Deformation Error Compensation in 5-Axis Milling Operations of Turbine Blades

Abstract

The precision and performance of machined parts can be affected by deformation errors in flexible part end milling. So, deformation errors prediction and compensation during milling operations of flexible parts can provide a key tool in accuracy enhancement of part production. In this study, an improved virtual machining system is used to assess and correct deformation errors caused by cutting temperature and forces in flexible part 5-axis milling operations. The improved Johnson-Cook model is utilized to investigate the cumulative impact of strain rate and deformation temperatures on flow stress during turbine blade machining process. To estimate deformation errors caused by cutting forces and temperature on the workpiece and cutting tool, the finite element analysis is applied. Then, volumetric vectors of deformation error at each cutting location along the machining pathways are generated in order to be compensated within new machining tool paths. As a result, the deformation error created by cutting forces and temperature on the workpiece and cutting tool are compensated in order to enhance accuracy during 5-axis milling operation of flexible turbine blades. Experiments are carried out using a 5-axis CNC machine tool and errors are quantified using a CMM to verify the study's developed strategy. As a consequence, precision of machining operations on flexible turbine blades can be enhanced by employing the methods presented in the study.

Full text

Vol.:(0123456789)
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Journal of the Brazilian Society of Mechanical Sciences and Engineering (2023) 45:289
https://doi.org/10.1007/s40430-023-04230-w
TECHNICAL PAPER
Deformation error compensation in5‑Axis milling operations
ofturbine blades
MohsenSoori1
Received: 21 June 2022 / Accepted: 24 April 2023
© The Author(s), under exclusive licence to The Brazilian Society of Mechanical Sciences and Engineering 2023
Abstract
The precision and performance of machined flexible parts are under influence of deformation errors during end milling
operations. Thus, prediction and compensation of deformation errors during milling operations of flexible parts can provide
a key tool in accuracy enhancement of part production. In this study, an improved virtual machining system is proposed in
order to assess and compensate deformation errors caused by cutting temperature and forces in 5-axis milling operations
of flexible parts. The improved Johnson–Cook model is utilized to investigate the cumulative impact of strain rate and
deformation temperatures on flow stress during milling operations of turbine blade. To estimate deformation errors caused
by cutting forces and temperature on the workpiece and cutting tool, the finite element analysis is then applied. As a result,
volumetric vectors of deformation error at each cutting location along the machining pathways are then generated in order
to be compensated within new compensated machining tool paths. Thus, the deformation error created by cutting forces and
temperature on the workpiece and cutting tool are compensated in order to enhance accuracy during 5-axis milling operation
of flexible turbine blades. Experiments are carried out using a 5-axis CNC machine tool and errors are quantified using a
CMM to verify the developed strategy in the study. As a consequence, precision of machining operations on flexible turbine
blades can be enhanced by employing the developed virtual machining system in the study.
Keywords Deformation error· 5-Axis CNC machine tools· Modified Johnson–Cook model· Cutting forces· Cutting
temperature· Virtual machining
1 Introduction
In order to achieve tight tolerances of complex parts, such
as blisks, turbine blades, and impellers which are com-
monly utilized in aviation industries, accuracy of machin-
ing operations should be evaluated and enhanced. Due to
low rigidity of turbine blades during 5-axis machining
operations, errors are created which can decrease accu-
racy as well as quality of machined components. The
deformation errors caused by cutting temperature and
forces in low-rigidity and flexible parts milling opera-
tions are the main cause of inaccuracy in the machined
components in the aerospace and aeronautics sector.
Cutting temperatures and forces can cause inaccuracy in
machined thin-walled components, leading in deforma-
tion error of final products. Previous research works [1–4]
indicate that the deformation error caused by the cutting
forces and cutting temperatures is the prevailing issue in
achieving high-precision machining of thin-walled com-
ponents. As a result, monitoring and compensating the
machining deformation error is critical and has received
a lot of attention from scientists in different research
works [5–8]. Virtual machining systems are being devel-
oped to increase precision and productivity of the part
production process. The production method and effec-
tive parameters can be modeled and modified in virtual
environments. As a result, using the virtual machining
method, more added values can be obtained by applying
the modified parameters of production process in virtual
environments [9, 10]. To reduce cutting forces and tool
deformation errors during milling operations, Law etal.
[11] increased the accuracy of machined parts by reduc-
ing the tangential cut depth in corner milling processes.
Technical Editor: Adriano Fagali de Souza.
* Mohsen Soori
Mohsen.soori@gmail.com; Mohsen.soori@kyrenia.edu.tr
1 Department ofAeronautical Engineering, University
ofKyrenia, Via Mersin 10, Kyrenia, NorthCyprus, Turkey

