultrasonic amplitude. Moreover, Yu et al. [9] effectively reduced the viscous resistance of dielectric by applying both planetary movement and ultrasonic vibration to the electrode, so the debris and bubbles can be easily removed from the discharge gap. Xing et al. [10] investigated the effect of ultrasonic amplitude on the performance of UVEDM micro-hole machining, and the experimental results showed that the higher the ultrasonic amplitude, the higher the MRR, and the smaller the electrode wear rate and hole taper.
Directly observing the machining process is difficult in the UVEDM of micro-holes. Therefore, some scholars have used numerical simulation to simulate the variations of the flow velocity and the pressure and debris distribution in the interelectrode gap (the part between the electrode and the workpiece, and including the bottom gap and side gap) during micro-hole machining. Liu et al. [11] established a mathematical model of flow velocity to investigate the effects of ultrasonic amplitude, ultrasonic frequency, and aspect ratio of micro-holes on debris distribution and movement using simulation software Fluent. The simulation results prove that electrode vibration is beneficial to remove the debris in the interelectrode gap, which improves the machining performance. Moreover, Li et al. [12] conducted a fluid simulation on the effect of the ultrasonic circular vibration (UCV) electrode on flow field distribution and discharge debris movement. The simulation results show an increase in both the flow velocity and the efficiency of debris removal, which is beneficial to decreasing the abnormal discharge and improving the machining performance of the micro-hole. Mastud et al. [13] investigated the variation of the flow velocity of the dielectric and debris distribution during UVEDM microhole array machining. The simulation results show that the flow velocity of the dielectric and the efficiency of debris removal increase with an increase in ultrasonic amplitude and ultrasonic frequency. It is an effective method to enhance the machining performance of the micro-hole by improving the efficiency of the debris removal in UVEDM micro-hole machining [14]. The presence or absence of ultrasonic vibration has different influences on the distribution state of the dielectric and the debris. Therefore, exploring the distribution state of the dielectric and the debris in the interelectrode gap is essential.
In this paper, a mathematical model of flow velocity of the dielectric considering electrode vibration is first established. Based on this, the turbulence model will be developed to investigate the flow velocity law and the debris distribution law with various ultrasonic amplitudes using COMSOL Multiphysics. Finally, experiments are conducted on the machining performance (e.g., MRR, electrode wear, and micro-hole taper) during UVEDM of micro-holes on titanium alloy TC4.

Geometric model
In conventional EDM of micro-holes, the electrode usually just rotates around its axis. However, in this study, an additional axial ultrasonic vibration has been applied to the electrode. During machining, the debris is generated at the bottom of the micro-hole by electric discharge and is then discharged from the interelectrode gap by the dielectric flow. The debris is usually spherical with a diameter ranging from 90 to 180 nm [13]. Thus, in the geometric model of this study, 10 5 spherical debris with an average diameter of 130 nm are considered randomly distributed at the bottom of the micro-holes. The machining process of micro-holes by UVEDM is illustrated in Fig. 1.
Three assumptions have been first put forward before simulation: (a) The dielectric is incompressible and initially motionless; (b) the process is adiabatic with no thermal exchange or heat dissipate; and (c) the effect of pulse discharge on the flow velocity is not considered.

Kinematic model of the flow field
In UVEDM of micro-holes, the debris generated in the electrode-workpiece gap is removed due to the drag force caused by the dielectric resistance and the lift force generated by the velocity field of the dielectric [11]. Hence, the velocity of the flow field significantly affects the removal behavior of debris in processing micro-holes by UVEDM. In this article, both ultrasonic vibration and rotational motion are placed on the electrode. For any point o in the dielectric, the synthesis velocity v o (t) can be expressed as the vector sum of the vibration velocity v 1 (t) and the linear velocity v 2 (Fig. 2). Note that the direction and magnitude of the vibration velocity v 1 (t) constantly change with time periodically, and the same is true for synthesis velocity v o (t). Based on this model, the kinematic models of the flow field under pure electrode vibration and pure electrode rotation are established separately.

