Analytic calculation method and system for inherent frequency of dumping type slope dangerous rock mass

Abstract

The invention provides a dumping type slope dangerous rock body inherent frequency analytical calculation method and system, and relates to the technical field of geotechnical engineering and geological disaster prevention and control. The method comprises the following steps: acquiring basic parameters of the toppling slope dangerous rock mass; according to the basic parameters, the safety coefficients of the gravity center of the toppling type slope dangerous rock body inside and outside the toppling point are determined; based on a modified Timoshenko beam theory, constructing a beam model; according to the beam model, analytic calculation is conducted on the toppling type side slope dangerous rock mass, and the inherent frequency of the toppling type side slope dangerous rock mass is determined; constructing a dynamic model of the toppling type slope dangerous rock mass; determining the relationship between the inherent frequency and the relative depth of the crack according to the dumping type slope dangerous rock mass dynamic model; according to the relation between the safety coefficient and the inherent frequency and the relative depth of the crack, a dumping type slope dangerous rock mass stability evaluation model is constructed; and carrying out stability analysis on the dumping type slope dangerous rock mass through the dumping type slope dangerous rock mass stability evaluation model.

Description

Technical Field

[0001] The present invention relates to the technical field of geotechnical engineering and geological disaster prevention and control, and in particular to a method and system for analyzing and calculating the natural frequency of a dangerous rock mass on a dumping slope. Background Art

[0002] With rapid economic development and urbanization, slope engineering, as a crucial component of infrastructure development, is attracting increasing attention for its stability. Particularly in areas with complex geological conditions, such as mountainous and hilly areas, the collapse of dangerous rock masses on slopes can often cause severe economic losses and casualties. Therefore, studying the stability of dangerous rock masses on slopes, particularly their natural frequencies and dynamic responses, has become a crucial issue in ensuring slope safety.

[0003] Existing research on the stability of precarious rock masses in toppling slopes relies heavily on traditional limit equilibrium methods and numerical simulations. While these methods can assess the stability of precarious rock masses, they often fail to fully consider the dynamic properties of the rock mass, particularly the impact of crack depth on its vibration characteristics.

[0004] In addition, traditional stability evaluation methods mainly rely on static analysis and fail to effectively combine the dynamic response of the rock mass, resulting in insufficient early warning capabilities and response sensitivity in practical applications. Summary of the Invention

[0005] In order to solve the technical problems that the research on the stability of dangerous rock masses on toppling slopes in the existing technology mostly relies on the traditional limit equilibrium method and numerical simulation method, which cannot fully take into account the dynamic characteristics of the rock mass, and the traditional stability evaluation method mainly relies on static analysis and fails to effectively combine the dynamic response of the rock mass, thereby resulting in insufficient early warning capability and response sensitivity in practical applications, the present invention provides a method and system for analyzing the natural frequency of dangerous rock masses on toppling slopes.

[0006] The technical solutions provided by the embodiments of the present invention are as follows:

[0007] First aspect:

[0008] An embodiment of the present invention provides a method for analytically calculating the natural frequency of a dangerous rock mass on a toppling slope, comprising:

[0009] S1: Obtain the basic parameters of the dangerous rock mass of the dumping slope;

[0010] S2: Determine the safety factor of the tipping slope dangerous rock mass when the center of gravity is inside the tipping point and outside the tipping point according to the basic parameters;

[0011] S3: Based on the modified Timoshenko beam theory, the tipping slope dangerous rock mass is simplified and a beam model is constructed;

[0012] S4: performing analytical calculation on the dangerous rock mass of the tilting slope according to the beam model to determine the natural frequency of the dangerous rock mass of the tilting slope;

[0013] S5: Construct a dynamic model of the dangerous rock mass on the toppling slope;

[0014] S6: determining the relationship between the natural frequency of the tilting slope dangerous rock mass and the relative depth of the cracks according to the dynamic model of the tilting slope dangerous rock mass;

[0015] S7: constructing a stability evaluation model for the tipping slope dangerous rock mass according to the safety factors of the center of gravity of the tipping slope dangerous rock mass within the tipping point and outside the tipping point and the relationship between the natural frequency of the tipping slope dangerous rock mass and the relative depth of the cracks;

[0016] S8: Performing stability analysis on the tilting type slope dangerous rock mass using the tilting type slope dangerous rock mass stability evaluation model.

[0017] Second aspect:

[0018] An embodiment of the present invention provides a system for analyzing and calculating the natural frequency of a dangerous rock mass on a toppling slope, comprising:

[0019] processor;

[0020] A memory having computer-readable instructions stored thereon, wherein when the computer-readable instructions are executed by the processor, the method for analytical calculation of the natural frequency of the dangerous rock mass of the dumping slope as described in the first aspect is implemented.

[0021] The third aspect:

[0022] An embodiment of the present invention provides a computer-readable storage medium having a computer program stored thereon. When the program is executed by a processor, the method for analytically calculating the natural frequency of a dangerous rock mass on a dumping slope as described in the first aspect is implemented.

[0023] The beneficial effects brought about by the technical solution provided by the embodiment of the present invention include at least:

[0024] In an embodiment of the present invention, by simplifying the dangerous rock mass of the tipping slope based on the modified Timoshenko beam theory, constructing a beam model, and performing analytical calculations on the dangerous rock mass of the tipping slope according to the beam model, the natural frequency of the dangerous rock mass of the tipping slope is determined, thereby reducing the dependence on the traditional limit equilibrium method and numerical simulation method, and being able to fully consider the dynamic characteristics of the rock mass. By constructing a stability evaluation model for the dangerous rock mass of the tipping slope, and using the stability evaluation model for the dangerous rock mass of the tipping slope, the stability analysis of the dangerous rock mass of the tipping slope is performed, thereby avoiding the dependence on static analysis in the traditional stability evaluation method, being able to effectively combine the dynamic response of the rock mass, and improving the early warning capability and response sensitivity in practical applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0026] Figure 1 A schematic flow chart of a method for analytically calculating the natural frequency of a dangerous rock mass on a toppling slope provided by an embodiment of the present invention;

[0027] Figure 2 A schematic diagram of a semi-infinite space foundation and base provided by an embodiment of the present invention;

[0028] Figure 3 A schematic diagram of overall shear deformation of a foundation part provided by an embodiment of the present invention;

[0029] Figure 4 A schematic diagram of local deformation of a base part provided by an embodiment of the present invention;

[0030] Figure 5 A schematic diagram of calculating the natural frequency of a dangerous rock mass on a toppling slope provided by an embodiment of the present invention;

[0031] Figure 6 A schematic structural diagram of a dynamic model provided by an embodiment of the present invention;

[0032] Figure 7 A schematic structural diagram of a system for analyzing and calculating the natural frequency of a dangerous rock mass on a toppling slope provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0033] The technical solution of the present invention is described below in conjunction with the accompanying drawings.

