Stochastic optimal control of wind-excited tall buildings using semi-active MR-TLCDs
Introduction
Mitigating wind-induced vibration response of building structures by using passive tuned liquid column dampers (TLCDs) has been studied extensively [1], [2], [3], [4], [5], [6], [7], [8]. The TLCD consists of a U-tube container with an orifice in the middle. It dissipates the energy of structural vibration by a combined action of inertia force induced by the movement of the liquid, the restoring force due to gravity on the liquid, and the damping effect caused by an orifice. The TLCD has attracted interest for engineers due to its cost-effectiveness, simplicity in installation, and low maintenance costs. A recent application of the TLCD is its implementation to the 46-story One Wall Centre in Vancouver [9]. Two TLCDs, each consisting of a four-story high, 50,000-gal water tank, were placed at the top of this combined hotel and residential building for mitigating wind-excited vibration. Each TLCD contains two water columns connected by a sluice gate (orifice) to regulate water flow. The damping system is tuned to the natural frequency of the building by regulating water flow through the gates and also monitoring the water levels in the tanks.
For a TLCD system, energy dissipation in the water column is due to the passage of the liquid through an orifice with inherent head-loss characteristics. However, the damping force induced by the orifice is nonlinear. The equivalent linearized damping coefficient is response-dependent, so that optimum damping condition cannot be maintained for a wide range of disturbances. Thus, it is highly desirable to develop damping-variable or parameter-adjustable TLCDs to achieve optimal control performance for a wide range of loading conditions and to be tolerant of probable structural uncertainty. Some research efforts have been made to this end. Haroun et al. [10] proposed a concept of hybrid liquid column damper by actively controlling the orifice to produce variable orifice opening ratio; Yalla et al. [11] proposed to convert passive TLCDs into semi-active TLCDs by introducing a controllable valve to adjust the orifice opening. However, no physical devices of TLCDs with variable or controllable damping have been developed.
The authors have recently developed a practical semi-active TLCD by using the smart magneto-rheological (MR) fluids [12]. An essential characteristic of MR fluids is their ability to reversibly change from a free flowing, linear viscous liquid to a semi-solid having a controllable yield strength in milliseconds when exposed to a magnetic field [13]. They are thus used as damping fluids to devise semi-active magneto-rheological tuned liquid column dampers (MR-TLCDs) with alterable fluid viscosity. The sharply alterable fluid viscosity results in adjustable and controllable damping force in the MR-TLCD for structural vibration control under a wide range of loading conditions. As a dissipative damper, the MR-TLCD generates the controllable damping force by utilizing the relative motion between the liquid and container and thus, does not have the potential to destabilize the system. Two MR-TLCD prototypes using different types of MR fluids have been fabricated [14], [15] and laboratory experiments showed that the devised MR-TLCDs offered much better damping performance than passive TLCDs even in open-loop control mode. In the present paper, after briefing the MR-TLCD design, a semi-active control strategy for tall building structures incorporating MR-TLCDs is developed and its effectiveness for mitigating wind-induced vibration response of a real 50-story building is examined.
Section snippets
Design and modeling of MR-TLCDs
As shown in Fig. 1, Fig. 2, an MR-TLCD consists of a U-tube container filled with the MR fluid and having an orifice opening in the middle of the bottom tube. Magnetic field offered to the fluid is generated by an electromagnet of length Lp around the bottom tube (usually Lp≪B where B is the horizontal length of the liquid column). The MR-TLCD device is rigidly connected to the primary structure and capable of dissipating energy through oscillation of the liquid column. With properly designed
Description of controlled system
Consider a high-rise building structure with n stories and a semi-active MR-TLCD installed at the top floor under random wind loading. For a linear elastic shear-type structure, the equation of motion of the controlled system can be expressed aswhere X denotes the n-dimensional horizontal displacement vector of the structure; M,C and K are the n-dimensional mass, damping and stiffness matrices of the structure, respectively; FW(t) is the n-dimensional wind loading vector. FD
Optimal control law
Perfect and complete observation is assumed in this study and then the optimal control design of system (19) can be performed directly based on the dynamical programming principle. Optimal control law commanding the semi-active damping force component of the MR-TLCD is determined by minimization of the following performance index J in an infinite time interval [22]where L is a Lagrangian of continuous differential convex functional. The corresponding dynamical
Response evaluation
To evaluate the random response of the controlled structural system with semi-active MR-TLCD under wind loading, the clipped optimal control damping force given in Eq. (26) is first linearized statistically as [20], [21]where the equivalent coefficient vector isThen the augmented matrix equation for the controlled structural system with semi-active MR-TLCD becomeswhere and C″A=BpCseqT.
Eq. (33) expresses a
Case study
A 50-story residential building is now under construction in Hong Kong. The initial design of this building was found to not satisfy the wind-resistant requirement prescribed in the Hong Kong design code, and therefore use of various supplemental damping devices in this building has been studied. The height of the building is 161.65 m and the total mass tr(M)=2.774×107 kg. Modal properties of the building have been obtained from a precise three-dimensional finite element model. The natural
Conclusions
An optimal control method for tall building structures using semi-active MR-TLCDs has been developed based on the dynamical programming principle and the statistical linearization method. The MR-TLCDs combine the benefits of controllable smart materials and tuned liquid column dampers. The dissipative optimal damping force produced by MR-TLCDs through relative motion between the liquid and the container does not have the potential to destabilize the structural system. The proposed real-time
Acknowledgements
The work presented in this paper was supported by a grant from The Hong Kong Polytechnic University through the Area of Strategic Development Programme (Research Centre for Urban Hazards Mitigation) and by a grant from the Zhejiang Provincial Natural Science Foundation, China (Grant No. 101046). These supports are gratefully acknowledged.
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