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Monitoring-based performance analysis of tuned mass dampers in tower structures

Kosmas Dragos1,*, Maria Steiner1, Ina Reichert2, Volkmar Zabel2, and Kay Smarsly1

Abstract: Slender structures, such as towers and masts, frequently exhibit unfavorable responses

when subjected to dynamic loads, characterized by the non-negligible contribution of higher modes

of vibration as well as resonant phenomena between the loads and the structure. To mitigate such

responses, tuned mass dampers have been employed in several slender structures. A tuned mass

damper (TMD) is a structural sub-system fixed to a structure and “tuned” to counteract a response

frequency, which is anticipated to induce large oscillations to the structure, through its out-of-phase

response. The design of tuned mass dampers is based on modeling assumptions and on sparse obser-

vations of the dynamic behavior of structures, which changes due to structural ageing, thus rendering

the regular performance assessment of TMDs essential. In this paper, a methodology for assessing

TMD performance using structural health monitoring (SHM) data is presented. Specifically, the

TMD functionality is assessed upon obtaining the phase shift between the TMD response and the

response of the structure, which, according to TMD theory, indicates favorable dampening effect

when assuming values close to 90°. The proposed methodology is validated via a case study of a

radio tower equipped with a TMD and an SHM system, showcasing the ability of the methodology to

assess TMD performance.

Keywords: Tuned mass dampers; slender structures; structural health monitoring

1Bauhaus University Weimar, Chair of Computing in Civil Engineering, Coudraystraße 7,

99423 Weimar, kosmas.dragos@uni-weimar.de 2Bauhaus University Weimar, Institute of Structural Mechanics, Marienstraße 7A, 99423 Weimar

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1 Introduction

Standard practice in designing structures against dynamic loads typically dictates the adoption of

regular structural systems with even distributions of mass and stiffness as well as low aspect ratios.

The aim of this design practice is to reduce the dynamic structural response to a combination of a

few vibration modes, hence avoiding complex structural dynamic responses. However, the nature of

civil infrastructure, operational and space constraints, or even aesthetical requirements, often result

in employing irregular structural systems, whose structural behavior under dynamic loads is com-

plex. Specifically, the dynamic structural response of slender structures, such as towers and masts

with high height-to-length (or height-to-width) ratios, is characterized by the non-negligible contri-

bution of higher-order vibration modes. In addition, following the increasing application of high-

strength state-of-the-art materials in construction in the last decades, there has been a growing trend

towards reducing the dimensions of structural member sections, thus resulting in lightweight struc-

tures. The effect of reducing the structural mass is detrimental to the intrinsic viscous damping prop-

erties of structures, which, particularly in tower structures, exacerbates dynamic structural responses

to resonant phenomena, such as the “lock-in” effect [1]. Mitigating such unfavorable behavior in

tower structures is essential to ensure the structural integrity and the serviceability of structures.

To counteract excessive oscillations under dynamic loads, the damping of slender structures is artifi-

cially enhanced by incorporating structural subsystems, such as viscous dampers. Representing a

special case of structural subsystems designed to dampen the frequency components dominating the

dynamic structural response (i.e. the dominant vibration modes), tuned mass dampers have been

widely adopted as vibration control devices in slender structures. A tuned mass damper (TMD) is

“tuned” to split an initial modal frequency of the structure, which is anticipated to cause unfavorable

oscillations, to two frequencies (one lower and one higher than the initial modal frequency). In the

new frequencies, the TMD responds out-of-phase with the structure and, thus, reduces the corre-

sponding oscillation amplitudes of the structure. Tuned mass dampers are either included in design

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or installed in existing structures (usually following unfavorable oscillation incidents), based on the

dynamic properties of structures. The dynamic properties are estimated through structural health

monitoring (SHM), which has evolved to a reliable and cost-efficient field of non-destructive testing

[2, 3]. SHM has been increasingly employed for assessing the structural condition and for gaining

insights into structural properties, such as dynamic properties [4,5]. According to SHM practice,

structural response data, such as accelerations, displacements and strains, collected from SHM sys-

tems is processed via algorithms, either embedded into sensor nodes or offline on centralized servers,

implementing system identification methods for yielding structural properties [6-8]. Since the dy-

namic properties of structures may vary due to ageing and/or structural degradation, assessing the

performance of tuned mass dampers using information, e.g. from SHM systems, is necessary to

adapt the tuning parameters to the current structural state.