[0034] In the embodiments of the present invention, words such as "exemplarily" and "for example" are used to indicate examples, illustrations, or explanations. Any embodiment or design described as an "exemplary" in the present invention should not be interpreted as being preferred or advantageous over other embodiments or designs. Rather, the use of the word "exemplary" is intended to present concepts in a concrete manner. Furthermore, in the embodiments of the present invention, "and / or" can mean both or either of the two.

[0035] In the embodiments of the present invention, the terms "image" and "picture" may be used interchangeably. It should be noted that, when the distinction between them is not emphasized, their intended meanings are the same. The terms "of," "corresponding," and "corresponding" may be used interchangeably. It should be noted that, when the distinction between them is not emphasized, their intended meanings are the same.

[0036] In the embodiment of the present invention, sometimes a subscript such as W1 may be written as a non-subscript form such as W1. When the difference is not emphasized, the meanings to be expressed are the same.

[0037] In order to make the technical problems, technical solutions and advantages to be solved by the present invention clearer, a detailed description will be given below with reference to the accompanying drawings and specific embodiments.

[0038] Reference Manual Figure 1 , which shows a flow chart of a method for analytically calculating the natural frequency of a dangerous rock mass on a dumping slope provided by an embodiment of the present invention.

[0039] An embodiment of the present invention provides a method for analytically calculating the natural frequency of a dangerous rock mass on a toppling slope. The method can be implemented by a device for analytically calculating the natural frequency of a dangerous rock mass on a toppling slope. The device can be a terminal or a server. The process flow of the method for analytically calculating the natural frequency of a dangerous rock mass on a toppling slope can include the following steps:

[0040] S1: Obtain the basic parameters of the dangerous rock mass of the dumping slope.

[0041] Optionally, the basic parameters include: height of the dangerous rock mass of the dumping slope, depth of the rear edge crack, horizontal distance of the center of gravity, height of the center of gravity, tensile strength, bonding area, vibration load, relative depth of cracks, inclination angle of the rear edge crack and deadweight.

[0042] S2: Based on the basic parameters, determine the safety factor when the center of gravity of the dangerous rock mass of the overturning slope is inside and outside the overturning point respectively.

[0043] In this invention, by calculating the safety factor inside and outside the tipping point, potential risk areas can be identified during the design phase, allowing necessary protective measures to be taken in advance. This helps avoid engineering accidents and improves the safety and reliability of the structure.

[0044] In a possible implementation, the safety factor of the center of gravity of the dangerous rock mass of the overturning slope within the overturning point is specifically:

[0045]

[0046] Among them, F1 represents the safety factor of the center of gravity of the dangerous rock mass of the overturning slope within the overturning point, It represents the anti-tilting moment when the gravity center of the dangerous rock mass of the overturning slope is within the overturning point. represents the overturning moment of the center of gravity of the tipping slope dangerous rock mass within the overturning point, W represents the deadweight of the tipping slope dangerous rock mass, a represents the horizontal distance of the center of gravity of the tipping slope dangerous rock mass, f represents the tensile strength of the tipping slope dangerous rock mass, S represents the bonding area, H represents the height of the tipping slope dangerous rock mass, h represents the depth of the trailing edge crack, sin represents the sine function, β represents the inclination angle of the trailing edge crack, P represents the vibration load, h0 represents the height of the center of gravity of the tipping slope dangerous rock mass, and λ represents the relative depth of the crack.

[0047] The safety factor of the dangerous rock mass of the overturning slope with its center of gravity outside the overturning point is:

[0048]

[0049] Among them, F2 represents the safety factor when the center of gravity of the dangerous rock mass of the overturning slope is outside the overturning point. It represents the anti-tilting moment when the gravity center of the dangerous rock mass of the overturning slope is outside the overturning point. It represents the overturning moment when the center of gravity of the dangerous rock mass of the overturning slope is outside the overturning point.

[0050] In this invention, by calculating the safety factor for locations inside and outside the tipping point, a comprehensive assessment of the stability of a dangerous rock mass can be performed. The change in the center of gravity's position is crucial to the rock mass's ability to resist tipping over. Therefore, calculating the safety factor for locations inside and outside the tipping point can help identify the rock mass's stability under different conditions. The introduction of vibration loads can simulate the rock mass's stability under actual vibration conditions. This allows for a reasonable prediction of how an actual slope will perform under earthquakes, construction, or other external dynamic loads.

[0051] S3: Based on the modified Timoshenko beam theory, the dangerous rock mass of the dumping slope is simplified and a beam model is constructed.

[0052] It should be noted that the modified Timoshenko beam theory is a beam bending analysis method that improves on the classic Euler-Bernoulli beam theory by simultaneously considering both bending and shear deformations. It is particularly suitable for shorter beams or beams with high shear stiffness. Unlike traditional beam theory, the modified Timoshenko beam theory incorporates the effects of shear deformation, enabling more accurate predictions of the bending response of beams under load. The theory also incorporates a correction for the moment of inertia, making the analysis of shorter or thicker beams more precise and accounting for the rotation and deformation caused by shear forces. This makes it particularly suitable for high-frequency vibrations or beams with large cross-sections.

[0053] In this study, a modified Timoshenko beam theory is used to simplify the rock mass of a toppling slope and construct a beam model. This improves analytical accuracy, considers more deformation factors (such as shear deformation), and is particularly suitable for high-frequency vibration conditions, providing a more precise theoretical basis for stability analysis of the rock mass. This approach enhances the model's applicability and reliability, optimizes safety assessments, and helps better predict and prevent collapse risks.

[0054] S4: Based on the beam model, perform analytical calculations on the dangerous rock mass of the toppling slope to determine the natural frequency of the dangerous rock mass of the toppling slope.

[0055] By constructing a suitable beam model, this method allows complex, dangerous rock masses on toppling slopes to be analyzed using a simplified structural model. This not only reduces computational complexity but also enables rapid stability analysis while ensuring accuracy. By analytically calculating the natural frequency, the vibration characteristics of the rock mass can be accurately predicted, thereby determining whether the rock mass is at risk of instability.

[0056] In a possible implementation, S4 specifically includes sub-steps S401 to S406:

[0057] S401: Determine the differential equation of the beam based on the beam model.

[0058] Optionally, determine the differential equation for the beam according to the following formula:

[0059]

[0060] Where S' represents the cross-sectional area of ​​the beam, G represents the shear modulus of the material, and μ represents the coefficient related to shear stiffness and deformation. represents the partial differential operator, y represents the vertical position of the beam, x represents the horizontal position of the beam, represents the bending angle of the beam, E represents the elastic modulus of the material, I represents the moment of inertia of the beam, ρ represents the density of the beam material, t represents the time, and q represents the external force distributed per unit length of the beam.

[0061] In this paper, by establishing differential equations for beams, complex physical problems can be transformed into mathematical form, making calculations more systematic and simplified. These differential equations take into account physical properties such as the beam's cross-sectional area, shear modulus, and elastic modulus. They accurately describe the beam's deformation behavior under load, providing a foundation for subsequent vibration analysis.

[0062] S402: Determine the vibration mode function of the beam based on the differential equation of the beam through the elimination method and the separation of variables method.

