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Procedia CIRP 14 (2014) 113 – 118

6th CIRP International Conference on High Performance Cutting, HPC2014

Cutting forces and vibrations during ball end milling of inclined surfaces
Szymon Wojciechowskia*, Paweł Twardowskia, Marcin Pelic a
a
Poznan University of Technology,Piotrowo 3, Poznan 60-965, Poland
* Corresponding author. Tel.: +48-061 6652608; fax: +48-061 6652200.E-mail address: sjwojciechowski@o2.pl..

Abstract

This work concentrates on the analysis of cutting forces and vibrations generated during ball end milling process with variable surface
inclination angle (α). The cutting force and tool’s displacements (vibrations) model including surface inclination was formulated. Experiments
were carried out on hardened alloy steel X155CrVMo12-1 with sintered carbide (TiAlN coating) monolithic tool. Instantaneous values of
cutting forces were measured in the range of variable feed per tooth (fz) and surface inclination angle (α) values. The shear (Kic) and edge (Kie)
coefficients in function of analyzed factors were determined using the measured cutting force signals as an input data. The research revealed
that cutting forces and vibrations are strongly affected by the surface inclination, both in quantitative and qualitative aspect. This observation is
also confirmed by the developed model.
© 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
© 2014 The Authors. Published by Elsevier B.V.
(http://creativecommons.org/licenses/by-nc-nd/3.0/).
Selection and peer-review under responsibility of the International Scientific Committee of the 6th CIRP International Conference on High
Selection and peer-review under responsibility of the International Scientific Committee of the 6th CIRP International Conference
Performance Cutting.
on High Performance Cutting
Keywords: ball end milling, dynamics, surface inclination

mechanistic. Analytical methods are usually including

1. Introduction
thermomechanical phenomena occurring in cutting process
and modeling physical mechanisms during work piece
Ball end milling process is a very popular technology in the
decohesion process, e.g. slip stress and strain via intense
production of drop forging dies and casting molds made from
plastic deformation [8]. Mechanistic models assume that
hardened steel [1,2]. This kind of process includes also the
cutting forces are proportional to the cross-sectional area of
machining of curvilinear surfaces in high speed machining
cut and so called specific cutting force coefficients [9, 10].
(HSM) conditions. During ball end milling of curvilinear
These mechanistic models can be classified into two groups.
surfaces the inclination angle of the machined surface is
In the first one, the effects of shearing mechanism due to the
variable, which affects the active length of cutting edge and
chip generating process on the tool’s rake face and effects of
working angle values. Consequently, this phenomenon
ploughing mechanisms on the flank face are expressed as the
influences cutting forces [3,4] and vibrations generated during
one specific force coefficient for each cutting force component
machining process. From the literature survey it is also
[11]. In the second model, the shearing and ploughing effects
resulting, that surface inclination angle affects surface
are characterized separately by the respective specific shear
roughness [5,6] and the tool wear [7] during ball end milling.
and edge force coefficients [12, 13].
Therefore, the reliable prediction of milling forces including
In this study, the influence of the machined surface
surface inclination is significant for the simulation of the
inclination angle on the cutting forces and vibrations is
machinability, cutter wear, vibrations, as well as the surface
investigated, both in the quantitative and qualitative aspect.
quality.
The cutting force and vibration model including kinematic –
There has been much research in the past on many force
geometric parameters is also formulated. The developed
components models during ball end milling process. The work
model is validated empirically during ball end milling of
can be classified into two ground groups: analytical and
hardened steel with variable surface inclination.

2212-8271 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/).
Selection and peer-review under responsibility of the International Scientific Committee of the 6th CIRP International Conference
on High Performance Cutting
doi:10.1016/j.procir.2014.03.102
114 Szymon Wojciechowski et al. / Procedia CIRP 14 (2014) 113 – 118

