*Corr. Author’s Address: PSG College of Technology, Department of Mechanical Engineering, India, smr.mech@psgtech.ac.in 339
Strojniški vestnik - Journal of Mechanical Engineering 68(2022)5, 339-349 Received for review: 2022-01-07
© 2022 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2022-02-14
DOI:10.5545/sv-jme.2022.4 Original Scientific Paper Accepted for publication: 2022-03-01
Optimization of a Tuned Mass Damper Location  
for Enhanced Chatter Suppression in Thin-Wall Milling
Selvakumar, M. – Venugopa, P.R. – Velayudhan, G.
Mohanraj Selvakumar
*
 – Prabhu Raja Venugopal
 
– Gautham Velayudhan
PSG College of Technology, Department of Mechanical Engineering, India
In this paper, a method of optimizing the location of a tuned mass damper (TMD) for enhancing the chatter stability limits of a thin-walled 
workpiece by using numerical techniques is proposed with due consideration of the mass effect of TMD. The dominant mode of the workpiece 
is identified from the simulated and measured frequency response functions of the workpiece. Two TMDs each having a single degree of 
freedom are designed, and a finite element model to predict the response of the workpiece-damper system is developed. The effect of damper 
location on the chatter stability of the thin-walled workpiece is investigated. Furthermore, the locations for TMDs are optimized for enhanced 
chatter stability of the workpiece by using response surface methodology. Milling tests were performed on workpieces with and without TMDs. 
A three-fold improvement in the minimum stable depth of cut is realized after incorporating the TMDs at their optimal locations on the thin-
walled workpiece.
Keywords: chatter, thin-wall milling, tuned mass damper, response surface optimization
Highlights
• 	 The dynamics of a thin-wall workpiece is studied using numerical and experimental methods.
• 	 A single degree of freedom tuned mass damper is designed to suppress the dominant mode of the workpiece.
• 	 A numerical model is developed to investigate the effect of the location of the tuned mass damper on the chatter stability of the 
workpiece.
• 	 The location of the tuned mass damper is optimized to enhance the chatter stability of the workpiece by adopting the response 
surface method.
• 	 The milling test revealed a three-fold improvement in the critical depth of cut of the workpiece by introducing TMD.
0  INTRODUCTION
Regenerative chatter is a sort of self-excited vibration 
that occurs during milling and other machining 
processes. Chatter has several drawbacks, including 
poor surface finish, shorter tool life, reduced material 
removal rate, and increased machining noise. Chatter 
is usually caused by the most compliant element of 
the structural dynamic chain. Because workpiece 
is the most compliant part in thin-wall milling, 
investigations on the chatter stability of the thin-
walled part are gaining momentum. Typical thin-
walled parts used in aerospace structures are shown in 
Fig. 1.
Altintas et al. [1] proposed a frequency domain 
analytical solution for a dynamic milling model based 
on the zero-order approximation (ZOA) method to 
predict chatter stability. Budak et al. [2] proposed an 
analytical method for modelling varying workpiece 
dynamics and their effect on process stability.
Iglesias et al. [3] stated that stability lobe diagrams 
(SLDs) aid in the practical selection of the optimum 
cutting conditions determined either by time domain 
or frequency domain-based methods. Heng et al. [4] 
suggested a new tuned mass damper (TMD) that was 
designed and optimized using a sequential quadratic 
programming algorithm. Experiments showed that 
the critical depth of the cut had increased as observed 
from SLD. 
Fig. 1.  Typical thin-ribbed aerospace structures;  
