Dynamic compaction vibration space-time attenuation law prediction method based on pulse function

Abstract

The invention belongs to the technical field of highway engineering, and discloses a dynamic compaction vibration space-time attenuation law prediction method based on a pulse function, and the method comprises the steps: constructing a dimensionless function relation between a dynamic compaction vibration speed and a control parameter through a dimensional analysis method; representing vibration wave acceleration input caused by dynamic compaction vibration by using a pulse function, constructing a dynamic compaction vibration wave acceleration expression in combination with an exponential decay function and a time retention factor, and obtaining a horizontal direction seismic acceleration prediction coefficient and a vertical direction seismic acceleration prediction coefficient based on the dynamic compaction vibration wave acceleration expression; and drawing an acceleration and time prediction curve. The method solves the technical problem that the empirical formula in the prior art cannot meet the field actual construction requirement due to large deviation between the calculation result and the actual construction.

Description

Technical Field

[0001] This invention belongs to the field of highway engineering technology, and in particular relates to a method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on pulse function. Background Technology

[0002] In dynamic compaction, the impact of the hammer on the ground generates significant vibrations. These vibration waves decay rapidly over time and distance. To control the harmful effects of these vibration waves on the surrounding environment and buildings, a method for quickly and accurately predicting the decay of dynamic compaction vibration acceleration is urgently needed in engineering projects. Current theoretical research largely relies on experimental monitoring data or empirical formulas from seismic design codes for blasting, often resulting in discrepancies between the calculated results and actual engineering conditions. However, dynamic compaction is an instantaneous impact load, and its wave propagation process depends not only on the compaction energy level, the foundation medium, and wave propagation parameters, but also on its own wave characteristics. Therefore, traditional empirical formulas are insufficient to meet the requirements of actual on-site construction. It is necessary to establish a calculation method that better reflects the wave characteristics, decay characteristics, and time retention characteristics of dynamic compaction vibration acceleration, providing a reference for future dynamic compaction construction and vibration wave control. Summary of the Invention

[0003] Based on this, the present invention discloses a method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on pulse function, in order to solve the technical problem that the traditional empirical formulas do not meet the actual construction requirements on site due to large deviations between the calculation results and the actual engineering.

[0004] To solve the above-mentioned technical problems, the technical solution adopted by this invention is a method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on impulse functions, comprising the following steps: S1: Construct a dimensionless functional relationship between dynamic compaction vibration velocity and control parameters; S2: The vibration wave acceleration input caused by the dynamic compaction vibration is characterized by the response form of the impulse function. The dynamic compaction vibration wave acceleration expression is constructed by combining the exponential decay function and the time retention factor. Based on the dynamic compaction vibration wave acceleration expression, the horizontal direction seismic acceleration prediction coefficient and the vertical direction seismic acceleration prediction coefficient are obtained. S3: Plot acceleration and time prediction curves to analyze the spatiotemporal attenuation law of dynamic compaction vibration.

[0005] Furthermore, the specific control parameters in S1 are: impact energy. Compression modulus Horizontal distance from measuring point to ramming core Vibration dominant frequency .

[0006] Furthermore, the specific implementation method of S1 is as follows: Construct a dimensionless functional relationship between the dynamic compaction vibration velocity and the control parameters, including selecting the compaction energy, compression modulus, horizontal distance of the measuring point from the compaction core and the vibration dominant frequency as control parameters and determining their dimensions; construct the first and second dimensionless numbers, establish and solve a system of linear equations based on the consistency of dimensions to determine the undetermined exponent; substitute the solution into the dimensionless number expression, and establish the functional relationship between the dimensionless numbers according to Buckingham's π theorem, finally transforming it into a linear form characterizing the exponential decay law of vibration velocity and distance.

[0007] Furthermore, the specific implementation method of S2 is as follows: Combining the exponential decay function and the time retention factor, an expression for the acceleration of the dynamic compaction vibration wave is constructed; the motion equation of the particle is derived based on the one-dimensional viscoelastic wave equation, and its solution is expressed as an exponentially decaying wave in the propagation direction; the acceleration expression is obtained by taking the second time derivative of the displacement, and its dominant frequency component is extracted; finally, based on the acceleration expression, the earthquake acceleration prediction coefficients in the horizontal and vertical directions are obtained respectively.

