Train-Induced Vibration and Structure-Borne Noise Measurement and Prediction of Low-Rise Building

Abstract

The advancement of urban rail transit is increasingly confronted with environmental challenges related to vibration and noise. To investigate the critical issues surrounding vibration propagation and the generation of structure-borne noise, a two-story frame building was selected for on-site measurements of both vibration and its induced structure-borne noise. The collected data were analyzed in both the time and frequency domains to explore the correlation between these phenomena, leading to the proposal of a hybrid prediction method for structural noise that was subsequently compared with measured results. The findings indicate that the excitation of structure-borne noise produces significant waveforms within sound signals. The characteristic frequency of the structure-borne noise is 25–80 Hz, as well as that of the train-induced vibration. Furthermore, there exists a positive correlation between structural vibration and structure-borne noise, whereby increased levels of vibration correspond to more pronounced structure-borne noise; additionally, indoor distribution patterns of structure-borne noise are non-uniform, with corner wall areas exhibiting greater intensity than central room locations. Finally, a hybrid prediction methodology that is both semi-analytical and semi-empirical is introduced. The approach derives dynamic response predictions of the structure through analytical solutions, subsequently estimating the secondary noise within the building’s interior using a newly formulated empirical equation to facilitate rapid predictions regarding indoor building vibrations and structure-borne noises induced by subway train operations.

1. Introduction

In recent years, due to population growth and increased transportation demand, rail transit systems have emerged as the primary mode of transportation. Subways, trams, high-speed rails, and other modes of transport offer people swift and convenient travel options, effectively mitigating traffic congestion. However, this has also resulted in closer proximity between rail transit and buildings, leading to vibrations and structure-borne noise from train operations that impact the daily lives, education, and scientific activities of nearby residents [1,2]. In the context of train-induced vibrations, the auditory perception of rumbling caused by metro trains is more prominent compared to vibrations. However, the structure-borne noise generated by metro trains is nearly impossible to eliminate, unlike other types of environmental noise due to long-term transportation demand [3].

When a subway operates, the contact between the wheels and rails induces vibrations [4], which then propagate through the soil to the ground, leading to building vibration responses and structure-borne noise.

For environmental vibration, some researchers have conducted extensive studies on the subject. Hu et al. [5] proposed a model that integrates the train, track structure, and building structure into a unified system for trains running on building floors, which can be utilized to predict the base vibrations of buildings and further predict the floor vibration. Auersch [6] proposed a simple and fast prediction of train-induced track forces and ground and building vibrations in which a transfer matrix is utilized for characterizing the dynamic response of individual subsystems as vibration propagates. Sanayei et al. [7] proposed a one-dimensional impedance model in which columns are modeled as rods to account for axial motions only and floor slabs are treated as infinitely long mass-driven points in order to predict floor vibrations caused by trains passing through buildings, and Zou further improved the model and expanded it into two-dimensional [8] and three-dimensional [9] impedance models. Papadopoulos et al. [10] considered how the uncertainty of subsoil conditions could affect the building response by means of Monte Carlo. Edirisinghe et al. [11] modeled a three-dimensional numerical model based on a pipe-in-pipe model that accounts for the through-soil coupling between a tunnel and a piled foundation. Li et al. [12] studied the effect of prolonged train-induced vibrations with different frequencies on loess disintegration, which revealed the macro–micro association of disintegration properties after train-induced vibration.

Compared to the study of train-induced vibration, research on building noise commenced relatively late. Vehicle-induced building structure noise encompasses both air noise and structure-borne noise. Air noise refers to vibrational noise caused by solid structure contact and pneumatic noise induced by gas fluid movement during train operation, primarily occurring at medium and high frequencies [13,14], especially the air noise generated during train operation on ground and elevated lines. Structure-borne noise, which refers to the noise generated by vehicle-induced vibrations propagating through the building structure, is predominantly of medium and low frequency [3,15]. Trains operating on various types of tracks can induce structure-borne noise in conjunction with vehicle-induced vibration, and their acoustic and vibration characteristics can be considered as a one-way coupling [16], where the influence of internal sound pressure on the structure may be disregarded.