a) frame, b) rib, c) impeller, d) blisk, e) sample part,  
f) bulkhead, and g) fuselage skin [5]
Fei et al. [6] proposed a moving damper for 
chatter suppression while milling flexible components 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)5, 339-349
340 Selvakumar, M. – Venugopa, P.R. – Velayudhan, G.
wherein the damper is supported on the back surface 
of the workpiece. Shui et al. [7] proposed a tuneable 
stiffness vibration absorber that could be tuned by 
changing the position of the slider. Sims [8] presented 
a new analytical solution for tuning methodology and 
observed that the critical depth of cut improved by 40 
% to 50 % compared to using classical TMD. Yang et 
al. [9] suggested two degrees of freedom (DOF) TMD 
that has both translational and rotational motions. 
The proposed method was found to reduce vibration 
amplitude by 81 % and double the critical depth of 
cut. Yang et al. [10] proposed a passive damper with 
tenable stiffness to accommodate variations in part 
dynamics during thin-wall milling, which reduced the 
surface roughness by 80 %. Bolsunovsky et al. [11] 
developed a TMD based on spindle frequency, which 
had a 20-fold mitigation of vibration amplitude. Wang 
[12] proposed a new non-linear TMD with a friction-
spring element that improved the critical depth of cut 
by 30 % as compared to the optimally tuned linear 
TMD.
Yang et al. [13] designed a two-DOF magnetic 
TMD to mitigate workpiece vibration in multiple 
modes. Researchers [14] and [15] proposed a passive 
damping solution using tuned viscoelastic dampers to 
mitigate the vibration of thin-wall castings focusing 
on change in coupled interaction between tool and 
workpiece due to the addition of tuned dampers. 
Yang et al. [16] investigated numerically the real part-
based tuning by employing the minimax numerical 
approach to maximize the minimum real part of 
the primary structure under harmonic excitation. 
They demonstrated the effectiveness by raising the 
minimum stability limit with single and multiple 
TMDs. Burtscher and Fleischer [17] proposed an 
innovative adaptive TMD with variable mass to 
suppress chatter in the machine tool. 
Despite many studies on the application of 
TMD for chatter suppression, the literature review 
reveals that no extensive studies on the effect of TMD 
mounting location on chatter stability of thin-wall 
workpieces are carried out. Hence, the present work 
addresses the above gap and focuses on proposing a 
generalized methodology for optimizing the location 
of TMD, taking into consideration the mass effect of 
TMD.
1  METHODOLOGY
In the present work, numerical and experimental 
methods are used to study the dynamics of a thin-wall 
workpiece. A TMD is designed based on the modal 
parameters of the workpiece corresponding to the 
dominant mode. A numerical model of the workpiece 
with TMD is developed to investigate the effect of 
the location of TMD on chatter stability and optimize 
the same by using the response surface optimization 
method.
1.1 Milling Dynamic Model
Precise dynamic modelling of the milling system 
plays a vital role in the design and analysis of TMDs. 
In thin-wall milling, the dynamic rigidity of the 
workpiece rather than the cutter is one of the critical 
parameters that decide the occurrence of chatter. 
Moreover, the dynamic rigidity of the workpiece in the 
normal direction to the cutter axis limits the process 
stability and material removal rate. Hence, a single 
degree of freedom milling dynamic model developed 
by considering only the dynamics of the workpiece to 
be sufficient to investigate chatter stability in thin wall 
milling.
Fig. 2.  Single degree of freedom milling dynamic model
A single degree of freedom milling dynamic 
model proposed by Budak
 