[0008] Furthermore, the specific steps of S1 are as follows: S1.1: Take impact energy Compression modulus Horizontal distance from measuring point to ramming core Vibration dominant frequency As the fundamental unit of measurement, impact energy Vibration dominant frequency Compression modulus Horizontal distance from measuring point to ramming core The dimensions are shown in equations (1) to (4): (1) Where: [ ] is a dimensionless symbol, For impact energy, Mass is a unit of measurement. Length is a unit of measurement. The unit of measurement is time; (2) in: It is the compressibility modulus; (3) in: This is the dominant vibration frequency; (4) in: The distance between the measuring point and the tamping core is the horizontal distance. S1.2: Constructing the first dimensionless number With the second dimensionless number As shown in equations (5) to (6): (5) in: It is the first dimensionless number. The first undetermined exponent of the first dimensionless number. The second undetermined exponent of the first dimensionless number. The third undetermined exponent of the first dimensionless number. Vibration velocity; (6) in: It is the second dimensionless number. It is the first undetermined exponent of the second dimensionless number. The second undetermined exponent of the second dimensionless number. The third undetermined exponent of the second dimensionless number; S1.3: Substituting equations (1) to (4) into equations (5) to (6), we obtain the first dimensionless number. With the second dimensionless number The functional relationships are shown in equations (7) to (8); (7) (8); S1.4: Establish a system of linear equations based on the principle of dimensional consistency, as shown in equations (9) to (10): (9) (10) Solving the linear equations (9) and (10) yields the undetermined exponent of the first dimensionless number. , , The undetermined exponent of the second dimensionless number , , As shown in equations (11) to (12): (11) (12); S1.5: Substituting equations (11) and (12) into equations (5) and (6), we obtain the first dimensionless number. With the second dimensionless number As shown in equations (13) to (14): (13) (14); S1.6: Establishing functional relationships based on Bukingham's π theorem ,Will Written in the form shown in equation (15): (15) Substituting equations (13) and (14) into equation (15), we obtain equation (16): (16) in: The first dimensionless number With the second dimensionless number Dependencies between them; Based on the vibration attenuation law of dynamic compaction, equation (16) can be written as equation (17): (17) Where: k is the equivalent coefficient, and α is the decay exponent; Taking the logarithm of both sides of equation (17) and transforming it into a linear form as shown in equation (18): (18); Where: y is the normalized velocity, and x is the normalized distance.

[0009] Furthermore, the specific steps for constructing the dynamic compaction vibration wave acceleration expression in S2 are as follows: S2.1: The motion of soil micro-elements is expressed by the one-dimensional viscoelastic wave equation, and the motion equation of the particle is obtained as shown in equation (19): (19) in: For soil density, This represents the displacement of a soil particle during wave propagation. For the propagation time of the vibration wave, Let the spatial coordinates be along the direction of wave propagation. The viscous damping coefficient is... Shear modulus; S2.2: The solution to the particle's motion equation is expressed as an exponentially decaying function propagating along the +z direction, as shown in equation (20): (20) in: For particle displacement, The initial amplitude, Let ξ be the natural constant and ξ be the spatial attenuation coefficient. The time-domain attenuation coefficient, For wave number, Angular acceleration, The time retention factor; S2.3: Calculate the displacement The second time derivative is shown in equation (21): (twenty one) in: To accelerate the compaction, Let z be the acceleration of a particle at a certain position z and time t. Acceleration of the particle The dominant frequency component is written in the form shown in equation (22): (twenty two); S2.4: The horizontal and vertical accelerations of the dynamic compaction vibration wave are obtained as shown in equations (23) and (24): (twenty three) (twenty four) in: This refers to the horizontal seismic acceleration. Let k be the vertical seismic acceleration. h k is the horizontal seismic acceleration coefficient. v V is the vertical seismic acceleration coefficient, g is the gravitational acceleration, and V is the acceleration due to gravity. s This represents the propagation speed of transverse seismic waves in the fill. This represents the propagation speed of longitudinal seismic waves in the fill.

[0010] Furthermore, the horizontal and vertical seismic acceleration prediction coefficients obtained in S2 are as follows: The prediction coefficients for horizontal and vertical seismic acceleration are shown in equations (25) and (26): (25) (26) in: Here, C is the horizontal seismic acceleration coefficient, and C is a constant. The horizontal distance between the measuring point and the compaction point; This is the vertical distance between the measuring point and the compaction point.

