Mechanism for material removal in ultrasonic vibration helical milling of Ti6Al4V alloy

Accepted Manuscript Mechanism for material removal in ultrasonic vibration helical milling of Ti-6Al-4V alloy

Guang Chen, Chengzu Ren, Yunhe Zou, Xuda Qin, Lianpeng Lu, Shipeng Li PII:

S0890-6955(18)30232-3

DOI:

10.1016/j.ijmachtools.2018.11.001

Reference:

MTM 3382

To appear in:

International Journal of Machine Tools and Manufacture

Received Date:

25 June 2018

Accepted Date:

04 November 2018

Please cite this article as: Guang Chen, Chengzu Ren, Yunhe Zou, Xuda Qin, Lianpeng Lu, Shipeng Li, Mechanism for material removal in ultrasonic vibration helical milling of Ti-6Al-4V alloy, International Journal of Machine Tools and Manufacture (2018), doi: 10.1016/j.ijmachtools. 2018.11.001

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ACCEPTED MANUSCRIPT Mechanism for material removal in ultrasonic vibration helical milling of Ti-6Al4V alloy

Guang Chen1,2*, Chengzu Ren1,2**, Yunhe Zou1, Xuda Qin1,2, Lianpeng Lu1, Shipeng Li1 1 Key Laboratory of Equipment Design and Manufacturing Technology, Tianjin University, Tianjin 300350, China 2 Key Laboratory of Mechanism Theory and Equipment Design of the State Education Ministry, Tianjin University, Tianjin 300350, China

*Corresponding author. Email: [email protected] **Corresponding author. Email: [email protected]

ACCEPTED MANUSCRIPT Mechanism for material removal in ultrasonic vibration helical milling of Ti-6Al-4V alloy

Abstract High quality hole-making technology in the aviation industry is urgently needed due to the application of difficult-to-cut materials, such as titanium alloy, composite materials and the stacks in aircraft fuselage skins. To improve the hole-making quality, an ultrasonic vibration helical milling (UVHM) technology was developed for machining of Ti-6Al-4V alloy, meanwhile, comparison experiments were conducted between UVHM and conventional helical milling (HM) processes. Material removal mechanism of UVHM was investigated by modeling of cutting trajectories and the analysis of tool-workpiece contact behavior for bottom and peripheral cutting edges. The actual vibration frequency in UVHM was also determined by a theoretical-experimental combined method. Due to the vibration in UVHM, the bottom cutting edges generate discontinuous contact with workpiece. Unit forces considering material removal were modeled and applied to analyze the axial force reduction. The axial cutting forces of UVHM were reduced by 38-64% compared with HM at different cutting speeds. The cutting speed of peripheral cutting edge changes periodically. The cutting edges can separate with chips due to axial vibration, which will contribute to reducing the cutting forces and improving heat dissipation. Meanwhile, a friction effect was generated by the peripheral cutting edge which can improve the micro-scale surface roughness. Due to the effects of periodical friction and compression by ultrasonic vibration, UVHM increases the surface compression stresses by 85% and 99% at the hole surface for axial and circumferential directions, respectively. Keywords: Ultrasonic vibration helical milling; Ti-6Al-4V alloy; Hole-making; Material removal Nomenclature A The vibration amplitude (mm) ap The axial feed per orbital revolution (mm/r) Cm, Cn The circles where instantaneous trajectory speeds are vm and vn, respectively Dh The target diameter of machined hole (mm) Dt The diameter of cutting tool (mm) dFB-HM The unit force caused by bottom cutting edge in HM dFB-UVHM The unit force caused by bottom cutting edge in UVHM dFP-HM The unit force caused by peripheral cutting edge in HM

ACCEPTED MANUSCRIPT dFP-UVHM The unit force caused by peripheral cutting edge in UVHM dFT-HM The unit force caused by bottom and peripheral cutting edges in HM dFT-UVHM The unit force caused by bottom and peripheral cutting edges in UVHM dsz The size of element removed by bottom cutting edge along tangential feed in HM (shown in Fig. 9a) dsp The size of element removed by peripheral cutting edge along tangential feed in HM (shown in Fig. 9a) dδz The equivalent size of element removed by bottom cutting edge along tangential feed in UVHM dδp The equivalent size of element removed by peripheral cutting edge along tangential feed in UVHM e The radius of orbital revolution (mm) f The vibration frequency (Hz) fa The axial feed speed (mm/min) ft The tangential feed speed (mm/min) fza The axial feed per tooth (mm/r) fzt The tangential feed per tooth (mm/r) Fc The resultant force at the XOY plane (N) Fr The radial direction cutting force (N) Ft The tangential feed cutting force (N) Fx The measured force along OX direction (N) Fy The measured force along OY direction (N) Fz The axial cutting force (N) g The coordinate of point P along the axis of OZ´ in Fig. 2 H The tangential feed per orbital revolution (mm) k The length of O'P in the X'O'Y' plane (0≤k≤Dt/2) k* The specific cutting coefficient

kb* The specific cutting coefficient for bottom cutting edges kp*

The specific cutting coefficient for peripheral cutting edges ke A specific coefficient related with material removal for bottom cutting edge kp A specific coefficient related with material removal for peripheral cutting edge l The length of the trajectory between the circle Cm and Cn in Fig. 6 lr The unit length along the bottom edge direction  L The length of arc A1B1

ACCEPTED MANUSCRIPT N The number of periodic micro-trajectories nrev The orbital revolution speed (rpm) nrot The spindle rotation speed (rpm) r (=O1A1) the length of cutting edge (mm) (in Fig. 4b) rn The tool nose radius (mm) s The length of trajectory along the tangential direction of the trajectory generated by the out end of bottom cutting edge, like A1A2A3 (in Fig. 4a) sN The coordinate of trajectory of Nth tooth along os direction t

The cutting time (s)

Δt The time interval from the beginning of trajectory A1A2A3 to trajectory B1B2B3(s) t0 The time when the trajectory cutting speed vt equals to the self-rotation speed of tool v2

t0 A specific time during the helical feed process tm, tn The times when the instantaneous trajectory speeds are vm and vn, respectively vt The resultant cutting speed of a bottom trajectory (mm/s) v1 The orbital speed of the tool center in Fig. 5 (mm/s) v2 The self-rotation seed of cutting tool in Fig. 5 (mm/s) vm, vn Two instantaneous speeds defined in Eq. (17) (mm/s) vs The speed of peripheral cutting edge along the trajectory direction (mm/s) vT_h The speed of peripheral cutting edge along the cutting edge direction in Fig. 8

(mm/s) vT_t The speed of peripheral cutting edge normal to the cutting edge direction in Fig. 8

(mm/s) vz The speed of peripheral cutting edge along the axial direction (mm/s) ΔV The deformation volume of discrete element (x, y, z) The coordinate of point P at the workpiece coordinate system (xUVHM, yUVHM, zUVHM) The coordinate of point P in UVHM (x', y', z') The coordinate at the tool coordinate system (xo, yo)

The coordinate of tool center in the workpiece coordinate system

(xp1, yp1) The coordinate of point P1 in the cutting path B1B2B3 in Fig. 4c (xp, yp) The coordinate of point P in the workpiece coordinate system in Fig. 4 Ze The number of tool edges