The behavior of tuned mass dampers has been the subject of extensive research. The selection of

optimal TMD parameters has been first proposed by Den Hartog (1956) [9]; building on Den Har-

tog’s approach, studies on the design of tuned mass dampers have been conducted by Fujino and Abé

(1993) and Rana and Soong (1998) [10, 11]. TMD applications have been reported in several engi-

neering fields, such as in earthquake engineering (Sadek et al., 1997) [12]. For assessing the perfor-

mance of semi-active tuned mass dampers, Pinkaew and Fujino (2001) and Demetriou et al. (2016)

have proposed using numerical simulations [13, 14], while the combination of numerical simulations

with field data and wind tunnel tests for assessing the efficiency of a passive TMD has been present-

ed by Tuan and Shang (2014) [15]. Performance assessment for tuned mass dampers under wind-

induced vibrations has been conducted by Kwok and Samali (1995) [16], and the behavior of tuned

mass dampers under seismic ground motions has been studied by Matta (2013) [17]. For the long-

term assessment of TMD performance, Weber and Feltrin (2010) have used system identification

methods [18]. TMD performance indicators based on frequency response amplitude ratios between

the structure and the TMD have been suggested by Hazra and Sadhu (2013) [19]. Finally, in their

study of the dynamic behavior of a footbridge in Portugal, Caetano et al. (2010) demonstrated the

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activation level of a TMD through monitoring of the damping force variation and of the attenuation

of structural responses [20].

Despite the extensive research on assessing the performance of tuned mass dampers, utilizing long-

term structural health monitoring data for investigating the functionality of tuned mass dampers has

not received adequate attention. In this paper, a methodology for investigating the performance of

tuned mass dampers is presented. Considering a structure equipped with a TMD and an SHM sys-

tem, and drawing from the fundamental principles of TMD theory, the favorable effect of the TMD

on the dynamic behavior of the structure is highlighted through a phase shift between the TMD and

the structure, which results in reduced oscillation amplitudes. The TMD assessment methodology is

validated through a case study of a radio tower equipped with a TMD and a long-term SHM system.

This paper is organized as follows: First, the fundamental principles of TMD theory are discussed.

Next, the methodology for assessing the performance of tuned mass dampers is illuminated via simu-

lations on a simple cantilever structure. Then, the case study for validating the TMD assessment

methodology is presented. The paper finally concludes with a summary of the methodology and an

outlook on future research.

2 Fundamentals of tuned mass damper theory

The following discussion concerns the fundamentals of tuned mass damper theory, as described by

Connor (2002) through a simplified example [21], which constitutes the mathematical background of

the TMD assessment methodology. Considering a simple single-degree-of-freedom (SDOF) oscilla-

tor, consisting of a main mass m, a spring with stiffness k and a dashpot representing the intrinsic

damping c of the oscillator, a TMD is simply an additional mass md connected to the main mass, with

its own stiffness kd and its own damping cd, as shown in Figure 1. md connected to the main mass,

with its own stiffness kd and its own damping cd, as shown in Figure 1.

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Fig. 1: Single degree-of-freedom system with a tuned mass damper attached.

Assuming that the main mass is subjected into harmonic excitation P(t) with amplitude po, the equa-

tions of dynamic equilibrium for the SDOF system with the TMD as well as of the dynamic load are

given in Eq. 1.

tptP

xkxcxxm

xxmpxkxcxm

o

dddddd

dd

sin)(

0)(

(1)

According to [21], the near-optimal case for selecting the TMD parameters corresponds to

m

k

m

k

d

dd (2)

In Eq. 2, ω is the natural frequency of the SDOF system and ωd is the natural frequency of the TMD.

