1 Introduction

The metro project has emerged as a prominent trend in future urban rail transit due to its remarkable advantages, including high speed, large capacity, and minimal impact on surface transportation. However, despite these benefits, the underground location and operation of the metro system give rise to vibrations that can be transmitted directly to nearby buildings through the soil [1, 2]. As urbanization and urban rail transit construction continue to progress, certain structures such as opera theatres are increasingly positioned in close proximity to the metro lines [3, 4]. The operation of the metro can severely disrupt opera performances if the vibrations emanating from the adjacent metro are not adequately controlled. Hence, prior to the construction of both the metro system and the adjoining buildings, the assessment and effective management of metro-induced vibrations is imperative in order to establish a low-vibration environment conducive to the operations of the opera theatre.

The passage of a metro train generates vibrations along the tracks, characterized by an uneven longitudinal distribution [5]. The vibrations are transmitted through the station lining structure and the adjacent solum, exerting a dynamic impact on the theatre structure [6]. To capture the vibrations caused by the railway, field experiments are conducted, employing an experimental methodology to acquire precise and reliable data. Sanayei et al. [7] analyzed the building foundation vibrations induced by the metro. Xia et al. [8] conducted an examination of railway vibrations on nearby soil and structures. Toygar et al. [9] carried out onsite experiments to assess the efficacy of open trenches in reducing vibrations. However, it is important to acknowledge that the experimental method does have certain limitations in specific scenarios [10]; for instance, its capability to accurately predict vibrations in advance may be inadequate. In contrast, the numerical method offers distinct advantages and broader application possibilities in terms of predicting metro-induced vibrations [11]. Additionally, the numerical method is of vital importance in guiding the design process of vibration control measures [12,13,14]. Various forecasting approaches have been developed for the anticipation of metro-induced vibrations, including empirical rules [15] as well as analytical approaches [16,17,18,19]. Yang et al. [20, 21] employed the finite/infinite element method and the 2.5D approach to investigate the train-induced soil vibrations and their mitigation. Zhu et al. [22, 23] proposed a random approach for predicting ground vibrations induced by subway trains. Zou et al. [24, 25] conducted a study to understand the impact of train-induced vertical vibrations on buildings located over the tracks. However, it is important to acknowledge that the application of these models still faces limitations [26,27,28], primarily due to challenges in accurately simulating intricate track configurations. Furthermore, the two-step or multiple coupling methods [29,30,31] often fall short in providing precise predictions of the entire process involved in the generation and transmission of metro-induced vibrations. The longitudinal distribution of vibration sources and the spatiotemporal movement characteristics of train loads pose difficulties in being adequately reflected within the two-step model, which typically considers fastener forces or track vibrations as external inputs.

In order to provide a quantitative description of the vibration impact and assess the effectiveness of vibration reduction measures before construction, a unified train-track-station-solum-opera (TTSSO) numerical simulation model is developed as a digital twin for a newly constructed opera theatre. This model evolves in tandem with the construction progress, enabling timely identification and resolution of issues, thereby aiding in the modification and validation of the design project. The main focus of this research paper is to predict the metro-induced vibrations, especially at the auditorium of the main stage and the underground stage. Subsequently, practical measures for vibration reduction are proposed, drawing upon the principles of phononic crystal theory and resonance theory.

2 Opera vibration forecasting approach

The proposed opera vibration forecasting approach integrates various theories, including multibody dynamics, wheel/rail contact theory, finite element theory, wave propagation theory, and resonance theory. Through a collaborative effort involving multiple software tools, the coupled model incorporates essential parameters such as train characteristics, rail irregularities, dynamic soil properties, geometric configuration, and material properties of the opera theatre. By inputting these parameters into the model, the vibrations propagating to the reconstructed opera theatre can be accurately calculated and captured.

The research introduces a unified dynamic numerical forecasting approach known as the TTSSO model. The coupled mode of the TTSSO model is derived from the well-established train-track model [32], which has been extensively validated in numerous train-track coupled problem scenarios. In the approach, the interaction forces between the wheel and rail are computed, taking into account the change in track dynamic displacement, the stiffness of the station, and the mass of adjacent solum and buildings. The acceleration of the train and the substructure continuously interact with each other throughout the simulation.