In the numerical calculation methods for structural noise, finite elements and boundary elements are commonly employed discrete methods, while the statistical energy method is a frequently used energy method. The statistical energy method demonstrates high computational accuracy and efficiency in dealing with medium- and high-frequency noise [17,18]; however, it is not so suitable for predicting low-frequency noise [19]. Nagy et al. [16] proposed an improved method for predicting indoor structure-borne noise based on global Rayleigh reflectivity and the direct boundary element method. Fiala et al. [20] applied the spectral finite element method to analyze vibration propagation and structure-borne noise in buildings’ foundations. Colaço et al. [21] enhanced the calculation efficiency by combining a 2.5-dimensional FEM with the method of fundamental solutions (MFS) model for structure-borne noise analysis. Although both the finite element method and boundary element method offer higher prediction accuracy for low-frequency structure-borne noise compared to empirical formula methods, they are hindered by their high computational cost when predicting indoor noise.

In the majority of application scenarios, the empirical formula method is widely used for noise prediction due to its simplicity and efficiency. Kurzweil et al. [22], Vér [23], and Melke [24] proposed or enhanced empirical prediction methods based on measured data, aiming to predict indoor noise by integrating floor vibration acceleration and relevant parameter correction factors. In these methods, the floor is simplified as the primary excitation source of sound induced by vehicle-induced vibration. Tao et al. [25] employed the empirical formulation in a low-rise over-track building to estimate the structure-borne noise. The findings revealed a notable increase in error at higher frequencies, peaking at almost 10 dB within the 50–80 Hz range. Li et al. [3] combined measurement and deep-learning-based approaches [26,27,28] to consider the transmission between building vibration and structure-borne noise and compared the predicted results with different empirical models [22,23,24,29,30,31,32,33]. The findings indicated that various empirical models exhibit distinct fitting effects across different frequency bands.

At present, the majority of research on vibration and structure-borne noise utilizes numerical analysis or empirical formula methods. However, the former requires complex model construction, resulting in low prediction efficiency, while the latter’s formulas have a limited application range and insufficient accuracy. The current study involves on-site measurements of vibrations in a two-story frame building and the corresponding structure-borne noise generated by subway trains, with the objective of elucidating the propagation characteristics and response patterns of both vibrations and structural noise, as well as their interrelation. A hybrid prediction methodology that is both semi-analytical and semi-empirical is introduced. The approach derives the dynamic response predictions of the structure through analytical solutions, subsequently estimating the secondary noise within the building’s interior using a newly formulated empirical equation. It is important to highlight that this novel empirical formula builds upon existing predictive models and prior research findings. The fundamental component of this empirical formula is the vibration acceleration level, which distinguishes it from previous methodologies that utilized varying reference speeds for vibration velocity levels, thereby establishing a more standardized foundational term.

The contribution of this paper is the proposed fast, hybrid, and user-friendly prediction method utilizing fundamental terms of vibration acceleration levels, which eliminates the need for complex index conversions during the prediction process. This method demonstrates acceptable accuracy and efficiency, with a straightforward and clear formulation.

2. Measurement

2.1. Measured Site Descriptions

A two-story frame structure building located adjacent to a subway line was chosen for testing. Due to the operation of A-type subway cars (6-car organization), the building is positioned approximately 40 m horizontally from the subway tunnel, which has a buried depth of 17 m. The spatial relationship between the building and the tunnel is depicted in Figure 1.

As per the findings of the geological exploration report, the soil dynamic parameters are detailed in

Table 1. Dynamic parameters of soils.

Soil Name	Thickness (m)	Density (kg/m3)	Poisson’s Ratio	Young’s Modulus (MPa)
Plain fill	2.1	1600	0.45	130
Clay	1.3	1840	0.46	130
Silty soil	8.4	1750	0.47	300
Silty clay	10.6	1930	0.46	380

.

The building foundation is constructed as an independent foundation with a size of 1.2 × 1.2 m. The structural column size is 0.4 × 0.4 m, and the building has a height of 8 m with 2 layers. The dimensions and dynamic parameters of the building structure are detailed in

Table 2. Parameters of axial load-bearing structure of building.

Floor Number	Load-Bearing Structure	h (m)	t (m)	w (m)	E (Gpa)	Ρ (kg/m3)
1–2	Structural column	4	0.4	0.4	32.5	2500

,

Table 3. Parameters of building floor structure.

Floor Number	t (m)	E (Gpa)	l (m)	w (m)	ρ (kg/m3)
1–3	0.1	30	6	6	2500

and

Table 4. Parameters of building foundation structure.

