Zhenlong PENG, Deyuan ZHANG,c, Xiangyu ZHANG,c,*

a School of Mechanical Engineering and Automation, Beihang University, Beijing 100083, China

b Institute of Bionic and Micro-Nano Systems, Beihang University, Beijing 100083, China

c Beijing Advanced Innovation Center for Biomedical Engineering, Beihang University, Beijing 100083, China

KEYWORDS High-speed machining;Minimum chip thickness;Mode-coupling;Thin-walled cylinder;Ultrasonic vibration cutting

Abstract Titanium alloys are widely used in the aviation and aerospace industries due to their unique mechanical and physical properties. Specifically, thin-walled titanium (Ti) cylinders have received increasing attention for their applications as rocket engine casings, aircraft landing gear,and aero-engine hollow shaft due to their observed improvement in the thrust-to-weight ratio.However, the conventional cutting (CC) process is not appropriate for thin-walled Ti cylinders due to its low thermal conductivity,high strength,and low stiffness.Instead,high-speed ultrasonic vibration cutting (HUVC) assisted processing has recently proved highly effective for Ti-alloy machining. In this study, HUVC technology is employed to perform external turning of a thinwalled Ti cylinder, which represents a new application of HUVC. First, the kinematics, tool path,and dynamic cutting thickness of HUVC are evaluated.Second,the phenomenon of mode-coupling chatter is analyzed to determine the effects and mechanism of HUVC by establishing a critical cutting thickness model. HUVC can increase the critical cutting thickness and effectively reduce the average cutting force, thus reducing the energy intake of the system. Finally, comparison experiments are conducted between HUVC and CC processes.The results indicate that the diameter error rate is 10% or less for HUVC and 51% for the CC method due to a 40% reduction in the cutting force. In addition, higher machining precision and better surface roughness are achieved during thin-walled Ti cylinder manufacturing using HUVC.

1. Introduction

Titanium (Ti) alloys are increasingly applied in the aviation and aerospace industries due to their unique properties, such as a high strength-to-weight ratio, strong fracture resistance,and good anticorrosion performance.1-3However, there are two general problems related to the manufacturing of Ti alloy parts, particularly thin-walled Ti cylinders, which are classed as a typical difficult-to-cut material. Firstly, Ti alloys have low thermal conductivity, high heat strength, and hardness.As a result, the cutting temperature is high and can easily exceed 1000°C at the tool-chip interface. Therefore, the surface quality of the workpiece is severely compromised, resulting in poor cutting performance and low cutting speed.4-6Secondly, as key bearing components, thin-walled Ti cylinder parts such as solid rocket engine casings and nozzles, aircraft landing gear, and aero-engine hollow shaft are widely used in the aviation and aerospace industries.However,it is difficult to ensure their machining precision and machined surface quality due to their low rigidity and frequent deformation and chatter during machining.7-9Consequently, the lack of high-speed and high-precision thin-walled Ti cylinder machining hinders development in the aviation and aerospace industries.

In order to control chatter and enhance the machining precision of thin-walled parts,a large amount of research has been conducted in the past few decades,which can be classified into three categories: (1) machining deformation prediction and compensation,10-13(2) tool path optimization,14,15and (3)optimization of tool parameters and process parameters.16-18For example, Wang et al.10developed a method to provide higher order convergence and introduced it into a real-time compensation process to reduce the machining deformation of thin-walled parts. Liu et al.14proposed a tool path generation method, whereby five-axis side milling of the tool was studied by considering the constraints of cutting force and machine kinematics. In this method, multiple relationships between the tool path point and the tool axis vector were considered before finally generating a curvature continuity fiveaxis milling tool path. Li and Liu16established an analytical model of the instantaneous undeformed chip thickness that considers the dynamic regenerative effects. On this basis, they simulated the chatter stability lobes in the time-domain and concluded that higher machining efficiency can be achieved by selecting appropriate process parameters.The above prediction methods rely on the accuracy of the analytical model and have been successfully implemented in thin-walled parts machining. However, these methods are restricted to low cutting speeds due to the high cutting temperature, high cutting force, and prohibitive tool wear when applied to difficult-tocut materials, which limit the processing efficiency of thinwalled Ti parts.Hence,machining precision cannot be ensured during the high speed machining of thin-walled Ti parts.

