Analytical modeling of material removal mechanism in dry whirling milling process considering geometry, kinematics and mechanics

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Analytical modeling of material removal mechanism in dry whirling milling process considering geometry, kinematics and mechanics Lexiang Wang , Yan He , Yulin Wang , Yufeng Li , Chao Liu , Shilong Wang , Yan Wang PII: DOI: Reference:

S0020-7403(19)33446-0 https://doi.org/10.1016/j.ijmecsci.2020.105419 MS 105419

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

11 September 2019 13 December 2019 3 January 2020

Please cite this article as: Lexiang Wang , Yan He , Yulin Wang , Yufeng Li , Chao Liu , Shilong Wang , Yan Wang , Analytical modeling of material removal mechanism in dry whirling milling process considering geometry, kinematics and mechanics, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105419

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Highlights 

Analytical modeling approach of material removal mechanism in whirling milling is proposed.



Models to predict undeformed chip geometry, MRR, cutting forces and form errors are developed.



Analytical models are validated with the largest error 11.3% and average error 13% for force and surface roughness prediction, respectively.



Influences of cutting parameters on cutting forces, surface roughness and MRR are analyzed.



The proposed analytical modeling approach can be used to achieve good efficiency and machining quality for whirling milling.

1

Analytical modeling of material removal mechanism in dry whirling milling process considering geometry, kinematics and mechanics Lexiang Wanga, Yan Hea,*, Yulin Wangb,*, Yufeng Lia, Chao Liua, Shilong Wanga, Yan Wangc a

State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China

b

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China

c

Department of Computing, Mathematics and Engineering, University of Brighton, Brighton BN2 4GJ, UK

* Corresponding author. E-mail address: State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China, [email protected] (Yan He, Tel: +86 13594166161); School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China, [email protected] (Yulin Wang, Tel: +86 13451851103), State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China, [email protected] (Lexiang Wang).

ABSTRACT Whirling milling is a promising machining process that couples the tools and workpiece motions while both conventional turning and milling are carried out, which is widely used for machining precision screw parts made of hard materials, such as titanium alloy, quenched steel, etc. The challenge is due to the complex kinematics can lead to the varying tool-workpiece engagement and undeformed chip geometries which affect significantly the mechanics, quality and productivity. The existing studies are mainly based on the simplification of process mechanism to study the mechanics and quality however, the material removal mechanism and influences of cutting parameters are still not well understood. This paper presents an analytical approach to investigate the material removal mechanism in whirling milling, thus to predict the undeformed chip geometry, material removal rate (MRR), cutting forces and form errors. The varying tool-workpiece engagement geometry along the cutting trajectory is identified to model the varying undeformed chip geometry including instantaneous chip thickness, cross-section area and tool-workpiece contact length. The form errors including circularity error, scallop height and surface roughness are defined and predicted as a function of tools and workpiece motion, position and dimension parameters. The whirling milling experiments were conducted to validate the analytical modeling approach with the largest error 11.3% and average error 13.0% for force and surface roughness prediction, respectively. The influences of cutting parameters on surface roughness and MRR are finally analyzed to explore the potential of productive cutting conditions for whirling milling.

2

Keywords: Whirling milling; Material removal mechanism; Cutting force; Surface roughness; MRR

3

1. Introduction Whirling milling process (see Fig. 1), developed by a Germany company (Burgsmuller) [1], which is one of the most commonly used material removal processes to produce the long and slender screw parts (e.g., worm and ball screws) made of hard-to-machine materials (e.g., titanium alloy and quenched steel) for many mechanical industries such as machine tool, automobile and aviation. It offers numerous advantages such as better surface quality than milling [2, 3], high productivity [4], dry cutting without coolants [5, 6]. This process allows both turning and milling on one machine tool (see Fig. 1), which results in a complex kinematics that multiple rotating tools cut the rotating workpiece. The complex kinematics leads to the varying tool-workpiece engagement and undeformed chip geometries, and further significantly affects the material removal mechanism of whirling milling. Hence, this paper presents an innovative analytical modeling approach of material removal mechanism in whirling milling which can predict the chip geometry, material removal rate (MRR), cutting forces and form errors to explore the potential of productive cutting conditions. Many existing studies on the whirling milling have mostly focused on the surface quality of machined screw surfaces. Mohan and Shunmugam [2] simulated the whirling milling to analyze the tool and surface profiles of worm threads. They found that the tilt angle of tool head (namely tool holder) has a significant effect on worm surface profiles, and it is better to set the tilt angle equal to the helix angle to ensure the smooth surface. Lee et al. [7] simulated the tool-tip interference base on a mathematical model of the tool-tip trajectories in whirling milling of worms. They reported that the interference area is spread on both the root and crest of the formed worm surface. Merticaru et al. [8] developed a simulation modular program (named EPAX) in MATLAB environment to describe the flank profile form error of the thread surfaces formed by worm whirling milling. In a recent study, Zanger et al. [3] also studied the worm surface profile in whirling milling using the numerical dexel-based simulation model which simplified the thread surface as a triangulated surface. The results showed that the accuracy of their simulation model is limited by the discretization of surface, the relative motion and the number of dexels. The above studies on surface quality were mainly based on simulation methods by simplifying the process mechanism without providing detailed analytical modeling of material removal mechanism in whirling milling. A theoretical model of form errors for screw surface was proposed by Han and Liu [9], which assumed that the surfaces are formed by the ideal helical motion. This 4

theoretical model was further implemented in MATLAB environment to analyze the form error distributions of screw surface in whirling milling [10]. Although Han and Liu [9, 10] reported the form errors are generated in axial and cross section of screw shaft in whirling milling, the effects of cutting parameters (e.g. cutting speed, workpiece speed, etc.) are not considered, which limits the effective parameter selection to improve the surface quality. Furthermore, Wang et al. [11] presented a two-dimensional theoretical modeling of formed scallop height in whirling milling in which the tilt angle and axial feed motion of the tool holder are not taken into account. Zanger et al. [3] applied an analytical model to determine the tool and workpiece profile, assuming an ideal equation of meshing for the interactions between tool and workpiece without considering the scallop height led from the interrupted cuts from the combined motion in whirling milling. Guo et al. [12] developed a surface roughness model using the position coordinates of a number of discrete points on screw surfaces in whirling milling. However, these analytical studies for surface quality were based on the simplification of process mechanism in which the combined motion of tools and workpiece, varying engagements and the significant influences of cutting parameters such as speeds of tools and workpiece, feed rate and tilt angle on form errors are neglected. Hence, there is an urgent need to consider these factors in the further study of the analytical modeling of the material removal mechanism to reveal the influences of cutting parameters on the form errors (e.g., surface roughness) for improvement of surface quality in whirling milling. Besides studies about surface quality, the mechanics as an important influencing factor for machining high quality screw parts in whirling milling has attracted more research attention. Lee et al. [7] simulated the cutting forces in whirling milling using the measured specific cutting energy which was assumed as a constant. The problem is the specific cutting energy may be a variable as the effects of cutting parameters [13, 14], which limit the force evaluation accuracy. Furthermore, FEM commercial simulation software is extensively employed in a number of studies to simulate the cutting forces of whirling milling in which constant cutting conditions. Researchers have carried out FEM simulations using DEFORM and ADAMS software to simulate the cutting forces of whirling milling [15, 16, 17]. However, the cutting forces were simulated based on the equivalent assumed undeformed chip model without considering the actual varying undeformed chip geometry, which may lead to an inaccurate estimation. Song and Zuo [18] employed another FEM tool (AdvantEdge) to simulate the cutting forces of whirling milling by simplifying arc cutting trajectories as a straight line and curved undeformed chip as a flat bottom 5

chip. Wang et al. [6] simulated the cutting forces of whirling milling using the DEFORM software, yet the influence of the tilt angle of tool holder on tool-workpiece engagements in whirling milling is neglected. However, the above FEM methods for cutting force modeling have very limited accuracy and industrial applications due to the overall simplification of the whirling milling which does not present the complex material removal mechanism. On the other hand, using these FEM methods, too many simulations need to be performed to select proper cutting parameters for controlling cutting forces, which leads to a waste of much time to implement. To address the above disadvantages, the experimental and mechanical modeling of mechanics has been applied for prediction the cutting force in machining process. Elhami et al. [19] presented an experimental investigation, numerical and analytical methods based on chip thickness to predict cutting force in ultrasonic assisted milling. Nieslony et al. [20] reported the experimental studies of the cutting force and surface morphology in drilling process to determine the effect of tool design. Tao et al. [21] carried out the experimental and modeling study to investigate the influence of feed direction on cutting forces in ultrasonic vibration-assisted milling. Zhao et al. [22] presented a theoretical and experimental investigation to calculated cutting forces in ultraprecision machining of microgrooves. In addition to experimental works, the mechanical modeling of cutting force has been also studied. A mechanistic model has been established by Wang and Qin [23] for evaluating the cutting force according to the cutting principle of helical milling. Zhang et al. [24,25] developed the mechanistic model to predict the cutting forces by considering such as the instantaneous uncut chip geometry, trochoidal trajectory, entry and exit angles during the material removal mechanism in micro-end-milling. Zhou et al. [26] also proposed a mechanistic cutting force model micro end milling by using the instantaneous uncut chip thickness which considers trochoidal trajectory of tool, entry and exit angles, etc. Recently, Priyabrata et al. [27] presented a new and nomically viable hybrid modeling approach for predicting the cutting forces in micro end milling by incorporating the effects of tool geometry, minimum chip thickness, elastic recovery of material removal mechanism. It can be concluded based on the above studies that the material removal mechanism related to tool-workpiece engagement, chip geometry, tool trajectory is significant for the modeling of cutting force, and it can further affect the machining quality [28, 29] in machining process. Compared to the above machining process, whirling milling is a complex combined turning and milling result for material removal, which is realized by a combination of workpiece rotating motion, tool rotating and feed motions. It is therefore necessary 6

