Acta Polytechnica


doi:10.14311/AP.2019.59.0283
Acta Polytechnica 59(3):283–291, 2019 © Czech Technical University in Prague, 2019

available online at http://ojs.cvut.cz/ojs/index.php/ap

DEFLECTIONS AND FREQUENCY ANALYSIS IN THE MILLING
OF THIN-WALLED PARTS WITH VARIABLE LOW STIFFNESS

Serhii Kononenko, Sergey Dobrotvorskiy, Yevheniia Basova∗,
Magomediemin Gasanov, Ludmila Dobrovolska

National Technical University “Kharkiv Polytechnic Institute”, Kyrpychova str. 2, Kharkiv, Ukraine
∗ corresponding author: e.v.basova.khpi@gmail.com

Abstract. The disadvantage of the geometry of thin-walled parts, in terms of processing, is the
low ability to resist static and dynamic loads. It is caused by the elastic deformation of elements
with a low stiffness. Modelling approaches for the evaluation of deflections during machining are
presented. Mathematical models of deflections, cutting forces and harmonic response are proposed.
The processes of material removal and deflection of a thin-walled sample at the critical points are
modelled. A frequency analysis was performed, consisting of a modal analysis of natural frequencies
and a harmonic response analysis. As a result, a graph of the deflections amplitude from the frequency
of driven harmonic oscillations is generated. The analysis of the obtained values was performed. As a
result, the resonance frequency and maximum amplitude of oscillations for the operating parameters
are determined.

Keywords: Thin-walled parts, undesirable deflections, end milling, modal analysis, harmonic response
analysis.

1. Introduction
In several branches of the industry, such as mechanical
engineering, aircraft manufacturing, drive production,
etc., engineering products that belong to a number
of thin-walled parts with variable stiffness are par-
ticularly prevalent. The processing of such parts on
metal-cutting machines generally requires the appli-
cation of both the specially developed devices that
prevent the parts from being deformed by cutting
forces and fixing or using new mechanical treatment
techniques. The creation and using of special devices
is associated with additional costs and, as a result,
with an increase in the cost of production. The tran-
sition to an adoption of contemporary mechanical
treatment methods such as high-speed milling reduces
the number of special technological equipment. How-
ever, this mechanical treatment technique needs to
be studied as well and the positive impact of reduced
cutting forces influence on the thing-walled element
should be approved. The aim of the research is to de-
velop the technological solutions to reduce undesirable
deviations of the geometry of thin-walled elements of
parts by high-speed end milling.

2. Literature review
Parts, such as turbine blades, impellers, and simi-
lar that have thin-walled elements in their geometry,
are indispensable in the automobile, aerospace and
other industries [1–4]. Such responsible parts of the
mechanisms are used in a variety of drive units [5].

The formation of surfaces of thin-walled parts with
variable low stiffness requires a consideration of a
number of factors that could prevent the achievement

of technological requirements for the product [6, 7].
Such factors are: deviations from a given shape that
increases during processing, vibrations, thermal defor-
mations and errors caused by tools, tool-path accuracy,
technological equipment [8, 9] and fixtures [10, 11].

In modern industry and research activity, methods
for recording and analysing the parameters of the cut-
ting process for thin-walled parts are being actively
developed. One of the methods based on a prediction
of cutting parameters is discrete-time modelling of
dynamic milling systems. The discrete-time model is
general, and it can be used simultaneously for predict-
ing the stability as well as the time response of the
milling system [12].

An experimental based study provides a systematic
measurement of process response parameters, videlicet
the cutting force and surface roughness [13], dynamic
stiffness measuring methods [14]. In addition, a spe-
cial attention is paid to the system identification,
signal, modal analysis, vibration absorption [15–18]
using the frame structures as a test rig for numerical
identification techniques.

Another researches represent a methods based on a
variable correction designed to compensate the deflec-
tion effect of a non-rigid construction of blades [19]
and technological and other factors of the accuracy
characteristics of the blades processing. The correc-
tion values depend on the contact area of the tool
with the workpiece. The converted parametric view
of the machining program makes it possible to include
compensating values to the machine manually by the
operator. In the process, the operator checks the cor-
rectness of the processing. In the case of incorrect

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S. Kononenko, S. Dobrotvorskiy, Ye. Basova et al. Acta Polytechnica

values of correction, the program should be changed
and processing starts again [20].
A research based on a simulation of the distortion

due to machining of thin-walled components states
that the distortion of components is strongly related
to the residual stress state induced by manufacturing
processes like heat treatment, forming or machining
[21].
As a result of the review, it was decided to focus

on the forces in the process of the material removal,
FE end-milling modelling and oscillations analysis
to determine the undesirable deflections degree. For
basic estimation and further validation, a simplified
model of a part is considered as a cantilever beam
[22, 23] that corresponds to an earlier study [24] where
it was clarified that the maximum of deviation occurs
on the ending free point of the sample.

