Modeling and experimental analysis of ultrasonic vibration drilling force prediction model for tiny small holes in high body fraction aluminum-based silicon carbide composites

: A new theoretical mechanical model was proposed by the study, which takes into account ultrasonic vibrations and undeformed cutting thickness. By analyzing the kinematic characteristics of ultrasonic vibration, the undeformed cutting thickness under ultrasonic vibration drilling is calculated, and the theoretical analysis of axial forces and torques is carried out by using the model with simulation software. The results show that both the theoretical drilling force model and the 3D drilling simulation can predict the average axial force and average torque of SiCp/Al composite drilling process more accurately, and the simulation prediction results are better than the prediction results of the theoretical model. The average error of the simulated axial force is 7.11% and the average error of the torque is 7.19%, and the average error of the theoretical axial force is 10.49% and the average error of the torque is 13.05%.


Introduction
SiCp/Al, as one of the metal matrix composites, has excellent characteristics of both aluminum alloy and SiC, and has good physical properties such as low density, high specific strength and specific stiffness, wear and high temperature resistance, which can be widely used in aerospace, defense industry, electronic communication, medical equipment, optical engineering and other fields [1].performance of SiCp/Al composites.Chen et al [15] conducted ultrasonic vibratory drilling (UAD) experiments to investigate the chip breaking mechanism and chip breaking parameters and found that UAD is more advantageous than conventional drilling with more effective chip breaking.In reference to the ultrasonic-assisted drilling experiments conducted by Shao et al Error!Reference source not found.on CFRP/Ti composites, it was found that the drilling force and torque were reduced by about 20% and 32%, respectively, under ultrasonic drilling conditions, and the hole diameter accuracy and hole surface quality were significantly improved.Liang et al Error!Reference source not found.analyzed the vibration drilling process of Ti-6Al-4V by simulation and experiment, and the results showed that ultrasonic vibration machining could not only reduce the surface roughness of the hole wall and the exit burr height, but also significantly improve the wear resistance of the tool.He, J et al [18] used ultrasonic drilling tests on SiCp/Al composites and found that ultrasonic vibration drilling can effectively reduce drilling forces and torque compared to conventional drilling.Ultrasonic drilling is a composite processing method combining ultrasonic processing and general drilling processing.
Under the high frequency vibration of ultrasonic tool shank, the drill bit and the workpiece material are constantly in contact and separated at a certain frequency.In recent years, scholars at home and abroad have done a lot of research in order to get the suitable ultrasonic vibration processing parameters.Wang, J [19] et al. performed rotary drilling machining tests on SiCp/Al composites and the experimental results on quartz glass, sapphire and C/SiC composites verified the ability of the model to predict correctly and the critical cutting forces independent of machining conditions such as feed rate, spindle speed, material properties and even tool wear.M. A. Kadivar et al [20] performed ultrasonic vibratory drilling of 5% volume fraction SiCp/Al composites using coated drills and found that the burr height and burr width under ultrasonic vibratory drilling were significantly reduced by 83% and 24%, respectively, compared to those under normal drilling.ZAI P H et al [21] established a theoretical model of burr height for ultrasonic-assisted high-speed drilling of Ti-6Al-4V microhole exit considering the acoustic softening effect and performed experimental validation, which showed that amplitude and spindle speed were negatively correlated with burr height, and feed rate was positively correlated with burr height.
Wang, J et al [22] established a new RUD method using longitudinal and torsional coupling (LTC) vibration to further reduce cutting forces, and machining tests showed that LTC-RUD can reduce cutting forces by more than 50% compared to conventional rotary ultrasonic drilling (Con-RUD) with longitudinal vibration.Li, Z et al [23] investigated the application of rotational ultrasonic machining technology for advanced composites and developed models for abrasive penetration depth and axial drilling force for rotational ultrasonic drilling (RUD).The results of validation tests on C/SiC composites showed that the prediction error was within 15%.Huang, S et al [24] studied the drilling of thin-walled workpieces with high volume fraction of silicon carbide particle-reinforced aluminum matrix (SiCp/Al) composite and showed that there was a significant difference in the variation of drilling force when drilling thin-walled workpieces compared to thick-walled workpieces.
Compared with other drilling methods, ultrasonic vibratory drilling can significantly improve the machining efficiency, reduce tool wear, and maximize the control of defect formation, thus significantly improving the machining accuracy, but a large number of studies have not conducted theoretical analysis of the drilling force mode, and domestic and foreign scholars mainly focus on small-hole drilling of SiCp/Al composites with low volume fraction, and there are fewer studies on high volume The research on ultrasonic vibration drilling of SiCp/Al composites with high volume fraction is less.
Therefore, this paper investigates SiCp/Al composites using ultrasonic drilling processing technology, and establishes a longitudinal torsional ultrasonic vibration drilling force model by considering the undeformed cutting thickness under ultrasonic drilling, and verifies the model through simulation and experimental studies.

