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An analytical force mode applied to three-dimensional turning based on a predictive machining theory
Accepted Manuscript
An analytical force mode applied to three-dimensional turning based on a predictive machining theory Zhongtao Fu , Xubing Chen , Jincheng Mao , Tao Xiong PII: DOI: Reference:
S0020-7403(17)32175-6 10.1016/j.ijmecsci.2017.12.021 MS 4085
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
7 August 2017 24 November 2017 10 December 2017
Please cite this article as: Zhongtao Fu , Xubing Chen , Jincheng Mao , Tao Xiong , An analytical force mode applied to three-dimensional turning based on a predictive machining theory, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.12.021
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ACCEPTED MANUSCRIPT Highlights
An analytical force model for three-dimensional turning is proposed based on a predictive machining theory.
Detailed analysis of tool geometry and related parameters are studied.
The effects of tool geometry and cutting parameters on the global and local chip flow angles and cutting forces are investigated. The data of cutting forces obtained from the proposed analytical model and the experiments
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are in good agreement.
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Graphical Abstract Axis of workpiece
a kK1
…
k K
wk
k
ap
k
…
ex
rk
k 1
st
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k 0
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a kp
ft /2
r
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Fig. 3 Projection of discretized cutting edges in the reference plane Pr
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An analytical force mode applied to three-dimensional turning based on a predictive machining theory Zhongtao Fu1
Xubing Chen1,*
Jincheng Mao1
Tao Xiong2
1
School of Mechanical & Electrical Engineering, Wuhan Institute of Technology, Wuhan 430205, China
2
School of Electrical & Information Engineering, Wuhan Institute of Technology, Wuhan 430205, China *
Email: [email protected],
Tel.: +86-27-8162-4640.
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Corresponding author;
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Abstract: Cutting forces play an important role in the three-dimensional turning processes and are commonly calculated by the mechanistic or empirical models which are considered time-consuming and impractical for various cutting conditions and workpiece-tool pair. Therefore, this paper proposes an analytical force model in three-dimensional turning based on a predictive machining theory, which adopts the non-equidistant shear zone model and regards the workpiece material properties, tool geometry and cutting conditions as the input data. In this model, the real cutting edge of the turning tool is decomposed into a series of infinitesimal cutting elements and the cutting action of which is equivalent to the oblique cutting process with the imposed condition that all chip elements flow in the same direction on account of the interaction between adjacent chip ones. Consequently, the cutting force components applied on each cutting element can be calculated using a modified version of predictive oblique cutting model and the total cutting forces are obtained by summing up the forces contributed by all cutting elements. Finally, the global and local chip flow angles are investigated under various conditions of tool geometry (edge inclination angle, lead angle, nose radius) and cutting parameters (depth of cut, feed, cutting velocity) in this model. Furthermore, a detailed parametric study is provided by the proposed analytical model of cutting forces in order to analyze the influences of cutting parameters and tool geometry on cutting forces, which are verified respectively by the experimental data. Good agreement shows the effectiveness of the proposed analytical model. Keywords: Analytical force model; Predictive machining theory; Chip flow direction; Turning process
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Nomenclature Pr , Ps , Reference plane, cutting plane and Pn normal plane A Tool rake face n Normal rake angle (deg) s Edge inclination angle (deg) n Normal relief angle (deg) r Lead angle (deg) re Nose radius (mm) ap Depth of cut (mm) ft Feed (mm/r) V Cutting velocity (m/min) Fnck , Ffck Normal force (N) and friction
Psk , Pnk
nk , sk
k , wrk , a kp , Ak
ck , ck c0
Cutting plane and normal plane of k-th cutting element Normal rake angle (deg) and edge inclination angle (deg) of k-th cutting element Angular position (deg), width of cut (mm), depth of cut (mm) and chip load (mm2) of k-th cutting element Local chip flow angle (deg) and deflection angle (deg) of k-th cutting element Global chip flow angle (deg)
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ek , shk
, Cp kp
dFt , j , dFr , j , dFa , j
Fx, Fy, Fz
1
Shear force (N) and normal force (N) in the primary shear plane of k-th cutting element Equivalent plane angle (deg) and shear flow angle (deg) of k-th cutting element Thickness (mm) and portion (1) of non- equidistant shear band Heat capacity (J/(kgK)) Thermal conductivity (W/(mK)) Taylor-Quinney coefficient (1) Tangential, radial and axial forces (N) of k-th cutting element Cutting forces (N) in global Cartesian coordinate {x0,y0,z0}
Fmck
, , m
T, τsh 0 Tr , Tm
A, B, n, C, m
Mean friction angle (deg) and normal shear angle (deg) of k-th cutting element Resultant force (N) exerted by the neighboring chip elements of k-th cutting element Shear strain rate (s-1), shear strain (1) and maximum shear strain rate (s-1) in the primary shear zone Temperature (K) and Shear flow stress (Pa) in the primary shear zone Reference shear strain rate (s-1) Room and melting temperature (K) Yield strength (Pa), strength coefficient, strain hardening exponent, strain rate sensitivity coefficient, thermal softening coefficient of the workpiece material
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Fsk , Fnsk
nk , nk
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force (N) in Aγ of k-th cutting element
Introduction
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Turning operation is one of the most important processes used to machine parts in manufacturing industries. In three-dimensional turning processes, turning tools have a complex geometry and the involved cutting edge is normally constituted of main cutting edge, end cutting edge and a linked nose radius, which influences the cutting forces and improves the finish surface [1]. Modeling of cutting forces is the basis for the prediction of tool wear and breakage, machine-tool vibration, and power requirement. Therefore, a reliable force model must be developed to ensure the prediction accuracy of cutting forces. Literature review shows that there have been many force models developed to predict the cutting forces in three-dimensional turning processes. Parakkal et al. [2] developed a mechanistic modeling approach to predicting cutting forces in turning which assumed the existence of an equivalent orthogonal cutting operation for any oblique operation, and the effects of tool nose radius and chip flow were also incorporated in this model. To take into account the effect of nose radius, Colwell [3] first suggested a simple geometric method using a single equivalent cutting edge, in which an imaginary straight line was drawn between the two contact points of contact between the tool and workpiece and the chip flow direction was assumed to be perpendicular to this line. Young et al. [4] modified the equivalent cutting edge model by assuming that the resultant friction force coincides with the chip flow direction. Although the nose radius was included in this model, the rake angle inclination angles were assumed to be zero. Hu et al. [5] and Wang [6] extended this method later by considering the rake angle and edge inclination angles were not equal to be zero and obtained the chip flow angle according to Stabler’s rule. Huang et al. [7] presented the modeling of the cutting forces due to chip formation under three-dimensional turning conditions using the extension of a 2D mechanistic force model while considering the effect of nose radius. Koné et al. [8] analyzed the experimental cutting forces and the chip flow angle with a groove coated tool when machining AISI 304L steel, and then they identified the empirical equations of cutting forces. Czarnota et al. [9] estimated 3D cutting forces in machining process using an hybrid numerical and analytical modeling, in which the chip flow direction
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model is coupled with plane FEM simulations. These models needs numerous experiments which are fastidious due to high cost and hardware setup complexity and are only valid for a certain workpiece-tool material pair [10]. To avoid such limitations, related researchers have made many efforts to develop the analytical models for the prediction of cutting forces in three-dimensional turning. One of the earliest researches to the analytical models was proposed by Merchant [11] for predicting there-dimensional cutting forces by the orthogonal and oblique cutting mechanics, which was little dependent of the experimental data and can be applied to the complex turning process. Later, Usui et al. [12] proposed an analytical model of chip forming process in three-dimensional cutting with single point tool, in which the process was interpreted as a piling up of orthogonal cuttings along the cutting edge, and the three components of cutting force were calculated by using only the orthogonal cutting data. A force model of three-dimensional cutting was developed by Strenkowski et al. [13] for predicting tool forces, and this model consisted of coupling an orthogonal finite element cutting model with an analytical model of three-dimensional cutting. Weng et al. [14] presented an analytical model for predicting cutting forces in turning operation by round insert considering edge effect. Molinari et al. [15] used an analytical approach to model the thermomechanical process of chip formation in a turning operation, in which the engaged part of the rounded nose was decomposed into a set of cutting edge elements for the real cutting edge geometry, thus each elementary chip produced by a straight cutting edge element was obtained from an oblique cutting process. These analytical models provide a simple and efficient approach to predict the cutting forces based on the predictive machining theory, which depends only on for cutting conditions and workpiece-tool pair without requiring extensive cutting tests. As mentioned above, compared to other force models, the analytical force models present major advantages reviewed by Germain et al. [10] and Arrazola et al. [16]. In the authors’ research work [17], an analytical force model based on the predictive machining theory [18] for predicting cutting forces in ball-end milling was proposed. Similar to ball-end milling cutter, practical turning tools can be also decomposed into a set of cutting edge elements and the cutting action of which is equivalent to the oblique cutting process. However, from the observation of the chip formation process in turning operation, the chip appears to flow in a single direction. Thus interaction between adjacent chip elements must be considered to ensure the local chip to flow in the global chip flow direction [15]. In addition, the global chip flow direction affects chip control and machined surface quality significantly, whereas the current research relies mainly on experimental approaches [19]. Therefore, the purpose of this paper is to develop an analytical force model for three-dimensional turning using the oblique cutting approach based on the non-equidistant shear zone mode proposed by Li et al. [18]. In this model, the cutting edge of turning tool is decomposed into a set of cutting elements, and the cutting force components applied on each cutting element are calculated using a modified version of predictive oblique cutting model in which all chip elements flow in the same direction and are governed by the local cutting edge. The model can be also used to predict the global chip flow direction. The organization of this paper is as follows: After introducing the tool geometry and the expression of the related parameters in three-dimensional turning process in Section 2, an analytical force model for the three-dimensional turning is proposed by regarding the cutting action of each element as the oblique cutting process with an imposed chip flow direction in which the force components can be calculated from the modified version of oblique cutting model based on the predictive machining theory. In addition, the chip flow direction is further investigated in Section 3. The model validation is implemented and discussed in Section 4 by the experimental
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2
The tool geometry for three-dimensional turning View p0 p0 Reference plane Pr
Pn
Ps
V
Cutting direction
Cutting plane
n
Feed direction
+
n A
-
S
+ Feed direction
p0
r
r
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Pr
x0
y0
Ps
View S
p0
z0
Reference plane P
r
-
s
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+
Fig. 1 Tool geometry for three-dimensional turning
Generally, in a three-dimensional turning operation, the tool geometry are defined by the basic planes Pr, Ps, Pn and cutting angles n , s , n according to ISO regulations [20] as shown
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in Fig. 1. The planes are defined as: the reference plane Pr is perpendicular to the direction of cutting velocity V; the cutting plane Ps contains the main cutting edge and is perpendicular to Pr; the normal plane Pn is parallel to the feed direction and also perpendicular to Pr. The normal rake angle n which is measured in Pn is the one between the tool rake face Aγ and the reference
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plane Pr. The normal relief angle n which is also measured in Pn is the one between the tool flank face and the reference plane Pr. The edge inclination angle s which is measured in Ps is
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the one between the cutting edge and the reference plane Pr.