[0011] The beneficial effects of this invention are as follows: The method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on the pulse function proposed in this invention not only considers the wave characteristics, attenuation characteristics in time and space, and time retention characteristics at different measuring point distances of dynamic compaction vibration waves, but also combines different impact energies, foundation medium characteristics, and wave propagation parameters on site. The resulting spatiotemporal attenuation law of dynamic compaction vibration is more consistent with the actual dynamic compaction construction process, which is conducive to the prediction of dynamic compaction construction acceleration and the control of the fluctuation range, reduces interference to the surrounding environment and buildings, and provides a strong reference for dynamic compaction construction. Attached Figure Description

[0012] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0013] Figure 1 This is the linear regression curve of vibration velocity in an embodiment of the present invention; Figure 2 This is a comparative diagram of the predicted curve, the actual curve, and the traditional predicted curve of this invention. Detailed Implementation

[0014] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0015] The present invention proposes a method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on a pulse function. Starting from the wave equation of a particle, the invention expresses the attenuation of particle waves due to material damping through frequency domain and constant attenuation, using an exponential function. Then, by differentiating with respect to time and dimensionless transformation, the common form of the pulse function is obtained. Finally, the pulse function is combined with the particle wave equation to obtain the dynamic compaction acceleration attenuation pulse function. The resulting dynamic compaction vibration acceleration image can effectively describe the attenuation law of dynamic compaction vibration, providing a strong reference for vibration propagation prediction, environmental impact assessment, and safety control measures during dynamic compaction construction.

[0016] The challenge of this application lies in its starting point: Starting from the wave equation of a particle, this invention reasonably simplifies and approximates a wave problem with infinite degrees of freedom, originating from continuum mechanics, into a "point source impulse response" problem controlled by a few key parameters. Simultaneously, by using frequency domain and constant attenuation and expressing the attenuation of particle waves due to material damping using an exponential function, it unifies the two key factors of vibration energy attenuation—geometric attenuation and material attenuation. Next, using impulse and exponential functions to express the actual dynamic compaction vibration process, it derives a functional form that conforms to both physical laws and mathematically tractable forms. Furthermore, during the dimensionless process, a "compression modulus index" is added to consider the influence of the foundation medium on the attenuation law of dynamic compaction vibration, ultimately transforming it into a powerful and practical predictive tool that can truly guide dynamic compaction construction and conduct environmental impact assessments.

[0017] This invention provides a method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on impulse functions, comprising the following steps: S1: Construct a dimensionless functional relationship between dynamic compaction vibration velocity and control parameters using dimensional analysis; The specific control parameters in S1 are: impact energy. Compression modulus Horizontal distance from measuring point to ramming core Vibration dominant frequency .

[0018] The specific implementation method of S1 is as follows: A dimensionless functional relationship between the dynamic compaction vibration velocity and the control parameters is constructed by dimensional analysis, including selecting the compaction energy, compression modulus, horizontal distance of the measuring point from the compaction core and the vibration dominant frequency as control parameters and determining their dimensions; constructing the first and second dimensionless numbers, establishing and solving a system of linear equations based on dimensional consistency to determine the undetermined exponent; substituting the solution into the dimensionless number expression, and establishing the functional relationship between the dimensionless numbers according to Buckingham's π theorem, finally transforming it into a linear form characterizing the power-law decay law of vibration velocity and distance.

[0019] The specific steps for constructing the dimensionless functional relationship between dynamic compaction vibration velocity and control parameters in S1 are as follows: S1.1: Take impact energy Compression modulus Horizontal distance from measuring point to ramming core Vibration dominant frequency As the fundamental unit of measurement, impact energy Vibration dominant frequency Compression modulus Horizontal distance from measuring point to ramming core The dimensions are shown in equations (1) to (4): (1) Where: [ ] is a dimensionless symbol, For impact energy, Mass is a unit of measurement. Length is a unit of measurement. The unit of measurement is time; (2) in: It is the compressibility modulus; (3) in: This is the dominant vibration frequency; (4) in: The distance between the measuring point and the tamping core is the horizontal distance. S1.2: Constructing the first dimensionless number With the second dimensionless number As shown in equations (5) to (6): (5) in: It is the first dimensionless number. The first undetermined exponent of the first dimensionless number. The second undetermined exponent of the first dimensionless number. The third undetermined exponent of the first dimensionless number. Vibration velocity; (6) in: It is the second dimensionless number. It is the first undetermined exponent of the second dimensionless number. The second undetermined exponent of the second dimensionless number. The third undetermined exponent of the second dimensionless number; S1.3: Substituting equations (1) to (4) into equations (5) to (6), we obtain the first dimensionless number. With the second dimensionless number The functional relationships are shown in equations (7) to (8); (7) (8); In this embodiment, in S1.3, substituting equations (1) to (4) into equations (5) to (6) yields the first dimensionless number. With the second dimensionless number The mathematical basis is the dimensional method. Based on the dimensional method, there are functional relationships of equations (9) and (10). By combining equations (1) to (6), (9), and (10), we can obtain the functional relationships of equations (7) and (8).