Foundation Type	l (m)	w (m)	E (Gpa)	ρ (kg/m3)
Rigid foundation	1.2	1.2	32.5	2500

, where h, t, w, l, E, and ρ represent the height, thickness, width, length, elastic modulus, and density of the corresponding structural elements. Additionally, the damping ratio and Poisson’s ratio of the reinforced concrete structural elements are specified as 0.02 and 0.2, respectively.

In order to study the characteristics of structural noise caused by train operation, microphones were installed in the center of the floor and next to the structural columns (represented by green triangles) on the second floor. Accordingly, accelerometers (represented by yellow triangles) were placed below microphones in their respective locations on the floor as well as at the same location on the 1st floor and outside the building to obtain vibrations.

Figure 1 illustrates the measuring point locations. In a frame building structure, the structural components can be categorized into propagation and response components based on their distinct functions. Among these, columns serve as vertical vibration propagation components due to their significant vertical stiffness, while beams facilitate horizontal vibration transmission. Floor slabs exhibit a rich variety of vibration modes and demonstrate significant compliance, making them serve as vertical vibration response components. Vibrations propagate from the foundation column’s base to the structures and are transformed into curved waves at the plate–beam–column junctions, where they propagate, respond, and dissipate towards the far side of the floor. Consequently, considering the response characteristics of various building components and the transmission dynamics of vehicle-induced vibrations within the structure, four measurement points were established: at the column bases and floor center on both the first and second floors. Concurrently, an outdoor ground vibration measurement point was set to facilitate the analysis of train-induced ground vibrations originating from both nearby and distant tunnels, serving as a reference for assessing the energy associated with these vibrations. Testing occurred on a working day with minimal traffic and pedestrian flow, under sunny weather conditions that have little impact on climate.

2.2. Measurement Program

Instrumentation and Signal Processing

The instrumentation utilized for measurement is shown in Figure 2, encompassing a Rion UC-59 microphone (RION Co., Ltd., Tokyo, Japan) and JM 3873 wireless data acquisition system (Jing Ming Technology Co., Ltd., Yangzhou, China). The microphone was installed on the preamplifier to obtain a clear noise signal, and the windshield was connected with the head of the microphone to effectively mitigate wind noise interference during the recording process and enhance the clarity and purity of the recorded audio, thereby improving the overall recording quality. Finally, the instrumentation assembly was installed on the tripod, positioned at a height of 1.5 m above the floor surface. The accelerometer was positioned appropriately on the floor below the microphone. Both the microphone and accelerometer were equipped with time synchronizers and memory cards.

A flowchart that more clearly illustrates the measuring procedures is shown in Figure 3.

The measuring procedures can be categorized into initial reconnaissance, pre-processing, and post-processing.

(1)

Initial reconnaissance

During the initial reconnaissance, the measurement site environment was assessed to ascertain its readiness for evaluation and to identify the specific locations of individual test points. Additionally, sensors were positioned on the ground for oscilloscope measurements to quantify the range of vibration acceleration influenced by train-induced vibrations, thereby minimizing potential errors.

(2)

Pre-processing

(a)

All instruments were calibrated based on the laptop’s time prior to measurement initiation, ensuring simultaneous data recording.

(b)

According to the Chinese code HJ 453–2018 [30], the dominant frequency range requirement for structure-borne noise analysis is 16–200 Hz. The sampling frequency for each sound level meter was set at 48 kHz, and it was chosen to facilitate noise analysis up to 24 kHz, despite such high frequencies not being typically necessary for noise analysis but sufficient for the common frequency of interest at 16–200 Hz for train-induced structure-borne noise. The sampling frequency was 512 Hz, which captured vibrations resulting from train passage and offered spectral information up to the Nyquist frequency of 256 Hz, allowing for analysis within the frequency range of 16–200 Hz.

(c)

After setting the instrumentation and starting the measuring, train pass-by events were simultaneously documented at the metro station, including details of the specific tunnel from which each train originated.

(3)

Post-processing

The original time domain signal file was downloaded from the wireless data acquisition system. The acquired raw data were subsequently processed analytically using a custom script. This script was specifically tailored to calculate the power spectral density of the signals, focusing on every train operation signal. Consequently, the signal captured by a sensor must be intercepted in accordance with the vibration waves produced and their spectrogram during the train pass-by events, as shown in Figure 4. Due to the variations in the speed of each train, the time intervals between each intercepted vibration signal differ; however, their spectrograms exhibit similar trends.