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2023) 45:289
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To enhance the precision of extremely flexible aerospace
components in peripheral end milling operations, simula-
tion and reduction of static form defects in plate milling
operations is presented by Budak and Altintas [12]. Wang
etal. [13] presented a cutting pattern modification strat-
egy for reducing component displacement in thin-wall
milling, which minimizes the axial depth of cut to mini-
mize component displacement errors. To properly antici-
pate and evaluate the deformation error in impeller blades
5-axis milling operations, Wang etal. [14] proposed an
advanced model of deformation error in thin-wall sur-
face components during the milling metal cutting using
finite element approach. To increase cutting precision of
thin-walled parts, Du etal.[15] developed circumferential
machining force impacted error correction using an ana-
lytically model approach and APDL displacement calcu-
lation. In order to detect and compensate the volumetric
errors in the machining operations of thin-walled parts,
error compensation for machining of large thin-walled
part with sculptured surface based on on-machine meas-
urement is presented by Huang etal. [10]. To enhance the
accuracy of machined parts, Cho etal. [16] developed an
enhanced error detection and compensation methodology
in order to collect inspection data using the On-Machine
Measurement device employing a spindle-mounted touch-
trigger probe. To analyze and compensate the deformation
error of thin-walled web parts, an advanced on-machine
observations compensation system for machining opera-
tions of thin-web parts is presented by Ge etal. [17]. In
order to achieve the tight tolerances during machining
operations of flexible parts, an advanced iterative meas-
urement and experiment processes are needed to be used
in terms of accuracy enhancement of final products [18].
Multiple sensors are applied to the machining process
in real-time compensation methods to track signals and
substantial improvements to production variables and cut-
ting tool orientations [19]. As a consequence of the online
cutting process monitoring system, the deflection error
can be reduced [20]. To reduce deformation errors in end
milling operations, Wang etal. [21] proposed real-time
deformation correction in large thin-walled components
machining. Liu etal. [4] proposed a technique for com-
pensating for real-time machining defects based on edge
characteristics for cutting force generated deformations
in flank milling in order to improve machined compo-
nent precision. In-process compensation methodology of
deformations error by adding the piezo-electric actuator
is developed by Diez etal. [22] to improve the precision
of flexible part end milling operations. To enhance the
precision of CNC machine tools, Zhou etal. [23] provide
modeling and adjustment of thermal properties of lead-
screw for machine tool depending on CNC system actual
statistics data. A closed-loop error compensation method
for robotic flank milling is presented by Xiong etal. [24]
to increase surface accuracy of produced parts using mill-
ing operations. Thermal error compensation modeling for
CNC machine tool worktables is proposed by Wei etal.
[25] to provide high prediction accuracy and stability
in worktables of CNC machine tool during machining
operations. To improve the error compensation effect for
low-stiffness structure during milling operations, machin-
ing error compensation for thin-walled parts considering
time-varying cutting condition is proposed by Zhao etal.
[26]. Soori and Asmael [27] developed optimal machining
settings to reduce displacement inaccuracy in thin-walled
rotor blades 5-axis milling operations. To minimize sur-
face integrity and residual stress during grinding opera-
tions of Inconel 718, optimized machining parameters
using the Taguchi optimization approach is presented by
Soori and Arezoo [28]. To execute the technology, mod-
ern sensors need to be inserted in the machining process
in order to directly measure and alter machining param-
eters, which can increase the cost of milling accuracy
enhancement [29]. In addition, repeated 'float' cuts are
frequently employed in industry to gradually erase surface
dimensional inaccuracies caused by machining processes
deformation [30]. All of the presented strategies in error
detection and compensation would increase machining
time and costs, making it hard to fully utilize the machine
tool. One of the most extensively used strategies in com-
pensation of dimensional surface and deformation errors
during milling operations of flexible parts is tool path
modification methodologies [31–33].
Soori etal. described virtual machining approaches and
methodologies for evaluating and improving machining
operations in digital settings [34–39]. Soori and Arezoo
[40] proposed a review in machining caused residual
stress to assess and decrease residual stress throughout
metal cutting operations. Soori etal. [41] presented an
enhanced virtual machining approach for improving sur-
face characteristics of turbine blades during five-axis end
milling operations. Altintas and Merdol [42] described
a virtual machining system and application in order to
achieve optimal milling conditions. Altintas etal. [43]
describe a virtual adjustment of deflection error utilizing
a cutting tool pathways modification method to improve
accuracy in ball end milling of flexible turbine blades.
However, the impacts of cutting temperature on deflection
error throughout flexible turbine blade milling operations
are not examined in the study. Habibi etal. [32] adjusted
the location of cutting tool pathways throughout milling
operations of free form surfaces in order to improve five-
axis ball end milling precision.
According to recently published research works, the
area of deformation error compensation due to cutting
temperature and forces through using virtual machining

