Experimental Study on Electromagnetic Control Model for Unstable Rock Stability Based on Natural Vibration Frequency

Zhang Minrui¹, Kong Deheng¹, Xiao Feizhi², Li Bo¹
(1. Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China;
2. China Railway Construction Bridge Engineering Bureau Group Co., Ltd, Tianjin 300300, China)

Abstract

Rockfall disasters along China's highway slopes are characterized by widespread distribution, high frequency, and strong unpredictability, posing significant challenges for hazard prevention due to the lack of reliable early warning indicators. This study introduces an innovative electromagnetic control model experiment method, systematically investigating the stability and natural vibration frequency characteristics of three typical unstable rock types—sliding, toppling, and falling—using laser vibration measurement technology. The results reveal that unstable rock masses exhibit distinctive variations in natural vibration frequency prior to failure. When the slope angle exceeds 70°, the frequency shows a stepwise decline accompanied by dual- or multi-peak spectral phenomena. Compared with conventional displacement monitoring, vibration frequency indicators can identify precursor signals of failure significantly earlier, substantially improving warning timeliness. The electromagnetic control model experiment offers advantages of operational simplicity and data reliability, providing an effective technical approach for early warning of slope rockfall disasters.

Keywords: slope; rockfall disasters; natural vibration frequency; early warning

Introduction

With continuous socio-economic development, China's highway network has expanded rapidly, reaching a total length of 117,000 kilometers that covers 98.8% of urban populations and connects all cities with over 200,000 people and prefecture-level administrative centers, as well as 88% of county-level administrative regions nationwide, greatly facilitating public mobility. The Ministry of Transport's 2035 highway network planning objectives emphasize building a "modern, high-quality national highway network that is extensively covered, functionally complete, intensive, efficient, green, intelligent, and safe and reliable." This framework specifically highlights the "safety and reliability" of highway engineering, requiring enhanced engineering reliability of highway networks in key areas such as natural disaster-prone regions to ensure infrastructure safety and respond to major geological hazards [1]. Therefore, identifying safety hazards along highways and implementing timely remediation and protection measures are critical for achieving a safe and reliable national highway network and constitute important safeguards for reducing casualties and property losses.

Common geological hazards along highways include landslides, rockfalls, and debris flows, among which rockfall disasters are particularly prevalent, featuring strong suddenness and high destructiveness. In the hilly regions of southeast China and mountainous areas of western China, highway slopes often consist of exposed, steep rock masses. Due to precipitation, temperature variations between day and night, and other environmental factors, weathering effects are intense, facilitating the development of joints and fractures within slopes that progressively reduce rock mass strength and form unstable rock blocks. Disturbances from earthquakes, extreme weather, and engineering activities can trigger rockfall disasters, leading to a high-frequency occurrence of slope rockfall hazards that are difficult to comprehensively prevent and control. For instance, the MW5.8 earthquake in Lushan County, Ya'an City, Sichuan Province on June 1, 2022, triggered over 2,300 rockfalls and landslides, causing casualties and property damage while significantly impeding subsequent rescue efforts [2]. Over the past decade, the number of rockfall disasters has remained high. According to National Bureau of Statistics data, 2,176 rockfall disasters occurred in China in 2023 [3], representing an increase compared to previous years (Figure 1 [FIGURE:1]).

Unstable rock masses exhibit numerous failure modes with various definitions and classifications. Dorren [4] and Hoek et al. [5] categorized rockfalls into sliding, toppling, and falling types based on differences in rockfall motion patterns caused by varying average slope angles. Hu Houtian [6] classified unstable rock mass failures into toppling, sliding, bulging, tensile cracking, and shearing types, as well as transitional types such as bulging-sliding and bulging-toppling, considering factors including rock properties, structural plane characteristics, and initial motion forms. Chen Hongkai et al. [7] investigated unstable rock masses in the Three Gorges Reservoir area, macroscopically dividing them into individual and group unstable rocks, and further classifying individual unstable rocks into compressive shear sliding, tensile shear toppling, tensile cracking falling, and tensile cracking compressive shear falling types based on mechanical mechanisms and instability modes. Overall, according to mechanical mechanisms and motion patterns of rock failure, the three primary types of slope rockfall are sliding, toppling, and falling (Figure 2 [FIGURE:2]).