2. Cutting forces and vibrations model dFtj Ktedl j  Ktc dAzj (2a)
dFrj K redl j  K rcdAzj (2b)
In order to determine cutter’s instantaneous displacements
related to cutter’s deflections, induced by cutting forces Fi, dFaj K aedl j  K acdAzj (2c)
one should solve the following differential motion equations:
where: Kte, Kre, Kae are the edge specific coefficients [N/mm],
mx ˜ x(t )  cx ˜ x (t )  k x ˜ x(t ) Fx (t ) (1a) Ktc, Krc, Kac are the shear specific coefficients [N/mm2],
dlj is the infinitesimal length of cutting edge [mm],
my ˜ y(t )  c y ˜ y (t )  k y ˜ y(t ) Fz (t ) ˜ sin D  Fy (t ) ˜ cos D (1b) Azj is the cross sectional area of cut [mm2].
In order to calculate cutting forces acting on i-th
mz ˜ z(t )  cz ˜ z (t )  k z ˜ z (t ) Fz (t ) ˜ cos D  Fy (t ) ˜ sin D (1c) infinitesimal segment of the cutting edge, it is necessary to
determine cross sectional area of cut and active length of
During ball end milling process of inclined surfaces, cutting edge, as well as calibrate specific coefficients.
cutter’s displacements (vibrations) are determined in the On the basis of Figure 1 the instantaneous cutting forces in
directions: perpendicular to the tool’s rotational axis and machine tool’s coordinates can be expressed:
collinear to the feed motion vector (y(t)), perpendicular to the
zc
tool’s rotational axis and feed motion vector (x(t)), parallel to (3a)
tool’s rotational axis (z(t)).
Fx ¦F
j 1
tj ˜ sin M j  Frj ˜ sin M rj ˜ cos M j  Faj ˜ cos M rj ˜ cos M j ,

In the equation (1) mi, ci, ki denotes modal parameters (mi – zc

modal mass, ci- damping coefficient, ki – stiffness Fy ¦ F rj ˜ sin M rj sin M j  Faj ˜ cos M rj ˜ sin M j  Ftj ˜ cos M j , (3b)
j 1
coefficient), which can be determined using impact test, while zc
(3c)
Fx, Fy, Fz instantaneous cutting forces in the machine tool’s Fz ¦F rj ˜ cos M rj  Faj ˜ sin M rj .
coordinates. In order to determine these cutting forces, j 1

mechanistic cutting force model, developed by Lee and

Altintas [12] is applied. In this model, a set of curvilinear where: zc is the active number of teeth.
coordinate system normal to the ball envelope is used to Positioning angles φrj and φj of the j-th cutting edge found
in equations (3) are expressed by:
specify the resultant force acting on the i-th infinitesimal
segment of the cutting edge. Figure 1 depicts cutting forces
M r1M r 2 (4a)
acting on the cutter and tool’s coordinates for a ball end mill. M rj
2
π ˜ n ˜ t \ l1 \ l 2 § 2π · (4b)
Mj   ( j  1) ˜ ¨ ¸  2π( N  1)
30 2 © z ¹

where: φr1, φr2 are the initial and final positioning angles in
the reference plane [rad],
ψl1, ψl2 are the initial and final lag angles [rad],
j is the ordinal number of tooth,
N is the number of tool’s rotation,
n is the spindle rotational speed [rev/min],
t is the time [s].
Instantaneous cross sectional area of cut can be calculated
on the basis of equation:

Azj R ˜ f z ˜ 1 - cosMr 2 - D ˜ sin M j (5)

where: R is the tool’s radius [mm],

fz is the feed per tooth [mm/tooth],
α is the surface inclination angle.
Infinitesimal length of cutting edge can be formulated from
the expression proposed by [12] as:

2
Fig. 1. Geometry and tool coordinates for a ball milling cutter. § dr (\ l ) · R2 (6)
dl ¨¨ ¸¸  r (\ l ) 2  d\ l
© d\ l ¹ tan 2 Os
The elemental tangential dFtj, radial dFrj, and axial dFaj
cutting forces acting on the j-th tooth, are expressed by:
The r(ψl) expression in equation (6) can be formulated
from:
 Szymon Wojciechowski et al. / Procedia CIRP 14 (2014) 113 – 118 115

