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©Civil-Comp Press, 2012
Proceedings of the Eleventh International Conference
on Computational Structures Technology,
B.H.V. Topping, (Editor),
Civil-Comp Press, Stirlingshire, Scotland
Paper 275
The Continuation Method for Dynamic Problems of
Frames with Viscoelastic Dampers
R. Lewandowski
Institute of Structural Engineering
Poznan University of Technology, Poland
Abstract
The method for determination of the dynamic characteristics of structures with
viscoelastic dampers is presented in this paper. The generalized Maxwell model is
used to describe the dynamic behaviour of dampers. The dynamic characteristics are
determined as a solution to the nonlinear eigenvalue problem which is a few times as
small as the linear eigenvalue problem resulting in the usual formulation. The
continuation method is proposed for solving the nonlinear eigenvalue problem. The
results of typical calculations are presented in order to show the accuracy and
efficiency of the proposed method of analysis.
Keywords: structures with viscoelastic dampers, dynamic characteristics,
generalized Maxwell model, nonlinear eigenvalue problem, continuation method.
1
Introduction
Viscoelastic (VE) dampers are often used to mitigate excessive vibrations of
structures. The properties of VE materials or fluids used for manufacturing VE
dampers depend on the temperature and excitation frequencies of the forces, acting
on dampers. Many rheological models, including the classical rheological models
and rheological models with fractional derivatives, are proposed for a correct
description of the dynamical behavior of VE dampers [1, 2].
The dynamic characteristics of structures with VE dampers, such as natural
frequencies, non-dimensional damping ratios and modes of vibrations, can be
obtained by solving linear or nonlinear eigenvalue problems. As shown in [1], the
linear eigenvalue problem arises when the state-space approach is used whereas the
nonlinear eigenvalue problem must be solved if equations of motion are written in a
traditional form or the damper model is described using a fractional derivative [2].
The dimension of the linear eigenvalue problem is more than twice as great as the
dimension of the nonlinear eigenvalue problem. However, methods of solution to
1
the nonlinear eigenvalue problems, obtained for VE systems or structures are less
common. Some methods are presented in papers [2 – 6] where the Biot model [3],
the complex stiffness model [4, 5] and the rheological models with fractional
derivatives [2] are used to describe VE materials or dampers. Only in [3] is the
method for determination of real eigenvalues and eigenvectors presented. In the
method, the equations of motion are transformed to a modal form using the
eigenvectors of an undamped system and introducing the small viscoelasticity
assumption. In the paper [4] the asymptotic numerical method together with the
continuation procedure is proposed for solving the nonlinear eigenvalue problem
which appears in the analysis of vibrations of viscoelastic structures.
In the paper, the problem of determination of dynamic characteristics of
structures with VE dampers is reduced to the solution of a nonlinear eigenvalue
problem of which the dimension is equal to the number of degrees of freedom of the
structure. The continuation method with an artificial main parameter is proposed to
solve the problem mentioned above. Only complex solutions, which are most
important in practice, are determined in this way.
2
Problem formulation
2.1 Rheological model of VE damper
In this paper, the generalized Maxwell model, shown in Fig. 1, is used to describe
the behaviour of the VE damper. The model is rather general and contains a number
of particular models, such as the very popular viscous model, the simple Kelvin
model and the simple Maxwell model. The model considered contains the spring,
the dashpot in parallel, which both constitute the simple Kelvin element and m-th
Maxwell elements connected in parallel (see Fig. 1).
∼ ∼
R 1 q1
∼
R2
∼
q
2
∼
y
km
cm
ki
ci
c1
k1
∼ ∼
R3 q3
∼
R4 ∼
q4
∼x
c0
k0
Figure 1: Rheological model of VE damper
According to this model, the total force in damper u (t ) is a sum of forces in
elements u i (t ) , i.e.:
m
u (t ) = ∑ u i (t ) .
i =0
2
(1)
The force in the simple Kelvin element is given by
u 0 (t ) = k 0 Δq(t ) + c0 Δq (t ) ,
(2)
while the force in the i-th Maxwell element is governed by the following equation:
ν i u i (t ) + u i (t ) = k i Δq (t ) ,
(3)
where ν i = k i / ci and k 0 , k i , c 0 , ci , (i = 1,2,..., m) denote the stiffness and damping
coefficients (see Fig.1), respectively. Moreover, Δq(t ) = q~3 (t ) − q~1 (t ) is the relative
displacement of damper (see Fig. 1).