Ultrasonic assisted machining(UAM)has been widely used as an assisted strategy to machine difficult-to-cut materials for several decades,particularly Ti alloys.This is due to its ability to improve the cutting force,machining precision,and surface quality when compared to conventional cutting (CC).19,20Simultaneously, researchers have demonstrated that chatter stability in the machining process can be improved by using UAM.Xiao et al.21proposed a new cutting model comprising a vibration cutting process and reduced the range of work displacement amplitudes from 10-102 μm to 3-5 μm. Tabatabaei et al.22presented an experimental and theoretical study on the effect of UAM on chatter, revealing that UAM has different effects on stability.They then proposed an equation to predict the effect of UAM on chatter. Ma et al.23showed that regenerative chatter during CC can be effectively suppressed by applying ultrasonic elliptical vibration on the cutting tool.However, the chatter of thin-walled Ti parts during ultrasonic assisted machining has received little research attention.Moreover, most research focuses on regenerative chatter of the machining tool where the machine structure is simplified to single degree-of-freedom control theory due to the high stiffness of the machine tool. However, due to the low structure stiffness of thin-walled Ti parts,it is more meaningful to study mode-coupling chatter whereby the tool vibrates in at least two different directions. Despite this, theoretical and experimental analysis of the effect of ultrasonic vibration on the modecoupling chatter of thin-walled Ti parts remains inconclusive.

Recently, Sui et al.24analyzed the feasibility of high-speed ultrasonic vibration cutting (HUVC) for Ti alloys, indicating that HUVC achieves a higher cutting speed, lower cutting force,prolonged tool life,and increased material removal rate,which is difficult to achieve in UAM.Subsequently,they studied the separation effect and provided phase shift control by designing a closed-loop direct digital synthesis (DDS) system in HUVC.As a result,they built a cutting model that considers the tool geometry and decreased surface roughness for HUVC.25-27Moreover, an ultrasonic-frequency repetitive impulse cutting force signal was measured for the first time,which explained the reduction in the cutting force,and a thermocouple system was established for measuring the temperature of HUVC, which achieved a maximum temperature reduction of 30% at cutting speeds of 250-300 m/min with a duty cycle value of 0.55.28,29Until now, previous research on HUVC mainly focused on feasibility and theoretical analyses through simple experiments,concluding that HUVC decreases the cutting force,reduces the cutting temperature,and extends the tool life under high-speed machining. However,due to the widespread use of Ti alloys, it is crucial to ensure the highprecision manufacturing of thin-walled Ti parts,which are typical difficult-to-cut structural components. Therefore, this study aims to prove the vast potential of the HUVC method for solving real problems in the aviation field. The novel contribution of this study is that HUVC is applied to complex thin-walled Ti parts base on the kinematic analysis, separated cutting characteristics, and dynamic cutting thickness as well as mode-coupling chatter analysis of HUVC, instead of the simple parts evaluated in previous work.

In this study,the HUVC method is employed for the external turning of thin-walled Ti cylinders under high cutting speeds. First, the kinematics and transitional cutting process of HUVC are presented. Second, the mechanism by which HUVC improves the machining precision and suppresses chatter is explained. Third, comparative high-speed machining experiments are conducted on thin-walled Ti cylinders to determine the influence of HUVC on chatter stability and machining precision. Finally, the results are summarized and discussed.

Fig. 1 HUVC method for a thin-walled cylinder.