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2023) 45:289
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technologies in 5 axis CNC end milling of flexible tur-
bine blades has not been investigated. Error prediction
and compensation methodology is developed in this study
in order to predict and compensate deformation errors
due to cutting forces as well as cutting temperatures dur-
ing flexible turbine blade milling operations. Accurate
prediction of tool and workpiece deformation errors can
be calculated by applying simulated cutting temperature
and forces to the tool's cantilever model and the work-
piece's Finite Element Method (FEM) model. As a result,
it is possible to compensate the deformation errors dur-
ing machining operations without repeating machining
experiments using developed virtual machining system
in the study. A virtual machining system is proposed in
this work for predicting and compensating deformation
error caused by cutting tool and workpiece deformation
throughout turbine blade milling operations. Using an
adapted Johnson–Cook model, the impact of strain inten-
sity and deformation temperatures on flow stress through-
out turbine blade end milling is explored. As a result,
cutting temperatures along machining paths are analyzed
by applying the FEM in order to be used in deformation
error calculation of flexible blades. Also, the tool deflec-
tion errors along machining paths are obtained in order to
calculate the deformation error in the machined turbine
blade. Then, the FEM is implemented in order to obtain
the deformation error as a result of cutting temperatures
and forces along machining paths. Volumetric vectors of
deformation error at each cutting tool location along the
machining pathways are calculated by considering the
cutting tool and workpiece defection errors in order to
compensate the deformation error in the machined tur-
bine blade. Finally, new compensated machining paths
regarding the calculated volumetric error of deformation
error at each cutting tool location along the machining
pathways are generated to increase accuracy in 5-axis
milling operations of flexible turbine blades. To validate
the methodology developed in the analysis, experimental
machining operations of the flexible turbine blade are
implemented by the 5-axis CNC machine tool. To obtain
the errors in machined parts, the CMM machine tool is
then used. As a consequence, the precision of the 5-axis
milling process of flexible turbine blades can be improved
by employing the virtual machining system designed in
the study.
2 Cutting force model in5‑Axis milling
operations
Song etal. [44] established a cutting force approach to sim-
ulate cutting forces during 5-Axis CNC milling operations.
Geometry of 5-axis milling machines, global coordinate sys-
tem (GCS) as fixed coordinate system, feed cross–feed nor-
mal system (FCN) as moving coordinate, and tool coordinate
system (TCS) with lead angle (α) (a) and tilt angle (β) (b)are
shown in Fig.1.
The transformation matrix R from FCN to TCS can be
obtained using the rotation matrix [44],
where α and β are lead and tilt angles, respectively, as shown
in Fig.1.
(1)
R
=
⎡
⎢
⎢
⎢
⎣
cos𝛼
sin𝛼sin𝛽
−sin𝛼cos𝛽
0
0
cos𝛽
sin𝛽
0
sin𝛼
−cos𝛼sin𝛽
cos𝛼cos𝛽
0
0
0
0
1
⎤
⎥
⎥
⎥
⎦
Fig. 1 Geometry of five-axis milling machine tool with lead angle (α) (a) and tilt angle (β) (b)