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Workpiece Local top view
Fx x0 Chip
V
Tool
Fy
y0
z0
ft
Fz
ap r
re
Fig. 2 The geometric relationship in bar turning
For the typical bar turning in Fig. 2, the cutting conditions need to be given prior as: cutting parameters such as depth of cut ap, feed ft, cutting velocity V and tool geometry such as the lead angle r , edge inclination angle s , normal rake angle n and nose radius re . Furthermore,
ACCEPTED MANUSCRIPT the bar turning process is analyzed with the following assumptions: The feed ft, is supposed to be small so that the end cutting edge is not engaged. The cutting edge is assumed to be perfectly sharp. The rake face Aγ is assumed to be flat. Axis of workpiece
a kK1
a kp
wk
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k K
k
rk
k 1
st
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ap
k
…
ex
k 0
ft /2
r
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Fig. 3 Projection of discretized cutting edges in the reference plane Pr
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Inspired by the contributions of Molinari and Moufki [15], the discretization of cutting edge and coordinate transformation methods (CTM) are enhanced here to determine the involved geometrical parameters in the calculation of cutting forces. The cutting edge of the turning tool is discretized into K+2 cutting elements on the rake face Aγ, each which is indexed as k with k [0, K 1] in Figs. 3 and 4, where K is the numbers of discretized cutting elements in tool nose
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part. Besides, Fig. 4 shows the projection of the discretized cutting edges in the reference plane Pr. For the k-th cutting element, the cutting plane Psk and normal plane Pnk are perpendicular to Pr, see Fig. 4. The angle rk which is measured in the plane Pr is the one between the planes Ps0 and Psk . Some expressions about the discretized cutting elements are obtained by the geometric
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relationships in Fig. 3 as follows: sin 1 ((re -a p ) re )
st 2 r
a p re (1 cos r ) a p re (1 cos r )
(1)
ex 2 sin 1 (0.5 ft / re )
(2)
k st (k 1) (ex st ) K
(3) (4)
a 2sin 1 (0.5 ft / re )
(5)
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k-th cutting element
Ps0
ck
y kr y
k 0
rk
y0r
z 0r y00
c0
z
z
ck
k r
z 0k
0 c
z ck
Global chip flow direction
z 00 R ak e
x0
A
V
fzt.jpg
z
Feed dir0 ection
Pr
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y0
face
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Main cutting edge
Axis of workpiece
Fig. 4 Discretization of the engaged cutting edge and its projection in the reference plane Pr
As shown in Figs. 3 and 4, the k-th cutting element can be defined in Pr, by five parameters: the angular position k , the angle rk , width of cut wk , depth of cut
a kp
, cutting thickness tk
and the chip load Ak , which are expressed in the following form:
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For the 0-th cutting element,
(6)
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0 st 0 r 0 a p re (1 cos r ) 0 w0 (a p re (1 cos r )) sin r a p re (1 cos r ) 0 0 a p w sin r t f sin t r 0 A0 t0 w0
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For the k-th cutting element, 1 k K , k st (k 1) k r k 2 r 2 wk 2r sin( 2) e k k k a p w sin( r r ) t f sin( k ) t r r k Ak tk wk 2re f t sin( 2) sin( r rk )
For the (K+1)-th cutting element,
(7)
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(8)
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The geometric relationship between the discretized cutting edge on the rake face Aγ and its projection on the reference plane Pr can be obtained using the CTM as shown in Fig. 4 .Therefore, the responding cutting angles (edge inclination angle sk , normal angle nk , deflection angle ck ) for each cutting element are also determined. The edge inclination angle sk and the normal angle nk for the k-th cutting element are measured in Psk and Pnk , respectively. The angles sk and nk are determined using the CTM as:
(9)
nk sin 1 (cos rk sin n0 sin rk cos n0 sin s0 )
(10)
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sk sin 1 (cos ck sin s0 sin ck sin n0 cos s0 )
where, for k=0 cutting edge, the cutting angles s0 , n0 is known. And the deflection angle ck is measured between the main cutting edge and the k-th cutting edge element in the rake angle Aγ and its projection in Pr corresponds to rk , which is derived using the CTM as: cos s0 tan rk 0 k 0 0 cos n tan r sin n sin s
3
(11)
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ck tan 1
Calculation of cutting forces
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3.1 The chip flow direction The chip flow direction has a critical effect on the friction force direction and chip control. In the chip flow model shown in Fig. 4, similar with the approach of Molinari et al. [12] and Dorlin et al. [25] about chip flow analysis, it is assumed that all the discretized cutting elements have the same chip flow direction and velocity Vc in the rake face Aγ as observed in experimental results [6]. The global flow direction in this chip flow model is represented by the angle c0 . Therefore, the local chip flow angle ck for the k-th cutting element in Fig. 4 can be expressed as:
ck c0 ck
(12)
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3.2 The equilibrium equation of chip elements During the chip formation process, all the chip elements have the same flow direction. Consequently, the k-th chip element is constrained by the adjacent chip elements. In addition, the formation of the k-th chip element is regarded as the shear effect in the primary deformation zone. Therefore, the equilibrium equation of the forces applied on the k-th chip element can be written as: Fnck x c Ffck zc Fmck yc Fsk x ksh Fnsk z ksh =0 Tool effect
Chip effect
Shear effect
(13)
where, xc is normal to the rake face Aγ; zc is the chip flow direction; yc is normal to the flow direction in Aγ; Fnck , Ffck are the normal force and friction force respectively in Aγ; Fmck is the
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(14)
where, the angle nk is the mean friction angle between the k-th chip element and Aγ. The
Fsk .
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k k k unkown Ffc , Fs , Fns can be solved by the Newton-Raphson iteration algorithm with the known
Furthermore, according to the condition that the internal forces among the chip elements vanish, the equilibrium equation of the forces applied on the all chip elements be written as: K 1
F
k nc
k 0
x c Ffck zc Fsk x ksh Fnsk z ksh 0
(15)
Combining Eqs. (13) and (15), the following implicit relation is obtained as: K 1
F
0
(16)
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k mc
k 0
The global chip flow angle c0 can be calculated by solving Eq.(16).
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3.3 Oblique force model of chip elements For the k-th cutting element, the cutting action can be regarded as the oblique cutting process with the imposed condition that chip flow velocity is equal to global chip flow velocity. The modified version of oblique cutting model is used to calculate the shear force Fsk and normal force Fnsk exerted by the shear effect of the primary deformation zone, and the relevant equations derived are established and summarized [17].
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Pnk
nk
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z
Pr
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y
y 0k
z
k n
k 0
nk
Vc
Chip
Vc
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Frk
xek
sk
Pshk
z ek
nk
V x
Fak
Tool
H
P
P
k 0
Ft k
nk
A
k xkn Pe
k n
k sh
k c
V
Workpiece
Psk
(a) Oblique cutting model and chip formation
(b) Non-equidistant shear model
Fig. 5 Oblique cutting model and chip formation of infinitesimal cutting edge
According to the coordinate transform and oblique geometric relationship in Fig. 5(a), The equivalent plane angle ek , which determines the equivalent plane Pek (the mechanism of oblique cutting is regarded as a two dimensional cutting process), is [21] : tan ck cosshk + sin(nk nk )sin shk cos(nk nk )
ek tan 1
(17)
The shear flow angle shk which characterizes the shear direction in primary shear zone, are given by [17]:
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shk tan 1
(18)
In Fig. 5(b), the primary shear zone is modeled as a parallel and non-equidistant shear band which consists of two non-equidistant thickness (1 )
with thickness
and
,
characterized by the portion [0,1] The governing equations for the shear strain rate , the shear strain , the temperature T and the maximum shear strain rate m in the primary shear zone, are detailed in [17], [22] and are given as: m k q [(1 ) ]q [ ze (1 ) ] m ( z k )q e q ( )
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zek [ (1 ) , 0]
(19)
z [0, k e
]
cosek T zek m C pV cos sk sin nk
(q 1)V cos sk cos nk (q 1)Vs cosshk cos(nk nk )
cosshk cos(nk nk )(cos nk cosshk tan sk sin shk ) cos nk
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cosek m k q 1 zek [ (1 ) , 0] V cos k sin k (q 1)[(1 ) ]q [ ze (1 ) ] s n k k q 1 cosek cos nk cose m ( ze ) z k [0, ] k k q k V cos s sin n (q 1)( ) sin n cos shk cos(nk nk ) e
(20)
(21)
(22)
(23)
The normal friction angle nk can be derived from the work of Budak et al. [23] using an
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analytical dual-zone model of the tool-chip interface as follows: 1/ s 1 1 s s P0 P0
(24)
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nk tan 1
where P0 4
1 cos 2 nk cosshk is the normal pressure; is the distribution exponent 2 sin[2(nk nk nk )] s
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for normal pressure taken as 3 based on the experimental data [24]; s is the sliding friction coefficient on the tool-chip interface and a power function of the chip velocity. The normal shear angle nk is determined by a modified Merchant equation [11] as:
nk C1 C2 (nk nk )
(25)
where C1 , C2 are constant depending on the workpiece-tool material. In order to determine the shear flow stress sh in primary shear zone, Johnson-Cook constitutive model, which describes the flow stress of the workpiece material by considering strain, strain rate, and temperature effects [15], is adopted in the present work because it is well-accepted, numerically robust and utilized widely in cutting modelling and simulations studies [16] and given as follows:
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m n T Tr 1 1 C ln 1 A B 3 3 0 Tm Tr
(26)
The characteristics of material behavior are determined by yield strength A, strength coefficient B, strain hardening exponent n, strain rate sensitivity coefficient C and thermal softening coefficient m. Furthermore, the shear force Fsk , which is proportional to the shear stress sh in primary shear plane, along with the normal force Fnsk can be expressed as:
Fnsk
Ak cos sk sin nk
(27)
[tan(nk nk )+ tan nk cosck ]cosshk k Fs 1 tan nk cosck tan(nk nk )
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Fsk sh
(28)
For the k-th cutting element, the three force components dFt , j , dFr , j , dFa , j (tangential, radial, axial) due to the shear effect, are evaluated from the following matrix form.