2. Chatter suppression mechanism of HUVC

2.1. Kinematics of HUVC

HUVC is a method in which high-frequency vibration is added on the cutting tool.The schematics of the HUVC cutting thinwalled Ti cylinder are shown in Fig. 1. In CC, the relative movement between the tool and the workpiece is composed of workpiece rotation around its axis and the tool feed along the axial direction (spindle rotation and axial feed), as illustrated in Fig. 1. A random point, P, is chosen to describe the kinematic equation of the tool. The equation of point P can be written as follows:

where r is the cutting radius (mm), θ is the rotation angle (degrees), and fzis the feed speed (mm/s). In HUVC, the tool vibrates along the axial direction with a sinusoidal signal according to the specific amplitude(A)and frequency(FHUVC)of the ultrasonic vibration. If point P is located at the peripheral cutting edge,the coordinate of point P along the oz direction in HUVC is:

HUVC enables the tooltip to vibrate along with the feed direction with ultrasonic frequency and achieves a noncutting duration during one single vibration cycle to achieve intermittent cutting at a high cutting speed under certain conditions.If the initial values of parameters θ in Eq. (1) are assumed to be 0, the coordinate of point P at the cutting edge in HUVC can be expressed as:

In HUVC, the feed speed and feed acceleration of the cutting edge in the feed direction change periodically. The feed speed equations and feed acceleration equations for HUVC and CC can then be obtained by deriving Eqs. (1) and (3),respectively, as shown in Eqs. (4) and (5):

Fig. 2 Feed speed and feed acceleration of CC/HUVC.

The feed speed and feed acceleration of CC/HUVC are simulated using MATLAB with the following parameters: A=10 μm, FHUVC=20000 Hz, and fz=0.1 mm/s (Fig. 2). The feed speed and feed acceleration of the CC process are not changed, whereas those of HUVC are changed periodically.The impact effect caused by this periodic shifting and variable acceleration characteristics is significant for material removal.At large accelerations, the cutting edge with greater kinetic energy impacts the material to be processed, which can cause local concentration of stress and deformation in the cutting zone, accelerate chip formation, and improve the material removal rate. At the same time, the negative speed and acceleration in the feed direction also make it possible to separate the tool from the material being processed in the feed direction, which enables the separated cutting characteristics of HUVC.

Then, zHUVCcan also be expressed by the following equation:

where f is the feed rate (mm/r); n is the rotation speed of the spindle(r/s).The ratio of vibration frequency FHUVCand spindle speed n should be noted,which is defined as the cutting frequency ratio,ε.The cutting frequency ratio can be divided into an integer part, Λ, and a fraction part, λ, which can be expressed as:

The separation criteria of HUVC is given by24:

Three parameters; i.e., the amplitude A, feed rate f, and fraction part λ, are used to determine whether the tooltip can separate from the workpiece.

The duty cycle (Dc) is an important parameter, which is used to describe the tool-workpiece separation of HUVC.During an ultrasonic vibration period(T)of HUVC,the tool starts to contact with the workpiece from t1then cuts into the workpiece with increasing feed speed. The tool then separates from the workpiece at t2before making contact with it again in the next vibration cycle. t2-t1is the cutting duration during an ultrasonic vibration cycle. Thus, the duty cycle of HUVC is given by:

2.2. Analysis of mode-coupling chatter stability

Mode-coupling chatter occurs when there are vibrations in two directions in the cutting plane. Regenerative chatter results from a phase shift between the vibration waves remaining on both sides of the chip and occurs earlier than mode-coupling chatter in most machining cases.17In order to facilitate the analysis of mode-coupling chatter stability, a typical weak stiffness thin-walled mode-coupling chatter cutting system is used, as shown in Fig. 3. h0is the intended chip thickness,Δh is the chip thickness, κris the tool cutting edge angle, and x1,x2are two rigid axes perpendicular to each other,the stiffness and damping of which are k1, k2(k1

Fig. 3 Mode-coupling chatter system.

Eq.(10)is the equation of motion for two independent single degree-of-freedom systems with no coupling between them.Under the action of the cutting force,the motions of these two degrees of freedom are coupled and cause vibration. The cutting process still plays a role in the control mechanism and displacement feedback. The above two equations of motion are coupled due to displacement feedback in the load.Meanwhile,in the previous research on mode-coupling chatter, the damping effect of the system is usually omitted.30,31So, due to the action of the cutting force, the equation of motion becomes:

where ΔFxixjis the increase in cutting force caused by the vibration displacement in the positive direction of the xiaxis on the xjaxis(i ∈{1,2},j ∈{1,2}).For example,ΔFx1x1is the increase in the cutting force caused by the vibration displacement in the positive direction of the x1axis along the negative direction of the x1axis.

