Research on the Dynamic Response Characteristics of Soft Coal Under Impact Disturbance Based on Hamilton

Abstract

To address the limitations of traditional elasticity theory in analyzing the dynamic response of soft coal under external impact, this study establishes a vibration control equation with an analytical solution based on Hamiltonian mechanics. Key control parameters within the equation were solved to determine the theoretical dominant vibration modes and natural frequencies of the weakest coal layer. Triangular and rectangular waves were transformed via FFT to analyze their harmonic components, and the superposition of the first four harmonics was selected as the input impact signal. The modal and natural frequency changes during the fragmentation of the central weak zone under external impact were simulated, and the dynamic displacement response was analyzed. The results indicate a strong response frequency range of 4.4–5.2 Hz, with the rectangular wave identified as the most effective response waveform. A similarity simulation platform was constructed, and experimental data showed that the velocity and displacement response trend of the coal mass aligned closely with theoretical predictions. Therefore, in actual underground operations, emphasis should be placed on monitoring low-frequency vibrations in mines, minimizing rectangular wave disturbances in the low-frequency range, and implementing pressure relief measures in high-risk zones to reduce the likelihood of coal and gas outbursts. By separately modeling high-risk zones and analyzing their dynamic response under external impact, this study explains the outburst mechanism of the weakest layer in soft coal from a dynamic perspective. Combining theoretical and experimental approaches, it provides a new theoretical basis for understanding and preventing coal and gas outbursts.

1. Introduction

Coal and gas disasters remain critical challenges to coal mine safety, with coal and gas outbursts being the combined result of in situ stress, gas pressure gradients, and the physical structure of the coal mass [1]. Such outburst accidents occur frequently; major and above-grade outburst incidents still account for about one-third of all coal mine accidents, often with severe consequences. For instance, on 12 January 2024, a major outburst in Pingdingshan, Henan Province, China, resulted in 16 fatalities and 5 injuries, causing significant personal and economic losses [2]. More than 95% of high-gas outburst-prone mines in China involve low-permeability coal seams, a considerable proportion of which consist of soft coal [3]. Such coal seams are typically characterized by heterogeneity, fine pores, and low strength. Under external dynamic loading, they often fail to release pressure promptly, leading to internal stress accumulation and a high risk of coal and gas outbursts [4,5].

Thus, accurately identifying and managing outburst-prone zones remains an urgent issue. Previous studies have predominantly applied static theories to investigate outburst-prone areas, which only describe the static load-bearing state. In reality, however, the initiation–evolution–termination process of an outburst is dynamic. Existing static theories cannot adequately capture dynamic damage processes, such as vibration frequency and amplitude modes of coal masses. Moreover, dynamic damage is closely associated with the dynamic response characteristics of soft coal masses. Traditional static elastic models fail to account for the influence of external dynamic excitation on the region of interest and cannot satisfy all boundary conditions during computation. As a result, the predicted response under static excitation significantly deviates from actual observations [6,7]. It is therefore imperative to develop new theoretical frameworks to interpret the dynamic processes involved in coal and gas outbursts. In contrast to conventional Lagrangian mechanics, the Hamiltonian mechanics model strictly adheres to the principle of energy conservation during dynamic response, thereby improving computational accuracy under dynamic excitation and providing a more accurate description of the behavioral characteristics of the studied system [8]. Hence, a Hamiltonian-based mechanical model can be effectively employed to analyze the excitation response and behavioral properties of coal masses.

Numerous studies have been conducted on the forced response of coal masses. Laboratory uniaxial compression tests on sampled coal blocks have revealed dynamic failure patterns: coals with higher impact propensity fail more quickly and release substantial energy upon fragmentation, providing an experimental basis for understanding vibration-induced failure and outburst prevention [9]. To further investigate the mechanical response and failure characteristics of deep coal measure strata (CMS), studies under uniaxial compression examined the mechanical properties, stress evolution, and failure modes of CMS with different lithological combinations and thickness ratios. It was found that soft coal primarily exhibits shear-dominated failure, while harder sandstone/mudstone layers show tension-dominated failure under external impact [10]. Subsequently, researchers such as Feng Li and Zhang Yaguang studied stress wave propagation in coal under dynamic loading, taking the elastoplastic coal and rock mass in front of a tunnel face as an example. Their results indicated that compressive stress waves induced by instantaneous impact can cause severe damage to the coal matrix, leading to a sharp increase in gas pressure and potentially triggering outbursts [11]. This finding also corroborates that outburst development initiates inside the coal mass before propagating toward the excavation face. Ma Jianguo conducted mechanical vibration experiments on 32 natural rock cores and summarized the effects of vibration frequency and intensity on core parameters [12]. Additionally, Ren Weijie et al. studied the influence of vibration on the fracture distribution and mechanical properties of coal, finding that forced vibration reduces overall structural strength and accelerates gas desorption [13]. Li Xiangchun theoretically analyzed the effect of external disturbances on coal structure, showing that vibrations induce internal fractures that can propagate extensively once a certain intensity threshold is reached [14]. Meanwhile, Li Chengwu designed shaking table tests and observed that the natural frequency of coal decreases with increasing damage, and the failure mode gradually shifts from shear to tension [15]. Wang Chenchen et al. used similar materials to prepare multi-layer composite coal models and conducted drop-weight impact tests. By analyzing failure patterns and dynamic strain responses, they revealed that impact loads propagate in the form of spherical waves inside coal, generating axial compressive strain and lateral tensile strain, thereby clarifying the transmission mechanism of impact loads within coal masses [16].