sin nk cos sk k Fs cos nk Fk sin nk sin sk ns
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Ft k cosshk cos nk cos sk sin shk sin sk k cosshk sin nk Fr k cos k cos k sin k sin k cos k sh n s sh s Fa
(29)
Finally, the cutting forces contributed by all cutting elements are summed to obtain the total forces on the tool as：
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k Fx 0 0 1 Ft K 1 k k k Fy 0 sin( r r ) cos( r r ) Fa F k 0 0 cos( k ) sin( k ) F k r r r r r z
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(30)
Model Validation and Discussion
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In order to validate the effectiveness of the proposed analytical model of cutting forces for three-dimensional turning, a series of turning experiments are carried out. The corresponding computer programs in MATLAB 8.5 are developed to implement on the analysis of cutting force model. The analyses are divided into two parts: one section is used to analyze the influence of tool geometry: tool nose radius re and lead angle kr and cutting parameters: depth of cut ap, feed ft, cutting velocity V on chip flow direction; another section is to analyze how the cutting parameters and tool geometry affect cutting forces and compare with the experimental data. In addition, the used workpiece is NAB (Nickel Aluminum Bronze) material which presents the properties of excellent wear resistance and stress corrosion resistance and is extensively used for marine propellers. In our preceding work [26], combining SHPB tests, predictive force model and cutting experiment, constitutive parameters of NAB are identify accurately in machining and given in Table 1. Table 1. Properties of workpiece material NAB[26] (1) Johnson-Cook parameters A
B
C
n
m
0
Tm
Tr
295MPa
759.5MPa
0.011
0.405
1.09
0.001s-1
1311K
298K
(2) Thermo-physical properties
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Heat capacity C p
7530 kg/m3
419 J/(kgK)
Thermal
Taylor-Quinney
conductivity k p
coefficient
41.9 W/(mK)
0.85
4.1 Chip flow direction analysis The chip flow direction has a significant influence on the calculation of cutting forces and surface quality in three-dimensional turning. When the actual cutting edge is divided into a series of cutting elements, the local chip flow direction (given by ck ) of each chip element is imposed
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to follow the global chip direction (given by c0 ) due to the interaction between adjacent chip elements. In addition, the analysis of global chip flow angle could be illustrated distinctly
according to the one of main cutting edge. Thus, the effects of tool geometry and cutting parameters on global and local chip flow directions are analyzed respectively for the subsequent calculation of cutting forces. In addition, the global chip flow angle is calculated by
solving implicit equation in Eq.(16), so chip flow angles are not influenced by the value for K.
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(1) Global chip flow direction
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(a) Effect of edge inclination angle s0 on c0 .
(b) Effect of lead angle r on c0 . Cutting conditions are: n0 00 , s0 00 , re 0.8mm,
a p 0.4mm, ft 0.1mm / r ,V 90m / min
a p 0.4mm, ft 0.1mm / r ,V 90m / min
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Cutting conditions are: n0 00 , r 750 , re 0.8mm,
(c) Effect of nose radius re on c0 . Cutting
(d) Effect of depth of cut a p on c0 . Cutting
conditions are: n0 00 , s0 00 , r 750 ,
conditions are: n0 00 , s0 00 , r 750 ,
a p 0.4mm, ft 0.1mm / r ,V 90m / min
re 0.8mm, ft 0.1mm / r,V 90m / min
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(e) Effect of feed f t on c0 . Cutting conditions are:
(f) Effect of cutting velocity V on c0 . Cutting
n0 00 , s0 00 , r 750 , re 0.8mm,
conditions are: n0 00 , s0 00 , r 750 ,
a p 0.4mm,V 90m / min
re 0.8mm, a p 0.4mm, ft 0.1mm / r
Fig. 6 Effect of tool geometry and cutting parameters on global chip flow direction
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The curves in Fig. 6 show the variations of the predicted global chip flow angle with the tool geometry (edge inclination angle, lead angle, nose radius) and the cutting parameters (depth of cut, feed, cutting velocity) respectively with given cutting conditions. It can be seen that the global chip flow angle increases with an increase in edge inclination angle, lead angle and nose radius, decreases with an increase in depth of cut, and changes slightly with feed and cutting velocity. Fig. 6(a) shows a linear relationship approximately between the predicted global chip flow angle and edge inclination angle which accords with the results of Wang et al. [6]. This is due to the reason that the component V sin s0 of cutting velocity in the paralleled cutting edge direction increases with an increase of edge inclination angle, which will cause the increase the component Vc sinc0
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of chip velocity in the same direction [17], i.e., the approximate linear increase of global chip flow angle. Fig. 6(b) gives a upwards curve containing concave and linear parts between the predicted c0 and lead angle, which may be explained that the cutting thickness increases with the increase
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of lead angle in Eq. (6) and produces the large plastic work, thus causing the increase of global chip flow angle .In addition, the phenomenon of curve saltation is that main cutting edge is engaged because of a p re (1 cos r ) for the values of r less than 60 deg. Fig. 6(c) depicts a
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convex upwards curve between the predicted c0 and nose radius while Fig. 6(d) gives a concave downwards one between the predicted c0 and depth of cut, which are in accordance with the experimental results of Young et al. [4], [13]. What is more, c0 varies very rapidly when the
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main cutting edge part is engaged, i.e. a p re (1 cos r ) . These reasons for Fig. 6(c) and Fig. 6(d) are illustrated that as the nose radius becomes small or the depth of cut becomes large, the proportion of engaged main cutting edge part increases, thus causing the decrease of c0 . Fig. 6(e) and (f) gives a constant global chip flow angle which is approximately equal to 36.5 deg with the feed or cutting velocity, for the cutting velocity or feed has little influence on the proportion of the engaged main cutting edge part as observed from experimental results[4]. (2) Local chip flow direction
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(b) Effect of lead angle r on ck . Cutting
Cutting conditions are: n0 00 , r 750 , re 0.8mm,
conditions are: n0 00 , s0 00 , re 0.8mm,
a p 0.4mm, ft 0.1mm / r ,V 90m / min
a p 0.4mm, ft 0.1mm / r ,V 90m / min
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(a) Effect of edge inclination angle s0 on ck .
(d) Effect of depth of cut a p on ck .
conditions are: n0 00 , s0 00 , r 750
Cutting conditions are: n0 00 , s0 00 , r 750
a p 0.4mm, ft 0.1mm / r ,V 90m / min
re 0.8mm, ft 0.1mm / r,V 90m / min
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(c) Effect of nose radius re on ck . Cutting
(e) Effect of feed f t on ck . Cutting conditions
(f) Effect of cutting velocity V on ck . Cutting
are: 0 , 0 , r 75 , re 0.8mm,
conditions are: n0 00 , s0 00 , r 750 ,
a p 0.4mm,V 90m / min
re 0.8mm, a p 0.4mm, ft 0.1mm / r
0 n
0
0 s
0
0
Fig. 7 Effect of tool geometry and cutting parameters on local chip flow direction
Fig. 7 gives the effect of tool geometry (edge inclination angle, lead angle, nose radius) and cutting parameters (depth of cut, feed, cutting velocity) on the predicted local chip flow angle respectively with the given cutting conditions. K=20 is chosen here for the chip flow angle is not affected by the value for K. It can be seen in Fig. 7 (a) that the local chip flow angle increases with an increase of edge inclination angle, which are explained that the local edge inclination angle of
ACCEPTED MANUSCRIPT discrete cutting element increases with an increase of edge inclination angle in Eq.(9), and then causes the stronger interaction of chip elements. In addition, the local chip flow angle is little affected by the edge inclination angle in the range of -20 deg to 0 deg, while large in the range of 0 deg to 20 deg. The discontinuity of local chip flow angle between the cutting elements k=0 and k=1 is that the main cutting edge is not engaged for a p re (1 cos r ) . Fig. 7(b) shows the curves of local chip flow angle with lead angle that is an increase of ck with an increase of the lead angle. The interactions of chip elements become stronger with the increase of the lead angle. Fig. 7(c)
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shows the curves of local chip flow angle with nose radius that is a decrease of ck with an increase of the nose radius, which is explained that with the decrease of the nose radius, the variation of chip element flow direction zck is larger and then causes the stronger interaction of
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chip elements. For the cutting element k=9, the chip element is free flow, but the flow becomes more constrained for cutting elements k<9 or k>9. Fig. 7(d) shows that the local chip flow angle decreases with an increase of the depth of cut, which is that the engaged main cutting edge increases with the increase of the depth of cut and causes the lower interaction of chip elements. Fig. 7(e) and (f) shows that the effects of the feed and cutting velocity on the local chip flow angle are little as the same in Fig. 6(e) and (f).
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4.2 Cutting force validation A series of three-dimensional turning tests have been performed on a CNC turning center without cutting fluid and the experimental setup is shown as in Fig. 8. The cutting inserts mounted on tool holder are used to machine the NAB bars of 120 mm diameter. The tool holder is mounted on a three-component Kistler table dynamometer (model 9523B) which is used to measure cutting forces. In addition, every test is repeated three times to ensure the reliability of cutting forces data, and Figs. 9 and 13 shows the predicted and experimental results of cutting forces in lines and discrete symbols, respectively.
