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Modeling and experimentation for threedimensional dynamics of endmills
Article in International Journal of Machine Tools and Manufacture · February 2012
DOI: 10.1016/j.ijmachtools.2011.09.005
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International Journal of Machine Tools & Manufacture 53 (2012) 39–50
Contents lists available at SciVerse ScienceDirect
International Journal of Machine Tools & Manufacture
journal homepage: www.elsevier.com/locate/ijmactool
Modeling and experimentation for three-dimensional dynamics of endmills
Bekir Bediz a, Uttara Kumar b, Tony L. Schmitz c, O. Burak Ozdoganlar a,n
a
b
c
Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA
Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
a r t i c l e i n f o
abstract
Article history:
Received 11 May 2011
Received in revised form
12 September 2011
Accepted 13 September 2011
Available online 22 September 2011
This paper presents a model for the three-dimensional (3D) dynamic response of endmills while
considering the actual fluted cross-sectional geometry and pretwisted shape of the tools. The model is
solved using the spectral-Tchebychev (ST) technique. The bending and the coupled torsional-axial
behavior of four different fluted endmills is compared to finite element model (FEM) predictions and
experimental results obtained using modal testing under free-free boundary conditions. For the first
eight modes, including six bending and two torsional/axial modes, the difference between the 3D-ST
and experimental natural frequencies is shown to be 3% or less for all four tools tested during this
study. For the same modes, the 3D-ST and FEM predictions agree to better than 1%. To demonstrate its
application, the 3D-ST model for the fluted section of a commercial endmill is coupled to the spindle–
holder to predict the tool-point dynamics using receptance coupling substructure analysis (RCSA) with
a flexible connection. The coupled model is validated through experiments.
& 2011 Elsevier Ltd. All rights reserved.
Keywords:
Endmill dynamics
Three-dimensional spectral Tchebychev
Modal analysis
Receptance coupling substructure analysis
1. Introduction
The dynamic behavior of the tool–holder–spindle–machine
assembly, as reflected at cutting points along a milling tool, often
determines the achievable process efficiency and quality that can
be obtained during milling. The combination of the cutting
mechanics, which describe the machining forces for the selected
machining conditions, tool geometry, and relative tool-workpiece
motion; and the structural dynamics, which dictate the relative
tool-workpiece motion at the cutting point in response to the
machining forces, determine the dynamic behavior of machining
operations [1–3]. The structural dynamics include the dynamic
behavior of the entire structural assembly, including the tool,
holder, spindle, and machine. Unless cutting conditions and tool
geometry are selected carefully, the mechanics–dynamics interaction can cause unstable cutting conditions (chatter) due to the
over-cutting of the surface left by one tooth of the vibrating cutter
by subsequent teeth. Even under stable cutting conditions, forced
vibrations of the flexible tool can lead to errors in the location
of the machined surface (in milling). If the dynamic response is
well-known, however, models can be applied to guide selection of
cutting conditions that increase stability and reduce surface
location errors.
In order to accurately predict the tool–holder–spindle–machine
(THSM) assembly dynamics, two fundamental capabilities are
n
Corresponding author. Tel.: þ1 412 268 9890; fax: þ1 412 268 3348.
E-mail address: ozdoganlar@cmu.edu (O. Burak Ozdoganlar).
0890-6955/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmachtools.2011.09.005
needed: (1) accurate and numerically efficient dynamic models of
realistic, arbitrary tool and holder geometries that capture nonsymmetric bending and coupled torsional/axial dynamics; and (2) a
means to join the experimentally determined spindle–machine
dynamics to the model-based tool–holder dynamics to obtain the
THSM assembly dynamics at any point along the tool [4–13].
This paper describes new results for modeling three-dimensional dynamic behavior of macro-scale milling tools using the
spectral-Tchebychev1 (ST) technique while considering the actual
fluted geometry. The 3D-ST model results for both bending and
torsion are compared to finite element computations and experimental data obtained using impact testing with free–free boundary conditions. The 3D-ST model for the actual geometry of the
fluted portion of the tool is then combined with the holder model
and the measured spindle dynamics to predict the THSM assembly dynamics through receptance coupling substructure analysis
(RCSA).
2. Modeling and spectral-Tchebychev solution of macroendmill dynamics
This section briefly describes the derivation of the endmill
dynamics model and the associated 3D-ST technique used to
solve the model numerically. The approach outlined here follows
1
The name of the Russian mathematician P.L. Tchebychev can be alternatively
transliterated as Chebychev or Chebyschev.
40
B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50
that presented (and experimentally validated) by Filiz and
Ozdoganlar [14,15].
A traditional macro-scale endmill can be represented by the
diameter of the shank section (commonly referred to as the tool
diameter, ds), helix angle C, the shank length Ls, the flute length Lf,
number of flutes, and the cross-sectional geometry of the fluted
region. Due to its simple circular cross-section, the axial, torsional, and bending deflections of the shank region are uncoupled.
Thus, an appropriate one-dimensional (1D) beam models (e.g., the
Timoshenko beam model) is sufficient to describe the dynamics of
the shank. However, the fluted section with its pre-twisted
geometry and complex cross-section causes the axial and torsional deflections to be coupled. Furthermore, unlike those of the
shank portion, bending deflections of the fluted portion are not
symmetric. As a result, accurately capturing the dynamic behavior
of the fluted section necessitates use of 3D modeling techniques.
