## 1 Introduction  
Marine structure safety, especially in the deep sea, is an important concern. The deep-sea floating platform is one of the most important large structures used in ocean energy exploitation. Dynamic responses due to the wind and wave effects must be considered to ensure a safe design [1]. The combined action of complex multiple loads could cause a large vibration amplitude in marine structures [2; 3]. Large amplitude and alternating vibration are the main detrimental effects that reduce the safety and fatigue life of a deep-sea platform. In the field of civil engineering, engineers use a wide variety of energy-absorbing devices to reduce the vibration of building structures. These energy-absorbing devices may also be applied to the problem of vibration suppression of deep-sea marine structures. The vibration of a floating platform could be reduced using active or passive controls [4; 5] and various devices have been proposed. A Tuned Liquid Column Damper (TLCD), a kind of passive vibration control device to suppress movement, was proposed by [PERSON] [6] in 1989. [PERSON] _et al._[7; 8] first used the TLCD to suppress the surge and sway motions vibration of a Tension Leg Platform (TLP). [PERSON] _et al._[9] further developed a simulation-based method for the design of mass dampers applied for the response mitigation of tension lag platforms. [PERSON] _et al._[10] experimentally studied the harmonic responses of the TLCD for wind excitations. [PERSON] _et al._[11] studied the effective performance of TLCD and another passive vibration-mitigating device, a tuned liquid column ball damper for the control of wave-induced vibration. [PERSON] _et al._[12] invented a new type of energy-absorbing device, the S-shaped TLCD. The S-shaped TLCD could effectively suppress the horizontal movement and vertical in-plane rotation of a TLP. Although these devices were effective for the suppression of surge and sway motion, TLCD was ineffective at the suppression of vertical movement, and sometime even enhanced this movement [10]. The vertical movement of the floating structures significantly impacts the strength and fatigue of the tension leg and mooring system, and the inability of these devices to suppress this movement could decrease the safety and service life of the structure. Vibration-induced vertical motion of TLPs could be balanced by using absorbers with air springs. Dual-chamber air springs are an effective energy-absorbing device used as a key component for vibration isolation. [PERSON], [PERSON], and [PERSON] _et al._[13; 14; 15] applied a system of vibration absorbers using dual-chamber air springs and water columns to suppress resonant motions and studied the effects of the orifice ratio for structural damping. [PERSON] and [PERSON] [16] studied the complex dynamic stiffness of the damper spring, a function of the excitation frequency. The experimental results measured by [PERSON] and [PERSON] [17] for a dual-chamber pneumatic spring exhibited significant amplitude-dependent nonlinear behavior. [PERSON] _et al._[18; 19; 20] proposed a characteristic output spectrum nonlinear (nCOS) method. The nCOS method is based on the theory of Volterra series expansion. They used the nCOS method for analysis and design of an air spring in a nonlinear vehicle suspension system. These studies focused on traditional forms of a dual-chamber air spring. A dual-chamber air spring device applied to marine structure motion suppression requires a gas-liquid coupling air spring structure. This differs from the traditional air spring both in structure type and in the vibration characteristics. The compression of the gas in the traditional air spring was controlled by a piston, which acted on the pressure surface of the air spring [21; 22; 23]. As shown in Figure 1a, the displacement of the piston, \(x_{0}\), causes a compression of the air in the upper chamber. The pressure variation of the air spring chamber is of the same phase and amplitude as the external load.