# Damping Estimation of an Eight-Story Steel Building Equipped with Oil Dampers
## 1 Introduction
These predictions could be modal properties, time histories, or frequency response functions. The objective of model updating is to reduce such discrepancy, improving the correlation between numerical models and their real-world counterparts. Generally, model updating belongs to the family of inverse problems, and, in most practical applications, it can be approached from an optimization perspective. In recent decades, a significant number of model updating methods have been developed. Literature review for this topic is not presented here since is beyond the scope of this study. The interested reader is referred to references [19; 20; 21]. This paper is concerned only with the so-called direct updating methods because of their particular features, the resulting updated model reproduces exactly the target/measured modal data, including natural frequencies and mode shapes. By taking advantage of such features, the damper stiffness effect on structural dynamic behavior can be easily captured in model updating. The direct updating methods are essentially analytical methods which do not require iterations. In those methods, modal orthogonality, symmetry, and eigenvalue equation are usually imposed in correcting mass matrix or stiffness matrix or both, and minimum possible changes in updated matrices are achieved by using Lagrange multipliers or generalized inverse. [PERSON] and [PERSON] [22] first formulated a procedure using Lagrange multipliers to correct stiffness matrix to satisfy modal orthogonality and the eigenvalue equation with nominal mass matrix. [PERSON] [23] questioned the assumption that nominal mass matrix is correct and developed a new procedure to modify mass matrix under modal orthogonality constraint. Later, [PERSON] and [PERSON] [24] integrated these two procedures into the so-called analytical model improvement technique to update a long-exposure facility structure. Based on this research, [PERSON] and [PERSON] [25] further described two direct methods for model updating using modal data. [PERSON] [26] considered the interaction between mass and stiffness matrices in their updating processes and proposed a united procedure to modify them simultaneously using the vibration test data. [PERSON] [27] developed an algorithm turning model updating to a forward problem, in which a transformation matrix is only required to update mass and stiffness matrices. [PERSON] and [PERSON] [28]employed the Moore-Penrose inverse, rather than Lagrange multipliers, to reformulate the model updating problem. But the resulting corrected stiffness matrices lose symmetricity. [PERSON] and [PERSON] [29] established a simple procedure to maintain the physical meanings in mass and stiffness matrices corrections, which allows the updated model to be compatible with the measured natural frequencies. All of these methods discussed above are computationally efficient and can be easily applied to large structural models [24; 27; 29]. However, for applications in earthquake engineering, a problem arises that the predicted mode participation factors by updated models may not be consistent with those identified from earthquake measurements. To address this problem, an improved Berman-Nagy method is developed, using a new procedure to correct nominal mass matrix under the constraints of both modal orthogonality and mode participation factors. A large earthquake event is chosen to demonstrate the effectiveness of the proposed framework for the damping estimation of the investigated building. The enhanced strain-energy method is first employed to calculate the equivalent damping ratios and natural frequencies provided by oil dampers, which are modeled by a linear Maxwell model consisting of a spring element connected in series with a dashpot element [7; 15]. Then, the modal characteristics extracted from earthquake measurements are modified accordingly. The modified parameters are ultimately the data utilized to update an initial finite element model of the building by using the improved Berman-Nagy method.