a)
b)
c)
Dynamic properties ofstructures with dampers modelled
using fractional order derivatives
Zdzisáaw Pawlak, Roman Lewandowski
Institute of Structural Engineering
Poznan University of Technology
60-965 PoznaĔ, Poland
Abstract—The focus of this paper is on determination of the dynamic parametersof structural systems with viscoelastic(VE)
dampers described by Maxwell rheological models. Such parameters could be obtained after solvingthe appropriately defined
nonlinear eigenvalue problem for frames with VE dampers. The solution to the nonlinear eigenvalue problemis obtained by
equating tozero thedeterminantof theconsidered systemof equations.Apart fromcomplexconjugate eigenvalues,the real ones
occurred when dampers that are described by the classic Maxwell model, are also determined.
Keywords: viscoelastic damper; rheological model; fractional derivative; nonlinear eigenvalue problem; dynamic properties;
1. Introduction
In civil engineering passive damping systems are mounted
on structures in orderto reduce excessive vibrations caused by
winds and earthquakes [1-3]. Different kinds ofmechanical
devices, suchas viscousdampers, viscoelasticdampers,tuned
mass dampers, or base isolation systems, are used in the
passive systems.Inthe past, several rheologicalmodels were
proposed for describing the dynamicbehaviourof VE dampers
and materials[1-3].In recentyears,the fractionalcalculushas
received considerable attention and has been used in modelling
the rheologicalbehaviourof VE materialsanddampers [4, 5].
The fractional models have an abilitytocorrectlydescribethe
behaviour of VEmaterials and dampers using a small number
of model parameters. However, in this case, the VEdamper
equation ofmotion isa fractionaldifferential equation [6].It is
the aim of this paper to find the dynamic properties (i.e.,
natural frequenciesand non-dimensionaldamping factors)for
structures with VE dampers. The above-mentionedproperties
aredefinedonthebasisofeigenvalues,obtainedfromthe
nonlinear eigenproblem.Theapproach,as presentedin this
work, differs fromthe standard one whichmostly uses the
state-space variablesand thedynamicparameters arederived
from the lineareigenproblem [7] or the non-linear
eigenproblem [8], dependingontheassumed model of damper.
One of the most important achievements of the proposed
formulationis thedimension ofthe problem,which ismuch
smaller, compared with the standard approach. The solution to
the nonlinear eigenvalue problem is obtained by equating to
zero thedeterminantoftheconsideredsystem of equations.
The results of sample numerical calculations arepresented and
discussed. It is shownthat the results of nonlinear
eigenproblem which correspond to the classic models differ
qualitatively from the results obtainedfor the fractional model.
2. RheologicalModelof Damper
The rheologicalproperties of VE dampers weredescribed
using three different Maxwellmodels, i.e., classic model (Fig.
1a), fractional model (Fig. 1b)andgeneralized model (Fig. 1c).
TheclassicMaxwell model consists of a dashpot with the
constant d
c, connected in series with a spring of the stiffness
d
k.
Figure 1.Rheologicalmodels of damper.
In the caseofthefractionalMaxwellmodelofdamper,
instead of the dashpot we havea fractional dashpot (see Fig. 1b)
with theconstants:d cand D(10 dD ), which denotesthe
order of fractionalderivative [6].
In the generalized Maxwellmodel (Fig. 1c),thereisan
additional element of the stiffness0 k, whichisconnected in
parallel with the other elementsof the system, described
cd
qk
qj
kd
ui
k
j
ui
ui ui
cd,
D
qk
qj
k
dk
j
k0
k1c1
k1c1
kpcp
qk
k
ui
uiqj
j
Open Journal of Applied Sciences
Supplement：2012 world Congress on Engineering and Technology
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qs,k
k
kd
cd
qs,k+1
pk
pk+1
k+1
damper i
fk
fk+1
pk
pk+1
respectively by stiffnessl kand dampingl c,( pl ,...,2,1 ).
The equations ofmotionforthe classic or fractionaland
generalized Maxwell models could be written as follows:
pltqktu
tqDktuDtu
tqDktutuD
ii
itllitlil
itdiit
,...,2,1)()(
)()()(
)()( )(
00
11
'
' 
' 
Q
Q
DD