Therefore, to simultaneously achieve numerical efficiency and
modeling accuracy, a 1D model for the shank and a 3D model for
the fluted section are used. The extended Hamilton’s principle is
applied to obtain the integral boundary value problem for the
each of the shank and fluted sections. The complete model is then
obtained by combining the shank (1D) and fluted (3D) portions
through component mode synthesis.
given as
q ni ¼ Q n q i
and
Z
L
^
AðzÞqðzÞqðzÞ
dz ¼ q T V A q^ ,
ð3Þ
0
where Qn is the nth derivative matrix, VA is the inner product
matrix and the underline indicates the sampled functions based
on Gauss–Lobatto sampling. The deflections associated with each
of the six degrees-of-freedom for each of the sample points can be
described in terms of q, e.g.
u ¼ ½I O O O O Oq ¼ Iu q,
ð4Þ
cx ¼ ½O O O I O Oq ¼ Icx q:
ð5Þ
Substituting Eqs. (2)–(4) in Eq. (1) and using the weighted
residual method, the mass, stiffness and forcing matrices can be
obtained as
M ¼ P T rðITu V A Iu þ ITcy V Ix Icy þ ITv V A Iv þ ITcx V Iy Icx V A
þ ITw Iw þ ITcz V Ip Icz ÞP,
ð6Þ
T
T
K ¼ P T ½EQcy V Ix Qcy þ ks GðQu Icy ÞT V A ðQu Icy Þ þ EQcx V Iy Qcx
T
T
þ ks GðQv þ Icx ÞT V A ðQv þ Icx Þ þEQw V A Qw þ GQcz V Js Qcz P,
ð7Þ
F ¼ PT VF q P T rðITu V A Iu þ ITcy V Ix Icy þ ITv V A Iv þ ITcx V Iy Icx
2.1. Solution of the 1D-ST for the dynamics of the shank
T
To insure the accuracy of the dynamic model, particularly for
higher modes, the Timoshenko beam model, which includes the
shear and rotary inertia effects, is used to model the 1D dynamics
of the shank section [16,17]. The model includes bending,
axial, and torsional motions of the shank section. Following the
extended Hamilton’s principle, the integral boundary value problem for the 1D model can be expressed as
Z L"
@2 c
@2 u
@2 v
@2 c
rA 2 u^ þ rIx 2y c^ y þ rA 2 v^ þ rIy 2x c^ x
@t
@t
@t
@t
0
@2 w
@2 cz ^
^ þ rIp
cz
w
2
@t 2
@t
^
^
@cy @c
@u
@u^ ^
@cx @c
y
x
cy
c y þ EIy
þ ks GA
þ EIx
@z
@z
@z @z
@z @z
^
@v
@v^
@w @w
^
þ cx
þc
þ ks GA
x þEA
@z
@z
@z @z
#
Z L
^
@cz @c
z
^ dz,
þ GJs
fF q gT fqg
dz ¼
@z @z
0
þ rA
ð1Þ
^ f
^ gT are the functions representing
^ f
^ T ¼ fu^ v^ w
^ f
where fqg
x
y
z
the variation terms, L is the length of the beam, r is the density,
A(z) is the cross-sectional area along the beam axis, E is Young’s
modulus, G is the shear modulus, Ip(z) is the polar moment of
inertia, Js(z) is the torsion constant, Ix(z) and Iy(z) are the second
area moments, ks is the shear constant, and x and y are the
angular deflections of the beam due to shear deformations.
The solution of Eq. (1) is obtained using the 1D spectralTchebychev technique described by Yagci et al. [18]. The generalized coordinates are sampled in the spatial domain, and
expressed as a truncated series expansion of orthogonal Tchebychev polynomials as
qi F
Nz
X
aqi ðtÞT nz 1 ðzÞ,
ð2Þ
nz ¼ 1
where qi is the generalized coordinate, T nz 1 are the scaled
Tchebychev polynomials, Nz is the number of polynomials used
in expanding the generalized coordinate, and aqi are the timedependent expansion coefficients. The derivative and inner product operations are defined in the sampled domain and can be
þ ITw V A Iw þ ITcz V Ip Icz ÞRq€b PT ½EQcy V Ix Qcy
T
þ ks GðQu Icy ÞT V A ðQu Icy Þ þ EQcx V Iy Qcx
T
T
þ ks GðQv þ Icx ÞT V A ðQv þ Icx Þ þEQw V A Qw þ GQcz V Js Qcz Rqb :
ð8Þ
2.2. Solution of the 3D-ST for the dynamics of the fluted section
The fluted section and the transition region (from the flutes to
the shank sections) are modeled using 3D dynamics. For a 3D
structure, axial, torsional, and bending motions are coupled. To
obtain the boundary-value problem (BVP) for the 3D deflections
of the fluted section, 3D linear elasticity theory is used to obtain
the strain (elastic) energy. Using the extended Hamilton’s principle and applying integration by parts, the integral BVP for the
fluted section can be obtained as
Z t2 Z
T
€ T fqgfqg
^
^ þ fF q gT fqg
^ dx dy dz dt ¼ 0,
½r fqg
½BT ½BC fqg
t1
Volume
ð9Þ
where fF q g ¼ fF x F y F z gT . In arriving at the BVP given in Eq. (9), the
stress–strain (constitutive) equations were fsg ¼ fCgfeg, where C is
the constitutive matrix, and the strain-displacement relations
were feg ¼ fBgfqg, where B is the differential operator matrix.
In order to solve the 3D fluted section problem defined by Eq. (9)
using the 3D-ST method, coordinate transformations and a mapping
of the complex cross-sectional flute geometry into a rectangular
domain are required. The pretwist effect can be accommodated by
using a local coordinate frame ðx, y, zÞ, and the complex crosssection defined by the ðx, y, zÞ domain is mapped onto a rectangular
cross-section domain defined by the coordinates ðx, Z, zÞ. Combining
both operations, the relation between the final coordinates ðx, Z, zÞ
and the physical domain ðx,y,zÞ can be written as
@
@
¼ Jxi jx
@x i
@xj
and
@
@
¼ J ijx x
,
@xi
@x i
i,j ¼ 1; 2,3, . . . ,
ð10Þ
xx
where J xx
ij and J ij are the elements of the Jacobian matrices of the
cross-section transformation and mapping, respectively.