§ \l ·
2
(7) a p ˜ tan Os
r (\ l ) R 1  ¨¨  1¸¸ \ l1 0 ; \ l 2 (9b)
© tan Os ¹ R
§ R  ap ·
In this study, two different cases of ball end milling M r1 0 ; M r 2 arccos ¨¨ ¸¸ (9c)
process were investigated, namely upward ramping with α >0 © R ¹
(Figure 2a) and slot milling with α =0 (Figure 2b).
Phase 3:
a) b)
a p ˜ tan Os
Ω3 d Ω  Ω4 π \ z ( j  1)   2π( N  1) (10a)
R
a p ˜ tan Os
\ l 2 0;\ l 2  Ω  π \ z ( j  1)  2π( N  1) (10b)
R
§ R  a p (Ω ) · § R  ap · (10c)
M r1 arccos ¨¨ ¸¸ ; M r 2 arccos ¨¨ ¸¸
© R ¹ © R ¹
2( j  1) π
R ˜ (Ω  π   2π( N  1))
a p (Ω ) z (10d)
tan(Os )
Fig. 2. Selected cutting modes of ball end milling: a) upward ramping, b) slot
milling. In equations (8d) and (10d) ap(Ω) denotes instantaneous
depth of cut which depends on tool rotation angle.
In this step the border conditions ψl1, ψl2, φr1, φr2 for the Figure 4 depicts phases of tool immersion into the work
investigated cutting modes will be discussed. During slot piece for the upward ramping.
milling, three phases of tool immersion into the work piece in
function of tool rotation angle Ω are distinguished (Figure 3).

Fig. 3. Border conditions for the slot milling process.

Phase 1:

π ˜ n˜t
Ω t Ω1 \ z ( j  1)  2π( N  1)
30 (8a)
a p ˜ tan Os
Ω  Ω2 \ z ( j  1)  2π( N  1)
R Fig. 4. Border conditions for the upward ramping process.
\ l1 0 ; \ l 2 Ω \ z ( j  1)  2π( N  1) (8b)
§ R  a p (Ω ) ·
Phase 1:
M r1 0 ; M r 2 arccos ¨¨ ¸¸ (8c)
© R ¹ 2( j  1) π
Ω t Ω1  2π( N  1) (11a)
R § 2 j  1 ˜ π · (8d)
z
a p (Ω ) ˜¨Ω   2π( N - 1) ¸ π 2( j  1) π § ap ·
tan Os © z ¹ Ω  Ω2   arccos ¨¨1  ¸¸  (1  cos D ) ˜ tan Os  2π( N  1)
© R ˜ sin D ¹
2
2 z

Phase 2: \ l1 0 ; \ l 2 0 ; M r1 0 ; M r 2 0 (11b)

Ω2 d Ω  Ω3 π \ z ( j  1)  2π( N  1) (9a)
116 Szymon Wojciechowski et al. / Procedia CIRP 14 (2014) 113 – 118

Phase 2: The piezoelectric force dynamometer was used to measure

total cutting forces components. Its natural frequency is equal
π 2( j  1) π § ap · to 1672 Hz. In order to avoid disturbances induced by
Ω2 d Ω  Ω3   arccos ¨¨1  ¸¸  (12a)
2 z © R ˜ sin D ¹
2
proximity of forcing frequency to gauge natural frequency,
 (1  cos D ) ˜ tan Os  2π( N  1) the band – elimination filter was applied. Cutting force
components were measured (in machine tool’s coordinates –
\ l1 1  cos D ˜ tan Os ; \ l 2 1  cos Mr 2 ˜ tan Os (12b)
Figure 5), in following directions: direction X – feed normal
force Fx [N], direction Y – feed force Fy [N], direction Z –
§ R  a p (:) · (12c)
M r1 D ; M r 2 D  arccos ¨¨ ¸¸ thrust force Fz [N].
© R ¹
ª 2π ˜ j  2π º
a p (Ω ) sin 2 D ˜ R sin «cosD ˜ tan Os  tan Os   2π ˜ N  Ω »  (12d)
¬ z ¼
 R sin 2 D  a p

Phase 3:

2( j  1) π
Ω3 d Ω  Ω4 2π   2π( N  1)  (1  cos D ) tan Os (13a)
z
\ l1 0 ; \ l 2 0 ; M r1 0 ; M r 2 0 (13b)