3. Results

A total of 40 trips were collected, and time–frequency conversion processing was conducted. The signals from both the far tunnel and the near tunnel comprised 20 train pass-by events each, thereby providing ample data support for subsequent analysis.

3.1. Vibration in Time Domain

The effective train vibration signal recorded at the scene lasted approximately 14.5 s, with the train running at a speed of about 50 km/h. In Figure 5, the time history signal depicts the vibration acceleration of the outdoor ground measuring points and two-story structural column feet under tunnel vibration. The peak value of soil vibration acceleration on-site was approximately 0.028 m/s², while that of two-story structural column feet was about 0.016 m/s², indicating a significantly greater soil vibration compared to the building structure. This is because soil–structure coupling loss occurs when a vibration propagates from soil to structures [6]. The underlying cause of train-induced vibrations stems from wheel–rail interaction, with the most significant vibrations occurring at the interface between the wheelset and the rail. Consequently, as an entire train traverses a given point, multiple wheelsets generate a comprehensive vibration wave. This results in an uneven distribution of vibrational energy within the time domain, as shown in Figure 4. Specifically, when a wheelset is perfectly aligned perpendicular to the measurement point, it exhibits heightened vibration energy. During propagation, factors such as soil damping attenuation, soil–structure coupling loss, and building response lead to notable alterations in the frequency domain. However, in terms of time domain analysis, various forms of attenuation diminish the overall wave amplitude without substantially altering its waveform. We have added the corresponding description.

Figure 6 depicts the time history curve of vibration acceleration at the outdoor ground measurement points under the influence of near-side and far-side tunnel vibrations. The peak soil vibration acceleration in the near-side tunnel measured approximately 0.028 m/s², while that in the far-side tunnel was about 0.022 m/s². Due to the similarity of the track bed, sleeper, and other tunnel structures, there is minimal distinction between the two when a train passes through. Furthermore, since the distance from the tunnel to the measurement point was approximately 40 m, in this case, the vibrations transmitted by both tunnels to the measurement point were both far-field vibrations, existing in the form of Rayleigh waves. Therefore, these two vibrations had already been subjected to a certain attenuation due to the soil damping effect at high frequencies and therefore have a similar order of magnitude. Nevertheless, it is noteworthy that the soil vibration levels were higher in the vicinity of the near tunnel compared to those near the far tunnel.

3.2. Structure-Borne Noise in Time Domain

Figure 7 shows the time history curve of sound pressure at the central measuring point on the second floor inside the building near the tunnel for a train. As depicted in the figure, the peak sound pressure induced by vehicle-induced vibration from this particular vehicle was approximately 0.04 Pa, exhibiting more distinct acoustic characteristics resembling a vibration waveform. The peak resulting from wheel–rail interaction is clearly discernible.

3.3. Vibration in Frequency Domain

The measured acceleration was processed into vertical vibration acceleration levels through 1/3 octave analysis. Given that the predominant vibration frequencies of the subway fall within the range of 20–80 Hz, a frequency band spanning from 16 to 200 Hz was selected for assessment [3,30]. To enable a clear comparison of vibration propagation in soil and structures, the envelope diagram illustrates the measured ranges of vibration acceleration levels recorded during various train trips. Spectrum analysis was performed by selecting test results with higher magnitudes of vibration to compare acceleration levels near tunnels at different measurement points, as shown in Figure 8.

Compared to the outdoor vibration G1 and indoor vibration C1, the energy loss of vibration propagating from the soil to the structure was more pronounced. The attenuation tended to be significant at medium and high frequencies, while low-frequency attenuation was less apparent. Vibration acceleration levels between 25 Hz and 100 Hz were notably reduced due to increased attenuation cycles experienced by medium- and high-frequency vibrations at the soil–structure interface, resulting in more pronounced losses. As vehicle-induced vibration primarily distributes its main energy within the range of 25–100 Hz, there were minimal energy components beyond 100 Hz. Consequently, vibration energy experienced only minor losses during propagation, leading to less noticeable attenuation above 100 Hz. An amplification occurred in the frequency band of 100–125 Hz due to elastic building resonance in anti-phase [31].

$$f_a={\displaystyle \frac{v_c}{4H_b}}\approx 109 \mathrm{Hz}$$

(1)

Compared to C1 on the first floor and C2 on the second floor, the characteristic frequency of vibration acceleration in the structural columns remained consistent during the propagation process as the building height increased, with minimal variation. This is attributed to the relatively low floor height and the building’s two-story configuration. When subjected to train-induced vibrations, the building system tended to vibrate as an entity, resulting in negligible disparity in vibration acceleration between the structural columns on the first and second floors.