Applying the Laplace transform, Equations (1) – (3) take the following form:
u 0 ( s ) = k 0 Δq ( s ) + s c0 Δq ( s ) ,
ui (s) =
ki s
Δq ( s ) ,
vi + s
`
m
m
⎛
ks ⎞
u ( s ) = ∑ u i ( s ) = ⎜⎜ k 0 + s c0 + ∑ i ⎟⎟Δq ( s ) ,
i =1 v i + s ⎠
i =0
⎝
(4)
(5)
where such quantities as u (s ) denote the Laplace transforms of u (t ) and s is the
Laplace variable.
2.2 Equations of motion of structure with VE dampers
The equation of motion of structure with VE dampers could be written in the
following form:
(t ) + Cq (t ) + Kq(t ) = p(t ) + f (t ) ,
Mq
(6)
where M, C, K are the (n × n) mass, damping and stiffness matrices of structure,
respectively. Moreover, q(t ), p(t ), f (t ) are the (n × 1) vectors of displacements,
excitation forces and the vector of interaction forces between the structure and
dampers. The dot denotes differentiation with respect to time t . The vector of
interaction forces is a sum of vectors f k (t ) which is caused by k-th damper, i.e.:
r
f (t ) = ∑ f k (t ) ,
(7)
k =1
where r is the total number of dampers.
Applying the Laplace transform, the equation of motion (6) can be written as:
( s 2 M + sC + K ) q ( s ) = p ( s ) + f ( s ) ,
(8)
and
r
f ( s) = ∑ f k ( s) .
k =1
3
(9)
First of all, the interaction forces caused by the k-th damper were considered. In
the global coordinate system, the vector of interactive forces caused by a single
damper could be written in the form:
mk
⎞
⎛
f k ( s ) = −⎜⎜ k 0 k + s c0 k + s ∑ Gik ( s ) ⎟⎟ L k q ( s ) ,
i =1
⎠
⎝
(10)
where L k is the (n × n) global location matrix of k-th damper and
Gik ( s ) =
k ik
.
vik + s
(11)
If a planar frame is the structure under consideration and the finite element
method is used to model the structure, then the matrix L k is built on the base of the
~
matrix L k which is the transformation matrix of damper from a local to the global
coordinate system, i.e.:
⎡ c~ 2 c~~
s − c~ 2 − c~~
s⎤
⎢ ~~ ~ 2 ~~ ~ 2 ⎥
cs s − cs − s ⎥
~
,
L k = ⎢⎢
s c~ 2 c~~
s ⎥
− c~ 2 − c~~
⎥
⎢
~
⎢⎣ − c~~
s −~
s 2 c~~
s
s 2 ⎥⎦
(12)
s = sin α and α is the angle between the global and local
where c~ = cos α , ~
coordinate systems. If the share frame is a model of structure and the damper is
located between the storeys number j + 1 and j , then L k = e k e Tk , where the ( n × 1 )
allocation vector is e k = col (0,..., e j = 1, e j +1 = −1,...,0) .
The total vector of interactive forces is
r
mk
f ( s ) = −(K d + sC d ) q ( s ) − s ∑∑ Gik ( s ) L k q ( s ) ,
(13)
k =1 i =1
where
r
r
K d = ∑ k0k L k ,
C d = ∑ c0k L k
k =1
(14)
k =1
The final form of equation of motion, written in the frequency domain, is:
r mk
⎞
⎛ 2
⎜⎜ s M + sC + sC d + K + K d + s ∑∑ Gik ( s ) L k ⎟⎟ q ( s ) = p ( s ) .
k =1 i =1
⎠
⎝
(15)
From Equation (15), the dynamic characteristics of structures with VE dampers,
such as natural frequencies, non-dimensional damping ratios and modes of
vibrations, can be determined after assuming that p ( s ) = 0 . Equation (15) is the
nonlinear eigenvalue problem. The number of eigenvalues and eigenvectors is
4
r
2n + ∑ mk . In the case of small damping, 2n eigenvalues and eigenvectors are
k =1
complex conjugate numbers and vectors, respectively. The remaining eigenpairs are
real numbers and vectors. The complex eigenvectors together with their complex
conjugates are sometimes called vibration modes while the real eigenvectors are
called viscous modes or overdamped modes (see [7]).
Given the complex conjugate eigenvalues s j = μ j + i η j and s j + n = μ j − i η j ,
natural frequencies of vibration ω j and non-dimensional damping ratios γ j can be
calculated from the following formulae:
γ j = −μ j / ω j .
ω 2j = μ 2j + η 2j ,
(16)
Introducing the so-called internal variables and applying the state space approach,
the considered problem can also be formulated in such a way that the linear
eigenvalue problem is obtained instead of the nonlinear one. The approach was
presented in [1]. However, the dimension of the linear eigenvalue problem is
approximately three times as great.