where, k
ji qqtq  ')( ,dd ck
Q
,lll ck
Q
. Moreover,
i
u,li uand j
q,k
qdenote thedampersforce, theforce inthe
j-thMaxwellunit andnodaldisplacements,respectively.The
symbol )(x
D
t
Ddenotes the Riemann-Liouville fractional
derivative ofthe orderDwith respect totime, t. More
information concerning the fractionalderivative can befound
in [6].Forconsistentnotation,we introduce)()(
1tqqDt
.The
equation of motion for the classic Maxwell model could be
obtained after introducing into(1)1 D.
3. Structuralsystemwithdampers
A.The equation of motion
In this paper,thestructure with VE dampersistreated as an
elastic linearsystemmodelled as ashear frame. Themassof
the system is lumped at thelevelof storeys.Viscoelastic
dampers areinstalled between twosuccessive storeys. The
equation ofmotion ofthe structurewith damperscanbe written
as follows:
)()()()()(ttttt ssssss pfqKqCqM   
where the symbolss M,s Cand s Kdenote the mass, the
damping,and thestiffnessmatrices,respectively. Moreover,
T
nsjsss qqqt ],...,,...,[)( ,,1,
qdenotes the vector of
displacements ofthe structure andT
nj pppt ],...,,...,[)( 1
pthe
vector of excitation forces. The components of vector
T
n
ffft ],...,,[)( 21 farethe interactionforces betweenthe
frame and thedampers (Fig.. 2).
Figure 2.Diagram of frame and the interaction forces.
If a structure with onlyone damper denoted as the damper
number i, mounted between two successivestoreys, kand k+1,
isconsidered, thenthe vectorofdamper forcescouldbe written
in the followingform:
)(]0,.....,,,...,0[)( 1tuufuft ii
T
ikikief  
where, T
kki ee ]0,...,1 ,1,...,0[1 
e. For a structure with
mdampersthe vector ofinteraction forces isthesum of vectors
)(t
i
f, i.e.:
¦
m
i
itt
1
)()( ff 
B.The Laplace transform
After applying the Laplace transform and taking into
account that:
 >@ qq tL
>@
qq DD stDt)(L
>@
qqstDt )(
1
L
the equation of motion (2) can be written as:
 )()()(
2sssss ssss pfqKCM  
The vectors)(s
s
q,)(tfand )(spdenote the Laplace
transforms of displacements and forces, respectively.
According to(3), for damperi, the forcetransform is
)()( sus iii ef . The Laplace transformconverts (1)into one
relationship which is validfor each consideredmodel of
damper:
>@ )()()( sqsGksuiivii ' 
The quantitiesvi kand )(sGiare defined as:
¦




p
lli
li
iivi
i
di
ivi
s
ks
sGkk
s
ks
sGk
i
i
1
0)(
)(0
Q
Q
D
D

for classic (1 Di) or fractional models and for generalized
Maxwell model, respectively. Nowthe secondindex inthe
symbols i k0,li k,li
Q
refers to the damper’s number. Moreover,
)()( ssq s
T
ii qe '.
Finally, one may rewrite (6) inthe following form:
>@ )()()(
2sssss sss pqGKCM 
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y
x
where, vs KKK  ,¦ m
i
T
iiviv k
1
eeK ,¦ m
i
T
iiisGs
1
)()( eeG ,
mstands for thenumber of dampers.
C.Nonlineareigenproblem
For 0p )( s, theequation ofmotion (9)expresses a
nonlinear eigenproblem from which the eigenvalues and
eigenvectors canbe determined [8].In the caseofthe fractional
Maxwell model itis possible to write the relationship:
 0qBKCM
»
»
¼
º
«
«
¬
ª

 ¦
si
m
ii
ss i
i
s
s
ss
1
2
)(
D
D
Q

where, T
iiii keeB . For the classic Maxwell model1 Di.
Thesolution tothe system(10)yieldsa set ofcomplexand
conjugate eigenvalues.The numberof pairs ofeigenvalues
equals the number of degreeof freedomfortheconsidered
system. Moreover, for a structural system with theclassicor
thegeneralized Maxwell models ofdampers weobtain a set of
real eigenvalues of which the number equals thenumber of all
dashpots occurring inthe damper models. The calculation
carried out bythe authors suggests thatfor thedampers
described by the fractional Maxwell model, real solutions do
not exist.
A nonlineareigenproblem canbe solvedusing the
continuation methodwhichis similar to the one described in
the paper[8].Anotherpossibility to obtain the valuesi s is a
method of equating to zero the determinant of the system of
equations[9]. Itis to benoted thatfor1 Dithevalueexisting
in the denominator in (10) leads to the singularity when
i
sQ . In order to eliminate these singularities, we transform
the systemof equations (10)into the following form:
 0qBKCM
»
»
»
»
¼
º
«
«
«
«
¬
ª
 ¦ z
si
m
i
m
ik
k
kss
m
i
ikii sssss
11
2
1
)()(
DDD
QQ