The integral BVP given in Eq. (9) is then solved using the 3D-ST
technique. First, each deflection term is expressed using a three-
41
B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50
dimensional Tchebychev (truncated) series expansion
qi ðx, Z, z,tÞF
Nx X
Nz
NZ X
X
aqi ðtÞT l1 ðxÞT m1 ðZÞT n1 ðzÞ,
ð11Þ
l¼1m¼1n¼1
where the T terms represent the orthogonal (scaled) Tchebychev
polynomials, aqi are the polynomial coefficients, and Nj is the
number of polynomials associated with the coordinate j ¼ ðx, Z, zÞ.
Similar to the 1D approach, the derivative and inner product
operations are defined in the sampled domain as
Z LZ
ð12Þ
q ni,x ¼ Q xni Ii q and
q q^ dx dy dz ¼ q T V xyz q^ ,
i
0
Area
where q ni,x is the nth spatial derivative and Q xni is the nth extended
i
derivative matrix with respect to coordinate xi, and Vxyz is the
inner product matrix. Similarly as in the 1D solution, the mass,
stiffness, and force matrices can be found from Eq. (9) as
M ¼ rðITu V xyz Iu þ ITv V xyz Iv þ ITw V xyz Iw Þ,
ð13Þ
K ¼ B T Vxyz B C ,
ð14Þ
F ¼ V xyz F q :
ð15Þ
3a
1
2b 2a
3b
4a
4b
Measured
Timoshenko beam model
3D-ST model
Fig. 2. RCSA components with coordinates.
3b
1
3a
2.3. Combining 1D and 3D solutions
Fig. 3. Component coordinates for flexible coupling.
The 1D formulation for the shank section provides six deflection components (three displacements and three angles) for a
given axial position. The 3D formulation for the fluted section, on
the other hand, provides three deflection components for each
sampling point at given three-dimensional position within the
solid. To combine the 1D-ST solution for the shank with the 3D-ST
solution for the fluted (and transition) regions, and thus, to form a
global boundary value problem, compatibility conditions are used
at the boundaries shared by the two sections. The compatibility
equations relate the three displacement degrees of freedom
ðu, v, wÞ from the 3D model and the six degrees of freedom
ðu, v, w, cx , cy , cz Þ for the 1D model for each point ðx,yÞ within
the (boundary) cross-section. Assuming small angles, this compatibility condition can be expressed mathematically as
fu, v, wg ¼ fuycz ,
v þ xcz ,
w þxcy þ ycx g:
ð16Þ
the archived spindle–machine dynamics and this result was
subsequently coupled (using a flexible connection) to the free–
free response of the tool. The component coordinates are identified in Figs. 2 and 3.
The receptance matrices for the assembly, Gij, and substructures, Rij, can be represented as shown in Eq. (17), where o is the
frequency, Xi and xi are the assembly and substructure displacements at the coordinate location i, Yi and yi are the assembly and
substructure rotations, Fj and fj are the assembly and substructure
forces applied at the coordinate location j, and Mj and mj are the
assembly and substructure and moments:
"
# 2 Xi Xi 3
"
# 2 xi xi 3
mj
fj
hij lij
Hij Lij
Fj
Mj
4
5
and Rij ðoÞ ¼
Gij ðoÞ ¼
¼ Yi Yi
¼ 4 yi yi 5:
nij pij
N ij P ij
m
Fj
Mj
fj
j
ð17Þ
3. Receptance coupling substructure analysis
To demonstrate the application of the 3D-ST model for
obtaining the tool-point dynamics, the RCSA technique is used.
Fig. 1 shows the schematic of the THSM assembly used for
experimental validation. The free–free response of the fluted
portion of the tool was obtained analytically by the 3D-ST
technique. Using the three-component RCSA technique for tool
point dynamics prediction [7], the modeled holder was coupled to
Spindle-Machine
Flange
Holder
Tool
Shank
Flutes
Using RCSA, the assembly dynamics at coordinate 1 is obtained
in two steps. First, the modeled free–free receptances of the
holder-shank component are coupled to the free–free receptances
of the remainder of the tool (fluted part) using the following:
G11 o ¼ R11 oR12a ðoÞ½R2b2b ðoÞ þ R2a2a ðoÞ1 R2a1 ðoÞ,
ð18Þ
G3a1 ðoÞ ¼ R3a2b ðoÞ½R2b2b ðoÞ þR2a2a ðoÞ1 R2a1 ðoÞ,
ð19Þ
G3a3a ðoÞ ¼ R3a3a ðoÞR3a2b ðoÞ½R2b2b ðoÞ þ R2a2a ðoÞ1 R2b3a ðoÞ,
ð20Þ
G13a ðoÞ ¼ R12a ðoÞ½R2b2b ðoÞ þ R2a2a ðoÞ1 R2b3a ðoÞ:
ð21Þ
This coupling result represents the tool model with end
coordinates 1 and 3a ðG11 ,G13a ,G3a3a ,G3a1 Þ. Second, the modeled
free–free receptances of the holder with portion of the shank
inside it is coupled to the spindle machine receptances:
G3b3b ðoÞ ¼ R3b3b ðoÞR3b4a ðoÞ½R4b4b ðoÞ þ R4a4a ðoÞ1 R4a3b ðoÞ:
2
3
4
Fig. 1. THSM assembly and coordinates.
1
ð22Þ
The free–free receptances are then considered as a component,
using the receptance matrices R11, R13a, R3a3a and R3a1 . G3b3b is
considered as R3b3b . Third, the holder–spindle–machine component is then flexibly coupled to the tool using translational and
rotational spring constants assembled in the stiffness matrix k.
42
B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50
sectioned by electrical discharge machining (EDM), and the crosssection was imaged at several locations, as shown in Fig. 4(b). The
s
Matlab image-processing toolbox was then used to identify the
boundaries (periphery), and sample points along the periphery
were extracted to define the cross-section.