According to traditional cutting theory,17,32the cutting fore is proportional to the cutting constant kc(N/mm2),the cutting width b (mm), and the cutting thickness h (mm).

Assuming that the displacement of the tool tip in the positive direction of the x1axis is x1, the projection of the tool tip in the negative direction of the Y-axis is x1cosα; i.e., the variation of the cutting thickness Δh is:

Thus, the critical cutting thickness of HUVC is larger than the critical cutting thickness of CC.This indicates that HUVC can have a larger cutting thickness than that of CC at the critical chatter point.The smaller the Dcvalue,the larger the multiple of the critical cutting thickness, and the less likely the system is to cause mode-coupling chatter. In general, HUVC increases the critical cutting thickness and suppresses modecoupling chatter compared with CC.

It is assumed that the system energy expenditure is a constant, which is W-. When the cutting forces do more positive work on the system than the system consumes (W+＞W(wǎng)-),the energy will accumulate and cause chatter. The W+ of CC and HUVC can be expressed as:

Eqs. (28) and (30) can be solved simultaneously to give:

Based on Eqs. (30) and (31), the energy intake W+HUVCof HUVC is always smaller than W+CCof CC.This indicates that HUVC can effectively reduce the energy intake of system and suppress chatter appearance compared to CC so as to improve the chatter stability and machining precision, and coincides with the Eq. (29) is correct.

As shown in Fig.4,the critical cutting thickness of CC(Dc=1)is hmax-CCand the critical cutting thickness of HUVC(Dc=0.8) is hmax-HUVC1. When the Dcof HUVC is further reduced to 0.5 and 0.2, the critical cutting thicknesses of the system will be further increased to hmax-HUVC2（=2hmax-CC）and hmax-HUVC3（=5hmax-CC）, respectively. HUVC (0

2.3. Dynamic cutting thickness of HUVC

As shown in Fig. 2, the cutting thickness is constant because the feed rate and cutting speed are constants in CC. The cutting thickness of CC can be given by:

When the separation condition(Eq.(8))is met,the tool has a cut-in and a cut-out relative to the workpiece in each cycle.The cutting thickness of HUVC can be written as:

where f（zHUVC，t） is a function of zHUVCand t.

At each cycle of HUVC,the feed speed and acceleration are periodically changed. The effect caused by this periodic shifting and variable acceleration characteristics is significant for material removal. Therefore, HUVC exhibits a transition cutting process in which the cutting thickness is increased from zero to the maximum cutting thickness then reduced to zero again in each ultrasonic vibration cycle. In this transition cutting process, the feed rate is smaller than the bilateral amplitude of the tool vibration (Eq. (8)), whereas the amplitude of the HUVC is generally on the order of microns. The cutting thickness is also on the order of microns. The obtuse circle of the tool rncannot be ignored; the point P on the cutting edge should be regarded as an arc with a radius. In this case,actual participation in the cutting involves not only the rake face and the flank face in the macroscopic sense, but also the blunt round part of the cutting edge.

Fig. 4 Relationship between system energy intake and expenditure.

Fig.5 illustrates chip formation with respect to the dynamic cutting thickness of HUVC.There are three cutting states in the transition cutting process at each ultrasonic vibration cycle:elastic state, plastic state, and cutting state. The tE1, tP1, and tS1represent the start time of elastic state,plastic state,and cutting state, respectively. And the tE2, tP2, and tS2represent the end time of elastic state,plastic state,and cutting state,respectively. Chip formation is generally associated with the cutting thickness.33When the cutting thickness is less than the minimum chip thickness: h（t）＜hlim, elastic deformation occurs,and the tool does not remove any material, as shown in Fig. 5(a). As the cutting thickness approaches the minimum chip thickness: h（t）≈hlim, chips are formed by shearing and elastic deformation still occurs,as illustrated in Fig.5(b).In this state, the removed depth of the workpiece is less than the desired depth. However, when the cutting thickness increases beyond the minimum chip thickness: h（t）＞hlim, the elastic deformation phenomena decreases significantly,and the entire depth of the cut is removed as a chip, as shown in Fig. 5(c).

