Structural Engineering and Mechanics, Vol. 41, No. 1 (2012) 113-137

113

Dynamic analysis of frames with viscoelastic dampers:
a comparison of damper models
R. Lewandowski*, A. Bartkowiaka and H. Maciejewskia
Department of Civil Engineering, Poznan University of Technology, ul. Piotrowo 5, 60-965 Poznan, Poland(Received November 12, 2010, Revised July 20, 2011, Accepted December 13, 2011)
Abstract. Frame structures with viscoelastic (VE) dampers mounted on them are considered in thispaper. It is the aim of this paper to compare the dynamic characteristics of frame structures with VE
dampers when the dampers are modelled by means of different models. The classical rheological models,the model with the fractional order derivative, and the complex modulus model are used. A relatively
large structure with VE dampers is considered in order to make the results of comparison more
representative. The formulae for dissipation energy are derived. The finite element method is used toderive the equations of motion of the structure with dampers and such equations are written in terms ofboth physical and state-space variables. The solution to motion equations in the frequency domain isgiven and the dynamic properties of the structure with VE dampers are determined as a solution to theappropriately defined eigenvalue problem. Several conclusions concerning the applicability of a family ofmodels of VE dampers are formulated on the basis of results of an extensive numerical analysis.
Keywords: dynamics of frames; viscoelastic dampers; dynamic characteristics; classical rheologicalmodels; Kelvin model with fractional order derivative; complex modulus model

1. Introduction
Viscoelastic (VE) dampers are energy dissipation devices of passive type, frequently used tomitigate excessive vibration of structures due to winds or earthquakes. A number of applications of
VE dampers in civil engineering are listed in the book by Christopoulos and Filiatrault (2006). The
properties of VE dampers, such as the possibility of energy dissipation and stiffness, are frequency
and temperature dependent and are commonly defined in terms of experimentally obtained storageand loss modules. Good understanding of the dynamic behaviour of dampers is required for the
analysis of structures supplemented with VE dampers. The frequency dependence of the properties
of VE dampers can be described by means of rheological models or using the complex modulus
model.
In the past, several rheological models were proposed to describe the dynamic behaviour of VE
dampers. Both the classical and the so-called fractional-derivative models of VE dampers areavailable. These models are presented by Chang and Singh (2009) and by Lewandowski andChor aç zí yczewski (2010). Very frequently the classical Maxwell or Kelvin models are used.
*Corresponding author, Professor, E-mail: roman.lewandowski@put.poznan.plaResearch Scholar

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R. Lewandowski, A. Bartkowiak and H. Maciejewski

Each group of models have their own advantages and disadvantages. The generalized rheologicalmodels, like the generalized Kelvin model or the generalized Maxwell model, contain moreparameters than the fractional ones but lead to the traditional differential equations of motion insteadof to the fractional differential equations when the fractional-derivative model of VE dampers is used, as
shown by Chang and Singh (2002). In general, the fractional differential equations are more difficult
to solve than the ordinary differential equations. The dynamic characteristic of structures with VE
dampers, like the frequencies and modes of vibration and the non-dimensional damping ratios, areobtained as the solution to the linear or quadratic eigenvalue problem when the classical rheologicalmodels of VE dampers are used. The fractional-derivative models of dampers require a solution to
the nonlinear eigenvalue problem, as proposed by Lewandowski and Pawlak (2010), or to theextremely large linear eigenvalue problem derived, for example, by Chang and Singh (2002) or by
Sorrentino and Fassana (2007). The modal strain method is also used to determine the nondimensional damping ratios in the papers by Xu (2007) and by Shen et al. (1995).
The dynamic analysis of frame or building structures with dampers are presented in many papers(Chang and Singh 2002, 2009, Lewandowski and Pawlak 2010, Hatada et al. 2000, Singh andMoreschi 2002, Matsager and Jangid 2005, Singh et al. 2003, Zhang and Xu 2000, Shukla andDatta 1999, Park et al. 2004, Lee et al. 2004, Tsai and Chang 2002, Okada et al. 2006, Mazza andVulcano (2007)). The simple Maxwell model was chosen by Hatada et al. (2000), Singh andMoreschi (2002) and by Singh et al. (2003). Moreover, in the papers by Singh and Moreschi(2002), Matsager and Jangid (2005) and by Lee et al. (2004) the simple Kelvin model is used todescribe the dynamic behaviour of dampers. These simple models are used by Singh and Moreschi(2002), Park et al. (2004) and by Lee et al. (2004), to solve the problem of optimal design ofstructures with VE dampers. In the papers by Chang and Singh (2002), Lewandowski and Pawlak(2010), and by Tsai and Chang (2002), the three-parameter fractional-derivative rheological model isused to model the dampers behaviour. Moreover, in the paper by Okada et al. (2006), rational
polynomial approximation modelling is used for the analysis of structures with VE dampers.
Up to now, the dynamic analysis of structures with VE dampers modelled by means of thegeneralized rheological models is rarely considered or discussed in the literature. The in-time
analysis of structures with dampers modelled by means of the models mentioned above can only befound in the papers by Singh and Chang (2009) and Mazza and Vulcano (2007).
Despite their popularity, the complex modulus model is not used in the context of dynamic
analysis of structures with VE dampers.
Design of structures with VE dampers together with optimization of damper locations andparameters are important problems from a practical point of view. Often the concept of performance-based
design is adopted in the process of designing of such structures. A detailed review of all the papers
is beyond the scope of this paper. Among other ones, we would like to mention the books by
Takewaki (2009) and by Christopoulos and Filiatrault (2006) and recent papers by Fujita et al.
(2010) and by Mazza and Vulcano (2007). Moreover, the concept of motion-based design isintroduced by Connor and Klink (1996).

2. Motivation, aims and scope of the paper
The review of existing literature suggests that there are many possible rheological models which
are or could be chosen to describe the properties of VE dampers. These models are usually chosenDynamic analysis of frames with viscoelastic dampers: a comparison of damper models

115

intuitively and the choice of the best model is not an obvious one. There is no comparison between
the results obtained for different models of dampers. In particular, the effect of VE dampers modelon the behaviour of a structure with dampers as a whole is not known. The dynamic characteristics
of structure (i.e., natural frequencies, non-dimensional damping ratios, modes of vibration, andfrequency response functions) depend on the properties of structure and dampers only while they do
not depend on earthquake. These quantities are general and appropriate for the analysis of influenceof VE damper models on the overall behaviour of structure. In particular, the non-dimensional
damping ratios measure the possibility of dissipation of energy introduced to structures by excitation
forces. However, this approach can be used only when structures is elastic.
Of course it is also possible to compare the responses of structures with dampers given in time
domain. In this case, we can compare the responses of structures which behave in a nonlinear
manner but conclusions which can be drawn from such an analysis are restricted to the earthquakes
which are taken into account. The comparison of transient response of simple dynamic systems withdifferent damping models is presented in the paper by Barkanov et al. (2003).
The aim of this paper is to compare the dynamic characteristics of structures with VE dampers
when the dampers are modelled by means of different models. A comparison in time domain isbeyond the scope of the present paper. A relatively large structure with VE dampers is considered inorder to make the results of comparison more representative. Planar frame structures, treated aslinear elastic systems, with the VE dampers mounted on the structures are considered. The simple
and generalized Kelvin and Maxwell models, the fractional-derivative Kelvin model, and thecomplex modulus model are used as models of VE dampers.
The outline of the paper is as follows. A description of the rheological models of VE dampers ispresented in Section 2. Moreover, formulae for dissipation energy are derived. In Section 3 theequations of motion of the whole system (the structure with dampers) are derived using the finite
element method and they are written in terms of both physical and state-space variables. A solution
to the motion equations of a whole system in the frequency domain is derived and the dynamic
properties of the structure with VE dampers are determined. The frequencies of vibration, the nondimensional damping ratios together with the corresponding eigenvectors are determined as asolution to the appropriately defined eigenvalue problem. The frequency response functions are also
determined. In Section 4 a comparison of the effect of damper models on the dynamic properties ofstructures with VE dampers is made on the basis of results of numerical calculation obtained for arepresentative frame with VE dampers. Section 5 contains concluding remarks concerning theinfluence of the damper model on the dynamic characteristics of the structure with VE dampers andsuggestions concerning the number of elements of generalized rheological models. Some usefulformulae are given in appendices.

