Shock and Vibration

Volume 2015, Article ID 627852, 13 pages

http://dx.doi.org/10.1155/2015/627852

## Modal Parameters Estimation of Building Structures from Vibration Test Data Using Observability Measurement

^{1}School of Architecture, Chonnam National University, Gwangju 500-757, Republic of Korea^{2}School of Architecture & Civil Engineering, Kyungpook National University, Daegu 702-701, Republic of Korea^{3}School of Architecture, Ajou University, Gyeonggi-do 443-749, Republic of Korea

Received 13 October 2014; Revised 21 January 2015; Accepted 26 January 2015

Academic Editor: Roger Serra

Copyright © 2015 Jae-Seung Hwang et al. 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

The load distribution to each mode of a structure under seismic loading depends on the modal participation factors and mode shapes and thus the exact estimation of modal participation factors and mode shapes is essential to analyze the seismic response of a structure. In this study, an identification procedure for modal participation factors and mode shapes from a vibration test is proposed. The modal participation factors and mode shapes are obtained from the relationship between observability matrices realized from the system identification. Using the observability matrices, it is possible to transform an arbitrarily identified state space model obtained from the experimental data into a state space model which is defined in a domain with physical meaning. Then, the modal participation factor can be estimated based on the transformation matrix between two state space models. The numerical simulation is performed to evaluate the proposed procedure, and the results show that the modal participation factor and mode shapes are estimated from the structural responses accurately. The procedure is also applied to the experimental data obtained from the shaking table test of a three-story shear building model.

#### 1. Introduction

The modal participation factors and mode shapes are coefficients that represent how the ground acceleration is distributed to each mode of a building structure. This is because the inertial force, which is generated by the ground acceleration and the mass of each floor, is distributed to each mode through the modal participation factor for typical building structure. Ground vibrations can be affected by natural processes like earthquakes as well as by human events such as transportation [1], civil work [2], blasting [3], and industrial activities. For this reason, the method to calculate the modal participation factor and mode shape is presented in many references in the area of dynamics of building structures [4, 5].

Kim and Choi [6] presented a nonlinear static analysis procedure for the design of supplementary dampers that uses the modal participation factor of the fundamental mode to obtain base shear versus roof-story displacement capacity curve of a structure from pushover analysis. Park et al. [7] proposed a factored modal combination method for accurate prediction of the inelastic earthquake response of a structure by pushover analysis. The modal combination factors for each mode in their proposed method are calculated based on the modal participation factor. Reinoso and Miranda [8] presented a method to estimate lateral acceleration demands in high-rise buildings. The acceleration demands are obtained by approximating the dynamic behavior of the building with that of a continuous beam which is characterized by mode shapes, period ratios, and modal participation factors. Bracci [9] presented a simplified procedure for evaluating the seismic performance and retrofit of existing low to midrise reinforced concrete buildings whose story demands are estimated using the modal participation factor and mode shape.

The modal participation factors and mode shapes of an idealized analytical model, however, are different from the actual one due to modeling and construction error. Therefore, there exist limits on the estimation of actual behavior. Due to the discrepancies between an idealized shear building model and a real structure, the modal participation factors and mode shapes may be calculated with errors resulting in incorrect estimation of structural behavior. Zhou et al. [10] reported that the excitation on the time-varying structures is unknown and random in many real-life applications so that output-only methods are appropriate.

Research on the identification of the dynamic parameters of a structure such as natural frequency and damping ratio using the output-only methods has been widely performed utilizing the system identification theory [11–13]. System identification in time and frequency domains have been studied actively [14, 15] and the identification of linear system as well as nonlinear system has been studied in various fields of engineering [16–18]. Cho et al. [19] and Kang et al. [20] extended the application of the output-only method to the identification of the secondary mass dampers such as tuned mass dampers and tuned liquid dampers installed on tall buildings to suppress wind-induced motion.

In order to estimate the modal participation factor from a vibration test, it is required to know the mass matrix and mode shape by the definition. That is, the more reliable modal participation factors can be obtained if sensors are installed on every floor so that the mode shape is measurable. However, it is not feasible to measure the dynamic behaviors of every floor in practice due to installation problem and laborious data processing procedure. Meanwhile, in the case of the estimation of damping ratio and modal frequencies that are defined by mass, stiffness, and damping matrices, the damping and frequency values for each mode can be directly estimated from the test without calculating those matrices [14]. Likewise, if the modal participation factor can be obtained directly from the test, not indirectly from the estimated mass matrix and mode shape, the dynamic characteristics and actual behavior of structures can be more precisely understood and modal participation factors can be more effectively utilized.

In this study, an identification procedure for modal participation factors and mode shapes directly from the measured response is proposed. The modal participation factors and mode shapes are obtained from the relationship between observability matrices realized from the system identification. Because the observability matrices can be easily constructed without knowledge of mass, stiffness, and damping matrices of the structure, it is possible to identify the modal participation factor directly from the measured response using the proposed method.

For the numerical derivation to estimate the modal participation factor, the single-input single-output (SISO) system in continuous time domain is considered. Accordingly, the proposed procedure has an advantage that the modal participation factor of the modes that are normalized to a certain element can be estimated from the response of the corresponding floor without knowing responses of other floors. That is, the proposed method can be utilized for the estimation of modal participation factor even when the sensors are not installed on every floor and thereby the mode shape is not available. Further, the mode shapes can also be estimated directly from the experimentally estimated modal participation factors. The numerical simulation is performed to evaluate the proposed procedure, and the procedure is also applied to the experimental data obtained from the shaking table test of a three-story shear building model.

#### 2. Modal Parameters Estimation

##### 2.1. Modal Participation Factor

The equation of motion for an -story shear building subjected to earthquake iswhere , , and are mass, damping, and stiffness matrices of the structure, respectively, is an column vector with all elements equal to one, is an column vector of the relative displacement of the structure to ground, and is ground acceleration. Equation (1) can be expressed in the modal coordinate system as (3) using the transformation expressed in (2)where is the generalized modal coordinate and is the mode shape matrix. Using the normal coordinate transformation, the equation of the th mode is given by where is the mode shape, is the modal displacement, is the modal damping ratio and is the frequency of the th mode. The modal participation factor of the th mode, , is defined from (4) by [4]

It is apparent from (5) that the modal participation factor is not a unique value but varies depending on the normalization method of mode shapes as well as floor mass distribution.

##### 2.2. Modal Participation Factor Estimation Using System Identification

For a SISO system with an acceleration output, the output for th floor is obtained from (2) aswhere is the th element of the th mode shape. If the th element of each mode is normalized to unit value, (6) becomes

The equation of motion presented in (4) and the output for th floor of (7) can be transformed into the state space form aswhere is the state variable, is the relative acceleration of the th floor to ground, and are the zero and identity matrices of size , respectively, is the zero vector of size , is the vector consisting of the modal participation factors of modes normalized to the th element, is the th element of , and diagonal matrices and , respectively, are

Equations (8a) and (8b) can be simplified aswhere is the input to the SISO system, that is, the ground acceleration, and , and are , , , and system matrices, respectively.

The state space model presented in (10a) and (10b) has a physical sense because it is derived from (1) which is defined in terms of a second order differential equation whose variable and its derivatives have physical meaning. In this study, the state space model defined in physical domain is denoted as a* typical state space model*. It is observed from (8a), (8b), (10a), and (10b) that the modal participation factor for each mode can be estimated if the system matrix of the typical state space model is obtained.

The state variable and state space model realized from the vibration test using the system identification method have arbitrary values because they are determined such that the relationship between input, , and output, , is simply satisfied. The state variable serves as intermediate variables that connect between the input and output, and there are infinite possible state variables that satisfy the relationship between input and output. Therefore, the corresponding state matrices have no physical meaning. In this study, the state space model realized from the vibration test is denoted as an* arbitrary state space model*.

