Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 2, pp. 467-476, Warsaw 2015
DOI: 10.15632/jtam-pl.53.2.467

OPTIMAL POSITIONS OF TUNABLE TRANSLATIONAL AND

ROTATIONAL DYNAMIC ABSORBERS IN GLOBAL VIBRATION

CONTROL IN BEAMS

Waldemar Łatas

Cracow University of Technology, Institute of Applied Mechanics, Kraków, Poland

e-mail: latas@mech.pk.edu.pl

The paper discusses the problem of vibration of an Euler-Bernoulli beamwith translational
and rotational dynamic absorbers attached. The beam is subjected to concentrated and
distributed harmonic forces.The equation ofmotion is solvedusing theFouriermethod.The
timeLaplace transformationallowsone todetermine theamplitude-frequency characteristics
of the beam deflection. The aim of the study is to examine the influence of positions of the
translational and rotationaldynamic absorbersonvibration suppression in the global control
problem in beams. Numerical examples present reduction of kinetic energy of the cantilever
beam with tunable absorbers over a wide range of frequencies. Optimal positions of the
absorbers are obtained.

Keywords: dynamic vibration absorber, beam vibration, vibration reduction

1. Introduction

Themain purpose of dynamic vibration absorbers (DVA) is suppression of motion at the point
of attachment (Harris and Piersol, 2002; Korenev and Reznikov, 1993; Mead 1999). The most
commonDVAs attached to the vibrating structure subjected to harmonic excitation are passive
tuned mass dampers (TMD). They are used to attenuate both the longitudinal and torsional
vibration.Many works have been devoted to the optimization of dynamic absorbers parameters
in both the linear and non-linear problems (Bisegna and Caruso, 2012; Krenk and Høgsberg,
2008; Lee et al., 2006; Mohtat and Dehghan-Niri, 2011; Rüdinger 2006; Sgobba and Marano,
2010; Tigli, 2012).

In civil engineering, TMDs are often used in structures susceptible to vibration induced
by wind flow or seismic ground motion: suspension and cable-stayed bridges (Abdel-Rohman
andMariam, 2006; Chen and Cai, 2004; Chen andWu, 2008), high-rise buildings (Bekdas and
Nigdeli, 2011; Liu et al., 2008; Moon, 2011; Nagarajaiah andVaradarajan, 2005;Wang and Lin,
2007], chimneys (Ricciardelli, 2001; Brownjohn et al., 2010), masts, wind turbine towers (Łatas
and Martynowicz, 2012). Tuned mass dampers are used in road and railway bridges (Li et al.,
2005; Luu et al., 2012; Shi and Cai, 2008; Yau and Yang 2004a,b), footbridges (Caetano et al.,
2010; Li et al., 2010)where the types of loading dependon the traffic of vehicles andpedestrians.

Because of multiple possible applications, a lot of attention has been put to the choice of
parameters of dynamic absorbers in beams (Brennan andDayou, 2000; Esmalizadeh and Jalili,
1998;Yang andSedaghati, 2009;Younesian et al., 2006). In continuous structures such as beams,
usually thebestabsorber locations are thepoints of loadingapplication, but itmaybetechnically
difficult to do so. Theproblem ismore difficultwhennon-collocated control and global vibration
control are considered. In such cases or when the distributed loading is assumed, the main task
is to find the appropriate positioning of damping devices on the structure (Brennan andDayou,
2000; Cheung andWong, 2008).



468 W. Łatas

Themain drawbacks of passive TMDs are their limitations to narrow frequency bands only,
ineffectiveness for non-stationary vibration and sensitivity for inaccurate tuning. To overcome
these limitations, an active force is introducedbetween thedamper and the structure (Lim, 2008;
Wang and Lin, 2007). A disadvantage of active systems is, in turn, high energy consumption
and dependence on the fault-free energy delivery. Their use can also cause instability of the
structure. Semi-active systems which can alter stiffness or damping (Keye et al., 2009; Kim and
Kang, 2012; Nagarajaiah and Varadarajan, 2005; Ricciardelli et al., 2000) in real time, improve
the efficiency of passive systems, do not require large amounts of energy and cannot destabilize
the system.