3. Description of models of dampers
3.1 Equation of motion of the generalized rheological models
The frequency dependence of the properties of VE dampers can be captured using generalized
rheological models. First of all, the generalized Kelvin model and the generalized Maxwell model
are considered. The generalized Kelvin model is built from the spring and a set of the Kelvinelements connected in series (see Fig. 1). The serially connected spring and dashpot will be referred

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Fig. 1 Scheme of the generalized Kelvin modelFig. 2 Scheme of the generalized Maxwell model

to as the Kelvin element. The generalized Maxwell model is built from the spring and a set of theMaxwell elements connected in parallel (see Fig. 2). The Maxwell element is the classical Maxwell
rheological model, i.e., the spring and dashpot connected in series. The number of the Kelvin and
the Maxwell elements is denoted by m.
Using the concept of internal variables, the dynamic behaviour of the Kelvin damper can be
described by means of the following equations
u0 ( t ) = k0 ( q̃w, 1 ( t ) – q̃1 ( t ) )

(1)

·
·
ui ( t ) = ki ( q̃w, i + 1 ( t ) – q̃w, i ( t ) ) + ci ( q̃w, i + 1 ( t ) – q̃w, m ( t ) )

(2)

·
·
um ( t ) = km ( q̃3 ( t ) – q̃w, m ) + c m ( q̃3 ( t ) – q̃w, m )(3)where ui ( t ) is the force in the i-th element of the model ( i = 0, 1, …, m ), ki and ci are the spring
stiffness and the damping factor of the dashpot of the i-th element of model, respectively, andsymbols q̃1 ( t ) and q̃3 ( t ) denote the external nodes displacements given in the local coordinate
system (compare Fig. 1). Moreover, the dot denotes differentiation with respect to time t and thesymbol q̃w, i ( t ) denotes the internal variable ( i = 1, …, m ).
After introducing the vector of external reactions R̃z ( t ) = col ( R̃1 ( t ), R̃2 ( t ), R̃3 ( t ), R̃4 ( t ) ) (seeFig. 1) and utilizing the equilibrium conditions of the external nodes: R̃1 ( t ) = –u0 ( t ) , R̃2 ( t ) = 0 ,
R̃1 ( t ) = um ( t ) and R̃4 ( t ) = 0 we can write the following matrix equation

Dynamic analysis of frames with viscoelastic dampers: a comparison of damper models

117

·
·
R̃z ( t ) = K̃zz q̃z ( t ) + K̃ zw q̃w ( t ) + C̃zz q̃z ( t ) + C̃zw q̃w ( t )(4)where q̃z ( t ) = col ( q̃1 ( t ), q̃2 ( t ), q̃3 ( t ), q̃4 ( t ) ) , q̃w ( t ) = col ( q̃w, 1 ( t ), …, q̃w, m ( t ) ) .
Moreover, the equilibrium conditions of the internal nodes i.e., ui – 1 ( t ) – u1 ( t ) = 0 for i = 1, …, mlead to the following matrix equation··(6)
K̃wz q̃z ( t ) + K̃ ww q̃w ( t ) + C̃wz q̃z ( t ) + C̃ww q̃w ( t ) = 0
TT

where K̃wz = K̃zw , C̃wz = C̃zw .
The VE damper Kelvin model equation written in the local coordinate system can be finallypresented in the form·
(7)R̃d ( t ) = K̃d q̃d ( t ) + C̃d q̃d ( t )where R̃d ( t ) = col ( R̃z ( t ), 0 ), q̃d ( t ) = col ( q̃z ( t ), q̃w ( t ) )K̃d = K̃zz K̃zw , C̃ d = C̃zz C̃zw
K̃wz K̃ww
C̃wz C̃ww(8)

In this paper, the unusual transformation of nodal parameters to the global coordinate system isused. The displacements of the damper external nodes are transformed as usually but the internalvariables of damper are still defined in the local coordinate system. It means that the transformation
matrix is
Td = T̃d 0
0 I

(9)

T̃d = T̃ 0 , T̃ = c s
–s c
0 T̃

(10)

where

c = cosα , s = sinα , α is the angle between the global and local coordinate system and I is the( m × m ) identity matrix.
The generalized Kelvin model, which is a set of elements connected together by hinges, is ageometrically movable system. The stiffness matrix of such a model system written in the global
coordinate system and using the usual transformation matrix is singular and the global stiffness
matrix of a structure with dampers is also singular. The above-mentioned unusual transformation
guarantees that the global matrix of the system is not singular.
After transformation of Eq. (7) to the global coordinate system we obtain(11)
Rd ( t ) = Kd qd ( t ) + Cd q· d ( t )Twhere Rd ( t ) = col ( Rz ( t ), 0 ) = Td R̃d Td, Rz ( t ) = col ( R1 ( t ), R2 ( t ), R3 ( t ), R4 ( t ) ) , qd ( t ) = col ( qz ( t ),T

qw ( t ) = q̃w ( t )) = Td q̃Td , qz ( t ) = col ( q1 ( t ), q2 ( t ), q3 ( t ), q4 ( t ) ) , are the vector of nodal reactions
and the vector of nodal parameters, respectively in the global coordinate system. The explicit formsof matrices K d and Cd are given in Appendix A.
The dynamic behaviour of the Maxwell damper could be described in a similar way. In terms of

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nodal displacements and internal variables, as defined in Fig. 2, the following equations could be
written··u0 ( t ) = k0 ( q̃3 ( t ) – q̃1 ( t ) ), uis ( t ) = k i ( q̃w, i ( t ) – q̃1 ( t ) ), uid ( t ) = ci ( q̃3 ( t ) – q̃w, i ( t ) )(12)
for the spring element and for the i-th Maxwell element ( i = 1, …, m ). The symbols u0 ( t ) , uis ( t )and uid ( t ) denote the force in the spring element, the force in the spring of the i-th Maxwell
element, and the force in the dashpot of the i-th Maxwell element, respectively.
The nodal reactions in the local coordinate system aremm

R̃1 ( t ) = – u0 ( t ) – ∑ uis ( t ), R̃2 ( t ) = 0, R̃3 ( t ) = u0 ( t ) + ∑ uid ( t ), R̃4 ( t ) = 0
i=1

(13)

i=1

After introducing relationships (12) into Eq. (13) we obtain again the matrix Eq. (4). In the global
coordinate system, Eq. (11) is also valid with the matrices Kd and Cd given in Appendix B.
Many particular rheological models existing in the literature can be obtained by varying thenumber of elements in the generalized models mentioned above.
3.2 Equation of motion of the simple rheological models, the fractional-derivative Kelvinmodel, and the complex modulus model
The simple Kelvin and Maxwell models, which contain only one Kelvin or Maxwell element,
respectively, are not particular instances of the discussed generalized models because the springelement with stiffness k0 is not present. The simple Kelvin model and the simple Maxwell model
are shown in Fig. 3.
The matrix Eqs. (7) and (11) are equations of simple models treated as the finite elements. The
matrices K d and Cd are given in Appendix C.
The fractional-derivative Kelvin model is shown in Fig. 4. The equation of motion of the model
can be written in the following formαu ( t ) = k1 ( q̃3 ( t ) – q̃1 ( t ) ) + c1 Dt ( q̃3 ( t ) – q̃1 ( t ) )

(14)

α

where the symbol Dt ( • ) denotes the Riemann-Liouville fractional-derivative of the order α( 0 < α ≤ 1 ) with respect to time, t. For additional information concerning the Riemann-Liouvillefractional-derivative, Podlubny (1999) may be consulted.
The matrix equation of the fractional-derivative Kelvin damper written in the local coordinate
system could be written in the formαR̃d ( t ) = K̃d q̃d ( t ) + C̃d Dt q̃d ( t )(15)Fig. 3 Scheme of (a) the simple Kelvin model and (b) the simple Maxwell Fig. 4 Scheme of the fractional
derivative Kelvin model
model