At the same feed rate, hCC＜hlimmay occur during the CC process.At this time,no chip is formed and the ploughing condition is dominant.However,for HUVC,the following conditions hHUVC＞hlimexist in a certain period tS1＜t ＜tS2because of the dynamic cutting thickness. In general, HUVC adds ultrasonic vibration in the feed direction, which increases the feed rate and cutting thickness for a certain period.Ultimately,HUVC enhances the fine cutting ability compared with CC.

2.4. Effect of HUVC on chatter stability

HUVC can machine high-precision thin-walled cylinders based on the separated cutting characteristics, and there are two keys:

(1) As described in Section 2.2, When Dcis less than 1, the HUVC can effectively reduce the energy intake of system as well as enlarge the critical cutting thickness compared with the CC.

Fig. 5 Dynamic cutting thickness of HUVC.

Fig. 6 HUVC/CC machining while cutting thick/thin cylinder (v represents cutting speed).

(2) As described in Section 2.3,a chip will not be generated if the cutting thickness is less than the minimum chip thickness.However,owing to the cutting thickness variability resulting from the ultrasonic vibration, even though the cutting parameters are set to the same conditions, HUVC will show a larger cutting thickness than CC because of the dynamic cutting process, as shown in Fig. 6(a) and (b).

Generally, when machining an extremely thin workpiece using CC, a small cutting thickness will lead to ploughing and extrusion effects, which means that no material will be removed. Nevertheless, chatter will occur with CC if the cutting thickness is increased. However, with HUVC, the critical cutting thickness for chatter stability is larger for HUVC than that for CC.Moreover,dynamic cutting thickness characteristics of HUVC makes material removal possible in a certain period when cutting thickness is greater than the minimum chip thickness (Fig. 5(c)). Therefore, HUVC can machine extremely thin workpiece without chatter so as to improve the chatter stability and machining precision, as shown in Fig. 6(c) and (d).

3. Experimental setup and procedure

An illustration of the experimental setup, including the spindle, workpiece, ultrasonic power, and force measurement device, is shown in Fig. 7. Thin-walled cylinder machining using CC and HUVC processes was performed on the Hardinge HLV-H precision machine tool. The ultrasonic vibration tool, which is mounted on the carriage apron of the machine tool, converts a 50 Hz alternating current supply to an ultrasonic-frequency vibrational output.The system can generate 20 μm amplitude of vibration with a 22 kHz frequency.The material used for the thin-walled cylinder was Ti-6Al-4V,with an inner diameter of 36 mm, cutting length of 30 mm, and a thickness range of 0.6-2.0 mm. Triangular cemented carbide tooltips from SANDVIK(TCMT06T102)were set at the specified tool position of the vibrator for each cutting condition.The detailed experimental and machining conditions are summarized in Table 1.

The CC process could be switched to HUVC by turning on the ultrasonic power. The machining parameters used for CC and HUVC processes are listed in Table 2, with the premise of ensuring the same material removal rate (MRR). In orderto eliminate random error,all experiments were repeated three times, and a new tool was used for each set of repetitions.

Table 1 Experimental conditions for machining the Ti-6Al-4V workpiece.

To investigate the machining mechanism and evaluate the precision, the chips, cutting forces, diameters, amplitude, surface roughness,and surface microstructures were tested.Chips were detected by a microscope (Olympus OLS4100). Average and dynamic cutting forces were measured by a dynamometer(Kistler 9254A)and a force sensor(PCB208C02),respectively.The diameter was measured by digital calipers with a resolution of 0.01 mm. The amplitude was measured by a laser displacement sensor (KEYENCE LK-H020). The surface roughness was measured by a roughness tester (Mahr M300C).When the roughness of the machined surface was larger than 0.4 μm,the thin-wall cylinder turning experiment with smaller thickness was not performed because the precision machining standard was exceeded. The surface microstructure of work-pieces was observed by a white light interferometer(ZYGO Nexview) and an ultra-depth microscope (HIROX RH-2000).All experiments were performed under a dry cutting condition.