et al. [3], presented in 
Fig. 2, is adopted for the study. The model consists 
of a spring-mass-damper system, where m, k, and 
c respectively represent the modal mass, modal 
stiffness, and modal damping coefficient at a dominant 
mode of vibration of the workpiece. The dynamic chip 
thickness that modulates the cutting force is denoted 
as h(t). The governing equation of motion of the 
system shown in Fig. 1 can be written as 
 my cy ky Kb ht
s
     , (1)
where, K
s 
and b are the specific cutting resistance 
of the workpiece material and axial depth of cut, 
respectively. According to the regenerative chatter 
theory, the axial depth of cut at the stability limit of 
the milling process is given by

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)5, 339-349
341 Optimization of a Tuned Mass Damper Location for Enhanced Chatter Suppression in Thin-Wall Milling 
 b
NK Re i
lim
ty yt c







2
 
, (2)
where b
lim
 is the limiting depth of cut, N
t
 is the 
number of teeth in the cutter, α
yy
 is the directional 
coefficient in the y-direction K
t
 is tangential cutting 
force coefficient, Re[Λ( iω
c
)] is the negative real part 
of receptance (displacement per unit force) of the 
workpiece in the y-direction and ω
c
 is the chatter 
frequency. The directional dynamic milling coefficient 
in the y-direction, α
yy
 is given by
  


yy rr
cosK Ks in
s
e
 




1
2
22 2. (3)
In Eq. (3), θ is the tool engagement angle, K
r
 is the 
radial milling force coefficient, and ϕ
s
 and ϕ
e
 are the 
start and exit angles of the tool during the machining. 
The relation between the phase shift ( ϵ), the integer 
number of oscillations (n) between each tooth pass 
and the chatter frequency ( ω
c
) can be expressed as
 
60
2


c
n

 , (4)
where Ω is the spindle speed of the machine tool. 
Using Eqs. (2) and (4), the stability lobe diagram can 
be plotted for different values of n.
1.2  Prediction of Thin-Rib Dynamics
Modal parameters, namely the modal frequency, 
modal mass, and modal damping ratio corresponding 
to the dominant mode of the workpiece, play a 
significant role in the design of TMD for enhanced 
chatter stability. These parameters are determined 
with the help of the frequency response function of the 
workpiece obtained at its critical location. The critical 
location is the location on the workpiece where the 
vibration amplitude is maximal. Numerical modal 
analysis is used to identify the critical location from 
the mode shapes of the workpiece. A representative 
thin-wall workpiece (Fig. 3) made of Aluminium 6061 
alloy with the following properties is considered for 
the study: Young’s modulus was 68.9 GPa, Poisson’s 
ratio 0.33, and density 2700 kg/m
3
. The bottom face 
of the base plate is constrained in all directions and 
the analysis is performed using a commercial finite 
element code ANSYS Workbench
®
. A chatter range of 
0 Hz to 533 Hz is considered for the analysis, which is 
the tooth passing frequency range associated with end 
milling using four flute cutters with a spindle speed 
range of 0 rpm to 8000 rpm.
Fig. 3.  Thin-walled workpiece
Fig. 4.  Mode shapes of workpiece
Fig. 4 depicts the mode shapes of the workpiece 
obtained from the modal analysis, where the bending 
mode about the x-axis and twisting mode about the 
y-axis are observed at 266 Hz and 353 Hz, respectively. 
The mode shapes are symmetrical to the YZ plane 
and the locations L1 and L5 (Fig. 5) are identified 
as critical locations where relative amplitudes of 
vibration are maximal. Hence, due to the symmetric 
nature of mode shapes, L1 is considered to be the 
critical location for subsequent analysis.
Fig. 5.  Experimental setup for impact hammer test
An impact hammer test (Fig. 5) was performed 
on the workpiece at L1 to determine the dynamic 
parameters of the workpiece through frequency 
response function (FRF) measurements. An impact 
hammer (Make: Brüel & Kjær, sensitivity: 2.097 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)5, 339-349
342 Selvakumar, M. – Venugopa, P.R. – Velayudhan, G.
mV/N) was used to excite the workpiece at L1 with an 
impulse force of 5 N to 10 N applied in +Y direction 
and vibration response was recorded right behind L1 
using a Laser Doppler Vibrometer (Make: Polytec, 
model: NLV-2500, sensitivity: 500 μm/V). The signals 
were obtained using a DAQ system (Make: National 
Instruments, model: PXIe-1078) at a sampling rate of 
5 kHz and receptance was estimated using LabVIEW
®
 