[0063] It's important to note that elimination is a common mathematical technique used to solve systems of linear or differential equations. It involves performing a series of transformations on the equations, eliminating one or more unknowns, thereby simplifying the problem into a lower-dimensional equation and ultimately finding a solution to the unknowns. During the solution process, elimination gradually eliminates variables, transforming the original problem into a more manageable form, often resulting in a more direct solution.

[0064] It's important to note that the separation of variables method is a commonly used approach for solving differential equations, particularly those involving separable variables. The basic idea is to separate the different variables in the equation through variable transformation, so that each variable appears on a separate side. This transforms the originally complex differential equation into two simple integration problems. By integrating these two parts, the solution to the equation is ultimately obtained. The separation of variables method is very effective in solving problems in fields such as physics and engineering, especially equations involving phenomena such as heat conduction and wave motion.

[0065] In the present invention, the elimination method and separation of variables method are used to solve the differential equation of the beam, which can decompose the originally complex multi-dimensional problem into a one-dimensional problem, thereby greatly simplifying the calculation process.

[0066] In a possible implementation, S402 specifically includes sub-steps S4021 to S4023:

[0067] S4021: Simplify the differential equations of a beam by elimination.

[0068] Optionally, simplify the differential equation for the beam according to the following formula:

[0069]

[0070] In this invention, the elimination method effectively simplifies the differential equation system by eliminating variables, transforming the originally complex multidimensional problem into a low-dimensional problem that is easy to solve. By eliminating variables, it is possible to focus on the key variables and avoid unnecessary complexity in the solution process.

[0071] S4022: Solve the free vibration characteristics of the beam through the separation of variables method and determine the vibration solution of the beam.

[0072] Alternatively, let q(x, t) = 0 and solve the free vibration characteristics of the beam using the separation of variables method to determine the vibration solution of the beam:

[0073] y(x,t)=Y(x)T(t)

[0074] Where q(x,t) represents the external force distributed on the unit-length beam corresponding to the horizontal position x at time t, y(x,t) represents the vertical position of the beam corresponding to the horizontal position x at time t, Y() represents the vibration mode function of the beam, and T() represents the time function of the beam.

[0075] In this paper, the separation of variables method is used to separate the originally coupled variables, significantly reducing the amount of computation required. Each variable is processed independently, making the calculation process more efficient. This is particularly important for solving the vibration characteristics of beams, as it allows accurate vibration solutions to be obtained in a relatively short time.

[0076] S4023: Determine the vibration mode function of the beam based on the simplified differential equation of the beam and the vibration solution of the beam.

[0077] Optionally, determine the mode shape function of the beam according to the following formula:

[0078]

[0079] It should be noted that S4023 specifically includes:

[0080] Combining the simplified differential equation of the beam and the vibration solution of the beam, we get:

[0081]

[0082] where ζ1 represents a constant related to material and geometric properties, and d represents a differential operator.

[0083] Arranged:

[0084]

[0085] Expanding Y(x) and T(t) yields:

[0086]

[0087] Where the characteristic equation is:

[0088] ζ1 2 r 4 +ω 2 ζ2 2 r 2 -ω=0

[0089] Here, r represents the eigenvalue associated with the vibration mode of the beam.

[0090] Solving for r yields:

[0091]

[0092] Among them, r 1,2 Represents parameters r1 and r2 related to foundation geometry and foundation properties, r 3,4 Indicates parameters r3 and r4 related to the imaginary part of the vibration mode.

[0093] Solve the general solution of the vibration equation:

[0094]

[0095] Where b is a constant. Alternatively, the general solution can be written as:

[0096]

[0097] The vibration mode function of the beam is specifically:

[0098] Y(x)=c1 coshg1x+c2 sinhg2x+c3 cosg2x+c4sinhg2x

[0099] Among them, g1=r1, g2=-ir3.

[0100] In the present invention, the mode function of the beam determines the vibration mode of the beam at different frequencies. Accurately solving the mode function can help engineers understand the dynamic behavior of the beam. The general solution of the vibration equation obtained by solving the characteristic equation can clarify the response of the beam under different boundary conditions and external forces. This provides theoretical support for the subsequent analysis of the impact of the movement and vibration of the beam on the stability of the dangerous rock mass, which is helpful for engineering design and safety assessment. By solving the eigenvalues ​​related to the geometric characteristics and foundation properties of the foundation, as well as the eigenvalues ​​related to the imaginary part of the vibration mode, the impact of the physical properties of the foundation and the foundation on the vibration characteristics of the beam can be fully considered.

[0101] S403: Determine the bending angle and deformation angle of the beam according to the mode shape function of the beam.

[0102] Optionally, the bending angle and deformation angle of the beam can be determined according to the following formula:

[0103]

[0104] in, represents the bending rotation angle of the beam corresponding to the horizontal position x at time t, β(x,t) represents the deformation rotation angle of the beam corresponding to the horizontal position x at time t, Φ() represents the bending rotation function of the beam, Β() represents the deformation rotation function of the beam, represents the phase angle, sinh() represents the hyperbolic sine function, cosh() represents the hyperbolic cosine function, sin() represents the sine function, cos() represents the cosine function, ω represents the natural frequency of the dangerous rock mass of the dumping slope, c1, c2, c3 and c4 represent unknown coefficients, g1 and g2 represent parameters related to the natural frequency, ρ represents the density of the beam material, ζ2 represents the parameter related to the shear stiffness, density and natural frequency, and ζ3 represents the coefficient related to the moment of inertia of the beam and the cross-sectional area of ​​the beam.

[0105] In the present invention, by obtaining the bending angle and deformation angle of the beam, the deformation of the beam under vibration and external force can be specifically understood, and then its stability and stress distribution can be analyzed.

[0106] Further, let The bending angle and deformation angle of the above beam are solved, and the expressions of the bending angle and deformation angle of the beam after solution are obtained:

[0107] Φ(x)=c1(g1+g3)sinhg1x+c2(g1+g3)coshg1x-c3(g2-g4)sing2x+c4(g2-g4)cosg2x

[0108] Β(x)=-c1g3sinhg1x-c2g3coshg1x-c3g4sing2x+c4g4cosg2x

[0109] Here, g3 and g4 both represent parameters related to natural frequency, i represents an imaginary unit, and r3 represents a parameter related to the imaginary part of the vibration mode.

[0110] By further calculating the bending and deformation angles of beams and introducing parameters related to natural frequencies and vibration modes, this method can more accurately describe the deformation behavior of beams under complex loads, improving the accuracy of dynamic response analysis and providing a more reliable theoretical basis for stability assessment of dangerous rock masses in toppling slopes. This refined analysis contributes to greater safety and accuracy in practical engineering applications, particularly under vibrating loads or complex foundation conditions.

[0111] Reference Manual Figure 2 , showing a schematic diagram of a semi-infinite space foundation and base provided by an embodiment of the present invention.

[0112] S404: Based on the semi-infinite space foundation theory, determine the anti-sway stiffness and anti-lateral displacement stiffness provided by the foundation to the entire foundation.