The RCSA equation for the flexible coupling tool point FRF [9] is
provided as
1 1
G11 ðoÞ ¼ R11 ðoÞR13a ðoÞ R3b3b ðoÞ þ R3a3a ðoÞ þ
R3a1 ðoÞ: ð23Þ
k
Identification of the stiffness matrix k is discussed in Section 7.
4.2. Experimental setups and conditions
4. Experimental methods
To determine the natural frequencies of the aluminum endmills, the frequency response functions (FRFs) of the endmills
were obtained by impact testing. Fig. 5 shows the experimental
procedure used to obtain the bending and torsional natural
frequencies of the endmill. To approximate the unconstrained
boundary conditions, the endmills were placed on soft foam.
The very low stiffness of the foam base relative to the endmills
provided a reasonable approximation to unconstrained (free–
free) boundary conditions. The FRFs were measured by exciting
them using a miniature impact hammer (PCB 0841A17, sensitivity 46.95 N/V) and recording the corresponding vibration using
a low mass accelerometer (PCB 352C23, sensitivity 1727 (m/V s2).
To assess the effect of the miniature accelerometer in the
measured dynamic response, a set of finite elements simulations
were conducted. It was seen that the added mass due to accelerometer does not change the natural frequencies by more than
0.05% for the first eight natural frequencies of the tool. Therefore,
the effect of the miniature accelerometer is deemed negligible.
Depending on the locations of the accelerometer and hammer
impact, direct and cross FRFs, Hij, were obtained, where i and j
represents the measurement and force locations, respectively. For
instance, in the configuration shown in Fig. 5(a), the accelerometer was placed at the shank end, and the force was applied at
the same location (from the other side of the endmill). Therefore,
the direct FRF at the endmill’s shank end was obtained.
Fig. 5(b) shows the experimental procedure used to identify
the torsional natural frequencies. The endmill was excited by
applying the force to one flute in the tangential direction and the
response was recorded in the same direction using an accelerometer placed on the opposite flute. In these experiments both
the bending and the torsional natural frequencies were excited.
The results for the first eight natural frequencies were recorded.
The resonant frequencies of the solid carbide endmill span a
considerably wider range than those of the aluminum endmills.
Therefore, the hammer impact test described above cannot be
effectively used to determine the first eight natural frequencies of
endmill 4.
Fig. 6(a) shows the experimental setup used for obtaining the
natural frequencies of the solid carbide endmills. The endmill was
suspended using flexible elastic bands from two sections along its
length to approximate unconstrained boundary conditions. The
dynamic excitations within a frequency range of 0–40 kHz were
provided using a 3 mm by 4 mm piezoelectric element with a
This section describes the experimental setup and procedures
used for validating the presented 3D-ST solution. For this purpose,
a set of modal tests were conducted on the four different endmills
described in Table 1. These four-flute endmills each had different
geometries (shank diameter, shank length, flute length, and helix
angle). Endmills 1–3 were machined from aluminum blanks so
that the cross-sectional geometry was explicitly known; the
geometry for the commercial endmill (identified as endmill 4)
was determined by sectioning and measurement.
4.1. Description of endmill geometries
The accuracy, and thus effective application, of the 3D-ST
solution for macro-endmills requires accurate knowledge of the
geometry and material parameters of the endmills. In particular,
cross-sectional geometry, twist rate, and the geometry of the
transition region (from the shank to the fluted portion) must be
well known. To facilitate evaluation of the 3D-ST model for wellknown geometries and for a range of geometric parameters, a set
of aluminum endmills (test endmills) were fabricated using fiveaxis machining. Since the aluminum endmills were described
s
using a 3D solid model (SolidWorks ), their geometry was
accurately defined within the accuracy of the manufacturing
technique. The cross-sectional geometry of the endmills are given
in Fig. 4(a).
To enable further assessment of the 3D-ST solution, a commercial (solid) carbide endmill (endmill 4) was selected for
testing. However, the detailed information about the cross-sectional geometry was not available. Therefore, the endmill was
Table 1
Properties of the endmills used for validation experiments (all parameters except
the helix angle were measured using digital calipers with a resolution of
0.01 mm).
Properties
Endmill 1
Endmill 2
Endmill 3
Endmill 4
Shank diameter (mm)
Shank length (mm)
Flute length (mm)
Helix angle (mm)
Material
38.20
101.35
201.14
30
Aluminum
38.17
102.98
175.23
30
Aluminum
38.20
77.66
200.44
30
Aluminum
12.66
66.40
84.41
30
Carbide
2 4
3
1
1
2
3
6
5
4
5
7
6
Fig. 4. Cross-section of (a) endmills 1–3 (coordinates are provided in mm), (b) endmill 4.
7
43
B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50
Foam Base
Miniature Impact
Hammer
Endmill
Miniature Impact
Hammer
Endmill
Accelerometer
Foam Base
Accelerometer
Fig. 5. Impact hammer experimentation setup (a) direct FRF measurement at the endmill shank end and (b) torsional measurement.
Microscope Head
Piezo
Tool Tip View
Metal
Stripes
Laser Source
Microscope
Elastic
Bands
Tool
Piezo
Metal
Stripes
Laser
Spot
Piezo
Cables
Base Mount
Fig. 6. Experimental setup for endmill 4 free–free boundary condition measurements.
125 mm thickness. As shown in Fig. 6(b), the piezoelectric element
was glued onto the macro endmill shank. When a voltage
is applied, the piezoelectric element expands and contracts along
its polarization axis. A pseudo-random excitation having a range
of 0–40 kHz with 140 V amplitude was supplied to the piezoelectric element. For each experiment, 50 measurements of
0.320 s each were completed and averaged to eliminate unbiased
noise.
The response is measured using a laser Doppler vibrometer
s
(LDV) system (Polytec MSA-400) with two fiber-optic laser
sources. Each fiber-optic laser source splits into a pair of channels
and, therefore, both relative and absolute measurements can be
performed. During the experiments, the fiber-optic laser sources
were fed through a microscope to reduce the laser-beam spot
sizes. For the measurements presented in this study, a
5 objective is used to obtain a spot size of 2:8 mm was used.