2.1 Integration theory of coupling

Drawing upon the principles of d'Alembert’s theory, the dynamic equilibrium equation is formulated by considering the force vector balance relationship for each component of the dynamic system [33]. Under the assumption of small deformation amplitudes for the train, track, station, solum body, and opera, the dynamic coupling formula for the TTSSO model is presented in Eq. (1):

$$\left[ {\begin{array}{*{20}c} {M_{vv} } & 0 & 0 & 0 & 0 \\ 0 & {M_{tt} } & 0 & 0 & 0 \\ 0 & 0 & {M_{ss} } & 0 & 0 \\ 0 & 0 & 0 & {M_{bb} } & 0 \\ 0 & 0 & 0 & 0 & {M_{oo} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\ddot{\delta }_{v} } \\ {\ddot{\delta }_{t} } \\ {\ddot{\delta }_{s} } \\ {\ddot{\delta }_{b} } \\ {\ddot{\delta }_{o} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} {C_{vv} } & {C_{vt} } & 0 & 0 & 0 \\ {C_{tv} } & {C_{tt} } & {C_{ts} } & 0 & 0 \\ 0 & {C_{st} } & {C_{ss} } & {C_{sb} } & 0 \\ 0 & 0 & {C_{bs} } & {C_{bb} } & {C_{bo} } \\ 0 & 0 & 0 & {C_{ob} } & {C_{oo} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\dot{\delta }_{v} } \\ {\dot{\delta }_{t} } \\ {\dot{\delta }_{s} } \\ {\dot{\delta }_{b} } \\ {\dot{\delta }_{o} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} {K_{vv} } & {K_{vt} } & 0 & 0 & 0 \\ {K_{tv} } & {K_{tt} } & {K_{ts} } & 0 & 0 \\ 0 & {K_{st} } & {K_{ss} } & {K_{sb} } & 0 \\ 0 & 0 & {K_{bs} } & {K_{bb} } & {K_{bo} } \\ 0 & 0 & 0 & {K_{ob} } & {K_{oo} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\delta_{v} } \\ {\delta_{t} } \\ {\delta_{s} } \\ {\delta_{b} } \\ {\delta_{o} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {P_{v} } \\ {P_{t} } \\ {P_{s} } \\ {P_{b} } \\ {P_{o} } \\ \end{array} } \right\}$$
(1)

where \(\left[ M \right]\),\(\left[ C \right]\), and\(\left[ K \right]\) represent the mass, damping, and stiffness array of the TTSSO model, \(\left\{ \delta \right\}\),\(\left\{ {\dot{\delta }} \right\}\), and\(\left\{ {\ddot{\delta }} \right\}\) represent displacement, velocity, and acceleration, \(\left\{ P \right\}\) represents load, and the index v, t, s, b, and o are the abbreviations of train, track, station, solum body, and opera theatre. The model is calculated through the ABAQUS dynamic explicit procedure, and the iteration method is the central difference method. In explicit analyses, the automatic incrementation time step is chosen to satisfy the Courant–Friedrichs–Lewy (CFL) stability criterion to ensure numerical stability.

2.2 Numerical prediction

A prediction model known as TTSSO is installed to accurately forecast vibrations. Figure 1 depicts the structure of the opera vibration forecasting approach, providing a visual representation of the approach. The irregularities present in the rail are incorporated as factors influencing the activation procedure [13, 34, 35]. The train is modeled as a multi-rigid-body structure, while the track, station, solum body, and opera theatre are meticulously represented with flexible elements [33]. These components are interconnected through contact theory, deformation coordination, and springs. This coupling methodology leads to the creation of a unified rigid–flexible coupled model that comprehensively considers the components of train, track, station, solum, and opera. With simulation calculations, the vibrations experienced by the train, track, solum, and opera are determined. By comparing the predicted vibrations of the train and track with existing literature and field tests, the validity and reliability of the model can be confirmed [36, 37]. Once the model's accuracy is established, the predicted theatre vibrations serve as a basis for assessing the train-induced vibrations and provide essential insights for implementing effective vibration reduction approaches.

