Earthquake Engineering Research - HKU Libraries - The University ...

454
In this paper, a stochastic optimal control method is developed for seismic response control of cablestayed
badges with use of active mass drivers (AMDs). Numerical simulation studies of applying the
proposed method to the cable-stayed Ting Kau Bridge (TKB) are then carried out to demonstrate the
control effectiveness and efficacy. Based on a precise three-dimensional finite element model of the
bridge, a control-oriented reduced-order modal model suitable for control design is developed.
Different AMD installation configurations (number and location) are designed for the bridge, and the
random earthquake excitation is assumed to act in the longitudinal, transverse and 45° directions,
respectively. The stochastic optimal control strategy based on the dynamical programming principle
and stochastic averaging method is devised to command the operation of AMDs. Twelve evaluation
cnteria are formulated to evaluate the control effectiveness and efficiency. Structural deflection and
internal force responses of the bridge without and with active AMD control are obtained and compared
under different excitation conditions.
PROPOSED CONTROL STRATEGY
Consider an rc-degree-of-freedom structure incorporated with active mass drivers. The governing
equation of motion of the integrated system can be expressed as
MX + CX + KX = ~x s ME - PDU (1)
in which M, C , K are mass, damping and stiffness matrices of the structure, respectively; E is the
n-dimensional vector associated with external excitation; P is an nxm matrix indicating the location
of control devices; D is a diagonal matrix comprising the masses of AMDs used. V denotes a vector
of relative accelerations where AMDs are attached; x g is the random ground acceleration excitation,
its spectrum being taken herein as the non-white Kanai-Tajimi power spectral density function with the
expression
where % g and & g represent the damping coefficient and predominant frequency, respectively, of the
ground motion; 5 0 is a constant power spectral density.
Based on the mode superposition method, displacement response of the structure can be expressed in
terms of modal transformation as
where & c , Q c are the dominant modal matrix and displacement vector, respectively.
Equation (1) can then be transformed into the following reduced modal coordinate equations
a+^^a+^G^-A^w-v, (i=i,2,...,o - (4)
where / is the reduced modal order; Q l9 CD I and £ are modal displacement, frequency and damping