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After substituting equation (3) into the above equation, it gives:
u[k +1] = Jfc b D( Az[&] + b Q u[k] + b&k +1] + E 0 w[£] + Ej w[fc +1]) + w[&] - & b Dz[&] (11)
After rearrangement, the shear force at the next time step can be estimated as:
u[k -f 1] = Pz[k] + Qu[&] + R 0 w[fc] + R t w[£ +1] (12)
where P = (l-* b Db i r l fc b D(A-I); Q = (l~£ b Db i r l (£ b Db 0 +1); R 0 =(l-*J>bO -I U>E 0 and
There are two possible cases for the estimated value of the shear force at the next time step u[k +1]:
When the estimated shear force is less than the maximum friction force, the interfaces stick together
and the assumption stated in equation (6) is satisfied. The shear force at the next time step is the same
as the estimated one:
(2) \u[k 4-1]| >it [Wi ~/tN
When the estimated force is greater than the maximum friction force, the friction between the
interfaces is overcome and the interfaces slide relative each other. The assumption stated in equation
(6) is violated. Therefore the magnitude of the shear force is the same as the maximum friction force
and the direction of the shear force is the same as the estimated one:
«[* +1] = «m« sgn02[* +1]) = /.iNsgn(u(k +1]) (14)
where sgn(») = •/[«! is the signum function. The procedures for the dynamic analysis of the frictiondamped
structure are shown in the flow chart (Figure 1).
NUMERICAL VERIFICATION
In order to illustrate the effectiveness of friction damper, a single-degree-of-freedom structure, with
mass m = 100 ton, stiffness k = 4000 kN/m, and damping coefficient c = 40 kN/(m/sec), is investigated.
Thus, the natural period and damping ratio of the structure are T = 0.993 sec and £=3.16 %,
respectively. The top of the friction damper is mounted underneath the beam while the bottom of the
damper is connected to the inverted V-shaped bracing through the T-shaped beam. Under earthquake
excitation, shear force between the interfaces of the friction damper is induced. If the friction is
overcome with the shear force, the damper slides. During the excitation, the motion of the damper
switches from the non-sliding phase to the sliding phase, and vise versa. The location vector of the
damper is b = -1 and the output vector for the story drift is D = [-b 7 oj= [l o]. The ratio between