The axis of the endmill is located in the vicinity of the nodal
line of the torsional motions. Therefore, measurements along the
axis of the endmill do not provide accurate determination of the
torsional modes. To facilitate measuring the torsional modes,
metal strips (0.22 mm by 3.12 mm) were glued to the sides of
the macro endmill. The response measured from the metal strips
amplifies the torsional motions, thereby enabling accurate determination of torsional/axial natural frequencies.
5. Alternative modeling techniques for comparative analysis
In this section, two alternative approaches for modeling the
dynamics of Table 1 endmills are described. The main purpose of
this study is to provide a comparative analysis of the 3D-ST technique
with two other popular solution approaches. First, a finite element
(FE) solution of the endmill dynamics using a commercial solver is
outlined. Next, simplified 1D solution of the endmill dynamics using
four different equivalent cross-section formulations is provided.
5.1. Finite element modeling (FEM) of macro-endmills
Using the geometry of the four endmills, finite element (FE)
models were constructed. Analysis was performed on the meshed
solid model of each endmill using the commercial software
ANSYSs Workbench (SOLID186 elements). Table 2 lists the material properties (Young’s modulus E, density, r, and Poisson’s ratio,
n). Modal analysis was used to identify the natural frequencies.
The convergence of the FE model was evaluated for all endmills based on the changes in the second torsional natural
frequency to ensure that a sufficient number of elements were
included. Table 3 summarizes the convergence study for the FE
model of endmill 1. The convergence study was performed in a
consecutive way, where the percent change in natural frequencies
44
B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50
were calculated between the two consecutive FE simulations
taking the finer meshed model as reference.
Table 2
Endmill material properties.
Endmill material
E (GPa)
Aluminum
Carbide
r
70
580
kg
m3
n
2700
14 500
5.2. Equivalent diameter approach
0.33
0.24
One of the common approaches to describe the dynamics of
the fluted section is to approximate it as a uniform circular crosssection beam with an equivalent diameter. Four methods were
applied to calculate this diameter: (1) the cross-sectional area;
(2) the area moment of inertia of the cross-section; (3) the
volume; and (4) the mass.
Table 3
Convergence study for endmill 1 from Table 1.
Second torsional natural
frequency
(Hz)
Number of
nodes
Number of
elements
% Change in natural
frequencies with
increasing number
of nodes
7776.5
7756.3
7746.3
7742.4
7739.3
3056
4077
7226
20 689
22 454
578
792
1485
4488
4828
–
0.26
0.13
0.05
0.04
1. The cross-sectional area of the fluted portion, Af, was obtained
from the solid model (endmills 1–3) or analysis of crosssection image (endmill 4) and the corresponding equivalent
diameter, deqA , was calculated using
rffiffiffiffiffiffiffiffiffi
4 Af
:
ð24Þ
deqA ¼
p
2. The area moment of inertia for the modeled cross-section of
the fluted portion, If, was determined and the equivalent
diameter, deqI , was calculated using
64 If 1=4
deqI ¼
:
ð25Þ
Table 4
Equivalent diameter of endmills.
p
Endmill
deqA (mm)
deqI (mm)
deqV (mm)
deqM (mm)
Aluminum
Carbide
30.40
9.90
31.10
10.10
30.96
9.95
31.55
9.92
3. The volume of the fluted section, Vf, was obtained and the
equivalent diameter was determined as
sffiffiffiffiffiffiffiffiffiffi
4 Vf
deqV ¼
:
ð26Þ
p Lf
Table 5
Comparison of the experimental and predicted natural frequencies.
Endmill 1
B11
B12
B21
B22
TA1
B31
B32
TA2
Experiment (Hz)
FE (Hz)
3D-ST (Hz)
1426
1400
1404
1430
1403
1407
4059
4000
4007
4068
4002
4023
4218
4115
4091
7530
7484
7486
7543
7492
7497
7777
7739
7750
Difference (%) between ST and FEM
Difference (%) between FE and experiment
Difference (%) between ST and experiment
0.28
1.81
1.54
0.32
1.92
1.60
0.17
1.45
1.28
0.53
1.62
1.10
0.60
2.42
3.00
0.03
0.61
0.58
0.06
0.68
0.62
0.14
0.49
0.34
Endmill 2
B11
B12
TA1
B21
B22
TA2
B31
B32
Experiment (Hz)
FE (Hz)
3D-ST (Hz)
1710
1691
1690
1717
1691
1695
4716
4606
4583
4802
4776
4772
4811
4776
4790
8558
8523
8563
8698
8673
8665
8719
8674
8680
Difference (%) between ST and FEM
Difference (%) between FE and experiment
Difference (%) between ST and experiment
0.06
1.12
1.18
0.22
1.47
1.26
0.51
2.32
2.81
0.08
0.55
0.63
0.30
0.72
0.43
0.47
0.40
0.06
0.09
0.28
0.37
0.07
0.52
0.45
Endmill 3
B11
B12
TA1
B21
B22
TA2
B31
B32
Experiment (Hz)
FE (Hz)
3D-ST (Hz)
1634
1611
1612
1638
1611
1615
4414
4349
4315
4588
4523
4529
4596
4524
4551
8412
8286
8293
8602
8556
8561
8615
8558
8568
Difference (%) between ST and FEM
Difference (%) between FE and experiment
Difference (%) between ST and experiment
0.07
1.40
1.34
0.21
1.64
1.43
0.79
1.46
2.24
0.13
1.42
1.30
0.60
1.57
0.98
0.09
1.49
1.41
0.06
0.54
0.48
0.13
0.66
0.54
Endmill 4
B11
B12
B21
B22
TA1
B31
B32
TA2
Experiment (Hz)
FE (Hz)
3D-ST (Hz)
2487
2463
2486
2487
2471
2495
7343
7242
7271
7343
7248
7272
11273
11365
11384
13515
13283
13337
13515
13336
13393
20071
20143
20163
Difference (%) between ST and FEM
Difference (%) between FE and experiment
Difference (%) between ST and experiment
0.93
0.96
0.05
0.98
0.64
0.33
0.40
1.37
0.98
0.33
1.29
0.96
0.17
0.82
0.99
0.41
1.72
1.32
0.43
1.33
0.91
0.10
0.36
0.46
45
B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50
4. The mass of the endmill, m, can be expressed as the sum of the
mass of the shank (first expression on the left hand side of
Eq. (27)) and the mass of the fluted portion (second expression)
r
p
2
ds
Ls
4
þr
p
2
deqM
4
Lf
¼ m,
ð27Þ
where deqM is the equivalent diameter of the fluted portion. By
weighing the endmill and substituting nominal value for the
density and geometry of the endmill, the equivalent diameter was
calculated according to Eq. (28). The benefit of this approach is
that no tool model is required, unlike the first three methods:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!