to investigate the material removal mechanism of whirling milling for help in a better understanding of the prediction and analysis of process states such as chip geometry, cutting force and surface quality. Some research studies have been reported for analysis of the material removal mechanism of whirling milling. Serizawa et al. [1, 30] studied the material removal to calculated undeformed chip thickness in whirling milling. Wang et al. [14] studied the modeling and analysis of specific cutting energy of whirling milling based on material removal. Besides, He et al. [31, 32] analyzed the material removal to predict the temperature distribution and specific cutting energy in whirling milling. The limitation of the above studies is that the material removal in whirling milling is performed in the 2D plane without the tilt angle of the tool holder. However, whirling milling has been known as a 3D machining process (see Fig. 1) having a tilt angle between tool holder and workpiece. The material removal mechanism cannot be analyzed by simply using the tool-workpiece engagement model of 2D cutting. The 3D whirling milling with the tilt angle can lead to variations of tool-workpiece engagement and chip geometries, entry and exit angles, etc. These variations significantly influence the chip formation process [27, 33], which are not included and addressed in the existing studies of material removal mechanism in whirling milling. In summary, the material removal mechanism in whirling milling are still unclear, limiting the quality and productivity improvement of this promising technology for the manufacturers. Especially, there has not been any study which combines the geometry, kinematics and mechanics in one model to clarify the material removal mechanism of whirling milling. The objective of this study is to present an analytical modeling approach of material removal mechanism in whirling milling with tilt angle by considering the kinematics, varying tool-workpiece engagement and undeformed chip geometries in one integrated model. This proposed approach can be implemented to predict the whirling milling process states such as chip geometry, material removal rate (MRR), cutting forces and form errors, etc. The remaining parts of this paper arranged as follows. In Section 2, the material removal mechanism of whirling milling is modeled by considering the kinematics, varying tool-workpiece engagement and undeformed chip geometries, MRR and cutting forces. The form errors related to the machined screw surface quality of whirling milling including the circularity error, scallop height and surface roughness are formulated in Section 3. The proposed analytical models are experimentally validated in Section 4. The influence of cutting parameters on surface roughness and MRR is analyzed in Section 5, and conclusions are given in 7

Section 6.

2. Material removal mechanism of whirling milling The machined surfaces on screw parts in whirling milling can be generated by a combination of workpiece rotating motion, tool rotating and feed motions. As shown in Fig. 1, the OTXTYTZT and OWXWYWZW denote the whirling tool holder (WTH) coordinate system (i.e., cutting tool coordinate system) and the workpiece coordinate system, respectively. The WTH is mounted on the B-Axis (i.e., ZT -Axis) and set to incline to the workpiece with a tilt angle of φ (equal to the helix angle), and it is eccentric to encompass the workpiece with an offset of e. Furthermore, the WTH rotates around the ZT -Axis at a high cutting speed of nt (e.g., 1000 r/min), and the workpiece with a radius r (mm) mounted on the C-Axis rotates synchronously around ZW -Axis at a very low workpiece speed of nw (e.g., 2 r/min). Z number of cutting tools with zero rake angle and at a tool nose rotation radius R (mm) are evenly clamped on the WTH. These cutting tools follow a circular path around the center (i.e. origin OT in Fig. 1) of the WTH, and the tools travel with the WTH along the feed path (i.e. green dashed line paralleled to C-Axis) at an axial feedrate of f (mm/s). The combined motion of the tools and the workpiece leads to the cutters plunging into the workpiece material to generate the pitch of screw p (mm), where the cutting tools travel one lead or pitch of screw p along the feed path when the workpiece completes one revolution. The axial feedrate f can be expressed as: f [mm/s]  pnw /60 f z [mm/tooth/rev] 

pnw p  (1) Znt Zrs

where fz is the feed per tooth of the cutting tool along the feed path as the WTH completes one revolution; Z is the number of cutting tools; nt is the cutting speed of the cutting tool, and rs = nt/nw is the ratio of the cutting speed nt to the workpiece speed nw. In addition, the offset of e (mm) between the workpiece and whirling tool holder as shown in Fig. 1 and Fig. 2 is determined as: e=R 

d2 (2) 2

where d2 (mm) is the root diameter of the screw (e.g., Fig. 2) according to the specification of screw parts.

8

Workpiece C-Axis

XW , XT

B-Axis

p e

r OW

φ YW φ YT

nw ZW ZT nt

OT Feed path

R

f

Cutting tool

Whirling tool holder Fig. 1. Schematic of the whirling milling. 2.1. Kinematics of whirling milling To understand the material removal mechanism in whirling milling, the modeling of kinematics is presented firstly. The analytical kinematics of whirling milling is illustrated in Fig. 2. The coordinate transformation between the WTH and workpiece frames is established to model the kinematics. The workpiece coordinate system (OWXWYWZW) is set as the reference coordinate system, and the workpiece is assumed to be static regarding the WTH [1]. Instead, the WTH rotates around the workpiece axis (ZW -Axis) at the radius equal to the offset e with the workpiece rotational speed of nw in the reverse direction as shown in Fig. 2. Simultaneously, it moves along the feed path (see Fig. 1) with a feedrate f. Furthermore, the undeformed chip is periodically cut by the previous cutting (i.e., i-1 th cutting) and the current cutting (i.e., i th cutting) due to combined motion of tools and workpiece in whirling milling. In Fig. 2, a coordinate system

OTi1 X Ti1 YTi1 ZTi1 is attached to the WTH in the i-1 th cutting, and a coordinate system OTi X Ti YTi ZTi is attached to the WTH in the i th cutting. The coordinates ( xTi , yTi , zTi ) in OTi X Ti YTi ZTi and coordinates ( xTi1 , yTi1 , zTi1 ) in OTi1 X Ti1 YTi1 ZTi1 can be transformed to the workpiece coordinates ( xTWi , yTWi , zTWi ) and ( xTWi1 , yTWi1 , zTWi1 ) in OWXWYWZW by the transformation matrixes Ti and Ti1 , respectively, as shown in Eq. (3) and Eq. (4) [2, 7].

9

Workpiece XW , XTi Undeformed chip Cutting tool

XTi-1

d2

YW

OW θTi-1 e nw OTi-1

YTi

OTi

nt

(i) th cutting YTi-1

(i-1) th cutting

Fig. 2. Analytical kinematics of the whirling milling in XW-YW plane.

 xTWi  0  xTi  1  W  Ti    yTi   y  0 cos   W   Ti ( , t )  z Ti   0 sin   zTi     1  0  1  0  

0  sin  cos  0

 xTWi1   xTi1  cos T  W   Ti1    yTi1  i 1  y  =  sin T  T (  , t ,  ) i 1 Ti 1  W   zTi1   0  zTi1      1  1     0  

e   xTi    0   yTi  (3)  ft   z Ti    1   1 

i 1

 sin T cos 

sin T sin 

i 1

cos T cos  i 1

 cos T sin 

sin 

cos 

0

0

i 1

i 1

i 1

e cos T   xTi1  i 1





e sin T   yTi1  (4) T  ft i 1   z i1  i 1

1

0 0 1 0 cos   sin  where Ti ( , t )  T2 T1 ; Ti 1 ( , t i 1 ,Ti1 )  T3T2i 1T1 ; T1   0 sin  cos   0 0 0

    1 

0 0  is the rotation 0  1

transformation matrix resulted by the tilt angle  (see Fig. 1) of the B – Axis in (YW –ZW) plane