To improve the efficiency of vibration reduction, there are systems ofmass dampers tuned to
a single (Li andNi, 2007; Li et al., 2005; Yau andYang, 2004a,b) or several resonant frequencies
for broadband excitation (Caetano et al., 2010; Luu et al., 2012). A special type of these systems
are continuous absorbers (Thompson, 2008).

In this article, amodel of thebeamsubjected to concentratedanddistributedharmonic forces
is given, with a system of translational and rotational dynamic vibration absorbers attached.
The effect of rotational absorbers for the effectiveness of vibration reduction is investigated.
Optimization of positions of the tunable absorbers in a cantilever beam subjected to a uniform
distributed harmonic force with kinetic energy taken as the criterion is presented.

2. Theoretical model

Figure 1 presents a structure considered in the paper – a beam subjected to concentrated and
distributed harmonic forces, with a system of translational and rotational dynamic vibration
absorbers attached. The beam is of length L, with the following constant parameters: mass
density ρ, cross-section areaA, geometrical moment of inertia I, Young’s modulusE.

Fig. 1. Beamwith a system of translational-rotational vibration absorbers

The equation of motion of the beam presented in Fig. 1 – assuming the Euler-Bernoulli
theory and internal damping described by the Voigt-Kelvin rheological model (parameter α) –
is given as (Cheung andWong, 2008)

ρA
∂2w

∂t2
+EIα

∂5w

∂x4∂t
+EI

∂4w

∂x4
= q(x,t)+

p
∑

j=1

Pj(t)δ(x−xOj )

+
r
∑

j=1

Fj(t)δ(x−xEj )+
r
∑

j=1

∂Mj(t)δ(x−xEj )
∂x

(2.1)



Optimal positions of tunable translational and rotational... 469

where: q(x,t) – distributed force;Pj(t) – j-th concentrated force applied at the pointx
O
j ;Fj(t) –

j-th concentrated force applied at the point xEj , acting on the beam from the translational

dynamic vibration absorber; Mj(t) – j-th concentrated torque applied at the point x
E
j , acting

on the beam from the rotational dynamic vibration absorber; mj,cj,kj – mass, damping and
stiffness coefficients of the j-th translational dynamic vibration absorber; Jj,γj,κj – moment
of inertia, damping and stiffness coefficients of the j-th rotational dynamic vibration absorber;
p – number of concentrated forces; r – number of translational-rotational vibration absorbers.
The term translational-rotational absorber represents two dampers, one rotational and one

translational, mounted in the same place.
Solution to equation (2.1) is assumed in form of the series

w(x,t) =
∞
∑

i=1

qi(t)ϕi(x) (2.2)

where ϕi(x) are the modes of vibration of the beam without absorbers attached, which are de-
pendent on the specific boundary conditions. The functions qi(t) are time-dependent generalized
co-ordinates that should be determined. It is assumed that the distributed loading has the form:
q(x,t)=h(t)g(x).
Substituting series (2.4) into equation (2.1) and performing the time Laplace transformation

(with initial conditions equal to zero) leads to

∞
∑

i=1

[

ρAs2Qi(s)+EIαβ
4
i sQi(s)+EIβ

4
iQi(s)−aiH(s)−

p
∑

j=1

djiPj(s)

−
r
∑

j=1

bjiFj(s)−
r
∑

j=1

ejiMj(s)
]

ϕi(x)= 0

(2.3)

Assumingorthogonality of the functionsϕi(x)with theweight functionη(x), the following values
of the coefficients appearing in equation (2.3) are obtained

ai =
1

K2i

l
∫

0

g(x)ϕi(x) dx dji =
ϕi(x

O
j )

