Study on vibration response and isolation of substation structure under subway load excitation

2.1. Project overview

The indoor substation features a reinforced concrete frame structure, comprising one underground level (3.3 m high) and two above-ground levels (both 5.3 m high), the total height of building is 11.4 m. The specific layout is as follows: the underground level accommodates the cable area, the first above-ground level houses the high-voltage distribution equipment area where the main electrical equipment is stored, and the second above-ground level contains the secondary equipment room and the duty room. The substation foundation utilizes a raft foundation, with a base elevation of –4.34 m. The material parameters used for various components of the building are detailed in Table 1. The main structure of the substation has a plan dimension of 41.8 m×24 m. Approximately 60 m to the east of the substation is a section of elevated subway line, with each span of the viaduct measuring 8 meters, a bridge deck width of 12.6 m, a pier diameter of 0.8 m, and a height of 8 m.

2.2. Design of measurement scheme

2.2.1. Vibration testing instruments and evaluation standards

The current study employs the 2D001 magnetic-electric transducer and the DH5922D dynamic signal testing and analysis system. Prior to testing, each instrument is calibrated and parameterized with a sampling frequency of 500 Hz. The main technical specifications of the transducer are as follows: sensitivity of 18.4V/m·s^-2 and a maximum range of 0.125 m/s. Due to the presence of precision equipment in indoor substation, this study adheres the control criteria for micro-vibration of precision instruments and equipment in Class C experimental buildings as specified in the internationally widely used VC standard for vibration of precision equipment (VC-C: 12.5 μm/s) [14]. This experiment collects the vibration velocity data of the structure as an evaluation index, as shown in Fig. 1.

Fig. 1International general vibration VC standard

2.2.2. Layout of measurement points and arrangements

Two measurement groups are established to conduct vibration response tests at the subway vibration source and indoor floor. Measurement points will be set up at the elevated bridge piers of the subway to measure vibration acceleration, which is used to determine the accuracy of the subsequent finite element model loading. Four measurement points are placed indoors, at the basement, first, second, and top floors, to measure the vibration velocity at the midspan of each floor. The layout of the relevant measurement points is illustrated in Fig. 2, and the corresponding test conditions are presented in Table 2.

Fig. 2Layout of measuring points

Table 2Vibration measurement arrangement

Measurement sequence	Measurement content
Ⅰ	A1 measurement of three-dimensional acceleration
Ⅱ	The vertical velocities V2, V3, and V4 are measured, while V1 measures the three-dimensional velocity.
Ⅲ	The vertical velocities V1, V3, and V4 are measured, while V2 measures the three-dimensional velocity.
Ⅳ	The vertical velocities V1, V2, and V4 are measured, while V3 measures the three-dimensional velocity.
Ⅴ	The vertical velocities V1, V2, and V3 are measured, while V4 measures the three-dimensional velocity.

2.3. Measurement results

Based on the vibration data measured in the indoor substation, the vibration velocity amplitude distribution of each floor under subway load in different directions is shown in Fig. 3. The results indicate that the vibration propagation law in the longitudinal track, vertical track, and vertical direction is the same, i.e., the vibration velocity slightly increases with the increase of the floor. The vertical vibration velocity response is the largest, and the maximum vibration response occurs at the top floor of the structure, reaching 7.33 μm/s. The VC-C level vibration velocity allowable limit is 12.5 μm/s. Currently, the vibration levels in the indoor substation are within the permissible range set by the VC-C standard, ensuring the safety of precision instruments indoors. The time history curve of vertical vibration velocity at the mid-span of the first and second floor slabs of the substation structure is shown in Fig. 4 (L: Longitudinal; T: Transverse; V: Vertical).

Fig. 3Velocity amplitude in three directions on each floor

Fig. 4Typical vertical vibration velocity time history curve of floor

a) First floor

b) Second floor

3.1. Finite element model of rail-soil-substation-equipment

Based on the collected geological data, the soil at the site of the indoor substation and the subway viaduct is simplified into four layers according to the soil properties [15]. Each layer represents a major shear wave velocity range, and the main physical parameters of the soil layers are shown in Table 3.