In summary, the response of coal and rock masses to impact disturbance is closely related to their intrinsic properties and environmental conditions. Shock disturbances can disrupt the stress equilibrium in certain regions, leading to local fragmentation, rapid energy release, and accelerated gas desorption, ultimately resulting in coal and gas outbursts. Therefore, investigating the dynamic response of soft coal under external impact is of great practical significance, providing a theoretical basis for the identification, monitoring, and prevention of outburst-prone areas. Based on Hamiltonian mechanics, this paper establishes a dynamic vibration model for coal, solving for natural frequency, amplitude, and mode shapes of soft coal under external impact. The response characteristics under different waveform excitations are analyzed, offering important theoretical support for the prevention of coal and gas outbursts in low-permeability coal seams.

2. Materials and Methods

2.1. Determination of Coal and Gas High Outburst Risk Area in Front of Impact

To analyze the response characteristics of coal and gas outburst-prone zones under dynamic impact, it is essential to first identify the high-risk regions within the coal mass ahead of the impact. As illustrated in Figure 1, a model of the coal measure strata ahead of the impact was established. Analyses of in situ stress, gas pressure gradient, and permeability within the coal mass were conducted. The results indicate that both in situ stress and gas pressure gradient initially increase and then decrease with increasing depth, while permeability exhibits an opposite trend, first decreasing and then increasing. Notably, the extreme values of all three parameters occur at the same depth, where in situ stress and gas pressure reach their maxima, and permeability is at its minimum.

According to the variations in stress, gas pressure, and permeability, the coal mass ahead of the impact front can be divided into the fractured zone, the damaged zone, and the elastic recovery zone. In the fractured zone, the in situ stress gradually increases, and the gas pressure gradient rises synchronously after a certain distance behind the impact surface. Upon entering the damaged zone, both the in situ stress and gas content increase with depth until reaching their peak values, while the permeability decreases to its minimum. Beyond the damaged zone lies the elastic recovery zone, where the in situ stress and gas pressure gradient gradually decline, and the permeability progressively increases. Across these three zones, both the fractured and elastic recovery zones exhibit relatively low in situ stress and gas pressure gradient, along with relatively high permeability, resulting in a lower risk of coal and gas outburst. In contrast, the damaged zone is characterized by high in situ stress and gas pressure gradient, low permeability, and significant stress concentration, making it highly susceptible to coal and gas outbursts. Therefore, the coal mass in this region is longitudinally segmented and discretized into multiple coal plates perpendicular to the impact direction. Among these, the first coal plate in the damaged zone experiences the highest peaks in in situ stress, gas pressure gradient, and permeability, making it particularly vulnerable to damage under dynamic impact due to its disruption of the balanced condition. Consequently, this coal plate is identified as a weak zone and subjected to in-depth analysis regarding its response characteristics under external impacts.

2.2. Establishment of Hamilton Mechanics Model

When an external impact acts on the coal surface, a stress wave originates from the point of impact and propagates radially along the surface in a circular pattern while advancing into the coal interior as a spherical wave [17]. Upon reaching the surface of the weak zone, the stress wave first contacts the projection of the impact center on that surface. Radially, the wave expands annularly outward from the center until it fully covers the surface; axially, it penetrates the coal mass and continues propagating inward as a spherical wave. During this process, the coal undergoes complex loading and unloading effects until the wave reaches the weak layer. The duration from impact initiation to arrival at the weak zone is extremely short (∆t → 0), meaning the stress wave may be considered to act nearly simultaneously across the entire surface of the weak zone.

By analogy with engineering cases such as explosion problems, shock waves also propagate in the form of spherical waves within a medium. Due to the high-frequency and rapid nature of shock transmission, the stress is commonly approximated as uniformly distributed over the cross-section upon impact [18]. This simplification, known as the plate theory, is applicable when surface deformation remains small enough to neglect spherical wave effects [19].

Therefore, to simplify the impact model of the coal plate, the spherical load induced by external impact is equivalently treated as a uniformly distributed planar load acting on the surface of the weak zone. In actual mining conditions, due to the squeezing of surrounding rock under in situ pressure, vibrational deformation at the boundary of the coal mass under impact disturbance is minimal. However, a certain degree of bending moment still exists along the boundary. To reduce errors and better reflect the actual scenario, the boundary of the weak-layer coal plate is modeled as a fixed support. The simplified mechanical calculation model is shown in Figure 2.

2.3. Coal Plate Impact Vibration Control Theory Based on Hamilton

To analyze the vibration response of a coal plate under impact disturbance, this study employs a Hamiltonian mechanics model for the mathematical modeling of the coal mass, transforming the dynamic response problem into the solution of eigenvalues and eigenvectors within the Hamiltonian framework. The Hamiltonian mechanics model introduces the concept of conjugate momenta, replacing the n second-order equations of Lagrangian mechanics with 2n first-order equations. By incorporating dual variables from symplectic geometry, the model ensures the preservation of phase space volume during evolution and reduces the order of the high-order differential equations through canonical transformation, thereby simplifying the solution process.

The instant at which the stress wave generated by a single impact reaches the upper surface of the coal seam is taken as the research scenario. Investigating the instantaneous mechanical equilibrium state of the coal plate, the governing partial differential equation for the coal mass under external impact vibration is derived as Equation (1):

$$D{\nabla }^{4}w\left(x,y,t\right)+\rho h{\displaystyle \frac{{\partial }^{2}w\left(x,y,t\right)}{\partial t^2}}=q\left(t\right)$$

(1)

In the equation, w(x,y,t) denotes the deflection, H represents the thickness of the weak layer, D represents the flexural rigidity of the material, and ρ is the material density; q(t) corresponds to the external impact load. At the instant when the single impact ceases, q(t) = 0. The time-dependent variation in deflection induced by the inherent vibration of the coal mass, in the vicinity of the vibrational equilibrium position, is described by the following relation:

$$w\left(x,y,t\right)=W\left(x,y\right)e^{i\omega t}$$

(2)