1 0 T  of the workpiece coordinate system; 2  0  0

0 1 0 0

0 e  1 0  0 0  i 1  T  and 2 0 1  ft    0 1  0

0 1 0 0

0 e  0 0  is the 1  ft i 1   0 1 

translation transformation matrices which can be evaluated based on the offset e and feedrate f; t 10

and t i 1 (s) is the cutting time of cutting tool in the i th and i-1 th cutting, respectively; the cos Ti1  sin Ti1 rotation transformation matrix T3    0   0

 sin Ti1 cos Ti1 0 0

matrix obtained by the rotation angle Ti1 

0 0  0 0 is the rotation transformation 1 0  0 1 

2 nw 2  Znt Zrs

of the WTH rotated around the

workpiece axis at the workpiece rotational speed of nw in the reverse direction. 2.2. Tool - workpiece engagement geometry The material removal mechanism of whirling milling is very complex due to the varying tool-workpiece engagement geometry caused by the complex combined motion of the tools and workpiece. Therefore, the tool-workpiece engagement geometry of whirling milling is investigated in this section as a basis for the modeling of the undeformed chip thickness, MRR, cutting forces and form errors. Based on the kinematics of whirling milling, the engagement conditions for material removal are divided into two zones (see Fig. 3): Zone 1: The workpiece material has not been cut by the i-1 th cutting tool. The engagement of the cutting tool increases as shown in Fig. 3d, while the tool tip point D moves from point A to E along the i th cutting path as the tool plunges into the materials. Point A and B are the i th and i-1 th cutting entry point on the workpiece surface, respectively. Point S is the intersection point of the workpiece outside circle outline and center line of the tool rake face. The undeformed chip thickness at engagement zone 1 is defined as h1 (t )= | SD | (see Fig. 3b), which increases from zero to its maximum value. Point E is a critical point of the undeformed chip thickness. Once the tool tip point D arrives at point E, the point S coincides with the point B and the undeformed chip thickness reaches its maximum value, i.e. hmax (tmax ) =|BE| where tmax is the maximum chip thickness arrival time. As shown in Fig. 3d, the cutting entry (  st ), zone 1 region ( 1 ) and instantaneous (  ) angles of the tool immersions are identified from the tool-workpiece engagement geometry, which is calculated in Section 2.3. The i th instantaneous cutting angle

 = nt t / 30 .

11

Cutting tool Ft(t) Workpiece Fr(t)

a)

b) Undeformed chip

f

b-b

Zone 1 Zone 2

nt

XW, X T OW

YW e OT

a-a

Fa(t) ZW

φ YT

nw ZT φ

h2(t)

c) Zone 2

X W, X T

nt

XTi-1

b

Workpiece surface

Cutting tool Undeformed chip

E D´ S D

Zone 2 θex θ2

a-a section S

h1(t) D D Chip cross-section

B

b C

b-b section S D´

A

θ θ1 r

θst

OW

YW

nw

e

OTi-1

OTi

YTi φ R

ZW

YTi-1

ZTi-1

φ

(i) th cutting (i-1) th cutting

ZTi

12

nt B a

XW, X T

d) Zone 1

XTi-1

Undeformed chip Workpiece surface

E a

Cutting tool

S D A

C

θ1

θ

θst

r

OW

YW nw

e

OTi-1

OTi

YTi φ R

ZW

YTi-1

ZTi-1

φ

(i) th cutting (i-1) th cutting

ZTi Fig. 3. Tool-workpiece engagement geometry of the whirling milling: (a) combined motion and cutting force; (b) varying undeformed chip and corresponding cutting zones; c) engagement in Zone 2; (d) engagement in Zone 1.

Zone 2: A part of the workpiece material has been removed by the i-1 th cutting tool. The remaining part of the workpiece material is cut by the i th cutting tool. In Fig. 3(c), the engagement of the cutting tool decreases, while the tool tip point D moves from point E to the exit point C. Point C is the intersection point of the i th and i-1 th cutting path, and point D’ is the intersection point of the i-1 th cutting path and center line of the tool rake face. The undeformed chip thickness is defined as h2 (t )= | DD | (see Fig. 3(b)), which decreases from the maximum value to zero. The zone 2 region angle  2 , cutting exit angle  ex and instantaneous angle  of tool immersion are defined in Fig. 3(c), and calculated in the following section. The instantaneous cutting angle  = nt t / 30 . 2.3. Modeling of undeformed chip geometry and material removal rate The cutting entry angle  st of tool immersion can be determined by the coordinates of point 13

D ( xWD , yWD , zWD ) and point A ( xWA , yWA , zWA ) in the workpiece system OWXWYWZW. As shown in Fig. 3d, when the tool starts cutting the workpiece material, the tool tip point D coincides with point A and point S where the cutting time t =0 , i.e.  = nt t / 30=0 . The relationship between the coordinates of point D and point A in workpiece system can be expressed as:

( xWD )2  ( yWD )2  ( xWA )2  ( yWA )2  r 2 (5) where the coordinates of point D ( xWD , yWD , zWD ) in the system OWXWYWZW can be calculated as a function of its coordinates ( xDTi , yDTi , zTDi )  ( R sin st , R cosst ,0) in the i th cutting system

X Ti  YTi  ZTi based on Eq. (3) [2]: R sin  st  e  xDTi    xWD    Ti    W   yD   T ( , t )  yD     R cos  cos  st  (6) i  z TDi    R sin  cos  st  ft   zWD        1   1   1  

The cutting entry angle  st is then obtained in Eq. (7) by substituting Eq. (6) into Eq. (5). In addition, the cutting exit angle  ex is also calculated as shown in Eq. (7a).  ( R sin  st  e)2  ( R cos  cos  st ) 2  r 2   e  e2 cos 2   sin 2  ( R 2 cos 2   r 2 ) (7)  st  arcsin R sin 2  

ex  st  2 (7a) where  2 is determined in the following part of this section. As shown in Fig. 3d, the undeformed chip thickness h1 (t )= | SOTi |  R in zone 1 can be formulated based on Eq. (8) [2] where | SOTi | can be calculated using the coordinates of point S ( xWS , yWS , zWS ) in the workpiece system OWXWYWZW, and ( xWS , yWS , zWS ) can be transformed using the T T T coordinates of point S ( xSi , ySi , zSi )  (| SOTi | sin(  st ),  | SOTi | cos(  st ),0) in the system

X Ti  YTi  ZTi .  xSTi    xWS   SOTi sin(   st )  e  Ti    W   yS   T ( , t )  yS     SOTi cos  cos(   st )  (8) i  zSTi    SOTi sin  cos(   st )  ft   zWS        1   1   1  

The | SOTi | is then determined by Eq. (8a) due to the point S on the workpiece surface. Therefore, the undeformed chip thickness h1 (t ) in zone 1 can be formulated in Eq. (8b). 14

 ( xWS )2  ( yWS )  r 2  ( SOTi sin(   st )  e) 2  (cos  SOTi cos(   st )) 2  r 2   e sin(   st )  r 2  cos 2 (   st )(r 2 sin 2   e2 cos2  ) (8a)  SOTi = 1  sin 2  cos 2 (   st ) 

h1 (t ) | SD || SOT |  R (8b) i

where  = nt t / 30 . The zone 1 region angle 1 (see Fig. 3d) of tool immersion is determined using the coordinates of point B in Eq. (9) and Eq. (9a) in the workpiece system OWXWYWZW which can be transformed

by

the

coordinates

( xBTi , yBTi , zTBi )  (|BOTi |sin(1  st ), |BOTi |cos(1  st ),0)

in

of the

system

point

OTi X Ti YTi ZTi

B and

( xBTi1 , yBTi1 , zTBi1 )  ( R sin st , R cosst ,0) in the system OTi1 X Ti1 YTi1 ZTi1 , respectively. When the cutting point D reaches point E, the tool tip point D and S coincide with point E and point B, respectively. Using the transformations in Eq. (3) and Eq. (4), the coordinates of point B in workpiece system are expressed as follows [2]: |BOTi |sin(1   st )  e   xBTi    xWB   Ti   |BO |cos  cos(   )   W Ti 1 st   yB   T ( , t )  yB    (9) i  z TBi   |BOT |sin  cos(1   st )  ft   zWB  i        1  1   1    xBTi1   R cos Ti1 sin  st  R cos  sin Ti1 cos  st  e cos Ti1   xWB    Ti1    W  yB   T ( , t i 1 , )  yB  =  R sin Ti1 sin  st  R cos  cos Ti1 cos  st  e sin Ti1  (9a) i 1 Ti 1   zTBi1    zWB   sin  R cos  st  ft i 1        1    1  1 

Furthermore, the feed per tooth per revolution fz = pnw / Znt (see Eq. (1)) of WTH along ZW -Axis could be neglected to determine the undeformed chip thickness due to its small value according to the study by Serizawa et al. [1]. The zone 1 region angle 1 of tool immersion and

|BOTi | are then solved in the X W  YW plane by Eq. (9b) as follows: |BOTi |sin(1   st )  e  R cosTi1 sin  st  R cos  sin Ti1 cos st  e cosTi1  |BOTi |cos  cos(1   st )  R sin Ti1 sin  st  R cos  cosTi1 cos st  e sin Ti1 (9b)

15

2  R cos Ti1 sin  st  R cos  sin Ti1 cos  st  e cos Ti1  e   |BO |  2  R cos  cos Ti1 cos  st  e sin Ti1  R sin Ti1 sin  st   Ti    cos      R cos Ti1 sin  st  R cos  sin Ti1 cos  st  e cos Ti1  e )   st (9c) 1  arcsin( BOTi 





Then, the cutting time t1 (s) of zone 1 and the maximum undeformed chip thickness

hmax (tmax ) can be calculated based on Eq. (9c):

t1  tmax  301 /  nt  hmax (tmax ) | BE || BOTi |  R (10) The undeformed chip thickness h2 (t )= | DOTi |  R | DD | (see Fig. 3c) in zone 2 is calculated using the coordinates of point D’ ( xWD , yWD , zWD ) in the system OWXWYWZW as a part of workpiece material has been cut by the i-1 th cutting tool. To obtain | DOTi | , ( xWD , yWD , zWD ) can be transformed

in

Eq.