software. 
Fig. 6.  Magnitude of receptance at location L1
The magnitude of receptance obtained from the 
test is depicted in Fig. 6, which shows two distinct 
peaks at 259 Hz and 346 Hz that correspond to the 
frequencies of 266 Hz and 353 Hz, respectively, as 
predicted by numerical analysis. A mode at 346 Hz 
is termed a dominant mode, which has a maximum 
magnitude of 1.72 mm/N. The damping ratio is 
identified as 0.0032 for the dominant mode using 
the half-power bandwidth method. The frequencies 
obtained by numerical analysis are in close agreement 
with experimentation with a maximum deviation of 
4.2 %, thereby validating the numerical model.
Fig. 7.  Excitation locations on the workpiece
The validated model is tuned with the 
experimentally obtained damping ratio to predict the 
receptances of the workpiece. It is evident from Eq. (2) 
that Re[Ʌ( iω
c
)] of the workpiece is a decisive factor 
for chatter stability in thin wall milling of a given 
workpiece material and tool combination. However, 
Re[Ʌ( iω
c
)] of the workpiece, in turn, depends on the 
location of excitation during machining. The harmonic 
response analysis is performed on the tuned model 
using ANSYS Workbench
®
 to predict the Re[Ʌ( iω
c
)] 
of the workpiece at critical locations. A force of unit 
magnitude is applied at L1 along the +Y direction, 
and the frequency of the applied force is swept 
between 300 Hz to 400 Hz to completely capture the 
Re[Ʌ( iω
c
)] of the dominant mode.
Fig. 8.  Comparison of real parts of receptance of L1
A comparison of the real part of receptances of 
the workpiece at L1 obtained with numerical and 
experimental methods is shown in Fig. 8. Re[Ʌ( iω
c
)] 
predicted using the numerical method is in close 
agreement with that obtained by experimentation 
with a frequency shift of 7 Hz corresponding to a 
2 % deviation in natural frequency. Furthermore, 
Re[Ʌ( iω
c
)] obtained with the numerical method is 
–0.9525 mm/N, which is in close agreement with 
the experimental result of –0.9902 mm/N with a 
maximum deviation of 4 %. 
1.3  Design of Tuned Mass Damper (TMD)
The modal stiffness (k) of the workpiece is calculated 
from the negative maximum of the imaginary part of 
receptance Im[Ʌ( iω
c
)] and damping ratio measured at 
L1 by the Eq. (5) [18].

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)5, 339-349
343 Optimization of a Tuned Mass Damper Location for Enhanced Chatter Suppression in Thin-Wall Milling 
 k
minI mi
c








1
2 
. (5)
The modal stiffness of the workpiece 
corresponding to Im[Ʌ( iω
c
)]  of –1.71 mm/N is 
calculated as 127.13 kN/m and the modal mass is 
found to be 0.0269 kg. A mass ratio ( μ) of 5 % [8] is 
used to design the TMD. The stiffness of the TMD is 
computed using Eq. (6) [18].
 f
f
f
r
d
w

 
 
 

