Keywords: supplemental dampers, Maxwell model, Kelvin model, passive control, response reduction, storey drift.

For structures that are built in earthquake prone areas, damping is an important component of structures in reducing the response during severe earthquakes. However, the inherent damping in structures is generally low so that supplemental dampers are required in order to reduce the vibration. The dynamic analysis of buildings with a supplemental dampers is considered in this paper. The added damper is connected to the structures through bracing systems using either a V or inverted V shaped bracing [1,2]. In this case bracings and added dampers are considered as the Maxwell models while the structure are considered as the Kelvin or Voigt model as shown in Figure 204.1. The combined Maxwell and Kelvin or Voigt models act together so that the response of the structure can be reduced.

Although the theoretical analysis of the Maxwell model as time dependent elasticity has been well known [3], the use of this model in the dynamic analysis of structures involving inertial forces has not been further explored.

Numerical simulations were conducted using a 25-storey building similar to that in reference [4]. It was also reported that the damping ratio achieved in the first mode was 12.5%. In reference [2] the finite element method was proposed to analyse the building with supplemental dampers connected with inverted V-bracings. The proposed analytical model was also compared to the one when additional degrees of freedom were added at mass point between damper and floor mass, known as exact method in reference [2]. Numerical simulations of the 25-storey building confirmed that the method was in a good agreement with the exact method.

In this paper, the Maxwell model is analysed from the constitutive equation relationship. The rate of forces in the Maxwell model can be obtained accordingly. By defining the state vector as the displacements of the structure, velocities of the structure and forces in the Maxwell model, a state space equation can be obtained. The proposed method is then applied to a 30-storey building with supplemental dampers. Results compared with the numerical simulation show that the proposed method is in very good agreement with the exact method, where additional degrees of freedom are added in the damper location. However, if the exact method is converted to a first order differential equation as the state space equation, the sizes of the matrices in the proposed method in general are three-quarters of the one of the exact method.

1

N. Kurata, T. Kobori, M. Takahashi, N. Niwa, and H. Midorikawa, "Actual Seismic Response Controlled Building with Semi-active damper System", Earthquake Engineering and Structural Dynamics, Vol. 28, pp. 1427-1447, 1999. doi:10.1002/(SICI)1096-9845(199911)28:11<1427::AID-EQE876>3.0.CO;2-#

2

T. Hatada, T. Kobori, M. Ishida, and N. Niwa, "Dynamic Analysis of Structures with Maxwell Model", Earthquake Engineering and Structural Dynamics, Vol. 29, 159-176, 2000. doi:10.1002/(SICI)1096-9845(200002)29:2<159::AID-EQE895>3.0.CO;2-1

3

W.N. Findley, J.S. Lai, and K. Onaran, "Creep and Relaxation of Nonlinear Viscoelastic Materials", North-Holland Publishing Co., Amsterdam, 1976.

4

N. Niwa, T. Kobori, M. Takahashi, T. Hatada and K. Haruhiko, "Passive Seismic Response Controlled High-Rise Building with High Damping Device", Earthquake Engineering and Structural Dynamics, Vol. 24, pp. 655-671, 1995. doi:10.1002/eqe.4290240504

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