## 1 Introduction  
Underground structures are essential infrastructure of modern metropoles that can be vulnerable to strong earthquakes [1; 2; 3]. During the Great Hanshin earthquake in 1995, Daikai subway station suffered catastrophic damage [4; 5]. Additionally, damages were also observed in underground structures after the Tang-Shan (Tang-Shan, China, 1976) [6], Parnitha (Parnitha, Athens, 1999) [7], Chi-Chi (Chi-Chi, China, 1999) [8] and Wen-Chuan earthquakes (Wen-Chuan, China, 2008) [9]. There is a broad need to improve the seismic safety of underground structures and develop innovative seismic-vibration control devices to protect underground structures from destruction during strong earthquakes [10; 11; 12]. Among the developed seismic-vibration control technologies, the seismic-isolation approach is definitely effective, involving installing an isolation system to reduce the seismic forces transferring from the foundation to the structure, and this approach has been used several decades for the earthquake protection of aboveground structures [13]. In recent years, seismic-isolation systems have been employed in underground structures to reduce or avoid excessive responses and seismic damage [14]. Existing studies on underground structures have reported that the central column is a key structural component whose failure leads to the collapse of the underground structure. Therefore, it was proposed that isolators be inserted at the top of central column to avoid excessive responses, especially with regard to shear forces. In employing conventional isolation systems (CISs) in underground structures, the implementation of flexible rubber joints [12], lead-rubber bearings [10] and sliding isolation bearings [11] has been proposed as an alternative selection. Despite the effectiveness of reducing shear force, the installation of the above CISs (flexible rubber joints, lead-rubber bearings, shear-panel dampers or sliding isolation bearings) cannot achieve high-level and multiperformance control, particularly the simultaneous control of both shear force and the relative displacement of underground structures. Different from CISs with positive stiffness that resists the movement of the device, negative-stiffness control devices have the potential to improve structural dynamic performance by using their negative-stiffness mechanism. In an ideal scenario, a negative-stiffness device can produce negative-stiffness elements, active [15], semiactive [16; 17] and passive [18; 19] negative-stiffness devices were developed, of which the passive type has attracted more attention, due to their stability and convenience. By utilizing precompressed springs [20; 21], prebuckled columns [22] and friction pendulum sliding components [23], passive negative-stiffness devices yield satisfactory reaction forces to assist in the motion of the device [24]. Departing from the widely analyzed negative-stiffness effect, variants of negative-stiffness-based systems and the potential for improved energy-dissipation efficiency both remain unclear, especially in the vibration control of underground structures, which is of concern in this study. Resulting from the basic characteristic of a negative-stiffness spring, whereby it can assist in spring deformation, two spring devices with negative and positive stiffness in a series layout yielded an amplification effect [25]. As a consequence, a dashpot in parallel with a positive-stiffness spring can obtain amplified deformation for increased energy-dissipation efficiency and capacity, which provides improved seismic performance for the structures. Therefore, a novel isolation system is proposed for the seismic-vibration control of underground structures, by incorporating a negative-stiffness amplification system (NSAS) and an isolator. The mechanical model, physical realization and energy-dissipation-enhancement mechanism of the NSAS are introduced and analyzed.