Increasing the building height did not significantly affect the difference in vibration acceleration between F1 on the first floor and F2 on the second floor in the frequency domain. Throughout the propagation process, characteristic frequencies remained consistent, leading to overall excitation of the building system and minimal variation in vibration acceleration between floors because the building system tends to vibrate as an entity.

In the two-layer slab, when comparing F2 and C2, the vibration amplitude tended to be greater at the floor center than at the structural column within the frequency range of 20–100 Hz. However, above 100 Hz, it was observed that the vibration amplitude was greater at the structural column than at the floor center. This phenomenon can be attributed to the fact that the structural column serves as a primary conduit for vertical vibration propagation, while the floor primarily responds to and dissipates energy from vertical vibrations. Consequently, in the characteristic frequency band of 20–100 Hz, there is a particularly noticeable vibration response in the floor. In contrast, when examining frequencies that exceed those impacting structural columns, their response diminishes in prominence. Consequently, within the frequency range of 20 Hz to 100 Hz, floor vibrations were more pronounced than those of structural columns; however, this dynamic reversed at elevated frequencies. The vibration exhibited prominent peaks at 50 Hz, attributed to the modal characteristics of the floor, resulting in resonance at this frequency.

To facilitate comparisons between distant and nearby tunnels, spectra of the vibration acceleration levels from similar measurement points for both the near and far tunnels were chosen for comparison, as depicted in Figure 9. In comparison with the outdoor vibration G1 from the near and far tunnels, the amplitude of G1 tends to decrease across the entire frequency spectrum. Furthermore, the disparity at medium to high frequencies is significantly more pronounced than that observed at low frequencies. Due to the shorter period of high-frequency waves, they undergo more cycles than low-frequency waves, resulting in greater dissipation and thus faster attenuation of high-frequency vibrations compared to low-frequency ones. Furthermore, within the same frequency range, a greater distance leads to higher energy consumption, resulting in a larger vibration acceleration level at the near tunnel compared to the far tunnel at identical measurement points.

3.4. Structure-Borne Noise in Frequency Domain

The measured sound pressure was converted into 1/3 octave bands to derive the structural noise sound pressure level. Given that the frequency of structural noise aligns with vibration, the frequency band of 16–200 Hz was selected for analysis. To facilitate a clear comparison of noise propagation within the building, the envelope diagram presents the measured noise pressure level range for different train trips. Furthermore, spectrum analysis was conducted by selecting test results with higher sound pressure levels to obtain a comparison of near-tunnel noise pressure levels at different measuring points, as depicted in Figure 10.

The comparison of structural noise between a room panel and its corner revealed that the corner exhibited significantly higher levels of structural noise, particularly above 31.5 Hz. This phenomenon can be attributed to the wall’s position, which functions as both a radiant and reflective surface for sound, resulting in the highest observed sound pressure level [3]. In contrast, the center of the floor experiences less structural noise due to its distance from the walls, resulting in reduced reflection superposition effects.

To compare structural noise between the far and near tunnels, respective comparisons were made using the noise pressure level spectra from under the floor center B1 and corner B2 for both tunnels, as illustrated in Figure 11. By comparing the structural noise at the same measuring point in a room caused by trains running in different tunnels, it was observed that as the distance between the tunnel and the building increased, there was no significant difference in the strength of structural noise between the center and corner of the floor in the room within the frequency domain; rather, it tended to be consistent. This can be attributed to structural noise being generated by building vibrations transmitted through solids into rooms. Although this vibration gradually attenuates, it is influenced by factors such as building materials and structural design. Therefore, a slight increase in distance between tracks and buildings may not necessarily have a pronounced impact on structural noise within rooms.

3.5. Transfer Function from Structural Vibration to Structure-Borne Noise

In order to investigate the correlation between structural vibration and structure-borne noise, the transfer difference (the disparity between the sound pressure level and vibration acceleration level) was utilized to quantify the relationship of transmission between structural vibration and structure-borne noise. As the distance of the tunnel does not impact the generation mechanism of structure-borne noise in the room, the transmission difference was computed for all test samples at the center of the floor of the two-story interior, as depicted in Figure 12.

