Shock and Vibration

Volume 2016 (2016), Article ID 6731928, 22 pages

http://dx.doi.org/10.1155/2016/6731928

## Identification of Torsionally Coupled Shear Buildings Models Using a Vector Parameterization

^{1}Facultad de Ingeniería Mecánica y Eléctrica, Universidad de Colima, 28400 Coquimatlán, COL, Mexico^{2}Instituto de Ingeniería, Universidad Nacional Autónoma de México, 04510 Coyoacán, DF, Mexico

Received 3 March 2016; Revised 16 May 2016; Accepted 17 May 2016

Academic Editor: Michael Todd

Copyright © 2016 Antonio Concha and Luis Alvarez-Icaza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A methodology to estimate the shear model of seismically excited, torsionally coupled buildings using acceleration measurements of the ground and floors is presented. A vector parameterization that considers Rayleigh damping for the building is introduced that allows identifying the stiffness/mass and damping/mass ratios of the structure, as well as their eccentricities and radii of gyration. This parameterization has the advantage that its number of parameters is smaller than that obtained with matrix parameterizations or when Rayleigh damping is not used. Thus, the number of spectral components of the excitation signal required to identity the structural parameters is reduced. To deal with constant disturbances and measurement noise that corrupt acceleration measurements, Linear Integral Filters are used that guarantee elimination of constant disturbances and attenuation of noise.

#### 1. Introduction

The parameter identification of torsionally coupled shear building models has been a topic of interest in the last three decades [1–13]. Its identification is important because it allows verifying the structural health, or designing control law techniques that attenuate the vibration of the building when it is excited by external forces as earthquakes or wind [14–16]. This kind of model more closely approximates a shear building than the planar frame model, which is widely used in the literature [17–19]. The reason is that most buildings present torsional movements under purely translational excitations since their centers of resistance and mass do not generally coincide [20]. Although in some identification schemes buildings are modeled as two independent planar frames where torsional motion is neglected, the focus in this paper is in recovering the model of buildings where the centers of mass and the centers of torsion in each floor do not coincide and, therefore, significant torsional motions are expected. There are significant differences between the responses of these two approaches, whose comparison is beyond the scope of this paper.

References [1–8] propose techniques to estimate the modal parameters of torsionally coupled buildings using acceleration measurements. Li and Mau [1] present a methodology that identity the modal parameters by minimizing the error between the measured accelerations and the ones predicted by the solution of the Duhamel integral. Ueng et al*.* [2], Lin et al. [3], and Nayeri et al. [4] obtain the parameters of a structure excited by means of ambient vibrations; [2, 3] combine an extended decrement random method and the Ibrahim Time Domain estimation technique, whereas [4] identifies a full-scale 17-story building using the NExT/ERA (Natural Excitation Technique in conjunction with the Eigensystem Realization Algorithm) and a time domain identification technique for chain-like MDOF systems. In [5, 6], identification techniques that employ the Eigensystem Realization Algorithm (ERA) in order to generate a building state space model are proposed. Hegde and Sinha [6] use only acceleration measurements of the top and first floor levels, but their method is applied only to structures whose masses and eccentricities are equal for all stories. Antonacci et al. [7] identify the modal parameters by means of the following methods: Enhanced Frequency Domain Decomposition (EFFD), ERA, Stochastic Subspace Identification (SSI), and Time-Frequency Instantaneous Estimators (TFIE). The algorithms in [8, 9] estimate the stiffness matrix of torsionally coupled buildings; Torkamani and Ahmadi [8] firstly identify the natural frequencies and modal shapes by means of the Fourier spectra of the acceleration of the floors; then, a parameter identification technique, which assumes knowledge of the building floors masses, is used to obtain the stiffness matrix. Omrani et al. [9] obtain the stiffness matrix from a methodology that uses the structure response to ambient excitation and the knowledge of all the masses and eccentricities of the floors. Wang et al. [10] proposed a procedure that combines the identification technique SRIM (System Realization using Information Matrix) and a damage index in order to estimate the damage of torsional coupled buildings. Angeles-Cervantes and Alvarez-Icaza [13] propose a technique that combines the online Least Squares Method with a parametrization of the building, which in the sequel will be called matrix parameterization, that is used for estimating the complete matrices and of the structure, where , , and are the mass, stiffness, and damping matrices, respectively; however, the identified matrices are overparameterized and, as a consequence the stiffness/mass and damping/mass ratios of the structure cannot be uniquely identified; in addition, with this parameterization, the zeros entries of the matrices and are also identified, which greatly increases the number of parameters to be estimated. Finally, [11, 12] propose an identification approach, based on the unscented Kalman filter, that identifies the stiffness and damping parameters of a torsionally coupled building, as well as the Bouc-Wen model parameters that represent the hysteretic response of each lateral load resisting elements of the structure; the algorithm in [12] is an extension of [11] and also estimates the mass eccentricities of the building; the unscented Kalman filter, proposed in these two references, uses only acceleration measurements and also estimates the velocity and displacement of the structure, thus avoiding the numerical integration of the acceleration measurements, which leads to velocity and displacement histories that drift away linearly and quadratically with the time, respectively [21]. The approach in [11] attenuates measurement noise; however, the constant disturbance voltage in accelerometers output cannot be eliminated by a Kalman filter. The usual approach to eliminate these constant disturbances is the offline processing of the accelerometer signals.