In the equation, ω denotes the angular frequency, representing the rate of change in vibrational phase per unit time. By introducing the complex exponential term e^iωt, the original vibrational deflection is transformed into a time-harmonic complex exponential form. Assuming θ = ∂W/∂x and defining the state vector Tv = [W,θ,−Vx,Mx]^⊤, the dual equations in the Hamiltonian system can be expressed in matrix form as follows [20]:

$$⌈\begin{array}{c}{\displaystyle \frac{\partial W}{\partial x}}\\ {\displaystyle \frac{\partial \theta }{\partial x}}\\ {\displaystyle \frac{\partial \theta }{\partial x}}\\ {\displaystyle \frac{{\partial M}_x}{\partial x}}\end{array}⌉=\left[\begin{array}{cccc}0& 1& 0& 0\\ -v{\displaystyle \frac{{\partial }^2}{\partial y^2}}& 0& 0& -{\displaystyle \frac{1}{D}}\\ -D\left(1-v^2\right){\displaystyle \frac{{\partial }^4}{\partial y^4}}+\rho w^{2}h& 0& 0& v{\displaystyle \frac{{\partial }^2}{\partial y^2}}\\ 0& 2D\left(1-v\right){\displaystyle \frac{{\partial }^2}{\partial y^2}}& -1& 0\end{array}\right]\ast \left[\begin{array}{c}W\\ \theta \\ -V_x\\ M_x\end{array}\right]$$

(3)

Based on the symplectic geometry method, the matrix is solved [21] to obtain the deflection equation. As illustrated in Figure 1, the four boundaries of the coal mass are fixed supports, meaning both the deflection and its first-order partial derivatives are zero along the boundaries. Thus, the frequency equation for vibration in the boundary directions can be expressed as follows:

$${\displaystyle \frac{1-\mathit{cos}{\alpha }_{1}b\mathit{cos}h{\alpha }_{2}b}{\mathit{sin}{\alpha }_{1}b\mathit{sin}h{\alpha }_{2}b}}={\displaystyle \frac{{{\alpha }_1}^2-{{\alpha }_2}^2}{2{\alpha }_1{\alpha }_2}}$$

(4)

In the equation, α₁, α₂, β₁, and β₂ are corresponding directional eigenvalues, and satisfy the following conditions:

$${\alpha }_1^2+{\alpha }_2^2={\beta }_1^2+{\beta }_2^2=2k^2,{\alpha }_1^2+{\beta }_1^2=k^2,{\alpha }_2^2+{\beta }_2^2=k^2$$

(5)

In the equation, k denotes the wave number. For thin plate vibrations, $k=\sqrt[4]{{\displaystyle \frac{\rho h{\omega }^2}{D}}}$, and the deflection equation can be expressed as follows:

$$\sum _{m=1}^{\infty }\rho h{W}_m\left(x,y\right)\left[{w^2}_m{\phi }_m\left(t\right)+{\phi }_m^{″}\left(t\right)\right]=q\left(t\right)$$

(6)

In the equation, φ_m(t) denotes the position equation of the mode as a function of time, and q(t) represents the stress distribution per unit area. Its orthogonal form can be expressed as follows:

$${\iint }_{\Omega }\rho h{W}_m\left(x,y\right)W_n\left(x,y\right)dxdy=0,\left(m\ne n\right)$$

(7)

Subsequently, multiplying both sides of Equation (7) by W_m(x,y), the governing partial differential equation of the coal mass under external impact vibration can thereby be transformed into:

$${\iint }_{\Omega }\rho h{W^2}_m\left(x,y\right)\left[{w^2}_m{\phi }_m\left(t\right)+{\phi }_m^{″}\left(t\right)\right]dxdy={\iint }_{\Omega }q\left(t\right)W_m\left(x,y\right)dxdy$$

(8)

Assumption:

$$M_m={\iint }_{\Omega }\rho h{W^2}_m\left(x,y\right)dxdy,P_m\left(t\right)={\iint }_{\Omega }q\left(t\right)W_m\left(x,y\right)dxdy$$

(9)

When the duration of the external impact is vanishingly short, when ∆τ → 0, Equation (8) reduces to the general form of Duhamel’s integral [22], expressed as follows:

$$M_m={\iint }_{\Omega }\rho h{W^2}_m\left(x,y\right)dxdy,P_m\left(t\right)={\iint }_{\Omega }q\left(t\right)W_m\left(x,y\right)dxdy$$

(10)

Thus, the Hamiltonian-based governing equation for the impact-induced vibration of the coal plate, which admits an analytical solution, is obtained. A computational program is developed to implement the derived formulations, establishing a theoretical foundation and technical support for solving the dynamic response parameters of the model under external vibration in subsequent sections.

2.4. Pre-Conditioning of External Impact Waveforms for Hamiltonian Formulation

Owing to the relatively soft nature of friable coal mass, its response undergoes significant temporal variations when subjected to external impact disturbances. For a Hamiltonian model system, directly applying an untreated input waveform during external excitation would lead to highly complex solution conditions, making it difficult to obtain valid results. Therefore, pre-conditioning of the input vibration waveform is necessary to ensure its coupling with the intrinsic vibration of the coal plate in the computational model. Assuming that all time-varying terms and constraints on the coal mass are functions of the same frequency, and that the boundary conditions and physical properties remain invariant under external impact, the harmonic vibration expression for the soft coal mass under external excitation can be written as follows:

$$u(x,t)=\tilde{u}\left(x\right)\mathit{exp}(j\omega t)$$

(11)

In the equation, u = u(x,t) denotes the vibrational state of the coal mass in both time and space domains; u~(x) characterizes the spatial distribution pattern of the vibration; and exp(jωt) represents the harmonic vibration factor.

For soft coal masses under external impact, the induced dynamic response can be categorized into two distinct phases. The first phase is the initiation stage, during which the unconstrained body begins to undergo harmonic motion following impact. In this stage, the amplitude increases progressively while the waveform remains essentially unchanged. The second phase is the steady-state stage, characterized by the displacement amplitude reaching its maximum and the oscillation frequency remaining stable over time.

