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N t)
= N + C V tanh( β v&
( ) )
(4)
( 0 z max
s t
The last equation implies that, when using the control law given in Eq. (3), the normal force of the PFD has a
lower and an upper bound, i.e.,
N ≤ N t)
≤ ( N + C )
(5)
0 ( 0 zVmax
MODELING OF A STRUCTURE WITH A SAF-TMD
Figure 3 shows the mathematical model used for numerical analysis. The figure shows that the sliding platform
of the SAF-TMD is modeled by a spring of stiffness k i and a friction element with friction coefficient μ i . The
former simulates stiffness due to the resilient mechanism whereas the latter models the friction effect of the
guide rail of the sliding platform. Figure 3, however, shows that the PFD is modeled by a variable friction
element with a friction coefficient μ d , and the symbol k d denotes the axial stiffness of the PFD. The mass of the
SAF-TMD and the primary structure are denoted by m s and m p , respectively. Moreover, the damping and
stiffness of the primary structure are denoted by c p and k p , respectively. Relative-to-the-ground displacements of
the SAF-TMD and the primary structure are denoted by x s and x p , respectively. Based on the mathematical
model, the dynamic equation of the system can be rewritten as
where
⎧x
p ( t)
⎫
x ( t)
= ⎨ ⎬ ,
⎩vs(
t)
⎭
M & x
t)
+ Cx&
( t)
+ Kx(
t)
= D ( u ( t)
+ u ( t))
+ E & x
( )
(6)
( 2 d i 1 g t
⎧ 1 ⎫
D = ⎨ ⎬ ,
2
⎩−1⎭
⎧−
m p ⎫
E = ⎨ ⎬ ,
1
⎩−
ms
⎭
⎡mp
0 ⎤
M = ⎢ ⎥
,
⎣ms
ms
⎦
c p
⎡ 0⎤
C = ⎢ ⎥
,
⎣ 0 0⎦
⎡k
p − ki
⎤
K = ⎢ ⎥
(7)
⎣ 0 ki
⎦
In Eq. (6), x(t) denotes the vector containing the system responses consisting of structural displacement x p (t) and
TMD stroke v s (t); & x g (t)
is the ground acceleration due to an earthquake; D 2 and E 1 denote the force placement
matrices for the SAF-TMD system and the excitation, respectively. The matrices M, C and K represent the mass,
damping and stiffness matrices of the SAF-TMD control system, respectively. The friction forces (damper
forces) of the PFD and sliding platform are denoted by u d (t) and u i (t), respectively. Moreover, the forces denoted
by u d (t) and u i (t) on the right-hand side of Eq. (6) are the nonlinear friction effects of the SAF-TMD. Note that
u d (t) is the controllable damper force provided by the PFD, u i (t) is an uncontrollable nonlinear force. Therefore,
the dynamic response of the SAF-TMD system can be attenuated by altering the force u d (t) in real time. For the
purpose of numerical simulation in this study, the Coulomb friction law is assumed to govern friction material
behavior in the damper and sliding platform, and the friction materials are assumed to have an equal static and
dynamic friction coefficient. When friction force terms u d (t) and u i (t) are included in the model of a structure
controlled by a SAF-TMD, the model becomes nonlinear, and numerical methods are generally needed to
analyze the dynamic behavior of the system. This study therefore applies shear balance method (SBM), a
numerical method of simulating a structure equipped with a friction-type TMD. The SBM is widely used to
simulate dynamic response in structures with friction-type devices such as friction damper (Lu et al. 2006) or
friction isolator (Wang et al. 1998). The SBM has proven efficient and generally accurate in analyzing structures
with frictional devices. Lin et al. 2010c comprehensively discussed the formulation and application of an SBA
algorithm in analysis of friction-type TMDs.
v s(t)
k d
k i
μ i
μ d
N(t)
m s
(SAF-TMD)
x s(t)
x p(t)
m p
(primary structure)
c p
k p
Figure 3 Mathematic model of a SDOF structural system with the SAF-TMD.
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