Kinematic analysis of the cutting edge of longitudinal-torsional composite ultrasonic vibration drilling
In conventional drilling, the drill bit rotates with the tool shank and is accompanied by a feed motion at a certain speed, with the Z-axis direction being the direction of the feed motion.The longitudinal-torsional compound ultrasonic vibration drilling drill not only rotates and feeds, but also performs high-frequency longitudinal vibration and torsional vibration in the Z-axis direction.
Take one unit of the drilling edge and establish the equation for the trajectory of the conventional drilling cutting edge.
The following equation is shown: r f is the feed per revolution (mm/r), R is the tool radius (mm), n is Spindle speed(r/min), f is Ultrasonic vibration frequency (Hz).
The equation of the main cutting edge trajectory corresponding to conventional drilling during longitudinal torsion ultrasonic vibration drilling is： A is Longitudinal vibration amplitude(μm), B is Torsional vibration amplitude(μm).
The conventional drilling trajectory and the longitudinal torsional ultrasonic vibration drilling trajectory were plotted using MATLAB analysis software, and the trajectory diagram is shown in Figure 1.Among them, Fig. 1   From Fig. 1 (a), it can be seen that longitudinal torsional ultrasonic vibration drilling has obvious periodic changes compared with conventional drilling, and the longitudinal vibration is more obvious compared with torsional vibration, which is because when the spindle speed exceeds the critical speed of torsional vibration, the drill bit will not show the rotation phenomenon, but will achieve the effect of fast and slow, and this effect will make the SiC particles in the workpiece more easily broken, so as to achieve the effect of reducing This effect will make the SiC particles in the workpiece more easily broken, so as to reduce the cutting force.When the SiC particles are broken more severely, there will not be serious chipping at the hole exit due to the dislodging of large particles.As can be seen in Fig. 1 (b), when the longitudinal amplitude is 0.8 μm, no trajectory crossover occurs, i.e., the undeformed chip thickness is greater than 0, and the separation of the tool from the workpiece has not yet been achieved.

Analysis of undeformed chip thickness variation in longitudinal torsion compound ultrasonic vibration drilling
In longitudinal torsional ultrasonic vibration drilling, the cutting thickness of the drilling process changes because the trajectory of the cutting edge changes periodically.The change in cutting thickness has a large impact on drilling force and hole quality, so this section analyzes the change in cutting thickness of longitudinal torsional compound ultrasonic vibration drilling.
From equation (2), the trajectory of one main cutting edge 1 Z is as follows: ( ) When the drill has rotated half a circumference, the other main cutting edge rotates to that main cutting edge position and the trajectory on the other main cutting edge  As can be seen from the graph, the thickness of undeformed chips in longitudinal torsion ultrasonic vibration drilling varies periodically with time, and the trajectory of the two cutting edges appears to overlap, and the closed area between the two cutting edges is the theoretical chip area, and the overlap phenomenon becomes more serious when the ultrasonic amplitude increases or the feed per revolution decreases, and the theoretical chip area becomes smaller, making it easier for the chips to fracture and be discharged from the hole.When the ultrasonic amplitude is reduced or the feed per revolution is increased, the overlap phenomenon disappears and the theoretical chip thickness is constantly greater than 0. However, the chips periodically appear as weak points and are more likely to fracture, the occurrence of which is mainly related to the deformation of the chips and the impact of the tool.
In order to provide a clearer analysis of the undeformed chip thickness for longitudinal torsional ultrasonic vibration drilling, the theoretical drilling undeformed chip thickness was calculated using equations ( 3) and (4)： ( ) ( ) p a is thickness of undeformed chips(mm).