3. Research methodology
The technological solution for determining the values
of undesirable deflections, in this paper, is the pre-
liminary calculation of the forces; a comparison of
the obtained values with the automated finite element
calculation; insertion of the directional milling force
into the deflection model; obtaining the deflections
amplitude through considering the milling tool as a
driven harmonic oscillator.

3.1. Deflection model
During the processing of the thin-walled element by
the end-mill, undesirable deviations occur in the direc-
tion of the force of the cutter pressure on the surface.
An additional complexity of calculating deflections is
the variable distribution of stiffness in each section.
It is caused by the geometry of a thin-walled element,
similar to an impeller blade. The considered model
with a larger thickness at the base and tapering to-
wards the end is presented in Fig 1. The model is
shear, undeformable and represents the static force
interaction. It is used for a further rough deflections
value validation.

The basis for the equation of the deflection model
is the Castigliano’s second theorem, which states that
the displacement ∆ of the point of application of the
generalized force is equal to the partial derivative of
the complementary strain energy Uc with respect to
this generalized force F:

∆ =
∂Uc
∂F

(1)

The complementary strain energy of bending:

Uc =
∫ L

0

M2

2EI
dx (2)

where Young’s modulus E is a measure of the material
stiffness [25]. The bending moment M and moment
of inertia I are functions of x:

M = F · x (3)

Figure 1. Model of a sample under the directional
force.

I =
b · (y0 + αx)3

12
(4)

where b is a beam width.
The increasing thickness measure α is represented

as (y1 − y0)/L. Combining the expressions into the
complementary strain energy of bending (2):

Uc =
6 · F 2

Eb

∫ L
0

x2

(y0 + αx)3
dx (5)

Combining the expressions into the Castigliano’s equa-
tion (1), the deflection:

∆ =
∂Uc
∂F

=
12 · F
Eb

∫ L
0

x2

(y0 + αx)3
dx (6)

As a result of calculations, where a Young’s modulus
for aluminium E = 69 GPa, y1 = 9.75 mm, y0 = 4.75
mm, b = 40 mm, L = 70 mm, F = 184 N - obtained
from the section 3.2, the deflection ∆ = 0.1642 mm.

3.2. Directional cutting force model
To determine the forces acting in the cross section of
the sample, as the most influencing factors of undesir-
able deflections, it is necessary to define the compo-
nents of the cutting forces Fig. 2.

The scheme of the cutting forces during the milling
depends on the machining method and the type of
the cutter. For the processing of thin-walled parts,
such profiles of blades, end mills are used. The fin-
ishing climb milling of an aluminium alloy sample is
presented. Climb milling provides a surface with a
lower roughness and a higher accuracy. The tooth
cuts into the material at a point A Fig. 2, causing a
number of forces directed towards the surface. But, in
the case of thin-walled elements that are not deprived
of the freedom degree from the back side, the effect
of deflection by the cutter of an element leads to the
processing errors.

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vol. 59 no. 3/2019 Deflections and Frequency Analysis in the Milling. . .

Figure 2. Components of cutting forces in the end
milling.

In accordance with the reference book [26], the
total tangential force Fz and the radial force Fy have
a resultant Fyz, which can be decomposed into two
forces - the lengthwise Fh and the transverse force Fv.
In the case of the processing model, the lengthwise
force Fh is directed along the line of the material
removal. The force Fv is directed perpendicularly to
the force Fh, in the direction of the supposed blade
sample deflection.
To determine the transverse force Fv, which is re-

sponsible for the occurrence of deviations, the total
tangential force Fz is needed:

Fz =
10 · Cp · tx · Syz · Bn · z

Dq · nw
· KMP (7)

where Cp is the coefficient that takes into account the
physical and mechanical properties of the material,
tabulated value; x, y, w, q - exponents that depend
on the type of machining, the material of the part and
the material of the cutting tool; t - depth of cutting,
mm; Sz - feed, mm/rev; B - cutting width, mm; z
- number of cutter teeth; D - diameter of the mill,
mm; n - rotational speed, min−1; KMP - the general
correction factor, which takes into account the quality
of the processed material [26].
As a result of calculations, where a cutting depth

t = 0.25 mm, cutting width B = 3 mm, diameter of
the contact area of the end spiral three-blade mill D
= 7.5 mm, and the material of the part - aluminium
alloy, the value of the tangential force is Fz = 263.2 N.
Forces Fz and Fv are in the ratio [26], as Fz:Fv =
0.7-0.9. Based on the ratio, the calculated value of the
transverse force is Fv = 184 N. The values obtained
are necessary for a further comparison with the finite
element analysis.