From the above deliberations and Figure 4 it is resulting

Fig. 5. Cutting force components in machine tool coordinates.
that in case of upward ramping, tool cuts only when the phase
2 of tool immersion into the work piece occurs. It means that
3.3. The calibration of specific cutting force coefficients
in milling with surface inclination angle, the active number of
teeth can be less than one, and thus pulsating forces can occur. In the investigations carried out it was assumed, that
maximum instantaneous forces in X and Z direction (Fx, Fz)
3. Experimental details and minimum instantaneous forces in Y direction (Fy) per tool
revolution are corresponding to maximum instantaneous
3.1. Work and tool materials values of cross sectional area of cut and active length of cut.
In order to calibrate specific cutting force coefficients,
Investigations have been carried out on hardened alloy steel maximum and minimum instantaneous forces per tool
X155CrVMo12-1 plate with hardness approx. 56 HRC and revolution in each direction (Fx, Fy, Fz) were acquired. These
dimensions: 125 x 230 x 160 mm. Monolithic ball end mill measured forces were substituted into equations for specific
made of sintered tungsten carbide (WC) with diameter cutting force coefficients:
d=16 mm and number of teeth z=2 was selected as milling
cutter. Milling tool made of fine-grained tungsten carbide had 2Ftcal  Ftp  2Facal  Fap  2Frcal  Frp  (14a)
K tc , K ac , K rc
anti-wear TiAlN coating and the following geometry: Az max Az max Az max
γo = -15°, λs = 30°, rn = 5 μm, αo = 6°.
In order to solve differential motion equation (1), modal 2 Ftp 2 Fap 2 Frp
K te , K ae , K re (14b)
parameters (m, c, k) were determined using impact test, and
lmax lmax lmax
thus the following parameters were received:
m = 0.079 Ns 2/m, c = 40.8 Ns/m, k = 19492469 N/m.
where: Ftcal, Frcal, Facal are the cutting forces in tool’s
3.2. Research range and method coordinate system applied in calibration [N],
Ftp, Frp, Fap are the ploughing forces in tool’s
The measured quantities in the carried out research were coordinate system applied in calibration [N],
cutting forces (Fx, Fy, Fz), fixed in machine tool’s coordinate
Azmax is the maximum value of cross sectional area of
system. Cutting parameters applied in the research are
cut per 1 tooth [mm2],
presented in the Table 1. Experiments were conducted on
lmax is the maximum length of cut per 1 tooth [mm].
5-axes CNC milling workstation (DMU 60monoBLOCK), in
Cutting forces in tool’s coordinate system, applied in
upward ramping and slot milling conditions. In all
calibration can be calculated on the basis of equations:
investigated cases tool’s effective diameter was lower than
the value of pick feed – Def<b r.
Ftcal
1
Fx sin Mcal  Fy cos Mcal  (15a)
Table 1. Cutting parameters applied in the research 2
α [°] fz [mm] ap [mm] vc [m/min] n [rev/min] Facal
1
 Fx cos Mcal cos Mrcal  Fy sin Mcal cos Mrcal  Fz sin Mrcal  (15b)
2
0–60
interval 0.15
0.02-0.1
interval 0.02
0.2 100 2297 -
8953 Frcal
1
 Fx cos Mcal sin Mrcal  Fy sin Mcal sin Mrcal  Fz cos Mrcal  (15c)
2
 Szymon Wojciechowski et al. / Procedia CIRP 14 (2014) 113 – 118 117

Positioning angles: φcal, φrcal applied for calibration, found values of measured forces per consecutive teeth are not
in equations (15) can be calculated from the equations: uniform. Alterations of these instantaneous maximal forces
produce the envelope, which has a period equal to tool
§ M max  Mmin · § Mr max  Mr min · (16) revolution time. This is probably caused by the cutter’s radial
Mcal Mmin  ¨ ¸, Mrcal Mr max  ¨ ¸
© 4 ¹ © 2 ¹ run-out phenomenon, related directly to the tool revolution
period. However, run-out component was not included in the
where: φmin, φmax are namely, minimal and maximal model. Figure 9 reveals that α angle has significant influence
positioning angles per 1 tooth [rad], also on cutter’s displacement form and amplitude.
φrmin, φrmax are namely, minimal and maximal
a)
positioning angles in the reference plane per tooth
[rad],
Estimated values of specific cutting force coefficients are
expressed in function of cutting parameters (Table 2).