K2i
bji =

ϕi(x
E
j )

K2i
eji =

−ϕ′i(xEj )
K2i

(2.4)

where: K2i =
∫L
0 η(x)ϕ

2
i(x)dx, and additionally, in equation (2.3) the notation is introduced

β4i = [ρA/(EI)]ω
2
i ;ωi is the i-th natural frequency of the beamwithout the vibration absorbers

attached andwith the internal dampingneglected (α=0). In equation (2.3), the symbols:Qi(s),
H(s), Pj(s), Fj(s),Mj(s) denote the Laplace transforms of the: qi(t), h(t), Pj(t), Fj(t), Mj(t),
respectively.
Taking into account the linear independence of the functions ϕi(x), from equation (2.3) an

expression for the Laplace transformW(x,s) of the beam deflection w(x,t) can be obtained

W(x,s)=
∞
∑

i=1

aiH(s)+
p
∑

j=1
djiPj(s)+

r
∑

j=1
bjiFj(s)+

r
∑

j=1
ejiMj(s)

ρAs2+EI(1+αs)β4i
ϕi(x) (2.5)

and its derivative with respect to x

∂W(x,s)

∂x
=
∞
∑

i=1

aiH(s)+
p
∑

j=1
djiPj(s)+

r
∑

j=1
bjiFj(s)+

r
∑

j=1
ejiMj(s)

ρAs2+EI(1+αs)β4i
ϕ′i(x) (2.6)



470 W. Łatas

The transforms of the force Fj(s) and torque Mj(s) acting on the beam from the j-th
translational-rotational vibration absorber attached at the pointxEj are given by the expressions
(Cheung andWong, 2008)

Fj(s)=−W(xEj ,s)
(cjs+kj)mjs

2

mjs2+ cjs+kj
Mj(s)=−Θ(xEj ,s)

(γjs+κj)Jjs
2

Jjs2+γjs+κj
(2.7)

where the notation Θ(x,s)=−∂W(x,s)/∂x is introduced.
The transforms given by formulas (2.7) should be inserted into expressions (2.5) and (2.6).

The resulting transforms of the beamdeflection and slope shouldbe satisfied at the pointswhere
the absorbers are attached to the beam. These conditions yield a system of linear algebraic
equations to determine the unknownsW(xEk ,s) andΘ(x

E
k ,s) (k=1,2, . . . ,r).

To simplify the expressions, the following notations are introduced

W(xEj ,s)=Wj Θ(x
E
j ,s)=Θj

ϕi(x
E
j )=ϕij ϕ

′
i(x
E
j )= εij

aiH(s)+
p
∑

j=1
djiPj(s)=Ai ρAs

2+EI(1+αs)β4i =Bi

(cjs+kj)mjs
2

mjs2+ cjs+kj
bji =Dji

(γjs+κj)Jjs
2

Jjs2+γjs+κj
eji =Eji

(2.8)

The system of 2r linear equations for the unknownsWk,Θk (k=1,2, . . . ,r) takes the form

Wk

(

1+
∞
∑

i=1

Dki
ϕik
Bi

)

+
r
∑

j=1,j 6=k

∞
∑

i=1

WjDji
ϕik
Bi
+
r
∑

j=1

∞
∑

i=1

ΘjEji
ϕik
Bi
=
∞
∑

i=1

Ai
ϕik
Bi

Θk

(

∞
∑

i=1

Dki
εik
Bi
−1

)

+
r
∑

j=1,j 6=k

∞
∑

i=1

ΘjEji
εik
Bi
+
r
∑

j=1

∞
∑

i=1

WjDji
εik
Bi
=
∞
∑

i=1

Ai
εik
Bi

(2.9)

Having solved the system (2.9), the transforms of the forces Fj(s) and torques Mj(s) may
be calculated from expressions (2.7) and used to calculate from formulas (2.5) and (2.6) the
transforms of the deflection and slope of the beam. Assuming steady-state vibration, after sub-
stituting s = jω (j =

√
−1), the expressions for the deflection and slope of the beam in the

frequency domain may be obtained.