Du Yan et al. introduced dynamic monitoring indicators such as kurtosis index and impact energy to study early monitoring and warning of rockfall failure processes. The results demonstrated that impact energy and vibration parameters could identify early signs of accelerated rock mass failure, with impact energy showing greater sensitivity [8]. Du Yan et al. also conducted various model experiments, including remote sensing monitoring warning tests [9], rock bridge length measurement tests [10], unstable rock safety monitoring tests [11], and cumulative damage evaluation tests [12]. Based on these experimental results, the slope rock mass failure process was broadly divided into stable, separation, and accelerated failure stages, with the separation stage identified as the critical early warning period for unstable rock recognition.

According to indoor model test results, monitoring natural vibration frequency indicators can provide criteria for early warning and effectively transform the passive prevention approach to brittle failure disasters such as rockfalls. However, current indoor model tests based on natural vibration frequency suffer from small differences, low controllability, and poor continuity, and related research cannot cover different failure types of unstable rock masses. To address these issues, this study employs an electromagnetic control model experiment based on unstable rock vibration theory to investigate the relationship between unstable rock mass stability and its natural vibration frequency.

2.1 Experimental Overview

As shown in Figure 3 [FIGURE:3], this study simulates unstable rock masses using gypsum specimens with six electromagnets placed equidistantly on the bottom surface. A laser Doppler vibrometer (LDV) and linear variable differential transformer (LVDT) measure the specimen's natural vibration frequency and displacement. By adjusting the power supply voltage from top to bottom to change the electromagnetic attraction force, the process of gradual crack propagation to failure in unstable rock masses can be simulated.

Figure 2 [FIGURE:2] illustrates the three primary types of unstable rock masses (from left to right: sliding type, toppling type, and falling type). Currently, the engineering community primarily uses displacement indicators to characterize precursor signals of accelerated failure for rockfall monitoring and slope stability assessment. However, displacement changes are minimal and insensitive during the cracking-to-failure process of unstable rock masses, resulting in low disaster monitoring foresight with existing equipment precision and potentially missing the critical prevention window. To address this limitation, scholars have proposed using natural vibration frequency indicators as characteristic parameters for identifying precursor signals of separation failure in unstable rock masses, which can improve the sensitivity of rockfall disaster early warning [8].

As shown in Figure 4 [FIGURE:4], adjusting the test platform inclination angle enables simulation of the failure process for sliding-type unstable rocks at different slope angles. Setting the platform angle to 90° allows falling-type tests, and further installation of blocking devices enables toppling-type tests.

μ — friction coefficient; θ — slope inclination angle, °; Fm — electromagnetic force, N; m — specimen mass, kg; g — gravitational acceleration, 9.81 m/s².

When θ reaches 90°, the test transforms into a falling-type test, as shown in Figure 6 FIGURE:6.

2.2 Calculation Principles

To conveniently describe specimen stability, the limit equilibrium method is adopted, using the ratio of sliding-resisting factors to sliding factors as a reference index, denoted as the theoretical factor of safety (TFS). When calculating TFS, the electromagnets and pressure sensors within the specimen are ignored, and the gypsum specimen is treated as a homogeneous block with uniform stress distribution.

As shown in Figure 5 [FIGURE:5], for sliding-type specimens, based on force analysis, the sliding-resisting force (Fsr) is friction, and the sliding force (Fs) is the gravity component.

For falling-type tests, the theoretical factor of safety is:

Based on the falling-type test, after installing blocking devices, the test transforms into a toppling-type test, as shown in Figure 6(b). The theoretical factor of safety then becomes the ratio of the sliding-resisting moment to the sliding moment.

It can be observed that the TFS in this case is related to the height of the bottom block, a. The sliding-resisting moment is:

H — specimen height, m; L — specimen length, m; a — bottom block height, 0 < a < H/2.

The sliding moment is:

Substituting equations (4), (5), and the force equilibrium relationship into equation (3) and simplifying yields:

After each test completion, the slope inclination angle is increased by 5°, and the test is repeated until the slope angle reaches 90°. The relationships between natural vibration frequency, displacement, and theoretical factor of safety for each test are shown in Figure 8 [FIGURE:8].