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119

In the global coordinate system we can write the following equationα

Rd ( t ) = Kd qd ( t ) + Cd Dt qd ( t )(16)where, again, matrices Kd and Cd are given in Appendix C.
The equation of motion of dampers modelled using the complex modulus model isu ( t ) = K ( q̃ 3 ( t ) – q̃1 ( t ) )(17)

where K is now the damper dynamic stiffness.
For the damper executing the harmonic motion, i.e., when u ( t ) = u0 exp ( iλt ) , q̃1 ( t ) = q̃10 exp ( iλt ) ,q̃3 ( t ) = q̃30 exp ( iλt ) the following equation could be written
u0 = ( K′ ( λ ) + iK″ ( λ ) ) ( q̃30 – q̃10 )(18)

in the frequency domain. Above i = –1 is the imaginary unit, K′ ( λ ) and K″ ( λ ) are the storage
modulus and the loss modulus, respectively.
The finite element equations of this model written in the frequency domain and in the local andglobal coordinate systems areR̃d0 = ( K̃′d ( λ ) + K̃ d″ ( λ ) )q̃d0, Rd0 = ( K′d ( λ ) + K d″ ( λ ) )qd0

(19)respectively, where matrices K′d ( λ ) K″d ( t ) are given in Appendix C. In Eq. (19) the symbols
R̃ d0 ( t ) = col ( R̃10 ( ( t ), R̃20 ( t ), R̃30 ( t ), R̃40 ( t ) ) ) , R d0 ( t ) = col ( R10 ( t ), R20 ( t ), R30 ( t ), R40 ( t ) ) , q̃d0 ( t ) =col ( q̃10 ( t ), q̃20 ( t ), q̃30 ( t ), q̃40 ( t ) ) and qd0 ( t ) = col ( q10 ( t ), q20 ( t ), q30 ( t ), q40 ( t ) ) denote the vector ofcomplex amplitudes of reaction and nodal displacements in the local and global coordinate systems,
respectively.
The above-mentioned storage and loss modulus are important characteristics of VE dampers. For
all of the considered models of dampers the appropriate formulae are collected in Appendix D.
3.3 Energy dissipated in the harmonically excited damper
The energy dissipated by the damper is an important characteristic of the VE damper. Usually theenergy dissipated by a damper making harmonic motions is used as a measure of this property. To
determine such energy, for the generalized Kelvin damper, it is assumed that the left external nodeof damper is fixed while the right node is harmonically excited, i.e.
q1 ( t ) = q2 ( t ) = q4 ( t ) = 0, q3 ( t ) = a3 sin λt + b3 cos λt(20)

where λ is the excitation frequency, symbols a3 and b3 denote the known amplitudes of harmonic
excitation. In this subsection, the equation of motion in the local coordinate system is used but
waves over the symbols are omitted to simplify the notation.
In the considered case, the dynamic behaviour of damper is described by(21)
Kww qw ( t ) + C ww q· w ( t ) = fd ( t )where fd ( t ) = col ( k0 q1 ( t ), 0, …, 0, km q3 ( t ) + cm q· 3 ( t ) ) .
The damper steady state motion is harmonic, i.e.
qw ( t ) = a sin λt + b cos λt(22)where a = col ( aw, 1, …, aw, m ) and b = col ( bw, 1, …, bw, m ) are the unknown vectors which could be

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R. Lewandowski, A. Bartkowiak and H. Maciejewski

determined from the following set of algebraic equations
Kww a – λCww b = fs, λCww a + K ww b = fc

(23)

where fs = col ( 0, 0, …, 0, k m a3 – λc m b3 ) and fs = col ( 0, 0, …, 0, km b3 + λcm a3 ) .
The energy dissipated by the damper can be calculated from the following formulaT
T
Ed = ∫ u ( t )x· ( t ) dt

(24)

0

where T = 2π ⁄ λ is the period of excitation, u ( t ) = col ( u1 ( t ), u2 ( t ), …, um ( t ) ) is the vector offorces in the Kelvin elements and x· ( t ) = col ( q· w, 2 ( t ) – q· w, 1 ( t ), …, q· 3 ( t ) – q· w, m ( t ) ) is the vector of
relative velocities of the Kelvin elements.
Taking into account relationships (1) - (3) and (22) and making the necessary integration the following
formula2

T

T

Ed = πλ ( ∆a Cd ∆a + ∆b Cd ∆b )(25)

is finally obtained, where Cd = diag ( c1, …, cm ) , ∆a = col ( aw, 2 – aw, 1, …, a3 – aw, m ) , ∆b = c ol
( b w , 2 – b w , 1 , …, b 3 – b w , m )The energy dissipated by the Maxwell damper can be obtained in a similar way. As previously, it
is assumed that the motion of the left and the right external nodes is described by relationship (20).
In this case, the motion of internal variables can be determined irrespective of each other. The
motion of the i-th Maxwell element is described by the equation
qw, i ( t ) + τi qw, i ( t ) = q1 ( t ) + τi q· 3 ( t )(26)

The damper steady state vibrations are also described by Eq. (22). After introducing Eq. (22) into
Eq. (26) the following result is obtained ( i = 1, …, m )2

2( τi λ ) b3 + τi λa3( τi λ ) a3 – τi λb3-----------------------------------(27)aw, i = -----------------------------------,
b
=
w
,
i
2
2
1 + ( τi λ )
1 + ( τi λ )The dissipation energy is also calculated from relationship (24) from which the final formula (25)
is obtained but the definitions of vectors ∆a and ∆b are slightly different, i.e.

∆a = col ( a3 – aw, 1, …, a3 – aw, m ), ∆b = col ( b3 – bw, 1, …, b3 – bw, m )(28)

The energy dissipated by the simple Kelvin damper and by the simple Maxwell damper can becalculated in a similar way. The final formulae are given in Appendix C.
Some explanation is needed for the fractional-derivative Kelvin model. In this case, x ( t ) = q3 ( t ) – q1 ( t ) ,the relationships (20) are also valid and after introducing (20) into (14) and taking into account that(compare Podlubny (1999))
α

α

α

α

Dt sin λt = λ sin ( λt + απ ⁄ 2 ), D t cos λt = λ cos ( λt + απ ⁄ 2 )(29)

From (24) it is obtained the following final result22

Ed = πλc ( ab + b3 )sin ( απ ⁄ 2 )(30)

The energy dissipated by the damper, modeled using the complex modulus model is2
Ed = πK″ ( λ )q30

(31)

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121

4. Determination of dynamic characteristic of structures with VE dampers
The considered frame structures with VE dampers are modeled using the finite element method.
The structure is treated as the linear system. The typical two node bar element with the sixth nodal
parameters is used to describe the structure. The mass and stiffness matrices together with thevector of nodal forces of the element can be found in many books. The equation of motion of thestructure with VE dampers modeled using the generalized rheological models can be written in the
form
(32)
Mss q·· s ( t ) + Css q· s ( t ) + Csd q· d ( t ) + Kss qs ( t ) + Ksd qd ( t ) = ps ( t )·(33)
Cds q ( t ) + Cdd q· d ( t ) + K ds qs ( t ) + Kdd qd ( t ) = 0
TT

where symbols Mss, Css, Csd = Cds , Cdd, Kss, K sd = Kds and Kdd denote the mass, damping andstiffness matrices of the structure with dampers, respectively, written in the global coordinate
(s)
(d )
(s )
(d )
system. The dimension of matrices Mss, Css = Css + Css and Kss = K ss + K ss is (n × n). Matrices(s )(s)Mss, Css and Kss describe the inertia, damping and elastic properties of the structure without
(d)
(d)
T
T
dampers, while the matrices Css , K ss and the (n × r) matrices Csd = Cds , Ksd = K ds represent
the effect of the coupling of dampers with the structure. The (r × r) matrices Cdd and Kdd describe
the damping and stiffness properties of dampers with braces, respectively. Moreover, qs(t), qd(t) andPs(t) are the global vectors of nodal generalized displacements, internal variables and nodal excitation
forces, respectively. The concept of proportional damping is used to model the damping properties
(s )
(s )
of the structure, i.e.: Css = αMss + κKss where α and κ are proportionality factors.
The equation of motion written in terms of state variables will also be useful. After introducing
the following state vector x ( t ) = col ( qs ( t ), q· s ( t ), qd ( t ) ) and adding the equation(34)
Mss q· s ( t ) – Mss q· s ( t ) = 0
to a system of Eqs. (32) and (33) the following state equation could be written
Ax· ( t ) + Bx ( t ) = s ( t )