ACCEPTED MANUSCRIPT z The displacement along the axial direction of tool zN, zN+1 The coordinate of trajectory of Nth and (N+1)th tootth along oz direction, respectively Δz The distance of the lowest points of two adjacent vibration trajectories (Fig. 12a) α The helical angle of the tool feed trajectory (º) α' The rotated angle of the orbital revolution in Fig. 2b (º) β The angle between O'P and O'X' in Fig. 2b (º) γ The angle between O1B1 and O1B1 in Fig. 4b (º) γB The ratio of unit forces caused by bottom cutting edges in UVHM and HM γP The ratio of unit forces caused by peripheral cutting edges in UVHM and HM δ The helical angle of peripheral cutting edge (º) θ The angle around the origin of workpiece coordinate in Fig. 4b (θ1, θ2, θ3) The relevant angles used for defining the speeds in Fig. 5 ξ A constant of the equivalent thickness of the unit removed by bottom cutting edge in UVHM φ The intersection angle between the two cutting speeds in Fig. 5 ω1 The angle speed of orbital revolution (rad/s) ω2 The rotated angle speed of milling tool (rad/s)

1. Introduction Titanium alloys are widely used in the aviation industry because of the low density, high strength-to-weight ratio and high corrosion resistance [1][2]. In aviation industry, titanium alloys are commonly used as structural materials and aircraft fuselage skins [3]. It is reported that about 14% of the fuselage used in Boeing 787 is made by titanium alloys [4]. In recent decades, carbon fiber reinforced plastics (CFRP)/titanium alloy stack was widely used in the aircraft fuselage, due to the combined properties of high strength and ductile [5]. No matter the structural parts or aircraft fuselage, large number of connection holes need to be machined in titanium alloy to conduct the riveting and bolting assembling. It was reported that about 55000 boreholes will be machined during the fabrication of an Airbus A350 aircraft [6]. As fasten holes are easy to generate stress concentration which may lead to crack initiation and propagation [7], the quality of hole-making in aircraft structures is essential to the safety and reliability of the aircraft. Normally, manufacturing process not only affects the machining accuracy but also affects the deformation behavior and microstructure which can induce evolution of residual stress and

ACCEPTED MANUSCRIPT fatigue life. Conventional drilling is a common process which has been widely used to machine boreholes. Cantero et al. [8] carried out dry drilling experiments for Ti-6Al-4V alloy. The tool wear, surface roughness, micro-hardness and hole subsurface microstructure were investigated. It was noted that the end of tool life was caused by catastrophic failure in the drill, and the machined surface roughness Ra ranged from 1 to 1.5µm. Rahim and Sharif [9] conducted drilling experiments for Ti-6Al-4V and Ti-5Al-4V-Mo/Fe alloy. Non uniform and chipping tool wear was reported in drilling of the titanium alloys. Meanwhile, severe plastic deformation was observed from the micrograph of subsurface microstructure. Although drilling was considered as a traditional hole-making process to machining titanium alloy, several drawbacks have been addressed, such as high load along axial direction of machined hole, poor heat dissipation, low dimensional accuracy and poor chip evacuation [10]. Compared to conventional drilling, helical milling (HM) is an emerging process for holemaking which involves milling with a helical feed path. In HM, the peripheral cutting edge experiences discontinuous cutting, while, the axial cutting edge generates continuous cutting [11]. Although there is zero speed point at the bottom edge of tool center, due to the helical feed, material near the hole center is removed by cutting rather than extrusion occurred in drilling [12]. Therefore, lower cutting forces, temperature, tool wear, and improved borehole geometrical accuracy can be achieved by HM process. It was reported that H7 quality holes with a surface roughness Ra 0.3µm were obtained in hole-making of AISI D2 steel by HM [13]. HM process was also applied to machine holes for CFRP–titanium stacks. Compared with conventional drilling, less fiber delamination and lower burr formation were observed in CFRP and Ti-6Al-4V alloys, respectively [14]. Eguti and Trabasso [15] designed a device and installed it onto an industrial robot to realize orbital drilling for aircraft assembling structures. It was reported that the machined circularity (or roundness) error of 7050-T6 sheet layers reached the accuracy requirements of aircraft parts, that is, roundness error is less than 26µm [16]. More details about the geometrical accuracy of hole-making by HM, such as roughness, roundness were presented in the recently published review work [10]. Zhao et al. [12] compared the micro-hardness and the residual stresses in hole-making of Ti-6Al-4V alloy by drilling and helical milling. It was presented that the drilled hole surface generates tensile stress while the hole surface made by HM generates compressive residual stress. Therefore, helical milling can generate better level of hole surface quality, including the geometrical and physical behaviors compared with conventional drilling. However, in aircraft assembling manufacture, hole-making can be achieved by drilling with subsequent finishing operations, such as reaming, countersinking, vibration-assisted reaming [17][18]. HM is only one operation which may present lower quality compared with drilled holes with subsequent finishing process like reaming [10]. Besides, it was reported the Ti-6Al-4V workpiece machined by HM can achieve longer fatigue life compared with the drilled pieces[19], which is beneficial to increase the safety of aircraft assembling. Ultrasonic vibration, as an old assisted process, has been used in different manufacturing processes for more than 50 years [20]. Recently, it addressed more attention and was applied as

ACCEPTED MANUSCRIPT assisted strategy for conventional machining processes, such as drilling, turning, due to its effects on improvement of machining accuracy and surface quality. Pujana et al. [20] applied ultrasonic assisted drilling in machining of Ti-6Al-4V. Comparing with conventional drilling, the feed forces were reduced by 10-20%. Meanwhile, it was reported that the increasing cutting temperature may lead to strain softening effect in ultrasonic assisted drilling. Similarly, the thrust forces of ultrasonic assisted drilling of CFRP and Ti-6Al-4V were about 30% of those by drilling [21]. It was reported that the intermitted character of vibration can lead to disengagement of the tool from workpiece, which caused the force reduction. Sui et al. [22] developed a high-speed ultrasonic vibration cutting process for Ti-6Al-4V alloy machining. High speed cutting was realized at macro scale while, intermitted cutting existed in micro level, which is useful to enable the cooling efficiency and dissipate the cutting heat. Besides, the ultrasonic vibration can make the flank face of tool “iron and press” the machined surface, leading to better surface roughness [23]. Patil et al. [24] carried out ultrasonic assisted turning experiments and modeling for Ti-6Al-4V alloy. It was presented that the reduction of toolworkpiece contact ratio during ultrasonic machining leads to lower forces and friction heat, generating smooth and regular surface with better quality level than that of conventional turning. Furthermore, cutting temperature was investigated in rotary ultrasonic elliptical machining (RUEM) of CFRP. The chip removal condition was improved and cutting temperature was reduced in RUEM, which resulted in less damage in the hole surface of CFRP [25]. Ishida et al. [26] developed a hybrid ultrasonic vibration helical milling (UVHM) with cryogenic tool cooling process for the machining of CFRP. It was reported that the intermittent cutting increases the flow of cutting fluid, generating the lubrication and cooling effects, which lead to the decrease of the thrust force in CFRP machining. Overall, ultrasonic vibration has been proven to be an effective method to improve the geometrical accuracy and physical behavior of machined surface; however, the basics of micro-scale material removal are still not clear, limiting the application of the processes in aviation industry. In this work, ultrasonic vibration helical milling (UVHM) process is developed for holemaking of Ti-6Al-4V alloy. As a new process of hole-making for Ti-6Al-4V alloy, the material removal mechanism in UVHM, including the surface trajectories at the bottom and surface of machined holes will be studied. The effect of ultrasonic vibration on hole-making surface quality will be also discussed. 2. Kinematics of UVHM and material removal analysis To analyze the material removal mechanism in UVHM, the kinematics of HM is presented firstly. Afterwards, the geometrical trajectories of bottom and peripheral cutting edges in UVHM will be analyzed. 2.1 Kinematics of helical milling In helical milling (HM), the relative movement between tool and workpiece is composed of tool rotation around its axis, the circular rotation around the axis of the machined hole and the

ACCEPTED MANUSCRIPT axial feed along the axial direction, that is, the spindle rotation, the orbital revolution and the axial feed, as shown in Fig. 1. The movements of the axial feed and the circular rotation form the helical feed path. Therefore, compared with drilling, there is more space to remove chips and dissipate cutting heat in helical milling process.

