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maximum friction force and there is no change in the relative displacement between the interfaces.
The non-sliding conditions are:
(4a)
(4b)
where /j. is the faction coefficient; N is the normal force applied to the interfaces; s[k -f 1] and s[k]
are the relative displacement between interfaces at the current and the next time steps. In the nonsliding
phase, the shear force is unknown but the relative displacement remains unchanged. During the
sliding phase, the shear force is large enough to make the damper slide and there is change in the
relative displacement. The sliding conditions are:
(5a)
(5b)
In the sliding phase, the shear force is known while the relative displacement is unknown. Therefore,
no matter the damper slides or not, either the shear force or the relative displacement between the
interfaces is known. With this additional condition, the shear force at the next time step can be
determined, and the response of the system can in turn be solved.
First, the friction damper is assumed in the non-sliding phase, so that the relative displacement of the
interfaces remains unchanged:
s[k + l] = y b [k+l]-y(k + l]~y b [k]-y[k] = S[k] (6)
where y b [k] is the relative displacement of the bracing system by which the friction damper is
mounted; y[k] is the relative displacement between the two stories where the friction damper is
located. The story drift y[k] is a linear combination of the state vector z[&] as:
y[*] = Dz[*] (7)
where D = [-& T 0 T J is the output row vector; T denotes the transpose of vector or matrix. The shear
force of the damper and the relative displacement of the bracing system are related as:
u[k] = k b y,[k] (8)
where & b is the stiffness of the bracing system. After pre-multiplying the stiffness & b to equation (6),
it gives:
Utilizing equations (7) and (8), the estimated shear force at the next time step is expressed as:
u[k + 1] = Jt b Dz[* + 1] + u[k] - * b Dz[*] (10)