[0113] It should be noted that the semi-infinite foundation theory is a theory used to analyze the interaction between soil and foundation. It is particularly suitable for describing the behavior of foundation layers that are physically semi-infinite, meaning that the foundation depth is large enough to ignore boundary effects. The theory assumes that when the foundation surface contacts the foundation, the load applied by the foundation is isotropically propagated through the foundation, and that the underlying soil or rock layer effectively distributes and bears the load. Unlike finite soil layer models, the semi-infinite foundation theory does not consider boundary effects at the foundation base, focusing primarily on the response of the upper foundation. It is applicable to foundation problems where the bottom depth is not a constraint. By modeling the elastic properties of the soil layer, it calculates how the load applied by the foundation propagates through the foundation and analyzes the foundation's settlement, deformation, and stiffness. The theory is widely used in foundation analysis for buildings, bridges, and other structures, and is particularly valuable in analyzing the stability of large-scale or deep foundations.

[0114] It should be noted that the foundation is the soil or rock layer on which the building or structure relies. It is located below the structure and is responsible for transferring weight and external loads to the underground. The stability, bearing capacity and stiffness of the foundation directly affect the safety of the structure above. In the analysis of dangerous rock masses on tilting slopes, the foundation is the soil or rock layer that supports the dangerous rock mass, and its properties (such as density, compressibility, etc.) determine the stability of the rock mass. The foundation is the part of the building structure that is in direct contact with the foundation and evenly distributes the load to the foundation. The foundation is usually made of concrete, steel, etc., and can be a shallow foundation (such as a strip foundation, an independent foundation) or a deep foundation (such as a pile foundation). In the stability analysis of dangerous rock masses, the foundation supports the dangerous rock mass and is connected to the foundation to ensure stability. The bearing capacity of the foundation and the dynamic response of the rock mass must be considered during the design.

[0115] In this paper, the semi-infinite space foundation theory enables more realistic foundation modeling and more accurately reflects the support provided by the foundation to the foundation. Calculating the foundation's anti-sway stiffness and anti-lateral displacement stiffness helps analyze the foundation's response to uneven settlement or other dynamic loads, ensuring that foundation effects are fully considered in the stability analysis of dangerous rock masses.

[0116] Specifically, based on the semi-infinite space foundation theory, the initial anti-translation stiffness and initial anti-sway stiffness of the foundation to the foundation are determined:

[0117]

[0118] Among them, K y Indicates the initial translational stiffness of the foundation to the foundation, K ψrepresents the initial anti-sway stiffness of the foundation, G' represents the shear modulus of the material, r1 and r2 represent parameters related to the geometric characteristics of the foundation and the nature of the foundation, B represents the width of the foundation, H represents the height of the foundation, h represents the depth of the trailing edge crack, α1 and α2 represent dimensionless parameters related to the natural frequency, v s represents the propagation velocity of transverse waves in the material, F1() represents the first frequency function, F2() represents the second frequency function, and v' represents the Poisson's ratio of the material.

[0119] Reference Manual Figure 3 , showing a schematic diagram of the overall shear deformation of a base part provided by an embodiment of the present invention.

[0120] Considering the overall shear deformation of the foundation, the anti-sway static stiffness of the foundation to the beam (rock beam) is determined as:

[0121] K ψ1 =GB(Hh)L

[0122] Among them, K ψ1 It represents the anti-sway static stiffness of the foundation to the beam (rock beam) determined by considering the overall shear deformation of the foundation part, and L represents the length of the beam (rock beam).

[0123] Reference Manual Figure 4 , showing a schematic diagram of local deformation of a base part provided by an embodiment of the present invention.

[0124] Considering the local deformation of the foundation, the anti-sway static stiffness of the foundation to the beam (rock beam) is determined as:

[0125]

[0126] Among them, K ψ2 It represents the anti-sway static stiffness of the foundation beam (rock beam) determined by considering the local deformation of the foundation part.

[0127] Considering the constraint of the weak rock layer at the bottom on the foundation deformation, the constraint stiffness is determined as:

[0128]

[0129] Among them, K ψ3 ′ represents the constraint stiffness determined by considering the constraint of the weak rock layer at the bottom on the foundation deformation, d represents the differential operator, and l represents the length related to the constraint.

[0130] K ψ1 , K ψ2 , K ψ3 ′ is equivalent to a series connection of springs. The separated beam (rock beam) has rotational motion. Considering the parametric mass, let the simple harmonic motion be Θ:

[0131]

[0132] Among them, Θ represents simple harmonic motion, θ represents the amplitude of simple harmonic motion, Represents the phase angle.

[0133] The total anti-sway dynamic stiffness K provided by the foundation is determined based on the anti-sway static stiffness of the foundation to the beam (rock beam) determined by considering the overall shear deformation of the foundation part, simple harmonic motion, the anti-sway static stiffness of the foundation to the beam (rock beam) determined by considering the local deformation of the foundation part, and the constraint stiffness determined by considering the constraint of the weak rock layer at the bottom on the foundation deformation. ψ3 for:

[0134]

[0135] Among them, K ψ3 Represents the total anti-sway stiffness provided by the foundation.

[0136] The equivalent circle area method is used to determine the total anti-sway stiffness provided by the foundation and the corresponding static restraint stiffness provided by the foundation:

[0137]

[0138] Among them, K ψ4 It represents the total anti-sway stiffness provided by the foundation and the corresponding static restraint stiffness provided by the foundation, and v represents the Poisson's ratio of the material.

[0139] According to the static constraint stiffness provided by the foundation and the initial translational stiffness of the foundation to the foundation, the dynamic-static stiffness ratio γ1 is determined:

[0140]

[0141] Among them, γ1 represents the ratio of dynamic to static stiffness.

[0142] Determine the static sway restraint stiffness provided by the foundation:

[0143]

[0144] γ1'=-0.0001χ 4 +0.002χ 3 -0.0148χ 2 +0.139χ+0.363

[0145]

[0146] Among them, K ψ5 represents the static rocking restraint stiffness provided by the foundation, γ1' represents the correction factor of the dynamic-static stiffness ratio, and χ represents a dimensionless parameter related to the foundation size and the depth of the trailing edge crack.

[0147] The dynamic constraint stiffness provided by the foundation is determined by combining the constraint stiffness determined by considering the constraint of the weak bottom rock layer on the foundation deformation and the static sway constraint stiffness provided by the foundation:

[0148] K ψ6 =K ψ5 γ1+K ψ3 '

[0149] Among them, K ψ6 Represents the dynamic restraint stiffness provided by the foundation.

[0150] According to the dynamic restraint stiffness provided by the foundation, considering the influence of foundation rotation on the beam (rock beam) constraint, the anti-sway stiffness provided by the foundation to the foundation part is determined:

[0151]

[0152] Among them, K ψ7 It indicates the anti-sway stiffness provided by the foundation to the foundation part.

[0153] It should be noted that K ψ3 and K ψ7 Equivalent to springs in series, according to relevant theories, the anti-sway stiffness provided by the foundation to the entire foundation is determined:

[0154]

[0155] in, It indicates the anti-sway stiffness provided by the foundation to the entire foundation.