3. Numerical results – optimization of position of the tunable vibration absorbers

The numerical algorithm allows one to calculate in the s-domain the transforms of the beam
deflectionand slope for arbitraryboundaryconditions.For aharmonic excitation, theamplitude-
-frequency characteristics of the bendingmoment, transverse force, time-averaged kinetic energy
of the whole beam or its part may be further calculated.

3.1. Optimizationof thepositionof the tunable translational-rotational vibration absorber

Acantilever steel beam is excited byauniformdistributedharmonic force (Fig. 2).Thebeam
is of length l=1.0m,mass density ρ=7800kg/m3, Young’s modulusE =2.1 ·1011N/m2, with
rectangular cross-section of width b = 0.05m and height h = 0.005m. The internal damping
of the beam is neglected. There is only one translational-rotational absorber attached at the
point xE1 . The aim of the absorber is to attenuate vibration of the beam. The time-averaged
kinetic energy of the whole beam is used as the global measure of vibration.



Optimal positions of tunable translational and rotational... 471

Fig. 2. Cantilever beam of length l excited by a uniform distributed harmonic force with a dynamic
translational-rotational vibration absorber attached

From the practical point of view, it is preferable to use tunable dampers (Brennan and
Dayou, 2000; Dayou and Brennan, 2002) because a simple control algorithm can be used. It is
assumed in further calculations that the absorbers attached are tuned so that they are resonant
at each single frequency and do not have energy dissipating appliances (c1 =0,γ1 =0). The first
four natural frequencies of the presented beam are: f1 =4.19Hz, f2 =26.26Hz, f3 =73.54Hz,
f4 =144.11Hz.
Themain aim is to find the optimal position of the tunable translational-rotational vibration

absorber for a given frequency band. For comparison, in Fig. 3 the calculated time-averaged
kinetic energy for different positions of the single translational absorber (J1 = 0) in the range
of frequency 〈0.0Hz,30.0Hz〉 is shown.

Fig. 3. Kinetic energy of the cantilever beam: without any absorber; with a single translational absorber
placed at different points of the beam. The absorber attached is tuned to be resonant at each frequency.

The numbers denote the dimensionless position of the absorber xE
1
/l

The best location of the single translational absorber in the range 〈0.0Hz,20.0Hz〉 is
xE1OPT =0.71l.
For other locations, the results obtained are worse. The calculations performed for the

translational-rotational absorber (translational and rotational absorbers are attached at the sa-
me location) give the same optimal position in the range 〈0.0Hz,40.0Hz〉 equal xE1OPT =0.71l.
In Fig. 4, plots of kinetic energy calculated for the optimal locations of the translational and
translational-rotational absorbers are shown for comparison.
The drawback of the tunable absorbers is that they cause an increase in the global vibration

at the new natural frequencies of the resulting structure (Brennan and Dayou, 2000). It is cle-
arly seen in Figs. 3 and 4. In the case of a single translational absorber, there is an additional



472 W. Łatas

Fig. 4. Kinetic energy of the cantilever beam: without any absorber; with the translational and
translational-rotational absorbers placed at the optimal position xE

1OPT
=0.71l. The absorbers attached

are tuned to be resonant at each frequency

resonant frequency equal to 23.49Hz. The addition of the rotational absorber enhances vibra-
tion attenuation. There is no additional resonance at the frequency equal to 23.49Hz and the
translational-rotational absorber works efficiently up to frequency 40Hz.

Inbothcases, theoptimal solutionsarevery sensitive to theaccuracyof theabsorber location.
Arelatively small change in it can significantlydecrease theeffectiveness ofvibration suppression.
A further improvement may be obtained by installing a few absorbers in different places of the
beam.