For a single coal mass system under external impact disturbance, the amplitude progressively increases within each cycle before the system stabilizes, making it impossible to deterministically describe its instantaneous state. However, upon reaching the steady state, the behavior of the resonant system becomes consistent across successive cycles. At this stage, a stable solution can be obtained by fitting the impact waveform to the resonant system of the coal mass. Therefore, by differentiating with respect to the temporal variable within the system and combining it with the governing vibration equation given in Equation (8), the frequency expression of the vibration model after Fourier transformation [23] is derived as follows:

$$\begin{gathered} -{\omega }^{2}M\tilde{u}+j\omega C\tilde{u}+\nabla ·\left(-K\nabla \tilde{u}\right)=\tilde{F} \\ \begin{array}{c}n·\left(K\nabla \tilde{u}\right)+A\tilde{u}=\tilde{f}\end{array} \\ \tilde{u}=\tilde{g} \end{gathered}$$

(12)

In the equation, C denotes the damping operator (unit: N·s/m); M represents the mass density operator (unit: kg); kk signifies the stiffness operator (unit: N/m); $\tilde{u}$ is the complex amplitude (unit: m); $\tilde{F}$ corresponds to the complex amplitude of the impact force applied to the system (unit: N/m); A indicates the boundary impedance coefficient (unit: N/m³); and $\tilde{g}$ refers to the displacement at the boundary (unit: m).

Within a specified period T, a periodic signal is decomposed through Fourier transformation into a linear combination of trigonometric functions and their integrals. The standard expression of the Fourier series expansion for the vibration is then given by:

$$f\left(t\right)\sim {\displaystyle \frac{a_0}{2}}+{\displaystyle \frac{2\sqrt{2}}{T}}\sum _{n=1}^{\mathrm{\infty }}{\int }_{t_0}^{t_0+T}f\left(t\right)·\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{2\pi nt}{T}}+{\displaystyle \frac{\pi }{4}}\right)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{1}{T}}{\int }_{t_0}^{t_0+T}f\left(t\right)·e^{-i\cdot {\displaystyle \frac{2\pi nt}{T}}}$$

(13)

In practical underground mining operations, blasting and roadway driving are identified as the primary triggering factors for coal and gas outburst incidents, accounting for nearly 79% of such accidents in China over the past five years [24].

According to research by Li Y, stress waves generated under explosive conditions are commonly described using triangular or exponential distributions [25]. Between these two, the triangular wave more accurately captures the time-history characteristics of blast-induced shock waves and is easier to express mathematically. Therefore, during underground operations, the shock waves produced by blasting are predominantly triangular. During coal mining operations, due to the large size of mining equipment, the influence of AC signals in the power supply lines is extremely minor compared to the stress waves generated by the drill bit impacting the coal mass and can be neglected. Furthermore, based on studies by Cao Anye, the waveform response during coal mining is related to the underground geological conditions. The impact of the drill bit on the coal mass produces rectangular and sinusoidal response waveforms, with rectangular waves occurring more frequently than sinusoidal waves. Thus, during coal mining, the waveform generated by the drill bit impacting the coal mass is approximated as a rectangular wave [26,27]. Accordingly, rectangular and triangular waves were selected for further investigation.

As illustrated in Figure 3, when external dynamic loading is applied to the coal mass, it undergoes cyclic loading and unloading states. Within a single impact cycle, the weak layer surface initially contacts the shock wave and enters the forward loading phase. At t = T/4, influenced by the unloading wave, it transitions into the forward unloading phase, which continues until t = T/2. During this interval, a closed region is formed above the x-axis by the waveform, and the dominant stress within the coal mass is compressive. Subsequently, the coal mass enters the reverse loading phase, primarily affected by reflected loading waves. After a duration of T/4, under the influence of reflected unloading waves, it proceeds into the reverse unloading phase until the end of the cycle. Throughout this stage, tensile stress becomes the predominant form of stress in the coal mass. While considerable attention has been devoted in current research to the forward loading and unloading stages, the response characteristics of coal during reverse loading and unloading remain relatively underexplored. Therefore, a complete loading–unloading cycle is selected for analysis. By performing Fourier expansion on Equation (13), the infinite series expansion of the rectangular wave is derived as follows:

$$f\left(t\right)~{\displaystyle \frac{4A_{max}}{\pi }}\sum _{n=1}^{\infty }{\displaystyle \frac{\mathit{sin}((2n-1)\omega t)}{2n-1}}\hspace{1em} n=\mathrm{1,3},\mathrm{5,7}\cdots$$

(14)

The infinite expansion series of triangular waves as follows:

$$f\left(x\right)={\displaystyle \frac{8A_{max}}{{\pi }^2{\sum }_{n=1}^{\infty }\frac{\left(-1{)}^{n-1}\mathit{sin}(\left(2n-1\right)wt\right)}{(2n-1{)}^2}}}\hspace{1em}n=\mathrm{1,2},3,\cdots$$

(15)

Within a single impact cycle, the first four terms of Equations (14) and (15) were evaluated and solved. The expanded waveforms for each term, along with the superimposed waveform, are plotted in Figure 4.

As observed, with the number of terms in the Fourier series expansion increases, the corresponding waveform progressively converges toward the infinite-term expansion. For the triangular waveform, the superposition after the second-order expansion closely approximates the waveform resulting from the infinite expansion. In the case of the rectangular wave, the superposition beyond the third-order expansion exhibits high similarity to the infinite-term waveform. Therefore, to simplify computation while ensuring that the applied vibration accurately replicates the original waveform, the external impact disturbance was simulated by superposing the first four terms of the corresponding Fourier expansion.