[0156] Furthermore, similar to the above analysis, the static stiffness of the foundation against translation provided by the foundation is specifically:

[0157]

[0158] Among them, K y1 It represents the static stiffness against translation provided by the foundation.

[0159] Taking into account the anti-sway stiffness provided by the foundation to the foundation as a whole and the anti-translation static stiffness provided by the foundation to the foundation, the dynamic-static stiffness ratio γ2 is determined:

[0160]

[0161] It should be noted that the static stiffness against translation provided by the foundation is specifically:

[0162]

[0163] γ2'=-0.0036χ 2 +0.1231χ+1.9932

[0164] Among them, Ky2 represents the static stiffness against translation provided by the foundation, and γ2' represents the corrected static-dynamic stiffness ratio.

[0165] The translational stiffness provided by the foundation is determined by combining the dynamic and static stiffness ratio γ2 and the anti-translational static stiffness provided by the foundation:

[0166] K' y2 =K y1 γ2

[0167] Among them, K' y2 Represents the translational stiffness provided by the foundation.

[0168] Considering the constraint of the weak rock layer at the bottom on the foundation translation, the constraint stiffness is determined as:

[0169]

[0170] Among them, K y3 It represents the constraint stiffness determined by considering the constraint of the bottom weak rock layer on the foundation translation.

[0171] According to the translational stiffness provided by the foundation and the constraint stiffness determined by considering the constraint of the weak rock layer at the bottom on the translation of the foundation, the lateral displacement stiffness provided by the foundation to the entire foundation is determined:

[0172] K h =K y2 +K y3 -ρS(1-λ)ω 2

[0173] Among them, K h It indicates the stiffness against lateral displacement provided by the foundation to the entire foundation.

[0174] By comprehensively considering the foundation's shear deformation, local deformation, and the influence of weak underlying rock layers, this method can more realistically simulate the actual response of the foundation under complex subgrade conditions. By also considering the foundation's anti-sway stiffness and anti-lateral displacement stiffness, the dynamic response of the subgrade and foundation can be more accurately predicted. This is particularly true in stability analysis of dangerous rock masses on toppling slopes, where accurate stiffness calculations help better assess the rock's stability under external loads. By considering the dynamic-static stiffness ratio and vibration modes, a more refined analysis can be provided for different subgrade and foundation characteristics.

[0175] S405: Based on the anti-sway stiffness and anti-lateral displacement stiffness provided by the foundation to the entire foundation, the bending angle and deformation angle of the beam are solved to determine the motion boundary conditions of the dangerous rock mass of the toppling slope.

[0176] In this paper, combining foundation stiffness with the vibration characteristics of the foundation can more accurately simulate the movement of dangerous rock masses on toppling slopes in real-world environments. Through precise boundary condition calculations, the stability of the rock mass under the combined effects of the foundation and the foundation can be comprehensively assessed, potentially avoiding risk points and improving safety.

[0177] In a possible implementation, S405 specifically includes sub-steps S4051 and S4052:

[0178] S4051: Determine the boundary conditions at the free end based on the stress characteristics of the dangerous rock mass of the dumping slope.

[0179] Optionally, the boundary conditions at the free end are determined according to the following formula:

[0180]

[0181] It should be noted that x=h at the free end.

[0182] In this paper, the boundary conditions at the free ends are key to describing the dynamic behavior of the structure. Accurately determining these boundary conditions allows for precise simulation of the vibration characteristics of the beam, particularly the dynamic response of the slope's precarious rock mass. In practical applications, such an accurate model can help better assess the stability of the rock mass and foundation under external loads (such as vibration), thereby mitigating potential collapse risks.

[0183] S4052: Based on the boundary conditions at the free end, the anti-sway stiffness and anti-lateral displacement stiffness provided by the foundation to the entire foundation, the bending angle and deformation angle of the beam are solved to determine the motion boundary conditions of the dangerous rock mass on the toppling slope.

[0184] Alternatively, the motion boundary conditions of the dangerous rock mass on the toppling slope can be determined according to the following formula:

[0185]

[0186] Where Y() represents the vibration mode function of the beam, K ψ7 It indicates the anti-sway stiffness provided by the foundation to the foundation part. Indicates the anti-sway stiffness provided by the foundation to the foundation as a whole, K h It indicates the stiffness against lateral displacement provided by the foundation to the entire foundation.

[0187] By considering the foundation's anti-sway and anti-lateral displacement stiffness, this method comprehensively assesses the synergistic effect between the foundation and the subgrade, ensuring that the impact of each stiffness on the dynamic response of the rock mass is fully considered in the stability analysis. This comprehensive calculation more accurately reflects the actual situation. By solving the bending and deformation angles of the beam, the deformation of the dangerous rock mass under external loads or vibration can be accurately assessed. This not only provides necessary data for stability assessment but also helps optimize design and reduce potential safety hazards caused by excessive deformation.

[0188] S406: Determine the natural frequency of the dangerous rock mass on the toppling slope according to the motion boundary conditions of the dangerous rock mass on the toppling slope.

[0189] In this paper, natural frequency is a key indicator for determining rock mass stability. By calculating the natural frequency, we can accurately assess the response characteristics of a dangerous rock mass on a toppling slope under external loads (such as vibration or earthquake). Calculating the natural frequency helps determine whether a dangerous rock mass will become unstable, providing a key parameter for design and early warning.

[0190] In a possible implementation, S406 specifically includes sub-steps S4061 to S4063:

[0191] S4061: Determine the linear algebraic determinant based on the motion boundary conditions of the dangerous rock mass on the toppling slope.

[0192] Alternatively, let q=ρS'hω 2 , determine the linear algebraic determinant:

[0193]

[0194] Where m' represents the coefficient related to the foundation geometry, anti-sway stiffness, foundation anti-sway stiffness and crack depth; n represents the parameter related to the mass, geometry and foundation stiffness of the beam; g1, g2, g3 and g4 represent the parameters related to the natural frequency; c1, c2, c3 and c4 represent the coefficients to be determined; K ψ7 It indicates the anti-sway stiffness provided by the foundation to the foundation part. Indicates the anti-sway stiffness provided by the foundation to the foundation as a whole, K h It indicates the stiffness against lateral displacement provided by the foundation to the entire foundation.

[0195] In this invention, by constructing coefficients related to foundation geometry, anti-sway stiffness, foundation stiffness, and crack depth, the complex dynamics of the system can be transformed into a mathematical form, making the solution more systematic and structured. This allows all factors to be accurately incorporated into the model, reflecting a more realistic engineering situation. By unifying all influencing factors within a single equation, any factors that may affect stability are avoided, thereby improving the accuracy of the calculation results.

[0196] S4062: Combine the undetermined coefficients in linear algebraic determinants to construct matrices.

[0197] Specifically, combine the undetermined coefficients c1, c2, c3, and c4 to construct the matrix D:

[0198]

[0199] Wherein, D represents a matrix, and D1, D2, D3, and D4 all represent elements in the matrix D.