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Therefore, undeformed chip thickness can be calculated in
TCS as [44],
where
ft
is feed per tooth,
nTCS
is the surface normal vector
of milling cutter in TCS, and
FTCS
is the feed direction vector
in TCS and can be evaluated by
where R is obtained by Eq.(1). During 5-axis end milling,
the cutting tool and workpiece interaction is shown in Fig.2.
The cutter tool and workpiece interaction regions are con-
densed to a few fundamental tool tip in order to create a math-
ematical simulation of end milling cutting forces. As a conse-
quence, the dz element can be written in the following format,
where
Zmax
and
Zmin
are the cutting tool's maximum and
minimum z coordinates, as well as engagement of work-
piece, respectively, and
Nz
is the discretization number of
regarding engaged areas. So, the radial, tangential and axial
cutting forces in the lth time interval,
mth
element on the
jth
cutting tooth can be presented by the differentiated forms as,
where
Krc
,
Ktc
, and
Kac
are the cutting force coefficients that
are donated by the shearing action in radial, tangential, and
axial directions, respectively. Also,
Kte
,
Kre
, and
Kae
are the
coefficients of edge force, which are derived from the experi-
mental results. The selected element's uncut chip thickness
in the cutting tool's normal direction is
w
. During the chip
generation process, the difference in chip thickness is meas-
ured as db which can be calculated as
db=
d
z∕
sin
𝜅
where
𝜅
is axial immersion angle. So, the cutting edge's differential
cutting length can be expressed as,
(2)
w
=ftFTCS.
n
TCS
‖nTCS‖
(3)
FTCS,1 =(1,0,0,1).R
(4)
d
z=
Z
max
−Zmin
N
z
(5)
⎧
⎪
⎨
⎪
⎩
dFr,lmj(z,t)=𝛿j(𝜑j(z,t))(Krcwdb + Kred
s)
dFt,lmj(z,t)=𝛿j(𝜑j(z,t))(Ktcwdb + Kteds)
dFa,lmj(z,t)=𝛿j(𝜑j(z,t))(Kac wdb + Kaed
s)
(6)
z
=
(
m−
1
2)
dz+Z
min
where
R(z)
is the local radius of cutting tool along Z axis
during milling operations.
The rectangular window function of cutting tooth j
throughout cutting operations may be represented as,
where the cutting tool edges' start and exit angles regrading
to the Z coordinate are as
𝜑st (z)
and
𝜑ex(z)
respectively. As
a consequence, the following equation can be used to deter-
mine differential cutting forces,
where
𝜅(z)
is axial immersion angle which is the angle
between cutting tool axis and edge normal and
𝜑j(z,t)
is
radial immersion angle.
Ultimately, the contribution of differential cutting forces
was calculated by the
Nz
discretized elements and the
(7)
d
s=dr=
√(
�
R(z)
)
2
+(R(z))2+d
z
(8)
𝛿
j
(
𝜑j(z,t)
)
=
{
1𝜑st (z)<𝜑
j(z,t)<𝜑
ex(z)
0𝜑
j
(z,t)>𝜑
ex
(z),𝜑
j
(z,t)<𝜑
st
(z
)
(9)
⎡
⎢
⎢
⎢
⎣
d
Fx,lmj
(
z,t
)
dFy,lmj(z,t)
dFz,lmj(z,t)
1
⎤
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎣
−sin𝜅(z)sin𝜑j(z,t)
−sin𝜅(z)cos𝜑j(z,t)
cos𝜅(z)
0
−cos𝜑j(z,t)
sin𝜑j(z,t)
0
0
−cos𝜅(z)sin𝜑j(z,t)
−cos𝜅(z)cos𝜑j(z,t)
−sin𝜅(z)
0
0
0
0
1
⎤
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎣
d
Fr,lmj
(
z,t
)
dFt,lmj(z,t)
dFa,lmj(z,t)
1
⎤
⎥
⎥
⎥
⎦
Fig. 2 The cutter tool and workpiece interaction during 5-axis end
milling

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2023) 45:289
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engaged flute of cutter, The cutting tool's total cutting force
can be determined as,
where
Nt
is the cutting-edge numbers which are engaged in
the cutting.
3 Johnson–Cook model
The Johnson–Cook model is employed in the calculation
of the pressure distribution of a material as a mix of
strain influences, thermal properties, and rate of strain
due to theoretical flexibility and precision of model.
The three variables describe the effect of hardness due
to stress, rate of strain, and heat relaxing on the stress of
the component's flow throughout deformation. Due to the
obvious method's versatility in FEM analysis, it is used to
assess the deformation inclinations of different materials.
The Johnson–Cook model is [45].
where ε is equivalent plastic strain, ε˙ and ε˙0 are the equiva-
lent and basis plastic strain rates, T,
Tm
, and
T0
are the cutting
zone temperature, melting and experimental room tempera-
tures, respectively. The m is index of softening with heat,
while the hardening index of the strain is N. A, B, and C
are the material's rate of elastic modulus, strain, and strain
responsiveness, respectively.
(10)
⎡
⎢
⎢
⎣
Fx(t)
Fy(t)
F
z
(t)
⎤
⎥
⎥
⎦
=
Nt
�
j=1
Nz
�
m=1
⎡
⎢
⎢
⎣
dFx,lmj(z,t)
dFy,lmj(z,t)
dF
z,lmj
(z,t)
⎤
⎥
⎥
⎦
(11)
𝜎
=(A+B𝜀n)(1+Cln 𝜀

𝜀0
)
[
1−
(
T−T0
T
m−
T
0)m]
According to the Johnson–Cook model, the three influ-
encing elements of strain, strain amplitude, and tempera-
ture are completely independent of each other, limiting any
one of them from having a cumulative effect. Such strain
rate dependency is difficult to anticipate using the usual
J–C constitutive model. The updated Johnson–Cook model
studies the interplay between deformation temperature and
strain rate on flow stress, significantly improving the model's
prediction accuracy over the original Johnson–Cook model
[46].
Lin etal. [47] propose an updated Johnson–Cook model
to solve the Johnson–Cook model’s limitations as Eq.(12),
where
A1
,
B1
,
B2
,
C1
,
𝜆1
and
𝜆2
are characteristics of material,
and the other parameters have the same quantities as in the
Johnson–Cook model.
Wang etal. [48] acquired the modified model of John-
son–Cook for the Al-7075 alloy as Eq.(13),
4 Analysis ofdeection errors inthin‑wall
workpieces duringmachining processes
Dimensional inaccuracy of machined surfaces refers to the
discrepancy between the real machined surface and theo-
retical surface of component as CAD model. In thin-walled
(12)
𝜎
=(A
1
+B
1𝜀
+B
2𝜀
2)(1+C
1
ln
𝜀
)
exp[(𝜆1+𝜆2ln 𝜀 )(
T
−
T
ref )]
(13)
𝜎=(
450.821
+
108.537
𝜀n)(
1
−
tanh(𝜀)
exp(𝜀p)
.0.197
ln(
T
T0
))