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NAB workpiece
Cutting insert Tool holder
Chuck
Dynamometer
Fig. 8 The experiment setup
(1) The effect of cutting parameters ( a p , ft ,V ) In order to investigate the effect of cutting parameters on cutting forces, the cutting inserts referenced Sandvik Coromant SNMG 12 04 08-MR 2025 and the geometrical parameters of this
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Fig. 9 Comparison of cutting forces versus depth of cut from predicted model and experimental data: (a) ft=0.1mm/r, V=90m/min; (b) ft=0.2mm/r, V=90m/min; (c) ft=0.2mm/r, V=150m/min.
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The four groups of cutting parameters: feeds (0.1mm/r, 0.2mm/r), and cutting velocities (90mm/min, 150mm/min) are chosen to investigate the effect of depths of cut (0.4mm, 0.8mm, 1.2mm, 1.8mm) on cutting forces. Variations of cutting force with depth of cut are shown in Fig. 9 and then comparison is carried out between the predicted model and experimental data, in which there is good agreement between the predicted and experimental data. The main cutting edge is engaged when the depth of cut ap is more than re (1 cos r ) 0.593mm which causes the cutting
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forces to increase proportionally with the increase of the depth of cut. While the depth of cut ap is small and less than 0.593mm, the engaged tool nose part becomes dominant, which causes the cutting forces to vary nonlinearly.
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Fig. 10 Comparison of cutting forces versus feed from predicted model and experimental data: (a) ap=0.4mm, V=90m/min; (b) ap=1.2 mm, V=90m/min; (c) ap=1.2mm, V=150m/min.
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The four groups of cutting parameters: depths of cut (0.4mm, 1.2mm), and cutting velocities (90mm/min, 150mm/min) are chosen to investigate the effect of feeds (0.05mm/r, 0.1mm/r, 0.15mm/r, 0.2mm/r) on cutting forces. Fig. 10 give the variations of cutting force with feed and the comparison of cutting forces between the predicted mode and experimental data, in which the predicted and experimental data have in good agreement and the cutting forces increase with the increase of the feed. Different from the effect of depth of cut, the cutting forces varies with feed linearly, which is due to the reason that the undeformed chip thickness t ft sin( r ) increases linearly with the increase of feed, i.e., which causes the cutting forces to increase.
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Fig. 11 Comparison of cutting forces versus cutting velocity from predicted model and experimental data: (a)
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ap=0.4mm, ft=0.1mm/r; (b) ap=1.2mm, ft=0.1mm/r; (c) ap=1.2mm, ft=0.2mm/r.
(2) The effect of tool geometry ( kr , re ) (b)
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The four groups of cutting parameters: depths of cut (0.4mm, 1.2mm), and feeds (0.1mm/r, 0.2mm/r) are chosen to investigate the effect of cutting velocities (60mm/min, 75mm/min, 90mm/min, 150mm/min) on cutting forces. In Fig. 11, as the cutting velocity is increased, the three force components, both predicted and experimental, which are in good agreement. The cutting forces decrease nonlinearly with the increase of the cutting velocity which conforms to the tendency of the experimental results by Young et al. [4]. This can be explained that the friction coefficient decreases with the increase of cutting velocity which results in the higher temperature in tool-chip interface [15], thus causes the decrease of cutting forces. What is more, the change rates of cutting forces are very slow when the cutting velocity is small.
Fig. 12 Comparison of cutting forces versus nose radius from predicted model and experimental data: (a)
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ap=0.4mm, ft=0.1mm/r, V=90mm/min; (b) ap=1.2mm, ft=0.1mm/r, V=90mm/min; (c) ap=1.2mm, ft=0.2mm/r, V=150mm/min.
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The three nose radii (0.4mm, 0.8mm, 1.2mm) by three different cutting inserts are chosen to investigate the effect of the nose radius on cutting forces with the same lead angle 75deg, edge inclination angle -6deg, normal rake angle 0deg and three groups of cutting parameters in Fig. 12. Variations of cutting force with nose radius are shown in Fig. 12 where the predicted and experimental data have in good agreement. It can be seen in Fig. 12(a) that the main cutting edge does not engages in cutting, i.e., a p re (1 cos r ) ), and the area of total undeformed chip is independent of the nose radius, the force component Fx is determined by the variation of chip velocity caused by the nose radius, however, the chip velocity keeps constant with nose radius, which further keeps Fx nearly constant. Conversely, Fy and Fz are affected significantly by the global chip flow angle, Fz is decreased with increase of the global chip flow angle which varies synchronously with the nose radius in Fig. 6 (c), while Fy is increased with increase of the nose
ACCEPTED MANUSCRIPT radius. Fig. 12(b) and (c) show that the main cutting edge is engaged, i.e. a p re (1 cos r ) ,). With the increase of the nose radius, the length of nose radius and width of cut is also increased, which causes the global chip flow angle to increase and the cutting forces further increase. (b)
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Fig. 13 Comparison of cutting forces versus lead angle from predicted model and experimental data: (a) ap=0.4mm, ft=0.1mm/r, V=90mm/min; (b) ap=1.2mm, ft=0.1mm/r, V=90mm/min; (c) ap=1.2mm, ft=0.2mm/r, V=150mm/min.
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The three lead angles (45 deg, 75 deg, 90 deg) by three different tool holders are chosen to investigate the effect of the lead angle on cutting forces with the same nose radius 0.4mm, edge inclination angle -6deg, normal rake angle 0deg and three groups of cutting parameters. Fig. 13 shows the predicted and experimental cutting forces plotted against the lead angle. The data show the linear increase in force components Fx and Fz but decreased Fy with the increase of lead angle, this trend conforms to the experimental results by Young et al. [4]. The effect of lead angle on the force component Fx is determined by the undeformed chip thickness normally. With the increase of lead angle, the engaged length of nose radius thus is increased, the interaction between chip elements is strengthened, which causes Fx to increase. While the effect of lead angle on the force components Fy and Fz is determined by the projection of the resultant force in y- and z- direction respectively, therefore, the cutting component Fz is increased and Fy is decreased with the increase of lead angle.
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In this paper, an analytical force model for three-dimensional turning has been developed and discussed based on a modified predictive machining theory and the analysis of turning tool geometry is the extension to the work of Molinari et al. [15]. The predicted cutting forces are verified the effectiveness by the experimental data. The contributions of the proposed analytical model are drawn as follows: (1) The analytical force model can predict the cutting forces and chip flow angles without the need for cutting tests in the three-dimensional turning. The cutting forces of each cutting element on the cutting edge are calculated using the modified version of the predictive oblique cutting model. (2) The proposed model has also been used to investigate the effects of cutting conditions and on the tool geometry on chip flow directions. The global and local chip flow angles are affected significantly by the tool geometry (edge inclination angle, lead angle, nose radius) and depth of cut, while little effect caused by feed and cutting velocity. (3) The proposed force model has been validated using a series of three-dimensional turning experiments. The data obtained from the proposed model and the experiments have been shown to be in good agreement. However, the influences of tool-chip interface and flank wear on predicted forces in actual three-dimensional turning process, need to be studied for more accurate prediction of cutting
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Acknowledge This work was partially supported by the Scientific Research Foundation of Wuhan Institute of Technology (Grant No. K201707), the Nature Science Foundation of Hubei (Grant No. 2017CFB346) and the National Natural Science Foundation (Grant No. 51705385).
Reference
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[1] Shaw M C, Cookson J O. Metal cutting principles. New York: Oxford university press, 2005. [2] Parakkal G, Zhu R, Kappor S G, DeVor R E. Modeling of turning process cutting forces for grooved tools. International Journal of Machine Tools Manufacture, 2002, 42:1 79–191. [3] Colwell L V. Predicting the angle of chip flow for single-point cutting tools. Transactions of the ASME, 1954, 76, 199–204. [4] Young H T, Mathew P, Oxley P L B. Allowing for nose radius effects in predicting the chip flow direction and cutting forces in bar turning. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 1987, 201(3): 213-226. [5] Hu R S, Mathew P,. Oxley P L B, Young H T. Allowing for end cutting edge effects in predicting forces in bar turning with oblique machining conditions, Proceedings of the Institution of Mechanical Engineers, 1986, 200(C2), 89–99. [6] Wang J. Development of a chip flow model for turning operations. International Journal of Machine Tools Manufacture, 2001, 41: 1265–1274. [7] Huang Y, Liang S Y. Modeling of cutting forces under hard turning conditions considering tool wear effect. Transactions of the ASME-B-Journal of Manufacturing Science and Engineering, 2005, 127(2): 262-270. [8] Koné F, Czarnota C, Haddag B, et al. Modeling of velocity-dependent chip flow angle and experimental analysis when machining 304L austenitic stainless steel with groove coated-carbide tools. Journal of Materials Processing Tech, 2013, 213(7):1166-1178. [9] Czarnota C, Koné F, Haddag B, et al. A predictive hybrid force modeling in turning: application to stainless steel dry machining with a coated groove tool. International Journal of Advanced Manufacturing Technology, 2016, 79(1):1-15. [10] Germain D, Fromentin G, Poulachon G, et al. From large-scale to micromachining: A review of force prediction models. Journal of Manufacturing Processes, 2013, 15(3):389-401. [11] Merchant M E. Basic mechanics of the metal-cutting process. ASME J. of Applied Mechanics, 1944, 11: A168. [12] Usui E, Hirota A, Masuko M. Analytical prediction of three dimensional cutting process. Part 1: basic cutting model and energy approach. Journal of Engineering for Industry, 1978, 100(2): 222-228. [13] Strenkowski J S, Shih A J, Lin J C. An analytical finite element model for predicting three-dimensional tool forces and chip flow. International Journal of Machine Tools and Manufacture, 2002, 42(6): 723-731. [14] Weng J, Zhuang K, Chen D, et al. An analytical force prediction model for turning operation by round insert considering edge effect. International Journal of Mechanical Sciences, 2017, 128: 168-180. [15] Molinari A, Moufki A. A new thermomechanical model of cutting applied to turning operations. Part I. Theory. International Journal of Machine Tools & Manufacture, 2005,
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45(2):166-180. [16] Arrazola P J, Özel T, Umbrello D, et al. Recent advances in modelling of metal machining processes. CIRP Annals-Manufacturing Technology, 2013, 62(2): 695-718. [17] Fu Z, Yang W, Wang X, et al. An analytical force model for ball-end milling based on a predictive machining theory considering cutter runout. The International Journal of Advanced Manufacturing Technology, 2016, 84(9-12): 2449-2460. [18] Li B, Wang X, Hu Y, et al. Analytical prediction of cutting forces in orthogonal cutting using unequal division shear-zone model. The International Journal of Advanced Manufacturing Technology, 2011, 54(5): 431-443. [19] D’Acunto A, Le Coz G, Moufki A, et al. Effect of cutting edge geometry on chip flow direction–analytical modelling and experimental validation. Procedia CIRP, 2017, 58: 353-357. [20] Astakhov V P. Basic definitions and cutting tool geometry, single point cutting tools. Geometry of Single-point Turning Tools and Drills: Fundamentals and Practical Applications, 2010: 55-126. [21] Oxley P L B. The mechanics of machining: an analytical approach to assessing machinability. New York: Ellis Horwood Limited, 1989. [22] Fu Z, Yang W, Wang X, et al. Analytical modelling of milling forces for helical end milling based on a predictive machining theory. Procedia CIRP, 2015, 31: 258-263. [23] Budak E, Ozlu E. Development of a thermomechanical cutting process model for machining process simulations. CIRP Annals-Manufacturing Technology, 2008, 57(1): 97-100. [24] Ozlu E, Budak E, Molinari A. Thermomechanical modeling of orthogonal cutting including the effect of stick-slide regions on the rake face. 10th CIRP International Workshop on Modeling of Machining Operations, Calabria, Italy, 2007. [25] Dorlin T, Fromentin G, Costes J P. Generalised cutting force model including contact radius effect for turning operations on Ti6Al4V titanium alloy. International Journal of Advanced Manufacturing Technology, 2016, 66(9-12):1-17. [26] Fu Z, Yang W, Zeng S, et al. Identification of constitutive model parameters for nickel aluminum bronze in machining. Transactions of Nonferrous Metals Society of China, 2016, 26(4):1105-1111.
