# Damping Estimation of an Eight-Story Steel Building Equipped with Oil Dampers
## 1 Introduction  
In the past few decades, many damping devices have been developed and applied in civil engineering structures, especially in earthquake-prone regions [1; 2; 3]. Viscous dampers generally represent a broad class of passive energy dissipation devices and have emerged as one of the most popular ones. By providing additional damping and energy dissipation, they can efficiently suppress earthquake-induced vibrations and, therefore, limit the damage to structural and non-structural components [4; 5; 6]. To better understand the damping properties of such dampers, many experimental studies have been conducted, including damper element tests [7; 8] and shaking table tests of small-scale and full-scale structures with viscous dampers [4; 5]. However, studies focusing on in-service structures with such dampers when subjected to large earthquakes are relatively rare [9; 10]. This practical issue requires accurate state quantification for dampers and structure itself, which holds a key position in structural health, safety, and risk assessments [11; 12]. This paper presents a study on the damping estimation of an eight-story building structure equipped with oil dampers through a proposed framework, which consists of an enhanced strain-energy method and an improved direct method for model updating. The strain-energy method is usually adopted at the design stage of passively controlled structures to assess the damping effect of added dampers quantitatively. Specifically, it incorporates thesupplemental damping provided by dampers into modal damping ratios, which are frequently referred to as equivalent damping ratios and are defined as the ratios of energy dissipated by dampers and stored in structure system. [PERSON] and [PERSON] [13] provided a procedure to quantify the damping effect of viscoelastic dampers and showed their effectiveness in attenuating structural seismic response. [PERSON] et al. [14] carried out a comprehensive experimental program on a 2/5-scale five-story steel frame with viscoelastic dampers and applied this procedure to assess the equivalent damping ratios for measured modes. [PERSON] et al. [15] conducted a series of shaking table tests on a full-scale five-story steel building with three types of passive dampers, including viscous dampers and oil dampers. Using the strain-energy method and system identification, they investigated the damping characteristics of those dampers and presented simplified estimations for their supplemental damping ratios. In this study, the strain-energy method is adopted to estimate the supplemental damping ratios provided by oil dampers for the investigated building under earthquake excitations. Additionally, it is notable that, except for the supplement damping, dampers can contribute to the increased lateral stiffness of the entire structure [14; 15; 16; 17]. This stiffness contribution, usually not large, may affect structural dynamic behavior, such as increasing natural frequencies. For evaluating such modal changes, an enhanced strain-energy method is proposed herein, quantifying the damper stiffness in the form of equivalent natural frequencies, i.e., the natural frequency changes of structures with and without dampers. Besides, note that the strain-energy method adopts assumptions of harmonic motion and mode-proportional deformation for dampers and structure itself, which violate their realistic situation during an earthquake event [15]. Consequently, the estimated equivalent damping ratios and natural frequencies may be questionable and, therefore, need to be verified under earthquake situations. In this study, it is achieved through model updating and prediction validations. Model updating using dynamic measurements has been a topic of extensive focus for many engineering principles, including civil engineering [18]. In structural dynamics community, it is well-understood that the predictions from an initial structural model are frequently inconsistent with those observed from its real-world counterpart [19].