The characteristic frequency band of vibration ranges from 25 Hz to 100 Hz, while that of noise spans from 25 Hz to 80 Hz; both waves share a similar characteristic frequency range. The sound pressure level and vibration acceleration level within this frequency band exhibit a specific decibel difference relationship, leading to a reduction in transmission ratio within the range of 25 Hz to 80 Hz. Frequencies outside this characteristic band become dominant environmental noise and environmental vibrations, causing an overall upward shift in curves above 80 Hz.

The figure reveals a spike at 25 Hz. This may be attributed to resonance phenomena occurring either when the frequency of the vibration wave matches the natural frequency of the building structure or when the sound wave frequency aligns with the acoustic natural frequency of the building room, resulting in an increased vibrational amplitude and consequently a spiking transmission difference. Therefore, there exists a positive correlation between structural vibrations and structure-borne noise; greater structural vibrations lead to more pronounced structure-borne noises. The room’s natural frequency [32] can be calculated based on the dimensions of the room, and the minimal natural frequency can be obtained using the following equation:

$$f_r={\displaystyle \frac{c}{2L}}\approx 28.4 \mathrm{Hz}$$

(2)

where c is the speed of sound and L is the maximum length of the room. The standing wave resonance established at this frequency results in the emergence of a wave peak within the 25–31.5 Hz range of the transfer function.

4. Hybrid Approach for Prediction

4.1. Structural Vibration Estimation Model

Given that the vibration propagation of low-floor buildings tends to affect the overall building movement, we considered the axial wave propagation within the structure. Consequently, a dynamic stiffness model for the structure was developed, treating each column as a single Bernoulli Euler beam and deriving the structural dynamic stiffness matrix for force and displacement:

$$\left[k_c\right]={\displaystyle \frac{{\overline{E}}_c{A}_c{\beta }_c}{\sin \left({\beta }_c{l}_c\right)}}\left[\begin{array}{cc}\cos \left({\beta }_c{l}_c\right)& -1\\ -1& \cos \left({\beta }_c{l}_c\right)\end{array}\right]$$

(3)

where A_c is the cross-section area of the structural column, ${\overline{E}}_c$ is the Young’s modulus of the rod material, ${\beta }_c=\omega \sqrt{{\rho }_c/{\overline{E}}_c}$ is the axial wavenumber, ${\rho }_c$ is the density of the rod material, $\omega$ is the angle frequency, and $l_c$ is the length of the structural column.

The floor and the vertical bearing structure are assumed to be rigidly connected, so when vibration propagates through the floor, it will also propagate to the exterior of the floor, resulting in energy dissipation. Therefore, in this context, the floor can be considered as an element that consumes energy with a fixed value. However, during structural response, it can be viewed as a frequency-dependent mass point; thus, the mass point becomes the primary component of the structural response, and its mass is not fixed during this process. Based on these concepts, the floor was modeled as a mass point primarily responsible for providing energy dissipation:

$$m_s={\displaystyle \frac{8}{i\omega }}\sqrt{{\displaystyle \frac{{\overline{E}}_f{h}_f^3}{12\left(1-{\nu }_f^2\right)}}{\rho }_f{h}_f}$$

(4)

where ${\rho }_f$, $h_f$, and $v_f$ are the material density, thickness, and Poisson’s ratio of the floor and ${\overline{E}}_f$ is the complex Young’s modulus of the floor.

In accordance with the concept of the structural dynamic stiffness cascade, the system equation in the frequency domain can be formulated as follows:

$$\left[K_c\right]\left\{U\right\}-{\omega }^2\left[M_s\right]\left\{U\right\}=\left\{F\right\}$$

(5)

where $\left[K_c\right]$ is the global stiffness matrix, $\left[M_s\right]$ is the global dynamic mass matrix, $\left\{U\right\}$ is the displacement vector of the system response, and $\left\{F\right\}$ is the input force vector of the system. The structural response can be efficiently obtained by constructing the global matrix and solving the system’s dynamic equation through inversion. Numerical integration poses no difficulty in this process, enabling rapid calculation of the structural response. The overall acceleration response {A} of the system can be determined using the following formula:

$$\left\{A\right\}=-{\omega }^2\left\{U\right\}$$

(6)