An analytical force mode applied to three-dimensional turning based on a predictive machining theory Zhongtao Fu , Xubing Chen , Jincheng Mao , Tao Xiong PII: DOI: Reference:
S0020-7403(17)32175-6 10.1016/j.ijmecsci.2017.12.021 MS 4085
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
7 August 2017 24 November 2017 10 December 2017
Please cite this article as: Zhongtao Fu , Xubing Chen , Jincheng Mao , Tao Xiong , An analytical force mode applied to three-dimensional turning based on a predictive machining theory, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.12.021
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An analytical force model for three-dimensional turning is proposed based on a predictive machining theory.
Detailed analysis of tool geometry and related parameters are studied.
The effects of tool geometry and cutting parameters on the global and local chip flow angles and cutting forces are investigated. The data of cutting forces obtained from the proposed analytical model and the experiments
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are in good agreement.
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Graphical Abstract Axis of workpiece
a kK1
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k K
wk
k
ap
k
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k 0
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a kp
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Fig. 3 Projection of discretized cutting edges in the reference plane Pr
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An analytical force mode applied to three-dimensional turning based on a predictive machining theory Zhongtao Fu1
Xubing Chen1,*
Jincheng Mao1
Tao Xiong2
1
School of Mechanical & Electrical Engineering, Wuhan Institute of Technology, Wuhan 430205, China
2
School of Electrical & Information Engineering, Wuhan Institute of Technology, Wuhan 430205, China *
Email: [email protected],
Tel.: +86-27-8162-4640.
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Corresponding author;
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Abstract: Cutting forces play an important role in the three-dimensional turning processes and are commonly calculated by the mechanistic or empirical models which are considered time-consuming and impractical for various cutting conditions and workpiece-tool pair. Therefore, this paper proposes an analytical force model in three-dimensional turning based on a predictive machining theory, which adopts the non-equidistant shear zone model and regards the workpiece material properties, tool geometry and cutting conditions as the input data. In this model, the real cutting edge of the turning tool is decomposed into a series of infinitesimal cutting elements and the cutting action of which is equivalent to the oblique cutting process with the imposed condition that all chip elements flow in the same direction on account of the interaction between adjacent chip ones. Consequently, the cutting force components applied on each cutting element can be calculated using a modified version of predictive oblique cutting model and the total cutting forces are obtained by summing up the forces contributed by all cutting elements. Finally, the global and local chip flow angles are investigated under various conditions of tool geometry (edge inclination angle, lead angle, nose radius) and cutting parameters (depth of cut, feed, cutting velocity) in this model. Furthermore, a detailed parametric study is provided by the proposed analytical model of cutting forces in order to analyze the influences of cutting parameters and tool geometry on cutting forces, which are verified respectively by the experimental data. Good agreement shows the effectiveness of the proposed analytical model. Keywords: Analytical force model; Predictive machining theory; Chip flow direction; Turning process
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Nomenclature Pr , Ps , Reference plane, cutting plane and Pn normal plane A Tool rake face n Normal rake angle (deg) s Edge inclination angle (deg) n Normal relief angle (deg) r Lead angle (deg) re Nose radius (mm) ap Depth of cut (mm) ft Feed (mm/r) V Cutting velocity (m/min) Fnck , Ffck Normal force (N) and friction
Psk , Pnk
nk , sk
k , wrk , a kp , Ak
ck , ck c0
Cutting plane and normal plane of k-th cutting element Normal rake angle (deg) and edge inclination angle (deg) of k-th cutting element Angular position (deg), width of cut (mm), depth of cut (mm) and chip load (mm2) of k-th cutting element Local chip flow angle (deg) and deflection angle (deg) of k-th cutting element Global chip flow angle (deg)
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ek , shk
, Cp kp
dFt , j , dFr , j , dFa , j
Fx, Fy, Fz
1
Shear force (N) and normal force (N) in the primary shear plane of k-th cutting element Equivalent plane angle (deg) and shear flow angle (deg) of k-th cutting element Thickness (mm) and portion (1) of non- equidistant shear band Heat capacity (J/(kgK)) Thermal conductivity (W/(mK)) Taylor-Quinney coefficient (1) Tangential, radial and axial forces (N) of k-th cutting element Cutting forces (N) in global Cartesian coordinate {x0,y0,z0}
Fmck
, , m
T, τsh 0 Tr , Tm
A, B, n, C, m
Mean friction angle (deg) and normal shear angle (deg) of k-th cutting element Resultant force (N) exerted by the neighboring chip elements of k-th cutting element Shear strain rate (s-1), shear strain (1) and maximum shear strain rate (s-1) in the primary shear zone Temperature (K) and Shear flow stress (Pa) in the primary shear zone Reference shear strain rate (s-1) Room and melting temperature (K) Yield strength (Pa), strength coefficient, strain hardening exponent, strain rate sensitivity coefficient, thermal softening coefficient of the workpiece material
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force (N) in Aγ of k-th cutting element
Introduction
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Turning operation is one of the most important processes used to machine parts in manufacturing industries. In three-dimensional turning processes, turning tools have a complex geometry and the involved cutting edge is normally constituted of main cutting edge, end cutting edge and a linked nose radius, which influences the cutting forces and improves the finish surface [1]. Modeling of cutting forces is the basis for the prediction of tool wear and breakage, machine-tool vibration, and power requirement. Therefore, a reliable force model must be developed to ensure the prediction accuracy of cutting forces. Literature review shows that there have been many force models developed to predict the cutting forces in three-dimensional turning processes. Parakkal et al. [2] developed a mechanistic modeling approach to predicting cutting forces in turning which assumed the existence of an equivalent orthogonal cutting operation for any oblique operation, and the effects of tool nose radius and chip flow were also incorporated in this model. To take into account the effect of nose radius, Colwell [3] first suggested a simple geometric method using a single equivalent cutting edge, in which an imaginary straight line was drawn between the two contact points of contact between the tool and workpiece and the chip flow direction was assumed to be perpendicular to this line. Young et al. [4] modified the equivalent cutting edge model by assuming that the resultant friction force coincides with the chip flow direction. Although the nose radius was included in this model, the rake angle inclination angles were assumed to be zero. Hu et al. [5] and Wang [6] extended this method later by considering the rake angle and edge inclination angles were not equal to be zero and obtained the chip flow angle according to Stabler’s rule. Huang et al. [7] presented the modeling of the cutting forces due to chip formation under three-dimensional turning conditions using the extension of a 2D mechanistic force model while considering the effect of nose radius. Koné et al. [8] analyzed the experimental cutting forces and the chip flow angle with a groove coated tool when machining AISI 304L steel, and then they identified the empirical equations of cutting forces. Czarnota et al. [9] estimated 3D cutting forces in machining process using an hybrid numerical and analytical modeling, in which the chip flow direction
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model is coupled with plane FEM simulations. These models needs numerous experiments which are fastidious due to high cost and hardware setup complexity and are only valid for a certain workpiece-tool material pair [10]. To avoid such limitations, related researchers have made many efforts to develop the analytical models for the prediction of cutting forces in three-dimensional turning. One of the earliest researches to the analytical models was proposed by Merchant [11] for predicting there-dimensional cutting forces by the orthogonal and oblique cutting mechanics, which was little dependent of the experimental data and can be applied to the complex turning process. Later, Usui et al. [12] proposed an analytical model of chip forming process in three-dimensional cutting with single point tool, in which the process was interpreted as a piling up of orthogonal cuttings along the cutting edge, and the three components of cutting force were calculated by using only the orthogonal cutting data. A force model of three-dimensional cutting was developed by Strenkowski et al. [13] for predicting tool forces, and this model consisted of coupling an orthogonal finite element cutting model with an analytical model of three-dimensional cutting. Weng et al. [14] presented an analytical model for predicting cutting forces in turning operation by round insert considering edge effect. Molinari et al. [15] used an analytical approach to model the thermomechanical process of chip formation in a turning operation, in which the engaged part of the rounded nose was decomposed into a set of cutting edge elements for the real cutting edge geometry, thus each elementary chip produced by a straight cutting edge element was obtained from an oblique cutting process. These analytical models provide a simple and efficient approach to predict the cutting forces based on the predictive machining theory, which depends only on for cutting conditions and workpiece-tool pair without requiring extensive cutting tests. As mentioned above, compared to other force models, the analytical force models present major advantages reviewed by Germain et al. [10] and Arrazola et al. [16]. In the authors’ research work [17], an analytical force model based on the predictive machining theory [18] for predicting cutting forces in ball-end milling was proposed. Similar to ball-end milling cutter, practical turning tools can be also decomposed into a set of cutting edge elements and the cutting action of which is equivalent to the oblique cutting process. However, from the observation of the chip formation process in turning operation, the chip appears to flow in a single direction. Thus interaction between adjacent chip elements must be considered to ensure the local chip to flow in the global chip flow direction [15]. In addition, the global chip flow direction affects chip control and machined surface quality significantly, whereas the current research relies mainly on experimental approaches [19]. Therefore, the purpose of this paper is to develop an analytical force model for three-dimensional turning using the oblique cutting approach based on the non-equidistant shear zone mode proposed by Li et al. [18]. In this model, the cutting edge of turning tool is decomposed into a set of cutting elements, and the cutting force components applied on each cutting element are calculated using a modified version of predictive oblique cutting model in which all chip elements flow in the same direction and are governed by the local cutting edge. The model can be also used to predict the global chip flow direction. The organization of this paper is as follows: After introducing the tool geometry and the expression of the related parameters in three-dimensional turning process in Section 2, an analytical force model for the three-dimensional turning is proposed by regarding the cutting action of each element as the oblique cutting process with an imposed chip flow direction in which the force components can be calculated from the modified version of oblique cutting model based on the predictive machining theory. In addition, the chip flow direction is further investigated in Section 3. The model validation is implemented and discussed in Section 4 by the experimental
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2
The tool geometry for three-dimensional turning View p0 p0 Reference plane Pr
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Fig. 1 Tool geometry for three-dimensional turning
Generally, in a three-dimensional turning operation, the tool geometry are defined by the basic planes Pr, Ps, Pn and cutting angles n , s , n according to ISO regulations [20] as shown
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in Fig. 1. The planes are defined as: the reference plane Pr is perpendicular to the direction of cutting velocity V; the cutting plane Ps contains the main cutting edge and is perpendicular to Pr; the normal plane Pn is parallel to the feed direction and also perpendicular to Pr. The normal rake angle n which is measured in Pn is the one between the tool rake face Aγ and the reference
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plane Pr. The normal relief angle n which is also measured in Pn is the one between the tool flank face and the reference plane Pr. The edge inclination angle s which is measured in Ps is
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Workpiece Local top view
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Fig. 2 The geometric relationship in bar turning
For the typical bar turning in Fig. 2, the cutting conditions need to be given prior as: cutting parameters such as depth of cut ap, feed ft, cutting velocity V and tool geometry such as the lead angle r , edge inclination angle s , normal rake angle n and nose radius re . Furthermore,
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wk
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…
k K
k
rk
k 1
st
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ap
k
…
ex
k 0
ft /2
r
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Fig. 3 Projection of discretized cutting edges in the reference plane Pr
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Inspired by the contributions of Molinari and Moufki [15], the discretization of cutting edge and coordinate transformation methods (CTM) are enhanced here to determine the involved geometrical parameters in the calculation of cutting forces. The cutting edge of the turning tool is discretized into K+2 cutting elements on the rake face Aγ, each which is indexed as k with k [0, K 1] in Figs. 3 and 4, where K is the numbers of discretized cutting elements in tool nose
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part. Besides, Fig. 4 shows the projection of the discretized cutting edges in the reference plane Pr. For the k-th cutting element, the cutting plane Psk and normal plane Pnk are perpendicular to Pr, see Fig. 4. The angle rk which is measured in the plane Pr is the one between the planes Ps0 and Psk . Some expressions about the discretized cutting elements are obtained by the geometric
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relationships in Fig. 3 as follows: sin 1 ((re -a p ) re )
st 2 r
a p re (1 cos r ) a p re (1 cos r )
(1)
ex 2 sin 1 (0.5 ft / re )
(2)
k st (k 1) (ex st ) K
(3) (4)
a 2sin 1 (0.5 ft / re )
(5)
ACCEPTED MANUSCRIPT Psk
k-th cutting element
Ps0
ck
y kr y
k 0
rk
y0r
z 0r y00
c0
z
z
ck
k r
z 0k
0 c
z ck
Global chip flow direction
z 00 R ak e
x0
A
V
fzt.jpg
z
Feed dir0 ection
Pr
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y0
face
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Main cutting edge
Axis of workpiece
Fig. 4 Discretization of the engaged cutting edge and its projection in the reference plane Pr
As shown in Figs. 3 and 4, the k-th cutting element can be defined in Pr, by five parameters: the angular position k , the angle rk , width of cut wk , depth of cut
a kp
, cutting thickness tk
and the chip load Ak , which are expressed in the following form:
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For the 0-th cutting element,
(6)
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0 st 0 r 0 a p re (1 cos r ) 0 w0 (a p re (1 cos r )) sin r a p re (1 cos r ) 0 0 a p w sin r t f sin t r 0 A0 t0 w0
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For the k-th cutting element, 1 k K , k st (k 1) k r k 2 r 2 wk 2r sin( 2) e k k k a p w sin( r r ) t f sin( k ) t r r k Ak tk wk 2re f t sin( 2) sin( r rk )
For the (K+1)-th cutting element,
(7)
ACCEPTED MANUSCRIPT K 1 ex a K 1 r r w K 1 f t K 1 ap 0 t f t K 1 AK 1 re2 sin 1 (0.5 ft / re ) 0.5 f t re2 f t 2 / 4
(8)
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The geometric relationship between the discretized cutting edge on the rake face Aγ and its projection on the reference plane Pr can be obtained using the CTM as shown in Fig. 4 .Therefore, the responding cutting angles (edge inclination angle sk , normal angle nk , deflection angle ck ) for each cutting element are also determined. The edge inclination angle sk and the normal angle nk for the k-th cutting element are measured in Psk and Pnk , respectively. The angles sk and nk are determined using the CTM as:
(9)
nk sin 1 (cos rk sin n0 sin rk cos n0 sin s0 )
(10)
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sk sin 1 (cos ck sin s0 sin ck sin n0 cos s0 )
where, for k=0 cutting edge, the cutting angles s0 , n0 is known. And the deflection angle ck is measured between the main cutting edge and the k-th cutting edge element in the rake angle Aγ and its projection in Pr corresponds to rk , which is derived using the CTM as: cos s0 tan rk 0 k 0 0 cos n tan r sin n sin s
3
(11)
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ck tan 1
Calculation of cutting forces
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3.1 The chip flow direction The chip flow direction has a critical effect on the friction force direction and chip control. In the chip flow model shown in Fig. 4, similar with the approach of Molinari et al. [12] and Dorlin et al. [25] about chip flow analysis, it is assumed that all the discretized cutting elements have the same chip flow direction and velocity Vc in the rake face Aγ as observed in experimental results [6]. The global flow direction in this chip flow model is represented by the angle c0 . Therefore, the local chip flow angle ck for the k-th cutting element in Fig. 4 can be expressed as:
ck c0 ck
(12)
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3.2 The equilibrium equation of chip elements During the chip formation process, all the chip elements have the same flow direction. Consequently, the k-th chip element is constrained by the adjacent chip elements. In addition, the formation of the k-th chip element is regarded as the shear effect in the primary deformation zone. Therefore, the equilibrium equation of the forces applied on the k-th chip element can be written as: Fnck x c Ffck zc Fmck yc Fsk x ksh Fnsk z ksh =0 Tool effect
Chip effect
Shear effect
(13)
where, xc is normal to the rake face Aγ; zc is the chip flow direction; yc is normal to the flow direction in Aγ; Fnck , Ffck are the normal force and friction force respectively in Aγ; Fmck is the
ACCEPTED MANUSCRIPT resultant force exerted by the neighboring chip elements; Fsk , Fnsk are the shear force and normal force in the primary shear plane, which can be derived from the model proposed in Sect. 3.3. Using the CTM notation, Eq. (14) can be derived from Eq. (13) as: Fnck [cos(nk nk ) tan nk sin(nk nk ) cosck ] Fmck sin(nk nk )sin ck Fsk cosshk 0 k k k k k k k Fnc tan n sin c Fmc cosc Fs sin sh 0 k k k k k k k k k k k k Fnc [sin(n n ) tan n cos(n n ) cosc ] Fmc sin(n n )sin c Fns 0
(14)
where, the angle nk is the mean friction angle between the k-th chip element and Aγ. The
Fsk .
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k k k unkown Ffc , Fs , Fns can be solved by the Newton-Raphson iteration algorithm with the known
Furthermore, according to the condition that the internal forces among the chip elements vanish, the equilibrium equation of the forces applied on the all chip elements be written as: K 1
F
k nc
k 0
x c Ffck zc Fsk x ksh Fnsk z ksh 0
(15)
Combining Eqs. (13) and (15), the following implicit relation is obtained as: K 1
F
0
(16)
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k mc
k 0
The global chip flow angle c0 can be calculated by solving Eq.(16).
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3.3 Oblique force model of chip elements For the k-th cutting element, the cutting action can be regarded as the oblique cutting process with the imposed condition that chip flow velocity is equal to global chip flow velocity. The modified version of oblique cutting model is used to calculate the shear force Fsk and normal force Fnsk exerted by the shear effect of the primary deformation zone, and the relevant equations derived are established and summarized [17].