1+0.031

1−( T
Tm
)
2.271

ln( 𝜀

𝜀0
)

1−

T−T0
Tm−T
0
0.981

Fig. 3 The impact of cutting
temperature and forces on sur-
face dimensional inaccuracies

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2023) 45:289
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component machining, deformation error is also caused by
cutting temperature and forces. Cutting pressures and tem-
peratures cause the workpiece to deform throughout metal
cutting, resulting in geometrical error as well as inaccu-
racy. The impacts of thermal errors and forces-induced
errors are depicted in Fig.3.
The surface dimensional inaccuracy can be presented
as Eq.(14) [49],
where
𝛿t,p
and
𝛿f,p
are deflection error for the Point P was
established by normal projections of the cutting temperature
and cutting forces, respectively.
The nominal radial depth of cut, indicated by RN, is
the spacing between the original surface to be machined
and the intended machined surface. To guarantee that the
surface dimensional inaccuracy does not exceed the toler-
ance throughout the milling process,
RA
is often specified
to be different from
RN≠RA
. So, the surface dimensional
inaccuracy can be calculated as [49],
(14)
ep=𝛿t
,
p+𝛿f
,
p
Note that
RN
and
RA
are the cut's nominal and specified
radial depth of cut, respectively.
5 Tool deection error
Modeling the cutting tool to the elastic supports on a
cantilever beam can be used to compute the cutting tool's
deflection error which is illustrated in Fig.4 [50].
The contact point CC of the cutting tool is considered
in the deflection error prediction in the Xc, Yc and Zc axes.
The deflection error in the Z-axis can be neglected since
the cutting tool is highly rigid in the direction during mill-
ing operations. Thus, deflection error of cutting tool by
considering the cutting forces in X and Y directions can
be presented as [50],
where δ is the deflection of cutter; F is the applied force
during machining; D is the overhang of tool; is the cutting
depth; E is Young’s ap modulus; I is the area moment of
inertia of the cutter; Z is the deflection position. As a result,
the cutter's deflection error in the X and Y axes can be com-
puted as [50],
where
𝛿X
,
Y
is the deflection error of tool in X and Y axes; F,
D are cutting force and tool overhang, respectively.
ap
is the
depth of cutting; E is Young’s modulus; I is the area moment
of inertia of the cutter; Z is the position of the deflection.
6 Analysis ofthin‑wall workpiece
deformation errors
The typical deviation of the actual machined surface
from the ideal machined surface in machining operations
is surface dimensional inaccuracy. Cutting pressures and
cutting temperatures cause a deflection inaccuracy when
machining thin-walled workpieces. The Cutting forces and
generated heat in the cutting zone displace the workpiece
into a new position when the cutting tool is engaged to
(15)
ep=𝛿t,p+𝛿f,p+RN−RA
(16)
⎧
⎪
⎨
⎪
⎩
𝛿=FZ2(3(D−0.5ap∕cos𝜃B)−z)
6EI ,0≤Z≤D−0.5ap∕cos𝜃B
𝛿=F(D−0.5ap∕cos𝜃B)2(3z−(D−0.5ap∕cos𝜃B))
6EI ,D−0.5ap∕cos𝜃B≤Z≤
D
(17)
⎧
⎪
⎨
⎪
⎩
𝛿X,Y=FX,YZ2(3(D−0.5ap∕cos𝜃B)−z)
6EI ,0
≤
Z
≤
D−0.5ap∕cos𝜃B
𝛿X,Y=FX,Y(D−0.5ap∕cos𝜃B)2(3z−(D−0.5ap∕cos𝜃B))
6EI ,D−0.5ap∕cos𝜃B≤Z≤D
Fig. 4 Cutting toll deflection error due to cutting forces [50]