[0020] v=LT -1 (9) R=L (10) S1.4: Based on the principle of dimensional consistency, establish a system of linear equations, as shown in equations (11) to (12): (11) (12) In this embodiment, the construction principle of equations (11) and (12) is as follows: because and All of them are dimensionless numbers, and due to the principle of dimension consistency (M, L, T are essentially dimensions), when the exponents of M, L, T in equations (7) and (8) are all 0, they conform to the definition of dimensionless numbers, and we get equations (11) and (12).

[0021] Solving the linear equations (11) and (12) yields the undetermined exponent of the first dimensionless number. , , The undetermined exponent of the second dimensionless number , , As shown in equations (13) to (14): (13) (14).

[0022] S1.5: Substituting equations (13) and (14) into equations (5) and (6), we obtain the first dimensionless number. With the second dimensionless number As shown in equations (15) to (16): (15) (16).

[0023] S1.6: Establishing functional relationships based on Bukingham's π theorem ,Will Written in the form shown in equation (17): (17) In this embodiment, the construction principle of equation (17) is as follows: according to Bukingham's π theorem, and All are dimensionless numbers. and There must exist some kind of relation that can be calculated to equal 0, that is, It can be written as about An equation can be expressed as follows: to indicate .

[0024] Substituting equations (15) and (16) into equation (17), we obtain equation (18): (18) in: The first dimensionless number With the second dimensionless number Dependencies between them; Analysis of the vibration attenuation law of dynamic compaction shows that the vibration velocity v of the ground particles caused by dynamic compaction has a power-law relationship with the horizontal distance R between the measuring point and the impact point. Equation (18) can be written as equation (19): (19) Where: k is the equivalent coefficient, and α is the decay exponent.

[0025] Taking the logarithm of both sides of equation (17) and transforming it into a linear form as shown in equation (18): (18).

[0026] Where: y is the normalized velocity, and x is the normalized distance.

[0027] S2: The vibration wave acceleration input caused by the dynamic compaction vibration is characterized by the response form of the impulse function. The dynamic compaction vibration wave acceleration expression is constructed by combining the exponential decay function and the time retention factor. Based on the dynamic compaction vibration wave acceleration expression, the horizontal direction seismic acceleration prediction coefficient and the vertical direction seismic acceleration prediction coefficient are obtained. In this example, the reason for using an impulse function to characterize the vibration wave acceleration input caused by dynamic compaction is that the impulse function can effectively simulate such intense vibrations that occur within a short period of time, making the results more realistically reflect the actual situation. Simultaneously, by combining an exponential function to describe the acceleration decay caused by material damping during dynamic compaction and adding a time-dependent time retention factor to the impulse function, the amplitude and duration characteristics of the instantaneous construction process are better preserved compared to traditional empirical formulas, making the resulting calculation method more realistically reflect the actual construction situation.

[0028] The specific implementation method of S2 is as follows: The vibration wave acceleration input caused by dynamic compaction is characterized by the response form of the impulse function. The acceleration expression of the dynamic compaction vibration wave is constructed by combining the exponential decay function and the time retention factor. The motion equation of the particle is derived based on the one-dimensional viscoelastic wave equation, and its solution is expressed as an exponentially decaying wave form along the propagation direction. The acceleration expression is obtained by taking the second time derivative of the displacement, and its dominant frequency component is extracted. Finally, based on the acceleration expression, the earthquake acceleration prediction coefficients in the horizontal and vertical directions are obtained respectively.