Defining the ratio of gravity to electromagnetic force as α, equation (6) becomes:

2.3 Experimental Equipment

The laser vibrometer employed in this study is the RSV-150 long-range laser vibrometer with a maximum working distance of 150 m. After acquiring frequency data, time-domain and frequency-domain vibration analyses are performed using fast Fourier transform (FFT) to obtain the specimen's vibration frequency distribution image and subsequently determine its natural vibration frequency.

The cast gypsum specimens have a mass of 10.5 kg and dimensions of 250 × 150 × 150 mm, with surfaces ground smooth. Six small electromagnets rated at 24V with 4 kg holding force each are arranged equidistantly on the bottom surface, with electromagnet dimensions of 30 × 20 × 20 mm. Testing confirmed that electromagnetic attraction force is linearly and positively correlated with voltage. High-precision LVDT is used to measure specimen displacement along the sliding direction, with a resolution of 0.1 μm.

3.1 Sliding-Type Tests

Testing confirmed that the friction angle between gypsum specimens and the test platform surface is approximately 32°; therefore, the initial slope angle for sliding-type tests was set at 35°. At the test start, all electromagnets were controlled at 24V, after which the voltage was reduced from top to bottom by 3V increments until specimen sliding occurred.

Using the 35° slope angle as an example, the relationships among natural vibration frequency, displacement, and theoretical factor of safety are presented in Table 1 [TABLE:1] and Figure 7 [FIGURE:7]. The results show that as the specimen's factor of safety gradually decreases, the frequency initially declines slowly, then drops sharply after the safety factor reaches 0.998; displacement increases slowly at first, then rises rapidly after the safety factor reaches 0.980. The natural vibration frequency predicts the sliding inflection point earlier than displacement.

The internal stress distribution within the specimen exhibits significant non-uniformity, with localized stress concentration zones forming near shear bands. The energy dissipation mode transitions from uniform viscoelastic dissipation to non-uniform plastic and brittle dissipation. The low-frequency peak corresponds to elastic vibration of the overall structure, reflecting the elastic energy storage-dominated stage where energy is slowly released through internal friction and viscous deformation. The high-frequency peak originates from local instability and friction within the shear band, characterizing the nonlinear failure stage of the local shear band, manifested as high-frequency vibration signals.

3.2 Falling and Toppling-Type Tests

The slope inclination angle was increased to 90° to conduct falling-type and toppling-type tests, following the same procedure as sliding-type tests. In toppling-type tests, block lengths of 1, 2, 4, and 6 cm were used.

The calculated relationships among natural vibration frequency, displacement, and theoretical factor of safety are shown in Figure 10 [FIGURE:10].

From Figures 7 and 8, for sliding-type specimens, natural vibration frequency predicts the sliding inflection point earlier than displacement, demonstrating superior early warning effectiveness. As the slope inclination angle increases, the initial natural vibration frequency decreases, indicating reduced specimen stability. When the slope angle is less than 70°, the frequency variation trend shows slow decline → sharp drop, forming a single-inflection pattern. When the slope angle is greater than or equal to 70°, the frequency variation trend exhibits slow decline → sharp drop → slow decline → sharp drop, forming a stepwise pattern.

Observation of specimen frequency distribution reveals that when the slope angle is greater than or equal to 70° and the specimen approaches sliding failure, its frequency distribution presents a "dual-peak" pattern, as shown in Figure 9 [FIGURE:9]. This phenomenon can serve as another indicator of impending instability. The appearance of "dual peaks" may result from stress differentiation and energy dissipation mechanism transitions within the specimen. As electromagnetic attraction force decreases from top to bottom, the frequency variation trend shows slow decline → sharp drop → slow decline → sharp drop, maintaining a stepwise pattern. However, compared with sliding-type test results, the frequency change rate is significantly faster, with the first stage shortened and the third stage extended.

For both falling and toppling types, the frequency distribution exhibits "dual-peak" or "multi-peak" patterns when the specimen approaches failure. Compared with sliding-type tests, the frequency distribution is more chaotic (Figure 11 [FIGURE:11]).