u
2
u 4
m p ds Ls
:
ð28Þ
deqM ¼ t
p Lf r
4
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
In this section, the natural frequencies obtained using modal
testing, the ST method, and FE model (FEM) simulations are
compared to assess the accuracy of the 3D-ST technique for
modeling endmill dynamics. Table 5 provides the natural frequencies from 3D-ST, FEM, and experimentation, as well as the
percent differences between the modes obtained from different
methods. The labeled mode shapes (as bending or torsional/axial)
are identified according to the calculated deformations. The
bending modes are represented by Bij where i is the mode number
and j is the principal direction of deformation. The coupled
torsional–axial modes are represented by TAi where i is the mode
number.
For each of the endmills, the percent errors between the ST
method and experimentation are seen to be less than 3.0%. This
4
11
9 10
7 8
es
6
d
o
5
TM
3 4
3D-S
1 2
12 13
14
MAC
3
4
5
FE
6
7
M
8
M
9
od
es 10
11
12
13
14
11
9 10
7 8
6
s
e
d
5
T Mo
3 4
3D-S
1 2
12 13
14
3
4
5
FE 6
M
7
M 8
od 9
es 10
11
12
13
14
11
9 10
7 8
es
d
o
6
5
TM
3D-S
3 4
1 2
12 13
14
12 13
14
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
2
5
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
3
FE 6
M 78
M
od 9
es 10
11
12
13
14
MAC
6.1. Comparison of tool modeling techniques
MAC
MAC
Table 4 lists the equivalent diameter (inmm) calculated by the
four methods for the four endmills.
6. Modal assessment
2
3
4
5
FE 6
M 78
M
od 9
es 10
11
12
13
14
10 11
8 9
7
des
5 6
T Mo
3 4
3D-S
1 2
Fig. 7. Modal Assurance Criteria (MAC) analysis for (a) endmill 1; (b) endmill 2; (c) endmill 3; and (d) endmill 4.
46
Cross-sectional Area
Area Moment of Inertia
Volume
Cross-sectional Area
Area Moment of Inertia
Volume
Mass
Mass
14
12
10
Absolute Percent Error (%)
8
6
4
2
0
1
2
3
compared
to
experimental results
compared
to
FE results
B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50
4
5
6
7
8
10
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
14
12
8
10
6
8
4
6
4
2
2
0
1
2
3
4
5
6
7
8
0
Mode Number
Fig. 8. Absolute percent errors of the approximate methods for (a) endmill 1; (b) endmill 2; (c) endmill 3; (d) endmill 4.
small difference between the 3D-ST and experimental natural
frequencies could be the result of differences between the
modeled and actual geometry and the non-uniform/inaccurate
material properties. Furthermore, it is observed that the largest
errors occur in the first torsional mode, which suggests that it is
the most sensitive mode to geometric uncertainties. A comparison between natural frequencies from the 3D-ST and FEM results,
on the other hand, shows a difference less than 0.8% for all four
endmills and modes. This good match between the 3D-ST and
FEM results, which use identical geometries and material properties, also supports this hypothesis. It should be noted that the
dynamic response is very sensitive to the geometry of the
transition region. Therefore, it is critical to accurately capture
this geometry.
For a more in-depth assessment of the 3D-ST model, the modal
assurance criterion (MAC) was used [20,21]. The modal assurance
criterion is a measure of consistency of the modal vectors
obtained through two different methods. If the MAC value for a
given mode is above 0.8 (i.e., 80%), it is considered that the modes
are consistent [21]. Therefore, this method was used to compare
the mode shapes (other than rigid body modes) obtained from the
3D-ST method and FEM analysis. To perform the MAC analysis,
the position and associated displacement at each FEM node was
obtained. Second, the 3D-ST method was used to determine the
displacements at the same positions for the given mode shape.
The MAC value for each mode was then calculated from
31.7
43
7
12.7 29.3
126.8
0.88
24.08
25.12
39.95
47.99
Fig. 9. Holder and blank dimensions for 127.68 mm overhang length (all dimensions are in mm).
Table 6
Carbide endmill/blank and holder material properties.
Component
E ðGPaÞ
Tool
Holder
580
200
r
kg
m3
14 500
7800
n
0.24
0.29
2
MAC ¼
9fFA gT fF3DST gT 9
fFA gT fFA g þfF3DST gT fF3DST g
,
ð29Þ
where FA and F3DST are the modal matrices obtained through FE
analysis (ANSYSs ) and the 3D-ST method, respectively.
The calculated MAC numbers are given in Fig. 7. A MAC number
closer to unity indicates good agreement between the mode shapes
for the two methods; the first 14 modes of each endmill were
analyzed. A good match between the 3D-ST and FEM modes is
B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50
observed. A minimum MAC value of 0.855, 0.993, 0.966, and 0.931
and the average MAC values of 0.975, 0.995, 0.992, and 0.989 are
obtained for endmills 1, 2, 3, and 4, respectively.