Modelling of longitudinal-torsional composite ultrasonic vibration drilling forces
The thickness of undeformed chips in drilling processes varies cyclically with time and the cutting forces also change with the thickness of the undeformed chips, in order to investigate this variation, a modelling analysis of the cutting forces is required.In the drilling of small holes in SiCp/Al composites, the addition of SiC particles results in a significant improvement in the material properties, but also makes drilling more difficult and causes severe tool wear.As the presence of SiC particles can make the ordinary drill bit processing wear more serious, this paper uses the diamond PCD drill bit for drilling SiCp/Al composites, but because the structure of the PCD drill bit is different from the ordinary twist drill bit, many scholars have only studied the drilling force of the ordinary twist drill, and less research has been done on the drilling force of the PCD drill bit, so this paper establishes a cutting force model considering the tool shape of the cutting force model.
In the drilling process, the axial forces and torques consist mainly of the forces on the main cutting edge and the cross cutting edge.Therefore, in order to analyze the axial forces and torques on the drill throughout the process, the geometry of the main cutting edge and the cross-cutting edge needs to be analyzed.The cutting of the drill is divided into two main categories: beveled cutting with the main cutting edge and orthogonal cutting with the cross cutting edge.In ultrasonic vibration machining, the workpiece is separated from the tool so that the drill bit has an impact on the workpiece, which in turn creates an axial impact force.As shown in Fig. 3, Fig. 3 (a), (b) and (c) represent the bevel cutting machining of the main cutting edge, the orthogonal machining of the transverse edge and the ultrasonic impact of the drill bit, respectively.

Fig.3 Geometric description of the cutting theory
In longitudinal torsional ultrasonic vibration drilling, the cutting thickness of the drilling process is changed due to the cyclic change in the machining trajectory of the cutting edge, which will have a very significant effect on the cutting forces.The longitudinal-torsional composite ultrasonic vibration drilling force modeling will be derived using the microelement method.
Firstly, the general drilling modeling was carried out using the micro-element method to obtain the model equations [25][25]， [26]: F is axial force, T Torque; C F , C T represent the cutting force and torque on the cross edge respectively; l F , l T represent total cutting force and torque; nd  , nd  represent dynamic friction angle and dynamic cutting front angle respectively, nd  is the dynamic shear angle, n  is the normal phase friction angle, AB k is the shear flow stress along the shear surface; r is the distance of any cutting unit M from the axis on the cutting edge;  is the angle between the two velocities perpendicular to the cutting edge plane; d is the diameter of the drill bit; ' d is cross blade length;2t is drill core thickness.
Due to the longitudinal vibration in the axial direction and the torsional vibration in the tangential direction, the axial and tangential velocities of each unit are constantly changing, so the angle between the cutting speed and the tangential velocity is also constantly changing.It is known by equation ( 3) that the axial velocity in the longitudinal torsional ultrasonic vibration drilling process is and the tangential velocity are equations : ( ) So the angle between axial and tangential velocity in longitudinal torsion ultrasonic vibration drilling process is: '  ,the angle between axial velocity and tangential velocity.
The normal forward angle during its processing is: ' n  , the normal forward angle.
For longitudinal torsional ultrasonic vibration drilling, the uncut thickness for machining any unit on the main cutting edge is the following equation: ' c h , undeformed cutting thickness.
For a cell with dl on the cutting edge, the shear force ' S dF can be expressed by the following equation: / 2 sin 2 sin 2 30 / cos ' ' ' sin sin ' S dF is shear force.
The magnitudes of the partial forces ' UC dF and ' UT dF are given by the following equations, respectively: ( ) ( ) The relationship between From the above, it can be concluded that for a unit with dl dl on the cutting edge, the drilling force increment ' l dF and the torque increment ' l dC can be expressed by the following equation： The total cutting force and torque are obtained by integrating the ' l dT and ' l dF expressions from the boundary between the cutting edge and the transverse edge to the periphery of the drill.and substituting ' s dF to obtain: ' l F and ' l T stand for drilling force and torque respectively.
The dynamic cutting front angle ' nd  at M for longitudinal torsional ultrasonic vibration drilling of the point on the transverse edge can be expressed by the following equation: For any cell M on the transverse edge, the cell shear force ' s dF is calculated as the following equation: T represent the cutting forces and torques applied to the cross-edge respectively.
In ultrasonic vibration machining the drill bit creates an impact effect on the workpiece due to vibration, creating an axial impact force.The axial impact force can be expressed by the following equation [27]: where M is the equivalent mass of the drill bit and a is the axial acceleration resulting from the vibration of the drill bit.According to equation ( 3), the value of a can be expressed by the following equation: The longitudinal torsional ultrasonic vibration axial force and torque of the whole drill can be expressed by equations ( 26), ( 27), (34), ( 35) and (36).
' F , the longitudinal torsional ultrasonic axial force; ' T , the longitudinal torque ultrasonic torque.