Processing with pure ultrasonic vibration on electrode
The schematic diagram of processing micro-holes with pure ultrasonic vibration on the electrode is shown in Fig. 3.  The electrode vibrates up and down along its axis with the given ultrasonic amplitude A u and ultrasonic frequency f. Under pure ultrasonic vibration conditions, the electrode trajectory y u (t), velocity v u (t), and acceleration a u (t) can be expressed as: In the side gap, the velocity of the flow filed v 1 (t) can be expressed as [15]: where L 1 is the length of the electrode immersed in the dielectric, q 1 is the flow rate which refers to the volume of liquid passing through the flow passage section per unit time, and A 1 is the area of the flow passage section. Therefore, q 1 and A 1 can be expressed as Eqs. (3) and (4), respectively. When the velocity v u (t) direction is similar to the pressure difference ΔP 1 , a negative sign is used in Eq. (3); otherwise, a positive sign is used: where μ is the viscous resistance of the dielectric, d is the electrode diameter, D is the micro-hole diameter, h is the width of the side gap, ΔP 1 is the pressure difference between inlet pressure P 1 and the bottom pressure P 2 for micro-holes, and the inlet pressure P 1 is zero. Therefore, the pressure difference ΔP 1 could be expressed as Eq. (5).
where A e is the radial cross-sectional area of the electrode, and m is the mass of the free overhanging part (refers to the part from the tip of the horn to the end of the electrode) of the electrode. Thus, the expression of L in Eq. (3) is as follows: but L 1 is usually larger than A u , and L can be considered equal to L 1 . When the electrode vibrates, the velocity v i direction is always opposite to that of pressure difference, so the sign in Eq. (3) needs to be negative, and Eq. (7) can be obtained by combining Eqs. (1)-(6).

Processing with pure rotation on the electrode
The geometrical model of the flow field in Fig. 1 is expanded along the circumferential direction and projected onto the XY plane, and the model of the electrode pure rotation in Fig. 4 can be obtained.
The above model is a flow field model of the parallel plate, so its flow rate q 2 and area of the flow passage section A 2 can be expressed in Eqs. (8) and (9) [15]: where v r is the linear velocity of the electrode surface, q 2 is the flow rate (refers to the volume of liquid passing through section A 2 per unit time), and A 2 is the area of the flow passage section (refers to the area of the ring with width h). ΔP 2 in the first term on the right side of Eq. (8) is zero, and the velocity v 2 of the flow field between the interelectrode gap in the electrode pure rotation can be expressed as Eq. (10): where n is the rotational speed of the electrode.
In UVEDM micro-hole processing, the electrode has both ultrasonic vibration and rotational motion, so the scalar value of velocity v o (t) can be expressed as: An example calculation of Eq. (11) was performed using MATLAB to find the relationship between ultrasonic amplitude A u and the velocity v 1 (t) of the flow field, where the ranges of the ultrasonic amplitude A u are from 0 to 10 μm, and the time t is in the range of two cycles of ultrasonic vibration. The remaining variables in Eq. (11) are selected as theoretical values, and we align these variables (such as D, d, n, etc.) with those in the experiment in order to better match the calculated results with the actual results. The result of the example calculation is shown in Fig. 5.
The velocity v o (t) of the flow field increases with the increase of the ultrasonic amplitude A u (Fig. 5). As time t increases, the velocity v o (t) of the flow field changes periodically instead of blindly increasing or decreasing.

