Keywords: active controlled structures, dynamic behaviour, differential equations of motion, discretization methods.

Modern structural control design methods for seismically excited structures require computer simulations in order to obtain the optimal control law and to check the effectiveness of the control system. Various control devices have been proposed recently for control law implementation. Battaini et al. [1] verified experimentally structural control strategies and showed that appropriately constructed bench-scale models can be used to study important aspects of full-scale structural control implementations.

The mathematical description of a common civil structure and its control system is a hybrid because of the recurrent equations describing the digital control system and differential equations describing other parts. Formal conjunction of these two parts is usually performed automatically when the SIMULINK software is used. However the problem of differential equations integration can be problematic for so-called stiff differential equations see Beskos and Brebbia [2].

In order to model the dynamic system of the building together with actuators and the measurement equipment, the differential equation of its dynamic equilibrium should include a part describing their behavior. Battaini et al. [1] showed that these factors have significant effect in overall dynamic properties of the system. In other words the state space vector of the system's continuous part, includes a state-space vector of the building and the state vectors of the actuators and the measurement subsystems

Usually the linear model of the controlled structure includes fast and slow variable parts. The fast variable part represents the measurement and electrical equipment, the actuators and the high vibration modes of the building. The slow variable part represents the lower modes of the building and the mechanical part of the actuators. Because of these two parts these equations belong to the class of stiff differential equations and cannot be solved numerically by applying known effective methods, as described by Beskos and Brebbia [2]. Solution of such equations can be obtained by Gear and Rosenbrock methods modified and realized by Shampine et al. [3]. However application of these methods requires long computation time, which is undesirable for design and simulation purposes.

To intensify the integration process and to decrease the required computation time, a method based on equivalent discrete model is proposed. According to the proposed method the system of differential equations is transformed into an equivalent system of difference equations. To demonstrate the proposed method a simulation of a linear two-story structure model with active mass driver described by Battaini et al. [2] subjected to 35% El Centro (1940) earthquake was performed. The structure is controlled by an LQG-optimal digital controller with sampling period of 0.01 sec.

The example showed that the seismic response of the structure obtained by applying the proposed method was close to that obtained using the method of differential equations integration. However, applying the proposed method yields significant reduction in computation time, required for the analysis, making it more efficient. Hence, the proposed fast simulation method may be successfully used in computer programs for seismic analysis of linear structures.

1

M. Battaini, G. Yang, B.F. Spencer Jr., "Bench-scale experiment for structural control", Journal of Engineering Mechanics, 126 (2), 140-148, 2000. doi:10.1061/(ASCE)0733-9399(2000)126:2(140)

2

D.E. Beskos, C.A. Brebbia, "Computer Analysis and Design of Earthquake Resistant Structures: a handbook", Southampton: Computational Mechanics Publications, 1997.

3

L.F. Shampine and M.W. Reichelt, "The Matlab ODE Suite", SIAM Journal on Scientific Computing, 18, 1-22, 1997. doi:10.1137/S1064827594276424

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