Figure.5 Theoretical calculation results of torque with different parameters
From Figs. 4 and 5, it can be seen that the larger the feed per revolution, the larger the axial force and torque; with the increase of ultrasonic amplitude, the axial force first decreases and then becomes larger, while the torque keeps decreasing; the larger the spindle speed, the smaller the axial force and torque.Compared with conventional drilling, under the same conditions, the axial force and torque of ultrasonic vibration drilling are smaller than those of ordinary drilling.

Finite element simulation analysis of longitudinal-torsional composite ultrasonic vibration drilling
3.1 3D ultrasonic vibration finite element simulation and analysis The drill bit as well as the workpiece were modeled by Solid works software and imported into ABAQUS software to establish the correct assembly relationship with the workpiece.In the simulation, the drill used is the same as the one used in the test, which is a PCD drill.In this section of the 3D simulation, the diameter of the workpiece is 2.6 mm, the height is 0.5 mm, and the taper angle is 60°, as shown in Fig. 6  ( ) The Johnson-Cook intrinsic parameters of the SiCp/Al composites are represented in Table 1.
The specific values of the equations are shown in Table 2, and the physical properties of the SiCp/Al composites are shown in Table 3. Reference strain rate ̇ 0.005

Analysis of drilling force results of 3D finite element simulation
The finite element simulation mainly studied the influence law of spindle speed, feed rate and ultrasonic amplitude on axial force and torque, and the specific simulation parameters are shown in Table 4. Since the longitudinal torsional ratio of the longitudinal ultrasonic toolholder used in the test is 1:0.8, the same longitudinal torsional ratio is used for the finite element simulation, and the ultrasonic amplitudes listed below are all longitudinal ultrasonic amplitudes.The single-factor simulation results were analyzed by EXCEL software, and the average values of axial force and torque were obtained for different machining parameters and acoustic parameters, as shown in Fig. 7 and Fig. 8.With the increase of ultrasonic amplitude, both drilling axial force and torque showed a trend of decreasing first and then increasing, and when the ultrasonic amplitude was 0.8 μm, the drilling axial force and torque reached the minimum value.Compared with ordinary drilling, the average axial force was reduced by 18.13% and the average torque was reduced by 21.74% for longitudinal-torsion compound ultrasonic vibration drilling.Compared with the average axial force, the reduction of the average torque was greater, indicating that ultrasonic vibration had a greater effect on the average torque.
The axial force and torque showed a decreasing trend with the increase of spindle speed.When the spindle speed reached 6000 r/min, the drilling axial force and torque reached the lowest, compared with the spindle speed at 3000 r/min, the average axial force decreased by 45.56% and the average torque decreased by 22.82%.Compared with the average torque, the decrease in average axial force was greater, indicating that the spindle speed had a greater effect on the average axial force.With the increase of feed speed, both drilling axial force and torque showed an increasing trend.When the feed rate is 0.002mm/r, the drilling axial force and torque are the smallest, and the average axial force is 38.81% smaller and the average torque is 49.57% smaller compared to the feed rate of 0.008mm/r.The increase of the average torque is greater compared to the average axial force, which indicates that the feed rate has a greater effect on the average torque.