The variation trends of frequency and displacement in this study are similar to those obtained by Du Yan et al., demonstrating the applicability of the electromagnetic control model experiment for unstable rock stability research. Compared with previous experiments, the electromagnetic control model experiment offers better process continuity and repeatability, more intuitive data, and greater controllability. However, fewer data points were recorded under large-angle conditions, likely because at large angles, increasing slope inclination makes it increasingly difficult for electromagnetic attraction to maintain specimen stability. Specifically, under large-angle conditions, specimens tend to slide even when all electromagnets operate at full voltage, which may be related to surface smoothness, air humidity, and specimen installation errors.

Under large-angle conditions, specimens exhibit "dual-peak" or "multi-peak" phenomena when approaching instability. Typically, the low-frequency peak has high amplitude while the high-frequency peak has lower amplitude. As the slope inclination angle increases, the frequency distribution gradually transitions from "dual-peak" to "multi-peak," with the dominant frequency corresponding to the low-frequency peak. This indicates that increasing slope angles intensify internal stress differentiation and gradually shift the energy dissipation mode toward non-uniform plastic and brittle dissipation.

Under the experimental conditions of this study, the critical slope angle distinguishing stepwise frequency variation and "dual-peak" ("multi-peak") frequency distribution is 70°. This boundary may be influenced by specimen material properties and frictional characteristics between the specimen and slope surface. Integrating numerical and graphical analysis, stepwise frequency variation and "dual-peak" ("multi-peak") frequency distribution can serve as criteria for determining impending instability of unstable rock masses on large-angle slopes.

Conclusion

This study proposes an improved electromagnetic control model experiment for unstable rock mass stability research, demonstrating its feasibility and result credibility. Compared with previous experiments, the electromagnetic control model experiment offers shorter testing time, more accurate control, more intuitive data, and better process controllability.

As slope angle increases, the frequency variation trend of unstable rock masses transitions from a single-inflection pattern (slow decline → sharp drop) to a stepwise pattern (slow decline → sharp drop → slow decline → sharp drop), with increasingly chaotic frequency distributions exhibiting "dual-peak" or "multi-peak" patterns. These features appear earlier than displacement surges and can serve as criteria for determining impending instability of unstable rock masses on large-angle slopes.

References

[1] Central People's Government of China. National Highway Network Planning [EB/OL]. https://www.gov.cn/zhengce/zhengceku/2022-07/12/5700633/files/1ca992f13a81434b89e222eb000acada.pdf, 2022-07.

[2] Ma Siyuan, Xu Chong, Chen Xiaoli. Comparison of emergency assessment models for seismic rockfall and landslide hazards: A case study of the June 1, 2022 MW5.8 Lushan earthquake [J]. Seismology and Geology, 2023, 45(04): 896-913.

[3] National Bureau of Statistics of China. China Statistical Yearbook [M]. Beijing: China Statistics Press, 2024.

[4] Dorren L K A. A review of rockfall mechanics and modelling approaches [J]. Progress in Physical Geography, 2003, 27(1): 69-87.

[5] Hoek E, Bray J D. Rock slope engineering [M]. CRC press, 1981.

[6] Hu Houtian. Study on rockfall [J]. Journal of Railway Engineering Society, 2005, (S1): 387-391.

[7] Chen Hongkai, Tang Hongmei. Classification and macro-criteria of unstable rocks in the Three Gorges Reservoir area [J]. The Chinese Journal of Geological Hazard and Control, 2005, (04): 57-61+82.

[8] Liu Ran, Luo Changwei, Sun Tao, et al. Study on design scheme of passive protection net based on rockfall motion characteristics analysis [J]. Journal of Water Resources and Architectural Engineering, 2023, 21(03): 74-83.

[9] Du Yan, Huo Leichen, Xie Mowen, et al. Experimental study on monitoring and early warning of unstable rock mass collapse disasters [J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(04): 1212-1221.

[10] Du Yan, Huo Leichen, Zhang Hongda, et al. Experimental study on remote sensing monitoring and early warning of rock block collapse disasters [J]. Journal of China University of Mining & Technology, 2022, 51(06): 1201-1208.

[11] Du Yan, Lu Yongdu, Xie Mowen, et al. Measurement of rock bridge length in unstable rock mass based on laser Doppler vibration [J]. Journal of China Coal Society, 2019, 44(S2): 560-565.

[12] Du Yan, Xie Mowen, Jiang Yujing, et al. Experimental study on safety monitoring of unstable rock based on natural vibration frequency [J]. Rock and Soil Mechanics, 2016, 37(10): 3035-3040.