3. Results

3.1. Analysis of Dynamic Response Characteristics of Soft Coal Under Impact Loading

3.1.1. Dominant Vibration Modes and Frequency Calculation in the Dynamic Response of the Weak Layer

Assume the dimensions of the weak outburst-prone layer model are length (x) × width (y) × height (z) = 4.5 m × 3.5 m × 0.6 m. Using the computational procedure established in Section 2, the vibrational governing equations of the outburst model are solved via Newton’s iteration method [28]. The resulting vibrational mode parameters for the first ten orders of the weak layer are obtained as follows

Table 1. Vibrational Mode Parameters of the First 10 Orders of the Outburst-Prone Weak Layer.

Parameter	1st Order	2nd Order	3rd Order	4th Order	5th Order	6th Order	7th Order	8th Order	9th Order	10th Order
β1	1.250	2.624	2.210	1.063	2.140	2.073	3.125	1.024	3.084	2.023
β2	1.769	1.130	2.487	3.566	3.103	3.908	3.311	4.529	3.775	4.794
α1	0.885	1.674	0.807	2.407	1.589	2.342	0.774	3.119	1.539	3.073
α2	1.977	2.314	3.228	2.838	3.418	3.753	4.486	3.440	4.625	4.199
k	1.531	2.020	2.352	2.631	2.665	3.128	3.219	3.283	3.446	3.679
${\iint }_{\Omega }{W}_m(x,y)dxdy$	20.47	8.3 × 10−12	6.91 × 10−12	11.57	7.79 × 10−7	8.41 × 10−7	16.75	3.51 × 10−9	1.23 × 10−10	9.91 × 10−11

:

Through double integration of the modal amplitude coefficients, it is determined that all coefficients except those of the 1st, 4th, and 7th orders are approximately zero. Hence, the 1st, 4th, and 7th orders are identified as the dominant vibration modes among the first ten orders of the coal plate. Meanwhile, standard geotechnical triaxial tests were conducted on soft coal samples collected underground using the experimental setup shown in Figure 5.

As shown in Figure 5, the triaxial testing system consists of an axial pressure loading device, a confining pressure loading device, a triaxial test chamber, and a control platform. The collected coal samples were machined and assembled. To minimize experimental errors, three groups of coal samples with identical dimensions were prepared for testing, and the average values of the obtained data were used as the parameter results. The dimensions of the machined coal samples were d × h = 50 mm × 100 mm. Following the triaxial tests, the physical properties of the specimens obtained from the control platform are as follows: density ρ = 1300 kg/m³, Poisson’s ratio μ = 0.39, and elastic modulus E = 0.4 GPa. Based on these parameters, the angular frequency of the dominant vibration mode is calculated as follows

Table 2. Natural Frequencies of the Dominant Modes in the Outburst-Prone Weak Layer under Impact Disturbance.

Parameter	1st Order	4th Order	7th Order
Oscillation/Rad·s−1	31.3372	70.2104	118.7548

:

As indicated in Table 2 and the governing vibration equation, the natural frequency of the weak layer increases consistently with the mode order, which aligns well with practical observations. When the natural vibration frequency of the coal mass satisfies nω = ωm (where n = 1, 2, 3, 4, ⋯) following an external impact, abrupt energy transitions may occur in critically stressed regions of the coal mass, thereby increasing the risk of outburst. Consequently, further analysis of the response characteristics of the dominant vibration modes under external impact is essential.

3.1.2. Analysis of Modal Variations in Dominant Modes of an Outburst-Prone Weak Layer Under Impact Disturbance

To accurately analyze the response of soft coal to different impact waveforms and frequencies, in situ mine parameters from a soft coal-containing site in Shanxi, China, were selected for model construction. The overburden pressure on the model was set to 10.12 MPa. Roller supports were applied on both lateral sides of the model to impose constraints. Parameters including the compressive strength, tensile strength, and porosity of the coal were assigned within the model.

Based on the dominant modes calculated in the front Table 1, a modal analysis [29] was performed on the first, fourth, and seventh orders of the original fracture-free coal plate model. Code was developed to extract the eigenfrequencies of the model, and the results were compared with theoretical values. The characteristic frequencies of the intact coal mass are summarized in

Table 3. Eigenfrequency Extraction of the Outburst-Prone Weak Zone Model.

Order	1	4	7
Frequency (Hz/Rad·s−1)	4.98/31.2744	11.15/70.022	18.92/118.8176

:

The modal shapes corresponding to the first, fourth, and seventh orders of the outburst-prone weak zone model are shown in Figure 6.

As shown in Table 3, the simulated natural frequencies of the dominant vibrational modes are in close agreement with the theoretically calculated values, which serves as a validation of the reliability of the theoretical results. Under actual conditions, however, when the coal mass is subjected to external impact, internal forced vibrations may cause a certain degree of damage, such as fracture development. The internal failure characteristics can be investigated through impact tests on multi-layered composite coal plates [30,31].

As shown in Figure 7, after external impact, the weak layer experiences a certain degree of damage due to the influence of the incident stress wave: severe material fragmentation occurs in the central area, while multiple fractures form simultaneously around the periphery. These fractures can be categorized into primary and secondary fractures. The primary fractures further consist of short-edge and long-edge fractures, which run approximately parallel to the boundaries of the coal plate and intersect near the center. Together with the central fragmented zone, they form an “O—+” shaped region. Secondary fractures develop within the areas enclosed by the primary fractures, exhibiting no clearly observable patterns.

To investigate the failure response characteristics of the weak layer under realistic conditions, a circular zone with a diameter of 100 mm was defined at the center of the model to simulate the fragmented area at the impact center. Additionally, rectangular zones with a width of 1 mm and a length of 600 mm were introduced along the east, west, north, and south directions to represent fractures induced by the impact. The model was then resolved again, and the resulting characteristic frequencies of the weakened zone after impact are summarized in

Table 4. Eigenfrequency Extraction of the Outburst-Prone Weak Zone Model After Impact-Induced Fracturing.