In this work,real and complexeigenvalues are obtained by
searching the value of determinantof the system (11) and by
evaluating the roots ofdeterminant function.
D.Dynamic properties of structure
The dynamic behaviourofa framewith viscoelastic
dampers is characterized by the natural frequenciesi Zand the
non-dimensionaldampingparametersi
J
. Similarlyto viscous
damping, the above-mentioned properties are defined as
follows:
iiiiii ZP JKP Z/,
222 
where )Re(iis
P
,)Im( ii s
K
. For the real eigenvaluesi
s
relationships (12) are not valid. The real eigenvalues
correspond to the rheological properties of the considered
dampers.
4. ResultsofCalculation
In thenumericalexample, atwo-storeybuilding structure
modelled asa shear planeframe with therigid beamsis
considered. Themass is lumped and same at everyfloor:
kgms2000 .The bending rigidity of each storey is
kN/mks4000 . The viscoelastic damper with the stiffness
parameter kN/mk2500
1 and damping/m kNsc
D
50
1 is
mounted on the first storey. Thus, the coefficient50
1 Q.On
the second floor thereis a damper,characterized by the
parameters: kN/mk600
2 ,/m kNsc
D
30
2 ,20 2
Q
.
Firstly, the calculationswere carried out for aframe with
dampers describedby thefractional Maxwellmodel, for which
the valueof thefractional parameter was6.0
D
. After
equating to zero the determinant of thesystemofequations
(11), weobtainthe characteristicequationwhich enables four
complex and conjugate eigenvaluesi s to be derived (see
Table I).
TABLE I.THE EIGENVALUES–F
RACTIONAL MAXWELLMODEL
Rootnumberi)( i s)( i s
1, 2-0.7254±28.3351
3, 4-1.8983±74.8284
The value determinantbiasZ  )(det(A) is a
complex number which dependson thecomplex variable
yixs  , where1 i. Thus, the value of the
determinant equals zeroonly if its real and imaginary part
simultaneously isequalto zero,0),( yxa and 0),( yxb .
The rootsiii yixs  of a characteristicequationare in a
complex planeat the intersectionof lines,along whichthe real
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y
x
Im(Z)
y
x=-60
x=-50
x=-40
part isequal tozero(solid line inFig.3) and theimaginary part
is equal to zero(dashed line inFig. 3).
Figure 3.Plot of functions(Z) ,(Z) - fractional Maxwellmodel.
In Fig. 3 one may observe foursuchintersectionpointsof
which the coordinates coincide with the values given in Table I.
Next, the dampers were modeled using the classic Maxwell
model. The eigenproblem derived in the form of(11) was
solved byequatingthe determinant ofthesystemof equations
to zero.This leads to a characteristicequation of which the
solution yields four complex, conjugate eigenvaluesi s and
four real eigenvalues (seeTable II).
TABLE II.THE EIGENVALUES –C
LASSIC MAXWELL MODEL
Rootnumberi)( i
s)( i s
1, 2-2.9931±29.7644
3, 4-3.6167±80.3574
5 -17.6578 0
6-200
7 -39.1227 0
8-500
The rootsof thenumber 6 and8correspondto the solutions
ii
sQ , thatmeans a singular solutionof (10), which should
not be treated as the eigenvalues. Forthesepoints, the value of
the determinant,as derivedfromeigenproblem (10), tends to
infinity.
The discussed solutions are presented in Fig.4, as the
points of intersectionof the zero level lines of the surface
y))(Z(x, andsurfacey))(Z(x, derived from(11).
Figure 4.Plotof functions(Z) and (Z) - classic Maxwell model.
The real solutions5 sand 7 sgiven in TableII coincide with
the rheological properties of dampers mountedin structure.
5. Concluding Remarks
Comparing theresults of calculations fora frame with
dampers modeled using the classic Maxwell model and the
results obtained for the fractional Maxwell model, we may
observe qualitative differences. Thesolution to the nonlinear
eigenproblemleads toa numberof pairsofcomplex and
conjugate eigenvaluesiii yixs r . Moreover, in the case of
the classic Maxwellmodel of damper we obtain some real
eigenvalues for0
i
xand 0
i
y. For the fractional Maxwell
model, real solutionsdo not existbecauseof discontinuity in
the imaginarypartofthe determinant),(det(A) yxZ (see
Fig. 5).
Figure 5.Diagram of imaginary partof determinant function.
6. Acknowledgment
The authorswish to acknowledgethe financial support
received fromthe Poznan University ofTechnology (Grant No.
DS 11-088/12)in connection with thiswork.
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