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Pnk
nk
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z
Pr
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y
y 0k
z
k n
k 0
nk
Vc
Chip
Vc
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Frk
xek
sk
Pshk
z ek
nk
V x
Fak
Tool
H
P
P
k 0
Ft k
nk
A
k xkn Pe
k n
k sh
k c
V
Workpiece
Psk
(a) Oblique cutting model and chip formation
(b) Non-equidistant shear model
Fig. 5 Oblique cutting model and chip formation of infinitesimal cutting edge
According to the coordinate transform and oblique geometric relationship in Fig. 5(a), The equivalent plane angle ek , which determines the equivalent plane Pek (the mechanism of oblique cutting is regarded as a two dimensional cutting process), is [21] : tan ck cosshk + sin(nk nk )sin shk cos(nk nk )
ek tan 1
(17)
The shear flow angle shk which characterizes the shear direction in primary shear zone, are given by [17]:
ACCEPTED MANUSCRIPT tan sk cos(nk nk ) tan ck sin nk cos nk
shk tan 1
(18)
In Fig. 5(b), the primary shear zone is modeled as a parallel and non-equidistant shear band which consists of two non-equidistant thickness (1 )
with thickness
and
,
characterized by the portion [0,1] The governing equations for the shear strain rate , the shear strain , the temperature T and the maximum shear strain rate m in the primary shear zone, are detailed in [17], [22] and are given as: m k q [(1 ) ]q [ ze (1 ) ] m ( z k )q e q ( )
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zek [ (1 ) , 0]
(19)
z [0, k e
]
cosek T zek m C pV cos sk sin nk
(q 1)V cos sk cos nk (q 1)Vs cosshk cos(nk nk )
cosshk cos(nk nk )(cos nk cosshk tan sk sin shk ) cos nk
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m
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cosek m k q 1 zek [ (1 ) , 0] V cos k sin k (q 1)[(1 ) ]q [ ze (1 ) ] s n k k q 1 cosek cos nk cose m ( ze ) z k [0, ] k k q k V cos s sin n (q 1)( ) sin n cos shk cos(nk nk ) e
(20)
(21)
(22)
(23)
The normal friction angle nk can be derived from the work of Budak et al. [23] using an
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analytical dual-zone model of the tool-chip interface as follows: 1/ s 1 1 s s P0 P0
(24)
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nk tan 1
where P0 4
1 cos 2 nk cosshk is the normal pressure; is the distribution exponent 2 sin[2(nk nk nk )] s
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for normal pressure taken as 3 based on the experimental data [24]; s is the sliding friction coefficient on the tool-chip interface and a power function of the chip velocity. The normal shear angle nk is determined by a modified Merchant equation [11] as:
nk C1 C2 (nk nk )
(25)
where C1 , C2 are constant depending on the workpiece-tool material. In order to determine the shear flow stress sh in primary shear zone, Johnson-Cook constitutive model, which describes the flow stress of the workpiece material by considering strain, strain rate, and temperature effects [15], is adopted in the present work because it is well-accepted, numerically robust and utilized widely in cutting modelling and simulations studies [16] and given as follows:
ACCEPTED MANUSCRIPT sh =
m n T Tr 1 1 C ln 1 A B 3 3 0 Tm Tr
(26)
The characteristics of material behavior are determined by yield strength A, strength coefficient B, strain hardening exponent n, strain rate sensitivity coefficient C and thermal softening coefficient m. Furthermore, the shear force Fsk , which is proportional to the shear stress sh in primary shear plane, along with the normal force Fnsk can be expressed as:
Fnsk
Ak cos sk sin nk
(27)
[tan(nk nk )+ tan nk cosck ]cosshk k Fs 1 tan nk cosck tan(nk nk )
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Fsk sh
(28)
For the k-th cutting element, the three force components dFt , j , dFr , j , dFa , j (tangential, radial, axial) due to the shear effect, are evaluated from the following matrix form.
sin nk cos sk k Fs cos nk Fk sin nk sin sk ns
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Ft k cosshk cos nk cos sk sin shk sin sk k cosshk sin nk Fr k cos k cos k sin k sin k cos k sh n s sh s Fa
(29)
Finally, the cutting forces contributed by all cutting elements are summed to obtain the total forces on the tool as：
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k Fx 0 0 1 Ft K 1 k k k Fy 0 sin( r r ) cos( r r ) Fa F k 0 0 cos( k ) sin( k ) F k r r r r r z
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(30)
Model Validation and Discussion
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In order to validate the effectiveness of the proposed analytical model of cutting forces for three-dimensional turning, a series of turning experiments are carried out. The corresponding computer programs in MATLAB 8.5 are developed to implement on the analysis of cutting force model. The analyses are divided into two parts: one section is used to analyze the influence of tool geometry: tool nose radius re and lead angle kr and cutting parameters: depth of cut ap, feed ft, cutting velocity V on chip flow direction; another section is to analyze how the cutting parameters and tool geometry affect cutting forces and compare with the experimental data. In addition, the used workpiece is NAB (Nickel Aluminum Bronze) material which presents the properties of excellent wear resistance and stress corrosion resistance and is extensively used for marine propellers. In our preceding work [26], combining SHPB tests, predictive force model and cutting experiment, constitutive parameters of NAB are identify accurately in machining and given in Table 1. Table 1. Properties of workpiece material NAB[26] (1) Johnson-Cook parameters A
B
C
n
m
0
Tm
Tr
295MPa
759.5MPa
0.011
0.405
1.09
0.001s-1
1311K
298K
(2) Thermo-physical properties
ACCEPTED MANUSCRIPT Density m
Heat capacity C p
7530 kg/m3
419 J/(kgK)
Thermal
Taylor-Quinney
conductivity k p
coefficient
41.9 W/(mK)
0.85
4.1 Chip flow direction analysis The chip flow direction has a significant influence on the calculation of cutting forces and surface quality in three-dimensional turning. When the actual cutting edge is divided into a series of cutting elements, the local chip flow direction (given by ck ) of each chip element is imposed
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to follow the global chip direction (given by c0 ) due to the interaction between adjacent chip elements. In addition, the analysis of global chip flow angle could be illustrated distinctly
according to the one of main cutting edge. Thus, the effects of tool geometry and cutting parameters on global and local chip flow directions are analyzed respectively for the subsequent calculation of cutting forces. In addition, the global chip flow angle is calculated by
solving implicit equation in Eq.(16), so chip flow angles are not influenced by the value for K.
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(1) Global chip flow direction
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(a) Effect of edge inclination angle s0 on c0 .
(b) Effect of lead angle r on c0 . Cutting conditions are: n0 00 , s0 00 , re 0.8mm,
a p 0.4mm, ft 0.1mm / r ,V 90m / min
a p 0.4mm, ft 0.1mm / r ,V 90m / min
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Cutting conditions are: n0 00 , r 750 , re 0.8mm,
(c) Effect of nose radius re on c0 . Cutting
(d) Effect of depth of cut a p on c0 . Cutting
conditions are: n0 00 , s0 00 , r 750 ,
conditions are: n0 00 , s0 00 , r 750 ,
a p 0.4mm, ft 0.1mm / r ,V 90m / min
re 0.8mm, ft 0.1mm / r,V 90m / min
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(e) Effect of feed f t on c0 . Cutting conditions are:
(f) Effect of cutting velocity V on c0 . Cutting
n0 00 , s0 00 , r 750 , re 0.8mm,
conditions are: n0 00 , s0 00 , r 750 ,
a p 0.4mm,V 90m / min
re 0.8mm, a p 0.4mm, ft 0.1mm / r
Fig. 6 Effect of tool geometry and cutting parameters on global chip flow direction
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The curves in Fig. 6 show the variations of the predicted global chip flow angle with the tool geometry (edge inclination angle, lead angle, nose radius) and the cutting parameters (depth of cut, feed, cutting velocity) respectively with given cutting conditions. It can be seen that the global chip flow angle increases with an increase in edge inclination angle, lead angle and nose radius, decreases with an increase in depth of cut, and changes slightly with feed and cutting velocity. Fig. 6(a) shows a linear relationship approximately between the predicted global chip flow angle and edge inclination angle which accords with the results of Wang et al. [6]. This is due to the reason that the component V sin s0 of cutting velocity in the paralleled cutting edge direction increases with an increase of edge inclination angle, which will cause the increase the component Vc sinc0
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of chip velocity in the same direction [17], i.e., the approximate linear increase of global chip flow angle. Fig. 6(b) gives a upwards curve containing concave and linear parts between the predicted c0 and lead angle, which may be explained that the cutting thickness increases with the increase
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of lead angle in Eq. (6) and produces the large plastic work, thus causing the increase of global chip flow angle .In addition, the phenomenon of curve saltation is that main cutting edge is engaged because of a p re (1 cos r ) for the values of r less than 60 deg. Fig. 6(c) depicts a
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convex upwards curve between the predicted c0 and nose radius while Fig. 6(d) gives a concave downwards one between the predicted c0 and depth of cut, which are in accordance with the experimental results of Young et al. [4], [13]. What is more, c0 varies very rapidly when the
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main cutting edge part is engaged, i.e. a p re (1 cos r ) . These reasons for Fig. 6(c) and Fig. 6(d) are illustrated that as the nose radius becomes small or the depth of cut becomes large, the proportion of engaged main cutting edge part increases, thus causing the decrease of c0 . Fig. 6(e) and (f) gives a constant global chip flow angle which is approximately equal to 36.5 deg with the feed or cutting velocity, for the cutting velocity or feed has little influence on the proportion of the engaged main cutting edge part as observed from experimental results[4]. (2) Local chip flow direction
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(b) Effect of lead angle r on ck . Cutting
Cutting conditions are: n0 00 , r 750 , re 0.8mm,
conditions are: n0 00 , s0 00 , re 0.8mm,
a p 0.4mm, ft 0.1mm / r ,V 90m / min
a p 0.4mm, ft 0.1mm / r ,V 90m / min
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(a) Effect of edge inclination angle s0 on ck .