The equivalent diameter simplification for the twisted fluted
section of the tool was also evaluated. Fig. 8 provides the percent
errors for each of the approximate methods with respect to the
results of the FEM results for each mode.
47
The general trend of the percent errors was similar for each tool
model. Comparing the different approximation methods, the most
accurate equivalent diameter was obtained using the cross-sectional area and the least accurate approach was the method based
on the endmill mass. Also, considering all four approximation
methods for each mode, the largest errors (410%) was observed
for the first torsional mode. Furthermore, the approximation errors
depend strongly on the particular geometric parameters of the
endmills. Therefore, these approximations should be used carefully.
The 3D-ST modeling approach presented in this study avoids such
errors while retaining numerical efficiency.
7. Tool point measurements and predictions
V
IV
III
II
I
Fig. 10. Blank-holder model components.
Table 7
Component dimensions for 127.68 mm overhang length.
Dimensions (mm)
Component
Outer diameter (d0), left
Outer diameter (d0), right
Inner diameter (di), left
Inner diameter (di), right
Length, L
III
II
I
43
31.7
7
7
8.04
31.7
31.7
12.7
12.7
14.83
31.7
31.7
12.7
12.7
1.04
31.7
29.3
12.7
12.7
24.08
12.7
12.7
0
0
127.68
x 10−5
Real (m/N)
x 10−5
0
−5
200
x
Imag (m/N)
IV
400
600
800
1000 1200 1400 1600 1800 2000
0
−2
−4
−6
−8
200
400
600
800
1000 1200 1400 1600 1800 2000
5
0
−5
200
10−5
Imag (m/N)
Real (m/N)
5
V
Tool point measurements were performed and predictions
were completed on a Mikron UCP-600 Vario five-axis machining
center. Predictions and measurements of the tool point FRFs are
presented here for three different overhang lengths (118.7 mm,
124.78 mm, and 127.59 mm) of endmill 4 (150.8 mm total length)
inserted in a steel tapered thermal shrink fit holder. The tool point
FRFs were measured by exciting the tool tip with a miniature
impact hammer (PCB 0841A17, sensitivity 46.95 N/V) and recording the corresponding vibration with a low mass accelerometer
(PCB 352C23, sensitivity 1727 (m/V s2). Also, measurements were
performed with the same shrink fit holder using a carbide blank
of approximately the same length as the endmill (152.8 mm). The
overhang lengths of the blank were adjusted to match the three
overhang lengths of the tool. Fig. 9 shows the dimensions of the
holder and blank for an overhang length of 127.68 mm. Table 6
x
2
0
−2
−4
−6
−8
200
400
Frequency (Hz)
Real (m/N)
4
600
800
1000 1200 1400 1600 1800 2000
600
800
1000 1200 1400 1600 1800 2000
Frequency (Hz)
10−5
400
x 10−5
2
0
−2
−4
200
400
600
800
600
800
1000 1200 1400 1600 1800 2000
Imag (m/N)
x 10−5
0
−2
−4
−6
−8
200
400
1000 1200 1400 1600 1800 2000
Frequency (Hz)
Fig. 11. Measured (solid line) and predicted (dotted line) tool point FRFs for the carbide tool blank with overhang lengths of: (a) 127.68 mm; (b) 124.35 mm; and
(c) 118.62 mm (rigid connection).
48
B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50
holder at coordinate 2 in Fig. 12. Since no connection is rigid in
reality, a flexible connection between the tool and the holder was
implemented.
The flexible coupling of the components is carried out in two
steps: (1) the spindle–machine is first rigidly coupled to the
holder and the portion of the shank inside the holder; (2) the
holder–spindle–machine component is then flexibly coupled to
the blank outside the holder using translational and rotational
spring constants assembled in the stiffness matrix k. The RCSA
equation for the flexible coupling tool point FRF in case of the
blank is provided in Eq. (30). The stiffness matrix [9] is given
by Eq. (31), where kxf, kyf , kxm, and kym are the displacement-toforce, rotation-to-force, displacement-to-moment, and rotationto-moment stiffness values, respectively (kyf and kxm were
assumed equal due to reciprocity).
lists the tool and holder material properties used for modeling
both the endmill and blank. The component model for one blankholder combination is shown in Fig. 10, and the corresponding
dimensions are listed in Table 7. Fig. 11 shows the comparison of
the measured and the predicted carbide blank tool-point FRF for
three different overhang lengths.
It is observed in Fig. 11 that the predicted natural frequencies
are higher than those from the experiments. This is attributed to
the assumption of a rigid connection between the tool and the
1 1
G11 ðoÞ ¼ R11 ðoÞR12a ðoÞ R2b2b ðoÞ þ R2a2a ðoÞ þ
k
1
2a
2b
Fig. 12. Component coordinates for flexible coupling in case of blank.
k¼
kyf (N/rad)
kym (Nm/rad)
3.05 106
3.07 106
0
Imag (m/N)
Real (m/N)
0
−5
x 10−5
4
2
0
−2
−4
−6
−8
200
400
600
800
1000
1200
1400
1600
1800
kyf
kym
#
:
ð31Þ
600
800
1000
1200
1400
1600
1800
5
0
2000
−5
200
400
600
800
1000
1200
1400
1600
1800
2000
2000
x 10−5
4
2
0
−2
−4
−6
−8
200
400
600
800
1000
1200
1400
1600
1800
2000
Imag (m/N)
Real (m/N)
5
400
kxm
x 10−5
x 10−5
200
kxf
ð30Þ
To identify the stiffness matrix, one overhang length of the blank
(124.35 mm) was considered and an optimization procedure
based on a genetic algorithm [19] was implemented. The variables were the three stiffness values, and the objective function to
be minimized is given by Eq. (32), where the difference between
the imaginary parts of the measured (m) and predicted (p) tool
Table 8
Stiffness matrix values.
kxf (N/m)
"
R2a1 ðoÞ,
Frequency (Hz)
Frequency (Hz)
Real (m/N)
x 10−5
5
0
−5
200
400
600
800
1000
1200
1400
1600
1800
2000
600
800
1000
1200
1400
1600
1800
2000
Imag (m/N)
−5
x 10
0
−2
−4
−6
−8
200
400
Frequency (Hz)
Fig. 13. Measured (solid line) and predicted (dotted line) tool point FRFs for the carbide tool blank with overhang lengths of: (a) 127.68 mm; (b) 124.35 mm; and
(c) 118.62 mm (flexible connection).