4. Results and discussion

4.1. Chips

The chips obtained by CC and HUVC at a cutting speed was 200 m/min are shown in Fig. 8. The chips of CC are typically continuous strip-shaped long chips, whereas the chips of HUVC are typically discontinuous unit chips or needleshaped chips.During processing,large chips require larger cutting energy, resulting in greater cutting forces, higher cutting temperatures, faster tool wear, and more frequent chatter.Therefore, larger chips produced in CC are not conducive to improving the quality of the machined surface and suppressing chatter.Compared with CC,the separated cutting characteristics in the HUVC process exhibits an obvious chip breaking effect, which can effectively reduce the chip winding phenomenon and improve the stability of the cutting process.

Fig. 7 Experimental setup.

Fig. 8 Chips of CC and HUVC.

4.2. Cutting force

The cutting force is important for evaluating the cutting process. The average principal cutting forces were measured for both CC and HUVC at different thin-walled cylinder thicknesses, as shown in Fig. 9. The principal cutting forces of HUVC are much smaller than those of CC. Compared to CC,the principal force of HUVC is reduced by approximately 40% under the same parameters (v=200 m/min, f=0.005 mm/r). As discussed in Section 4.1, the material removal of HUVC can change from continuous cutting to intermittent cutting (Fig. 8), which will reduce the principal cutting forces generated by the cutting edge.

Fig. 9 Relationship between cutting force and thickness at different cutting conditions and the same MRR.

To further study the cutting forces of HUVC and CC under the same parameters (v=200 m/min, f=0.005 mm/r), the average cutting forces and dynamic cutting force were measured for a thin-walled cylinder thickness of 2 mm.

(1) The average cutting forces of HUVC and CC under the same thickness are shown in Fig. 10(a). During the CC process, the ultrasonic power is turned on to convert the system to HUVC processing. The entire measurement operation is held for 30 s. The principal force Fpand thrust force Ftin HUVC are significantly lower than the principal force Fpand thrust force Ftin CC,by 40%and 30%, respectively.

(2) The dynamic cutting force (ultrasonic-frequency repetitive impulse cutting force) of HUVC was measured using the latest dynamic cutting force measurement method,27,28as shown in Fig. 10(b). As is described in this method, when the resonant frequency of the force sensor satisfies the measurement requirement and the clamp circuit is applied, the dynamic cutting force can be obtained.A force sensor(PCB208C02)with an upper measuring frequency of 36 kHz is utilized because of the 22 kHz dynamic cutting force signal. And a constant current conditioner (PCBF484B02) that used clamp circuit can provide a continuous positive polarity signal.The period of the dynamic cutting force is 45 μs because the ultrasonic frequency is 22 kHz.The maximum value of the dynamic cutting force signal is 20 N, which is almost equal to the average cutting force of CC.During one ultrasonic cycle, the cutting force is maintained at zero for a certain period because the tool and the workpiece are separated. It is then gradually increases due to the transitional cutting process caused by the gradual increase in cutting thickness. From the perspective of the cutting force signal, HUVC exhibits separated cutting characteristics and dynamic cutting thickness characteristics. Hence, HUVC has a distinct advantage over CC method in decreasing the cutting force.

Fig. 10 Cutting force under the same parameters and cylinder thickness.

4.3. Machining precision and chatter stability

Machining errors are unavoidable during the cutting process,and machining precision is the most important evaluation indicator for thin-walled cylinder machining.Due to the low rigidity and thin thickness of the thin-walled Ti cylinder parts, the parts will be deformed under the action of the cutting force;thus,it is difficult to ensure machining precision.In this study,the diameter error rate and diameter difference between the two ends are used to evaluate the machining precision during thin-walled cylinder machining, as shown in Fig. 11.