The structural response for the unfavorable case of resonance (ω = Ω) is obtained from the theory of

forced vibration, as shown in Eq. 3.

tAx

tAx

dd sin

sin (3)

In Eq. 3, φ is the phase angle of the response of the main mass, θ is the phase difference between the

TMD and main mass, while A and Ad are the displacement response amplitudes of the main mass and

Mainmass

TMD

k

c m

kd

cd md

P(t)

x x+xd

6

the TMD, respectively. For harmonic excitation, φ and θ are obtained from the expressions of Eq. 4.

2arctan

2

12arctan

ddm

m

(4)

In Eq. 4, ξ and ξd are the ratios of critical damping for the main mass and the TMD, respectively.

From Eq. 4, it is evident that the oscillation mitigation effect of the TMD is expressed through a

phase difference of π/2 (90°), which results in the reduction of the response amplitude A. In multi-

degree-of-freedom (MDOF) systems, the TMD analysis and design focuses on one mode of vibra-

tion; once the target mode of vibration n is selected, the corresponding generalized mass mn, general-

ized stiffness kn and generalized damping cn are calculated and the same procedure as in the SDOF

oscillator is followed.

3 A methodology for assessing the performance of tuned mass dampers

In this section, the methodology for assessing the performance of tuned mass dampers using struc-

tural response data from an SHM system is presented. The proposed methodology is based on the

expected phase shift between the response of the damper and the response of the structure. In the

following subsections, the mathematical basis of the TMD assessment methodology is formulated,

and a simulation example of a 5-DOF oscillator equipped with a TMD is presented to illuminate the

methodology concept.

3.1 Mathematical basis

First, acceleration response data collected from both the TMD and the structure are integrated to

obtain the corresponding displacement response data. For integrating, the Newmark-β algorithm is

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employed in this study (Eq. 5) [22].

12

1

11

221Δ2

1Δ

ΔΔ1

nnnnn

nnnn

xxttxxx

xtxtxx

(5)

In Eq. 5, ẍ, ẋ, and x are points of acceleration, velocity, and displacement response data sets, respec-

tively. Δt is the integration time step, while γ and β are the Newmark integration coefficients, typical-

ly holding the values γ = 0.5 and β = 0.25, which are also adopted in this study. Upon performing the

integration, the displacement response data is transformed into the frequency domain by applying the

fast Fourier transform (FFT) [23], as shown in Eq. 6.

tNkNNkexX

N

n

N

nik

nk Δ),0[1

0

2

(6)

In Eq. 6, Xk is the complex kth Fourier value of the N-point displacement response data set at fre-

quency ω. The modulus of Xk expresses the amplitude A of the displacement response data at fre-

quency ωk, while the argument of Xk represents the phase angle θ, as shown in Eq. 7.

k

kkk

kkkk

X

XX

XXXA

Re

Imarctanarg

ImRe 22

Ntk k Δ (7)

Drawing from the theory of signal processing, the cross-spectrum between two signals (in this case,

two sets of displacement response data) is typically employed to study the relations between the sig-

nals [19]. The cross-spectrum G, between a set i and a set j of displacement response data is derived

from Eq. 8.

NNkXXG jkikijk ),0[*,,, (8)

8

In Eq. 8, the superscript “*” denotes complex conjugate. Finally, the phase shift between the TMD

and the structure is obtained from the phase angles of the cross spectrum at specified frequencies.

The selection of frequencies is based on the highest (peak) amplitudes, which typically correspond to

vibration modes (modal peaks). The efficiency of the TMD is assessed upon the deviation of the

phase shift from 0 or 2π, thereby indicating out-of-phase response of the TMD with respect to the

structure.