Fig. 1
figure 1

Structure of the numerical forecast methodology

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Figure 2 provides a comprehensive representation of the completed model, showcasing front, side, vertical, and aerial views. The finite element (FE) model meticulously incorporates the degrees of freedom (DOFs) of the train [13], in addition to the intricate configuration of the track, station, opera, and layered solum. The ballastless track configuration [38] comprises rails, a fastening system, and a track bed. In light of the primary origin of vibrations, the intricate simulation of the track is of utmost importance and significantly influences subsequent vibration analysis and reduction efforts. The horizontal and vertical irregularities of the rail model are obtained from measured data acquired within the Beijing metro system. The wheel interface unevenness is set in accordance with EN15313 and the Wheel-Rail Interface Handbook [34, 39]. When the train moves, relative displacement occurs between the wheel and rail, generating corresponding interaction forces. In the process of grid division for the soil, the 30-m soil is divided into seven layers based on soil parameters, and each layer is further divided into grids ranging from 0.25 to 3 m. Similarly, the building's grid is determined by the structural dimensions of the beams, slabs, and columns, with sizes ranging from 0.5 to 1 m. The TTSSO model comprises a total of 391,693 elements, including elements of type R3D4, C3D8R, B31, B32, S4R, and S3R. The model was established and computed using the ABAQUS finite element software.

Fig. 2
figure 2

Prototype of the track-station-solum-opera

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The selection of parameters is based on the specific project requirements and engineering expertise [40], as shown in Table 1. To mitigate the influence of vibration wave boundary reflection, artificial boundaries were defined around the solum in the simulation [41]. The North Kunqu Opera Theatre, the subject of this study, is a multilevel building consisting of nine aboveground floors with a total area of 23,616 m2 and three underground floors with a total area of 6585 m2. The underground portion is buried at a depth of 20.5 m and a distance of 42 m, as shown in Fig. 3. The theatre comprises three stages, with the main stage located on the first floor and another stage in the cellar. To model the opera, an interface program called YJK-ABAQUS is utilized. The parameters of the opera, including the sections, materials, and loads applied to beams, slabs, columns, walls, and other components, can be input into the TTSSO model.

Table 1 Parameters of the TTSSO model
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Fig. 3
figure 3

The location relationship of the metro and the opera

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A comprehensive investigation of the solum conditions at the proposed project site is conducted, employing various techniques. Based on the drilling statistical and geotechnical data, the solum is segmented into several layers. Additionally, the dynamic elastic modulus and dynamic Poisson ratio are calculated using Eqs. (2) and (3), respectively. The shear stress yield criterion is represented by Eq. (4) [42]. Table 2 provides a comprehensive overview of the solum characteristics.

$$E_{d} = \frac{{\rho C_{s}^{2} \left( {3C_{p}^{2} - 4C_{s}^{2} } \right)}}{{C_{p}^{2} - C_{s}^{2} }}$$
(2)
$$V_{d} = \frac{{C_{p}^{2} - 2C_{s}^{2} }}{{2\left( {C_{p}^{2} - C_{s}^{2} } \right)}}$$
(3)
$$\tau_{f} = c + \sigma \tan \varphi$$
(4)

where ρ represents the density, Cs and Cp represent the shear and compression wave velocity, Ed and Vd represent the dynamic elastic modulus and dynamic Poisson ratio, τf represents the ultimate shear stress, c represents the cohesion, φ represents the internal friction angle, and σ represents the normal stress.

Table 2 Multilayer solum body parameters
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The integration of each substructure within the system establishes a cohesive body. The wheel–rail contact takes into account the nonlinear parameters. The track and the station are unified through the interface spring, ensuring their mutual influence is captured. Moreover, the opera is tied in the solum and the deformation coordination condition is adopted, enabling the accurate representation of their interaction and response.

2.3 Model validation

The coupling theory employed in this study has undergone extensive verification and has been successfully applied in numerous investigations related to track dynamics. The forecasting approach adopted in this research takes into account the specific characteristics of the TTSSO model. To enhance the credibility of the approach, the calculated vibrations are compared with measurements, serving as a validation process to assess the accuracy and reliability of the simulation results.

Vibration tests were conducted on the subway platform, ground vibration attenuation tests in the vertical line direction, and surface vibration tests in the area to be constructed, as shown in Fig. 4. The subway platform and the ground vibration attenuation test in the vertical line direction are used for model verification. The ground vibration distribution test in the area to be constructed is used to evaluate the environmental vibration in the area to be constructed.

Fig. 4
figure 4

On-site experiments of platform and ground vibrations

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As can be seen from Fig. 5, the peak values of vibration acceleration in the simulation and test time history curves of platform vibration are close, and the line shapes are consistent. The vibrations in the station exhibit a close similarity between the calculated and measured values. Both show a variation of approximately 40 dB in frequencies below 32 Hz, with peak values hovering around 80 dB at 64 Hz. Ground vibrations gradually diminish as the distance from the line increases, ranging from 80 dB at 0 m to 65 dB at 25 m. It is noteworthy that there is a slight increase near 35 m due to the enlarged vibration phenomenon.