Fig.1 Schematic of helical milling process

The schematic of helical milling process is illustrated in Fig. 1. The diameter of cutting tool is Dt (mm), the diameter of machined hole is Dh (mm), and the orbital revolution speed is nrev (rpm). Thus, the feed speed in tangential direction ft (mm/min) can be defined as

f t =Hnrev

(1)

where H (mm) is the tangential feed per orbital revolution, and it is calculated as

H  2πe=π( Dh  Dt )

(2)

meanwhile, the tangential feed speed ft can also be calculated based on the tangential feed per tooth fzt (mm/r) as (3) f t  f zt Z e nrot where Ze is the number of tool edges, nrot (rpm) is the spindle rotation speed. The axial feed speed fa (mm/min) can be calculated according to the axial feed per orbital revolution ap (mm/r) and the orbital revolution speed nrev (rpm).

f a  ap nrev

(4)

Similarly, it also can be calculated by the axial feed per tooth fza (mm/r) as

f a  f za Z e nrot

(5)

To analyze the speeds and movement in helical milling, the tangential, axial speeds and the geometries are shown in a cylindrical plane which contains the helical trajectory (Fig. 1). The helical angle of the tool feed trajectory α is given as

ACCEPTED MANUSCRIPT   arctan

ap H

 arctan

fa f  arctan za ft f zt

(6)

where fa (mm/min) is the feed rate along axial direction. The parameters Dh, Dt, ap, fza, and fzt are the main cutting parameters applied in helical milling. In addition, the trajectory of helical milling process will be determined based on these parameters. 2.2 Geometrical trajectory of cutting edge in UVHM Normally, the trajectory of helical milling can be determined through workpiece and tool coordinate systems [14][27]. In this work, the trajectory of UVHM is analyzed in these two coordinate systems, as shown in Fig. 2. The workpiece coordinate system is fixed on the workpiece, while the tool coordinate system is fixed on the axis of tool and the origin of tool coordinate is attached on a point in the axis of milling tool. In detail, the tool coordinate system does translational motion in the workpiece coordinate system. Meanwhile, the tool coordinate system moves with orbital revolution in the two-dimensional coordinate system, as shown in Fig. 2b.

Fig. 2 Workpiece and tool coordinate systems for UVHM (a) three-dimensional coordinate (b) twodimensional coordinate

A random point P of the milling tool locates on the XOY plane. β is the angle between O'P and O'X'. α' is the rotated angle of the orbital revolution. The coordinate value of point P in the workpiece coordinate (X, Y, Z) system can be calculated in terms of the coordinate in the tool coordinate system (X', Y', Z') according to coordinate transformation. Thus, the coordinate of point P is defined as

 x  k cos   e cos     y  k sin  +e sin   z   f t  g a 

(7)

ACCEPTED MANUSCRIPT where k is the length of O'P in the X'O'Y' plane (0≤k≤Dt/2); Dt is the diameter of milling tool; g is the position of point P along the axis of O'Z', and e=OO'= (Dh-Dt)/2, is the radius of orbital revolution. In UVHM machining, milling tool vibrates along the axial direction (OZ) with sinusoidal signal in terms of specific amplitude (A) and frequency (f) of ultrasonic vibration system. If point P locates at the peripheral cutting edge, i.e., k=Dt/2, the coordinate of point P along OZ direction in UVHM is

zUVHM  z +A sin(2πft )

(8)

If the initial values of parameters β and α' in Eq. (7) are assumed as 0, the coordinate of point P at the peripheral cutting edge in UVHM process can be expressed in the workpiece coordinate system as

2πnrevt Dt 2πnrot t   xUVHM  e cos( 60 )  2 cos( 60 )  2πnrevt Dt 2πnrot t  )  sin( )  yUVHM  e sin( 60 2 60   zUVHM   f a t  g  A sin(2πft )  

(9)

Thus, the trajectory of point P at the peripheral cutting edge in UVHM process at specific condition is calculated and shown in Fig. 3. It can be seen that periodical vibration exists in the trajectory of point P, and the amplitude is much smaller than the periodical movement of tool rotation and helical feed.

Fig. 3 Movement trajectory of point P at peripheral cutting edge in UVHM with vibration frequency f=35kHz and amplitude A=4μm (a) macro-scale curve (b) partial magnified curve

2.3 Determination of vibration frequency in UVHM Vibration frequency affects tool movement and material removal process. It was reported that the vibration frequency affects the plastic deformation behaviors, including the residual yield stress and the strain hardening rate, for the forming of lightweight materials [28]. Sajjady et al. [29] applied ultrasonic vibration assisted turning to generate micro surface texture on the surface of Al7075-T6. They found that the vibration frequency has the greatest influence on shrinking time increment which affects the generation of surface texture. Therefore, the

ACCEPTED MANUSCRIPT vibration frequency is an important factor that can affect the micro topography of surface texture and also the material plastic behaviors in machining. However, it is difficult to measure the actual vibration frequency during the UVHM. This section will present a method to calculate the actual vibration frequency in UVHM according to experimental results. 2.3.1 The trajectory of the bottom cutting edge in UVHM As shown in Fig. 3, the cutting edge of tool generates sinusoidal vibration movement in UVHM. The vibration frequency affects the micro-scale material removal mechanism, such as instantaneous cutting speed, force, heat generation, etc. Therefore, to model the material removal in UVHM, the vibration frequency should be obtained. Vibration frequency is normally dependent on the frequency of ultrasonic power. Normally, the vibration frequency is measured before machining since it is difficult to measure the vibration frequency in machining process. Besides, due to the limitation of control and display precision of the ultrasonic power, it is difficult to keep frequency constant at different machining conditions. In this work, to calculate the actual frequency in machining process, two series of UVHM experiments were carried out. Through holes were machined in one test, besides, blind holes (with depth of 1mm) were machined in the same conditions to obtain the texture generated by the bottom cutting edge to calculate the vibration frequency. At the bottom of the blind holes, the vibration frequency can be observed by identifying the number of the vibration periods and the cutting time of specific length of trajectory. Firstly, the trajectories at the bottom of blind holes were analyzed. Fig. 4 shows the schematic of cutting paths of bottom cutting edges at the two-dimensional coordinate system. At the bottom of blind holes, milling tool rotates with angle speed ω2 (rad/s) at the clockwise direction, meanwhile, the axis of tool moves along the anti-clockwise orbital revolution with angle speed ω1 (rad/s). The parameters ω1 and ω2 can be calculated according to the experimental conditions. During helical milling process, when the tool rotates an angle of 180°, one path of bottom texture generated by the outside end of one bottom cutting edge like O1A1 is shown in Fig. 4a. In detail, when the tool rotates 90°, the cutting edge O1A1 rotates to the position of O2A2, meanwhile, the center of cutting tool moves along the orbital revolution from point O1 to point O2. After same time interval, the cutting edge rotates from O2A2 to O3A3, and the center of cutting tool moves from point O2 to point O3 along the circular feed direction. As the cutting depth of every point in the cutting edge is identical, the bottom cutting edge generates an arc texture of A1A2A3 at the surface of blind hole by the outside end of cutting edge. In addition, the trajectories generated by the tool when it rotates an angle from 180° to 360° were omitted because they will be removed by the following trajectories like A1A2A3 during the helical feed process. In the trajectory, the micro-periodic texture generated by ultrasonic vibration will be discussed in section 2.3.2.

