Earthquake Engineering Research - HKU Libraries - The University ...

499
numerical approach. However, one can also formulate a more convenient self-adjoint system of state
equations with the help of the physical model shown in Fig. 3.1. To describe the dynamic deformation
of each damper completely, we introduce auxiliary displacement coordinates at the interface of each
damping element as shown in Fig. 3.1. In terms of these auxiliary displacement coordinates and the
displacement coordinates that define the motion of each structural degree of freedom, one can develop
the kinetic energy function, potential energy function, and dissipation function. Using the Lagrange
equations, the equations of motion can then be explicitly obtained. Using appropriate auxiliary
equations, and after some re-arrangements of terms, these equations can be written in the self-adjoint
state form. For a shear building with viscoelastic dampers installed in each story, these state equations
can be written as follows:
(4.1)
where the state matrices are defined as:
C
0
0
-c 4
M
0
C 2
0
0
0 -(
C+C
0
Ml
0
0
0
0 J
,B =
~~K
Kfc
0
K 4
_ 0
K b
-K,-K
K 2
0
0
K 2 0 0
K 4 Ol
ic 3 o
-K 3 -K 4 0
0 Mj
(4.2)
withC = C + C 4 , K=K + K 4 +K fl , C, ^
and 0 elsewhere, and the state vector is defined as {y} = {{*},fa},{z 2 },{zi},{x}] . It is noted that the
state matrices are symmetric, and the system is self-adjoint. It is also noted that, for the third order
model used here, the state vector is of size 5N. Usually a third order model would have enough
parameters to change to obtain a close match with the experimentally obtained moduli.
The system of equations, Eqn. 4.1 can be uncoupled using the eigen properties of the system. A modal
analysis approach can be formulated, with all the advantages associated with such an approach. For
example, one can develop a response spectrum approach to calculate the maximum displacement, story
shear, floor acceleration, etc. for seismic design inputs defined in terms of response spectra. The
details of this formulation are provided by Chang (2002).
Equation Section 5
NUMERICAL RESULTS
In this section we present the numerical results of seismic analysis of a structure installed with
viscoelastic dampers. To show that the higher order linear model of Eqn. 3.5 is as good as the
fractional derivative model, we compare the numerical results obtained by the two formulations.
A 4-story shear building with story masses of m^ ~ m^ = 40 Mg, m^ = m 4 = 36 Mg, and story stiffness
coefficients of K : =18 MN/m , K 2 = £ 3 =12MN/m, K 4 = 10MN/m, was chosen to obtain the
numerical results. To introduce inherent energy dissipation in the system, a damping matrix providing