An analytical calculation method for cylindrical wave scattering by periodic barrier structures

Abstract

The present invention belongs to the field of barrier vibration isolation technology, specifically to an analytical calculation method for the scattering of cylindrical waves by periodic barrier structures. This proposed analytical calculation method for the scattering of cylindrical waves by periodic barrier structures first utilizes a wave function expansion method to perform a zero-order Hanke expansion of the incident cylindrical wave. The scattered wave fields of all piles are then solved using the boundary conditions of all piles. This theoretical method describes the vibration isolation effect of a finite group pile barrier on cylindrical waves, addressing the lack of consideration of the impact of incident wave curvature in previous theoretical analyses.

Description

Technical Field

[0001] The present invention belongs to the technical field of barrier vibration isolation, and in particular relates to an analytical calculation method for cylindrical wave scattering by a periodic barrier structure. Background Art

[0002] Numerous scholars, both domestic and international, have conducted research on barrier vibration isolation, achieving fruitful results. This research covers a wide range of topics and employs diverse methods, which can be summarized into three categories: experimental research, numerical calculations, and theoretical analysis. Experimental research can intuitively reveal the underlying principles, numerical calculations facilitate simulation of diverse models, and theoretical analysis, primarily based on analytical methods, offers irreplaceable accuracy and insight into the nature of the problem.

[0003] However, the incident waves targeted by relevant research are all plane waves, and no research considering the influence of the curvature of the incident wave has been reported. In pile driving construction, a common type of environmental vibration, most of the energy of the hammer is diffused outward through the soil in the form of elastic waves (including body waves and surface waves) at the pile tip and pile body (end-bearing piles and friction piles). Since body waves decay faster than surface waves, the impact of pile driving vibration in the far field is mainly due to surface waves, but in the near field it is mainly due to body waves, and the curvature of the incident wave in the near field cannot be ignored. Therefore, it is necessary to consider the influence of the curvature of the incident wave when studying barrier vibration isolation problems, especially near-field vibration isolation. Analytical calculation methods related to the scattering of cylindrical waves by periodic barrier structures are urgently needed to be developed. Summary of the Invention

[0004] In order to solve the problems and shortcomings existing in the prior art, the present invention takes into account the influence of the curvature of the incident wave field, adopts a vibrating single pile based on the Novak pile-soil plane strain model as the seismic source, establishes a pile row vibration isolation model with cylindrical waves as the incident wave field, and proposes an analytical calculation method for the scattering of cylindrical waves by a periodic barrier structure.

[0005] The technical solution adopted by the present invention to solve the technical problem is: an analytical calculation method for cylindrical wave scattering by a periodic barrier structure, comprising the following steps:

[0006] (1) Construct a pile row vibration isolation model with cylindrical waves as the incident wave field. The piles are located in isotropic homogeneous elastic soil and numbered by row and column.

[0007] (2) The cylindrical wave source is represented by the wave function expansion method, and the Graf addition theorem is used to obtain the representation of the incident wave field in the arbitrary scattered wave field;

[0008] (3) Solve the scattered wave field of the pile group; superimpose the incident wave field and all scattered wave fields to obtain the total wave field in the soil;

[0009] (4) The total wave field expression in the soil and pile is expressed as stress σrr , σ rθ and displacement u r 、u θ ,; Bring the total wave field in the soil and the scattered wave field in the pile into the boundary condition formula;

[0010] (5) Solve the equations to obtain all the unknown coefficients, substitute these coefficients into the expression of the scattered wave field of any pile in the local coordinate system of the pile to be determined, and obtain the scattered field of all piles;

[0011] (6) The scattered fields of all piles are superimposed on the incident field formula to obtain the total wave field expression in the soil, and then all the stresses and displacements are obtained.