Order	1	4	7
Frequency (Hz/Rad·s−1)	4.85/30.45	10.12/63.5536	16.61/104.3108

.

The modal shapes corresponding to the first, fourth, and seventh orders of the outburst-prone weak zone model are shown in Figure 8.

Comparison of the pre- and post-modification model results indicates that as the analysis order increases, the modal density gradually increases and the regions become continuously more connected, with an annular connected region emerging at the seventh order. When the model incorporates fractured zones and cracks, the regional morphology changes significantly compared to the original model, primarily manifested as a redistribution of modes near the fractured and cracked areas: At the first order, no significant change is observed in the region except for slight bending at the bottom boundary. The change in the modal density region before and after modification is approximately 5% based on image comparison, and the corresponding characteristic frequency decreases by 2.6%. At the fourth order, noticeable changes occur concentrated at the crack regions, characterized by interruptions along the longer edges of the fractured zones. Image comparison shows a change in the modal density region of about 25%, and the corresponding characteristic frequency decreases by 9.23%. At the seventh order, the overall modal pattern is reconfigured around the central area, forming an approximately inverted triangular region with side lengths of about 700 mm and distinct protrusions at the midpoints of each side. Comparison of the pre- and post-modification image regions reveals a change in the modal density region of up to 85%, and the corresponding characteristic frequency decreases by 12.21%.

Analysis of the characteristic frequencies before and after model modification shows that, compared to the original characteristic frequencies, the natural frequencies of the coal plate containing fractured zones are lower, and the reduction in characteristic frequency increases with the order, reaching a decrease of 2.31 Hz at the seventh order. In summary, under the dominant vibration orders, the modal characteristics of the softened coal plate with fractures differ significantly from those of the original intact coal plate. The characteristic frequency decreases progressively with increasing order, with a maximum reduction of 12.21%, and the rate of change in the modal density region gradually increases with the analysis order. These findings indicate that under dominant vibration modes, the equilibrium state of softened coal plates is prone to alteration under impact disturbances, which may subsequently trigger outburst incidents. Therefore, further analysis of the response of coal plates under various impact disturbances is essential.

3.1.3. Analysis of Response Frequency in Soft Coal Under Impact Disturbance

To further investigate the response of soft coal to external impacts, it is essential to analyze the frequency response characteristics of the coal mass under such disturbances. After applying an external impact to the soft coal plate, the displacement response across different frequencies was examined using inverse Fast Fourier Transform (FFT) analysis, enabling the evaluation of potential outburst-prone conditions. Under a single external impact, the mean displacement response of the coal plate within the 0–20 Hz range was compiled, resulting in the response frequency versus coal displacement relationship shown in Figure 9.

Analysis of the relationship between displacement response and frequency reveals that after applying a single external impact, the mean displacement amplitude of the coal plate increases gradually with frequency, reaching a maximum of 0.967 mm at 4.5 Hz. Subsequently, the displacement exhibits a sharp decline followed by a rapid rebound, recovering to 0.918 mm at 4.9 Hz. It then decreases rapidly, pausing briefly around 6.5 Hz before dropping abruptly below 0.1 mm. Beyond this point, minor fluctuations are observed near 9.6 Hz and 10.2 Hz, though the displacement remains under 0.2 mm, after which it continues to decline and eventually approaches zero.

Analysis across the entire frequency range indicates that the coal mass exhibits the strongest response when the excitation frequency falls within 4.4–5.2 Hz, referred to as the strong response interval, while the displacement response remains relatively small at other frequencies, termed the weak response interval. Within the strong response interval, the response amplitude of the coal mass is significantly higher, with the maximum response being 8–9 times greater than that in the weak response interval. During actual underground mining operations, prolonged low-frequency vibration is not only a major factor triggering coal and gas outbursts, but also serves as a critical indicator for predicting such disasters [32]. Specifically, when the vibration frequency is close to the natural frequency of the coal mass, the probability of failure increases. Research by Li Chengwu et al. on the variation characteristics of natural frequency during vibration-induced failure shows that under excitation at 5 Hz, the coal mass undergoes intense response, ejecting a large amount of coal particles [15]. In actual mine environments, field investigations conducted by Zhang Lifen et al. revealed that coal-rock masses generate substantial energy release when subjected to vibrations around 5 Hz, which corresponds to a high risk of coal and gas outbursts [33]. Therefore, within the strong response frequency interval, the equilibrium state of weak layers is easily disrupted by external disturbances. When combined with factors such as in situ stress, the coal is prone to fragmentation, releasing substantial energy within a short period and potentially leading to coal and gas outbursts. During operations, it is essential to monitor the vibration frequency of the coal and rock mass in the working area. If the natural frequency remains within the low-frequency range for an extended period, operations should be suspended. Measures such as regional pressure relief should be implemented before resuming work.

Additionally, the maximum displacement response of the coal mass and its corresponding frequency within both the strong and weak response intervals were selected. Inverse FFT was applied to these values to plot the time-history response diagrams of the coal mass at the two frequencies [34], enabling further analysis of the response characteristics of the soft coal mass under the corresponding frequencies.

As shown in Figure 10, the response frequencies of 4.5 Hz in the strong response interval and 9.6 Hz in the weak response interval were selected to plot the time-history curves of the maximum displacement response of the coal plate under the corresponding frequencies. The results indicate that the soft coal mass exhibits significantly different response levels at different frequencies: under 4.5 Hz, the maximum displacement response reaches 1.29 mm, while at 9.6 Hz, it is only 0.14 mm—a difference of 9.21 times. Consistent with the conclusions in Section 1, when the excitation frequency approaches the characteristic frequency of the soft coal mass, particularly the first-order dominant natural frequency, the displacement response amplifies markedly. When the natural vibration frequency of the coal mass reaches such critical values after impact, the risk of coal and gas outbursts increases substantially. Therefore, in practical mining operations, enhanced monitoring of low-frequency vibrations is essential. Additionally, to analyze the response characteristics of the soft coal plate at varying distances from the impact center, measuring points were arranged along the fracture periphery to collect and evaluate displacement response data.