Experimental study of ultrasonic vibration drilling force of SiCp/Al composites
In order to further investigate the influence law of ultrasonic vibration on the force magnitude of small hole drilling processing and optimize the drilling processing parameters.Based on the finite element simulation, drilling tests are conducted to study the variation law of drilling force under different machining parameters, and the theoretical model and finite element simulation are verified to finally achieve to the purpose of optimizing the machining parameters.

Longitudinal-torsional composite ultrasonic vibro-acoustic drilling system
The ultrasonic vibration drilling system mainly consists of an ultrasonic generator, transducer, variable amplitude rod and tool system [30].Fig. 9 shows the schematic diagram of the toolholder for longitudinal-torsional composite ultrasonic vibration drilling, which integrates the transducer, amplitude rod and tooling system.The parameters of the drill are shown in Table 5.The workpiece material used in the test is SiCp/Al metal matrix composite, the size of the workpiece is 80×40×5mm, the volume fraction of SiC particles in the material is 65%, and the particle size distribution of the particles is between 20~40μm.

Single-factor test design and analysis of drilling force
In order to study the variation of drilling force under different processing parameters, the degree of reduction of drilling force by ultrasonic amplitude, and to lay the foundation for determining the optimal processing parameters in subsequent orthogonal tests, the following single-factor tests were designed.The single-factor test parameters were the same as those of the 3D drilling simulation, and the test protocol is shown in Table 4.In order to reduce the impact of chips on the drilling quality, the test in this paper uses pecking drill processing with a step depth of 0.1mm.The pecking drill schematic is shown in Fig. 11.In order to explore the influence of ultrasonic vibration on drilling force, the axial force and torque under longitudinal torsional ultrasonic vibration drilling and conventional drilling were compared, and Fig. 12 showed the variation curve of axial force and torque of ordinary drilling and ultrasonic drilling with machining time under the same machining parameters, and the partial step process was analyzed separately, and the axial force and torque of ultrasonic vibration drilling were smaller than those of conventional drilling.As can be seen from Fig. 13, the average axial force and average torque both decrease first and then increase with the increase of ultrasonic amplitude, and the ultrasonic amplitude reaches the minimum when the ultrasonic amplitude is 0.8 μm, and the average axial force and average torque in ultrasonic vibration drilling processing are smaller than those in ordinary drilling processing, so in the subsequent research on ultrasonic vibration drilling, the ultrasonic amplitude is set to 0.8 μm. the average axial force in ultrasonic drilling is reduced by 24.41% and the average torque is reduced by 30.61% compared with The average axial force was reduced by a maximum of 24.41% and the average torque was reduced by a maximum of 30.61% for ordinary drilling.In contrast, the reduction of the average torque by the application of ultrasonic vibration is more obvious, which is the same as the 3D simulation results, mainly because the application of ultrasonic torsional vibration makes the drill have a repeated grinding effect on the hole, and the SiC particle debris near the cutting edge is more easily dislodged and does not obstruct the cutting edge, which makes the average torque further reduced.
When the ultrasonic amplitude increases, the impact effect of longitudinal ultrasonic vibration is more obvious, which makes the workpiece easy to produce fine cracks and easier to be removed by the tool, and has a facilitating effect on the chip discharge, and the SiC particles are more easily broken by the application of torsional vibration, so the application of ultrasonic vibration makes the drilling force have a tendency to decrease.However, when the ultrasonic amplitude is too large, the impact of the drill bit on the workpiece increases, and the SiC particle fragments are easily pressed inside the Al matrix and cut with the tool instead of the drill bit, causing an increase in the drilling force.When the amplitude is 1 μm, although the undeformed chip thickness is 0, the maximum undeformed chip thickness is also becoming larger, and the maximum cutting force value is also becoming larger, and the influence of cutting thickness on the drilling force has increased, which makes the average drilling force size increase.Therefore, the ultrasonic amplitude is not as large as possible.