(35)

where
⎧ p( t ) ⎫⎪⎪0 0 , B = 0 –Mss 0 , s ( t ) = ⎨ 0 ⎬⎪⎪Kds 0 Kdd
0 C dd⎩ 0 ⎭Css Mss Csd

A = Mss
Cds

Kss

0

Ksd

(36)

Please note that the matrices A and B are symmetric and the matrix B is non-singular.
The solution to the homogenous state equation, i.e., when s ( t ) = 0 , is assumed in the formx ( t ) = a exp ( st )(37)

( sA + B )a = 0(38)

The linear eigenvalue problem
must be solved to determine the (2n + r) eigenvalues si and eigenvectors ai. In the case of anundercritically damped structure the 2n eigenvalues (eigenvectors) are the complex and conjugate
numbers (vectors) while the remaining r eigenvalues (eigenvectors) are the real numbers (vectors).
The dynamic behavior of a frame with VE dampers is characterized by the natural frequencies ωi

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R. Lewandowski, A. Bartkowiak and H. Maciejewski

and the non-dimensional damping parameters γi. The above-mentioned quantities are defined as222

ω i = µ i + ηi , γi = –µi ⁄ ω i(39)

where µ i = Re ( si ) , ηi = Im ( si ) . The formulae (39) refer to complex eigenvalues only.
The third dynamic characteristic of the considered system are the frequency response functions.
To determine these functions the steady state harmonic responses of the system are considered. If
the excitation forces vary harmonically in time, i.e., whenp ( t ) = P exp ( iλt )(40)then the steady state response of the structure and the vector of state variables can be expressed asqs ( t ) = Qs exp ( iλt ), qd ( t ) = Qd exp ( iλt )(41)

After substituting relationships (40) and (41) into Eqs. (32) and (33), which can be rewritten in
the form(42)
Mq·· ( t ) = Cq· ( t ) + Kq ( t ) = p ( t )where⎧ qs ( t ) ⎫⎧ p( t) ⎫Kss Ksd
C C
M = Mss 0 , C = ss sd , K =, q(t) = ⎨⎬, p ( t ) = ⎨⎬Cds CddKds K dd
⎩ qd ( t ) ⎭
⎩ 0 ⎭
0 0

(43)

the following relationship describes the matrix of the frequency response functions
2

H ( λ ) = ( – λ M + iλC + K )

–1

(44)

If the structure with dampers modeled by the simple Maxwell model is considered, then all of therelationships presented above in this section are valid provided that the matrices given in Appendix
C are used to generate the global matrices appearing in Eqs. (32) and (33).
The vector of internal variables qd ( t ) and the equation of motion (33) are not present in the caseof the structure with dampers modeled by the simple Kelvin model, and the motion Eq. (32) takes
the form(45)
Mss q·· s ( t ) + ( Css + Cdd )q· s ( t ) + ( K ss + Kdd )qs ( t ) = ps ( t )The matrices Cdd and Kdd appearing in (45) are built from the matrices Cd and Kd, respectively,
given by formulae (C.2).
The state equation has the form of Eq. (35) where now⎧ qs ( t ) ⎫⎧p(t) ⎫
K + Kdd 0
Css + C dd Mssx( t) = ⎨, B = ss
, s(t) = ⎨
, A=
⎬
⎬
0
–Mss
Mss
0
⎩ q· s ( t ) ⎭
⎩ 0 ⎭

(46)

Moreover, the matrix of frequency response functions is defined as
2

H ( λ ) = [ – λ Mss + iλ ( Css + Cdd ) + Kss + K dd ]

–1

(47)

In the case of structure with dampers modeled using the fractional-derivative Kelvin model theequation of motion has the form (see paper by Lewandowski and Pawlak 2010)2

1

α

Mss Dt qs ( t ) + Css Dt qs ( t ) + Cdd Dt qs ( t ) + ( Kss + Kdd )qs ( t ) = ps ( t )(48)

Dynamic analysis of frames with viscoelastic dampers: a comparison of damper models

123

2
1
where symbols Dt qs ( t ) = q·· s ( t ) and Dt qs ( t ) = q· s ( t ) are used in order to be consistent with thenotation. Here, it is assumed that for all dampers the values of parameter α are identical.
The state equation is1

αADt z ( t ) + A1 Dt z ( t ) + Bz ( t ) = s ( t )1where z ( t ) = col ( qs ( t ), Dt qs ( t ) ),

A=

(49)

s ( t ) = col ( ps ( t ), 0 )Css Mss

K + Kdd 0, A1 = Cdd 0 , B = ss
0
0
–Mss
0 0

Mss

(50)

The eigenvalue problem which must be solved to determine the eigenvalue s and the eigenvector
a is the nonlinear one and is in the following formα

( sA + s A1 + B )a = 0(51)

The above nonlinear eigenvalue problem can be solved using the continuation method describedby Lewandowski and Pawlak (2010) and the relationships (39) are used to determine the natural
frequencies and non-dimensional damping ratios.
After applying the Laplace transformation on Eq. (48) and assuming zero initial conditions andps ( t ) = 0 , we obtain the following eigenvalue problem, alternative to (51)2α( s Mss + sCss + s Cdd + Kss + K dd )a = 0(52)

The matrix of frequency response functions is defined as
2αH ( λ ) = [ – λ Mss + iλCss + ( iλ ) C dd + Kss + Kdd ]

–1

(53)

The equation of motion of structure with VE dampers modeled using the complex modulus model
can be written in the form
(54)
Mss q··s ( t ) + ( Kss + K dd )qs ( t ) = ps ( t )where Kdd is the global, complex stiffness matrix of dampers.
Assuming that ps ( t ) = 0 and qs ( t ) = a exp ( ist ) we obtain, from Eq. (54), the following nonlinear
eigenvalues problem
2( Kss + K′dd ( ω ) + i K″dd ( ω ) – s Mss )a = 0(55)

where the global matrices K′dd ( ω ) and K″dd ( ω ) are built from the matrices K′d ( ω ) and K″d ( ω )defined in Appendix C and ω is the natural frequency of vibration calculated using eigenvalue sand Eq. (39.1).
The iterative procedure was used to solve the eigenvalue problem (55). At the beginning of theiteration process the matrices K′dd ( ω ) and K″dd ( ω ) are calculated for ω = 0 and the resulting
linear eigenvalue problem is solved. Next, for the chosen eigenvalues s i and the corresponding ωi,the matrices K′dd ( ω ) and K″dd ( ω ) are recalculated and the next linear eigenvalue problem issolved. The iteration process is considered to have converged when(k)(k – 1)(k)(k )(k – 1)(k)
ωi – ωi≤ εω i , γ i – γ i≤ εγ i(k – 1 )are the current and previous values
γi

(56)
where
and
of natural frequency anddamping ratio of interest, respectively, and ε is the convergence tolerance.
(k)(k – 1)(k )ωi , ωi , γ i

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R. Lewandowski, A. Bartkowiak and H. Maciejewski

In this case, the matrix of frequency response functions is defined by
2

H ( λ ) = [ Kss + K′dd ( λ ) + iK″dd ( λ ) – λ Mss ]

–1

(57)

One important remark can be made concerning the equivalency of the fractional-derivative Kelvinmodel and the complex modulus model of damper. If Css = 0 and assuming that all dampers haveidentical properties and the storage and loss modulus of the fractional-derivative Kelvin model andthe complex modulus model are also identical then, taking into account relationships (C.2), (C.9)
and (D.6), we can writeααK′dd ( ω ) = Kss + C dd ω cos ( απ ⁄ 2 ), K″dd ( ω ) = Cdd ω sin ( απ ⁄ 2 )

(58)

α

Since i = cos ( απ ⁄ 2 ) + i sin ( απ ⁄ 2 ) , we can conclude that Eqs. (52) and (55) will be identical ifααs = ω . This, however, is not true which means that eigenvalue problems (52) and (55) havedifferent solutions and the considered models of dampers are not identical.