[0029] The specific steps for constructing the dynamic compaction vibration wave acceleration expression in S2 are as follows: S2.1: The motion of soil micro-elements is expressed by the one-dimensional viscoelastic wave equation, and the motion equation of the mass point is obtained, as shown in equation (21): (twenty one) in: For soil density, This represents the displacement of a soil particle during wave propagation. For the time of dissemination, Let the spatial coordinates be along the direction of wave propagation. The viscous damping coefficient is... Shear modulus; S2.2: The solution to the particle's motion equation is expressed as an exponentially decaying function propagating along the +z direction, as shown in equation (22): (twenty two) in: For particle displacement, The initial amplitude, Let ξ be the natural constant and ξ be the spatial attenuation coefficient. The time-domain attenuation coefficient, The wave number is denoted as k = ω / V. s , Angular acceleration, V is the time retention factor. s This represents the propagation speed of transverse seismic waves in the fill. Equation (22) uses an exponential decay function and a time retention factor to describe the propagation and decay of vibration waves. This form is equivalent to the response form of an impulse function, which is used to simulate the acceleration input caused by dynamic compaction vibration. This expression combines the characteristics of impulse functions (such as instantaneity and decay) and simulates the amplitude decay caused by material damping through an exponential function, thereby characterizing the acceleration input of vibration waves caused by dynamic compaction vibration.

[0030] In this embodiment, the construction principle of equation (22) is based on the physical characteristics of dynamic compaction vibration, that is, the vibration propagates in the form of waves and decays exponentially with distance. This assumed solution takes into account the spatial decay and time periodicity of the vibration, and is a solution method for solving differential equations.

[0031] S2.3: Calculate the displacement The second time derivative is shown in equation (23): (twenty three) in: To accelerate the compaction, Let z be the acceleration of a particle at a certain position z and time t. In this embodiment, the construction principle of equation (23) is as follows: acceleration is the second derivative of displacement, and according to the definition, the functional relationship shown in equation (24) is: (twenty four) Based on this, calculate the displacement. The second time derivative can be obtained from equation (23).

[0032] Acceleration of the particle The dominant frequency component is written in the form shown in equation (25): (25) In this embodiment, let β≪ω, then equation (25) can be written in the form shown in equation (26): (26) make: (27) in: For amplitude coefficient, The normalization constant is This is the initial amplitude; In this embodiment, the normalization constant g is used to make K dimensionless. Then, the horizontal distance R from the measuring point to the tamping core and the vertical distance H from the measuring point to the tamping point are used to represent the distance from the measuring point to the tamping point. The common forms of the acceleration of the dynamic compaction vibration wave in the horizontal and vertical directions can be obtained, as shown in equations (29) and (30): (29) (30) In this embodiment, the horizontal spatial attenuation coefficient e -ξR Vertical spatial attenuation coefficient e -ξH The horizontal seismic acceleration coefficient k is obtained by incorporating it into the amplitude coefficient K. h Vertical seismic acceleration coefficient k v As shown in equations (31) and (32): (31) (32) S2.4: Substituting equations (31) and (32) into equations (29) and (30), the horizontal and vertical accelerations of the dynamic compaction vibration wave are obtained as shown in equations (33) and (34): (33) (34) in: This refers to the horizontal seismic acceleration. Let k be the vertical seismic acceleration. h k is the horizontal seismic acceleration coefficient. v V is the vertical seismic acceleration coefficient, g is the gravitational acceleration, β is the time-domain attenuation coefficient, ω is the angular acceleration, and V is the vertical seismic acceleration coefficient. s This represents the propagation speed of transverse seismic waves in the fill. This represents the propagation speed of longitudinal seismic waves in the fill. The horizontal and vertical seismic acceleration prediction coefficients obtained in S2 are as follows: The prediction coefficients for horizontal and vertical seismic acceleration are shown in equations (35) and (36): (35) (36) in: Here, C is the horizontal seismic acceleration coefficient, and C is a constant. The horizontal distance between the measuring point and the compaction point; This is the vertical distance between the measuring point and the compaction point.

[0033] Example 1: Example 1 takes the dynamic compaction test site in Longgang District, Shenzhen as an example. The soil at the test site is clay sand. A set of test data with a compaction energy of 3000 KN·m is used to verify the prediction method of the spatiotemporal attenuation law of dynamic compaction vibration based on the pulse function proposed in this application.