The diameter error rate is calculated by the following equation:

where Dcut-inis the measured diameter at the cut-in position and D0is the expected diameter at the cut-in position.In order to better describe the machining precision,the diameter difference between the two ends is defined as:

where Dcut-outis the measured diameter at the cut-out position. It can be concluded from Fig. 11 that the amount of deformation of the workpiece in CC is several times the amount of deformation in HUVC. Therefore, HUVC effectively improves the precision of thin-walled cylinder machining. An important point to note here is that the diameter error rate and diameter difference between the two ends are almost invariant in HUVC. The maximum and minimum diameter error rates of CC (v=50 m/min) are 56% and 15%, respectively. Consistent with previous research results,it is advantageous to improve the machining precision by increasing the cutting speed (v=200 m/min) and reducing the feed rate. However, as the thickness of the thin-walled cylinder is reduced, the machining error is gradually increased to 51%.Under the limitation of surface quality and other conditions, the cutting speed cannot be constantly promoted;therefore, high-speed machining is not an effective solution to the problem of thin-walled Ti cylinder machining. Conversely, using the HUVC method at the same cutting speed and feed rate results in maximum and minimum diameter error rates of 10%and 3%, respectively. The diameter error rate is decreased by 40%. Ultimately, HUVC can machine thinwalled Ti cylinders with a thickness of 0.6 mm by assuming a surface roughness Ra of less than 0.4 μm; this is compared to 1 mm for CC at cutting speeds of 200 m/min and 50 m/min, respectively. It can be concluded that better machining precision can be achieved by HUVC under appropriate machining parameters.

In this study, amplitude of the shank was used to express the chatter stability, and the amplitude of the shank was measured by laser displacement sensor (sampling frequency is set to 100 kHz). Amplitude range and amplitude mean deviation are used to evaluate chatter stability during thin-walled cylinder machining, as shown in Fig. 12 and Table 3.

The amplitude range is calculated by the following equation:

where Amaxand Aminrepresent the maximum and minimum amplitudes, respectively.

The amplitude mean deviation is defined as:

It can be concluded from Fig. 12 that the amplitude of the shank in HUVC is significantly smaller than that in CC. In Fig. 12(a), the amplitude range of HUVC is 1.55 μm and is kept in a favorable cutting state. Fig. 12(b)-(d) show the results with different cutting speed and thin-walled cylinder thickness obtained by CC and HUVC. In comparison with CC, the amplitude range of HUVC is much smaller and it is possible to suppress chatter. By using HUVC, the amplitude range of the shank is reduced from 4.22 μm to 2.05 μm and chatter is effectively suppressed when the cutting speed is 200 m/min and thin-walled cylinder thickness is 1.0 mm.

Fig. 11 Relationship between machining precision and thin-walled cylinder thickness.

Fig. 12 Comparsion of amplitude in CC and HUVC.

Seen from the Table 3, the amplitude range and the amplitude mean deviation of HUVC were significantly lower than those in CC. When the thin-walled cylinder thickness are1.4 and 10 respectively, the amplitude range and the amplitude mean deviation of HUVC have decreased by up to 51.42%and 54.17%, respectively. The results show that a relatively high cutting stability can be obtained by using HUVC.

The above experimental results indicated that compared with CC,the HUVC could effectively improve the chatter stability,suppress chatter as well as contribute to obtain high precision. The above experimental results also proved the chatter suppression mechanism analysis of HUVC in Section 2. The major reasons for the above excellent effects of HUVC were attributed to the separated cutting characteristics and dynamic cutting thickness characteristics.

Table 3 Amplitude of the shank.

4.4. Surface roughness

A comparison of the surface roughness values measured after CC and HUVC processes for different thicknesses of the thinwalled Ti cylinder is shown in Fig.13.For each condition,the surface roughness of machined surface was measured three times. As seen from the graph, the Ra value of the surface machined by CC increases rapidly with a decrease in cylinder thickness, while that of the surface machined by HUVC increases slightly with the cylinder thickness reducing from 2 mm to 0.6 mm. For different cylinder thicknesses, the surface roughness Ra values obtained from CC and HUVC processes fluctuate with similar tendencies. The Ra value of surface machined by HUVC was lower than that machined by CC,decreasing by 11.7%-20.4% of that of CC at high cutting speeds.Thus,it was clear that the surface quality after HUVC was better than that in CC.

As discussed in Sections 4.1 and 4.2,the HUVC process has advantages wherein it reduces cutting forces and the chip winding phenomenon.It achieves a better reduction of surface roughness values compared with that by CC.Moreover,ultrasonic vibrations introduce ‘‘iron and press” effects on the machined surface, further reducing the surface roughness.34All of these advantages can thus guarantee a better machined surface using HUVC since HUVC requires lower cutting forces and exhibits better cutting characteristics in machining a thin-walled Ti cylinder as compared to CC.