3.3. Harmonic response model
In classical mechanics, a harmonic oscillator is a sys-
tem that, when displaced from its equilibrium position,
experiences a restoring force F proportional to the

displacement x:

F = −k · x − b · v (8)

where, k is a positive real number, characteristic of the
spring; x - amount of the displacement; b - is a constant
that depends on the properties of the environment
and the dimensions of the object; v - is the velocity
of the object.

Driven harmonic oscillators are damped oscillators
further affected by an externally applied force F(t).
Let us assume a driving force is F = F0 · cos(ωextt)
[27], then the totals force:

F = F0 · cos(ωextt) − k · x − b · v (9)

The equation of motion, F = m · a, becomes:

m · d2x
dt2

= F0 · cos(ωextt) − k · x −
bdx

dt
(10)

After a steady state has been reached, the position
varies as a function of time:

x(t) = A · cos(ω · t + ϕ) (11)

where, ω = ωext is the angular frequency of the driving
force. The amplitude of oscillation:

A =
F0

m · ((ω20 − ω2)2 + (
b·ω
m

)2)1/2
(12)

where, ω20 = k/m. ω0 is the natural frequency of
the undamped oscillator. When the frequency of the
driving force is close to the natural frequency and the
drag force is small, then the denominator in the above
expression becomes very small and the amplitude
becomes very large. This increase in amplitude is
called resonance, ω = ω0 is resonance frequency.
Due to the geometrical complexity of the sample,

a calculation for a slab of a similar cross sectional
area and stiffness is performed. The size of the slab:
70 mm long, 40 mm wide, 8.25 mm thick, and stiffness
k = 1.14 · 106. According to the section 3.1, the
stiffness of the model is k = F/∆ = 1.12 · 106. To
show the response of the slab to the force in the graph,
the Duhamel’s integral is used:

u(F,t) =
1

mω0

∫ t
0
Fτe

−h(t−τ)sin(ω(t − τ))dτ (13)

The graph on the Fig. 3 displays the dependency of
the oscillation amplitude on time for the functions of
the force for resonance Fr(t) and for normal conditions
Fn(t). It is assumed that, under normal conditions,
the frequency fn = 35 Hz corresponds to the gener-
ated oscillations during a conventional machining at a
750 rpm by the three blade end-mill. Based on the cal-
culation results, the amplitude of the oscillation An =
0.2247 mm, the resonance amplitude Ar = 4.023 mm.

4. Results
Results are based on a finite element analysis. The
values obtained as a result of previously manual cal-
culated models and automated are compared.

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S. Kononenko, S. Dobrotvorskiy, Ye. Basova et al. Acta Polytechnica

Figure 3. The amplitudes of oscillations at resonance
u(Fr , t) and normal conditions u(Fn, t).

Figure 4. Model of analysis and forces probes.

4.1. End milling analysis
CAD models of a helical conical three-blade end mill
and a thin-walled sample element with a variable dis-
tribution of stiffness have been designed and imported
into CAE program. Based on the model defined in
the section 3.1, the length L of the sample is 70 mm.
Width b is 40 mm. The thickest y1 and the thinnest y0
regions of the middle section of the part are 9.75 mm
and 4.75 mm respectively. The thickness at the top
edges (point 1, point 3) Fig. 5 is 3.75 mm.
The material, contact surfaces of the sample and

cutters, boundary surfaces and a mesh for the finite
element calculation are set. The estimated allowance
for the finishing process is in the range of up to 500
µm. Therefore, the density of the mesh of the sample
is much smaller than that, and equals 0.5 mm.
The initial position of the milling cutter and the

sample is calculated at the CAD design step. A par-
ticular attention should be paid to the position of
the sample and the tool, both relative to each other,
and relative to the coordinate system. In the CAE
environment, both elements are tied to the global co-
ordinate system. A cylindrical coordinate system is
additionally assigned to the milling cutter. Having the
specified parameters above, the dynamic parameters
are set: axial rotation of the milling cutter and axial
movement of the part on the machining length.

One of the main parameters of interest in this article
are the forces acting on the thin-walled element in the
transverse direction. The simulated force Fv.fea can
be obtained by taking stress probes from the back of
the sample along the cutting area, as it can be seen

Figure 5. Set of critical points.