22
21
2
2
, (6)
where, f
r 
, f
d
 and f
w
 are the frequency ratio, the natural 
frequency of the damper and the dominant frequency 
of the workpiece, respectively. The design parameters 
of TMD are calculated and are presented in Table 1.
Table 1.  Design parameters of TMD
Frequency ratio 1.036
Natural frequency [Hz] 358
Effective mass [kg] 0.0013
Stiffness [kN/m] 6.58
Fig. 9.  Proposed design of TMD
Fig. 9 shows a proposed design of TMD, which 
consists of a ring mass, guide pin, base plate, and 
a helical compression spring. The base plate of 
the TMD is made of an aluminium sheet of 3 mm 
thickness. Steel dowel pins of 2.5 mm and 25 mm long 
are used as guide pins. An open coil helical spring of 
mean coil diameter, wire diameter, free length, and 
number of active coils of 10 mm, 1 mm, 10 mm, and 
3 turns, respectively, is designed to obtain the required 
stiffness.
1.4  Effect of Location of TMD on Chatter Stability 
Some of the applications of TMD based on 
conventional approaches may not yield optimum 
performance in thin wall milling. Therefore, the 
location on the workpiece that results in an optimum 
mass ratio needs to be identified by considering the 
stationary component of TMD. 
A TMD having a single degree of freedom is 
designed based on the design parameters obtained in 
the previous section. The TMD is mounted on one 
of the locations on the workpiece defined by X and 
Y coordinates about the coordinate system, as shown 
in Fig. 10. A COMBIN14 element with longitudinal 
stiffness of 6.58 kN/m is used between the ring mass 
and the base plate of the TMD. A frictionless contact is 
applied between the ring mass and the guide pin, and 
the base plate of the TMD is attached to the workpiece 
using a bonded connection.
Fig. 10.  TMD and its location on the workpiece
Parameterized harmonic response analysis was 
carried out with coordinates of damper location (X, 
Y) as input parameters and the magnitude of the 
negative maximum of imaginary parts of receptance  
(Im[Ʌ( iω
c
)]) as an output parameter. Thirty-six 
mounting locations of TMD on the workpiece were 
considered for the investigation.
Custom design of experiments with 36 design 
points as the coordinate of the damper is employed for 
the analysis. The X-coordinate of the damper is varied 
from 0 mm to 260 mm in steps of 65 mm along the 
length of the plate. Similarly, the Y-coordinate of the 
damper is varied from 25 mm to 100 mm in steps of 25 
mm. Harmonic response analysis is performed on the 
workpiece for location L1 and the negative maximum 
of the real part of receptance of the (Re[Ʌ( iω
c
)]) is 
noted for all the thirty-six locations of TMD. 
Similar analyses are carried out for the other 
two excitation locations L2 and L3 and the results 
presented as response surfaces as shown in Fig. 
11, which portrays the effect of TMD location on 
the negative (Re[Ʌ( iω
c
)]) of the workpiece three 
excitation locations.
Since the workpiece is geometrically symmetrical 
about the YZ plane, which passes through L3, the 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)5, 339-349
344 Selvakumar, M. – Venugopa, P.R. – Velayudhan, G.
contours of |(Re[Ʌ( iω
c
)])| of L5 and L4 will be the 
mirror images of the contours of that of L1 and L2, 
respectively. As L3 is the node point for mode 2, it 
exhibits the least value for |(Re[Ʌ( iω
c
)])| in almost all 
the locations of the damper except for L1 and L5.
a) 
b) 
c) 
Fig. 11.  Effect of TMD location a) L1, b) L2 and c) L3 on 
(Re[Ʌ( iω
c
)])
It is inferred from Fig. 11 that the damper is 
ineffective for all values of Y when it is located at 
X = 130 mm and most effective near the location of 
excitation. Hence, the optimum location of TMD 
needs to be identified for effective chatter suppression.
1.5  Optimization of TMD Location for Enhanced Chatter 
Stability
The optimum location of the damper for enhancing 
the chatter stability of the workpiece is carried out 
by employing response surface optimization. The 
response surfaces obtained for  are optimized using 
genetic algorithms. The basic flow chart for carrying 
out parameterized harmonic analysis followed by 
optimization is shown in Fig. 12. The optimization 
is performed for each of the five excitation locations 
to minimize the  of the workpiece. Three candidate 
points that represent the optimal locations for the 
damper on the workpiece are obtained for each 
excitation location. 
It is inferred from Table 2 that the Y-coordinates 
for the candidate points of all excitation locations 
remain the same irrespective of X-coordinates. The 
average Y-coordinate of damper for minimizing  is 
found to be 98.6 mm, which is 98 % of the height 
of the part from the base. The  corresponding to the 
critical location L1 is found to be higher in comparison 
to other locations as expected. Considering the critical 
locations L1 and L5, it is evident from Table 2 that the 
coordinates, as well as the objective equation values, 
are quite close to each other for all the three candidate 
points. Therefore, the average X-coordinate of the 
damper for minimizing chatter is found to be 65 mm, 
which is at 25 % of the length of the workpiece from 
the respective excitation locations. Hence, the TMD is 
placed on the workpiece at (X, Y) = (65, 99) mm and 
harmonic response analysis is performed for each of 
the excitation locations of the workpiece.
Table 2.  Optimal locations of damper for each excitation location
Location
Candidate 
point
Coordinates [mm]
Objective equation 
values
X Y
Max (Re[Ʌ( iω
c
)])
[mm/N]
L1
1 65.89 98.79 -0.4691
2 63.89 98.45 -0.4692
3 61.77 98.83 -0.4688
L2
1 47.30 98.81 -0.1942
2 51.33 98.95 -0.1924
3 53.70 98.77 -0.1908
L3
1 62.48 98.63 -0.0107
2 195.62 98.69 -0.0106
3 76.62 96.22 -0.0089