Fig.13 Relationship between surface roughness and thin-walled cylinder thickness.

4.5. Surface microstructure

The thin-walled Ti cylinders machined by HUVC are shown in Fig. 14(a), and a machined surface comparison between CC and HUVC is shown in Fig.14(b).From a macro perspective,due to the consistency of the tool path of CC,the light reflected by the CC-machined surface is brighter and more consistent(specular reflection). In contrast, due to the dynamic cutting thickness and the overlapping tool path, the light reflected by the HUVC-processed surface is darker (diffuse reflection).This is the expected response of the surface microstructure to the different cutting edge trajectories of HUVC and CC. The surface microstructure of the thin-walled Ti cylinders under different processing conditions are shown in Figs. 15 and 16.

The 3D surface microstructures of the workpiece machined with the same machining parameters (v=200 m/min, f=0.005 mm/r)using CC and HUVC are shown in Fig.15.From the results, the density and height of the surface texture in the case of HUVC were clearly lower than those in the case of CC.Moreover, the surface of HUVC was viewed as flat and smooth from a naked-eye view, whereas obvious surface defects,caused by machining chatter,and a rough surface were observed in CC.

The surface microstructure obtained for the same machining parameters (v=200 m/min, f=0.005 mm/r) after CC and HUVC is shown in Fig. 16. To obtain macro-scale and micro-scale surface features, the surface microstructure was observed under 200×, 800×, and 2000× magnification. For CC, the surface formed into regular arrayed grooves with a width of 5 μm (i.e., corresponding to the feed rate), and interference from adjacent cutting trajectories was also encountered in the micro scale. Compared with CC,we can clearly see that uniform surface microstructure was obtained in HUVC.From the 2000× view, the surface microstructure of HUVC showed an extruded ridge shape, whereas the microstructure of CC showed a straight strip shape. As discussed in Sections 2.1 and 2.4,the separated cutting characteristics and dynamic cutting thickness characteristics of HUVC are demonstrated.

5. Conclusions

(1) CC and HUVC cutting trajectories,feed speed,and feed acceleration were compared. The feed speed and feed acceleration fluctuate periodically in HUVC but remain constant in CC. By satisfying the separation conditions,overlapping tool paths can be obtained, thereby achieving chip breaking. Due to the vibration, material removal by the cutting edge is transformed from continuous cutting by CC to discontinuous cutting by HUVC under specific conditions.

Fig. 14 Photographs of the thin-walled Ti cylinders machined by CC and HUVC.

(2) The dynamic cutting thickness of HUVC leads to a higher cutting thickness when machining thin-walled cylinders. HUVC adds ultrasonic vibrations in the feed direction, which increases the feed speed and cutting thickness. HUVC therefore enhances the fine cutting ability compared to CC. Additionally, HUVC can increase the critical cutting thickness and effectively reduce the average cutting force, thus reducing the energy intake of the system and the vibration amplitude of mode-coupling chatter.

(3) The cutting force in HUVC is smaller than that in CC,and the principal force is reduced by 40% compared with CC. By studying the dynamic cutting forces, the maximum value of the dynamic cutting force signal is found to be 20 N, which is almost equal to the average cutting force of CC.HUVC therefore demonstrates clear advantages over CC method by decreasing the cutting force.

Fig. 15 Surface microstructure shown by white light interferometer images.

Fig. 16 Surface microstructure shown by ultra-depth microscope images.

(4) The machining precision and surface roughness were compared between CC and HUVC. Compared with CC, the machining precision of HUVC is improved due to the increase in the critical cutting thickness and substantial reduction in the average cutting force.Compared with CC, the Ra value of surfaces machined by HUVC is reduced by 11.7%-20.4%.

(5) This study reveals the chatter stability and precision of thin-walled Ti cylinder machining by HUVC, thereby presenting an effective processing strategy for thinwalled Ti parts machining in the aviation and aerospace industries.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This study was supported by the Defense Industrial Technology Development Program of China(No.JCKY2018601C209).