3.2 Simulation example

To showcase the performance of the TMD in terms of the phase shift between the TMD and the

structure, a simulation example is devised. More specifically, a numerical model of a MDOF oscilla-

tor with a TMD attached to it is created. The oscillator considered in this study is a 5-DOF cantilever

with a TMD attached to the 5th (uppermost) DOF designed to dampen the fundamental (first) vibra-

tion mode (Figure 2). Therefore, the analysis in this subsection focuses on the fundamental vibration

mode and the respective mass and stiffness properties (m1, k1). Since slender structures are typically

subjected to wind loads, and in accordance with the field of wind engineering, the excitation is as-

sumed to be random Gaussian [25]. For the load distribution, the Eurocode 1 profile for wind loads is

adopted [26]. The simulation is described in the following steps:

1. The mass and stiffness of the cantilever are assumed to be concentrated at the 5 degrees of

freedom, as shown in Figure 2. For the sake of simplicity, and given the nature of wind

loading, only degrees of freedom along the x-axis are considered.

2. The loading profile due to wind loads is calculated according to Eurocode 1, for terrain cat-

egory I, basic wind speed of vb = 35 m/s, air density of ρ = 1.25 kg/m3, and orography fac-

tor co = 1.

3. The fundamental eigenfrequency (frequency at the fundamental vibration mode) f1 of the

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cantilever along with the corresponding generalized mass m1 are calculated through modal

analysis.

4. Upon selecting a ratio μ between the TMD mass md and m1 based on engineering judgment,

the optimal TMD parameters kd,opt, md,opt are calculated according to [4], along with the

added critical damping ratio for the TMD (ξd), to be used as starting values.

5. For the loading profile defined in the second step, the response of the cantilever, in terms of

displacement, is obtained through linear time history analysis using the Newmark-β algo-

rithm for integration (Eq. 5).

6. The cross spectra between the displacement response data of each DOF and the displace-

ment response data of the TMD are obtained using Eq. 6 and Eq. 8.

7. The phase angles of the cross spectra are retrieved through Eq. 7.

8. The analysis is repeated by varying the TMD parameters (except ξd) so that a family of

cross spectrum phase angles θ for different values of ωd/ω is obtained.

Fig. 2: Simulation of the 5-DOF oscillator subject to wind loads.

The fundamental eigenfrequency of the cantilever is calculated at f1 = 2.13 Hz and the corresponding

generalized mass at m1 = 12.76 t. For estimating the optimal TMD parameters according to [9],

which are used as starting values, the ratio between the TMD mass and m1 is set equal to μ = 4%.

Therefore:

m1 = 20 t

60/60

50/60

40/50

40/40

30/30

m2 = 16 t

m3 = 18 t

m4 = 14 t

m5 = 12 tkd

cd

md

4.00

4.00

4.00

4.00

3.00

P(t)

0.95·P(t)

0.88·P(t)

0.78·P(t)

0.62·P(t)

z

x

Column sections:bx/by (cm)

10

kN/m 859615.0213.2 t5.0

9615.01

1%,12

18

304.0

,2

,,

1,

1

optddoptddoptd

doptoptd

d

mkm

fm

m

(9)

Finally, the inherent damping of the structure follows the Rayleigh damping paradigm and is as-

sumed constant for all frequencies [27]. To demonstrate the effect of the TMD on the displacement

response of the structure, the root mean square (RMS) of the displacement of the 5th DOF from each

time history analysis with respect to the ratio ωd/ω1 is plotted in Figure 3.

Fig. 3: Dampening effects of TMD reflected on the displacement on the uppermost DOF on the

structure.

It is evident from Figure 3 that the closer the ratio ωd/ω1 to the optimal value fopt given in Eq. 9 is, the

stronger the dampening effect gets. While the reduction of displacement amplitude serves as a clear

indication of the TMD effect, the trend appearing in Figure 3 is obtained only under wind loads with

the same excitation function and with the same load amplitude, which is an unrealistic scenario.

Therefore, for assessing the TMD effect using SHM data from real structures, the phase shift of the

cross spectrum is considered more reliable. The effects of varying the TMD parameters on the phase

shifts are shown in Figure 4.

0.000

0.005

0.010

0.015

0.020

0.025

0.00 0.50 1.00 1.50 2.00

Dis

plac

emen

t RM

S o

f th

e 5th

stor

y (m

)

ωd/ω1

11

Fig. 4: Phase shift of the cross spectrum between the structure and the TMD at the fundamental

mode of vibration.