5. Comparison of dynamic characteristics of structure with VE dampers modelledusing different rheological models
5.1 Description of a representative structure
An eight-storey RC frame with three bays and with insufficiently stiff beams is selected forcomparison of the influence of the different models of VE dampers on the dynamic characteristics
of a structure with dampers. The frame is designed according to the EC8 Part 1 for class B soils.
The height of the columns is 3.0 m, the span of the beams is 5.0 m and Young’s modulus forconcrete is 31.0 GPa. The dimensions of the cross-section of the structural elements are presented inTable 1 while the unit masses of the frame elements are given in Table 2. The data of the structure
Table 1 Dimensions of the eight-storey frame elements
Storey level

Lateral column[cm]

Central column
[cm]

Beams
[cm]

7, 8
5, 6
3, 4
1, 2

35×35
40×40
45×45
50×50

40×40
45×45
53×53
60×60

30×40
30×45
30×50
30×50

Table 2 Unit mass of the eight-storey frame elements
Storey level

Unit lateral column
mass[kg/m]

Unit central column
mass[kg/m]

Unit beam mass
[kg/m]

7, 8
5, 6
3, 4
1, 2

306.2
400.0
506.2
625.0

400.0
506.2
702.2
900.0

15000.015000.015000.015000.0

Dynamic analysis of frames with viscoelastic dampers: a comparison of damper models

125

Table 3 Natural frequencies of frame without dampers
Natural frequencies [rad/sec]
3.131142.12518.658251.1550

15.4268
52.3598

23.7804
57.6067

31.264765.6532

40.1148
69.9862(except the data concerning the mass per unit length of elements) are taken from the paper by
Ribakov and Agranovich (2010).
Table 3 shows the natural frequencies of vibration of the frame without dampers. The dynamic
properties of the structure are obtained by two-dimensional analysis of the frame; axial deformations
and internal damping of structure are neglected.
5.2 Parameters of VE damper models
The dampers are attached on the sixth and seventh floors of the structure. Such location of thedampers is chosen because, in this position, the maximal peak value of the bending moments in thelowest columns is minimal when the structure is subjected to forces produced by the El Centro
earthquake. The dampers’ location is determined as follows: The equation of motion of structurewith one damper connected at the first floor is solved using the Newmark method and the maximal
peak value of the bending moments is determined. The calculation is repeated for all the possible
positions of the dampers. The position which corresponds to the minimal value of the abovementioned maximal peak value of the bending moments in the columns is chosen as the optimal
one. When the position of the first damper is fixed, the same procedure is used to determine theposition of the second damper. Of course, the optimal positions of the dampers obtained for one
earthquake are not necessarily optimal for other earthquakes. However, optimization of dampers’
positions is beyond the scope of our paper. On the other hand, the dampers’ positions chosen in thispaper are, in some degree, similar to those determined by Agranovich and Ribakov (2010) who
found that, on the considered frame, dampers must be connected at the 5th, 6th and 7th floors.
The dampers are modeled using the following rheological models: i) the complex modulus model,ii) the fractional-derivative Kelvin model; iii) the simple Kelvin model; iv) the simple Maxwell
model; v) the generalized Kelvin model with three, five, and seven parameters, and vi) thegeneralized Maxwell model also with three, five, and seven parameters.
In this paper we do not adopt data from the real experiment. Instead, the storage and loss modulus
are calculated from the formulae (D6), which means that the fractional-derivative Kelvin model canbe considered as being exact. The chosen parameters of the fractional-derivative Kelvin model are:66α
α = 0.63 , k = 0.8 × 10 N ⁄ m and c = 7.2 × 10 N sec ⁄ m .
The parameters of the generalized Kelvin model and the generalized Maxwell model, both withseven parameters, are given in Table 4. In Fig. 5 the loss modulus, and the dissipation energy of theabove-mentioned models are compared. The dissipation energy is calculated assuming that theamplitude of vibration of damper is equal to 0.01 m in all of the considered cases. From thesefigures it can be concluded that the loss modulus and the dissipation energy of the fractionalderivative Kelvin model, the complex modulus model, and both generalized models are almost
identical in the considered range of excitation frequency, i.e., in the range 0-15.0 rad/sec. This rangeof frequency covers the range of the first three natural frequencies of vibration of the consideredstructure.

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R. Lewandowski, A. Bartkowiak and H. Maciejewski

Table 4 Parameters of generalized Kelvin and Maxwell models
Stiffness (×106) [Ν/m]
k0
k1
k2
k3

Damping factor (×106) [Ν/m]Kelvin model

Maxwell model

Kelvin model

Maxwell model

115.300
36.71012.3201.109

0.213
66.770
6.621
2.886

–
5.458
12.38017.350

–
2.957
3.463
16.610c1
c2
c3

Fig. 5 Comparison of loss modulus

Fig. 6 Comparison of dissipation energy

Fig. 7 Comparison of dissipation energy for threeparameter models

Fig. 8 Comparison of loss modulus for simple models

The values of three and five parameters of the generalized Kelvin and Maxwell models are also
given in Table 4. For three-parameter models the k0, k1 and c1 values are used while k0, k1, k2, c1
and c2 are the values of the five-parameter models. The comparison of energy dissipated in dampers
modeled by the above-mentioned three-parameters models and the fractional-derivative model are
shown in Fig. 6. We see that the accuracy of approximation of dissipated energy increases with theDynamic analysis of frames with viscoelastic dampers: a comparison of damper models

127

Fig. 9 Comparison of dissipation energy for simple models

number of parameters of generalized models.
7
7
The parameters of the simple Kelvin model are: k = 1.49275 × 10 N ⁄ m , c = 0.268839 × 10 N ⁄ m
7and the values of parameters of the simple Maxwell model are: k = 3.80783 × 10 N ⁄ m and7
c = 0.677338 × 10 N ⁄ m . These parameters are calculated by minimizing the mean square norm ofdifference between the target modules and the analytical modules of the respective model. Comparison
between the obtained loss module and the target ones is shown in Fig. 7. Moreover, the dissipationenergy of simple models and the dissipation energy of the fractional-derivative model are compared inFig. 8. As expected, the differences are significant, especially for the simple Maxwell model.
Chevron-braces connect the dampers with the structure. The braces are made of HEB 200 stainless steel
972
profiles of which the parameters are: EA = 1.60105 × 10 N and EJ = 1.1685 × 10 Nm .
5.3 Comparison of dynamic characteristics of considered frame
Results of the solution to the eigenvalue problems are presented in Tables 5-10. The real and thecomplex conjugate numbers are obtained as eigenvalues. In Tables 5 and 6 the values of the first
three complex conjugate eigenvalues are given for all of the considered models. Results concerning
the fractional-derivative Kelvin model and the complex modulus model are given in Table 5.
Table 5First three complex conjugate eigenvalues for a frame with dampers modeled using the fractional
derivative Kelvin model and the complex modulus model
Fractional derivative Kelvin model of damperst

1 eigenvalue

2nd eigenvalue

3rd eigenvalue- 0.065729 ± i 3.20576- 0.85650 ± i 9.46271- 0.492826 ± i 15.6919

Complex modulus model of damperst

1 eigenvalue

2nd eigenvalue

3rd eigenvalue

3.20630 ± i 0.064877

9.49174 ± i 0.808670

15.6958 ± i 0.482061

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R. Lewandowski, A. Bartkowiak and H. Maciejewski

Table 6 First three complex conjugate eigenvalues for frame with dampers
Kelvin model of damper with
7 parameters

5 parameters

3 parameters

2 parameters-0.06490 ± i 3.20697-0.86607 ± i 9.45473-0.49557 ± i 15.7308-0.05740 ± i 3.23237-0.83770 ± i 9.67058-0.45514 ± i 15.8313-0.02938 ± i 3.2751-0.66116 ± i 10.010-0.35155 ± i 15.964-0.03789 ± i 3.22543-0.89249 ± i 9.35764-0.67370 ± i 15.6689