Effect of spindle speed on drilling force
When the feed per revolution is 0.004mm, the trend line graph of the effect of spindle speed on the average axial force and average torque is plotted, as shown in Fig. 14.
As shown in Fig. 14, the average axial force and average torque both decrease with the increase of spindle speed, and the average axial force and torque of ultrasonic vibration drilling are smaller than the average axial force and torque of conventional drilling, which is consistent with the theoretical calculation results and 3D simulation results.When the amplitude of ultrasonic vibration is 0.8μm, the spindle speed increases from 3000r/min to 6000r/min, the axial force of ultrasonic drilling decreases by 43.91%, the axial force of conventional drilling decreases by 43.16%, the torque of ultrasonic drilling decreases by 35.60%, the torque of conventional drilling decreases by 19.52%, and the comparison shows that the reduction of torque by increasing the spindle speed when ultrasonic vibration is applied is more obvious.
The magnitude of torque reduction is more obvious.By analyzing the drilling force, it can be found that the reduction of the average axial force and torque decreases gradually with the increase of the spindle speed.This is because when the spindle speed increases, the number of collisions between the drill and the SiC particles per unit time increases, which makes the cutting force tend to increase, but the increase in spindle speed makes the particles more impacted and easier to be destroyed, and it is also accompanied by the increase in drilling temperature, which softens the aluminum substrate and reduces the force on the drill.In ultrasonic vibration drilling, increasing the spindle speed will suppress the process effect of longitudinal torsion ultrasonic vibration machining, thus slowing down the trend of reducing the drilling force.In summary, an increase in spindle speed still results in a tendency to reduce the overall force of the drilling process.As can be seen from Fig. 15, the average axial force and average torque of ultrasonic vibration drilling are smaller than the average axial force and average torque of ordinary drilling, and both increase with the increase of feed per revolution, which is consistent with the theoretical calculation results in Section 2 and the 3D simulation results in Section 3. When the ultrasonic amplitude was 0.8 μm, the average axial force increased by 39.36% and the average torque increased by 50.26% when the feed per revolution was increased from 0.002 mm/r to 0.008 mm/r, and the average axial force increased by 36.30% and the average torque increased by 29.00% for ordinary drilling.With the increase of feed per revolution, the cutting thickness is also increasing, the force of workpiece on the drill bit is also increasing, and the axial force and torque of drilling will also increase accordingly.From the results of single-factor analysis of drilling force, it can be seen that the theoretical model calculation results and simulation results of drilling force have a consistent trend with the test results, but because the theoretical calculation and simulation analysis are carried out under ideal conditions, the homogeneous material is applied to the workpiece, and the drill bit is set as a rigid body, and the influence of a series of external factors such as the breakage of SiC particles, tool wear, and defects of the workpiece itself is not considered, which makes the theoretical calculation results and simulation results have a certain error with the test results, and the error calculation results are shown in Table 6, and the histogram of the error calculation results is drawn according to the calculation results as shown in Fig. 16.As can be seen from Table 6, the average error of the simulated axial force is 7.11%, the average error of the torque is 7.19%, the average error of the theoretical axial force is 10.49%, and the average error of the torque is 13.05%, indicating that both the theoretical model of drilling force and the 3D drilling simulation can predict the average axial force and the average torque of the drilling process of SiCp/Al composites more accurately, and the simulation The prediction results are better than the prediction results of the theoretical model.
Fig. 16 Error analysis results under different parameters From Fig. 16(d), it can be seen that when the ultrasonic amplitude is 1.2 μm, the error is larger.From Table 6, it can be seen that when the ultrasonic amplitude increases, the theoretical calculated torque shows a decreasing trend, which is different from the test results and simulation results, mainly because the theoretical model does not take into account the influence of ultrasonic vibration on the workpiece impact and other reasons, which needs to be further improved.By comparing the drilling test results with the simulation results, it is seen that the magnitude of the simulated axial force and torque is always smaller than the axial force and torque measured by the test, mainly because of the following two reasons.
(1) In the 3D drilling simulation, the impact of tool wear is not considered because the drill is set as a rigid body.
However, in the actual machining, the drilling force is much larger due to the effect of tool wear.
(2) In the 3D drilling simulation, due to the influence of material damage parameters, some cell meshes in the chip and workpiece are removed due to failure, and the number of chips is reduced, so that the force it exerts on the tool is weakened.