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2023) 45:289
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the thin wall workpiece. Figure5 shows the amount of
deformation error generated by cutting tool and workpiece
deflections in a machined thin-walled component, where
𝛿t
and
𝛿w
are cutting tool and workpiece deformations,
respectively [51].
The created deformation errors in the machined parts
are different throughout machining surfaces due to dif-
ferences between cutting forces and workpiece stiffness
at each position of cutting tool along machining paths.
A cantilever beam can be used in order to simulate and
forecast the deflection inaccuracy of a cutting tool. Work-
piece's deformation error can be accurately calculated by
using a finite element model. Metal is removed from a
thin-walled component and the workpiece thickness are
decreasing which can reduce the stiffness of the workpiece
along the cutting tool's path during the milling opera-
tions. So, the workpiece stiffness during milling opera-
tions is achieved by suspending the contributions of the
removed material from the original workpiece by using an
efficient structural stiffness correction parameters. Thus,
the workpiece deformation error
𝛿w,j(W;Z,t)
is calculated
by combining the stiffness of the in process workpiece
and the dispersed cutting forces to derive the equations of
static equilibrium for the investigated machining system,
as illustrated in Eq.(18). [51].
where
uT
w
,
j
(W;Z,t
)
is the statics displacement of workpiece,
and
nj(W;Z,t)
is the unit normal of envelope surface. Work-
piece and cutter deformation errors are calculated indepen-
dently. The overall deformation error is then calculated by
adding the individual components at a particular time. As a
consequence, we can calculate the overall deformation error
as [51],
where
𝛿t
,
j(W;Z,t)and𝛿w
,
j(W;Z,t)
are deformation error of
cutting tool and machined part.
Ultimately, in order to locate the appropriate machined sur-
face, the whole deformation errors are imprinted on the flute's
envelope surface formed by the
jth
flute [51]. As a result, the
cutting tool's deformation error can be determined as Eq.(20)
[51].
where
Sj(W;Z,t)
is the surface of sphere centers of cutting
tool on the workpiece along the Z axis and feed direction.
Each flute of the cutting tool can provide a unique computed
(18)
𝛿
w,j(W;Z,t)=u
T
w
,
j
(W;Z,t)nj(W;Z,t
)
(19)
ej(W;Z,t)=𝛿t,j(W;Z,t)+𝛿w,j(W;Z,t)
(20)
Mj(W;Z,t)=Sj(W;Z,t)+(rj(z)+ej(W;Z,t))nj(W;Z,t)
Fig. 5 The machined thin-walled part's deformation inaccuracy [51]
Fig. 6 The volumetric error vector generation due to cutting tempera-
tures and forces during 5-axis milling operations of turbine blade

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2023) 45:289
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machined surface due to unique amount for each flute's
sphere radius function as
rj(z)
and deformation error
ej(W;Z,t)
.
7 Deformation error compensation
methodology
In order to derive volumetric errors at each position of cut-
ting tool along machining paths, error vectors related to cut-
ting temperatures and forces are constructed Using the idea of
closed loop at during the turbine blade's 5-axis milling opera-
tions [52]. Figure6 depicts the closed loop volumetric error
vector resulting of cutting forces and cutting temperatures
during turbine blade 5-axis milling operations. Throughout
milling operations, the point O is the point of contact between
the workpiece and the cutter.
As a result, the real and compensated volumetric error
vectors of cutting forces and cutting temperatures for the 20
sample points in 5-Axis operations of turbine blade can be
shown in Fig.7.
In order to compensate the volumetric error vectors of
contact points O along cutting tool paths during chip forma-
tion process, the algorithm of iterative compensation is used.
In this method, the surface error of machined part is moving
step by step to the nominal dimensions of the part in order to
compensate the volumetric error vectors during milling opera-
tions. The volumetric error vector at the point O is obtained
as
V1
. Then, the point O will be moved to the Point
O′
1
as the
Fig. 7 The real and compen-
sated volumetric error vectors
of cutting temperatures and
forces for the 20 sample points
in 5-Axis operations of turbine
blade
Fig. 8 The iterative method of
surface error compensation at
the contact point of O

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2023) 45:289
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Page 9 of 16 289
O�
1
=O−V
1
. The error in the point
O′
1
is
E1=V1
. The volu-
metric error vector at the point
O′
1
is obtained as
V2
. So, the
point
O′
1
is moved to the point
O�
2
=O−V
2
. Also, the error in
the point
O′
1
is
E2=V2−V1
. So, the new contact point
O′
1
the
interaction process can be presented as,
where E is the error in each new contact point during the
iteration process. Also, the volumetric error in the point
O′
2
is
obtained as
V3
which is used to move the point
O′
2
to the point
O�
3
=O−V
3
. The error in the point
O′
2
is
E3=V3−V2
. The
iteration compensation process will be repeated until the
desired result of error is
|
|
E
n|
|<𝜀
and then the contact points
O will be moved to
O�
n−1
.
The iterative method of surface error compensation at the
contact point of O is shown in Fig.8.
Figure9 shows a flowchart of the surface error compensa-
tion using the iterative technique at the contact point of O.
(21)
O�
1=O−E1
O
�
2=O�
1−E2
O
�
3=O�
2−E3
⋮
O
�
n
=O�
n−1
−E
n
8 Virtual machining system
The virtual machining method is developed in this study
using the programming language of Visual Basic. The sys-
tem can receive the nominal machining paths, the geometry
and material characteristics of the milling cutting tool, as
well as the CAD model of sample component. Based on
cutting tool attributes and machining operation parameters,
the generated virtual machining system can calculate cutting
Fig. 9 The flowchart of the surface error compensation at the contact
point of O by using the iterative method
Fig. 10 The flowchart of cutting force and cutting temperature calcu-
lation
Fig. 11 The flowchart Deformation error prediction