[0034] First, the calculation parameters and measured data in Tables 1 and 2 were determined through on-site investigation and testing, and used for the calculations in this application; Table 1 Calculation parameters

[0035] Table 2 Measured Data

[0036] Substitute the data from Tables 1 and 2 into equation (18), and the variables in equation (18) will be... The x and y values ​​are calculated using M (mast of the falling weight) × g (acceleration due to gravity) × D (fall distance). The results are shown in Table 3. A graph is plotted using the data in Table 3. Figure 1 As shown: Table 3 Parameter Summary Table

[0037] Figure 1 The slope of the curve between x and y is α, and the intercept is lnk. From this, the attenuation exponent α and the equivalent coefficient k in equation (19) can be calculated. Figure 1 It can be seen that the linear regression curve fitting effect of vibration velocity reaches 0.971.

[0038] Then, use Table 2 to obtain the horizontal distance between the measuring point and the compaction point. Vertical distance between the measuring point and the compaction point The horizontal distance between the measuring point and the compaction point Vertical distance between the measuring point and the compaction point Substituting the attenuation index α and the equivalent coefficient k from equation (19) into equations (31) and (32), the horizontal seismic acceleration prediction coefficient is obtained. Vertical seismic acceleration prediction coefficient .

[0039] Finally, the obtained horizontal seismic acceleration prediction coefficients were used... Vertical seismic acceleration prediction coefficient Substituting the vibration velocity v in Table 2 into equations (33) and (34), we obtain the predicted curve of acceleration versus time, i.e. Figure 2 The prediction curve of the present invention in the figure, from Figure 2 As can be seen from the comparison of the measured data and the calculated data when R=5m, the predicted curve and the measured data are highly consistent in terms of peak value and main fluctuation trend, with a goodness of fit of 0.903. This indicates that the method of the present invention can accurately predict the decay law and time history characteristics of dynamic compaction vibration.

Claims

1. A method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on impulse function, characterized in that, Includes the following steps: S1: Construct a dimensionless functional relationship between dynamic compaction vibration velocity and control parameters; S2: The vibration wave acceleration input caused by the dynamic compaction vibration is characterized by the response form of the impulse function. The dynamic compaction vibration wave acceleration expression is constructed by combining the exponential decay function and the time retention factor. Based on the dynamic compaction vibration wave acceleration expression, the horizontal direction seismic acceleration prediction coefficient and the vertical direction seismic acceleration prediction coefficient are obtained. S3: Plot acceleration and time prediction curves to analyze the spatiotemporal attenuation law of dynamic compaction vibration.

2. The method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on impulse function according to claim 1, characterized in that, The specific control parameters in S1 are: impact energy. Compression modulus Horizontal distance from measuring point to ramming core Vibration dominant frequency .

3. The method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on impulse function according to claim 1, characterized in that, The specific implementation method of S1 is as follows: Construct a dimensionless functional relationship between the dynamic compaction vibration velocity and the control parameters, including selecting the compaction energy, compression modulus, horizontal distance of the measuring point from the compaction core and the vibration dominant frequency as control parameters and determining their dimensions; construct the first and second dimensionless numbers, establish and solve a system of linear equations based on the consistency of dimensions to determine the undetermined exponent; Substituting the solution into the dimensionless expression and establishing the functional relationship between dimensionless numbers based on Buckingham's π theorem, the solution is ultimately transformed into a linear form characterizing the exponential decay law of vibration velocity and distance.

4. The method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on impulse function according to claim 1, characterized in that, The specific implementation method of S2 is as follows: Combining the exponential decay function and the time retention factor, an expression for the acceleration of the dynamic compaction vibration wave is constructed; the motion equation of the particle is derived based on the one-dimensional viscoelastic wave equation, and its solution is expressed as an exponentially decaying wave in the propagation direction; the acceleration expression is obtained by taking the second time derivative of the displacement, and its dominant frequency component is extracted; finally, based on the acceleration expression, the earthquake acceleration prediction coefficients in the horizontal and vertical directions are obtained respectively.