As shown in Figure 11, the maximum response frequency was applied to the central fractured zone of the outburst-prone weak layer. Displacement measurement points were established at distances of 75 mm, 150 mm, 300 mm, 450 mm, and 600 mm from the impact center along a single fracture to investigate the response of the coal mass at varying distances from the fragmentation center under this specific frequency.

As shown in Figure 12, under the applied frequency, the displacement responses along fractures in the soft coal mass vary significantly with distance from the central fractured zone. The peak response occurs at 75 mm from the center, reaching a maximum value of 1.293 mm, while the smallest peak response appears at the farthest point (600 mm), with a value of 1.074 mm. The maximum values in the peak regions of all measurement points were extracted to plot the peak displacement versus distance relationship. The results show that the peak displacement at the farthest point (Point 5) decreases by 16.9% compared to that at the nearest point (Point 1). This indicates that under external impact vibration, the maximum displacement response diminishes with increasing distance from the central fractured zone. Areas closer to the center are associated with higher outburst risks, highlighting the need for timely pressure relief measures in these regions to prevent potential hazards.

3.1.4. Response of the Outburst-Prone Weak Zone to Different Impact Waveforms

In addition to variations in displacement response due to changes in coal vibration frequency, the response of the outburst-prone weak zone in soft coal may also be influenced by alterations in the waveform of external impacts. It should be noted that inherent parameters such as the natural frequency of the coal mass depend solely on its geometry, dimensions, and boundary conditions, and are independent of external impact vibrations. Based on the conclusions drawn in the previous section, this section selects the frequency corresponding to the maximum coal response identified earlier as the external impact frequency to analyze the response characteristics of soft coal under different impact waveforms.

As established in Section 2, coal and gas outbursts in underground mining are primarily triggered by roadway driving and blasting operations, which generate rectangular and triangular waves, respectively. The Fourier series expansions for these two waveforms have been derived in Section 2. Therefore, this section employs signal simulation to analyze the vibrational response of the coal mass. Prior to conducting vibration simulations, the reliability of the method was verified by comparing analytical and computed values of the Fourier coefficients over the frequency range of 0–2300 Hz, as illustrated in Figure 13.

According to the validation results, the analytical and computed values of the Fourier transforms for both rectangular and triangular waves exhibit a high degree of agreement within the common frequency range. It can therefore be approximated that their excitatory effects on the coal mass are identical. Based on this, simulated waveforms from the software were used to apply impact loads to the soft coal plate. In addition to the triangular and rectangular waves, a harmonic wave was also included as a reference for comparison. After processing the three waveform types with a built-in FFT algorithm, they were applied to the outburst-prone weak zone model at the maximum response frequency. The resulting displacement responses of the model under these three waveforms were recorded. For comparative purposes, the negative values on the x-axis were converted using absolute value transformation, as shown in Figure 14.

As shown above, under consistent boundary conditions and at the same maximum response frequency, the soft coal mass exhibits distinct displacement responses to different impact waveforms. The largest displacement response occurs under rectangular wave excitation, reaching 1.66 mm, followed by the sinusoidal wave with 1.29 mm, while the triangular wave yields the smallest response of 1.08 mm. In terms of peak displacement response, the soft coal mass is most responsive to the rectangular wave and least to the triangular wave, with a peak ratio of 1.53. Furthermore, the response intensity of the soft coal under various waveform impacts was analyzed from an energy release perspective [35,36]. Under impact vibrations with the same peak value and frequency, the energy input to the coal plate system by different waveforms f(t) over a half-cycle T/2 is given by:

$$I=\int f\left(t\right)dt$$

(16)

Based on Equation (16), the energy inputs of the three waveforms were compared over an identical period, with the results shown in Figure 15.

As illustrated in Figure 15, the energy inputs of the three waveforms into the system were compared over a study period of π/2. The rectangular wave exhibits the highest energy input, with a total value of π/2 during this interval; the sinusoidal wave ranks second, with a total input of 1; and the triangular wave shows the lowest input, amounting to π/4. The energy input of the rectangular wave is 1.5 times that of the triangular wave, which aligns almost exactly with the ratio of their maximum displacement responses mentioned earlier. Therefore, it can be concluded that under impact loading at the same frequency, the soft coal mass responds differently to various waveforms, with the response magnitude ranking as follows: rectangular wave > sinusoidal wave > triangular wave. Consequently, under identical disturbance amplitudes, the rectangular wave contributes the highest energy input to the system, making it more likely to cause the weak zone to accumulate energy beyond the critical threshold, thereby increasing the risk of coal and gas outbursts.

3.2. Numerical Simulation of the Dynamic Response of Friable Coal Under Impact Loading

3.2.1. Design of a Similitude Simulation Test Rig for Impact Response in Soft Coal

To further validate the conclusions presented above, this section designs a similarity simulation response experiment for soft coal masses. The experiment employs a three-dimensional similarity simulation test platform. The material ratios and overall construction of the model are based on the principles of similarity simulation testing [37], which primarily include geometric similarity, material similarity, and kinematic similarity. The determination of the similarity ratio is based on the distance from the weakest layer to the working face, with a selected similarity constant α of 15. The similarity materials are proportioned according to mechanical similarity properties, as detailed in

Table 5. Mixing ratio of similar materials for soft media.

Material	Sand	Lime	Gypsum	Coal Powder	Water
Unit mass content/kg	0.47	0.06	0.13	0.14	0.19

.