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2023) 45:289
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forces for each point of cutting tool throughout machining
pathways. By using the calculated cutting forces along
machining paths, the amounts of ε as equivalent plastic
strain, ε˙ and ε˙0 as the equivalent and basis plastic strain
rates are calculated. As a result, the modified model of John-
son–Cook for the Al-7075 alloy as Eq.(13) is used to obtain
the information of cutting temperature (T) during milling
operations. To calculate the cutting temperature during the
chip generation process, the developed virtual machining
system is coupled to the FEM analysis software of Abaqus
R2016X. The CAD model of the product is then mesh-
generated, allowing it to be examined using finite element
methodology. As a result, the FEM technique can properly
forecast the cutting temperature during the chip production
process. Figure10 depicts the workflow and approach used
by the virtual machining system to determine cutting tem-
perature and force during machining operations.
The cutting temperature and force data are then submitted
to the Abaqus R2016X FEM analyzer, which calculates the
machined turbine blade's deformation error. To assess the
temperature and cutting force-induced deformation errors
Fig. 12 The flowchart of deformation error compensation
Fig. 13 Methodology for study
in deformation error compen-
sation during 5-Axis milling
operations of turbine blades

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during end milling, the mesh is then added to a thin-walled
CAD model of turbine blade. To compute node displace-
ment, the expected cutting temperature and forces at each
location of the cutting tool during machining pathways are
integrated to each part model's mesh node. So, the defor-
mation error due to cutting temperature and forces can be
calculated for each cutting tool location. Figure11 depicts
the method and approach for computing cutting force and
evaluating deformation error using the advanced virtual
machining system.
Then, volumetric error vectors of the deformation error at
each cutting tool location along the machining paths are cal-
culated. To compensate the obtained volumetric error vector
at each position of cutting tool along the machining paths,
the iterative method which is described in the Sect.7 is used.
Then, new machining paths regarding the calculated volu-
metric vectors of deformation error at each cutting tool loca-
tion along the machining pathways are generated in order to
compensate the deformation error in the machined turbine
blade. Figure12 shows the deformation error compensation
methodology in the study.
The flowchart of the study in deformation error compen-
sation during 5-Axis milling operations of turbine blades is
shown in Fig.13.
9 Validation
In order to assess the proposed techniques in the study, the
turbine blade is manufactured using a 5-axis CNC milling
machine tool Kondia HM 1060. AL 7075 is the material
used for the turbine blades. Dimensions of sample tur-
bine blade in the experiments are as 170mm length and
143.36mm width. Also, the average of sample turbine blade
thickness is 2.15mm. The Masterccam software is used to
obtain cutting tool pathways during 5-Axis CNC milling
operations of sample turbine blades. The turbine blades are
then manufactured using a 5-axis Kondia HM 1060 machine
tool. The experiment was conducted using a carbide ball
nose end mill with an 8mm diameter, helix angel 30°, flute
number 4, overall length 60mm, and flute length 35mm.
The spindle speed is 200m/min and the feed rate is 200mm/
Fig. 14 Machining operation of
test turbine blade
Fig. 15 The inspection procedure of machined turbine blade by using
the CMM machine

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289 Page 12 of 16
min. Figure14 depicts the milling operations of sample tur-
bine blade.
The proposed cutting force model of a ball nose end mill
throughout 5-axis CNC machining processes is utilized to
estimate cutting forces [44]. To obtain the cutting forces
coefficients for Al 7075 materials, the mean cutting forces of
twenty slot milling experiments with 1.5mm cutting depth
are recorded using the Kistler 9139AA dynamometer. The
feed rate 100mm/min and feed per tooth 0.5mm, and the
spindle rotates 3000rpm are selected for the machining
Fig. 16 The process of surface
generation from the measured
data of machined turbine blade
Fig. 17 Calculated deformation errors in the machined turbine blade in mm unite
Fig. 18 Calculated deformation
errors in the machined turbine
blade