(11)

and

Eq.

(11a)

by

the

coordinates

of

point

D’

( xDTi  , yDTi  , zTDi  )  ( DOTi sin(  st ),  DOTi cos(  st ),0) in the i th cutting system OTi X Ti YTi ZTi and point D’ ( xDTi1 , yDTi1 , zDTi1 )  ( R sin( i 1  st ), R cos( i 1  st ),0) in the i-1 th cutting system

OTi1 X Ti1 YTi1 ZTi1 , and  i 1 is the i-1 th cutting instantaneous angle which can be expressed as

 i 1 = nt t i 1 / 30 . Based on Eq. (11) and Eq. (11a) [2], | DOTi | is solved as Eq. (11b).   DOTi sin(   st )  e  xDTi     xWD    Ti    W    D O cos  cos(    ) Ti st  yD   T ( , t )  yD  =   (11) i T W  z Di     z D        D OTi sin  cos(   st )  ft    1 1       1 

 xDTi 1   xWD   Ti1   W  yD   T ( , t i 1 , )  yD  i-1 Ti 1  z TDi 1   zWD       1   1   R cos Ti1 sin( i 1   st )  R cos  sin Ti1 cos( i 1   st )  e cos Ti1    R sin Ti1 sin( i 1   st )  R cos  cos Ti1 cos( i 1   st )  e sin Ti1   =  (11a) i 1   sin  R cos(    )  ft st     1

16

 e(1  sin 2  cos 2 T  cos 2  cos T ) sin(   st )      e cos  sin T (cos 2   sin 2  cos T ) cos(   st )      R 2 (sin T sin(   st )  cos  cos T cos(   st )) 2     cos   R 2 cos 2  (cos T sin(   st )  cos  sin T cos(   st )) 2     e 2 (cos  cos(   st )(1  cos T )  sin T sin(   st )) 2   | D OT |   (sin T sin(   st )  cos  cos T cos(   st )) 2        2 2   cos  (cos  sin  cos(   )  cos  sin(   ))   T st T st      where  = nt t / 30 i 1

i 1

i 1

i 1

i 1

i 1

i 1

i 1

i 1

i 1

i

i 1

i 1

i 1

i 1

         ,   

(11b)

As analyzed above, the | DOTi | is then also solved in the X W  YW plane as shown in Eq. (11b) by using Eq. (11) and Eq. (11a), and the undeformed chip thickness h2 (t ) in zone 2 is evaluated as Eq. (11c)

h2 (t ) | DOTi |  | DOTi | = | DOTi | R (11c) Fig. 3c shows the undeformed chip thickness h2 (t )  0 in zone 2, while the tool tip point D arrives at point C. Both the point D and D’ coincide with point C. The zone 2 region angle  2 of tool immersion and the cutting time t2 (s) of zone 2 can be determined by Eq. (11c) when

h2 (t2 )=0 . Therefore, the undeformed chip thickness h(t ) or h( ) in whirling milling can be formulated as a function of cutting conditions as follows:

 h1 (t )，0  t  t1 h(t )  h (t ), t  t  t , or  2 1 2  h1 ( )，0    1   h( )  h2 ( ), 1     2  (12)  where  = nt t / 30 Using the Eq. (12), the undeformed chip thickness can be obtained. Fig. 4 shows the variation of undeformed chip thickness with rotational angle under various tilt angles. It can be observed from Fig. 4 that the undeformed chip thickness decreases with the increase of tilt angle φ, and the maximum thickness decreases from 78.6 μm to 66.7 μm as φ increases from 0° to 25°. Especially, it also can be seen that the cutting rotational angle θcutting for once cutting (i.e., the tool moves from point A to C in Fig. 3) increases with the increases of φ, for example θcutting increases from 0.4491 rad to 0.6292 rad when φ increases from 0° to 25°. Hence, the tilt angle φ in whirling milling could significantly affect the undeformed chip thickness and influence the tool-engagement geometries, thus it is necessary to consider the tilt angle φ for the analyses of the material removal mechanism of whirling milling.

17

Undeformed chip thicknesses [μm]

80.0 nt = 495 (r/min) nw = 3 (r/min) Z=3 φ = 0° φ = 10° φ = 15° φ = 20° φ = 25°

60.0

40.0

20.0

0.0 0.00 0.08 0.17 0.25 0.34 0.42 0.50 0.59 Rotational angle [rad] Fig. 4. Undeformed chip thicknesses for different tilt angles. To establish the MRR model in whirling milling, the material removal volume needs to be determined first. According to the material removal mechanism, the workpiece material in whirling milling is periodically removed by the cutting tools, and only one of the cutting tools is used for material removal during each cutting period. Hence, MRR in whirling milling can be obtained as the ratio of the material removal volume Vc of one chip to cutting time tc  60/Znt of per cutting period. Vc is calculated based on the instantaneous undeformed chip area A(t ) which is calculated based on the tool-chip engagement geometry details (see Fig. 5) as follows:

A(t ) 

AA ((tt),)，t0  tt  tt 1

1

2

1

2

  r  h (t ) )  ( r  h (t )) r  ( r  h (t ))  ,  r arccos(  r   r  SD  )  ( r  SD ) r  ( r  SD )  r arccos(   r   r  ( SD  h (t ))  )   r arccos( r        r  ( SD  h (t ))  r   r  ( SD  h (t ))  2

t

2

1

t

t

1

0  t  t1

2

t

t

1

t

2

2

t

t

t

2

t

t

t

2

t

2

t

t

2

t

2

t

t

2

2

(13)     ，t1  t  t2    

where A1 (t ) and A2 (t ) are the instantaneous undeformed chip cross-section area in zone 1 and zone 2, respectively; rt is the radius of the cutting tool (see Fig. 5), and SD in zone 2 can be expressed based on Eq. (8a) as SD  SOTi  R .

18

dVc

dVc

b-b

a-a

(i-1) th cutting (i) th cutting

(i) th cutting rt rt S

G

G

H

D´

h1(t) h2(t)

A2(t) D l(t)

Workpiece

Zone 2: b-b section

S

rt

H

A1(t) D l(t) Zone 1: a-a section

Fig. 5. Tool-chip engagement geometry details. The volume of chip Vc can be calculated by integrating the infinitesimal volume of the undeformed chip. The MRR is then evaluated as: t2 R n A(t )  t dt Vc   dVc  0 30  Vc 2 R t2   A(t )dt  MRR  tc Z 0  t2 2 R t1   A1 (t )dt   A2 (t )dt (13a)  0 t  1  Z





To model the cutting force in whirling milling in the following section, the instantaneous tool-workpiece contact length l (t ) is evaluated from the length of arc G-H (see Fig. 5) as:

  r  SD 2r arccos  l (t )  GH    r 0, t

t

t

  , 0  t  t2 (14)  t  t2

2.4. Cutting forces model Whirling milling is an intermittent cutting process which can cause periodic cutting forces during cutting. As analyzed above, the material is cut by only one cutting tool during each cutting period of whirling milling. As shown in Fig. 3, the instantaneous cutting forces acting on the cutting tool during each cutting period are separated into tangential, radial and axial directions:

Ft (t ) , Fr (t ) and Fa (t ) , respectively. Since the undeformed chip area A(t ) and tool-workpiece contact length l (t ) in Section 2.3 are calculated from the foregoing kinematics and 19

tool-workpiece engagement geometry of whirling milling, the cutting forces contributed by the cutting tool can be evaluated as [27, 34]:

 Ft (t )  Ktc  A(t )  Kte  l (t )   Fr (t )  K rc  A(t )  K re  l (t ) (15)  Fa (t )  K ac  A(t )  K ae  l (t ) where K tc , K rc and K ac (N/mm2) are the shearing force coefficients in the tangential, radial, and axial direction, respectively; K te , K re , and K ae (N/mm) are the edge force coefficients in the tangential, radial, and axial directions, respectively. The shearing and edge force coefficients can be determined by using the time domain cutting forces measured in whirling milling experiments and the least square fitting method [34, 35]. In general, the cutting force can be measured by the dynamometer which gives the force data in x, y and z directions. When the dynamometer frame is oblique to the tool frame, the force components in x (Fx), y (Fy) and z (Fz) directions can be obtained by transforming the force components in tangential (Ft), radial (Fr) and axial (Fa) directions from tool frame to the dynamometer (x-y-z) frame using the transformation matrix [27]. In this study, a triaxial piezoelectric force sensor (Kistler 9602A, see Fig. 6) was mounted on the tool apron and rotated with the cutting tool to measure the cutting force in whirling milling, and then both the tool and dynamometer are installed on the tool holder with the tilt angle. Hence, the dynamometer x-y-z frame coincides with the tool r-a-t frame, as shown in Fig. 6. Therefore, the transformation matrix is not employed to evaluate the cutting force of whirling milling in this study.

20

Fig. 6. Cutting force measurement details in whirling milling.