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)5, 339-349
345 Optimization of a Tuned Mass Damper Location for Enhanced Chatter Suppression in Thin-Wall Milling 
A comparison of the real parts of receptance of 
the plate with and without TMD is shown in Fig. 13. 
It is inferred from Fig. 13 that negative (Re[Ʌ( iω
c
)]) 
of the workpiece at L1 is reduced from -0.9542 mm/N 
to -0.4687 mm/N, which corresponds to about a 51 % 
improvement in dynamic stability. Also, the TMD is 
found to be effective for the locations L1, L2, and L3 
and least effective for L4 and L5 when located at the 
optimum location for L1. This implies that the use of 
a single TMD is effective when the cutter traverses 
between L1 and L3 and is least effective between 
L3 and L5. Therefore, one more TMD with the same 
Fig. 12.  Flow chart for parameterized harmonic analysis and optimization
a)          b) 
Fig. 13.  Comparison of Re[Ʌ( iω
c
)] a) with and b) without TMDs

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)5, 339-349
346 Selvakumar, M. – Venugopa, P.R. – Velayudhan, G.
corresponding to L1 and L5, respectively. Initially, the 
impact hammer test was carried out with one TMD 
mounted at an optimum location and then with both 
the TMDs mounted at their optimum locations. 
Fig. 15.  Thin-walled workpiece with TMDs
a) 
b) 
Fig. 16.  Comparison of real part of receptances;  
a) location L1, and b) location L5
The comparison of Reobtained at critical 
locations L1 and L5 are shown in Figs. 16a and b, 
respectively. It is inferred from Figs. 16a and b that 
modal parameters was designed and located at the 
optimum location for L5 and the harmonic response 
analyses were repeated for L1 and L5.  
Fig. 14.  Real parts of receptance at L1 and L5
Fig. 14 depicts the comparison of the real part of 
receptance obtained at L1 and L5 of the workpiece 
without TMD, with one TMD and two TMDs. It is 
evident from the figure that the use of two TMDs 
considerably minimizes Re[Ʌ( iω
c
)] at L5 from 
0.7417 mm/N to 0.511 mm/N, which correspond to an 
improvement of 31 % when compared to the use of 
one TMD and 46 % without TMD. Even though the 
negative maximum values of Re of the workpiece at 
L1 and L5 are minimized using two TMDs, there is 
a drop in the corresponding chatter frequencies from 
353 Hz to 336 Hz.
2  EXPERIMENTAL VALIDATION  
OF OPTIMUM LOCATION OF TMD
2.1  Impact Hammer Test on Workpiece with TMDs
TMDs were fabricated as per the design parameters 
obtained in the previous section. Ring masses are 
prepared by stacking and gluing the brass washers 
of outer diameter, inner diameter and thickness of 16 
mm, 5 mm, and 0.3 mm, respectively. Each washer 
weighs about 4.75 g, and three washers are used to 
achieve a total mass of 2.75 g, which is slightly larger 
than the required mass of 2.69 g. The increased mass 
of the ring mass alters the mass ratio from 5 % to 5.2 
%, which is negligible and insignificant. The prepared 
TMDs were mounted on the workpiece at optimum 
locations obtained in the previous section. Fig. 15 
shows the workpiece with TMDs in which TMD1 
and TMD2 were mounted at the optimum locations 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)5, 339-349
347 Optimization of a Tuned Mass Damper Location for Enhanced Chatter Suppression in Thin-Wall Milling 
the negative real parts of receptance of the workpiece 
are minimized at both critical locations of the 
workpiece after the introduction of TMDs. When one 
TMD is located at the optimum location for L1, there 
is a marginal improvement in negative Re of L5 as 