3
The continuation method
The continuation method, also termed as the path following method or the homotopy
method, is frequently used to solve nonlinear equations with parameter, occurring in
many problems of modern mechanics. The static analysis of geometrically or/and
physically nonlinear structures (see [8]) and the analysis of large-amplitude free and
steady state vibrations [9, 10] are examples of such problems. A general description
of the continuation method can be found, for example, in [11].
In the continuation method, the set of nonlinear equations with one parameter,
also called the main parameter, is considered. Here, an artificial main parameter κ ,
( 0 ≤ κ ≤ 1 ) is introduced and the above-mentioned set of equations is:
r mk
⎞
⎛
h 1 ( q , s ) = ⎜⎜ s 2 M + κ sC + κ sC d + K + K d + κ s ∑∑ Gik ( s ) L k ⎟⎟ q = 0 ,
k =1 i =1
⎠
⎝
(17)
h2 ( q , s ) = 12 q T H qs ( q , s ) − a = 0 ,
(18)
where a is given value and
H qs ( q , s ) =
r mk
r mk
⎞
∂G ( s )
∂h 1 ⎛
= ⎜⎜ 2 sM + κ C + κ C d + κ ∑ ∑ Gik ( s ) L k + κ s ∑ ∑ ik
L k ⎟⎟ q ,
∂s ⎝
∂s
k =1 i =1
k =1 i =1
⎠
∂Gik ( s )
k ik
.
=−
∂s
(vik + s ) 2
(19)
(20)
Equation (18) may be considered as a way of normalization of the eigenvector q .
Moreover, in this way, the symmetry of incremental equations which will be derived
later is preserved. A solution to the original problem (15) is obtained when κ = 1 .
5
The lower limit of κ is chosen to be equal to zero because, for this value of κ ,
Equation (17) is reduced to the following linear eigenvalue problem:
(s M + K + K ) q
2
0
d
0
=0 ,
(21)
which can be solved numerically using the standard procedure. Please note that the
matrix K d , which takes into account part of the stiffness properties of dampers,
appears in Equation (21). Moreover, the values of parameter a could be specified
from relationship a = s 0 q 0T M q 0 .
The solution to Equation (21) provides a good starting point for determination of
a complex conjugate solution to the original problem (15). One solution from a set
of solutions to the linear eigenvalue problem (21) is chosen for which the
continuation method has been applied and two curves s (κ ) and q (κ ) will be
numerically determined for a set of values of κ . The incremental – iterative
procedure will be used to determine the above mentioned curves. A notation like
s r (κ ) and q r (κ ) is used to denote the values of s (κ ) and q (κ ) for κ = κ r .
Based on the solution obtained for a certain value of parameter κ = κ r a solution
is sought for a new value of this parameter κ r +1 = κ r + Δκ , where Δκ is the
assumed increment of κ . The approximate solution to a new value of parameter κ ,
obtained at the iteration step i, will be denoted s r( i+)1 and q r( i+)1 . In the first iteration
step, the solution obtained for κ r is used, i.e., s s r(1+)1 = s r and q r(1+)1 = q r .
The incremental equations of the Newton method, associated with Equations (17)
and (18), are in the following form:
H qq δ q + H qs δ s = −h1 ,
H sq δ q + H ss δ s = − h2 ,
(22)
where
H qq
r mk
∂h1
2
=
= s M + κ s C + κ s C d + K + K d + κ s ∑∑ Gik ( s ) L k ,
∂q
k =1 i =1
H sq =
H ss
r mk
r mk
⎛
⎞
∂G ( s )
∂h2
= q T ⎜⎜ 2 s M + κ C + κ C d + κ ∑∑ Gik ( s ) L k + κ s ∑∑ ik
L k ⎟⎟ ,
∂q
∂s
k =1 i =1
k =1 i =1
⎝
⎠
r mk
r mk
⎞
∂Gik ( s )
∂ 2 Gik ( s )
∂h2 1 T ⎛
=
= 2 q ⎜⎜ 2M + 2κ ∑∑
L k + κ s ∑∑
L k ⎟⎟ q ,
2
∂s
∂s
∂s
k =1 i =1
k =1 i =1
⎝
⎠
∂ 2 Gik ( s)
2k ik
=
.
2
∂s
(vik + s) 3
(23)
The new approximation of the solution is obtained after solving the set of
Equations (22) with respect to δ q and δ s and using the following formulae:
s r( i+)1 = s r( i+−11) + δ s ,
q r( i+)1 = q r( i+−11) + δ q .
The iteration process is finished when
6
(24)
δ s < ε 1 s r(i+)1 ,
δq < ε 2 q r(i+)1 ,
(25)
where ε 1 and ε 2 are the assumed accuracies of calculations.
The continuation method has good convergence properties. For
ε 1 = ε 2 = 0.00001 , one or two incremental steps and three or four iterations in the
incremental step are enough to reach a solution.