49
B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50
x 10−5
5
5
Real (m/N)
Real (m/N)
x 10−5
0
−5
200
400
600
800
1000
1200
1400
1600
1800
0
−5
2000
200
0
−5
−10
−15
200
400
600
800
400
600
800
1400
1600
1800
1000
1200
1400
1600
1800
2000
x 10−5
Imag (m/N)
Imag (m/N)
x 10−5
400
600
800
0
−5
−10
−15
200
1000 1200 1400 1600 1800 2000
Frequency (Hz)
1000 1200 1400 1600 1800 2000
Frequency (Hz)
Real (m/N)
x 10−5
5
0
−5
200
400
600
800
600
800
1000
1200
2000
Imag (m/N)
x 10−5
0
−5
−10
−15
200
400
1000 1200 1400 1600 1800 2000
Frequency (Hz)
Fig. 14. Measured (solid line) and predicted (dotted line) tool point FRFs for the carbide endmill with overhang lengths of: (a) 127.59 mm; (b) 124.78 mm; and
(c) 118.7 mm (flexible connection).
Table 9
Percent error between measurement (M) and prediction (P) for Fig. 13.
Overhang length
(mm)
Table 10
Percent error between measurement (M) and prediction (P) for Fig. 14.
Mode 1
Mode 2
Mode 3
Mode 4
M
(Hz)
M
(Hz)
M
(Hz)
P
(Hz)
M
(Hz)
P
(Hz)
P
(Hz)
P
(Hz)
Fig. 13(a)
127.68
% error
569.5 570.7 2030
0.2
5.1
2133.5 3073
3.1
2979
3925
1.1
3880
Fig. 13(b)
124.35
% error
595.1 594.9 2045
0.0
4.8
2143
3111
0.9
3082.3 3800
2.4
3890
Fig. 13(c)
118.62
% error
644.8 643.2 2060
0.2
4.6
2155
3230
1.1
3264
3909
4087
4.4
point FRFs was squared and summed over all frequencies within
the range of interest.
qX
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
min
ðImðHm ÞImðHp ÞÞ2 :
ð32Þ
The k matrix values obtained by this approach (see Table 8)
were used to predict the tool point FRFs for other overhang
lengths of the blank as well as the endmill. The comparison of the
measurements and predictions for the three overhang lengths of
the carbide blank is shown in Fig. 13. Note that due to the linear axis
of the FRF plots in Figs. 13 and 14, the higher modes cannot be
Overhang length
(mm)
Mode 1
Mode 2
Mode 3
Mode 4
M
(Hz)
P
(Hz)
M
(Hz)
M
(Hz)
M
(Hz)
Fig. 14(a)
127.59
% error
666
2.5
682.2 2105
2.5
Fig. 14(b)
124.78
% error
688.4 716
4.0
Fig. 14(c)
118.7
% error
P
(Hz)
P
(Hz)
P
(Hz)
2052 2864
4.1
2745.4 3750
4.5
3920
2070
0.1
2072 2944
4.2
2819
3820
2.7
3922
739.3 766.8 2090
3.7
0.2
2094 3087
3.2
2987.3 3910
0.5
3928
observed, however, comparison of four dominant modes was
included in the analyses. Table 9 lists the percent errors between
the measured and predicted natural frequencies for the four
dominant modes within the 5000 Hz measurement bandwidth.
The stiffness values given in Table 8 were used to predict the
THSM assembly tool point FRFs for the carbide endmill with
different overhang lengths. Fig. 14 shows the measured and
predicted results. The percent errors between the predicted
natural frequencies and the measured values for the four dominant modes within the 5000 Hz measurement bandwitdh were
listed in Table 10 for three overhang lengths.
50
B. Bediz et al. / International Journal of Machine Tools & Manufacture 53 (2012) 39–50
8. Conclusion
This paper presented new modeling results for the threedimensional (3D) dynamic behavior of macro-scale milling tools
using the spectral-Tchebychev (ST) technique. The actual complex
cross-sectional geometry and the pretwisted shape of endmills
were taken into account during modeling. The bending and
torsional behavior of three endmills with known cross-section
was modeled and verified against both finite element models and
experiments (impact testing with free–free boundary conditions).
Model validation was also performed for a commercial carbide
endmill. The difference between the experiments and the spectral-Tchebychev method predictions was seen to be less than 3%
for all the four tools for the first six bending modes and first two
torsional/axial modes. The natural frequencies from the finite
elements model and the 3D-ST method were seen to match with
less than 1% difference.
To demonstrate the application of the modeling approach, the
3D-ST model of a commercial carbide endmill was coupled to the
Timoshenko beam model of a shrink fit holder and the measured
spindle receptances using receptance coupling substructure analysis (RCSA). The tool point measurements and predictions were
compared for three different overhang lengths. A flexible connection between the tool and the holder was implemented, where
the holder–tool interface stiffness values were determined using a
carbide blank and a genetic algorithm-based optimization technique. The stiffness values were used to predict the tool point
FRFs of other blanks with different overhang lengths, as well as
the endmill with various overhang lengths. The maximum error
between the natural frequency of the tool point measurement and
prediction was less than 1% for the carbide blank and less than 4%
for the endmill.
Acknowledgments
The authors gratefully acknowledge financial support from the
National Science Foundation (CMMI-0928393 and CMMI-0928211).
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