This paper presents an identification technique that estimates the parameters of a torsionally coupled building model, which exhibits torsionally movements under purely translational excitations due to a seismic event. A vector parameterization of the building is proposed that has the following characteristics: (1) it assumes that the structure has classical Rayleigh damping and (2) it contains a vector whose entries depend on the building stiffness/mass ratios, which are estimated using accelerations measurements of the ground and floors. Once that these ratios are identified, it is possible to estimate the building damping/mass ratios and the modal parameters; additionally, it is also possible to identify the radii of gyration and eccentricities of the floors and the stiffness, damping, and mass matrices of the structure.

In contrast to the estimation methods in [6, 8, 9], the proposed technique does not assume that the eccentricities of all the floors are in the same direction or that all the floor masses are equal or known. The proposed vector parameterization is combined with the offline or with the online Least Squares Method (LSM) and can be used for estimating the complete model of the structure if all the floors are instrumented or be employed for identifying a reduced model of the building, if only some floors are equipped with accelerometers.

Assuming Rayleigh damping for the structure permits reducing the number of parameters contained in the proposed vector parameterization, since it is composed only of stiffness/mass ratios instead of stiffness/mass and damping/ratios as in the matrix parameterization [13] or the work of Omrani et al. [12]. The number of parameters contained in the vector parameterization is thus reduced with the positive impact on the (1) reduction of the computational effort that allows implementing the online LSM and (2) the reduction of the spectral richness of the excitation signal required to uniquely estimate the structural parameters.

This paper extends the use of the Linear Integral Filters (LIF), first introduced in [19], to eliminate constant disturbances in the acceleration measurements and to attenuate measurement noise, to the parameter estimation of torsionally coupled building models, a more challenging problem when real-time estimation is desired, as the number of parameters involved in the torsion based model greatly increases.

The paper is organized as follows. Section 2 describes the model of a torsionally coupled multistory building. Section 3 presents the proposed vector parameterization and introduces the LIF. Section 4 shows both the offline and the online LSMs employed to estimate the stiffness/mass ratios and describes the methodology that allows estimating the damping/mass ratios, eccentricities, radii of gyration, and the mass, stiffness, and damping matrices of the structure. Experimental results in a five-story torsional building obtained with the vector parameterization and the LSM are presented in Section 5; three cases are considered, the first where all the floors are instrumented, the second where only the first, third, and fifth stories are equipped with accelerometers, and the third where only the first and the top floors are instrumented. Finally, Section 6 establishes the conclusions of this paper.

#### 2. Mathematical Model of a Torsionally Coupled Shear Building

Figure 1 shows a torsionally coupled shear building that is seismically excited, where the centers of mass and centers of resistance of the floors do not lie on one vertical axis. The building model is defined as [2, 10, 22, 23]where , , and are, respectively, the mass, stiffness, and damping matrices. Moreover, the variable denotes a zero matrix of size , and represents the absolute ground acceleration induced by an earthquake, which is given bywhere is the total number of floors and the terms and are the ground accelerations in the and directions, respectively. Furthermore, vector is defined aswhere , , is the displacement vector in the th coordinate and and are the first and the second time derivatives of . Vectors , , have the following structure: