Estimation of modal masses of a structure with a mass-type damping device

Highlights

• A method for estimating modal masses of higher modes is presented.

• This method was applied to a 39 story building structure with an active mass damper.

• The good agreement between the analytical and experimental resonance responses.

Abstract

In this paper, a method for estimating modal masses is presented by extending the procedure proposed by a previous study for estimating the first modal mass to a procedure which allows the estimation of modal masses of higher order modes. System matrices are first obtained by using a general system identification technique, and then the natural frequency and damping ratio of each mode are extracted from the obtained system matrices. Controllability and observability matrices are constructed by using both the identified system matrices and the modal space system matrices in which the modal masses are considered as unknown variables. The modal masses can be obtained based on the fact that the multiplication of the controllability and observability matrices does not change with the type of the system matrices, including the modal space ones. The advantage of the proposed method is that an accurate estimation of the modal mass of a higher mode is possible without the information on the modal vector which is difficult to experimentally identify. The proposed method was applied to a real-world 39 story building structure with an active mass damper and higher order modal masses of the building were identified. The dynamic responses analytically obtained by using the identified modal masses were found very close to the measured responses.

Introduction

One of the most widely used methods for obtaining the dynamic responses of a multi-degree-of-freedom (MDOF) system is the modal analysis approach which transforms the MDOF system into a series of single-degree-of freedom (SDOF) systems [1]. Recently, the identification of those modal parameters of a SDOF system such as damping ratio, frequency, and modal vectors has raised significant interest in the field of damage identification for buildings and civil structures [2], [3], [4], [5], [6], [7], [8], [9], [10]. Although the modal mass is one of the parameters constituting the governing equation of a SDOF system, its identification has not gained the same level of interest as the natural frequency and damping ratio. This may in part be that the modal mass has been considered as a parameter whose magnitude is arbitrarily scaled according to the corresponding modal vector. However, the magnitude of the modal mass assumes physical meaning when the mode vector is scaled to a specific engineering application, e.g., a mass type damper in the MDOF system. In this case, the modal vector is usually normalized so that the value of the modal vector is unity at the degree of freedom of the damper installation [11], and the modal mass is used as a representative value of the mass of the specific mode of the MDOF system. Also, the magnitude of the modal mass, which is obtained by normalizing the eigenvector to have unit value at the top floor, becomes important for an accurate estimation of the top floor responses of a tall building structure under wind load.

In a theoretical process, a mode vector is first calculated by using the specified mass and stiffness matrices, and then the modal mass is simply obtained by the calculated mode and the mass matrix. Therefore, the error in the mode vector due to the modeling error of the mass or the stiffness matrix generates the difference between the calculated modal mass and the actual one. Extensive research has been conducted for the accurate estimation of the mass, damping and stiffness matrices of a building or civil structure based on the modal test data, and integrated health monitoring systems based on full scale data have been suggested [12], [13]. Kim et al. detected damage in beam-type structure by using the modal parameters identified by frequency- and mode shape-based methods [14]. From an engineering view point, it is very difficult to precisely estimate the system matrices based only on measured signals due to limitations in the sensor installation and in the quality of the measured data, for instance in the case of highly non-stationary processes and presence of noise. Model updating techniques improving the analytical model based on the measured data have been developed to overcome this difficulty [15], [16]. Al-Hadid updated the modal mass based on the sensitivity method compensating errors in mode shapes by using data from two or more excitation frequencies [17]. Another issue in the field of system identification is modal parameter extraction from output-only data. Hermans and Auweraer presented a method a modal testing and analysis of structure under operational conditions that only response data are measurable while the actual loading conditions are unknown [18]. Output-only identification method is generally based on the assumption that the input is white noise and its restriction is that the scaling of the mode shape is very difficult or impossible. Devriendt and Guillaaume identified modal parameters from output-only transmissibility measurements without the assumption that the input is white-noise excitation [19]. Devriendt et al. proposed an ouput-only modal analysis for the structure under the unknown operational forces such as colored noise swept sine and impact [20]. Parloo proposed a sensitivity method for scaling of the mode shape by using output-only data [21]. Porras et al. estimated the modal mass or scale of a vibration mode by attaching an oscillating mass to a structure and using only output response [22]. A direct estimation of the modal mass that avoids the necessity of accurately identifying the mass and stiffness matrices and mode vectors, would be very helpful for optimally designing the mass type dampers. Hwang et al. [23] proposed a method to obtain the modal mass of the first mode by using the general system identification and H-infinity optimal model reduction techniques. It was shown that the method accurately estimates the first modal mass of an actual building with a tuned mass damper (TMD). However, the responses induced by the higher modes contaminated the accuracy of the first modal mass.

In this paper, the procedure proposed by Hwang et al. [23] is extended to estimate the modal masses of higher modes as accurately as for the first mode. In the previous research, the first modal mass of a structure with a TMD was estimated by using the relationship between controllability and observability matrices, and H-infinity model reduction technique. In the design process of a TMD whose frequency is tuned to one of the first mode of the structure, only the first modal mass is important and the modal masses of the other modes are not used for the design of the TMD. However, for the optimal design of an active mass damper (AMD) which can control higher order modal responses including the first modal one, the accurate identification of the higher order modal characteristics matters. In the previous study, if higher order modal effect shall be considered, the system matrix size becomes larger, the observability matrix becomes close to be singular and the error of the estimated modal masses increases. This singularity problem was solved by applying H-infinity model reduction. In this paper, a process for identifying the modal masses of higher order modes is proposed. First, the state space model is identified by applying a general system identification technique, and then natural frequency and damping ratio of each mode are extracted from the obtained system matrices. Controllability and observability matrices are constructed by using both the identified system matrices and the modal space system matrices in which the modal masses are considered as unknown variables. The modal masses can be obtained based on the fact that the product of the controllability and observability matrices is unique for the system matrices, including one in the modal space. Finally, it was theoretically shown that when the mode vector of the AMD input location is normalized to be unity, the inverse value of the extracted input matrix of the modal space system corresponds to the modal masses. The effectiveness of the proposed method is verified applying it to a real-world 39-story building structure with an active mass damper (AMD). The numerical instability reported by the previous study in identifying the higher order modal masses was not observed.