Critical points Deflection values, mm
1 0.17984
2 0.15186
3 0.17984
4 0.06180
5 0.05319
6 0.06180

Table 1. Deflections estimation.

in Fig. 4. It is necessary for the comparison with the
calculated force Fv.calc = 184 N.
Analysing the obtained values, it can clearly be

seen that the finite element analysis allows getting
the approximate values of forces.
However, it is necessary to take into account the

difference in values, depending on the location of the
probes, despite the constant cutting parameters. That
may be caused by inaccuracies in the context of this
type of analysis. The error of the applied method is
12 % on average.

Considering six points on the surface Fig. 5, five of
them are critical - they are at the edges of the sample
and far from the fixation site. Consequently, at these
points, the maximum deviation from the initial state
is presumably observed.
The point 5 is the test point where the lowest de-

flection is expected. The location of the points ap-
proximately matches with the lines of the end milling.

By loading the critical points by forces in sequence,
it is possible to estimate the magnitude of the devi-
ations in each region. The direction of the forces is
set along the Y axis of the local coordinate systems.
The Y axis is perpendicular to the sample surface.
The values of the deviations picked at different levels
are different; it is caused by variable a distribution of
stiffness of the sample [28, 29].

The values obtained by the analysis allow to clearly
evaluate the degree of the maximum deflection values
of the sample in different regions Fig. 6. The results
of the analysis are listed in Tab. 1. The maximum

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vol. 59 no. 3/2019 Deflections and Frequency Analysis in the Milling. . .

Figure 6. Sample deflection at point 1 and 2 by 0.17984 mm and 0.15186 mm respectively.

deflection value for the sample thickness 3.75 mm at
points 1 and 3 is 0.17984 mm.

At the section 3.1, the deflection ∆ = 0.1642 mm is
obtained. That simplified model geometrically repre-
sents the force interaction at the level of the second
point of the sample where the thickness is 4.75 mm.
The deflection at this point according to the FE model
is 0.15186 mm, which is quite similar. The different
calculation methods allow to be sure that in the pro-
cess of building of the FE model no errors were made.
Also, the simplified model allows to roughly estimate
what value is expected as a result of the FE analysis.

The FE analysis in the CAE software gives more
possibilities due to working with a complex 3D model
thus the more accurate calculations can be performed.

4.2. Modal analysis
Calculating of the natural frequency of the structure
should especially be analysed if there are tendencies
to the occurrence of oscillations in the process of treat-
ment. This calculation makes it possible to define the
Harmonic Response oscillations further, i.e. calculate
the maximum amplitude of oscillation.
The interaction of a rotating cutter with a certain

frequency and a thin-walled element can be repre-
sented as a mechanical system with cyclic loads. The
response of a system to a dynamic loading can be
among the following [30]:

(1.) the intensity of the response can converge;
(2.) the system can constantly oscillate;
(3.) the intensity of the system can diverge.

In all cases, the 3-rd type of the response should be
avoided since the system could collapse.

If the excitation load has a frequency which becomes
close to the natural frequency of the system, the os-
cillations increase, and if the frequencies coincide, a
resonance phenomenon occurs with subsequent nega-
tive consequences. Therefore, it is critical to be able
to calculate the natural frequency of the system. For
this purpose, the finite-element analysis of frequencies
and oscillations is presented.

This analysis calculates the natural frequency based
on the geometry and material of the sample, and
allows to determine the resonant frequency.
The modal analysis settings are: Max Modes to

Find - 6 (each of them has a tendency to oscillate in
different directions), Fixed Support - the base surface
of the model. The result is shown in Fig. 7, the output
listed in Tab. 2.
The first mode - 1728.4 Hz is the most significant,

since at this frequency, the sample is going to oscil-
late in the same direction as during the lengthwise
processing.

4.3. Harmonic Response Analysis
Having a modal analysis, the Harmonic Response
analysis can be performed. Any sustained cyclic load
produces a sustained cyclic response (a harmonic re-
sponse) in a structural system. The harmonic response
analysis gives the ability to predict the sustained dy-
namic behaviour of structures, thus enabling to verify
whether or not the designs will successfully overcome
the resonance, fatigue, and other harmful effects of
forced vibrations [31].

Harmonic Response analysis settings are: previously
defined Modal analysis, Fixed Support - base surface
of the model, analyzed frequency is in the range from
0 to 2000 Hz, input Nodal Force load at the critical

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S. Kononenko, S. Dobrotvorskiy, Ye. Basova et al. Acta Polytechnica

Figure 7. Oscillations under different natural frequencies.