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2023) 45:289
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Fig. 19 The calculated deforma-
tion error of machined turbine
blade
Fig. 20 The measured and
predicted deformation errors
in the machined turbine blade
without and with compensation
methodology for the L5 line in
Fig.18
Table 1 The measured and predicted deformation errors of 10 selected points of the machined turbine blade without and with compensation
methodology for the L5 line in Fig.18
No Before compensation After compensation Percentage of change
Measured errors
(mm)
Predicted errors
(mm)
Measured compensated
errors (mm)
Predicted compensated
errors (mm)
Measured errors Predicted errors
Point 1 0.043 0.041 0.011 0.012 74.41860 70.73170
Point 2 0.046 0.043 0.012 0.01 73.91304 76.74418
Point 3 0.051 0.045 0.015 0.013 70.58823 71.11111
Point 4 0.045 0.039 0.013 0.01 71.11111 74.35897
Point 5 0.043 0.04 0.014 0.011 67.44186 72.5
Point 6 0.042 0.041 0.012 0.012 71.42857 70.73170
Point 7 0.048 0.043 0.015 0.011 68.75 74.41860
Point 8 0.043 0.04 0.013 0.013 69.76744 67.5
Point 9 0.047 0.039 0.012 0.014 74.46808 64.1025
Point 10 0.045 0.041 0.014 0.012 68.88888 70.73170
Average 0.0453 0.0412 0.0131 0.0118 71.07758 71.29305

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parameters. As a consequence, the cutting forces coefficients
for Al 7075 materials are calculated,
The CMM machine is used to measure the deformation
inaccuracy of machined turbine blades. The used probe
in the measuring process is Renishaw SP25M while its
repeated accuracy in directions of touching is 1μm. The
inspection procedure of machined turbine blade by using
the CMM machine is shown in Fig.15.
Loft through the section curves is passed in order to gen-
erate a NURBS surface from the scan data. As a result, the
machined surfaces of turbine blade is generated in virtual
environments to be used in the deformation error calcula-
tion. The process of surface generation from the measured
data of machined turbine blade is shown in Fig.16.
The amount of error between the CAD model of turbine
blade and fitted NURBS surfaces from the measured data of
the CMM machine are calculated in each section of gener-
ated surface as the deformation error of machined turbine
blade. The calculated turbine blade deformation errors are
shown in Figs.17 and 18.
The Abaqus program is utilized to estimate and evaluate
the deformation errors of machined turbine blades caused
by cutting temperature and forces. Figure19 depicts the
predicted deformation error of the produced turbine blade
using the FEM approach.
As a consequence, the measured and predicted defor-
mation errors in the machined turbine blade without and
with compensation methodology for the L5 line in Fig.18
is presented in Fig.20.
The measured and predicted deformation errors of 10
selected points of the machined turbine blade without and
with compensation methodology for the L5 line in Fig.18
is shown in Table1.
10 Conclusion
The machined thin-walled parts are deflected during the
chip forming process of machining operations caused by
cutting temperature and forces. In order to develop the thin-
walled parts milling operations, the errors can be modeled
and studied in virtual environments. In the research work,
deformation error compensation methodology caused by
cutting temperature and forces is presented by applying vir-
tual machining system. By assessing the combined influence
of tensile rate and deformation temperature on flow stress,
the modified Johnson–Cook Model is employed in the study
(22)
K
tc
=810.42, K
te
=16.93
K
rc =406.1, Kre =17.68
Kac =
241.28,
Kae =
0.74
to investigate and examine the chip formation process. Then,
in order to obtain the cutting temperatures during chip for-
mation process, the FEM method is used. As a result, the
deformation errors caused by cutting temperature and forces
to the workpiece and cutting tool along machining paths
are calculated by using the FEM method. Finally, volumet-
ric vectors of deformation error at each cutting tool loca-
tion along the machining pathways are generated in order
to be compensated utilizing the study's established system.
Ultimately, cutting operations on the flexible turbine blade
are conducted utilizing a 5-axis CNC machine tool and the
deformation errors are then measured using CMM equip-
ment in order to validate the established procedure in the
study. The following summarizes the findings from the
study,
1. A 94.3% compatibility is obtained in comparison with
the results of experimental and virtual machining system
for the flexible turbine blade deformation error.
2. A 71.48% reduction in the measured errors of the
machined flexible turbine bale is obtained by the modi-
fied machining paths in the error compensation algo-
rithm.
3. The force-induced error vectors are bigger than cutting
temperature error vectors at each position of cutting tool
along machining paths due to effects of cutting forces to
the flexible turbine blades during 5-axis milling opera-
tions.
4. The amount of deformation error due to the cutting tool
as well as flexible workpiece deflection can be accu-
rately predicted and minimized. As a result, accuracy of
machined parts using the developed virtual machining
system in the study can be enhanced.
5. The precision of the machined flexible turbine blades
can be increase by using the modified machining codes
in the developed virtual machining system.
6. To enhance the accuracy of machined parts, The influ-
ence of cutting tool configurations, such as radius and
helix angle, tool wear, and edges on the thin-walled
components deformation error in machining processes
can be examined.
7. These are the ideas of author for future research projects.
Acknowledgements Not applicable.
Author’s contribution Not applicable.
Funding There is no funding for the submitted manuscript.
Data availability The used data and materials in the research works
are available.

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2023) 45:289
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Declarations
Conflict of interest There is no conflict of interest regarding the sub-
mitted manuscript.
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