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After the friction dampers are added to the structure, friction mechanism makes the structural system
hishly non-linear. When the structure is driven with external loads, the motion of the friction damper
may be shifted from the non-sliding mode to the sliding mode, and vise versa. In the non-sliding
mode, the relative displacement between the interfaces of the damper remains unchanged but the shear
force between the interfaces is unknown. In the sliding mode, the relative displacement is unknown
but the shear force is the maximum friction force. No matter in the non-sliding or sliding mode, either
the kinematic or kinetic condition is known. In this paper, a simple and efficient numerical method is
developed for the dynamic analysis of friction-damped structures. Only a single equation of motion is
used to describe the dynamic system for both non-sliding and sliding modes. The stability and
accuracy are guaranteed even though the time step of integration remains constant during the process
of the solution. Finally, the effectiveness of the friction dampers is studied.
MATHEMATICAL DERIVATION
When a discrete-parameter structural system with n degrees of freedom is subjected to environmental
loads w(r) and counteracted by a friction force w(r), its governing equation can be taken as:
Mx(r) + Cx(r) + Kx(r) = bu(0 + Ew(0 (1)
where x(f) is the displacement vector; M, C and K are the mass matrix, damping matrix and
stiffness matrix, respectively; b is the location vector of the friction damper; E is the location matrix
of the environmental loads. Represented in state-space form, the second-order differential equation (1)
is changed to the first-order differential equation as:
where z(r) = ., I, A =, . ,, Ur =,
I x(r) -M- 1 K -M- l C f c M' l b
and E c = | ,.__!„ I. Since the recorded
load functions are commonly discretized and the friction force are piece-wise linear in nature, it is
logical to assume linear variations of these loading functions between two consecutive sampling
instants. Therefore, With the sampling period &* , the solution of the state equation (2) becomes a
difference equation as:
z[k + 1] = Az[*] + b Q it[k] + b^u[k + 1] + E 0 w[fc] + E! w[& + 1] (3)
where A = e AA