3. Modeling of form errors Whirling milling generates non-ideal screw surfaces with form errors due to the aforementioned material removal mechanism. The form errors of the screw surfaces are generated by both the i th and i-1 th cutting path due to the combined motion including workpiece rotating motion, the cutting tools co-rotating motion and feed motion of. Fig. 7 represents the examples of form errors on screw surface in whirling milling including the 3D view in Fig. 7a, actual machined screw surfaces in Fig. 7b. Hence, the form errors generated in whirling milling are can be divided into two categories in this paper, including the circularity error (i.e. cutting mark) in the circumferential direction caused by the rotating motions of the tools and workpiece, and the scallop height (i.e. feed mark) in the axial direction contributed by the tool feed motion at the feed per tooth fz along the ZW -Axis.

21

Fig. 7. Form errors in whirling milling. a) 3D view of form errors; b) form errors on machined screw surface. 3.1. Circularity error According to the work by Karagüzel et al. [28], the circularity error in the circumferential direction of the screw surfaces in whirling milling is represented as the difference of the projections of the ideal and obtained screw surfaces on the X W  YW plane, as shown in Fig. 8. The obtained screw surfaces (green line), which consist of thousands of discrete envelope surfaces swept by each tool along the cutting path, are quite different from the ideal screw surfaces (red line). It can be seen from Fig. 8 the projection of the obtained screw surfaces is a polygon composed of lots of elliptical arcs, but the projection of the ideal screw surfaces is a circle. In addition, Fig. 8 also shows the circularity error is periodically generated on the screw surfaces at a period of angle  P equal to the rotation angle Ti1 =2 /Zrs (see Fig. 2) in screw circumferential direction, and  P decreases as the number of cutting tools Z and ratio rs increase. It can be seen in Fig. 8 that when  P decreases, the number of discrete envelope surfaces increases. Generally, since the screw surfaces are generated by more discrete envelope surfaces, the circularity error ec is smaller. Therefore, ec decreases with the increase of rs and Z. According to Fig. 8, the circularity error ec can be determined as ec  OW C  d2 /2 which is the maximum difference value between the ideal and obtained screw surfaces. Similar to Eq. (11) used for calculating the undeformed chip thickness in Section 2.3, the coordinates of point C ( xCW , yCW , zCW ) in the system OWXWYWZW can be calculated as [2]:

22

  xCW  R sin( 2   st )  e  xCTi     W   Ti      yC   T ( , t )  yC     R cos  cos( 2   st )  i  zCTi    R sin  cos( 2   st )  ft2    zCW        1   1     1  (16)    nt t2 /30

The circularity error eC in whirling milling is then evaluated at X W  YW plane based on the coordinates of point C as:

eC  OW C  d2 /2=

 R sin(2  st )  e    R cos  cos(2  st )  2

2

 d 2 /2 (16a)

where d 2 (mm) is the root diameter of the screw according to the specification of a screw part.

Obtained screw YW surfaces

Workpiece

Tool path

C

P d2

YW OW

Ideal screw surfaces

Fig. 8. Representation of circularity error on the X W  YW plane. 3.2. Scallop height The details of scallop height generation in whirling milling is presented in Fig. 9. It can be observed that the scallop height on a whirling milled screw surface is generated by the feed per tooth of the cutting tool due to the tool feed motion along ZW -Axis, and directly related to the feed per tooth f z , the radius rt of the cutting tool and the tilt angle  of the tool holder. According to Fig. 8, the scallop height in screw axial direction is also periodically generated on the screw surfaces at a period of angle  P . As seen from Fig. 9, the scallop height hS is calculated geometrically based on f z , rt , and  as follows:

23

hS  rt  rt 2  (

f Z cos  2 p cos  2 )  rt  rt 2  ( ) (17) 2 2Zrs

(i) th cutting hs

rt

(i-1) th cutting

φ fz Workpiece Fig. 9. Details of scallop height generation 3.3. Surface roughness of the screw surface The form errors such as circularity error and scallop height have significant effects on surface roughness as these errors are included in roughness measurement [28]. Following the previous study by Karaguzel et al. [28], the surface roughness of screw surfaces in whirling milling can be also contributed by both the obtained circularity error and the scallop height. Hence, the mean height (i.e. (1/m) i 1 Zti ) parameter is employed to describe the surface roughness Rscrew of the m

screw surface according to the amplitude parameters of ISO4287/ISO4288 [36, 37] and the study by Karaguzel et al. [28]. Then, Using the calculations of circularity error in Eq. 16a and scallop height in Eq. 17a, the surface roughness Rscrew of screw surfaces in whirling milling is formulated as follows [36]: Rscrew  =

1 2



eC  hS 2

 R sin(



  st )  e    R cos  cos( 2   st )   d 2 /2  2

2

2

1

r  r  (

2

2

t

t

p cos  2 Zrs

)

2

  (18) 

4. Experimental verifications In this section, a series of whirling milling experiments were carried out to verify the proposed analytical models, and these experiments have been conducted on a CNC whirling milling machine tool (HJ092 × 80, Hanjiang Machine Tool Co., Ltd, China), as shown in Fig. 10. GCr15 bearing steel (AISI 52100) with a high hardness approximately 62 HRC after heat treatment was employed as the workpiece material due to its widespread application in screw parts manufacturing. The geometrical parameters of workpiece are: outer diameter of screw d1 = 62.05 24

mm (workpiece radius r = d1/2 = 31.025 mm), root diameter of screw d2 = 57.95 mm. The chemical compositions and mechanical properties of the workpiece material are listed in Table 1. The tool nose rotation radius was adjusted as R = 43.5 mm, and then the whirling tool holder was mounted on the whirling machine and adjusted with the workpiece at an offset e = R- d2/2 = 14.525 mm, i.e., Eq. 2. Polycrystalline cubic boron nitride (PCBN) cutting tools (i.e., 27 tools) having rake angle 0 deg, clearance angle 9 deg and radius rt = 3.3 mm were employed, and all whirling milling experiments were conducted under dry cutting environment. The cutting conditions used in whirling milling tests are listed in Table 2. Each set of cutting parameters was tested three times, and new cutting tools were used for each test so that to reduce the effect of tool wear on the machined surface. Table 1 Chemical compositions and properties of workpiece material. Fe (basic); C (0.98); Cr (1.5); Mn (0.35); Si

Chemical compositions (wt. %)

(0.21); S (0.02); P (0.021); 3

Mechanical properties

Density (kg/m )

7810

Young's modulus (GPa)

201

Poisson's ratio

0.277

Thermal conductivity (W/ (m K))

46.6

Table 2 Cutting conditions for whirling milling tests. Test no.

Cutting speed nt (r/min)

Workpiece speed nw (r/min)

Ratio rs = nt/nw

Number of cutting tools Z

Tilt angle φ (°)

Lead p (mm)

1

354

3

118

3

2.8924

10

2

495

3

165

3

2.8924

10

3

637

3

212.33

3

2.8924

10

4

495

2

247.5

3

2.8924

10

5

495

4

123.75

3

2.8924

10

6

495

3

165

2

2.8924

10

7

495

3

165

4

2.8924

10

8

495

3

165

3

4.3341

15

9

495

3

165

3

5.7702

20

4.1. Verification of cutting force model The cutting forces including tangential Ft, radial Fr and axial Fa components were measured on the HJ092 × 80 whirling machine by a Kistler 9602A triaxial piezoelectric force sensor with the integrated charge amplifier electronics as seen in Fig. 10. The force signals were collected by the data acquisition system consisting of a NI cDAQ-9189 Compact DAQ data acquisition cassis, a NI-9239 data acquisition card and a LabVIEW programming tool. Based on the cutting conditions in Table 2 for whirling experiments, the highest tooth passing frequency is 33 Hz for 25

whirling experiments. Hence, the force signals were measured at a sampling frequency of 1 kHz. In force signal processing, the low pass filter in LabVIEW programming tool was set at a low pass frequency of 50 Hz due to the highest tooth passing frequency of 33 Hz, and the DC coupling was used in the NI-9239 card for force measurement. Both the cutting tools and force sensor were mounted on the rotating whirling tool holder. The cutting force measurement in whirling milling is a challenge since the force sensor rotates with the tool holder. Therefore, a slip ring with rotor and stator was employed to transmit the force signal, where the rotor and stator were connected with the force sensor (Kistler 9602A) and the data acquisition module (NI-9239), respectively.

Fig. 10. Experimental setup for whirling milling.