In the ABAQUS finite element model, the dimensions of the soil mass are set to 160 m × 100 m × 50 m. The soil model utilizes C3D8R solid elements for the calculation units, with a mesh size ranging from 0.3 to 1.0 m. Near the vibration source, the element size is set to 0.3 m, gradually increasing away from the source area. Infinite elements (CIN3D8) are used for the outermost elements of the model to simulate infinite boundary conditions. The soil model with the infinite element boundary set is depicted in Fig. 5.

Fig. 5Soil finite element model

The three-dimensional finite element model is developed according to the specifications outlined in Section 1.1, and the primary materials and their corresponding parameters are detailed in Table 4. Solid element (C3D8R) is employed for beams, slabs, and columns, while reinforcement is modeled as wire element (B31). The contact behavior between the substation and the track structure with the soil is Embed, effectively capturing the interaction between the structure and the soil. The finite element analysis model, based on the relevant parameters, is depicted in Fig. 6.

Fig. 6Three-dimensional finite element model of rail-soil-substation

Based on the measured vibration data of the equipment, this study develops a finite element model to investigate the interaction between the substation and the equipment (as shown in Fig. 7). The equipment is simplified as a single-degree-of-freedom (SDOF) system, its mass concentrates at the top and represents by point mass elements (MASS). The vibration frequency of the equipment is simulated using SPRING elements.

Fig. 7Substation-equipment interaction model

3.2. Numerical model accuracy verification

The validation work mainly involves three aspects: 1) the measured acceleration of the vibration source in the field is used as the model's loading input, and the response at the mid-span position of the structure's second floor slab is collected as the comparison data; 2) numerical simulation analyses are carried out to extract the response data at the model’s corresponding position; 3) the accuracy is evaluated by calculating the error between the measured results and the simulated results. The vibration measurements are utilized as the loading input in this study, as depicted in Fig. 8. Multiple-point input is employed to simulate the traveling effect of subway load. A comparison of the spectral curves of the vibration response at the mid-span position of the second-floor slab is shown in Fig. 9. The comparison results show that the spectral curves of the measured and simulated results are in good agreement. The frequency corresponding to the main peak of the simulated vibration velocity matches well with the measured data. Table 5 quantifies the average measured velocity response and simulated data for each floor, indicating that the error between the two is less than 8 %. This indicates that the numerical model effectively captures the dynamic characteristics of the actual structure and can be utilized for subsequent vibration isolation analysis.

Fig. 8Subway loading input

Fig. 9The comparison of the spectral curves of vibration response

3.3. Structural vibration response analysis at different source distances

The study investigates the safety distance of the substation under subway vibration [16], based on the calculated results of the model and in combination with the vibration evaluation criteria (VC-C standard). Six working conditions, with distances of 10 m, 20 m, 30 m, 40 m, 45 m, and 50 m, have been selected. The analysis indicator is the velocity response at the mid-span position of each floor slab. Fig. 10 displays the 1/3 octave frequency spectrum curves of the substation vibration velocity at different distances. The root-mean-square (RMS) velocity response of the substation structure decreases significantly with an increase in distance. When the distance between the substation and the subway line is 45 m or greater, the vibration velocity response of the structure is less than 12.5 μm/s, meeting the VC-C standard of the code. However, when the distance is less than 45 m, the numerical value of the structural vibration response exceeds the standard, failing to meet the safety distance requirements. Based on these findings, it is recommended that the distance for the construction of substations should be greater than 45 m when planning subway lines. Alternatively, when regional limitations necessitate a construction distance of less than 45 m, effective isolation measures should be implemented to reduce the impact of subway vibration on the structure and equipment.