Simulation model and boundary condition
The three-dimension simulation model is shown in Fig. 6. The electrode diameter d and the micro-hole depth H are set at 200 and 300 μm, respectively. The interelectrode gap h plays a crucial role in debris removal and is determined by the four groups of the pre-experiment (the pulse width t i is 0.6 μs, the pulse interval t o is 0.6 μs, the peak current i e is 6 A, and voltage u o is 120 V), and the results are shown in Fig. 7.
The average of the four micro-hole diameters is 230 μm; therefore, the interelectrode gaps (both the side and bottom gaps) are set as 15 μm. The size of the three-dimensional simulation model and other main parameters is listed in Table 1.
In terms of the physical fields of the fluid flow, all the motion conditions of the electrode will be set on the electrode wall and the electrode bottom, and the micro-hole wall and the micro-hole bottom will be set on a fixed wall. The ring top of the model is the pressure outlet, which has no velocity inlet and with zero as the initial velocity of the fluid. In the physical field of particle tracing for fluid flow, the micro-hole bottom is the debris inlet, which is used to generate the debris, and the ring top of the model is the outlet for debris to be discharged from the model; all other walls are set to bounce wall. The debris with a diameter of 90 nm-180 nm occupies the main part of all the debris, and its shape is mostly spherical [13]. To simplify the simulation model, the debris is considered to be spherical with a diameter of 130 nm, and its density is set the 4510 kg/m 3 similar to that of the TC4 workpiece material. In addition, the debris is randomly distributed at the micro-hole bottom, the number of the debris is set to 10 5 , and the debris velocity is defined as following the flow velocity field.
The debris removal process is divided into two stages: the first stage is the debris removal from the bottom gap of the micro-hole to the side gap, and the second stage is the debris removal from the side gap of the micro-hole to the outside. Figure 8 shows two sections (sections A and B) set on the model shown in Fig. 6. Section A is set at a distance of 7.5 μm from the bottom of the model to determine the working fluid in the bottom gap under different electrode motion states. To accurately observe the law of the changing of fluid velocity with the X-and the Y-coordinates, the flow field distribution is influenced by making two straight lines (lines X and Y) in section A. Section B was set at 150 μm from the bottom of the model and was used to calculate the number of debris passing through this position under different electrode motion states.

Effect of ultrasonic vibration amplitude on flow field
The average velocity of the dielectric in the bottom gap is shown in Fig. 9, and the x-and y-direction flow velocities are shown in Fig. 10. Moreover, Fig. 9a shows the average velocity under the condition that A u = 0 (the electrode without the ultrasonic vibration) is the smallest in all results and is closely near to 0. Similar results are also shown in Fig. 10a and b. Therefore, removing debris from the bottom of the micro-hole is difficult, which will affect the stability of the micro-hole machining, and it even halts the hole-making processing if the debris accumulates to a certain extent [12]. Furthermore, Fig. 10b-f, shows that the average velocity increases with the ultrasonic amplitude A u , and the velocity difference gradually becomes similarly more pronounced.
The maximum velocity of the dielectric has appeared at the 1/2 radius of the micro-hole under the condition that A u = 0, and the maximum velocity only occurs near the micro-hole wall under the condition that A u is larger than 0 (Fig. 9). In addition, Fig. 10 shows that both the velocity component of the x-and y-directions increase with the ultrasonic amplitude A u . The greater the ultrasonic amplitude A u , the larger the maximum flow velocity, and the greater the velocity difference. The debris prefers to move from a low-to a highvelocity area. Thus, the increase of the ultrasonic amplitude A u is beneficial to debris removal from the micro-hole bottom to the side gap.
As the depth of the micro-hole increases, the viscous resistance of the dielectric in the side gap increases gradually,  =120V   Fig. 7 The results of the pre-experiment which makes it more difficult to remove debris to the outside [11]. The high-frequency alternating pressure waves in the dielectric could be generated by the electrode vibration, thereby changing the flow condition of the dielectric and reducing the viscous resistance of the dielectric. The distribution of the flow field in the side gap over time is shown in Fig. 11. The solution time is ten cycles of the ultrasonic vibration, and five cycles of the ultrasonic vibration are selected here. In addition, Fig. 11 shows the velocity and pressure components in the z-direction in the dielectric do not change significantly under the condition of A u = 0, while the value of both is close to 0. Furthermore, Fig. 11a shows that the velocity component in the z-direction changes alternately with time, and the velocity amplitude increases continuously with the increase of the ultrasonic amplitude A u , which is consistent with the results shown in Fig. 5. In addition, the z-direction pressure is shown in Fig. 11b and also has similar laws to the velocity component in the z-direction. The ultrasonic vibration not only drastically makes the velocity and pressure of the dielectric change but also effectively facilitate the removal of the debris from the side gap to the outside.

Effect of ultrasonic vibration amplitude on removed debris
In EDM micro-hole machining, too much debris accumulated at the bottom of the micro-hole will cause abnormal discharge, which will not only reduce the machining The distribution of the debris at two cycles of the ultrasonic vibration (~ 46 μs) under the conditions of different ultrasonic vibrations is shown in Fig. 13. When the ultrasonic amplitude A u is 0, almost no z-direction velocity exists in the dielectric, which cannot provide sufficient driving force for the debris to be discharged from the side gap to the outside. Thus, the debris is more likely to accumulate at the bottom of the micro-hole, where little debris passes through section B. This will increase the secondary and concentrated discharge at the bottom gap of the microhole, thereby reducing the performance of the micro-hole machining [16]. As the ultrasonic amplitude A u increases, the number of debris passing through this section increases and reaches a maximum value of 3332 at A u = 13.32 μm. This indicates that the larger the ultrasonic amplitude, the higher the efficiency of debris removal.