[0200] In the present invention, the undetermined coefficients are combined into a matrix form, making the problem solution more concise and easy to calculate. The matrix form helps to quickly process complex nonlinear equations through numerical calculations, facilitating efficient solution. The construction of the matrix D helps to systematically integrate all required parameters. Through matrix algebraic operations, the calculation process can be simplified while ensuring mathematical correctness and improving computational stability.

[0201] Reference Manual Figure 5 , shows a schematic diagram of calculating the natural frequency of a dangerous rock mass on a dumping slope provided by an embodiment of the present invention.

[0202] S4063: Determine the natural frequency of the dangerous rock mass on the toppling slope by solving the matrix using MATLAB.

[0203] Alternatively, let |D|=0, and solve the matrix D through MATLAB to determine the natural frequency of the dangerous rock mass of the toppling slope.

[0204] Specifically, since the calculation involves multiple parameters and the relevant mechanical parameters of the rock mass need to be considered, manual calculation becomes very complicated. Therefore, MATLAB programming is used for numerical solution. Figure 5 ), the frequency corresponding to the first intersection with the X-axis is the natural frequency of the dangerous rock mass.

[0205] In this paper, manual calculations are very difficult and error-prone because the problem involves multiple variables and complex parameter relationships. MATLAB's numerical solution capabilities effectively handle complex equations through iterative methods, greatly reducing the complexity of the calculation process. MATLAB can quickly perform large-scale numerical calculations and, by displaying the results graphically, allows intuitive observation of the natural frequencies. The natural frequencies can be directly obtained by looking at the intersection points in the graph, greatly improving the accuracy and efficiency of the solution.

[0206] S5: Construct a dynamic model of dangerous rock mass on a toppling slope.

[0207] Reference Manual Figure 6 , which shows a structural schematic diagram of a dynamic model provided by an embodiment of the present invention.

[0208] S6: Based on the dynamic model of the dangerous rock mass on the toppling slope, determine the relationship between the natural frequency of the dangerous rock mass on the toppling slope and the relative depth of the cracks.

[0209] In this paper, by establishing a dynamic model that links the relationship between natural frequency and crack depth, the dynamic behavior of rock masses can be more accurately simulated and predicted. By directly linking natural frequency and crack depth, stability analysis can be made more comprehensive, accurately capturing the impact of crack evolution on dangerous rock masses and providing basic data for subsequent early warning and stability analysis.

[0210] In a possible implementation, S6 specifically includes sub-steps S601 and S602:

[0211] S601: Determine a dynamic equation of the dangerous rock mass on the toppling slope according to a dynamic model of the dangerous rock mass on the toppling slope.

[0212] Alternatively, the dynamic equation of the dangerous rock mass on the toppling slope can be determined according to the following formula:

[0213]

[0214] Where J represents the moment of inertia, represents the acceleration of the rotation angle, K represents the anti-sway stiffness of the bonding surface, θ0 represents the rotation angle, m represents the external moment acting on the beam micro-segment, L represents the length of the beam, E represents the elastic modulus of the material, H represents the height of the dangerous rock mass of the dumping slope, h represents the depth of the trailing edge crack, and d' represents the thickness of the bonding surface.

[0215] S602: Solve the dynamic equation of the dangerous rock mass of the toppling slope to determine the relationship between the natural frequency of the dangerous rock mass of the toppling slope and the relative depth of the cracks.

[0216] Optionally, the relationship between the natural frequency of the dangerous rock mass of the dumping slope and the relative depth of the crack is determined according to the following formula:

[0217]

[0218] Among them, ω represents the natural frequency of the dangerous rock mass of the dumping slope, λ represents the relative depth of the crack, and J1 represents the correction coefficient of the moment of inertia.

[0219] It should be noted that the correction coefficient J1 can be adjusted according to the geometric characteristics, material properties and depth of the rock mass, so as to more accurately reflect the real dynamic behavior of the rock mass.

[0220] In this paper, the dynamic model directly incorporates the relationship between relative crack depth and natural frequency into the analysis, helping to more accurately capture the impact of cracks on rock mass stability. By incorporating relative crack depth into the dynamic equation, it is possible to quantify the impact of crack extension on the vibration characteristics and stability of the rock mass. This allows stability assessment to be based not only on static conditions but also on the actual vibration characteristics of the rock mass, increasing the accuracy and scientific nature of the assessment.

[0221] S7: A stability evaluation model for the dangerous rock mass on the toppling slope is constructed based on the safety factors when the center of gravity of the dangerous rock mass on the toppling slope is within and outside the toppling point and the relationship between the natural frequency of the dangerous rock mass on the toppling slope and the relative depth of the cracks.

[0222] In a possible implementation, S7 specifically includes:

[0223] The solved dynamic equation of the dangerous rock mass on the toppling slope is substituted into the safety factor when the center of gravity of the dangerous rock mass on the toppling slope is inside and outside the toppling point, and a stability evaluation model for the dangerous rock mass on the toppling slope is constructed.

[0224] Optionally, a stability evaluation model for dangerous rock masses on toppling slopes can be constructed according to the following formula:

[0225] F0=F(ω)

[0226]

[0227] λ=f(ω)

[0228] Among them, F0 represents the output of the stability evaluation model, F(ω) represents the stability coefficient of the dangerous rock mass of the toppling slope, and ω represents the natural frequency of the dangerous rock mass of the toppling slope.

[0229] In this invention, by combining the natural frequency and crack depth, the model can simultaneously consider the dynamic response and static stability of the rock mass, which is more accurate and comprehensive than a simple static analysis. The impact of crack depth on rock mass stability is fully reflected in the dynamic analysis, thus avoiding the potential risks that traditional methods may ignore due to cracks. By correlating the natural frequency with the stability coefficient, a quantitative assessment standard for rock mass stability can be provided. The calculation of the stability coefficient takes into account the physical properties of the rock mass and the crack depth, making the stability assessment more objective and operational, and providing a scientific basis for the design of subsequent protective measures.

[0230] S8: The stability analysis of the dangerous rock mass on the dumping slope is carried out through the stability evaluation model of the dangerous rock mass on the dumping slope.

[0231] The present invention utilizes a stability evaluation model that comprehensively considers multiple factors, including the rock mass's natural frequency, crack depth, and foundation characteristics, providing a comprehensive stability assessment. This not only assesses the rock mass's current stability but also predicts potential stability risks, thereby preventing sudden collapse accidents. The stability analysis model can be dynamically updated based on real-time monitoring data, allowing rock mass stability assessments to be conducted in real time, on-site. This helps identify potential risks in advance and enable timely implementation of appropriate preventive measures, thereby improving safety.

[0232] Reference Manual Figure 7 , which shows a structural schematic diagram of a natural frequency analytical calculation system for a dumping type dangerous rock mass on a slope provided by the present invention.

[0233] The present invention further provides a system 20 for analyzing and calculating the natural frequency of a dangerous rock mass on a toppling slope, which is applied to the above-mentioned method for analyzing and calculating the natural frequency of a dangerous rock mass on a toppling slope, and comprises:

[0234] Processor 201;

[0235] The memory 202 stores computer-readable instructions. When the computer-readable instructions are executed by the processor 201 , the method for analytically calculating the natural frequency of a dangerous rock mass on a dumping slope as described in the method embodiment is implemented.