Fig. 10Maximum vibration response of substations at different distances

4.1. Working principle

The effects of low-frequency vibration on structures equipped with precision instruments cannot be ignored. This study proposes a nonlinear gas-spring quasi-zero stiffness isolator (NGS-QZSI) that incorporates gas-spring components, as depicted in Fig. 11. The isolator comprises four dual-chamber gas-springs arranged vertically and four single-chamber gas-springs arranged horizontally in parallel. The vertical gas-springs function as positive stiffness elements to uphold the upper structure’s weight, while the horizontal gas-springs contribute negative stiffness. The stiffness of the gas-springs within the isolator can be adjusted by altering the gas pressure to accommodate variations in the supported structure’s mass, thereby ensuring the isolator's quasi-zero stiffness characteristics.

Fig. 11Sectional diagram of NGS-QZSI

4.2. Theoretical analysis

The initial state of the NGS-QZSI is characterized by the isolation system being at a state of static equilibrium, as shown in Fig. 12. At this point, both the vertical and horizontal gas-springs are in a compressed state. The upper mass is fully supported by the vertical gas-spring, while the horizontal gas-spring experiences zero force in the vertical direction. Assuming the upper mass is denoted as $m$, the relationship can be expressed as follows:

The relationship between the pressure and volume of the gas-spring can be obtained from the ideal gas equation as follows:

where, $n=$ 1.38 is the gas polytropic index [17].

Fig. 12Schematic diagram of the initial state of NGS-QZSI

Fig. 13Schematic diagram of NGS-QZSI under external force disturbance

When the isolator is disturbed by static force $f$, the bearing platform of the isolator produces a vertical downward displacement $x$, as shown in Fig. 13. As a result, the lower chamber of the vertical gas-spring is compressed, leading to a decrease in volume from $V_l$ to $V_{l2}$ and an increase in pressure from $P_l$ to $P_{l2}$. Conversely, the upper chamber experiences tension, causing the volume to increase from $V_u$ to $V_{u2}$ and the pressure to decrease from $P_u$ to $P_{u2}$. Similarly, the horizontal gas-spring chamber undergoes tension, resulting in an increase in volume from $V_h$ to $V_{h2}$ and a decrease in pressure from $P_h$ to $P_{h2}$. By utilizing geometric relationships and Eq. (2), the changes in pressure and volume for the vertical and horizontal gas-springs is calculated. At this stage, the force on the quasi-zero stiffness isolator of the gas-spring can be expressed as:

Simplifying Eq. (3), the dimensionless force $\widehat{f}$ of the system is obtained as:

where, $\widehat{f}=f/P_l{A}_l$, $h_l=V_l/A_l$, $h_u=V_u/A_u$, $h_h=V_h/A_h$, ${\widehat{h}}_l=h_l/L$, ${\widehat{h}}_u=h_u/L$, ${\widehat{h}}_h=h_h/L$, $\alpha =A_u{P}_u/A_l{P}_l$, $\beta =A_h{P}_h/A_l{P}_l$, $\widehat{x}=x/L$.

By taking the partial derivative of $\widehat{x}$ in Eq. (4), the dimensionless stiffness $\widehat{k}$ of the system is obtained as:

According to the quasi-zero stiffness characteristic, the isolation system satisfies $\widehat{x}=0$ and $\widehat{k}=0$ at the static equilibrium position. In practical engineering applications, it is also necessary to ensure that the isolator does not exhibit negative stiffness. Therefore, at the static equilibrium position, the system stiffness should be at its minimum, i.e., ${\widehat{k}}^{\text{'}}=0$. Substituting $\widehat{x}=0$ into the equation and simplifying:

By substituting Eq. (6) and Eq. (7) into Eq. (4), the dimensionless force-displacement relationship expression for the NGS-QZSI is obtained as:

4.3. Numerical simulation

In this study, the connector element in the interaction module of ABAQUS is used for simulation. The section type of the connector is chosen as “axial + torsional”. The isolator is placed between the column and the foundation, and the connector element is used to simulate the force-displacement characteristics of the NGS-QZSI by inputting the performance curve based on the structural isolation design of the isolator. The corresponding connector element is shown in Fig. 14.