26

Ft-Predicted

Test No. 4

100 80

Fa-Predicted

60

Ft-Experiment Fr-Experiment

40

Fa-Experiment 20 0 0.00

0.04

0.07

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Cutting force (N)

Cutting force (N)

Fr-Predicted

0.22

180 Test No. 5 160 140 120 100 80 60 40 20 0 0.00 0.04 0.07

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Cutting force (N)

Fa-Experiment

0.11

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Ft-Predicted Fr-Predicted Fa-Predicted Ft-Experiment Fr-Experiment Fa-Experiment

80 60 40 20

0 0.00

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80 60 40 20

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Time (s)

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160 140

Ft-Predicted Fr-Predicted Fa-Predicted Ft-Experiment Fr-Experiment Fa-Experiment

Test No. 9

120

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Cutting force (N)

120

0.07

Fr-Experiment

Test No. 7

0.22

Ft-Predicted Fr-Predicted Fa-Predicted Ft-Experiment Fr-Experiment Fa-Experiment

Test No. 8

0.04

Ft-Experiment

100

160

0 0.00

Fa-Predicted

120

Ft-Predicted Fr-Predicted Fa-Predicted Ft-Experiment Fr-Experiment Fa-Experiment

Time (s) 140

Fr-Predicted

Time (s)

Time (s) 200 Test No. 6 180 160 140 120 100 80 60 40 20 0 0.00 0.04 0.07

Ft-Predicted

100 80

60 40

20 0 0.00

0.22

Time (s)

0.04

0.07

0.11

0.15

0.18

0.22

Time (s)

Fig. 11. Predicted and measured cutting forces in whirling milling for different conditions in Table 2.

Firstly, whirling milling tests under three cutting conditions (No. 1-3) as seen in Table 2 have been performed to obtain the cutting force coefficients in Eq. (15) using the time domain measured forces. The six force coefficients were identified as [Ktc, Krc, Kac] = [465.70, 362.78, 38.16] N/mm2 and [Kte, Kre, Kae] = [1.17, 0.52, 0.35] N/mm. Then, the predicted forces with different conditions (No. 4-9) are compared with the experimental forces to verify the proposed model and the identified force coefficients. As shown in Fig. 11, the model-based predicted results are in good agreement with the measured forces under various parameter conditions: with different cutting speeds, workpiece speeds, number of cutting tools, tilt angles, which validate the proposed integrated tool-workpiece engagement and undeformed chip geometries, MRR and cutting force analytical model of material removal mechanism in whirling milling. The experimental and predicted maximum resultant cutting forces for different cutting 27

conditions in Table 2 are shown in Fig. 12. According to the work by Priyabrata et al. [27], each maximum resultant force value was measured by the average value of 10 maximum peak forces during the measured force curve. The predicted and measured results are in a good agreement with the largest error of 11.3%. Fig. 12 shows for the given cutting conditions, it is worth noting that the maximum resultant forces decrease with the increase of nt and Z, but increase with the increase of nw. It can be seen that the forces change slightly with the increase of φ at the given three conditions.

Experimental

Predicted

Max. Resultant Force (N)

250 200

nw = 3 r/min nt = 495 r/min Z=3 Z=3 φ = 2.8924° φ = 2.8924°

nt = 495 r/min nt = 495 r/min nw = 3 r/min nw = 3 r/min φ = 2.8924° Z = 3

150 100 50 0

nt = 354 r/min nt = 495 r/min nt = 637 r/min

nw = 2 r/min nw = 3 r/min nw = 4 r/min

Z=2 Z=3 Z=4

φ = 2.8924° φ = 4.3341° φ = 5.7702°

Fig. 12. Comparison of measured and predicted peak resultant forces for different cutting conditions.

4.2. Verification of form errors To verify the proposed form error model, the predicted surface roughness Rscrew was compared with the measured results. According to the analytical surface roughness model in Eq. (18), the predicted surface roughness Rscrew was calculated for different machining conditions as listed in Table 2. The experimental surface roughness of the screw surfaces machined by whirling milling was measured by the Rtec multifunction tribology tester MFT-5000 (Rtec Instruments, USA) integrating a white light module, as shown in Fig. 13. The roughness parameter Rc was measured to analyze the surface roughness Rscrew of the screw surfaces in whirling milling. For each cutting condition, the surface roughness was measured at three positions with different angles (i.e., angle interval is about 120°) along the circumferential direction of the screw surface, and each measurement was repeated three times for each position.

28

Fig. 13. Surface roughness measurement of screw surface in whirling milling.

The proposed model was verified by a comparison between the predicted and measured results of surface roughness under different cutting conditions, as shown in Fig. 14. The predicted surface roughness agrees reasonably with the measured results under different conditions with the average error of 13.0% and largest error of 21.9%. The reason for this error might be explained by that the effects of the tool edge profile, material spring back and residual errors on surface roughness are not considered according to Section 3.3. Hence, the proposed model can be able to predict surface roughness of screw surfaces in whirling milling, which showed the effectiveness of the proposed form error model. It can be seen from Fig. 14 that the surface roughness Rscrew decreases with the increase of cutting speed nt and number of cutting tools Z, while it increases with the increase of workpiece speed nw and tilt angle φ. Especially, it is observed from Fig. 14a and 14b that Rscrew decreases with the increase of the ratio rs = nt/nw. The results can be explained by the model of the surface roughness Rscrew in Section 3.2 that both circularity error eC and scallop height hS decrease as rs increases according to the material removal mechanism of whirling milling.

29

0.25

0.20 0.15 0.10

0.05 0.00 354

Surface roughnessRscrew (μm)

c) 0.35

495

Measured Predicted

0.25

0.20 0.15 0.10 0.05 0.00 2

3

Measured Predicted

0.20 0.15

0.10 0.05

0.00 2

637

Cutting speed nt (r/min)

0.30

Surface roughnessRscrew (μm)

b) 0.25 Measured Predicted

d) 0.18 Surface roughnessRscrew (μm)

Surface roughness Rscrew (μm)

a) 0.30

3

4

Workpiece speednw (r/min)

0.16 0.14 0.12 0.10 0.08

Measured Predicted

0.06 0.04 0.02 0.00

4

2.8924

Number of cutting toolsZ

4.3341

5.7702

Tilt angle φ (° )

Fig. 14. Comparison of measured and predicted surface roughness in whirling milling at different a) cutting speeds nt (nw = 3 r/min, Z = 3, φ = 2.8924°); b) workpiece speeds nw (nt = 495 r/min, Z = 3, φ = 2.8924°); c) number of cutting tools Z (nt = 495 r/min, nw = 3, φ = 2.8924°); d) tilt angles φ (nt = 495 r/min, nw = 3, Z = 3,).

5. Analyses and discussion Using the proposed analytical model of the whirling milling, the influence of the cutting parameters, such as nt, nw, Z and φ, on the surface roughness Rscrew as well as MRR is analyzed in this section, which can provide insightful information for improving screw surface quality and machining efficiency. The dimensions of the workpiece (i.e. d1 and d2) and installation parameters (i.e. R and e) were selected as the same as given in Section 4. Rscrew was calculated employing the proposed model, i.e., Eq. (18), and MRRs were calculated by Eq. (13b). 5.1. Influence of cutting parameters on Rscrew and MRR As shown in Fig14, the influence of cutting parameters on Rscrew and MRR in whirling milling is investigated. For MRRs, Fig. 15a, c, and d show MRRs remain almost constant at about 142 mm3/s (black line) and 316 mm3/s (red line) as nt, Z and φ increase. However, in Fig. 15b, MRR increases linearly from about 28 to 316 mm3/s with a linear increase of nw from 1 to 11 r/min. 30

These results mean nt, Z and φ nearly have little influence on MRR, however, nw has a dominant influence. Thus, increasing nw means linearly increasing machining efficiency of whirling milling as the MRR increases. For Rscrew, Fig. 15a shows that Rscrew decreases nonlinearly with the increase of nt. When nt increases from 300 to 1300 r/min, Rscrew decreases from 0.429 to 0.023 μm (black line) and 0.736 to 0.039 μm (red line). Fig. 15c also shows nonlinear decreasing trends of Rscrew from 0.429 to 0.023 μm (black line) and 0.736 to 0.039 μm (red line) with the increase of Z. But, Rscrew increases nonlinearly from 0.003 to 0.382 μm (black line) and 0.0003 to 0.039 μm (red line) with the increase of nw in Fig. 15b. The great difference of the gradients (see Fig. 15b) of Rscrew with nw is due to the significant influence of the ratio rs as mentioned in Section 4.2. Similar to the influence of nw, Fig. 15d shows Rscrew increases nonlinearly with the increase of φ, but the influence of φ on Rscrew is less compared with the influence of nw. Therefore, it is worth noting from the above analyses that increasing nt or Z and decreasing nw or φ can improve the screw surface quality for whirling milling since Rscrew can be reduced. In summary, higher nt (e.g., 1300 r/min), lower nw (e.g., 1 r/min), bigger Z (e.g., 12), smaller φ (e.g., 1.4471°) are beneficial to Rscrew, and nt, Z and φ nearly have no influence on MRR, but nw has a dominant influence on MRR. Therefore, the selection of cutting parameters can be optimized to achieve both higher screw surface quality and machining efficiency. However, low nw will cut down the machining efficiency (i.e., MRR). Hence, when MRR is constant, it is necessary to analyze the influences of cutting parameters, especially the particular parameter (i.e., rs), which shall be investigated to explore the potential improvement of both Rscrew and MRR for whirling Rscrew

0.80 0.70

MRR 350

nw=11 r/min, Z=12, φ=8.6191°

0.60

nw=5 r/min, Z=6, φ=4.3341°

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b)0.45

Material remvoal rate MRR (mm3/s) Surface roughness Rscrew (μm)

Surface roughness Rscrew (μm)

a)

Cutting speed nt (r/min)

31

0.40

Rscrew

MRR

nt=1300 r/min, Z=12, φ=8.6191° nt=700 r/min, Z=6, φ=4.3341°

350 300

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Material remvoal rate MRR (mm3/s)

milling in the following Section 5.2.

350

1.40 1.20 1.00

300 nt=1300 r/min, nw=11 r/min, φ=8.6191°250

nt=700 r/min, nw=5 r/min, φ=4.3341°

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0.04

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50

0.00

0 1.4471 2.8924 4.3341 5.7702 7.1991 8.6191

Tilt angle φ (° )

Material remvoal rate MRR (mm3/s)

Surface roughness Rscrew (μm)

MRR

Material remvoal rate MRR (mm3/s) Surface roughness Rscrew (μm)

Rscrew

c)

Fig. 15. Influence of nt, nw, Z and φ on Rscrew and MRR.

5.2. Influence of rs and cutting parameters on Rscrew for constant MRR The influence of rs and cutting parameters on Rscrew with constant MRRs is analyzed in this section to explore the potential selection of cutting parameters to achieve Rscrew reduction and MRR improvement. nt used to analyze in this section are at five different speeds such as at nt = 400, 600, 800, 1000 and 1200 r/min. Then, the variations of Rscrew for the different constant MRR of 114 mm3/s (i.e., nw = 4 r/min), 228 mm3/s (i.e., nw = 8 r/min) and 342 mm3/s (i.e., nw = 12 r/min) are shown in Fig. 16-18. From these figures, it can be concluded that: 1) although MRR remains constant, Rscrew changes at a wide range under different parameter combinations; 2) the variation trends of Rscrew are similar at constant MRR while two of the parameters including rs, Z and φ are given.

b) nw = 4 r/min φ = 4.3341°

Rscrew (μm)

Rscrew (μm)

a)

nw = 4 r/min Z=6

Fig. 16. Influence of rs, Z and φ on Rscrew at a constant MRR = 114 mm3/s.

32

b) nw = 8 r/min φ = 4.3341°

nw = 8 r/min Z=6

Rscrew (μm)

Rscrew (μm)

a)

Fig. 17. Influence of rs, Z and φ on Rscrew at a constant MRR = 228 mm3/s.

b)

nw = 12 r/min φ = 4.3341°

Rscrew (μm)

Rscrew (μm)

a)

nw = 12 r/min Z=6

Fig. 18. Influence of rs, Z and φ on Rscrew at a constant MRR = 342 mm3/s.

For constant MRRs, Fig. 16a shows Rscrew decreases nonlinearly from 1.392 to 0.006μm at MRR = 114 mm3/s, when rs and Z increase. As seen from Fig. 16b, if rs increases and φ decreases, Rscrew also decreases nonlinearly from 0.193 to 0.015 μm. Fig. 17 and Fig. 18 show the similar change tendencies of Rscrew at MRR = 228 mm3/s and MRR = 342 mm3/s, respectively. The above analyses indicate that the wide ranges of Rscrew can be obtained by selecting different cutting conditions at constant MRRs such as 114 mm3/s, 228 mm3/s and 342 mm3/s. Therefore, there is a good potential to reduce Rscrew without cutting down MRR by the proper selection of parameter combinations, for example, Rscrew at constant MRR 114 mm3/s can be reduced from 1.392 to 0.155 μm (Fig. 16a) by selecting [rs =150, nw =4 r/min, Z =4, φ = 4.3341°] instead of [rs =100, nw =4 r/min, Z =2, φ = 4.3341°]. For different MRRs, Fig. 16-18 show the change trends of Rscrew at three levels. It can be observed that although MRR increases from 114 mm3/s to 342 mm3/s, Rscrew can keep constant by

33

using the appropriate selection of parameter combinations. For example, Rscrew = 0.618 μm at MRR = 114 mm3/s is obtained under [rs =150, nw =4 r/min, Z =2, φ = 4.3341°] (see Fig. 16a). The same Rscrew = 0.618 μm can also be generated at [rs =75, nw =8 r/min, Z =4, φ = 4.3341°] at MRR = 228 mm3/s (see Fig. 17a) and [rs =50, nw =12 r/min, Z =6, φ = 4.3341°] at MRR = 342 mm3/s (see Fig. 18a). Therefore, a potential of MRR improvement without sacrificing the requested surface quality can be achieved through proper selection of cutting conditions. On the other hand, it is noteworthy that compared with the change ranges of Rscrew at the three levels of MRR, there is a good potential to reduce Rscrew as well as to improve MRR. It can be found that Rscrew = 1.392 μm at MRR = 114 mm3/s under [rs =100, nw =4 r/min, Z =2, φ = 4.3341°] (see Fig. 16a), and it is easy to select parameter combinations in Fig 16 and in Fig 17 which result in Rscrew less than 1.392 μm, such as Rscrew = 0.501 μm at MRR = 342 mm3/s under [rs =83.4, nw =12 r/min, Z =4, φ = 4.3341°] (see Fig. 18a). For the analytical modeling of surface roughness in Section 3.3, a previous study [38] reported that surface roughness may be contributed by the geometric form error due to the material removal mechanism, tool edge profile errors, material spring back errors caused by material properties and other errors due to random factors such as tool wear, vibration, etc. However, the geometric form errors were identified as the basic factors affecting the machining quality, which need to be investigated for quality improvement [39]. In this study, the surface roughness of screw surfaces in Eq. (18) is therefore developed in this time to contribute with a fundamental understanding of the basic quality factor based on the material removal mechanism of whirling milling. Hence, the limitation of this surface roughness model is the influences of the tool edge profile errors, material spring back and residual errors are not taken into account, and a comprehensive surface roughness prediction will be a key topic in whirling milling for future work by the authors.

6. Conclusions Whirling milling is widely used in machining screw parts due to its high productivity and quality characteristics. In this paper, an innovative analytical modeling of material removal mechanism with the consideration of process geometry, kinematics and mechanics is proposed to predict the undeformed chip geometry, material removal rate (MRR), cutting forces, and form errors in whirling milling. The major conclusions of this research are drawn as follows: 1.

The varying tool-workpiece engagement geometry along the cutting trajectory in whirling milling was defined by dividing into two cutting zones based on a combination of workpiece 34

rotating motion, tool rotating and feed motions. Then, the varying undeformed chip geometry was studied to calculate the instantaneous chip thickness, cross-section area and tool-workpiece contact length which were used to develop the material removal rate and cutting force models of whirling milling. 2.

Whirling milling generates non-ideal screw surfaces with form errors due to the aforementioned material removal mechanism. The form errors were divided into the circularity error in the circumferential direction and scallop height in the axial direction. Both the circularity error and scallop height were combined to predict the surface roughness as a function of the tool and workpiece motion, position and dimension parameters.

3.

The proposed analytical modeling of material removal mechanism was validated in a good accuracy to predict the forces with the largest error of 11.3%, and to predict the surface roughness with the average error of 13.0% and largest error of 21.9%. Additionally, the cutting forces decrease with nt and Z, and increase with nw, but change slightly with φ.

4.

From the analyses of surface roughness Rscrew and MRR, this study showed that: (1) Rscrew decreases nonlinearly with nt, Z and rs, increases nonlinearly with nw and φ; (2) nw has a dominant influence that MRR linearly increases with the increase of nw; (3) Rscrew can vary with a wide range under different cutting conditions, which offers a good potential to reduce Rscrew and improve MRR, simultaneously.

5.

The proposed analytical modeling approach is a quick way to not only provide better understanding of the basics of the material removal mechanism but also highlight the influence of cutting conditions on cutting forces, MRR and surface roughness, which could be employed for the selection of machining parameters to achieve good efficiency and machining quality for whirling milling. Furthermore, the productivity and surface quality in machining processes is usually restricted by

dynamic issues such as the occurrence of chatter, vibration, etc. Although, the floating supports and holding devices have been used in this study to improve the rigidity of the workpiece in whirling milling process, the productivity and surface quality can be still limited by dynamic issues more or less. In the future study, a comprehensive modeling approach of productivity and surface quality integrating dynamic issues could be a meaningful research topic for the slender workpiece in whirling milling.

35

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.