4.2. Structure-Borne Noise Estimation Model

Utilizing the aforementioned model for addressing the system’s acceleration response, and drawing from a comprehensive review of multiple empirical models, including RIVAS [33], HJ 453–2018 [30], FTA Guidelines [29], Kurzweil [22], and Melke [24], the following corresponding formulations are as shown in Equations (5)–(9), respectively:

$$L_p=L_{V1}-27$$

(7)

$$L_p=L_{v1}+10{\log }_{10}\left({\sigma }_{rad}\right)-10{\log }_{10}\left(H_r\right)-20+10{\log }_{10}\left(T_r\right)$$

(8)

$$L_A\approx L_{V2}+K_{A-wt}-5$$

(9)

$$L_p=L_{acc}-20\times {\log }_{10}\left(f\right)+37$$

(10)

$$L_p=L_{v3}+10{\log }_{10}\left({\sigma }_{rad}\right)+10{\log }_{10}\left({\displaystyle \frac{4S_{sur}}{S_r}}\right)$$

(11)

where L_v1, L_v2, and L_v3 are the velocity levels, with reference velocities of 1 × 10⁻⁹ m/s, 1 × 10⁻⁶ in/s, and 5 × 10⁻⁸ m/s; L_p is the sound pressure level; L_A is the A-weighting sound pressure level; $K_{A-wt}$ is an A-weighting adjustment at the 1/3-octave band center frequency; $L_{acc}$ is the acceleration level (re: 1 × 10⁻⁶ m/s²); f is the frequency; ${\sigma }_{rad}$ is the average radiation ration for the floor bay; $S_{sur}$ is surface area of the room; H_r is the height of the room; and $T_r$ is the reverberation time, which can be calculated according to Sabine’s formula [34]:

$$T_r={\displaystyle \frac{0.16\times V}{S_r\times {\alpha }_{room}}}$$

(12)

where $S_r$, ${\alpha }_{room}$, and $V$ represent the sound absorption area, average sound absorption coefficient, and room volume, respectively. Generally, a higher reverberation time is achieved using Saibine’s formula; consequently, the methodology for calculating reverberation time is frequently expanded to include Eyring’s formula [35]:

$$T_r={\displaystyle \frac{0.16\times V}{-S_r\times \ln \left(1-{\alpha }_{room}\right)}}$$

(13)

In most contemporary empirical models, the relationship between velocity and pressure levels is the basis of establishing the model. However, in various environmental vibration specifications, the definition of velocity levels varies, often with distinct reference speeds [29,30,33]. This introduces significant uncertainty to the prompt assessment of structure-borne noise resulting from environmental vibrations. However, in the vast majority of vibration assessments, when using acceleration as an evaluation parameter, the standard reference acceleration is typically set at 1 × 10⁻⁶ m/s². Taking into account the expediency of rapid forecasting and the consistency of indicators, this paper presents the following empirical formula to forecast unweighted structure-borne noise from vibration acceleration levels based on the Equation (10) proposed by Kurzweil [22]. Other empirical formulas concluded, based on extensive field experiments, according to Equation (8), that an increase in the room’s height will lead to a reduction in the sound pressure level of structure-borne noise. Based on this, a correction term for room height has been introduced, and the frequency in the original equation has been adjusted to a circular frequency:

$$L_{\mathrm{p}}=L_{\mathrm{acc}}-20{\log }_{10}\left(\omega \right)+37-20{\log }_{10}\left(H_r\right)$$

(14)

where w is the circular frequency. The dynamic stiffness model of the structure considers the floor as a mass point responsible for damping energy dissipation, resulting in a mass point response. Therefore, obtaining the bottom floor response enables prediction of the superstructure response, and further substitution into the noise model allows for prediction of structure-borne noise in the frequency domain.

4.3. Model Validation

Considering ${\sigma }_{rad}$ = 1, $S_{sur}$ = $S_r$, and ${\alpha }_{room}$ = 0.1 [36], we compared the measured vibration acceleration level and sound pressure level data in the center of the two-story room with the predicted values calculated using the proposed structural vibration prediction model and structure-borne noise prediction model, as depicted in Figure 13. The results illustrate that the predicted outcomes of the model presented in this paper exhibit a strong fitting performance within the frequency range of 31.5 to 125 Hz, which is recognized as the frequency of interest for structure-borne noise generated from train-induced vibrations. In the 16–25 Hz and 160–200 Hz frequency bands, background noise predominates, leading to a decrease in model accuracy. However, the frequency ranges do not represent the frequencies of interest associated with train-induced vibrations, whose energy levels are minimal and insufficient to significantly impact human comfort. Consequently, the differences can be considered acceptable.