Experimental validation
The previous chapter discussed the effect of the ultrasonic amplitude on the flow fluid in the inter-electrode gap, and the simulation results show that increasing the ultrasonic amplitude A u is beneficial for improving the efficiency of debris removal. To further verify the effect of ultrasonic vibration on EDM micro-hole machining, an experimental study on micro-hole machining under different ultrasonic amplitude A u will be conducted.

Experimental details
The UVE micro-hole machining system in this study is shown in Fig. 14. Taking SE-WK008 EDM micro-hole machine as the basic motion platform, a self-developed ultrasonic transducer is fastened to the vertical column of   Fig. 12 The number of debris passing through section B under different amplitude A u the micro-hole machine, and the tungsten electrode with a diameter of 200 μm is clamped on the front end of the horn. In the experiment, the method of positive polarity processing was used to process the through holes in the TC4 with a thickness of 2 mm. In addition, the electrical machining parameters (the pulse width t i is 0.6 μs, the pulse interval t o is 0.6 μs, the peak current i e is 6 A, and the voltage u o is 120 V) during the entire experiment are consistent with that of the pre-experiment. In addition, the electrode wear that the electrode feed stoke is set to 2.7 mm to realize through-hole machining. The working frequency of the transducer measured by the impedance analyzer PV90A is 43,280 Hz (Fig. 15), and the ultrasonic amplitude under different ultrasonic generator power is measured by the vibrometer LV-FS01. First of all, the need to ensure that the axis of the sensor head and the transducer is coaxial, and the need to repeat four times to replace the measuring point after each measurement is completed, and take the average of the results; the result is shown in Fig. 16. In the experiment, the ultrasonic amplitude is controlled by adjusting the ultrasonic generator power. The detailed experimental parameters are shown in Table 2.

Time ( s)
The MRR is directly related to the machining efficiency of the micro-hole. The relative tool wear rate (RTWR ) affects the micro-hole accuracy, especially in the blind hole processing. The micro-hole taper θ is one of the important indicators to characterize the shape accuracy of the micro-hole. Therefore, the MRR, RTWR , and micro-hole taper θ are selected to be the evaluation indexes of the machining performance of the micro-hole. The MRR, RTWR , and micro-hole taper θ could be expressed in Eqs. (12)- (14), respectively. Moreover, Fig. 17 is a schematic cross-sectional view of the micro-hole: where D in is the diameter of the micro-hole inlet, D out is the diameter of the micro-hole inlet outlet, H is the micro-hole depth, t is the machining time, r is the electrode radius, and h is the length of the electrode wear.   Fig. 17 The schematic diagram of micro-hole cross-section