ACCEPTED MANUSCRIPT

Fig. 4 The schematic of cutting paths of bottom cutting edge for machined holes (a) generation of a path (b) calculation of trajectory (c) generation of second path (d) generation of the third path

During the rotation, the trajectory can be calculated in the two-dimensional coordinate system XOY, as shown in Fig. 4b. The coordinate of point P in the trajectory A1A2A3 can be defined in the tool coordinate system X'O1Y' as  x  r sin(2t ) (10)   y  r cos(2t ) where r=O1A1, is the length of cutting edge. Meanwhile, the coordinate of the center of cutting tool O1in the workpiece coordinate is  xo  e sin(1t )   yo  e cos(1t )

(11)

where e=OO1, is the radius of orbital revolution. Therefore, the coordinate of point P can be obtained by conversion of coordinate as   xp  r sin(2t )  e sin(1t ) (12)    yp  r cos(2t )  e cos(1t )

ACCEPTED MANUSCRIPT Note that each cutting edge will generate a trajectory at the bottom of blind hole during machining. The arc of A1A2A3 is generated by the cutting edge of O1A1, while, the next trajectory should be generated by the adjacent cutting edge O1B1 (Fig. 4b). The adjecent trajectory will start when the cutting edge rotates to O1B1 where B1 becomes the point of tangency of the the milling tool and the machined hole. When the cutting edge of O1B1 rotates an angle γ to O1B1 , the tool center rotates an angle θ from O1 to O1 , thus, these parameters can be defined as

    2π / Z e  (13)   2t    t 1  Therefore, the cutting path B1B2B3 formed by the adjecent cutting edge O1B1 (as shown in Fig. 4c) can be considered as the curve A1A2A3 revolves an angle of θ around the origin of workpiece coordintate system point O. Then, the coordinate of point P1 in curve of B1B2B3 can be defined as

 xp1  xp cos   yp sin  (14)   yp1  xp sin   yp cos  Similarly, other following curves (like C1C2C3 in Fig. 4d) at the bottom of the hole can be obtained by this method. 2.3.2 Determination of actual vibration frequency in UVHM process Due to the axial vibration, periodic micro-trajectories can be observed at the bottom of machined hole. Therefore, the vibration frequency can be calculated if the number of vibration periods and the time interval of certain length at bottom surface trajectory are determined. To determine the time interval, the instantaneous cutting speed of point P at any trajectory of bottom hole should be determined firstly.

Fig. 5 Schematic of cutting speed calculation

ACCEPTED MANUSCRIPT In Fig. 5, one curve starts from point P, after time interval t, O1P rotates an angle of θ2 around the tool center to O1P , meanwhile, the tool center rotates an angle of θ1 around the hole center O. Then, the cutting speed of point P can be considered as a resultant speed (vt) of the selfrotation speed (v2) of point P and the orbital speed (v1) of the tool center O1 . The intersection angle (φ) between the two cutting speed directions equals to θ3+π/2. Then, the instantaneous cutting speed of point P at any trajectory of hole bottom can be defined as

v  v 2  v 2  2v v cos  1 2 1 2  t v1  1e  v2  2 r   2πn 60 rev  1 2  2πnrot 60 

(15)

  3  π 2    π   +  1 2 (16) 2  3 1  1t   2  2t According to Eq. (15), the self-rotation speed v2 is much higher than the orbital speed v1(v2 ≫ v1). In this work, the cutting speeds of trajectory speed vt and tool self-rotation speed v2 during the formation of one trajectory (0≤θ2≤π) at condition No.1 (v=2000rpm, ap=0.15mm/r, fzt=0.03mm/r) are shown in Fig. 6a. Note that at time t=t0, the trajectory cutting speed vt equals to the self-rotation speed of tool v2. Then, two other instantaneous cutting speeds vm (at tm) and vn (at tn) are selected to calculate the frequency. In Fig. 6a, the speed v2 is in the middle of speeds vm and vn, that is, v v (17) v2  m n 2

ACCEPTED MANUSCRIPT

Fig. 6 Variation of cutting speed at the bottom cutting trajectory at condition No.1(v=2000rpm, ap=0.15mm/r, fzt=0.03mm/r) (a) the comparison of cutting speed v2 and vt (b) trajectory with the cutting speed range of vm and vn (c) the machined micro-scale texture

In this work, the cutting speeds of vm and vn were selected as vn˗v2= v2˗vm=1(mm/s) as shown in Fig. 6a. Submitted t0, tm and tn to Eqs.(12)-(14), the points of bottom trajectory with instantaneous speed of v2, vm and vn can be obtained (in Fig. 6b) as the circles of Cv, Cm and Cn, respectively. Then the length of a trajectory l between the circle Cm and Cn can be determined by the integral of trajectory speed vt (in Fig. 6a). Meanwhile, as the speed v2 is in the middle of speeds vm and vn, the length of l can be calculated as tn

l   vt dt  v2 (tn  tm ) tm

(18)

Therefore, the frequency f can be calculated during time interval between tm and tn if the number of vibration periods of curve l is obtained. The actual vibration frequency is calculated as Nv2 N (19) f   tn  tm l where N is the number of periodic micro-trajectories, and l is the length of measured trajectory which locates between two trajectory circles with speeds meet Eq. (17). Then, actual vibration frequency is calculated by measuring the micro-trajectory at each condition. It should be noted that the actual frequency will not be affected by the values of cutting speeds vm and vn when Eq. (17) is satisfied, because the variable cutting speed (vt) of trajectory l can be calculated by v2 according to Eq. (17). 2.4 Material removal for the bottom of machined hole In section 2.3, the trajectories at the bottom of machined holes were modeled at the XOY plane, however, the vibration of trajectory and axial feed were not considered in the modeling process. In this section, to analyze the material removal at the bottom of machined hole and the hole surface, the trajectory like A1A2A3 is analyzed at a coordinate system of soz, where os is the tangential direction of one trajectory generated by the out end of bottom cutting edge, such as trajectory A1A2A3 (as shown in Fig. 4c), oz is the axial direction of tool. In HM, the axial

ACCEPTED MANUSCRIPT position of any point of cutting tool can be expressed by the axial feed per orbital revolution (ap) as

z

ap nrevt

(20) 60 In each orbital revolution, the tangential feed distance in one orbital revolution can be expressed as f Zn 2πe  zt e rot (21) nrev then, the orbital revolution velocity nrev can be calculated by Zn f (22) nrev  e rot zt 2πe Therefore, using Eq.(20), Eq.(22) and the calculated velocity along the trajectory in Eq. (15) the tool tip trajectory of the HM at the soz coordinate can be described as  s  t v dt  0 t  Zn a f t  z   e rot p zt 60  2πe 

(23)

In addition, considering the vibration displacement Asin(2πft), the trajectory generated in UVHM at soz coordinate can be calculated as  s  t v dt  0 t  Zn a f t  z   e rot p zt  A sin(2πft ) 60  2πe 

(24)

Thus, the Nth trajectory of cutting edge in UVHM is expressed as  s  t v dt  NL  N 0 t (25)  Z e nrot ap f zt  zN    t  N t   A sin  2πf  t  N t   60  2πe   where L is the length of arc A1B1, as shown in Fig. 4c. As two adjacent curves (like A1A2A3 and B1B2B3 in Fig. 4c) can machine same area of hole surface, the os coordinate in the soz  coordinate system should calculate the arc A1B1 ( L ) along the os coordinate direction by Eq.(25). Δt is the time interval from the beginning of trajectory A1A2A3 to the beginning of trajectory B1B2B3. It can be calculated according to Eq. (13) as 2π t  (26) Z e (1  2 ) Using the developed equations, the coordinate of any point at the trajectory generated by the bottom cutting edge at the soz coordinate system can be calculated. 2.5 Material removal for the hole surface

