## 1 Introduction  
New metro lines are sometimes inevitably built close to highly vibration-sensitive areas owing to the rapid development of urban railway transits [1, 2, 3, 4, 5]. Therefore, train-induced vibrations have received increasing attention because of their negative impact on the long-term protection of historical buildings [6, 7], the operation of precision instruments [8, 9, 10] and the residents living in the buildings over the metro depots [11, 12, 13, 14], which are still urgent issues that need to be addressed today. In addition, how to predict train-induced environmental vibration more efficiently and accurately has also been a focus of attention in recent years [15, 16], for example, considering various types of irregularities [17], establishing analytical prediction models [18] and introducing neural networks [19]. Vibration mitigation measures are adopted to address these issues in the vibration source [20, 21], propagation path [22, 23] and vibration receivers [24] based on the vibration propagation route. Among these measures, vibration source mitigation is widely considered the most cost-effective [25]. Several vibration source mitigation techniques are used, such as rail pads [26], floating slab tracks (FSTs) [27, 28] and ladder tracks [29]. FSTs are widely used in metro lines passing through vibration-sensitive areas because of their excellent vibration-reduction performance [30]. Currently, the natural frequency of FST is approximately 10 Hz, and the effective vibration isolation frequency band exceeds \(\sqrt{2}\) times its frequency. However, its vibration reduction effect is limited to the low-frequency range and can amplify the vibration in the frequency band near its natural frequency [31], to which buildings and precise instruments are somewhat sensitive [32]. The contradiction between the amplification of surface vibrations caused by FSTs and the adverse effects of low-frequency vibrations on precision instruments and ancient buildings must be resolved urgently. [PERSON] et al. [33; 34; 35] established a three-dimensional (3D) coupled dynamic model of a metro train and FST with dynamic vibration absorbers equipped on slabs to investigate the low-frequency vibration isolation ability of this type of track. They found that the slab acceleration and supporting force significantly decreased at low frequencies of 10-20 Hz; however, the displacement of the absorbers was larger than that of the slabs. [PERSON] et al. [36] proposed a new supporting form using steel-spring isolators, whose vibration reduction frequency range is 1-25 Hz, but the resonance could not be eliminated successfully. [PERSON] et al. [37] developed an FST with nonlinear variable-stiffness isolators and semi-active magneto-rheological dampers, which can provide a 5 dB reduction of the vertical acceleration level at the resonance frequency. [PERSON] et al. [38] investigated the structure of a phononic crystal vibration isolator, which increased the number of bandgaps and broadened the FST bandgap. [PERSON] et al. [39; 40] proposed a phononic crystal steel-spring FST structure and introduced an elastic wave bandgap to improve the low-frequency vibration reduction performance of the FST. These studies were considered from the single perspective of vibration source reduction. In recent years, the bandgap theory has developed rapidly. Its basic principle is to design the parameters of periodic structures to prevent vibration waves in specific frequency bands from passing through periodic structures to achieve the goal of isolating vibration propagation [41], which makes it possible to minimise the negative impact of FST resonance on the ground vibration response. [PERSON] et al. [42] first proposed pile barriers for ground vibration reduction. Subsequently, several studies have been conducted in this area. [PERSON] et al. [43] proposed a periodic vibration isolation foundation with broad low-frequency bandgaps. [PERSON] et al.