4.4. Error Analysis

Figure 14 illustrates the errors between the measured and predicted results of vibration and noise. As illustrated in Figure 14a, the vibration prediction values generated by the proposed model demonstrate a high degree of accuracy, with errors at each frequency generally remaining within 5 dB. Notably, the errors at 125 Hz and 200 Hz are more pronounced, reaching approximately 10 dB. This discrepancy may be attributed to the predominance of environmental vibrations within these frequency bands, which can adversely affect the model’s accuracy. As illustrated in Figure 14b, the predicted structure-borne noise values generated by the proposed model demonstrate a high degree of accuracy, with errors at each frequency generally remaining within 8 dB. In comparison to other empirical models, the proposed model demonstrates superior fitting in the primary frequency bands associated with train-induced vibrations, particularly within the range of 31.5–80 Hz. Although the formulas of RIVAS and Melke exhibit a better fit at 16–31.5 Hz, this frequency band is not of primary concern for train-induced vibrations, which possess alower energy in this frequency band.

4.5. Model Limitation Discussion

A hybrid prediction methodology that is both semi-analytical and semi-empirical has thus been introduced. The approach derives the dynamic response predictions of the structure through analytical solutions and subsequently estimates the secondary noise within the building’s interior using a newly formulated empirical equation. This model enables the fast estimation of vibrations and structure-borne noise in low-rise buildings. However, the model is subject to certain limitations.

Firstly, the model’s input consists of the foundation basement vibrations, indicating that it is necessary to obtain the vibration values which have undergone soil–structure coupling loss when utilizing this model. By integrating various methodologies, the vibrations are modified during the propagation to enhance both accuracy and efficiency, which aligns with the objectives of the hybrid model proposed in this paper. Consequently, the model presented herein exclusively addresses the final stages of vibration propagation—specifically, propagation and response within the building—thereby excluding any variations in geological conditions. However, this can be accomplished through various soil–structure interaction models, including numerical integral methods [11], simple node-coupled systems [31], and deep learning approaches [37], etc. We posit that it is feasible to conduct vibration measurements for diverse foundations and geological conditions to summarize the law of soil–structure interaction, thereby proposing corresponding coupling loss corrections. Utilizing this idea, the model can maintain both high efficiency and acceptable accuracy. Most importantly, it is user-friendly and can be easily understood and applied by most engineers. We aspire to improve this approach in future work.

What is more, the proposed model is deemed appropriate for frame structures. However, its applicability remains uncertain for superstructure buildings featuring load-bearing wall structures and those with transfer floor configurations. This paper exclusively validated a frame building. Similarly, regarding the proposed structural secondary noise model, only the vibration acceleration levels within the board were considered as inputs for predictions; however, it is important to note that structural noise sources in a room are also generated from walls, which may limit the model’s applicability to buildings with special wall structures.

5. Conclusions

A two-story frame structure was subjected to testing for its vibration along a subway line and the resulting structure-borne noise. The study focused on examining the relationship between the vibration and noise caused by train operation in both far and near tunnels. A hybrid prediction method comprising a structural vibration prediction model and a structure-borne noise prediction model was proposed. The results obtained from this method were compared with field measurements. The main conclusions are as follows:

(1)

The vibration acceleration characteristics of building components are closely associated with the vibration frequency, showing significant differences in performance across different frequency bands.

(2)

Noise within the indoor structure of buildings is not uniformly distributed; corner areas exhibit more pronounced noise than central indoor floor areas.

(3)

There exists a positive correlation between structural vibration and structure-borne noise, where greater structural vibrations lead to more noticeable structure-borne noise. The characteristic frequency band of structure-borne noise falls within 25 Hz–80 Hz.

(4)

The hybrid prediction method effectively forecasted the impact of subway vibration and structural noise on surrounding buildings under specific working conditions. This method demonstrate high prediction accuracy with a simple and convenient calculation process and is suitable for the preliminary prediction and confirmation stage of specific engineering projects.