The arbitrary state space model obtained from vibration test using the system identification method can be expressed aswhere is the state variable and , and are system matrices realized by the system identification.

As described above, the state variable, , in (10a) and (10b) has physical meaning of modal displacement and velocity in modal coordinate, but the state variable, , in (11a) and (11b) has no physical meaning. Therefore, the system matrix realized by the system identification is generally different from the system matrix of the typical state space model. In order to estimate the modal participation factor from the vibration test, consequently, it is required to obtain the system matrices of the typical state space model from those of the arbitrary state space model. That is, it is required to find a relationship between the state variables, , of the typical state space model and of the arbitrary state space model.

If the transformation matrix transforms the state variable, , in (11a) and (11b) into the state variable, , in (10a) and (10b), the relationship is given bySubstituting (12) into (10a) and (10b) leads toComparing (13a) and (13b) and (11a) and (11b), the following relationships between the system matrices of two state space models are obtainedwhere and are the observability matrices of the typical and arbitrary state space models, respectively [21, 22]. The observability matrix is a measure whether the information on the state variable can be determined from the output of a system and is defined by

The transformation matrix in (12) is obtained from (14) asand the relationship between the system matrix realized by the system identification and the system matrix of the typical state space model is given by

Referring to (8a) and (8b), the values of first rows of vector are all zeros. Therefore, the observability matrix of the typical state space model in (17) can be rewritten in partitioned form aswhere , , , and are submatrices of and and are submatrices of . Finally from (18), the modal participation factors of modes normalized to th element are given byIt is noted that and are only used to calculate the modal participation factor in (19). This is to avoid mathematical ill-conditioning during the inverse matrix calculation.

##### 2.3. Mode Shape Estimation from Modal Participation Factors

The mode shape can be estimated directly from the experimentally estimated modal participation factors based on the simple relationship between factors. From (5), the modal participation factor of th mode, , obtained using the th mode shape normalized to th element is given bywhere is the th mode shape normalized to the th element. If is the th mode shape normalized to the th element, the is related to bywhere is the th element of the th mode shape normalized to the th element. Substituting (21) into (20) leads towhere is the modal participation factor of the th mode obtained using the th mode shape normalized to th element.

Accordingly, th element of the th mode shape normalized to the th element can be estimated using the identified modal participation factor as

#### 3. Numerical Verification

In this chapter, the proposed procedure is verified numerically using two examples: a 3-story building subjected to the white noise ground acceleration and an 8-story building subjected to the 1940 El Centro earthquake ground acceleration. The following step-by-step procedure is used to estimate the modal participation factors and mode shapes.

*Step 1. *Identify the arbitrary state space model for the SISO system using any system identification technique.

*Step 2. *Obtain the modal frequencies and damping ratios from the identified arbitrary state space model.

*Step 3. *Construct the system matrices and of the typical state space model using the modal frequencies and damping ratios obtained from Step 2.

*Step 4. *Calculate the observability matrices of the typical and arbitrary state space models using (15).

*Step 5. *Calculate the modal participation factor from (19) using the partitioned observability matrices and the system matrix b of the arbitrary state space model identified in Step 1.

*Step 6. *If more than one floor is measured, the mode shape can be calculated from (23) using the calculated modal participation factors in Step 5.

##### 3.1. 3-Story Building

###### 3.1.1. Building Description

A 3-story shear building structure is first used for numerical simulation. The mass, damping, and stiffness of each floor are 22.758 kg, 6.50 N·s/m, and 3,764 N/m, respectively. These values correspond to the experimental setup used in a laboratory. The natural frequencies are 0.900 Hz, 2.522 Hz, and 3.644 Hz and modal damping ratios are 0.50%, 1.40%, and 2.03%. The modal characteristics of the structure are summarized in Tables 1 and 2. The mode shapes are normalized to the third floor in Table 1, and the modal participation factor of each mode is obtained for three cases with different normalization floor in Table 2. As noted in (5), it is apparent that the modal participation factor varies depending on the normalization method of mode shapes.