Acknowledgements The authors would like to thank the support from the National Key R&D Program of China (Grant No. 2018YFB2002200), the National Natural Science Foundation of China (Grant No. 51575072, 51605058), the Fundamental Research Funds for the Central Universities, China (No. 2018CDQYJX033)

References [1] Serizawa M, Suzuki M, Matsumura T. Microthreading in whirling. ASME J Micro Nano-Manuf 2015;3:041001. https://doi.org/10.1115/1.4030704. [2] Mohan LV, Shunmugam MS. Simulation of whirling process and tool profiling for machining of worms. J Mater Process Technol 2007;185:191–7. https://doi.org/10.1016/j.jmatprotec.2006.03.115. [3] Zanger F, Sellmeier V, Klose J, Bartkowiak M, Schulze V. Comparison of modeling methods to determine cutting tool profile for conventional and synchronized whirling. Procedia CIRP 2017;58:222-7. https://doi.org/10.1016/j.procir.2017.03.216. [4] Bangalore HMT. Production Technology. 28th ed. India: Tata McGraw-Hill; 2008. [5] Son JH, Lee YM. Thermal analysis of whirling unit attached to thread whirling machine. Mater Sci Forum 2008;575-578:93-7. https://doi.org/10.4028/www.scientific.net/MSF.575-578.93. [6] Wang YL, Li L, Zhou CG, Guo Q, Zhang CJ, Feng HT. The dynamic modeling and vibration analysis of the large-scale thread whirling system under high-speed hard cutting. Mach Sci Technol 2014;18:522-46. https://doi.org/10.1080/10910344.2014.955366. [7] Lee MH, Kang DB, Son SM, Ahn JH. Investigation of cutting characteristics for worm machining on automatic lathe – comparison of planetary milling and side milling. J Mech Sci Technol 2008;22:2454-63. https://doi.org/10.1007/s12206-008-0713-1. [8] Merticaru V, Mihalache A, Nagîţ G, Dodun O, Slătineanu L. Some aspects about the significant parameters of the thread whirling process. Appl Mech Mater 2016;834:96-101. https://doi.org/10.4028/www.scientific.net/AMM.834.96. [9] Han QQ, Liu R. Theoretical model for CNC whirling of screw shafts using standard cutters. Int J Adv Manuf Technol 2013;69:2437–44. https://doi.org/10.1007/s00170-013-5214-4.

36

[10] Han QQ, Liu R. Theoretical modeling and error analysis for CNC whirling of the helical surfaces of custom screws using common inserts. Proc Inst Mech Eng Part C-J Eng Mech Eng Sci 2014;228:1948-57. https://doi.org/10.1177/0954406213513573 [11] Wang YL, Tao LJ, Guo Q, Feng HT. Study on the forming process of thread raceway surface under the hard whirling. In: Proceedings of the 2013 8th International Conference on Computer Science & Education (ICCSE 2013), SRI LANKA: Colombo APR 26-28, 2013.p.627-32. [12] Guo Q, Ye L, Wang YL, Feng HT, Li Y. Comparative assessment of surface roughness and microstructure produced in whirlwind milling of bearing steel. Mach Sci Technol 2014;18:251-76. https://doi.org/10.1080/10910344.2014.897843 [13] Nandy AK, Gowrishankar MC, Paul S. Some studies on high-pressure cooling in turning of Ti-6Al-4V. Int J Mach Tools Manuf 2009;49:182-98. https://doi.org/10.1016/j.ijmachtools.2008.08.008. [14] Wang LX, He Y, Li YF, Wang YL, Liu C, Liu XH, Wang Y. Modeling and analysis of specific cutting energy of whirling milling process based on cutting parameters. Procedia CIRP 2019;80:56-61. https://doi.org/10.1016/j.procir.2019.01.028. [15] Lee YM, Choi WS, Son JH, Oh KJ, Kim SI, Park GW, Jung HC. Analysis of worm cutting characteristics in whirling process. Mater Sci Forum 2008;575-578:1402-6. https://doi.org/10.4028/www.scientific.net/MSF.575-578.1402 [16] Hwan SJ, Woo HC, Ryong LS, Moon LY. Structural stability estimation of whirling unit. Trans Nonferrous Met Soc China 2009;19:S202-8. https://doi.org/10.1016/S1003-6326(10)60271-3. [17] Son JH, Han CW, Kim SI, Jung HC, Lee YM. Cutting forces analysis in whirling process. Int J Mod Phys B 2010;24:2786-91. https://doi.org/10.1142/S0217979210065635. [18] Song SQ, Zuo DW. Modelling and simulation of whirling process based on equivalent cutting volume. Simul Model Pract Theory 2014;42:98-106. https://doi.org/10.1016/j.simpat.2013.12.011. [19] Elhami S, Razfar MR, Farahnakian M. Analytical, numerical and experimental study of cutting force during thermally enhanced ultrasonic assisted milling of hardened AISI 4140. Int J Mech Sci 2015;103:158–71. https://doi.org/10.1016/j.ijmecsci.2015.09.007. [20] Nieslony P, Cichosz P, Krolczyk GM, Legutko S, Smyczek D, Kolodziej M. Experimental studies of the cutting force and surface morphology of explosively clad Ti-steel plates. Measurement 2016;78:129–37. https://doi.org/10.1016/j.measurement.2015.10.005. [21] Tao G, Ma C, Shen X, Zhang J. Experimental and modeling study on cutting forces of feed direction ultrasonic vibration-assisted milling. Int J Adv Manuf Technol 2027;90(1-4):709–715. https://doi.org/10.1007/s00170-016-9421-7. [22] Zhao ZJ, To S, Zhu ZW, Yin TF. A theoretical and experimental investigation of cutting forces and spring back behaviour of Ti6Al4V alloy in ultraprecision machining of microgrooves. Int J Mech Sci 2019;169:105315. 37

https://doi.org/10.1016/j.ijmecsci.2019.105315. [23] Wang H, Qin X. A mechanistic model for cutting force in helical milling of carbon fiber-reinforced polymers, Int. J Adv Manuf Technol 2016;82(9–12):1485-1494. https://doi.org/10.1007/s00170-015-7460-0. [24] Zhang X, Ehmann KF, Yu T, Wang W. Cutting forces in micro-end-milling processes. Int J Mach Tools Manuf 2016;107:21–40. https://doi.org/10.1016/j.ijmachtools.2016.04.012. [25] Zhang X, Yu T, Wang W. Prediction of cutting forces and instantaneous tool deflection in micro end milling by considering tool run-out. Int J Mech Sci 2018;136:124–33. https://doi.org/10.1016/j.ijmecsci.2017.12.019. [26] Zhou Y, Tian Y, Jing X, Ehmann KF. A novel instantaneous uncut chip thickness model for mechanistic cutting force model in micro-end-milling. Int J Adv Manuf Technol 2017;93:2305–19. https://doi.org/10.1007/s00170-017-0638-x [27] Priyabrata S, Tej P, Karali P. A hybrid modelling approach towards prediction of cutting forces in micro end milling of Ti-6Al-4V titanium alloy. Int J Mech Sci 2019;150:495–509. https://doi.org/10.1016/j.ijmecsci.2018.10.032. [28] Karaguzel U, Uysal E, Budak E, Bakkal M. Analytical modeling of turn-milling process geometry, kinematics and mechanics. Int J Mach Tools Manuf 2015;91:24–33. https://doi.org/10.1016/j.ijmachtools.2014.11.014. [29] G. Chen, C. Ren, Y. Zou, X. Qin, L. Lu, S. Li, Mechanism for material removal in ultrasonic vibration helical milling of Ti–6Al–4V alloy, Int J Mach Tools Manuf 2019;138:1–13. https://doi.org/10.1016/j.ijmachtools.2018.11.001. [30] Serizawa M, Matsumura T. Control of helical blade machining in whirling. Procedia Manuf 2016;5:417-26. https://doi.org/10.1016/j.promfg.2016.08.035. [31] He Y, Liu C, Wang YL, Li YF, Wang SL, Wang LX, Wang Y. Analytical modeling of temperature distribution in lead-screw whirling milling considering the transient un-deformed chip geometry. Int J Mech Sci 2019;157-158:619-32. https://doi.org/10.1016/j.ijmecsci.2019.05.008. [32] He Y, Wang LX, Wang YL, Li YF, Wang SL, Wang Y, Liu C, Hao CP. An analytical model for predicting specific cutting energy in whirling milling process. J Clean Prod 2019;240:118181. https://doi.org/10.1016/j.jclepro.2019.118181. [33] Pimenov DY, Guzeev VI, Patra K. A study of the influence of processing parameters and tool wear on elastic displacement of the technological system under face milling, Int J Adv Manuf Technol 2017; 92(9-12):4473–4486. https://doi.org/10.1007/s00170-017-0516-6. [34] Altintas Y. Manufacturing Automation. second ed. New York: Cambridge University Press; 2012. [35] Wang SB, Geng L, Zhang YF, Liu K, Ng TE. Cutting force prediction for five axis ball-end milling considering cutter vibrations and run-out. Int J Mech Sci 2015;96:206–15. https://doi.org/10.1016/j.ijmecsci.2015.04.007. [36] ISO 4287:1997. Geometrical Product Specifications (GPS) – surface texture: profile method – terms, 38

definitions and surface texture parameters. [37] ISO 4288:1996. Geometrical Product Specifications (GPS) – surface texture: profile method – rules and procedures for the assessment of surface texture. [38] Kurniawan R, Kiswanto G, Ko TJ. Surface roughness of two-frequency elliptical vibration texturing (TFEVT) method for micro-dimple pattern process. Int J Mach Tools Manuf 2017;116:77–95. https://doi.org/10.1016/j.ijmachtools.2016.12.011. [39] Liu XL, Zhang XD, Fang FZ, Liu SG. Identification and compensation of main machining errors on surface form accuracy in ultra-precision diamond turning. Int J Mach Tools Manuf 2016;105:45–57. https://doi.org/10.1016/j.ijmachtools.2016.03.001.

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