ACCEPTED MANUSCRIPT To analyze the tool-chip contact in the hole surface, the velocities of peripheral cutting edge in HM and UVHM are calculated. The schematic of peripheral cutting edge involved in HM and UVHM is illustrated in Fig. 7a and b, respectively. Due to the self-rotation and orbital revolution, the cutting edge has a velocity of vs along the trajectory s, meanwhile, it feeds along axis z with the speed vz in HM. Then, the cutting speed of short helical cutting edge which generates chips by HM can be calculated by the derivative of displacement in the soz coordinate system according to Eq. (23) ds  vs = dt  vt  v = dz   Z e nrot  ap f zt  z dt 60 2πe

(27)

For the UVHM, considering the movement caused by vibration, the cutting speed of peripheral cutting edge can be calculated according to the derivative of displacement defined by Eq. (24) ds  vs =  vt   dt  v = dz   Z e nrot  ap f zt  2πfA cos(2πft ) z  dt 60 2πe 

(28)

ACCEPTED MANUSCRIPT Fig. 7 Schematic of chip formation related with peripheral cutting edge (a) in HM (b) in UVHM (c) the velocity of peripheral edge in the soz coordinate system

Verma et al. [30] investigated the speed components along the helix and transverse directions in ultrasonic vibration assisted milling (UAM), and the intermitted cutting was observed by the instantaneous cutting speed in UAM. In this work, the cutting speed of peripheral cutting edge in UVHM is characterized in the coordinate system of soz, as shown in Fig. 7c. The short helical cutting edge can be considered as a short line (like oblique cutting) because os is the tangential direction of the trajectory of peripheral cutting edge. In addition, the uncut chip thickness along the cutting edge is identical. In soz coordinate system, another coordinate system x'o'y' is attached on the cutting edge, that is, o'x' is in the direction along helix cutting edge which has a helix angle δ=38°. Then, the instantaneous cutting speed along the helix cutting edge direction o'x' (vT_h) and the cutting speed along the transverse direction of helix cutting edge direction o'y' (vT_t) can be deterimined as  vT_h  vssin -vz cos  (29)   vT _t  vs cos +vz sin 

Fig. 8 Cutting speeds and the state of tool-chip contact at test of No.2 (v=2500rpm, ap=0.15mm/r, fzt=0.03mm/r) with amplitude of vibration 4.2µm

The calculated vT_h and vT_t with UVHM at one test condition (such as No.2) are shown in Fig. 8. It should be noted that the amplitude of vibration may change slightly due to the variation of cutting forces at different conditions. The actual amplitude can be obtained from the texture of hole surface. In test No.2, the measured amplitude is about 4.2µm. For one vibration cycle, the cutting speed of peripheral cutting edge can be divided into five stages: a~e. At stage a, vT_t >0 and vT_h <0, so the move direction of cutting edge is indicated as the dash line in stage a (Fig. 8). Similarly, the movement of cutting edge at stages b~e is also illustrated in Fig. 8. Note that, at stage c, vT_t <0, which means the cutting edge separates from the formed chip. That is, tool can separate with cutting chip periodically due to the vibration effect in UVHM. In addition, the relative speed between the tool and the chip at hole surface is not constant. Compared with

ACCEPTED MANUSCRIPT the constant speeds vT_h and vT_t in HM, cutting speeds vT_h and vT_t in UVHM vary as sinusoidal waves. The highest vT_t in UVHM is about twice of the speed in HM. Normally, the increase of instantaneous cutting speed can increase the cutting temperature which will lead to a thermal softening effect, reducing the cutting forces [20]. Besides, it was reported that the reduction of tool-chip can lead to reduction of cutting forces [31], however, the vibrationassisted milling does not always result in reduced forces due to dependence on vibration frequency [32]. Meanwhile, the speed vT_h in UVHM varies from positive to negative, indicating a movement forward and back along the direction of cutting edge, which will generate a friction effect to the machined surface. The influence of the effect on the machined surface will be discussed in section 3.4. 2.6 Modeling of the ratio of unit forces considering material removal Based on the force modeling theories, the cutting forces of bottom and peripheral cutting edges can be calculated based on the discrete chip volume [33, 34]. The local forces dF along axial direction can be calculated based on the material removal of unit element as follows: dF  k *V

(30) where is the specific cutting coefficient, ΔV is the discrete deformation volume. In the study, the unit force is classified as the force caused by bottom cutting edge and the force caused by peripheral cutting edge, respectively. The forces caused by bottom cutting edge in HM and UVHM are defined as dFB-HM and dFB-UVHM, respectively. k*

dFB-HM  kb* f za dsz lr

(31)

dFB-UVHM  k zd zlr * b

(32) where dsz is the size of element removed by bottom cutting edge along the direction of tangential feed in HM as shown in Fig. 9a. dδz is the equivalent size of element removed by

k*

bottom cutting edge along tangential feed in UVHM. b is the specific cutting coefficient for bottom cutting edges. lr is the unite length along the bottom edge direction. In this work, it is assumed that the equivalent size of element removed in UVHM, dδz, decreases with the increases of the vibration frequency at unit width as v ds d  z  ke z  ke t dsz f f vt (33) where ke is a specific coefficient.

ACCEPTED MANUSCRIPT

Fig. 9 Schematic of material removal under different conditions (a) variation of material removal under different cutting speeds (b) variation of material removal under different axial feeds (c) variation of material removal under different tangential speeds

According to Eqs. (1)-(5), the axial feed per tooth can be expressed as

f za 

ap f zt 2πe

(34) With respect to UVHM, the coordinates of the lowest points of two adjacent trajectories generated by bottom cutting edges are calculated according to Eq. (25) as Z e nrot ap f zt   t0  N t   A  z N   60  2πe  (35)  z   Z e nrot ap f zt t    N  1 t   A 0   N+1 60  2πe  Thus, the distance of the lowest points of two adjacent vibration trajectories (Δz) generated by bottom cutting edge can be calculated by

z  z N  z N 1 

 Z e nrot ap f zt  2π   60  2πe  Z e 1  2  

(36) As the angle speed of milling tool is much higher than that of orbital revolution, i.e., ω2>>ω1, then ω1+ω2≈ω2=2πnrot/60; then Δz is further calculated according to Eqs.(34) and (36),

z 

 Z e nrot ap f zt Z e nrot ap f zt  2π   60  2πe  Z e 1  2   60  2πe

 2π     f za Z   e 2

(37) In this work, the mill tool has four cutting edges, i.e., Ze=4. The ratio of unit forces caused by bottom cutting edges in UVHM and HM can be defined as dF zd z  ke vt  B  B-UVHM   dFB-HM f za dsz f (38) As ω2>>ω1, the cutting speed vt≈ω2r=2πnrot r/60. Therefore, the ratio of unit forces generated by bottom cutting edge in UVHM and HM meets the following relationship n  B  rot f (39)

ACCEPTED MANUSCRIPT With respect to the material removal by peripheral cutting edges, the unit forces caused by peripheral cutting edge in HM and UVHM are defined as dFP-HM and dFP-UVHM, respectively. Note that the unit element volume is related with the cutting time, that is, the material removal varies with time, then the unit forces caused by peripheral cutting edge in HM and UVHM are defined as follows:

dFP-HM (2t )  kp*dsp df ztt-HM (2t )dapt-HM (2t )

(40)

dFP-UVHM (2t )  kp*d p df ztt-UVHM (2t )dapt-UVHM (2t )