(d) Effect of depth of cut a p on ck .
conditions are: n0 00 , s0 00 , r 750
Cutting conditions are: n0 00 , s0 00 , r 750
a p 0.4mm, ft 0.1mm / r ,V 90m / min
re 0.8mm, ft 0.1mm / r,V 90m / min
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(c) Effect of nose radius re on ck . Cutting
(e) Effect of feed f t on ck . Cutting conditions
(f) Effect of cutting velocity V on ck . Cutting
are: 0 , 0 , r 75 , re 0.8mm,
conditions are: n0 00 , s0 00 , r 750 ,
a p 0.4mm,V 90m / min
re 0.8mm, a p 0.4mm, ft 0.1mm / r
0 n
0
0 s
0
0
Fig. 7 Effect of tool geometry and cutting parameters on local chip flow direction
Fig. 7 gives the effect of tool geometry (edge inclination angle, lead angle, nose radius) and cutting parameters (depth of cut, feed, cutting velocity) on the predicted local chip flow angle respectively with the given cutting conditions. K=20 is chosen here for the chip flow angle is not affected by the value for K. It can be seen in Fig. 7 (a) that the local chip flow angle increases with an increase of edge inclination angle, which are explained that the local edge inclination angle of
ACCEPTED MANUSCRIPT discrete cutting element increases with an increase of edge inclination angle in Eq.(9), and then causes the stronger interaction of chip elements. In addition, the local chip flow angle is little affected by the edge inclination angle in the range of -20 deg to 0 deg, while large in the range of 0 deg to 20 deg. The discontinuity of local chip flow angle between the cutting elements k=0 and k=1 is that the main cutting edge is not engaged for a p re (1 cos r ) . Fig. 7(b) shows the curves of local chip flow angle with lead angle that is an increase of ck with an increase of the lead angle. The interactions of chip elements become stronger with the increase of the lead angle. Fig. 7(c)
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shows the curves of local chip flow angle with nose radius that is a decrease of ck with an increase of the nose radius, which is explained that with the decrease of the nose radius, the variation of chip element flow direction zck is larger and then causes the stronger interaction of
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chip elements. For the cutting element k=9, the chip element is free flow, but the flow becomes more constrained for cutting elements k<9 or k>9. Fig. 7(d) shows that the local chip flow angle decreases with an increase of the depth of cut, which is that the engaged main cutting edge increases with the increase of the depth of cut and causes the lower interaction of chip elements. Fig. 7(e) and (f) shows that the effects of the feed and cutting velocity on the local chip flow angle are little as the same in Fig. 6(e) and (f).
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4.2 Cutting force validation A series of three-dimensional turning tests have been performed on a CNC turning center without cutting fluid and the experimental setup is shown as in Fig. 8. The cutting inserts mounted on tool holder are used to machine the NAB bars of 120 mm diameter. The tool holder is mounted on a three-component Kistler table dynamometer (model 9523B) which is used to measure cutting forces. In addition, every test is repeated three times to ensure the reliability of cutting forces data, and Figs. 9 and 13 shows the predicted and experimental results of cutting forces in lines and discrete symbols, respectively.
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NAB workpiece
Cutting insert Tool holder
Chuck
Dynamometer
Fig. 8 The experiment setup
(1) The effect of cutting parameters ( a p , ft ,V ) In order to investigate the effect of cutting parameters on cutting forces, the cutting inserts referenced Sandvik Coromant SNMG 12 04 08-MR 2025 and the geometrical parameters of this
ACCEPTED MANUSCRIPT coated cutting insert with chip breaker are: rake angle 6 deg, edge inclination angle -6 deg, nose radius 0.8 mm. The tool holder referenced DSBNR2525M12 is chosen herein. Therefore, for this combination in turning process, the lead angle is 75 deg and the normal rake angle is 0 deg. (c)
(b)
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Fig. 9 Comparison of cutting forces versus depth of cut from predicted model and experimental data: (a) ft=0.1mm/r, V=90m/min; (b) ft=0.2mm/r, V=90m/min; (c) ft=0.2mm/r, V=150m/min.
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The four groups of cutting parameters: feeds (0.1mm/r, 0.2mm/r), and cutting velocities (90mm/min, 150mm/min) are chosen to investigate the effect of depths of cut (0.4mm, 0.8mm, 1.2mm, 1.8mm) on cutting forces. Variations of cutting force with depth of cut are shown in Fig. 9 and then comparison is carried out between the predicted model and experimental data, in which there is good agreement between the predicted and experimental data. The main cutting edge is engaged when the depth of cut ap is more than re (1 cos r ) 0.593mm which causes the cutting
(a)
(c)
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(b)
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forces to increase proportionally with the increase of the depth of cut. While the depth of cut ap is small and less than 0.593mm, the engaged tool nose part becomes dominant, which causes the cutting forces to vary nonlinearly.
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Fig. 10 Comparison of cutting forces versus feed from predicted model and experimental data: (a) ap=0.4mm, V=90m/min; (b) ap=1.2 mm, V=90m/min; (c) ap=1.2mm, V=150m/min.
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The four groups of cutting parameters: depths of cut (0.4mm, 1.2mm), and cutting velocities (90mm/min, 150mm/min) are chosen to investigate the effect of feeds (0.05mm/r, 0.1mm/r, 0.15mm/r, 0.2mm/r) on cutting forces. Fig. 10 give the variations of cutting force with feed and the comparison of cutting forces between the predicted mode and experimental data, in which the predicted and experimental data have in good agreement and the cutting forces increase with the increase of the feed. Different from the effect of depth of cut, the cutting forces varies with feed linearly, which is due to the reason that the undeformed chip thickness t ft sin( r ) increases linearly with the increase of feed, i.e., which causes the cutting forces to increase.
ACCEPTED MANUSCRIPT (a)
(b)
(c)
Fig. 11 Comparison of cutting forces versus cutting velocity from predicted model and experimental data: (a)
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ap=0.4mm, ft=0.1mm/r; (b) ap=1.2mm, ft=0.1mm/r; (c) ap=1.2mm, ft=0.2mm/r.
(2) The effect of tool geometry ( kr , re ) (b)
(c)
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The four groups of cutting parameters: depths of cut (0.4mm, 1.2mm), and feeds (0.1mm/r, 0.2mm/r) are chosen to investigate the effect of cutting velocities (60mm/min, 75mm/min, 90mm/min, 150mm/min) on cutting forces. In Fig. 11, as the cutting velocity is increased, the three force components, both predicted and experimental, which are in good agreement. The cutting forces decrease nonlinearly with the increase of the cutting velocity which conforms to the tendency of the experimental results by Young et al. [4]. This can be explained that the friction coefficient decreases with the increase of cutting velocity which results in the higher temperature in tool-chip interface [15], thus causes the decrease of cutting forces. What is more, the change rates of cutting forces are very slow when the cutting velocity is small.
Fig. 12 Comparison of cutting forces versus nose radius from predicted model and experimental data: (a)
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ap=0.4mm, ft=0.1mm/r, V=90mm/min; (b) ap=1.2mm, ft=0.1mm/r, V=90mm/min; (c) ap=1.2mm, ft=0.2mm/r, V=150mm/min.
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The three nose radii (0.4mm, 0.8mm, 1.2mm) by three different cutting inserts are chosen to investigate the effect of the nose radius on cutting forces with the same lead angle 75deg, edge inclination angle -6deg, normal rake angle 0deg and three groups of cutting parameters in Fig. 12. Variations of cutting force with nose radius are shown in Fig. 12 where the predicted and experimental data have in good agreement. It can be seen in Fig. 12(a) that the main cutting edge does not engages in cutting, i.e., a p re (1 cos r ) ), and the area of total undeformed chip is independent of the nose radius, the force component Fx is determined by the variation of chip velocity caused by the nose radius, however, the chip velocity keeps constant with nose radius, which further keeps Fx nearly constant. Conversely, Fy and Fz are affected significantly by the global chip flow angle, Fz is decreased with increase of the global chip flow angle which varies synchronously with the nose radius in Fig. 6 (c), while Fy is increased with increase of the nose
ACCEPTED MANUSCRIPT radius. Fig. 12(b) and (c) show that the main cutting edge is engaged, i.e. a p re (1 cos r ) ,). With the increase of the nose radius, the length of nose radius and width of cut is also increased, which causes the global chip flow angle to increase and the cutting forces further increase. (b)
(c)
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(a)
Fig. 13 Comparison of cutting forces versus lead angle from predicted model and experimental data: (a) ap=0.4mm, ft=0.1mm/r, V=90mm/min; (b) ap=1.2mm, ft=0.1mm/r, V=90mm/min; (c) ap=1.2mm, ft=0.2mm/r, V=150mm/min.
5
Conclusions
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The three lead angles (45 deg, 75 deg, 90 deg) by three different tool holders are chosen to investigate the effect of the lead angle on cutting forces with the same nose radius 0.4mm, edge inclination angle -6deg, normal rake angle 0deg and three groups of cutting parameters. Fig. 13 shows the predicted and experimental cutting forces plotted against the lead angle. The data show the linear increase in force components Fx and Fz but decreased Fy with the increase of lead angle, this trend conforms to the experimental results by Young et al. [4]. The effect of lead angle on the force component Fx is determined by the undeformed chip thickness normally. With the increase of lead angle, the engaged length of nose radius thus is increased, the interaction between chip elements is strengthened, which causes Fx to increase. While the effect of lead angle on the force components Fy and Fz is determined by the projection of the resultant force in y- and z- direction respectively, therefore, the cutting component Fz is increased and Fy is decreased with the increase of lead angle.
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In this paper, an analytical force model for three-dimensional turning has been developed and discussed based on a modified predictive machining theory and the analysis of turning tool geometry is the extension to the work of Molinari et al. [15]. The predicted cutting forces are verified the effectiveness by the experimental data. The contributions of the proposed analytical model are drawn as follows: (1) The analytical force model can predict the cutting forces and chip flow angles without the need for cutting tests in the three-dimensional turning. The cutting forces of each cutting element on the cutting edge are calculated using the modified version of the predictive oblique cutting model. (2) The proposed model has also been used to investigate the effects of cutting conditions and on the tool geometry on chip flow directions. The global and local chip flow angles are affected significantly by the tool geometry (edge inclination angle, lead angle, nose radius) and depth of cut, while little effect caused by feed and cutting velocity. (3) The proposed force model has been validated using a series of three-dimensional turning experiments. The data obtained from the proposed model and the experiments have been shown to be in good agreement. However, the influences of tool-chip interface and flank wear on predicted forces in actual three-dimensional turning process, need to be studied for more accurate prediction of cutting
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Acknowledge This work was partially supported by the Scientific Research Foundation of Wuhan Institute of Technology (Grant No. K201707), the Nature Science Foundation of Hubei (Grant No. 2017CFB346) and the National Natural Science Foundation (Grant No. 51705385).
Reference
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