Mode Frequency, Hz
1 1728.4
2 5137.1
3 5839.5
4 7603.1
5 13097
6 17771

Table 2. Natural frequency analysis output.

Figure 8. Frequency Response location.

point (thin end of the sample) - 184 N. The Frequency
Response is taken from the critical point 3 location
Fig. 8.
The graphs of Oscillation Amplitude and Phase

Angle are presented on the Figure 9; values for each
frequency and corresponding amplitude, phase angle
are presented in the Tabl. 3.
The amplitude is the maximum value of the os-

cillation of the sample under the specified load and
corresponding frequency. The phase angle is a mea-
sure of the time by which the load lags (or leads) a
frame of reference.

The range from 0 to 2000 Hz is significant to analyse
as the natural frequency lies in this range - 1728.4Hz.
If the frequency of the cyclic load matches with the
natural frequency, the oscillations increase and a reso-
nance occurs.
In confirmation, from the data received Fig. 9,

Tab. 3, a large increase of the amplitude of the sample
oscillations - 2.7934 mm at the frequency - 1720 Hz

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Figure 9. Oscillation Amplitude of the deflection of the sample and Phase Angle.

Frequency, Hz Amplitude, mm Phase Angle, deg.
40 0.12408 -0.0503
120 0.12458 -0.1516
250 0.12649 -0.32113
500 0.13455 -0.68722
750 0.15072 -1.1669
1000 0.18183 -1.9077
1250 0.24956 -3.3521
1500 0.4692 -7.8414
1720 2.7934 -76.045
1875 0.61746 -165.95
2000 0.32309 -171.92

Table 3. Shortened list of Frequencies and corresponding Amplitudes and Phase Angles.

could be noticed. It shows that the system starts to
resonate with the frequency of the input loading.

The expected rotation of the milling tool for a con-
ventional machining is near 750 rpm. It corresponds
to the frequency 12.5 Hz. The milling tool has three
blades, so expected frequency of the contact of the
blades with the surface is tripled. Thus the frequency
of interest is near 37.5 Hz, which is lower than the
natural frequency, so milling processing can be safely
performed. Finally, according to the Table 3, the max-
imum oscillation amplitude of the sample at 40 Hz is
0.12408 mm. The same way, during the high speed
machining at 15 000 rpm the frequency of interest is
750 Hz, and the deflection is 0.15072 respectively.

As a result of the calculations in the section 3.3, the
osculation amplitude for the conventional machining
is 0.2247 mm, whereas according to the FE analysis,

it is 0.12408 mm for the conventional machining and
0.15072 mm for a high-speed one. The difference is
expected since, unlike the simplified model, the FE
model has a thick base thus it has a higher natural
frequency so it is more stable to oscillations.
The difference in results shows the disadvantage

of the simplified model calculation in the section 3.3
in contrast with the finite element simulation. The
complexity of the manual mathematical description
of the geometry imposes restrictions on the accuracy
of calculations for the specific model.

5. Discussion
In terms of processing of thin-walled parts, Harmonic
Response analysis brings the perspective to estimate
deflections at different frequencies. Modern high-
speed machining equipment allows the flexibility to

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S. Kononenko, S. Dobrotvorskiy, Ye. Basova et al. Acta Polytechnica

choose the most efficient processing speed. Negative
factors, such as large deviations or even resonance,
can be avoided by increasing or decreasing the number
of tool’s teeth and revolutions of the cutting tool.

6. Conclusion
The static and dynamic methods of loading of the thin-
walled sample with variable low stiffness are presented.
A deflection evaluation is performed. An analysis
of a sample is made, as a result, the cutting force
components are defined; the relative transverse force
Fv.calc is 184 N, which is similar to Fv.fea. The error
of the applied method is 12 % on average.

The modelling of the deflections is implemented. As
a result of a static analysis, the maximum deviation
at the thinnest region of the part is 0.17984 mm. As a
result of the frequency analysis, the maximum oscilla-
tion amplitude of the sample at a normal processing
frequency 40 Hz is 0.12408 mm and at a high-speed
one - 750 Hz, it is 0.15072 mm. In a further research,
the presented methods are considered for creating pro-
cessing programs that take into account the geometry
of the part and compensate most of the undesirable
deflections.

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	Acta Polytechnica 59(3):283–291, 2019
	1 Introduction
	2 Literature review
	3 Research methodology
	3.1 Deflection model
	3.2 Directional cutting force model
	3.3 Harmonic response model

	4 Results
	4.1 End milling analysis
	4.2 Modal analysis
	4.3 Harmonic Response Analysis

	5 Discussion
	6 Conclusion
	References