Fig. 5
figure 5

Experimental results and simulation comparison. a Simulation results of platform vibration. b Test results of platform vibration. c Comparison of platform vibration levels. d Comparison of ground vibration attenuation

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As can be seen from Fig. 6, the contours of ground vibration align reasonably well with the distribution observed in on-site tests. Moving from the proximal to the distal areas of the site, the vibrations gradually decrease. However, as the vibrations propagate further, an amplification area emerges where the vibrations slightly intensify. The location of this amplification area can vary slightly due to differing geological conditions. The vibration level within the theatre's construction area primarily ranges from 60 to 70 dB, with the peak vibration level nearing 80 dB.

Fig. 6
figure 6

Contour of ground vibration in the theatre area to be constructed

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Drawing on the comparative analysis of the transmission and distribution of vibrations in the station and on the ground, adequate verification and validity in predicting metro-induced vibrations can be demonstrated.

3 Prediction results

The focus of the study will be on the main theatre, which serves as the primary performance venue. Of particular interest is Theatre 1 located in the cellar. Figure 7 shows the structure of the opera and vibration-sensitive points. Note that the slab of the cellar has been removed to better illustrate the underlying configuration.

Fig. 7
figure 7

The vibration-sensitive points of the opera

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Figure 8 presents the time history of the opera accelerations. When examining the vibrations at Theatre 1, situated at location B1, the maximum amplitudes range from 0.08 to 0.14 cm/s2. The support columns have a great influence on the accelerations at Theatre 1. As the vibrations propagate to the main theatre, specifically at the F1 auditorium, the amplitudes increase and reach values between 0.10 and 1.00 cm/s2. The accelerations at F1 exhibit a strong correlation with the longitudinal beam length. In the case of the cantilever auditorium at location F2, the time-domain peak accelerations escalate to a range of 0.306–1.690 cm/s2, varying across different locations. The vibrations at F2 are influenced by the overhang length and beam constraints. Further analysis of the frequency domain characteristics of vibrations is necessary to gain a more comprehensive understanding.

Fig. 8
figure 8

The time history of the opera accelerations

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The frequency domain can be determined by employing a discrete Fourier transform (DFT) analysis. Figure 9 displays the frequency domain of opera accelerations. For B1, point 1 exhibits 0.05 cm/s2 at a frequency of 67 Hz, while point 5 demonstrates 0.06 cm/s2 at a frequency of 21 Hz. Turning to F1, point 5 exhibits 0.15 cm/s2 and 0.39 cm/s2 at frequencies of 26 Hz and 67 Hz. The significant fluctuation observed at point 5 at F1 coincides with the midpoint of the longest span of the longitudinal beams. As for F2, the largest amplitudes rise from 0.07 to 0.26 cm/s2. This augmentation is attributed to the joint impact of beam limitations and cantilever shape. The dominant frequency bands contributing to the vibrations are 20–40 Hz and 60–80 Hz, with the 60–80 Hz range playing a prominent role. Modal analysis of the theatre indicates a natural frequency range of 20–40 Hz. Experiments indicate that the vibration of the metro mainly falls in the frequency range of 60–80 Hz. Consequently, building vibrations can be controlled from two perspectives: firstly, by managing the vibration through the transmission approach, and secondly, by enhancing the building structure to reduce resonance effects.

Fig. 9
figure 9

The frequency domain of opera accelerations

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In accordance with the standard (JGJ/T170-2009) [43] and ISO 2631/1-1997 [44], the weighted root mean square (RMS) ranging from 4 to 200 Hz is used to calculate the total vibration level (VLZ), as shown in the formula. The maximum vibration levels of acceleration can be determined at various positions within the theatre, as presented in Table 3.

$${\text{VL}}_{Z} = 20\lg \frac{{a_{w} }}{{a_{0} }}{\text{(dB}})$$
(5)
$$a_{w} = \left[ {\sum\limits_{i} {\left( {w_{i} a_{i} } \right)^{2} } } \right]^{\frac{1}{2}}$$
(6)
$$a_{i} = \left[ {\frac{1}{T}\int_{0}^{T} {a_{i}^{2} } (t){\text{d}}t} \right]^{\frac{1}{2}}$$
(7)

where aw represents the frequency-weighted acceleration, ai is the root mean square acceleration of the i-th 1/3 octave band, wi is the weighting factor specified in the JGJ/T170-2009 standard, and T is the duration of the measurement.