Analysis of the micro-hole machining performance
When ultrasonic amplitude A u is 0 μm, Fig. 18 shows that the MRR is only 1.83 × 10 5 μm 3 /s, but the RTWR and microholes taper θ are as high as 14.7% and 0.3°, respectively. As the ultrasonic amplitude A u increases, the MRR increases accordingly and reaches the maximum value of 6.57 × 10 5 μm 3 /s at A u = 8.51 μm, which is 2.6 times higher than the MMR at A u = 0 μm. However, the RTWR reaches the minimum value of 6.5% at A u = 5.22 μm, which is only about half of the RTWR with A u = 0 μm. In addition, Fig. 18c shows that when the A u is larger than 1.86 μm, the micro-hole taper θ is maintained at about 0.2°, and the change in micro-hole taper θ is not significant with the increase of A u . Based on comprehensive consideration, the micro-hole machining performance with ultrasonic amplitude A u is believed to be at its best at 5.22 μm.
The simulation results show that the flow velocity and pressure of the fluid in the interelectrode gap are almost zero, so debris can easily accumulate at the bottom of microholes, which increases the probability of the abnormal discharge, thereby reducing the MRR and increasing the RTWR . The accumulation of debris at the inlet of the micro-hole causes the probability of the side discharge to increase, so the micro-holes taper θ will also become larger. However, the increase of ultrasonic amplitude A u can increase the difference in the velocity and pressure of the interelectrode gap fluid, thereby improving the efficiency of debris removal and enhancing the stability of micro-hole machining. The probability of the abnormal discharge and side discharge is reduced, which could increase the MRR and decrease the RTWR and micro-hole taper θ. A decrease in the MRR and an increase in the RTWR with A u = 13.32 μm, which are not in line with the simulation results. In addition, Fig. 19 shows that the quality and roundness of the micro-holes inlet and outlet, as well as the number and size of the attachments (the arrows in the Fig. 19) at the inlet and outlet increase and then decrease as the ultrasonic amplitude A u increases, the best quality and roundness of the micro-hole inlet and outlet are obtained with A u = 5.22 μm, and the worst for A u = 13.32 μm.
Two reasons for the above situation exist. On the one hand, the amplitude at the end of the electrode exceeds the width of the maximum discharge gap, which causes the electrode to contact the workpiece in a very short time, resulting in a short circuit between both [17,18]. Dong et al. [18] analyzed the situation from the perspective of discharge energy and further explained that when the ultrasonic amplitude is greater than the width of the maximum discharge gap, it will not only cause a short circuit between the electrode and the workpiece. However, the effective discharge energy is greatly reduced, which is the main reason for the lower MRR. On the other hand, when an excessively large ultrasonic amplitude is applied to the slender electrode, it is prone to radial swinging, and then a short circuit between the electrode and the micro-hole wall inevitably occurs. The state is undesirable in EDM micro-hole machining, which not only negatively affects the MRR and the RTWR but also damages the quality of the micro-hole inlet [19]. The frequent contact between the electrode and the workpiece causes the electrode to retreat frequently. Although the retreat time of the electrode is short each time, it still reduces the MRR. In addition, the occurrence of a short circuit between the electrode and the workpiece will also cause the RTWR to increase and the quality of the micro-hole inlet to decrease sharply.

Conclusions
The study was conducted to investigate the impact of the ultrasonic amplitude on the performance of EDM micro-hole machining. Under the ultrasonic vibration of the electrode, a mathematical model of the flow field in the interelectrode gap in EDM micro-hole machining was established, and the CFD simulation of the interelectrode gap fluid was performed. A single-factor experimental study of EDM microhole machining under different ultrasonic amplitudes was conducted, which verified the rationality of the results of the mathematical model and the CFD simulation. The main conclusions from this study are as follows: 1. The flow velocity of the fluid in the interelectrode gap is related to ultrasonic amplitude, ultrasonic frequency, and rotational speed; is not constant when the electrode vibration is used; and alternately changes over time.
In addition, the amplitude of the flow velocity also increases with the increase of the ultrasonic amplitude. 2. The electrode vibration can increase the difference in the flow velocity and pressure of the fluid in the interelectrode gap, and increasing the efficiency of debris removal is also beneficial. Under the condition of ultrasonic amplitude A u = 13.32 μm, the number of chips passing through section B is 61,535. Compared with traditional EDM micro-hole processing, the time for chips to reach section B for the first time is reduced by about 80%. 3. In this study, the micro-holes machining performance with ultrasonic amplitude A u is best at 5.22 μm. Currently, the MRR is increased by 2.44 times, and the RTWR and micro-holes taper θ were reduced by 56% Fig. 19 The micro-hole inlet morphology under ultrasonic amplitude with a A u = 0 μm, b A u = 1.86 μm, c A u = 3.35 μm, d A u = 5.22 μm, e A u = 8.51 μm, f A u = 13.32 μm and 24%, respectively, compared with A u being 0. The excessive ultrasonic amplitude cannot be used in the EDM micro-holes process; otherwise, it will cause some unfavorable factors (e.g., the short circuit between the electrode and the workpiece), which will eventually lead to a decrease in MRR.
Author contribution All the authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Peng Zhang. The first draft of the manuscript was written by Peng Zhang, and all the authors commented on previous versions of the manuscript. All the authors read and approved the final manuscript.
Funding Financial support for this work provided by the National Natural

Declarations
Consent to participate It is confirmed that all the authors are aware and satisfied of the authorship order and correspondence of the paper.