[0236] The natural frequency analytical calculation system 20 for the dangerous rock mass on a toppling slope provided by the present invention can execute the above-mentioned natural frequency analytical calculation method for the dangerous rock mass on a toppling slope and achieve the same or similar technical effects. To avoid repetition, the present invention will not elaborate on it.

[0237] The beneficial effects brought about by the technical solution provided by the embodiment of the present invention include at least:

[0238] In an embodiment of the present invention, by simplifying the dangerous rock mass of the tipping slope based on the modified Timoshenko beam theory, constructing a beam model, and performing analytical calculations on the dangerous rock mass of the tipping slope according to the beam model, the natural frequency of the dangerous rock mass of the tipping slope is determined, thereby reducing the dependence on the traditional limit equilibrium method and numerical simulation method, and being able to fully consider the dynamic characteristics of the rock mass. By constructing a stability evaluation model for the dangerous rock mass of the tipping slope, and using the stability evaluation model for the dangerous rock mass of the tipping slope, the stability analysis of the dangerous rock mass of the tipping slope is performed, thereby avoiding the dependence on static analysis in the traditional stability evaluation method, being able to effectively combine the dynamic response of the rock mass, and improving the early warning capability and response sensitivity in practical applications.

[0239] It should be understood that the processor in the embodiments of the present invention may be a central processing unit (CPU), and the processor may also be other general-purpose processors, digital signal processors (DSP), application-specific integrated circuits (ASIC), field programmable gate arrays (FPGA) or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor may be a microprocessor or any conventional processor, etc.

[0240] It should also be understood that the memory in the embodiments of the present invention may be a volatile memory or a non-volatile memory, or may include both volatile and non-volatile memories. Among them, the non-volatile memory may be a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), or a flash memory. The volatile memory may be a random access memory (RAM), which is used as an external cache. By way of example and not limitation, many forms of random access memory (RAM) are available, such as static RAM (SRAM), dynamic random access memory (DRAM), synchronous DRAM (SDRAM), double data rate synchronous dynamic random access memory (DDR SDRAM), enhanced synchronous dynamic random access memory (ESDRAM), synchronous link DRAM (SLDRAM), and direct rambus RAM (DR RAM).

[0241] The above embodiments can be implemented in whole or in part through software, hardware (such as circuits), firmware, or any other combination. When implemented using software, the above embodiments can be implemented in whole or in part in the form of a computer program product. The computer program product includes one or more computer instructions or computer programs. When the computer instructions or computer program are loaded or executed on a computer, the processes or functions described in accordance with the embodiments of the present invention are generated in whole or in part. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via a wired method (such as infrared, wireless, microwave, etc.). The computer-readable storage medium can be any available medium that can be accessed by a computer or a data storage device such as a server or data center that contains a collection of one or more available media. The available medium can be a magnetic medium (such as a floppy disk, hard disk, or magnetic tape), an optical medium (such as a DVD), or a semiconductor medium. The semiconductor medium can be a solid-state drive.

[0242] It should be understood that the term "and / or" as used herein simply describes a relationship between associated objects, indicating that three possible relationships exist. For example, "A and / or B" can represent: A alone, A and B together, or B alone. A and B can be singular or plural. Furthermore, the character " / " as used herein generally indicates an "or" relationship between the associated objects, but it may also indicate an "and / or" relationship. For specific understanding, please refer to the context.

[0243] In this disclosure, "at least one" means one or more, and "plurality" means two or more. "At least one of the following" or similar expressions refers to any combination of these items, including any combination of single or plural items. For example, "at least one of a, b, or c" can mean: a, b, c, ab, ac, bc, or abc, where a, b, and c can be single or plural.

[0244] It should be understood that in various embodiments of the present invention, the size of the serial numbers of the above-mentioned processes does not mean the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiments of the present invention.

[0245] Those skilled in the art will appreciate that the units and algorithm steps of each example described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are performed in hardware or software depends on the specific application and design constraints of the technical solution. Professionals and technicians can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of the present invention.

[0246] Those skilled in the art will clearly understand that, for the convenience and brevity of description, the specific working processes of the above-described equipment, devices and units can refer to the corresponding processes in the aforementioned method embodiments and will not be repeated here.

[0247] In the several embodiments provided by the present invention, it should be understood that the disclosed devices, apparatuses and methods can be implemented in other ways. For example, the device embodiments described above are merely illustrative. For example, the division of the units is merely a logical function division. In actual implementation, there may be other division methods, such as multiple units or components can be combined or integrated into another device, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be through some interface, indirect coupling or communication connection of the device or unit, which can be electrical, mechanical or other forms.

[0248] The units described as separate components may or may not be physically separate, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of these units may be selected to achieve the purpose of this embodiment according to actual needs.

[0249] In addition, each functional unit in each embodiment of the present invention may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit.

[0250] If the functions are implemented in the form of software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, or the part of the technical solution, can be embodied in the form of a software product. The computer software product is stored in a storage medium and includes several instructions for enabling a computer device (which can be a personal computer, server, or network device, etc.) to execute all or part of the steps of the method described in each embodiment of the present invention. The aforementioned storage medium includes various media that can store program codes, such as a USB flash drive, a mobile hard disk, a read-only memory (ROM), a random access memory (RAM), a magnetic disk, or an optical disk.

[0251] An embodiment of the present invention provides a computer-readable storage medium having a computer program stored thereon, wherein when the program is executed by a processor, the method for analytically calculating the natural frequency of a dangerous rock mass on a dumping slope as described in the method embodiment is implemented.

[0252] The computer-readable storage medium provided by the present invention can realize the steps and effects of the analytical calculation method for the natural frequency of the dangerous rock mass on the dumping slope in the above-mentioned method embodiment. To avoid repetition, the present invention will not elaborate on them.

[0253] The beneficial effects brought about by the technical solution provided by the embodiment of the present invention include at least:

[0254] In an embodiment of the present invention, by simplifying the dangerous rock mass of the tipping slope based on the modified Timoshenko beam theory, constructing a beam model, and performing analytical calculations on the dangerous rock mass of the tipping slope according to the beam model, the natural frequency of the dangerous rock mass of the tipping slope is determined, thereby reducing the dependence on the traditional limit equilibrium method and numerical simulation method, and being able to fully consider the dynamic characteristics of the rock mass. By constructing a stability evaluation model for the dangerous rock mass of the tipping slope, and using the stability evaluation model for the dangerous rock mass of the tipping slope, the stability analysis of the dangerous rock mass of the tipping slope is performed, thereby avoiding the dependence on static analysis in the traditional stability evaluation method, being able to effectively combine the dynamic response of the rock mass, and improving the early warning capability and response sensitivity in practical applications.

[0255] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in the present invention should be included in the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be based on the scope of protection of the claims.

[0256] There are a few points to note:

[0257] (1) The drawings of the embodiments of the present invention only relate to the structures related to the embodiments of the present invention. Other structures may refer to conventional designs.