Table 3 Vibration levels
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The maximum vibration level at various frequencies is subject to a limitation of 65 dB after frequency weighting. Upon comparison with this threshold, it is observed that point 5 at the F1 position experiences an amplification of nearly 2.0 dB, while point 3 at F1, point 4 at F2, and point 6 at F2 narrow to 1.0 dB. Vibration levels recorded at B1 remain under 60.0 dB. Heightened attention is warranted for the F1 and F2 auditorium points, particularly the long beam span and cantilever margin.

The contour plots provide insight into the pronounced fluctuations observed in the auditoriums of the main theatre, as depicted in Figs. 10 and 11. Specifically, Figs. 10 and 11 showcase selected frames from the B1, F1, and F2 auditoriums of the main theatre. In these plots, the red blocks correspond to vertical vibrations with positive amplitudes, while the blue blocks represent vertical vibrations with negative amplitudes. The intensity of the color within the contours indicates the magnitude of the vertical accelerations, with darker shades denoting larger absolute values.

Fig. 10
figure 10

B1 acceleration contours of Theatre 1

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Fig. 11
figure 11

Acceleration contours of the main theatre. a F1 acceleration contours. b F2 acceleration contours

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Adjacent to the contour plots, schematic diagrams of the beams and columns are provided for reference. In Fig. 10, the alternating colors of positive and negative amplitudes can be attributed to the presence of spaced column supports. Figure 11a highlights the distinct vibration modes exhibited by the long and short longitudinal beams in the first-floor auditorium. The extended section of the longitudinal beam experiences vibration as a cohesive unit, while the shorter section vibrates independently within the constraints imposed by the longitudinal beam. In Fig. 11b, the region situated in the middle of the cantilevered structure displays the most pronounced fluctuations, owing to its considerable distance from the side and back supports.

The analysis of the time-domain prediction results reveals that the opera theatre resides in the pre-resonant region. Among the different areas within the theatre, the B1 exhibits the minimum vibration, followed by the F1, while the F2 experiences the largest vibration levels. There are two distinct frequency ranges contributing to the overall vibration; the range of 20–40 Hz arises from the structural resonance, whereas the 60–80 Hz range propagates from the train-caused vibrations. Notably, the B1 vibration level adheres to the specified limits. However, the F1 vibration level exceeds the established threshold, and the F2 vibration level nearly reaches the threshold. Consequently, the implementation of vibration reduction measures becomes imperative for the auditoriums on F1 and F2. By analyzing the contours, the underlying causes of excessive vibration on each floor can be identified, thereby facilitating targeted optimization efforts for the building.

An analysis is conducted to determine the reasons behind the exceeded vibration levels at specific positions. The prominent frequency band of vibrations is primarily associated with both the metro and the opera. Targeted measures are required to address the vibrations in the pre-resonant and post-resonant regions. In the pre-resonant region, it is crucial to implement stringent measures to increase the natural frequency, such as reinforcing the building, in order to prevent resonance. In the post-resonant region, flexible measures such as the incorporation of damping tracks are recommended to decrease the natural frequency in order to attenuate the vibrations.

4 Vibration reduction analysis

Section 3 highlights that certain points within the theatre exhibit vibrations that exceed the prescribed limits outlined in the standards JGJ/T170-2009. These excessive vibrations not only compromise the audience's experience during opera performances but also contribute to secondary noise that hampers the overall quality of the production [45]. Consequently, implementing a vibration reduction design to mitigate these undesirable effects is imperative.

Building upon the analysis of vibration characteristics presented in Sect. 3, appropriate measures are devised to address the transfer path and receiver vibrations. The first approach involves mitigating the vibrations of the metro through the manipulation of the propagation path. The second approach focuses on reducing building resonance by enhancing the structural integrity of the edifice. Due to site conditions, the trench is located on the outside of the continuous wall of the building foundation, 42 m from the subway line. The continuous wall outside the foundation of the building is 18 m deep. To ensure effective vibration control, the filled trench extends downward by 5 m, with a depth set at 23 m. According to the principle that the trench depth reaches 1.33 times the Rayleigh wavelength, the vibration wave can be effectively blocked. This depth of the filled trench can block vibration waves above 9 Hz. The trench is constructed with each layer of material having a thickness of 160 mm. The project employs extruded polystyrene (XPS) panels as the vibration-damping material. Its elastic modulus and Poisson’s ratio are 16 MPa and 0.28, respectively. To optimize the building structure, specific measures are undertaken to reinforce the beams, as depicted in Fig. 12, with color legends providing annotations to signify the changes implemented.