Table 2. Equations of specific cutting force coefficients

Form of equation
Edge kte = 0.00005α -0.0073 α3 + 0.33 α2 -5.6 α+49;
4

coefficients: kre = 26e-0.026α;

kae = -0.00003 α4 +0.0039 α3 - 0.17 α2+2.58 α-10.6.
Shear kac = -2128.1 + 74.7α + 49541fz -0.872 α2 -408 α fz -2.99E5 fz2; b)
coefficients: krc =1782.1 - 53.9α + 11713.5fz +0.7 α2 -16.1 α fz -85602.6 fz2;
ktc = 8780.6 – 368.4α + 37397.9fz +4.6 α2 +39.2 α fz -2.8E5 fz2.

3.4. Results and discussion

Figures: 6–8 depict the comparison of measured and

estimated cutting forces (Fx, Fy, Fz) in function of time for the
different surface inclination angle α values. From the figures
6–8 it is resulting that surface inclination angle α significantly
affects cutting forces, both in quantitative and qualitative
aspect. The growth of α angle decreases the limitary area of Fig. 6. (a) measured time courses of cutting forces for α = 0; (b) calculated
time courses of cutting forces for α = 0.
tool immersion into the work piece (defined by the working
angle ψ and active number of teeth zc) - see Figure 3 and
a)
Figure 4. Consequently, pulsating forces can be generated
during finishing ball end milling, because for the surface
inclination angle α>0 number of active teeth is very often less
than unity (zc<1). It can be also seen, that surface inclination
angle growth induces the decline in cutting force amplitudes
(both estimated and measured ones). Furthermore, the Fx force
is the most sensitive for the surface inclination angle
variations. The highest cutting force amplitudes in three
investigated directions can be observed for the slot milling
(α=0) process. This dependency confirms the fact that in case
of plain milling, cutting speed near cutter’s free end is close to
b)
zero, and thus the appearance of ploughing mechanism
induces large elastic and plastic deformations of work
material. From the Figures 6–8 it can be also seen that cutting
forces estimated on the basis of the developed model stay in a
good agreement with measured ones. Nevertheless some
discrepancies between these courses are found. The highest
differences between the calculated and the measured forces
are observed for the slot milling (α=0). The maximal error in
this case is equal to about 35% (for the X direction). In the
upward ramping process (α>0), maximal errors of cutting
force estimation are lower than 8%. These discrepancies are
resulting probably from the accuracy of specific cutting force
Fig. 7. (a) measured time courses of cutting forces for α = 30°; (b) calculated
coefficients calibration. Furthermore, maximal instantaneous time courses of cutting forces for α = 30°.
118 Szymon Wojciechowski et al. / Procedia CIRP 14 (2014) 113 – 118

The highest displacement amplitudes appear for the slot This can be attributed to the distribution of cutting force
milling, which is induced by the highest cutting force values components along the cutting edge, which depends on surface
in this cutting mode, in comparison to the remaining modes inclination. It is worth indicating that, cutter’s displacements
with α > 0. affect surface texture. Therefore, the appropriate selection of
a) surface inclination angle can improve the surface quality.

4. Conclusions

In this work, the influence of the machined surface

inclination angle on the cutting forces and vibrations was
investigated. The cutting force and vibration model including
kinematic – geometric parameters was formulated.
The research revealed that cutting forces and vibrations are
strongly affected by the surface inclination angle, both in
quantitative and qualitative aspect. The lowest cutting force
b) and displacement amplitudes were generated during upward
ramping with the highest investigated surface inclination
angle. The cutting forces estimated on the basis of the
developed model stay in a good agreement with measured
ones. However some discrepancies between these courses
results from the appearance of cutter’s radial run-out
phenomenon. Therefore, in order to improve the accuracy of
the developed force model, one should include the radial run-
out phenomenon. Deliberations presented in this study can be
also the starting point to the formulation of surface texture
model for a ball end milling of inclined surfaces.

Fig. 8. (a) measured time courses of cutting forces for α = 60°; (b) calculated References
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