Orthogonal test design and analysis of drilling force
In order to determine the significance of the effect of each machining parameter on the drilling force under ultrasonic vibration drilling processing, a 3-factor, 4-level orthogonal test was designed and the results were analyzed to derive the optimal combination of parameters for reducing the axial force.The factor levels of the orthogonal test are shown in Table 7.The average axial force and average torque calculation results obtained by orthogonal tests are shown in Table 8.In order to clarify the significant degree of each factor on the average axial force under drilling processing, the test results were subjected to the extreme difference analysis, as shown in Table 9.By comparing the magnitude of K values in Table 9, the best combination of parameters to obtain the minimum axial force can be obtained.The trend of the influence of each factor level on the average axial force during ultrasonic vibration drilling was analyzed by using the results of the extreme difference analysis, as shown in Fig. 17.From the figure, it can be seen that the average axial force becomes smaller with larger spindle speed, smaller than larger with larger ultrasonic amplitude, and larger with larger feed per revolution, which is consistent with the conclusions obtained in Section 2. When the spindle speed is 6000r/min, the ultrasonic amplitude is 0.8μm, and the feed per revolution is 0.002mm/r, the average axial force can reach the minimum.9, the main order of influence of each factor on the axial force is obtained as follows: feed per revolution > spindle speed > ultrasonic amplitude.
Since ANOVA does not consider the influence of test error on the test results, ANOVA can more accurately analyze the error and can determine the significance of each factor by testing, so this section conducted ANOVA on the basis of ANOVA, as shown in Table 10.
From the ANOVA table, it can be seen that when the significance level α value is 0.05, the P values of the significance indicators of the three factors are smaller than the α value, indicating that all three factors have a significant effect on the average axial force.By comparing the magnitude of F values in the table, the order of the magnitude of the influence of each factor on the average axial force can be obtained as follows: feed per revolution > spindle speed > ultrasonic amplitude, which is consistent with the results of the ANOVA.The extreme difference analysis of the significance of each factor on the average torque under ultrasonic vibration drilling processing was performed using Excel software, as shown in Table 11.By comparing the magnitude of K values in Table 11, the best combination of parameters to obtain the minimum torque can be obtained, and the trend of the effect of the level of each factor on the average torque in the table was analyzed, as shown in Fig. 18.From the figure, it can be seen that the average torque becomes smaller as the spindle speed becomes larger, smaller then larger as the ultrasonic amplitude becomes larger, and larger as the feed per revolution becomes larger, and it can be seen that the trend of the average torque is consistent with the conclusions obtained in Section 2. When the spindle speed is 6000r/min, the ultrasonic amplitude is 0.8μm, and the feed per revolution is 0.002mm/r, the average torque can reach the minimum.By comparing the magnitude of the extreme difference value R in Table 11, the primary and secondary relationships of the factors affecting the average torque in the longitudinal torsional ultrasonic vibration drilling process of SiCp/Al composites can be determined, and the primary and secondary relationships of the factors affecting the average torque are: feed per revolution > ultrasonic amplitude > spindle speed.
Fig. 18 Trend of the effect of the level of factors on the average torque An analysis of variance (ANOVA) was performed on the torque based on the analysis of extreme variance, as shown in Table 12.From the ANOVA table, it can be seen that the significance of all three factors is less than the value of α when the significance level α is 0.05, indicating that all three factors have a significant effect on the average torque.By comparing the magnitude of F-values in the table, it was found that the order of magnitude of the influence of each factor on the average torque was: feed per revolution > ultrasonic amplitude > spindle speed, which was consistent with the results of the ANOVA and different from the degree of influence on the average axial force.
By analyzing the orthogonal test results of the average axial force and average torque, it was found that the optimal combination of parameters in longitudinal torsional ultrasonic vibration drilling of SiCp/Al composites was spindle speed n=6000r/min, ultrasonic amplitude A=0.8μm, and feed per revolution fr=0.002mm/r.
(a) shows the local enlarged view of the drilling trajectory.

Fig. 1
Fig.1 Conventional drilling trajectory and longitudinal torsional ultrasonic vibration drilling trajectory

Fig. 2
Fig.2 Thickness of undeformed chips under ultrasonic vibration drilling In order to analyze the effect of longitudinal torsional ultrasonic vibration drilling on the thickness of undeformed chips, the ultrasonic frequency f=33KHz, the longitudinal amplitude A=1μm, the torsional amplitude B=0.8μm, and the feed per revolution f=0.004mm/r, n=4000r/min, the local trajectory diagram of the two cutting edges was drawn by MATLAB software, as shown in Fig.2.

