## 1 Introduction  
Passive control technology has been proven to be an efficient way to achieve vibration control for aseismic structures, especially inerter-based vibration control devices. Such technology has attracted significant attention [1; 2; 3; 4] owing to its apparent mass amplification and negative stiffness effects. Inerters are inertial elements with two terminals that develop an acceleration-dependent force that is proportional to a constant (inertance), which assumes mass units [5; 6; 7]. Their apparent mass amplification effect refers to the fact that the inertance can be thousands of times greater than the gravitational mass of the inerter [8]. Typical inerters include ball-screw [9], fluid-based [10; 11], rack-pinion [12], and electromagnetic [13] mechanisms. Regardless of the physical mechanism employed, the most widely considered inerter systems are tuned viscous mass dampers (TVMDs) [8] and tuned inerter dampers [14]. Other novel mechanical layouts have also been proposed, such as particle inerter systems [15], tuned liquid inerter systems [16], and lightweight tuned inerter mass systems [17; 18]. [PERSON] et al. [8] proposed a TVMD consisting of an inerter incorporating a dashpot in parallel and then in series with a tuning spring. They also pointed out a dashpot-deformation amplification effect, that the deformation of the dashpot in a TVMD can be larger than the entire deformation of the TVMD for enhanced energy dissipation. [PERSON] et al. [19] derived a closed-form equation to quantify the theoretical relationship between the amplification of dashpot deformation and the key parameters of the inerter system. [PERSON] et al. [20] proposed a displacement demand-oriented design method for an inerter system to achieve the desired seismic performance for primary structures. [PERSON] et al. [21; 22] analyzed the benefits of an isolation system enhanced by a TVMD and developed a multiobjective-based optimal design method. Reviewing the existing studies referring to TVMDs, performance related to displacement and acceleration has been analyzed extensively [8]. However, from the energy perspective, the advantageous energy dissipation enhancement effect, by which the TVMD exhibits increased energy dissipation capacity compared with a single dashpot, remains unclear. For passive seismic control of structures, energy-based analysis and optimal design are effective solutions for the typical control of dampers [23; 24; 25; 26; 27] in earthquake-prone areas. [PERSON] et al. [28; 29; 30] investigated the energy performance of a typical frame and developed a design for hysteretic energy-dissipating devices. Energy spectrum and design curves are efficient approaches to designing such devices, while a rigid foundation is commonly assumed without considering the soil [25; 31; 32]. As for controlled structures built on deep cover soft deposits, the energy responses calculated by this assumption are not accurate because of the nonlinear soil behavior excited by the ground motion. Seismic ground motions pass from the bedrock through the soil to the ground surface, yielding significant changes in their characteristics and in the energy performance of the controlled structures. In the existing literature, typical representations to simulate the nonlinear dynamic behavior of soil include equivalent linearization [33], elastoplastic dynamic constitutive [34], and viscoelastic dynamic constitutive [35] models. Compared with the simple-form equivalent linearization model and the impractical elastoplastic dynamic constitutive model, a viscoelastic dynamic constitutive model, such as the Davidenkov model, is not only capable of describing the nonlinear behavior of the soil but also convenient for practical application [36]. To quantify the energy dissipation enhancement effect of TVMDs, we developed an energy spectrum analysis framework by considering the soil underlying the inerter-based structure.