The dimensions of the similarity simulation model were configured as 1800 mm × 160 mm × 1100 mm (length × width × height). Black pigment was added to the similarity material in the weak layer zone to visually emphasize the study area. External impact was simulated via single-point excitation at the top surface using a rectangular wave. Electromagnetic acceleration sensors and displacement sensors were embedded to collect data signals after impact [31]. After completion of the model construction, experiments were conducted and data were collected. The similarity model is shown in Figure 16:

3.2.2. Analysis of Dynamic Response in Similitude Coal Under External Impact

Upon completion of the experiment, the data acquired from the sensors were compiled and analyzed. As shown in Figure 17a, following the impact disturbance, the entire process can be divided into two distinct phases based on the response intensity. The first phase involves forced vibration of the coal mass induced by the transient impact, resulting in a sharply downward velocity on the upper surface of the coal mass, reaching a maximum value of −14.9 mm/s within an extremely short duration, followed by a rapid recovery. This leads to the second phase, identified as the vibration recovery stage, during which the coal mass gradually returns to its pre-impact state. Influenced by the accumulated elastic potential energy, the velocity initially rebounds to 9.5 mm/s, followed by a short-term fluctuation with a second peak of 8.5 mm/s. These variations correspond to the loading and unloading stages of the coal mass under impact. Subsequently, as time progresses, the velocity peaks fluctuate with a decreasing trend, continuously approaching zero.

Simultaneously, the instantaneous displacement data obtained from displacement sensors were compiled and analyzed to evaluate the displacement response. As illustrated in Figure 17b, following external impact disturbance, the displacement response exhibits an approximately sinusoidal variation, with the corresponding oscillation period and peak magnitude first increasing and then decreasing. Immediately upon impact, the displacement of the coal mass measures −0.003 mm. Over time, the displacement shifts in the positive direction, reaching an extreme value of 0.012 mm at the third peak, after which it oscillates periodically with gradually decaying amplitude toward zero.

Some discrepancies are observed between the similitude experimental results and the numerical simulations, which may be attributed to instrumental limitations of the sensors and objective environmental errors in the test setup. Therefore, further analysis was conducted on the relationship between the instantaneous frequency and the maximum response degree to validate the response characteristics under varying frequencies. The results are presented in Figure 18.

As shown above, the velocity–frequency and displacement–frequency response diagrams are plotted. Analysis of Figure 18a indicates that after impact, the coal mass exhibits the strongest velocity response within the 4–6 Hz range, identified as the strong response region. The peak velocity response of the weak-layer coal mass reaches 9.5 mm/s at 4.8 Hz. Beyond this frequency, the response decreases sharply, with brief upward shifts observed around 10 Hz and 18 Hz, reaching approximately 4 mm/s. Comparative analysis of Figure 18b reveals that the displacement response follows a trend similar to the velocity response in the initial stage, increasing rapidly and reaching a maximum of 0.012 mm at 4.9 Hz. This is followed by rapid fluctuations, with a second peak occurring near 6 Hz at approximately 0.01 mm. Subsequently, the response declines sharply, exhibits minor fluctuations around 7.5 Hz, and stabilizes thereafter until the end of data acquisition. Comparison of the experimentally acquired and theoretically simulated images reveals that, although certain discrepancies exist—which are attributed to material heterogeneity during model construction and unavoidable voids introduced during sensor embedding—both sets of results exhibit similar trends. This consistency supports the theoretical conclusions and further validates the findings of this study.

4. Discussion and Conclusions

To address the inability of traditional mechanics to characterize the dynamic response of coal in outburst-prone zones, a theoretical mechanics model based on Hamiltonian formulation was constructed. By introducing conjugate momentum, the matrix equations were canonically transformed to reduce the order of high-order differential equations, thereby improving solution accuracy, reducing computational difficulty, and decreasing the required computational effort. This provides a new theoretical foundation for studying the dynamic response of coal under impact. Based on Hamiltonian theory, the governing equations of the weak zone model under external impact vibration were derived. Using measured values from coal samples, the unknown parameters in the equations were solved. The results indicate that the dominant vibration modes of a single coal plate are the first, fourth, and seventh orders, with the first order having the most significant influence.

Rectangular and triangular waves were decomposed via FFT, and the external impact on the model was simulated by superposing the first four terms of the corresponding waveforms. Modal analysis was performed on both the intact model and the post-impact fractured model. The results show that the modal patterns of the impacted model changed significantly near the central region compared to the original model. The natural frequencies of the dominant modes decreased after impact, with the reduction magnitude increasing progressively with higher mode orders. Analysis of the displacement response of the soft coal under impact revealed a strong response within the frequency range of 4.4–5.2 Hz, with the peak displacement response occurring at 4.5 Hz. In other frequency ranges, the displacement response was weaker, with the average amplitude in the strong response zone being 8–9 times greater than that in the weak response zones. Additionally, the displacement response was more pronounced closer to the impact center. Under the optimal response frequency, the soft coal’s response to different impact waveforms was analyzed, yielding the following order of response intensity: rectangular wave > sinusoidal wave > triangular wave. By designing a similarity simulation experiment on the impact response of soft coal mass, it was found that under experimental conditions, the maximum response velocity of the coal mass was 9.5 mm/s and the maximum displacement response was 0.012 mm. The trends of velocity and displacement responses were consistent with theoretical analysis, further supporting the conclusions.

This study is based on soft coal mass with simple geology and homogeneous distribution, incorporating certain idealized assumptions. The model may not be fully applicable to complex geological conditions or heterogeneously distributed coal masses, and further analysis by other researchers is welcomed. Due to experimental limitations, the values may differ from field measurements in underground mines, but the response characteristics align with observations. Therefore, in practical production, it is necessary to enhance monitoring of low-frequency vibrations in coal masses, minimize rectangular wave-induced disturbances in the low-frequency range, and implement timely pressure relief operations in areas with prolonged low-frequency vibrations to reduce the risk of coal and gas outbursts in hazardous zones. In summary, investigating the dynamic response characteristics of soft coal masses under impact loading holds significant importance for preventing coal and gas outburst accidents in soft coal seams.