5. The method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on impulse function according to claim 1, characterized in that, The specific steps for S1 are as follows: S1.1: Take impact energy Compression modulus Horizontal distance from measuring point to ramming core Vibration dominant frequency As the fundamental unit of measurement, impact energy Vibration dominant frequency Compression modulus Horizontal distance from measuring point to ramming core The dimensions are shown in equations (1) to (4): （1） Where: [ ] is a dimensionless symbol, For impact energy, Mass is a unit of measurement. Length is a unit of measurement. The unit of measurement is time; （2） in: It is the compressibility modulus; （3） in: This is the dominant vibration frequency; （4） in: The distance between the measuring point and the tamping core is the horizontal distance. S1.2: Constructing the first dimensionless number With the second dimensionless number As shown in equations (5) to (6): （5） in: It is the first dimensionless number. The first undetermined exponent of the first dimensionless number. The second undetermined exponent of the first dimensionless number. The third undetermined exponent of the first dimensionless number. Vibration velocity; （6） in: It is the second dimensionless number. It is the first undetermined exponent of the second dimensionless number. The second undetermined exponent of the second dimensionless number. The third undetermined exponent of the second dimensionless number; S1.3: Substituting equations (1) to (4) into equations (5) to (6), we obtain the first dimensionless number. With the second dimensionless number The functional relationships are shown in equations (7) to (8); （7） （8）； S1.4: Establish a system of linear equations based on the principle of dimensional consistency, as shown in equations (9) to (10): （9） （10） Solving the linear equations (9) and (10) yields the undetermined exponent of the first dimensionless number. , , The undetermined exponent of the second dimensionless number , , As shown in equations (11) to (12): （11） （12）； S1.5: Substituting equations (11) and (12) into equations (5) and (6), we obtain the first dimensionless number. With the second dimensionless number As shown in equations (13) to (14): （13） （14）； S1.6: Establishing functional relationships based on Bukingham's π theorem ,Will Written in the form shown in equation (15): （15） Substituting equations (13) and (14) into equation (15), we obtain equation (16): （16） in: The first dimensionless number With the second dimensionless number Dependencies between them; Based on the vibration attenuation law of dynamic compaction, equation (16) can be written as equation (17): （17） Where: k is the equivalent coefficient, and α is the decay exponent; Taking the logarithm of both sides of equation (17) and transforming it into a linear form as shown in equation (18): （18）； Where: y is the normalized velocity, and x is the normalized distance.

6. The method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on impulse function according to claim 1, characterized in that, The specific steps for constructing the dynamic compaction vibration wave acceleration expression in S2 are as follows: S2.1: The motion of soil micro-elements is expressed by the one-dimensional viscoelastic wave equation, and the motion equation of the particle is obtained as shown in equation (19): （19） in: For soil density, This represents the displacement of a soil particle during wave propagation. For the propagation time of the vibration wave, Let the spatial coordinates be along the direction of wave propagation. The viscous damping coefficient is... Shear modulus; S2.2: The solution to the particle's motion equation is expressed as an exponentially decaying function propagating along the +z direction, as shown in equation (20): （20） in: For particle displacement, The initial amplitude, Let ξ be the natural constant and ξ be the spatial attenuation coefficient. The time-domain attenuation coefficient, For wave number, Angular acceleration, The time retention factor; S2.3: Calculate the displacement The second time derivative is shown in equation (21): （21） in: To accelerate the compaction, Let z be the acceleration of a particle at a certain position z and time t. Acceleration of the particle The dominant frequency component is written in the form shown in equation (22): （22）； S2.4: The horizontal and vertical accelerations of the dynamic compaction vibration wave are obtained as shown in equations (23) and (24): （23） （24） in: This refers to the horizontal seismic acceleration. Let k be the vertical seismic acceleration. h k is the horizontal seismic acceleration coefficient. v V is the vertical seismic acceleration coefficient, g is the gravitational acceleration, and V is the acceleration due to gravity. s This represents the propagation speed of transverse seismic waves in the fill. This represents the propagation speed of longitudinal seismic waves in the fill.

7. The method for predicting the spatiotemporal attenuation law of dynamic compaction vibration based on impulse function according to claim 1, characterized in that, The horizontal and vertical seismic acceleration prediction coefficients obtained in S2 are as follows: The prediction coefficients for horizontal and vertical seismic acceleration are shown in equations (25) and (26): （25） （26） in: Here, C is the horizontal seismic acceleration coefficient, and C is a constant. The horizontal distance between the measuring point and the compaction point; This is the vertical distance between the measuring point and the compaction point.

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