(41) where dfztt-HM and dfztt-UVHM are the instantaneous tangential feed per tooth in HM and UVHM, respectively. dapt-HM and dapt-UVHM are the instantaneous axial feed in HM and UVHM,

k *p

respectively. is the specific cutting coefficient for peripheral cutting edges. As shown in Fig. 9, material removal by peripheral cutting edge is related with axial feed per revolution ap and the tangential feed per tooth fzt. Similar with the material removal for bottom cutting edges, when ap and fzt remain constant, the equivalent specific chip tangential length dδp decreases with the increase of the vibration frequency at unit width as ds v d  p  kp p  kp t dsp f f vt (42) where kp is a coefficient as ke. Thus, the ratio of unit forces caused by peripheral cutting edges in UVHM and HM can be defined as

 P t  

dFP-UVHM kp vt df ztt-UVHM (2t )dapt-UVHM (2t )  dFP-HM fdf ztt-HM (2t )dapt-HM (2t )

Then, the ratio of total unit forces in UVHM and HM can be calculated as dFT-UVHM dFB-UVHM  dFP-UVHM  B dFB-HM   P dFP-HM   dFT-HM dFB-HM  dFP-HM dFB-HM  dFP-HM

(43)

(44)

3. Experimental investigation for hole making by HM and UVHM In this work, a series of experiments were presented to investigate the material removal in UVHM. The modeled trajectories were compared with the texture of machined surfaces. Cutting forces, surface roughness and residual stresses of machined surface were also presented and discussed based on the material removal analysis. 3.1 Experiment system Hole making of commercial grade Ti-6Al-4V alloy by HM and UVHM processes were performed on a DMU 60 monoBLOCK® vertical machining center. The experimental setup is shown in Fig. 10. The ultrasonic vibration tool holder is mounted on the spindle of the machining center. A non-contact electromagnetic unit was applied to transform the electrical

ACCEPTED MANUSCRIPT signal. An upper ring of the electromagnetic induction coil is fixed on the spindle shaft of the machine. Ultrasonic power (type: SZ12) was connected with the upper ring of the electromagnetic induction coil. In UVHM, the ultrasonic vibration tool holder with another electromagnetic induction coil can receive the electrical signal, which will drive the piezoelectric ceramics to generate high frequency vibration for tool holder. The milling tool fixed on the ultrasonic tool holder can generate ultrasonic vibration when the ultrasonic power is turned on. The system can generate about 4~6µm amplitude of vibration with about 35kHz frequency.

Fig. 10 Experimental setup for Ti-6Al-4V hole-making by UVHM

Ultrafine grained WC-12%Co cemented carbides milling tools with 6mm diameter and 4 peripheral edges were used in the experiments. The parameters of tool geometry are listed in Table 1. Commercial grad Ti-6Al-4V alloy workpiece was installed on a clamp which was fixed on the dynamometer, as shown in Fig. 10. Cutting forces were measured by a Kistler® 9257A three-component dynamometer and a 5070 charge amplifier. The dimension of workpiece is 258mm×125mm×5mm. The designed hole diameter that needs to be machined is 10mm. The distance between the axes of two adjacent holes is 15mm. All experiments were carried out under dry cutting condition. Table 1 Geometrical parameters for helical milling tool Helix angle (°)

Tool length (mm)

38

60

Axial cutting Length (mm) 7

Rake angle of bottom edge (°) 0

Rake angle of peripheral edge (°) 5

Clearance angle of bottom edge (°) 8

Clearance angle of peripheral edge (°) 10

Tool nose radius (mm) 0.5

In machining, the process of HM can be switched to UVHM by turning on the ultrasonic power. In this work, the experimental conditions for HM and UVHM processes were listed in Table 2. At each condition, two holes were machined by HM and UVHM processes, respectively. In order to eliminate the random error, all experiments were repeated three times, and a new tool was used for each set of repeated experiments.

ACCEPTED MANUSCRIPT

Test conditions (No.) 1 2 3 4 5 6 7 8 9 10 11 12 13

Table 2 Machining parameters for hole-making of Ti-6Al-4V alloy Spindle rotation Axial feed Tangential feed Axial feed per tooth speed nrot (rpm) per tooth fzt fza (mm/r) ap (mm/r) (mm/r) 2000 0.15 0.03 0.00035828 2500 0.15 0.03 0.00035828 3000 0.15 0.03 0.00035828 3500 0.15 0.03 0.00035828 4000 0.15 0.03 0.00035828 2500 0.15 0.02 0.000238854 2500 0.15 0.04 0.000477707 2500 0.15 0.05 0.000597134 2500 0.15 0.06 0.000716561 2500 0.1 0.03 0.000238854 2500 0.2 0.03 0.000477707 2500 0.25 0.03 0.000597134 2500 0.3 0.03 0.000716561

Actual vibration frequency f (Hz) 33603 33651 34658 34530 34798 33802 33454 33792 33504 33634 33610 33507 33680

To investigate the machining mechanism and machined hole accuracy, cutting forces, surface roughness and residual stresses were tested. In addition, in order to remove residual chips and impurities from hole surface, ultrasonic cleaning was performed on workpiece before the measurement of surface roughness. 3.2 Cutting trajectories of bottom and hole surface In this work, when the tool center rotates an angle of 360°, the curve formed by one cutting edge was calculated based on the method proposed in section 2.3.1. The calculated and experimental bottom trajectories (measured by KENYENCETM VHX-2000 microscopy) at different tangential feeds (in Table 2, test conditions: Nos. 2, 6, 7, 8, 9) are shown in Fig. 11. Due to the 0.5mm tool nose radius, a cyclic annular area (CA, as shown in Fig. 11a) exists in the bottom trajectory. It can be seen that the calculated trajectories agree well with the measured curves. The bottom trajectory is mostly affected by the orbital feed (i.e., tangential feed) speed. Note that the calculated curves in Fig. 11 did not consider the micro-periodic texture.

ACCEPTED MANUSCRIPT

Fig. 11 Comparison of the bottom texture (v=2500rpm, ap=0.15mm/r) at different tangential feeds (a) fzt=0.02mm/r (b) fzt =0.03mm/r (c) fzt =0.04mm/r (d) fzt =0.05mm/r (f) fzt =0.06mm/r

To calculate the bottom cutting trajectory in UVHM, the actual vibration frequency in UVHM is calculated based on the developed method in section 2.3. The calculated frequency values

ACCEPTED MANUSCRIPT are listed in Table 2. According to the trajectory modeling method in section 2.4, the bottom trajectories can be obtained. Fig. 12a shows eight paths generated by the end of bottom cutting edges (point P in Fig. 5) for cutting condition of test No.1 at the soz coordinate system. The shadow area covered by the 8th path represents the material removed by the bottom edge. The experimental texture of the machined bottom surface was measured by KENYENCETM VHX2000 microscopy and shown in Fig. 12b. The distance of adjacent shadow area agrees well with the experimental result (about 18µm). It can be seen that the material was removed periodically and interrupted, and it is affected by the vibration and cutting conditions. Similarly, microintermitted turning process was accomplished by high speed ultrasonic vibration turning process [22], and it was stated that the transition from continuous cutting to separation state in high speed ultrasonic turning is related with the flank face interference [35]. It should be mentioned that the intermitted cutting involved in vibration assisted machining processes is a combined kinematic result, which is normally realized by the relative movement of workpiece and cutting edge concerning vibration.