From Figure 4, it is clear that for TMD parameters close to the optimal parameters, the phase shift θ

is π/2 (cosθ ≈ 0), while deviating from the optimal parameters results in phase shifts of 0, π, or 2π

(cosθ ≈ 1 or cosθ ≈ -1), thus indicating in-phase or antiphase response with no dampening effect.

4 Case study: Assessing the performance of a TMD installed in a radio tower

To validate the TMD assessment methodology, a case study assessing the performance of a TMD

using real SHM data from a radio tower is devised. First, the radio tower and the SHM system are

briefly described. Next, the results from applying the proposed methodology are presented.

4.1 Description of the radio tower and the SHM system

The radio tower has a height of approximately 190 m and is composed of three structural segments,

as shown in Figure 5: (i) a lower 108 m tall reinforced concrete segment, (ii) a middle 60 m tall steel

segment, and (iii) an upper 22 m tall glass fiber segment. The interface connections between the

tower segments are bolted. The concrete segment features a tapered circular hollow section with

variable thickness (450 mm at the ground level and 270 mm at the interface with the steel segment).

cosθ

ωd/ω1

-1.00

-0.60

-0.20

0.20

0.60

1.00

0.00 0.50 1.00 1.50 2.00

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The outer diameter of the concrete segment is 8.6 m at the ground level and 4.8 m at the interface

with the steel segment. The steel segment consists of three parts of constant circular hollow sections

and constant thickness of 20 mm. The outer diameters of the three parts are 3.0 m, 2.0 m, and 1.8 m,

respectively, the accommodations of diameter change between the parts being achieved through

short tapered sections. The glass fiber segment of the tower is a constant circular hollow section of

1.6 m outer diameter and of 25 mm thickness. At the top of the glass fiber part, a tuned mass damper

(TMD) is installed, which consists of a concrete ring suspended from the top hatch of the tower,

connected to the tower shell through springs.

The SHM system installed in the radio tower comprises accelerometers, temperature sensors, and

anemometers placed at two instrumentation levels, one at the interface between the concrete segment

and the steel segment at 109 m height and one at the top of the glass fiber segment at 190 m. In each

instrumentation level the data from all sensors are collected by a data acquisition unit (DAQ). In this

study, the focus is placed on the second instrumentation level, which includes the TMD, where two

pairs of accelerometers are installed, one measuring the acceleration response of the tower shell and

the other measuring the acceleration response of the TMD. The accelerometers in each pair are per-

pendicular to each other, one measuring along the north-south direction and the other measuring

along the east-west direction. As shown in Figure 5, the data from each instrumentation level is

transferred to a centralized controller box located at the ground level of the tower.

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Fig. 5: Instrumentation overview of the radio tower.

4.2 Application of the TMD assessment methodology

The TMD assessment methodology is applied using acceleration response data from the SHM sys-

tem installed in the radio tower. From wind-related data retrieved from the anemometer of the sec-

ond instrumentation level, it is observed that the dominant wind direction is west/south-west. There-

fore, the acceleration response data from the accelerometers measuring along the east-west direction

is used for applying the methodology, as described in the following steps.

1. Sets of acceleration response data along the east-west direction are collected in hourly in-

tervals over a period of 6 months and integrated to obtain the corresponding sets of dis-

placement response data using the Newmark-β algorithm.

2. The average cross spectra of the sets of displacement response data at 10-minute intervals

are calculated using Eq. 6 and Eq. 8, along with the corresponding mean wind speed.

3. The phase angles of the cross spectra are obtained at four modal peaks, which are identified

in the Fourier amplitude spectra of the acceleration response data.

For illustration purposes, the phase angles of the cross spectra at the four identified modal peaks over

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a period of one month with respect to the mean wind speeds are plotted in Figure 6.

Fig. 6: Cosine of the phase angles at the modal peaks depending on the wind speed.