InSections 3.2 and3.3 situationswith twoabsorbers located at differentplaces are presented.
Referring to graphs in Fig. 4, the aim of optimization is to find the positions of the absorbers,
which give the best vibration reduction efficiency in the given range 〈0.0Hz,40.0Hz〉.

3.2. Optimization of positions of the tunable translational and rotational vibration

absorbers

Figure 5presents theanalyzedbeamwith twodynamicvibrationabsorberswithoutdamping:
one translational placed at the point xE1 and one rotational placed at the point x

E
2 . The aim is

to find the optimal locations of the absorbers x∗E1OPT , x
∗E
2OPT , which can give the best vibration

reduction in the range 〈0.0Hz,40.0Hz〉. The absorbers attached are tuned to be resonant at
each frequency. The results of numerical calculations are summarized in Section 3.4.

Fig. 5. Cantilever beam of length l excited by a uniform distributed harmonic force with tunable
dynamic translational and rotational vibration absorbers attached at different locations



Optimal positions of tunable translational and rotational... 473

3.3. Optimization of positions of the two tunable translational vibration absorbers

Figure 6 presents the studied beam with two dynamic translational vibration absorbers
without damping placed at the points xE1 , x

E
2 . The aim is to find the optimal locations of the

absorbers x∗∗E1OPT , x
∗∗E
2OPT , which give the best vibration reduction in the range 〈0.0Hz,40.0Hz〉.

The absorbers attached are tuned to be resonant at each frequency. The results of numerical
calculations are gathered in Section 3.4.

Fig. 6. Cantilever beam of length l excited by a uniform distributed harmonic force with two tunable
dynamic translational vibration absorbers attached

3.4. Summary of the results of numerical calculations

The calculated optimumpositions of the absorbers for the cases presented in Section 3.2 and
3.3 are as follows:
– two absorbers; one translational of position xE1 and one rotational of position x

E
2 (Fig. 5)

x∗E1OPT =0.60l x
∗E
2OPT =0.80l

– two translational absorbers of positions xE1 , x
E
2 (Fig. 6)

x∗∗E1OPT =0.41l x
∗∗E
2OPT =0.83l

In Fig. 7, plots of kinetic energy for the given above optimal positions of the absorbers in the
two investigated cases are shown. For comparison, the kinetic energy for the optimal position of
the translational-rotational absorber (xE1OPT =0.71l, Fig. 2) is also presented.
The option to use several absorbers in different locations greatly increases the efficiency

of vibration suppression. The calculations performed show that the use of two translational
absorbers gives a better result than the use of one translational and one rotational absorber.
But if there is only one place to locate the absorber, the addition of the rotational absorber to
the translational onemay improve vibration attenuation.

The results presented are valid for a harmonic force uniformly distributed over the length
of the beam. For another type of loading, the optimization can give different results, as for the
other frequency bands. Further improvement of thr performance may be obtained by adding
damping or de-tuning the absorbers (Brennan and Dayou, 2000).

4. Conclusions

Themodel presented in the paper can be used in local and global problems of optimal choice of
positions and physical parameters of translational and rotational vibrations absorbers in beams.



474 W. Łatas

Fig. 7. Kinetic energy of the cantilever beam: without any absorber; with the translational-rotational
absorber; with one translational and one rotational absorber; with two translational absorbers. The
absorbers attached are placed at the optimal positions and are tuned to be resonant at each frequency

Theoretical calculations are illustrated by optimization of positions of the tunable absorbers
in the global control of kinetic energy of the beam. The results of numerical calculations show
that by adding a rotational absorber, the effectiveness of vibration reduction can be improved,
as it can absorb rotational motion of the beam by applying torque to the beam. The addition
of rotational absorbers to translational ones allows one to move resonances and expands the
frequency band of effective vibration reduction. This effect can be also used in other structures
such as frames, curved beams and pipes.

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Manuscript received May 9, 2014; accepted for print December 2, 2014