Maxwell model of damper with
7 parameters

5 parameters

3 parameters

2 parameters-0.06429 ± i 3.2066-0.84920 ± i 9.5529-.40766 ± i 15.774-0.07035 ± i 3.192-0.89831 ± i 9.462-0.42859 ± i 15.74-0.06537 ± i 3.1553-0.97980 ± i 9.2092-0.47366 ± i 15.660-0.08317 ± i 3.2176-0.46152 ± i 9.7251-0.15489 ± i 15.701

Table 7 Real eigenvalues for frame with dampers
Kelvin model of damper with
7 parameters- 0.366069- 0.378653- 2.55708- 2.71947- 21.2562- 25.6171

5 parameters- 2.12886- 2.27026- 18.7723- 22.7386

3 parameters- 15.4856- 17.4200

2 parameters- 93.46470- 120.536- 30098.8- 30104.3-

Maxwell model of damper with
7 parameters

5 parameters

3 parameters

2 parameters- 0.161840- 0.166646- 1.64402- 1.74088- 12.7486- 14.3179- 1.62449- 1.73319- 12.5245- 14.1509- 11.8525- 13.6682- 3.09734- 3.67824-

Table 8 The natural frequencies and non-dimensional damping ratios of frame with dampers – the fractional
derivative model and the and complex modulus model
Natural frequencies [rad/sec]

Non-dimensional damping ratios

Fractional
Kelvin model

Complex modulus
model of damper

Fractional
Kelvin model

Complex modulus
model of damper

3.206449.50139
15.69960

3.20695
9.5261215.70320

0.020499
0.0901444
0.031391

0.020230
0.0848897
0.0306982

Dynamic analysis of frames with viscoelastic dampers: a comparison of damper models

129

Table 9 The natural frequencies of frame with dampers
Naturalfrequency
ω1ω2ω3Naturalfrequency
ω1
ω2
ω3

Kelvin model of damper with
7 parameters[rad/sec]

5 parameters[rad/sec]

3 parameters[rad/sec]

2 parameters[rad/sec]

3.207629.4943115.7386

3.232889.70680
15.8378

3.27528
10.031415.9674

3.22565
9.4001015.6834

Maxwell model of damper with
7 parameters[rad/sec]

5 parameters[rad/sec]

3 parameters[rad/sec]

2 parameters[rad/sec]

3.207299.59056
15.7789

3.19275
9.50468
15.7483

3.15601
9.26113
15.6673

3.218659.7360315.7020

Table 10 Non-dimensional damping ratios of frame with dampers
Kelvin model of damper with

Dampingratio

7 parameters

5 parameters

3 parameters

2 parameters

γ1
γ2
γ3

0.0202338
0.0912198
0.0314874

0.0177562
0.0863001
0.0287375

0.0089706
0.0659092
0.0220165

0.0117468
0.0949451
0.0429562

Maxwell model of damper with

Dampingratio

7 parameters

5 parameters

3 parameters

2 parameters

γ1
γ2
γ3

0.020046
0.088543
0.025836

0.022035
0.094512
0.027215

0.020712
0.105796
0.030232

0.025839
0.047404
0.009864

Because the solution to the equation of motion (54) is taken in the form qs ( t ) = a exp ( ist ) , the realor imaginary parts of s must be compared with the imaginary or real part of eigenvalue s of theeigenvalue problem (51), respectively. Results show small differences of which the source is
explained at the end of Section 3. The maximal differences reach 5.6% and 0.3% for the real andimaginary parts of s, respectively.
In Table 6 the first three complex eigenvalues are presented when the classical rheological modelsof dampers are used. The model with two parameters refers to the simple Kelvin or Maxwell model.
After comparing the results obtained for the Kelvin and Maxwell models with seven parameterswith the results obtained for the fractional derivative Kelvin model, they are observed to be in goodor satisfactory agreement. The maximal differences are 1.3% and 0.25% for the real and imaginary
part of the Kelvin model with seven parameters, respectively. For the Maxwell model with sevenparameters these differences are 17.3% and 0.95%, respectively. Similar differences are observed
when we compare results obtained for the Kelvin or Maxwell model with seven parameters and theresults obtained for the complex modulus model.
For the simple Kelvin model discussed, maximal differences are 42.0% and 1.1%, respectively.

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R. Lewandowski, A. Bartkowiak and H. Maciejewski

Moreover, when dampers are modeled using the simple Maxwell model, the above mentioned
differences are 68.0% and 2.8% and they are greater than for the simple Kelvin model.
Real eigenvalues are also solutions to the eigenvalue problems when dampers are modeled by theclassical rheological models. In the case of the generalized Kelvin and Maxwell models the real
eigenvalues reflect the dynamics of internal variables and the number of real eigenvalues is equal to
the number of internal variables. Comparing the values of real eigenvalues obtained for both
discussed models we see that these values are of the same order but the differences between them
are significant. It can be observed that by increasing the number of parameters in the generalized
Kelvin or Maxwell model we obtain values which show some convergence trend. If the simple
Maxwell model is used to damper modeling then we also obtain the real eigenvalues of which thenumber is equal to the number of internal variables and which also reflect the dynamics of internalvariables.
The real eigenvalues of the eigenvalue problem obtained for the frame with dampers modeled by
the simple Kelvin model, which has no internal variables, indicate that two modes of vibration are
overdamped. It is the qualitative difference in comparison with the results obtained for other modelsof dampers. The existence of the discussed real eigenvalues depends on the values of the parametersof the simple Kelvin model.
The first three obtained natural frequencies and non-dimensional damping ratios are presented inTables 8-10. In terms of these quantities, conclusions concerning differences between the obtained
results are similar. Natural frequencies and non-dimensional damping ratios obtained for the framewith dampers modeled using the fractional-derivative Kelvin model and the complex modulus are
given in Table 8. Relative differences are not greater than 0.26% (natural frequencies) or 5.8%
(damping ratios) which means they are in good agreement.
In Table 9 the natural frequencies are given for the frame when dampers are modeled using thesimple and generalized Kelvin and Maxwell models. In a similar way, in Table 10, the nondimensional damping ratios are presented. The most accurate results are for the seven-parameter
Kelvin and Maxwell models. Relative differences between results obtained for the discussed Kelvinmodel and the fractional-derivative Kelvin model are not greater than 0.25% (natural frequencies)
and 1.2% (damping ratios). For the seven-parameter Maxwell model the discussed differences are
not greater than 1.0% (natural frequencies) and 16.8% (damping ratios). In conclusion we can saythat the first three natural frequencies and the non-dimensional damping ratios of the frame show
good or satisfactory agreement when VE dampers are modeled by the fractional-derivative model,
the complex modulus model or the seven-parameter Kelvin and Maxwell models.
If the number of parameters of the generalized Kelvin model growth then the natural frequencies
slightly decrease and the non-dimensional damping ratios increase. Results for the generalized
Maxwell model show the opposite trends, i.e. natural frequencies slightly increase and the nondimensional damping ratios decrease with the number of model parameters.
The relative differences of natural frequencies are relatively small (not greater than 5.6%) for allof the damper models compared. However, the relative differences of non-dimensional damping
ratios could be of the order of 42.0% and 68.0% for the simple Kelvin model and the simple
Maxwell model, respectively. Consequently, the simple rheological models do not accurately
describe the dissipation property of the frame with VE dampers. The three parameter Kelvin model
also cannot be recommended as the appropriate one because the discussed differences reach 56.0%.
If we accept the relative differences of the order 18.0% then the remaining classical rheologicalmodel could be used to model VE dampers. The relatively large difference of γ3 (i.e. 17.7%) exists

Dynamic analysis of frames with viscoelastic dampers: a comparison of damper models

131

Fig. 10 Modulus of the frequency response function H 36, 36 ( λ ) for fractional derivative Kelvin model (thickline), the simple Kelvin model (thin line with crosses), and for the simple Maxwell model (thin linewith rhombs)

for the seven-parameter Maxwell model.
Finally the frequency response functions for the structure with differently modeled VE dampers
are compared. Here the element H ij ( λ ) of the matrix frequency response function is interpreted as
the displacement of the i-th degree of freedom of the structure subjected to the unit harmonicallyvarying force at the j-th degree of freedom. The frequency response function H36, 36 ( λ ) is
compared. The displacement number 36 is the horizontal displacement of the eight storeys. The
discussed frequency response function calculated for frame with dampers modeled by the fractionalderivative model, the complex modulus model and the seven-parameter Kelvin model are almost
identical. The maximal peak differences in the resonances are of the order of 1.0% for all of theconsidered resonance areas. These differences are of the order of 20.0% in the third resonance area
when results obtained for the seven-parameter Maxwell model are compared with those obtained forthe fractional-derivative Kelvin model.
For the simple Maxwell and Kelvin models, the discussed differences are the greatest. In Fig. 10,
comparison of modulus of H36, 36 ( λ ) obtained for the frame with dampers modeled using thefractional-derivative Kelvin model (the thick line), the simple Kelvin model (the thin line withcrosses), and the simple Maxwell model (the thin line with rhombs) are shown. Importantdifferences between the compared frequency functions are visible.