5.1. Isolator parameter design

Based on the actual structural parameters and isolation requirements of the substation, the size parameters of the vibration isolator have been preliminarily determined. The length $L$ is set at 100 mm, with ${\widehat{h}}_l$= 1.8, ${\widehat{h}}_u$= 1.0, and ${\widehat{h}}_h$= 1.0, to determine the chamber lengths $h_l$, $h_u$ and $h_h$ for the vertical and horizontal gas-springs. The piston areas $A_l$, $A_u$, and $A_h$ of each gas-spring chamber are calculated based on the isolator's working space in practical applications. Subsequently, the pressures $P_l$, $P_u$ and $P_h$ of each gas-spring chamber are determined according to the upper weight $mg$, using the equations $\alpha =A_u{P}_u/A_l{P}_l=1-mg/4A_l{P}_l$ and $\beta =A_h{P}_h/A_l{P}_l$.

In the case of the F/1 column of the substation, the isolator supports a weight of 263182 N. The NGS-QZSI parameters are detailed in Table 6. As the bearing weight ($\mathrm{}mg$) varies, the dimensional parameters in the table remain constant, with adjustments made only to the cylinder gas pressures ($P_l$, $P_u$ and $P_h$) to accommodate the changes in the upper weight.

Fig. 14Connector element

5.2. Isolation scheme

A vibration isolator is installed under each independent column, totaling 46 isolators, as depicted in Fig. 15. The static analysis is utilized to compute the weight supported by the isolator at the base of the column, as detailed in Table 7. The variation in pressure at the base of each column poses a challenge for traditional isolators. This necessitates the production of isolating elements with different parameter configurations, thereby increasing the complexity of isolation construction to some extent. In contrast, the quasi-zero stiffness gas-spring isolator proposed in this study can achieve varying isolation performance simply by adjusting the gas pressure, rendering its implementation straightforward.

5.3. Isolation effect analysis

The isolation performance of NGS-QZSI is evaluated by comparing the vibration acceleration responses of each floor before and after isolation in this section. The acceleration time history curves and 1/3 octave frequency bands at the mid-span of each floor slab are shown in Fig. 16 to 18, and the corresponding quantified data of peak acceleration and root mean square isolation effect are presented in Tables 8 and 9.

Comparison of the acceleration time history curves with and without the isolation system indicates a significant reduction in vibration response of the isolated structure, with isolation effects of 74.98 % for peak response and 71.9 % for RMS response. Although there are some differences in isolation effects among different floors, the overall isolation effect remains within the range of 55 % to 75 %. Furthermore, the comparison of the 1/3 octave reveals that NGS-QZSI exhibits good control effects on subway vibrations across the entire frequency range, fully leveraging the advantages of quasi-zero stiffness low-frequency isolation.

Fig. 15NGS-QZSI arrangement scheme

Fig. 16Comparison of first floor vibration response before and after isolation

a) Acceleration time history

b) One-third octave band frequency

Fig. 17Comparison of second floor vibration response before and after isolation

a) Acceleration time history

b) One-third octave band frequency

Fig. 18Comparison of top floor vibration response before and after isolation

a) Acceleration time history

b) One-third octave band frequency

In addition, the vibration velocity response of the substation structure isolated with NGS-QZSI at different distances from the vibration source is analyzed, as shown in Fig. 19. The comparison results demonstrate that the substation structure isolated with nonlinear gas-spring quasi-zero stiffness isolation exhibits a reduction in the RMS velocity response from 20.0 μm/s to 12.4 μm/s at a distance of 20 m from the subway line, which is less than 12.5 μm/s, meeting the VC-C standard. Compared with the non-isolated structure, the safety distance is reduced from 40 m to 20 m, which greatly improves the regional adaptability of the subway line.

Fig. 19Structural velocity response under different vibration source distances