observed as predicted by numerical analysis.
The marginal improvement in Re of L5 is 
attributed to the inertial effect of TMD located at L1. 
It is also revealed from Fig. 16b that a considerable 
improvement in the dynamic stability of the workpiece 
is observed on applying two TMDs at their optimum 
locations.
The Reof the workpiece at critical locations for 
the cases without TMD, with one TMD and with two 
TMDs are found to be about -0.9902 mm/N, -0.6747 
mm/N, and -0.3952 mm/N, respectively, which 
correspond to 32 % and 60 % improvements in the 
dynamic stability of the workpiece. 
2.2  Machining Test
The chatter stability of the undamped and damped 
workpiece was predicted and compared by using 
SLDs which were plotted (Fig. 17) by using the modal 
parameters and the identified cutting coefficients of 
the workpiece material, namely K
tc 
= 792.8 N/mm
2
, 
K
rc 
= 157.5 N/mm
2
, K
ac 
= 208.0 N/mm
2
, K
te 
= 22.9 
N/mm, K
re 
= 25.6 N/mm, and K
ae 
= 1.6 N/mm. A 
comparison of SLDs of the workpiece without TMD 
and that with two TMDs is depicted in Fig. 17. It is 
observed that the minimum stable depths of cut of the 
undamped and damped workpiece are in the order of 
0.08 mm and 0.24 mm, respectively, which correspond 
to a three-fold improvement in the chatter stability of 
the workpiece.
Fig. 17.  Stability lobe diagrams
The application of TMDs for rough milling may 
not be feasible as a large amount of material needs to 
be removed from both sides of the workpiece which 
hinders machining operation and greatly alters the 
structural dynamics. Also, the surface roughness is 
of least consideration in rough milling. However, 
chatter needs to be avoided during finish milling as it 
affects the surface quality of the machined part. The 
chatter stability during finish milling of thin walls 
can be improved by mounting the TMDs on a surface 
that is opposite to the surface of the thin wall being 
machined. 
Milling tests were carried out on a three-axis 
machining centre (Jyoti & Model: RDX20) shown in 
Fig. 18, using a carbide milling cutter of diameter 16 
mm with 4 flutes. The TMDs were mounted at their 
optimum locations on the surface of the thin wall just 
behind the surface being machined. Down milling was 
carried out on the damped and undamped parts on the 
part surface opposite to the surface where TMDs were 
placed. The test was conducted under similar cutting 
conditions to validate the damping characteristics of 
TMDs. Similar cutting parameters, namely a feed rate 
of 1200 m/min, spindle speed of 5000 rpm, axial depth 
of cut of 0.3 mm and radial depth of cut of 1 mm were 
selected from the SLDs. The above parameters were 
selected so that point P (Fig. 17) lies in the region of 
the SLD which results in unstable machining without 
TMDs. A uniaxial accelerometer (Dytran 3224A1 
with a sensitivity of 9.68 mV/g) was used to acquire 
vibration signals during milling. The time domain and 
frequency domain signals, along with the machined 
surface quality, with and without TMDs are shown in 
Fig. 19.
Fig. 18.  Experimental setup for machining tests
It is observed from the figure that the magnitude 
of vibration has reduced from 38 g to 18 g after 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)5, 339-349
348 Selvakumar, M. – Venugopa, P.R. – Velayudhan, G.
introducing TMDs. It can also be observed from the 
frequency spectrum that the spike at 340 Hz (chatter 
frequency of undamped workpiece) has vanished. 
Because of the above, the damped workpiece does 