Fig. 12 The end of bottom cutting edge movement trajectory during UVHM (Test No.1) (a) calculated curves (b) measured texture

In addition, as the angle speeds of the tool self-rotation and orbital rotation in UVHM are constant, the trajectory generated by the end of peripheral cutting edge can be calculated considering the tool nose radius (rn=0.5mm) by the same modeling method, that is, r should be changed as O1A1+rn=3mm. The calculated and experimental trajectories (measured by VHX-2000 microscopy) generated by peripheral cutting edge are shown in Fig. 13. The trajectory is mainly affected by the tool rotation speed and the vibration frequency. Note that the bottom of calculated curves (in Fig.13a-c) represents the texture of machined surface, and the calculated curves are consistent with the experimental textures of hole surface. Ultrasonic elliptical vibration cutting was applied for machining microgrooves on round-shaped AISI 1045. It was found that the cutting speed and the vibration frequency of elliptical locus are the two main factors which determine the morphology of microgrooves [36]. Therefore, the micro-texture of the machined surface varies with the tool rotation speed and vibration

ACCEPTED MANUSCRIPT frequency. The agreement between the calculated and measured micro-texture also indicates the accuracy of the calculated frequency by the method proposed in section 2.3.

Fig. 13 Comparison of calculated curves and measured texture of hole surface at test conditions of No.1(a and a'), No.4 (b and b') and No.11 (c and c').

3.3 Comparison of cutting forces and axial force reduction In helical milling, cutting forces were measured by dynamometer with sampling frequency of 20kHz. The cutting forces of Fx and Fy have similar variation trend since the tool rotates in the XOY coordinate system, as shown in Fig. 2b. Therefore, the total force in the XOY plane is normally used to express the overall resultant force [37][38]. The resultant force Fc in the XOY plane is defined as

ACCEPTED MANUSCRIPT

Fc 

Fx2  Fy2 

Fr2  Ft 2

(45)

where Ft and Fr are the tangential (feed) and radial (thrust) cutting forces, respectively. In this work, the calculated resultant force Fc and the axial force Fz by HM and UVHM at different cutting conditions are shown in Fig. 14. The resultant forces Fc are much smaller than axial forces Fz. As the four bottom cutting edges can remove material simultaneously at the XOY plane, most of the resultant force caused by the bottom edge should be balanced with the four edges along the X and Y direction. Therefore, the main part of the resultant force Fc should be caused by the cutting of peripheral cutting edges [39]. Both of peripheral and bottom cutting edges can contribute to the axial cutting force Fz. In HM, the axial forces increase with the increase of tangential and axial feeds. The cutting speed has little influence on the forces. In addition, the resultant forces of UVHM are close to the forces of HM. However, the axial forces of UVHM are much smaller than those of HM. The axial forces in UVHM at different cutting speeds are reduced by 38-64% compared to the forces in HM. Besides, the axial forces of UVHM are reduced by 19-48% and 28-47% at different tangential and axial feeds, respectively. As discussed in section 3.2, the material removal of bottom cutting edge with ultrasonic vibration can change from continuous cutting to discontinuous cutting as shown in Fig. 12. Then the axial cutting forces generated by bottom cutting edge will be reduced. Besides, based on the analysis of material removal of hole surface, the peripheral cutting edge involved in cutting can be considered as an oblique cutting process. In UVHM, the peripheral cutting edge moves along the helix cutting edge and the normal of cutting edge directions (Fig. 8). Due to the vibration, based on the analysis of speed component along the normal direction of peripheral cutting edge, the highest cutting speed in UVHM is about twice of that in HM. Meanwhile, the tool can separate from the chip at specific time interval during a vibration period. Besides, the forward and backward movement along the cutting edge direction will change the friction behaviors at tool rake face. It was reported that the length of sticking zone in cutting is strongly influenced by cutting speed. The tool-chip contact length and the cutting forces can be reduced with the increase of cutting speed in Ti6Al-4V orthogonal cutting [39]. Therefore, the axial vibration in UVHM can change the toolchip contact behavior and reduce the axial forces greatly compared with those in HM. Note that the mechanism of force reduction in UVHM machining of Ti-6Al-4V alloy is different from that using hybrid UVHM with cryogenic tool cooling in machining of CFRP, which is mainly caused by the increase of lubrication and cooling effects [26].

ACCEPTED MANUSCRIPT

Fig. 14 Comparison of cutting forces at different (a) cutting speeds; (b) tangential feeds; (c) axial feeds.

In this work, the force reduction ratio of the axial forces between HM and UVHM processes will be discussed based on the modeling of ratio of unit forces, concerning the material removal analysis in section 2.6. Normally, the instantaneous cutting forces are calculated by summation or integration of unit forces along the bottom and peripheral teeth involved in material removal during HM process [33, 40, 41]. Note that the ratio of unit forces between HM and UVHM for each unit element along the cutting edge is identical. Therefore, the ratio of axial forces between in HM and UVHM can be characterized by the ratio of unit forces. When the axial feed ap, tangential feed per tooth fzt and axial feed per tooth fza remain constant and the cutting speed nrot (i.e., spindle rotation speed) increases (as shown in Fig. 9a), γB is proportional to nrot/f according to Eq. (39). For peripheral cutting edge, when cutting speed increases, the ratio of unit forces in HM and UVHM caused by peripheral cutting edges can be defined by Eq. (43) as v n  P  kp t  rot f f (46) Both of γB and γP are proportional to nrot/f, then the ratio of total unit forces caused by bottom and peripheral cutting edges can be defined according to Eq. (44) as

ACCEPTED MANUSCRIPT dFT-UVHM  B dFB-HM   P dFP-HM nrot   dFT-HM dFB-HM  dFP-HM f

(47)

Then the force reduction will decrease linearly with the increase of the ratio of unit forces in UVHM and HM, as shown in Fig. 15a.

Fig. 15 The reduction ratio of axial force Fz using UVHM compared with HM (a) under different cutting speeds (b) under different axial feeds

When fzt and nrot remain constant and, ap and fza increase (as shown in Fig. 9b), for bottom cutting edge, γB is constant according to Eq. (39). With respect to peripheral cutting edge, the cutting width of the peripheral cutting edge dapt increases, however, the unit axial feed dapt-UVHM is equal to dapt-HM at any time. Additionally, the peripheral cutting process can be considered as an oblique cutting process, when ap increases, it can be considered as the cutting width increases, then the peripheral cutting forces will increase accordingly. Although the forces caused by peripheral cutting edge increase when ap increases at each moment, the ratio γP remains constant at different axial feeds. That is, the cutting condition will not affect the ratio of forces in HM and UVHM. Therefore, the total force reduction ratio remains constant, as shown in Fig. 15b. When ap and nrot remain constant and, fzt and fza increase (as shown in Fig. 9c), for bottom cutting edge, γB is constant. With respect to peripheral cutting edge, when the tangential feed fzt increases, the ratio of dapt-uvhm/dapt-hm keeps constant, however, the dfztt changes with time. Additionally, if the peripheral cutting process is considered as an oblique cutting process, when fzt increases, it can be considered as the uncut chip thickness increases. Normally, the cutting force does not change with the identical ratio with that of the uncut chip thickness. Therefore, the ratio of peripheral cutting forces γP will change with the increase of fzt. Therefore, the force reduction of axial force changes without obvious rule with the increase of the tangential feed. 3.4 Comparison of machined holes by HM and UVHM

ACCEPTED MANUSCRIPT To analyze the hole-making quality and mechanism, surface roughness and residual stresses by HM and UVHM were tested for different machining conditions. 3.4.1 Surface roughness Surface roughness Ra was measured by Talysurf® 2300A-R surface profiler instrument. The tester probe scans the surface profile with a cut-off length of 0.8 mm and the total test length 4.0 mm respectively. For each condition, the surface roughness Ra of hole surface was measured four times at different angles. The intersection angle between two adjacent measuring paths is 90° (as shown in Fig.16a). Comparison of the measured surface roughness by HM and UVHM at different conditions is shown in Fig. 16. The Ra value of hole surface machined by HM increases with the increase of cutting speed, while, the Ra value of hole surface machined by UVHM deceases slightly with cutting speed until 3000 rpm, afterward, it increases with cutting speed. At different axial and tangential feeds, the surface roughness Ra by HM and UVHM processes fluctuates with similar tendency. Ra values with UVHM are smaller than those with HM for all test conditions.