Fig. 4
Fig.4 Theoretical calculation results of axial force with different parameters From the axial force and torque models of ordinary drilling and longitudinal torsional ultrasonic vibration drilling, it can be seen that the spindle speed n, feed per revolution r f , ultrasonic amplitude A and B and the geometry of the tool all have an effect on the drilling force, but the trend of the effect on the drilling force cannot be seen directly.In order to clarify the trend of the influence of different machining parameters on the drilling force, the theoretical model of axial force and torque was analyzed by using MATLAB software, and the theoretical value at a certain moment was calculated, and the calculation results are shown in Fig.4 and Fig.5.

Fig. 6
Fig.6 Schematic diagram of the 3D model of drilling simulation In modeling of forming, fabrication and structural mechanics, the widely used image-only intrinsic model is commonly used to characterize the thermodynamic behavior of materials.Junfeng Xiang et al [28] determined the imageonly intrinsic model of SiCp/Al composites by quasi-static and dynamic compression tests and used the image-only intrinsic model to reflect the thermodynamic behavior of SiCp/Al composites during the drilling process.The material'sintrinsic model mainly reflects the relationship between the forces and deformation of the material, so it is crucial to choose a suitable intrinsic model when doing simulations[29].Compared with other models, the Johnson-Cook plasticity model has a comprehensive consideration of the strain and strain rate of the material, involves fewer parameters and is easier to obtain, and is adapted to a variety of different materials.Therefore, in order to simulate the stresses during drilling, the

Fig. 7
Fig.7 Simulation results of axial force under different parameters From Fig.7 and Fig.8, it can be seen that the axial force and torque show different trends as the amplitude, spindle speed and feed per revolution increase.Compared with normal drilling, the axial force and torque tend to become smaller with the application of ultrasonic vibration under the same machining conditions, but the amount of reduction constantly changes with the change of amplitude.

Fig. 8
Fig.8 Simulation results of torque with different parameters

Fig. 9
Fig.9 Schematic diagram of longitudinal torsion compound ultrasonic vibration drilling toolholder 4.2 Longitudinal-torsional composite ultrasonic vibration drilling test platformThe machine tool used for the longitudinal-torsional compound ultrasound-assisted drilling test is a VMC850E CNC machining center, which is capable of three-axis linkage automatic control function.Fig.10shows the longitudinal-torsion compound ultrasonic-assisted drilling test platform, and the drilling force is collected by Kistler 9257B force measuring instrument and the corresponding soft wire system.

Fig. 11
Fig.11 Schematic diagram of the pecking drill principle 4.4 Effect of ultrasonic amplitude on drilling force

Fig. 12
Fig.12 Curve of drilling force over time for a partial stepping process

Fig. 14
Fig.14 Trend of influence of spindle speed on average axial force and average torque 4.6 Effect of feeds per revolution on drilling forces When the spindle speed is 4000r/min, the trend line graph of the effect of feed per revolution on the average axial force and average torque is plotted, as shown in Fig.15.

Fig. 15
Fig.15 Trend of influence of feed per revolution on average axial force and average torque 4.7 Verification of theoretical analysis and simulation results Fig.17 Trend of the influence of the level of factors on the average axial force By comparing the magnitude of the extreme difference R in Table9, the main order of influence of each factor on the

Table 1
In this paper, the Johnson-Cook shear damage model is introduced to match the Johnson-Cook plasticity model used in the drilling simulation, under which the equivalent plastic strain at failure occurs as the following equation:

Table 2
Johnson-Cook failure parameters of SiCp/Al composites

Table 6
Error analysis of theoretical calculation results of drilling force and simulation results

Table 7
Table of orthogonal test factor levels

Table 8
Results of orthogonal test

Table 9
Average axial force extreme difference analysis results

Table 10
Average axial force ANOVA results

Table 11
Average torque extreme difference analysis results

Table 12
Average torque ANOVA results