However, at the present stage of development the proposed method has one
important drawback. The method fails in attempts to determine the response curves
s (κ ) and q (κ ) starting with real values of s 0 and q 0 .
4
Results of a typical calculation
A six-storey share frame is selected to determine the dynamic characteristics of a
structure with dampers. The mass of each storey of structure is 90000.0 kg. The
k1, s = k 2, s = 1.2 × 10 8 N/m ,
storeys are of the following stiffnesses:
k 3, s = k 4, s = 1.0 × 10 8 N/m , k 5, s = k 6, s = 0.8 × 10 8 N/m . The damping properties of
the structure are neglected.
Six VE dampers are on the structure, one on each storey. The generalized
Maxwell model with eight parameters is used as a model of damper. All dampers
have identical parameters, as shown in Table 1.
Stiffness
Damping factor
(×10 6 ) [N / m]
(×10 6 ) [N sec/ m]
k0
0.2130
c0
0.000
k1
66.770
c1
2.957
k2
6.6210
c2
3.463
k3
2.886
c3
16.610
Table 1 Parameters of the generalized Maxwell model
Results of calculation are summarized in Table 2. A significant influence of VE
dampers on natural frequencies is observed. The first three natural frequencies
increase by about 9%, 26% and 31%, respectively. Please note that nondimensional
damping ratios of first two modes are of the order of 10% and they are greater than
the remaining damping ratios.
The considered problem was also solved using the state space approach, in which
the problem could be reduced to the following linear eigenvalue problem (see [1] for
details):
7
(sA + B) a = 0 ,
(26)
where a is the eigenvector and the matrices A and B are appropriately built from
the mass, stiffness and damping matrices of the frame with VE dampers.
The solution to the linear eigenvalue problem is presented in Table 3.
Comparing the results in Table 2 and 3, it is concluded that almost identical
values of complex, conjugate eigenvalues are obtained as the solution to the
nonlinear (15) and linear (26) eigenvalue problems. The real solutions to linear
eigenvalue problem (26) reflect the dampers’ dynamics and could be divided into
three groups. The values of real eigenvalues in one group are of the order of inverse
relaxation times of Maxwell elements which is: k 3 / c3 = 0.174 , k 2 / c 2 = 1.91 and
k1 / c1 = 22.58 , respectively.
Frame with dampers
Frequencies of frame
without dampers
Frequency
Complex eigenvalues
Damping ratio
[rad/sec]
[rad/sec]
1
8.35405
-0.94354 + i 9.04467
9.09375
0.103757
2
23.3784
-3.50908 + i 29.2854
29.4949
0.118973
3
37.4414
-4.04305 + i 48.8769
49.0438
0.082438
4
49.6335
-4.14593 + i 65.0887
65.2206
0.063568
5
59.5951
-4.14252 + i 78.1702
78.2798
0.052919
6
68.0692
-3.93662 + i 87.4849
87.5734
0.044951
Table 2 Results of calculation – complex eigenvalues
5
Concluding remarks
In this paper a method of determination of the dynamic characteristics of structures
with VE dampers modelled with the help of the generalized Maxwell model is
presented. The Maxwell model considered contains, in particular, very popular
models such as the viscous model, the simple Kelvin model and the simple Maxwell
model among other ones.
The problem considered is reduced to the nonlinear eigenvalue problem which is
more than two times as small as the linear eigenvalue problem obtained using the
state space approach. The continuation method which is an incremental – iterative
method is adopted to find the complex eigenvalues and eigenvectors. The proposed
incremental – iterative procedure is very fast. However, in the present stage of
8
development only the complex solutions to nonlinear eigenvalue problem can be
found. The proposed method is quite general because it can be used to solve
nonlinear eigenvalue problems of different types, including the quadratic eigenvalue
problems which very often appear in many instances.
Complex eigenvalues
Real eigenvalues
− 0.94354 ± 9.04467 i
− 0.167633
− 1.76238
− 13.5729
− 3.50910 ± 29.2853 i
− 0.168832
− 1.79067
− 14.4332
− 4.04316 ± 48.8769 i
− 0.168833
− 1.79108
− 14.9484
− 4.14605 ± 65.0887 i
− 0.168834
− 1.79317
− 15.1556
− 4.14265 ± 78.1702 i
− 0.168839
− 1.81012
− 15.8474
− 3.93662 ± 87.4849 i
− 0.168840
− 1.81226
− 20.8232
Table 3 Solutions to the eigenvalue problem (26)
Acknowledgments
The authors wish to acknowledge the financial support received from the Poznan
University of Technology (Grant No. DS 11 - 088/12) in connection with this work.
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[2]
[3]
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[5]
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9
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