[0258] (2) For the sake of clarity, the thickness of layers or regions in the drawings used to describe the embodiments of the present invention are exaggerated or reduced, that is, these drawings are not drawn to scale. It is understood that when an element such as a layer, film, region, or substrate is referred to as being "on" or "under" another element, the element may be "directly" "on" or "under" the other element or intervening elements may be present.

[0259] (3) In the absence of conflict, the embodiments of the present invention and the features therein may be combined with each other to form new embodiments.

[0260] The above are only specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. The protection scope of the present invention shall be based on the protection scope of the claims.

Claims

1. A method for calculating the natural frequency of a dangerous rock mass on a dumping slope, characterized in that: include: S1: Obtain the basic parameters of the dangerous rock mass of the dumping slope; S2: Determine the safety factor of the tipping slope dangerous rock mass when the center of gravity is inside the tipping point and outside the tipping point according to the basic parameters; S3: Based on the modified Timoshenko beam theory, the tipping slope dangerous rock mass is simplified and a beam model is constructed; S4: performing analytical calculation on the dangerous rock mass of the tilting slope according to the beam model to determine the natural frequency of the dangerous rock mass of the tilting slope; S5: Construct a dynamic model of the dangerous rock mass on the toppling slope; S6: determining the relationship between the natural frequency of the tilting slope dangerous rock mass and the relative depth of the cracks according to the dynamic model of the tilting slope dangerous rock mass; S7: constructing a stability evaluation model for the tipping slope dangerous rock mass according to the safety factors of the center of gravity of the tipping slope dangerous rock mass within the tipping point and outside the tipping point and the relationship between the natural frequency of the tipping slope dangerous rock mass and the relative depth of the cracks; S8: Performing stability analysis on the tilting type slope dangerous rock mass using the tilting type slope dangerous rock mass stability evaluation model.

2. The method for calculating the natural frequency of a dangerous rock mass on a dumping slope according to claim 1, characterized in that: The basic parameters include: height of the dangerous rock mass of the dumping slope, depth of the rear edge crack, horizontal distance of the center of gravity, height of the center of gravity, tensile strength, bonding area, vibration load, relative depth of the crack, inclination angle of the rear edge crack and deadweight.

3. The method for calculating the natural frequency of a dangerous rock mass on a dumping slope according to claim 1, wherein: The safety factor of the center of gravity of the dangerous rock mass of the overturning slope within the overturning point is specifically: Among them, F1 represents the safety factor of the center of gravity of the dangerous rock mass of the overturning slope within the overturning point, It represents the anti-tilting moment when the gravity center of the dangerous rock mass of the overturning slope is within the overturning point. represents the overturning moment of the center of gravity of the tipping slope dangerous rock mass within the overturning point, W represents the deadweight of the tipping slope dangerous rock mass, a represents the horizontal distance of the center of gravity of the tipping slope dangerous rock mass, f represents the tensile strength of the tipping slope dangerous rock mass, S represents the bonding area, H represents the height of the tipping slope dangerous rock mass, h represents the depth of the trailing edge crack, sin represents the sine function, β represents the inclination angle of the trailing edge crack, P represents the vibration load, h0 represents the height of the center of gravity of the tipping slope dangerous rock mass, and λ represents the relative depth of the crack; The safety factor of the tipping slope dangerous rock mass with its center of gravity outside the tipping point is specifically: Among them, F2 represents the safety factor when the center of gravity of the dangerous rock mass of the overturning slope is outside the overturning point. It represents the anti-tilting moment when the gravity center of the dangerous rock mass of the overturning slope is outside the overturning point. It represents the overturning moment when the center of gravity of the dangerous rock mass of the overturning slope is outside the overturning point.

4. The method for calculating the natural frequency of a dangerous rock mass on a dumping slope according to claim 1, wherein: The S4 specifically includes: S401: Determine a differential equation of the beam according to the beam model; S402: Determine the mode function of the beam according to the differential equation of the beam by using the elimination method and the separation of variables method; S403: Determine the bending angle and deformation angle of the beam according to the mode shape function of the beam; S404: Based on the semi-infinite space foundation theory, determine the anti-sway stiffness and anti-lateral displacement stiffness provided by the foundation to the entire foundation; S405: Solving the bending angle and deformation angle of the beam based on the anti-sway stiffness and anti-lateral displacement stiffness provided by the foundation to the entire foundation, and determining the motion boundary conditions of the dangerous rock mass on the toppling slope; S406: Determine the natural frequency of the tilting type slope dangerous rock mass according to the motion boundary conditions of the tilting type slope dangerous rock mass.

5. The method for calculating the natural frequency of a dangerous rock mass on a dumping slope according to claim 4 is characterized in that: The S402 specifically includes: S4021: Simplifying the differential equation of the beam by elimination method; S4022: Solve the free vibration characteristics of the beam using the separation of variables method and determine the vibration solution of the beam; S4023: Determine the mode shape function of the beam according to the simplified differential equation of the beam and the vibration solution of the beam.

6. The method for calculating the natural frequency of a dangerous rock mass on a toppling slope according to claim 4 is characterized in that: The S405 specifically includes: S4051: Determine the boundary conditions at the free end based on the stress characteristics of the dangerous rock mass of the tilting slope; S4052: Based on the boundary conditions at the free end, the anti-sway stiffness and anti-lateral displacement stiffness provided by the foundation to the entire foundation, the bending angle and deformation angle of the beam are solved to determine the motion boundary conditions of the dangerous rock mass of the tilting slope.

7. The method for calculating the natural frequency of a dangerous rock mass on a dumping slope according to claim 4, characterized in that: The S406 specifically includes: S4061: Determine a linear algebraic determinant based on the motion boundary conditions of the dangerous rock mass on the toppling slope; S4062: Combining the undetermined coefficients in the linear algebraic determinant to construct a matrix; S4063: Solve the matrix using MATLAB to determine the natural frequency of the dangerous rock mass on the dumping slope.

8. The method for calculating the natural frequency of a dangerous rock mass on a dumping slope according to claim 1, wherein: The S6 specifically includes: S601: Determine a dynamic equation of the dangerous rock mass on the toppling slope according to the dynamic model of the dangerous rock mass on the toppling slope; S602: Solve the dynamic equation of the dangerous rock mass of the toppling slope to determine the relationship between the natural frequency of the dangerous rock mass of the toppling slope and the relative depth of the cracks.

9. The method for calculating the natural frequency of a dangerous rock mass on a toppling slope according to claim 3, wherein: The S7 is specifically: The solved dynamic equation of the tipping slope dangerous rock mass is substituted into the safety factor of the tipping slope dangerous rock mass when the center of gravity is inside and outside the tipping point, respectively, to construct a stability evaluation model for the tipping slope dangerous rock mass.

10. A natural frequency analytical calculation system for a dangerous rock mass on a dumping slope, characterized in that: include: processor; A memory having computer-readable instructions stored thereon, wherein when the computer-readable instructions are executed by the processor, the method for analytical calculation of the natural frequency of a dangerous rock mass on a dumping slope according to any one of claims 1 to 9 is implemented.

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