Fig. 12
figure 12

Opera-strengthening strategies

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The effectiveness of vibration reduction measures is assessed by comparing the vibration levels in the presence and absence of the implemented strategies, as depicted in Figs. 13 and 14. Since the attenuation coefficient of train-induced ground vibration is difficult to specify, insertion loss is used to describe the vibration control effect of the filled trench. Three distinct vibration reduction measures are employed: the XPS trench, opera theatre optimization, and the superposition of both strategies. These measures aim to mitigate vibrations and minimize their impact on the opera theatre.

Fig. 13
figure 13

Vibration level of different strategies at B1 Theatre

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Fig. 14
figure 14

Vibration level of different strategies at main theatre

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The vibration level of the XPS trench, opera theatre optimization, and the superposition are illustrated in Figs. 13 and 14. The vibration levels at B1 are influenced primarily by the implementation of the XPS trench. In the F1 auditorium, the peak vibrations occur at points 3 and 5, which correspond to the center location of the elongated beam. Conversely, in the F2 auditorium, the maximum is observed at points 4 and 6, representing the intermediate and edge position of the overhanging beam. All three strategies effectively decrease the vibration of F1. However, in the case of F2, while the vibrations decrease significantly in the intermediate position of the beam, the vibration reduction effect at the edge of the overhanging beam is relatively poor compared to the intermediate points.

The insertion loss is calculated by comparing the vibration levels in the presence and absence of the implemented vibration mitigation strategies. The insertion loss varies across different points, resulting in a range of 0.14–4.47 dB. The XPS trench provides an insertion loss of approximately 1 dB, while the building optimization measures yield an average insertion loss of around 2 dB. When the two measures are combined, the average insertion loss increases to approximately 2.5 dB. While the filled trenches are more suitable for addressing slight excessive vibrations, the building optimization measures are universally applicable and practical. However, it should be noted that the combined effect of the two measures does not exhibit linear superposition. Nonetheless, the combination demonstrates a more favorable performance compared to the individual measures.

Given the location of the project in the pre-resonant region, the implementation of measures to ensure that the theatre’s vibration remains within acceptable limits is crucial. In this regard, a rigid strategy like beam strengthening is adopted to prevent excessive vibration. Additionally, prior to construction, adequate space should be allocated for the implementation of filled trenches, which effectively mitigate metro-induced vibrations.

During the opera theatre project, careful consideration should be taken. Firstly, it is advisable to avoid employing long longitudinal beam structures, as they can exacerbate vibration-related issues. Furthermore, special emphasis should be placed on strengthening cantilever structures, which are susceptible to vibration effects.

The filled trench measure is suitable for addressing situations where the vibration slightly exceeds the specified limit. On the other hand, building optimization measures should be incorporated into all design aspects, given the overarching importance of vibration control.

5 Conclusions

Extensive computational modeling and experiments were undertaken to forecast and assess the effects of vibrations caused by moving trains on the opera theatre, as well as to evaluate the efficacy of various mitigation strategies. The predictions of auditorium vibrations were obtained through the utilization of a comprehensive dynamic coupling TTSSO model, considering independent and superposition of vibration mitigation strategies. The following conclusions can be drawn:

  1. 1.

    The proposed TTSSO model incorporates the interconnectedness of each component, allowing for a comprehensive analysis of their interrelationships. Through rigorous validation using experimental data collected at the station and along the metro line on the ground, the forecasting approach has demonstrated its remarkable credibility and reliability.

  2. 2.

    The vibrations induced by the metro system exert a substantial influence on the opera theatre, warranting careful consideration. The vibration predominantly manifests in two primary frequency bands: 10–40 Hz and 60–80 Hz. The former closely aligns with the natural frequency of the building, whereas the latter represents the primary frequency scope associated with train-caused vibrations.

  3. 3.

    The vibration mitigation measures of filled trenches and building optimization have proven to be effective in reducing the metro-induced vibrations in the opera theatre, achieving a reduction of 1–2.5 dB. The maximum vibration values remain below the established limit, ensuring compliance with the specified requirements.