Fig. 16 Comparison of surface roughness at different (a) cutting speeds (ap=0.15mm/r, fzt=0.03mm/r); (b) tangential feeds(v=2500rpm, ap=0.15mm/r); (c) axial feeds(v=2500rpm, fzt=0.03mm/r).

The decrease of Ra value by UVHM can be explained by the surface generation related with material removal in section 2.4 and 2.5. Besides, the schematic of cutting process of peripheral cutting edge in HM and UVHM is illustrated in Fig. 17. In HM, peripheral cutting

ACCEPTED MANUSCRIPT edge feeds along the helical feed direction with cutting speed vs (Fig. 17a). In UVHM, in addition to the helical feed, the cutting edge vibrates along axial direction and generates a periodic trajectory in the hole surface (Fig. 17b). As discussed in section 2.5, the vibration can generate a friction effect for the machined surface in UVHM, which is useful to rub the micropeaks of machined surface texture. It can be observed by the microscopic morphology of machined hole surface (measured by a WykoTM NT9300 optical profiler), as shown in Fig. 18. For the surface machined by HM, the major direction of texture is along the feed direction. For the surface machined by UVHM, some miro-peaks of machined surface texture are rubbed flat by the vibration combined movement, as shown in Fig. 18b. Therefore, the roughness values with UVHM are smaller than those with HM.

Fig. 17 Schematic of surface roughness generation by the process of (a) HM (b) UVHM

Fig. 18 The micro-scale morphology of machined surface (v=2000rpm, ap=0.15mm/r, fzt=0.03mm/r) by (a) HM (b) UVHM

Furthermore, the improvement of surface roughness by UVHM is related with the vibration trajectory at specific length along the circumferential direction of machined hole. According to the trajectories of hole surface in HM and UVHM defined by Eq.(23) and Eq.(24), respectively, the improvement of surface roughness by UVHM is related with the ratio of 𝑡

vibration effect Asin(2πft) and the trajectory length s=∫0𝑣𝑡𝑑𝑡. Meanwhile, the vibration effect was also indicated in Fig. 13. When the vibration amplitude and frequency keep constant, the

ACCEPTED MANUSCRIPT improvement of surface roughness is affected by the trajectory speed vt which is mainly dependent on tool self-rotation cutting speed. Therefore, the reducing rate of Ra with UVHM changes with rotation speed compared to HM. While, at different axial and tangential feeds, the variation tendencies of Ra of hole surfaces machined by UVHM and HM are similar with each other, due to the same cutting speed and similar vibration frequency and amplitude in the hole surface. 3.4.2 Residual stress To check the stress state of the machined surface, the axial and circumferential residual stresses of machined hole surface were tested by a μ-X360n® X-ray residual stress measurement system. To determine the residual stress profiles at different depths of machined surface, successive layers of material were removed by electro-polishing. The axial and circumferential residual stresses of hole surfaces machined by HM and UVHM are shown in Fig. 19.

Fig. 19 Comparison of residual stresses (v=2000rpm, ap=0.15mm/r, fzt=0.06mm/r) along the (a) axial and (b) circumferential directions

The residual stresses are compressive in both axial and circumferential directions. The axial and circumferential residual compressive stresses decrease with the increase of depth from machined surface, and they reach the initial state at approximately the depth of 100μm from machined surface. Besides, the compressive residual stresses of the UVHM are larger than that of HM, when the depth of machined surface is within 60µm and 30µm in axial and circumferential directions, respectively. The residual stresses of the hole surface at the axial direction are -285 MPa and -154 MPa for UVHM and HM, respectively. While, the residual stresses in the circumferential direction are -213 MPa and -107 MPa for UVHM and HM, respectively. Therefore, compared with HM process, UVHM increases the surface compressive stresses by 85% and 99% at the hole surface for axial and circumferential directions, respectively. Residual stress is commonly caused by plastic deformation of the surface and subsurface of workpiece owing to the mechanical and thermal loads. With respected to helical milling, it has

ACCEPTED MANUSCRIPT been reported that it generates compressive residual stress, which can extend the fatigue life of hole surface [12]. Compared with helical milling, the ultrasonic vibration generates impulsive sinusoidal trajectories by peripheral cutting edges on the machined surface. During the axial movement of the cutting edge, the micro-peaks of machined surface are rubbed by the friction effect. Material near the micro-peaks of machined texture is compressed by the periodic friction, leading to higher compressive residual stresses at the surface and subsurface of the machined holes. 4. Conclusions In this paper, a novel ultrasonic vibration helical milling process (UVHM) was developed for hole-making of Ti-6Al-4V alloy. The material removal mechanics of bottom and hole surfaces were analyzed according to the modeling of machined trajectories and cutting speeds. The quality of machined holes by HM and UVHM was also presented and discussed according to the material removal analysis. 1. The actual vibration frequency was calculated by the measured texture at bottom surface of machined holes in UVHM. With the calculated frequency, the movement of bottom cutting edges was modeled. Due to the vibration, the material removal of bottom cutting edge transmitted from continuous cutting by HM to discontinuous by UVHM at specific conditions. 2. The trajectory of hole surface was modeled and the cutting speed of peripheral edge was analyzed. The cutting speed in UVHM fluctuates periodically rather than keeps constant in HM. The variation of periodical cutting speed leads to higher cutting speed along the normal cutting edge direction periodically. The peripheral edge can separate with chips formed at hole surface, which can reduce the cutting forces and increase the heat dissipation. The modeled trajectories on the bottom and hole surfaces agreed well with the experimental results, which indicated the effectiveness of the calculation of the actual vibration frequency for studying the material removal in UVHM. 3. The ratio of unit forces in UVHM and HM were modeled based on the material removal analysis of the bottom and peripheral cutting edges. Axial forces in UVHM were much smaller than those of HM, and the maximum decrease of axial force was 64% compared with HM. Based on the modeling of unit forces in HM and UVHM, the force reduction decreases with the increasing rotation speed and remains stable for different axial feeds. 4. Surface roughness and residual stress were compared between the processes of HM and UVHM. Compared with HM, the surface roughness in UVHM was improved due to a friction effect which rubs the micro-peaks of machined surface texture. Hole surface machined by UVHM generates plastic deformation by the effects of periodical friction and compression. Compared with HM, UVHM increases the compressive stresses of hole surface by 85% and 99% for the axial and circumferential directions, respectively. The improvement of surface roughness and residual stresses were

ACCEPTED MANUSCRIPT analyzed according to the material removal mechanism concerning the cutting trajectories and the speed of cutting edge. 5. This study reveals the underlying material removal mechanism for hole-making of Ti6Al-4V alloy by UVHM, which provides an effective processing strategy for holemaking of aviation alloys.

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ACCEPTED MANUSCRIPT Acknowledgements The authors are very grateful for the support received from the Natural Science Foundation of China No. 51575384, the Natural Science Foundation of Tianjin Nos.16JCQNJC04600, 16JCZDJC38300. The authors would like to gratefully acknowledge anonymous reviewers for their careful work which helped improve an earlier version of the manuscript substantially.

ACCEPTED MANUSCRIPT Highlights

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Ultrasonic vibration helical milling process was developed for machining of Ti6Al-4V alloy.

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The actual vibration frequency was calculated by the cutting trajectories generated by bottom cutting edge.

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The trajectories and the cutting speeds of the bottom and peripheral edges were modeled.

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The axial cutting forces are reduced up to 64% due to the interrupted cutting effect in UVHM.

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Surface roughness and compressive residual stress were improved by periodic friction and compression in UVHM.