6. Conlusions
Several models of dampers, used to describe the dynamic behaviour of frame structures with VE
dampers, are considered and compared in this paper. The fractional-derivative Kelvin model, thecomplex modulus model and a family of rheological models, including the very often used simple
Kelvin and Maxwell models, are compared in full detail. The comparison is made in the frequency
domain for a carefully chosen frame structure with VE dampers. The formulae for the energy
dissipated in a damper modelled in different ways are derived.
Several conclusions can be formulated on the basis of the results of numerical analysis presented
above. The most important ones are listed below.
1. Different models are able to correctly describe the dynamic behaviour of VE dampers. In

132

R. Lewandowski, A. Bartkowiak and H. Maciejewski

particular, the fractional-derivative Kelvin model, the complex modulus mode, and the sevenparameter Kelvin model give us almost identical results. Results suggest that the generalized
Maxwell model could also be used if the number of model parameters is sufficient. This conclusion
is in agreement with the results presented by Singh and Chang (2009) where the generalized Kelvinand Maxwell models are used as models of VE dampers.
2. The simple Kelvin and the simple Maxwell model are not able to correctly describe, in thefrequency domain, the dynamic behaviour of frames with VE dampers. In particular, relative
differences concerning the non-dimensional damping ratios are great.
3. In the considered case, the differences between the results obtained for the generalized Maxwell
model are greater in comparison with the results obtained using the generalized Kelvin model ifboth models have an identical number of parameters.
4. The needed number of parameters of the generalized Kelvin and Maxwell models and thefractional-derivative model depends on the frequency range for which the storage and loss modulus
of VE dampers must be approximated. Smaller numbers of parameters are needed for the fractionalderivative model.
5. The nonlinear eigenvalue problems must be solved when the complex modulus model or thefractional-derivative model of VE dampers are chosen. The solution procedure for these problems ismore complicated than the solution procedure for the linear eigenvalue problems which are obtained
for the remaining models.
6. There are some qualitative differences between the results obtained. For the frame with a fixed
number of VE dampers the total number of eigenvalues and eigenvectors depends on the chosenmodel of dampers and the number of parameters of the models. Moreover, for the generalized
Kelvin and Maxwell models and the simple Maxwell model both the real and complex numbers are
eigenvalues while only the complex numbers are obtained (for sufficiently small damping) when theremaining models are chosen.
7. The dissipative energy is a good measure of equivalence of different models of VE dampers.
Acknowledgments
The first author wishes to acknowledge the financial support received from the Poznan Universityof Technology (Grant No. DS 11-068/12) in connection with this work.

References
Agranovich, G. and Ribakov, Y. (2010), “A method for efficient placement of active dampers in seismicalle
excited structures”, Struct. Control Hlth. Monit., 17, 513-531.
Barkanov, E., Hufenbach, W. and Kroll, L. (2003), “Transient response analysis of systems with different
damping models”, Computer Meth. Appl. Mech. Eng., 192, 33-46.
Chang, T.S. and Singh, M.P. (2009), “Mechanical model parameters for viscoelastic dampers”, J. Eng. Mech.,
135, 581-584.
Chang, T.S. and Singh, M.P. (2002), “Seismic analysis of structures with a fractional derivative model ofviscoelastic dampers”, Earthq. Eng. Eng. Vib., 1, 251-260.
Christopoulos, C. and Filiatrault, A. (2006), Principles of passive supplemental damping and seismic isolation,
IUSS Press, Pavia, Italy.
Connor, J.J. and Klink, B.S.A. (1996), “Introduction to motion-based design”, WIT Press.

Dynamic analysis of frames with viscoelastic dampers: a comparison of damper models

133

Fujita, K., Moustafa, A. and Takewaki, I. (2010) “Optimal placement of viscoelastic dampers and supporting
members under variable critical excitations”, Earthq. Struct., 1, 43-67.
Hatada, T., Kobori, T., Ishida, M.A. and Niwa, N. (2000), “Dynamic analysis of structures with Maxwell
model”, Earthq. Eng. Struct. Dyn. Earthq., 29, 159-176.
Lee, S.H., Son, D.I., Kim, J. and Min, K.W. (2004), “Optimal design of viscoelastic dampers using eigenvalue
assignment”, Earthq. Eng. Struct. Dyn., 33, 521-542.
Lewandowski, R. and Chor ça zí yczewski, B. (2010), “Identification of the parameters of the Kelvin-Voigt and theMaxwell fractional models, used to the modeling of viscoelastic dampers”, Compos. Struct., 88, 1-17.
Lewandowski, R. and Chor aç zí yczewski, B. (2007), “Remarks on modelling of passive viscoelastic dampers,
Proceedings of the 9th International Conference Modern Building Materials, Structures and Technique, Vilnius,
Lithuania, May.
Lewandowski, R. and Pawlak, Z. (2011), “Dynamic analysis of frames with viscoelastic dampers modelled byrheological models with fractional derivatives”, J. Sound Vib., 330, 923-936.
Matsagar, V.A. and Jangid, R.S. (2005), “Viscoelastic damper connected to adjacent structures involving seismic
isolation”, J. Civ. Eng. Mana., 11, 309-322,
Mazza, F. and Vulcano, A. (2007) “Control of the along-wind response of steel framed buildings by using
viscoelastic or friction dampers”, Wind Struct., 10, 233-247.
Okada, R., Nakata, N., Spencer, B.F., Kasai, K. and Kim, B.S. (2006), “Rational polynomial approximation
modeling for analysis of structures with VE dampers”, J. Earthq. Eng., 10, 97-125.
Park, J.H., Kim, J. and Min, K.W. (2004), “Optimal design of added viscoelastic dampers and supporting
braces”, Earthq. Eng. Struct. Dyn., 33, 465-484.
Podlubny, I. (1999), Fractional Differential Equations, Academic Press.
Ribakov, Y. and Agranovich, G. (2011), “A method for design of seismic resistant structures with viscoelastic
dampers”, Struct. Des. Tall Spec. Build., 20, 566-578.
Shen, K.L., Soong, T.T., Chang, K.C. and Lai, M.L. (1995), “Seismic behaviour of reinforced concrete framewith added viscoelastic dampers”, Eng. Struct., 17, 372-380.
Singh, M.P. and Chang, T.S (2009), “Seismic analysis of structures with viscoelastic dampers”, J. Eng. Mech.,
135, 571-580.
Singh, M.P. and Moreschi, L.M. (2002), “Optimal placement of dampers for passive response control”, Earthq.
Eng. Struct. Dyn., 31, 955-976.
Singh, M.P., Verma, N.P. and Moreschi, L.M. (2003), “Seismic analysis and design with Maxwell dampers”, J.
Eng. Mech., 129, 273-282.
Sorrentino, S. and Fassana, A. (2007), “Finite element analysis of vibrating linear systems with fractional
derivative viscoelastic models”, J. Sound Vib., 299, 839-853.
Shukla, A.K. and Datta, T.K. (1999), “Optimal use of viscoelastic dampers in building frames for seismic force”,
J. Struct. Eng., 125, 401-409.
Takewaki, I. (2009) Building control with passive dampers, Optimal performance-based design for earthquakes,
Wiley and Sons (Asia), Singapore.
Tsai, M.H. and Chang, K.C. (2002), “Higher-mode effect on the seismic responses of buildings with viscoelastic
dampers”, Earthq. Eng. Eng. Vib., 1, 119-129.
Xu, Z.D. (2007), “Earthquake mitigation study of viscoelastic dampers for reinforced concrete structures, Vib.
Control, 13, 29-43.
Zhang, W.S. and Xu, Y.L. (2000), “Vibration analysis of two buildings linked by Maxwell model defined fluid
dampers”, J. Sound Vib., 233, 775-796.