From Figure 6, it is observed that for the first three modal peaks no phase shift occurs between the

TMD and the structure, because cosθ ≈ 1 and, therefore, θ = 0 or θ = 2π for almost all wind speeds.

Some deviations of the cosine from unity are attributed to randomness and to approximation errors,

especially since no correlation between the results deviating from unity and the wind speed is ob-

served. Moreover, it is expected that the TMD is activated for high wind speeds, which does not

correspond to the results for the modal peaks at frequencies f1, f2, and f3. However, in the fourth

modal peak (at f4), there is non-negligible deviation of cosθ from unity for high mean wind speeds.

The cosines of the phase angles assume values between 0.45 and 0.80, which indicate out-of-phase

response. Nonetheless, only a few of the results approach the near optimal case of cosθ ≈ 0 (θ ≈ π/2).

Although it cannot be assumed that the TMD is tuned exactly at a frequency equal to f4 = 2.04 Hz,

there are strong indications that it is tuned at a frequency in the region of f4. Finally, investigating the

possibility that the TMD is tuned to a higher frequency is not possible due to the interference of ra-

cosθ

Wind speed (m/s)

f = 0.35 Hz

0.00

0.50

1.00

1.50

0 5 10 15 20

cosθ

Wind speed (m/s)

f = 0.62 Hz

0.00

0.50

1.00

1.50

0 5 10 15 20

0.00

0.50

1.00

1.50

0 5 10 15 20

cosθ

Wind speed (m/s)

f = 1.32 Hz

0.00

0.50

1.00

1.50

0 5 10 15 20

cosθ

Wind speed (m/s)

f = 2.04 Hz

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dio signals from the tower operation with the acceleration response data collected by the SHM sys-

tem.

5 Summary and conclusions

Unfavorable structural vibrations are typically a result of adopting irregular structural systems in

design or of using lightweight materials, thus reducing the inherent material damping. To mitigate

such unfavorable vibrations, structural subsystems, such as dampers, are usually attached to struc-

tures to artificially enhance the damping forces. In this context, tuned mass dampers, i.e. dampers

“tuned” to counteract vibrations at a specific frequency, have been widely employed. Assessing the

performance of tuned mass dampers is important to ensure the optimum functionality and to enable

preventive and retrofitting action in case of “de-tuning”. In this paper, a methodology to assess the

performance of tuned mass dampers using acceleration response data collected from a structural

health monitoring system has been presented. The proposed methodology builds upon the fundamen-

tals of tuned mass damper (TMD) theory, according to which a phase shift of approximately 90° is

expected between the displacement response data of the tuned mass damper and the displacement

response data of the structure at the frequency in which the tuned mass damper is tuned. The phase

shift is reflected in the cross spectrum between two sets of displacement response data, one from the

tuned mass damper and one from the structure, obtained from integrating the respective acceleration

response data.

The TMD assessment methodology proposed in this study has been validated via a case study of a

monitored radio tower equipped with a TMD. Acceleration response data has been collected from the

TMD and from the tower at the same location, and it has been integrated to yield the corresponding

displacement response data. The cross spectrum has been obtained from the displacement response

data and the phase angles at four identified modal peaks have been retrieved. As a result, it has been

observed that the phase angle of the fourth modal peak assumes values deviating from 0 or 2π, for

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high wind speeds (i.e. strong excitation), thus indicating that the tuning frequency is in the region of

the fourth modal peak. It is, therefore, concluded that estimates of the actual tuning frequencies of

tuned mass dampers can be obtained with the proposed methodology. Future work will be inclined

towards analyzing the performance of tuned mass dampers over a wider frequency band to extract

estimates of the tuning parameters and of the structural parameters of the tuned mass damper itself.

Acknowledgements

The authors would like to gratefully acknowledge the financial support offered by the German Re-

search Foundation (DFG) under grant GRK 1462 (“Evaluation of Coupled Numerical and Experi-

mental Partial Models in Structural Engineering”). Any opinions, findings, conclusions or recom-

mendations expressed in this paper are those of the authors and do not necessarily reflect the views

of DFG.

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