## 1 Introduction  
In the past years, the burgeoning population and spatial constraints in metropolitan areas have promoted the increase of high-rise buildings. Meanwhile, rising concerns about the global energy crisis have led to a greater demand for environmentally sustainable and energy-efficient designs in such buildings. Due to the excellent acoustic insulation and thermal performance, double-skin facades (DSF) are widely adopted in high-rise buildings [1], including the Shanghai Tower, the Shard, and the Central Park Tower. However, the low stiffness and damping of super-high-rise buildings often result in excessive wind-induced vibration (WIV) responses [2]. The excessive vibration-induced floor acceleration response will lead to discomfort among occupants on the upper floors and limit the usability of the building, which should be suppressed from both perspectives of building safety and daily use [3]. Generally, three primary strategies are employed to mitigate building vibrations [4]: (a) structural modification of stiffness and mass, (b) aerodynamic optimization of the building shape, and (c) the use of auxiliary active, semi-active, or passive dampers to increase structural damping [5, 6]. However, altering the structural characteristics is not cost-effective, as it typically increases construction costs, while optimizing aerodynamic shapes may conflict with architectural design principles. Recently, the development of cost-effective passive dampers has made passive control an increasingly popular approach to managing vibrations. These dampers operate with constant parameters, require no energy input, and eliminate the need for stiffer or heavier structural elements. Among passive dampers, the tuned mass damper (TMD) has proven effective in a few high-rise buildings [7]. A TMD consists of a mass, spring, and damper, and its natural frequency can be tuned to that of the structure to transfer vibrational energy to the TMD, which dissipates the energy through its damping mechanism [8]. However, TMDs struggle to handle off-tuning and multimodal vibrations, and in super high-rise buildings, they are typically bulky and require considerable internal space. To enhance the effectiveness and robustness of TMDs, the multiple tuned mass damper (MTMD) was introduced [8; 9; 10; 11; 12]. The MTMD consists of several smaller mass dampers, which can be installed on the top floor (series MTMD) or distributed across multiple floors (distributed MTMD, or d-MTMD), offering a flexible installation scheme. Each TMD in an MTMD system can either be tuned to different modes of the structure to control multimodal responses or distributed around a single mode to improve the effectiveness of a specific modal response. However, MTMD systems do not reduce the physical mass of dampers. In recent years, the inerter, a mechanical device with two terminals, has been introduced in civil engineering vibration control. The inerter generates an inertance force proportional to the relative acceleration of its terminals, creating a substantial virtual mass that greatly exceeds its physical mass [13]. Consequently, the physical mass required for TMDs and MTMDs can be significantly reduced. This advantageous property has led to the widespread application of inerters in enhancing traditional passive control devices, including the tuned inerter damper (TID) [14; 15; 16], tuned viscous mass damper (TVMD) [17; 18], and tuned mass damper inerter (TMDI) [19; 20]. These inerter-enhanced devices are now being extensively studied for applications in high-rise buildings [21], wind turbines [22], and long-span bridges [23]. However, to optimize WIV control performance, inerters must be installed on different floors and connected to damper masses, which can occupy considerable internal space in high-rise buildings [24; 25]. Consequently, the engineering application of inerter-enhanced dampers for WIV control in high-rise buildings remains limited.