134

R. Lewandowski, A. Bartkowiak and H. Maciejewski

Appendix A. Finite element matrices of the generalized Kelvin model of VE damper
In this appendix the explicit form of the matrices used to describe the generalized Kelvin modelof the VE damper is given.
Kzz Kzw

Kd =

Kww =

Kwz Kww, Cd =

Czz Czw(A.1)

Cwz Cwwk0 + k1–k1 ……0

0

0……0

0

–k 1

k1 + k 2 …0

0

0……0

0

………… ………… …………
0
0…… – ki – 1 ki – 1 + k1…………
–ki ……0

…………
0

…………
0

…………
0………0

…………
0… – k m – 1…………
km – 1 + km

c1

–c1 ……0

0

0……0

0

–c1

c1 + c2 …0

0

0……0

0

…………
0

Cww = ………… ………… ………… ………… …………
–ci …00
0
0… – ci – 1 ci – 1 + ci

(A.2)

(A.3)

………… ………… ………… ………… …………
0
0……0
0
0… – c m – 1 cm – 1 + cm2

c k0 csk0
Kzz =T

Kzw = K wz =00

csk0 s k0 0

0

2

2

0

0 c km cskm

0

0 cskm s km2

–ck0 0…0…

0–sk0 0…0…

0

0

0…0… –ck m

0

0…0… –skm, Czz =

0
0

0
0

0
0

0

0

c cm csc m

00

cscm csc mT, Czw = Cwz =

0
0

2

0
0
0

0…0…
0
0…0…
0
0…0… –ccm

0

0…0… –scm

(A.4)

(A.5)

Dynamic analysis of frames with viscoelastic dampers: a comparison of damper models

135

Appendix B. Finite element matrices of the generalized Maxwell model of VE damper
In this appendix the explicit form of matrices used to describe the generalized Maxwell model ofthe VE damper is given.
K=

K zw =

K zz K zw
Kwz Kww, Cd =–k1–k2 …… –k i ……

–km

0
0
0

0……
0……
0……

0
0
0

0……
0……
0……

Kww = diag ( k1, k2, ……, km ),(B.1)

Cwz Cww0 0…… 0…… 0
0 0…… 0…… 0–c1 –c2 … –ci … –cm, Czw =

0

0…… 0……

–c k0

–csk0

m
m
K zz = cs ⎛⎜ k + k ⎞⎟ s2 ⎛⎜ k + k ⎞⎟0
∑ 1⎠ ⎝ 0 ∑ i⎠
⎝
i=1
i=1

–csk0

–s k0

2

–csk0
2

–csk0
0
0
0

0
0
0

2

–s k0

0
0

0
0

m

m

c ∑ ci cs ∑ ci ,
i=1
m

0

0

2

i=1
m

2

2

c k0

csk0

csk0

s k0

T
Czw = Czw =

(B.3)

(B.4)

2

0
0…… 0……
0
0
0…… 0……
0–cc1 –cc2 … –cci … –ccm

(B.5)

–sc1 –sc2 … –sci … –scmcs ∑ c i s ∑ ci
i=1

2

(B.2)

0

Cww = diag ( c1, c2, ……, cm )

m
m
⎞ ⎛
⎞
2⎛
c ⎜ k 0 + ∑ k 1⎟ cs ⎜ k0 + ∑ ki⎟
⎝
⎠ ⎝
⎠
i=1
i=1

–c k0

Czz =

Czz Czwi=1–ck1 –ck2 … –cki … –ckm
T

K zw = Kzw =–sk1 –sk2 … –ski … –skm
0
0

0…… 0……
0…… 0……

0
0

(B.6)

136

R. Lewandowski, A. Bartkowiak and H. Maciejewski

Appendix C. Finite element matrices of the simple and fractional-derivative Kelvinmodel, the simple Maxwell model and the complex modulus model of the VE damper
The explicit form of the vectors and matrices used to describe the simple and fractional-derivative
Kelvin model of VE damper is
qd ( t ) = qz ( t ) = col ( q1 ( t ), q2 ( t ), q3 ( t ), q4 ( t ) )c

22

cs –c –cs
2

c

2

2

(C.1)
2

cs –c –cs22

Kd = Kzz = k1 cs s –cs –s , Cd = Czz = c1 cs s –cs –s2222–c –cs c cs
–c –cs c cs
2

–cs –s cs s22

–cs –s cs s

(C.2)

2

The explicit form of the matrices used to describe the simple Maxwell model of VE damper is
qd ( t ) = col ( qz ( t ), qw ( t ) ), qz ( t ) = col ( q1 ( t ), q2 ( t ), q3 ( t ), q4 ( t ) ), qw ( t ) = col ( qw, 1 ( t ) )Kzz K zw

Kd =K wz K ww, Cd =

Czz Czw
Cwz Cww2–s2
TKzz = k cs s 0 0 , Kzw = K zw = k1
, Kww = [ k1 ]0
0 0 0 00
0 0 0 00 0 0 00 0 0 02

0 0 c cs
0 0 cs s

(C.4)

–c

c cs 0 0

Czz = c1

(C.3)

(C.5)

0
T

, Czw = Czw = c 1

2

0
–c

, Cww = [ c1 ]

(C.6)

–s

The formulae for the energy dissipated in the damper modeled by the simple Kelvin model andfor the simple Maxwell models are22

Ed = πλc1 ( a3 + b3 )(C.7)

πc 1 λ22( a3 + b3 )Ed = -----------------2 2
1+τ λ

(C.8)

respectively.
The explicit form of matrices used to describe the complex modulus model of VE damper, in the
global coordinate systems isDynamic analysis of frames with viscoelastic dampers: a comparison of damper models

c

2

2

cs –c –c s
2

c

2

2

cs –c –cs22

2

K′d ( λ ) = K′ ( λ ) cs s –cs –s , K ″d ( λ ) = K″ ( λ ) cs s –cs –s2222–c –cs c cs
–c –cs c cs
2

–cs –s cs s

2

137

2

–cs –s cs s

(C.9)

2

Appendix D. The formulae for the storage modulus and the loss modulus of different models of VE damper
The formulae for the storage modulus K′ ( λ ) and the loss modulus K″ ( λ ) of the consideredrheological models of VE damper are:i) for the simple Kelvin model
K′ ( λ ) = k, K″ ( λ ) = cλ(D.1)ii) for the simple Maxwell model
2 2

τλ
τλ
K′ ( λ ) = k -----------------, K″ ( λ ) = k ----------------2 2
2 2
1+τ λ
1+τ λ
iii) for the generalized Kelvin model
L″ ( λ )L′ ( λ )-, K″ ( λ ) = -----------------------------------K′ ( λ ) = -----------------------------------22
2
2L′ ( λ ) + L″ ( λ )L′ ( λ ) + L″ ( λ )
where
m

(D.2)

(D.3)

m

τr λ
1
1
L′ ( λ ) = ---- + ∑ --------------------------, L″ ( λ ) = ∑ --------------------------2
2 2
2
k0
k (1 + τ λ )k (1 + τ λ )
r=1 r

r

r=1 r

2 2

m

(D.4)

r

iv) for the generalized Maxwell model
m

τr λ
τr λ
K′ ( λ ) = k0 + ∑ k1 -----------------, K″ ( λ ) = ∑ kr -----------------2 2
2 2
r = 1 1 + τr λ
r = 1 1 + τr λ(D.5)v) and finally for the fractional-derivative Kelvin ModelααK′ ( λ ) = k + cλ cos ( απ ⁄ 2 ), K″ ( λ ) = cλ sin ( απ ⁄ 2 )(D.6)