not exhibit chatter marks, as shown in Fig. 19b. The 
setup for surface roughness measurement is shown 
in Fig. 20. The surface roughness of the machined 
workpieces was measured using a surface roughness 
tester (Make: Kosaka Lab, Model: SE-1200) with a 
diamond stylus of 5 µm radius at a measuring speed 
of 0.2 m/s.
a) 
b) 
Fig. 19.  Results of experimentation; a) vibration signals obtained 
during milling tests, and b) surface quality of the machined parts
Fig. 20.  Setup for surface roughness measurement
The average surface roughness (R
a
) of the 
undamped and damped workpieces are found to be 
4.24 μm and 2.63 μm, respectively, which correspond 
to an improvement in surface finish by 38 %.
3  CONCLUSIONS 
A novel methodology for optimizing the TMD 
location towards enhancing chatter suppression in 
thin-wall milling considering the mass effect of TMDs 
is presented. The optimization of TMD location 
is carried out based on a parameterized harmonic 
analysis aided with a response surface methodology. 
The effectiveness of the proposed methodology is 
assessed by plotting SLDs followed by a series of 
milling tests. The results reveal that chatter in thin-
wall milling can be suppressed to a great extent with 
the proposed approach. The following are the major 
conclusions drawn from the research work:
• The location of TMD is a crucial factor that 
influences the dynamic stability of the workpiece 
at critical mode.
• The optimal X- and Y-coordinates of TMD 
location for effective chatter suppression of the 
thin-walled workpiece with dominant mode 
as torsional mode are about 25 % of the length 
from either end, and 99 % of the height of the 
workpiece, respectively. 
• Use of a single TMD is found to be effective only 
for the first half of the cutter path whereas two 
TMDs are required to improve chatter stability 
while machining along the entire path. 
• Milling tests revealed that the application of 
TMDs yielded a 60 % improvement in the 
overall dynamic stability of the workpiece which 
corresponds to a three-fold improvement in 
productivity with an enhanced surface finish of 
38 %.
The above results portray the effectiveness of 
the proposed method. Even though the method is 
validated using a simple thin-walled part, it can also 
be applied to improve the chatter stability of complex 
thin-wall structures, such as turbine blades, impellers, 
and other aircraft and automobile parts, where one just 
needs to design and locate TMDs at their optimum 
locations.
4  NOMENCLATURES
m modal mass, [kg]
c modal damping coefficient, [Ns/m]
k
 
modal stiffness, [kN/m]
ζ modal damping ratio, [-]
f
r 
frequency ratio, [-]
f
d 
frequency of TMD, [Hz]
f
w 
frequency of workpiece, [Hz]
Λ frequency response function, [mm/N]
ω
c
 chatter frequency, [rad/s]

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)5, 339-349
349 Optimization of a Tuned Mass Damper Location for Enhanced Chatter Suppression in Thin-Wall Milling 
K
tc 
tangential cutting force coefficient, [N/mm
2
]
K
rc 
radial cutting force coefficient, [N/mm
2
]
K
ac 
axial cutting force coefficient, [N/mm
2
]
K
te 
tangential edge force coefficient, [N/mm
2
]
K
re 
radial edge force coefficient, [N/mm
2
]
K
ae 
axial edge force coefficient, [N/mm
2
]
5  ACKNOWLEDGEMENTS
The authors would like to express their sincere thanks 
to the management of PSG College of Technology, 
Coimbatore and the Office of Principal Scientific 
Advisor to GoI, New Delhi for providing the necessary 
facilities for carrying out this research work.
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