[0012] Preferably, in the step (1), the pile vibration isolation model with the cylindrical wave as the incident wave field is an N*M array, and the center of the pile body in the 1st row and the 0th column is taken as the coordinate origin to establish an overall plane rectangular coordinate system (x, y); take any one of the pile bodies as the research object, and take its center as the coordinate origin to establish a local plane rectangular coordinate system (x c,j ,y c,j ); Among them, the polar coordinate system corresponding to the global plane rectangular coordinate system (x, y) is (r, θ), and the local plane rectangular coordinate system (x c,j ,y c,j ) corresponds to the polar coordinate system (r c,j ,θ c,j ).

[0013] Preferably, in step (2), in the local coordinate system (r0, θ0) of the cylindrical wave source, the incident cylindrical wave can be expressed as:

[0014]

[0015] in, is the amplitude of the incident cylindrical P wave, α1 is the wave number of the P wave in the soil, the positive form i represents the imaginary unit, and the superscript (i) represents the incident wave field; is the wave field representation of the incident cylindrical P wave in the local coordinate system (r0, θ0), is the wave field representation of the incident cylindrical SV wave in the local coordinate system (r0, θ0), ψ0 is the amplitude of the incident cylindrical SV wave, and β1 is the wave number of the SV wave in the soil; represents the first kind of zero-order Hankel function;

[0016] The incident cylindrical wave is expressed in the local coordinate system corresponding to each pile body. Here we use the Graf addition theorem applicable to the transformation between arbitrary coordinate systems, that is:

[0017]

[0018] in:

[0019]

[0020] Among them, C n (r1) represents the coefficient of the fluctuation at a specific location; cosnθ1 and sinnθ1 represent the angular distribution of the fluctuation. m (r0) is the first kind m-order Bessel function, which is used to describe the radial fluctuation distribution; and is a function describing the relationship between two coordinate systems, where and denote the radial wave components propagating in the forward and reverse directions, respectively, and It represents the angular wave components of forward and reverse propagation; e is the distance between the two coordinate systems, δ is the angle between the two coordinate systems, and the relative position relationship between the two coordinate systems can be uniquely determined by a set of e and δ; ε m is a sign factor, which takes the value of 1 when m=0 and takes the value of 2 when m≠0; C n+m (e) and C n+m (e) represents the coefficient function related to the coordinate system transformation, which is used to describe the contribution of fluctuations at different positions; n and m are the harmonic order of the angular distribution and the order of the Bessel function, respectively, indicating the contribution of fluctuations at different orders;

[0021] Substituting the expression of the incident cylindrical wave into the Graf addition theorem expression, we can obtain the representation of the free field in any coordinate system.

[0022] Preferably, in step (3), the process of solving the pile group scattered wave field is:

[0023] (1) Solve the scattered wave field of each pile in its corresponding local coordinate system:

[0024]

[0025] in, They represent the P wave and SV wave in the local coordinate system (r c,j ,θ c,j ) in the scattered wave field, the superscript (r) represents the scattered wave field; represents the first kind n-order Hankel function; and is the scattering coefficient of the pile numbered (c, j) in the soil domain;

[0026] (2) According to the relative positions (e, δ) between the local coordinate systems of different piles, the Graf addition formula between arbitrary coordinate systems is applied to transform the scattered wave fields of all piles into the polar coordinate systems of all the piles to be determined:

[0027]

[0028] Among them, (p,q) is the pile to be determined; (r p,q ,θ p,q ) is the local polar coordinate system of the pile to be determined; (c, j) is any other pile.

[0029] Preferably, in step (3), the total wave field in the soil is expressed as:

[0030]

[0031] Preferably, in step (4), the boundary conditions are that the stress and displacement at the pile-soil interface are continuous, and the stress at the inner diameter of the pipe pile is zero.

[0032] This paper proposes an analytical calculation method for the scattering of cylindrical waves by periodic barrier structures. First, a zero-order Hanke expansion of the incident cylindrical wave is performed using a wave function expansion method. Then, the scattered wave field of all piles is calculated using the boundary conditions of all piles. This innovative technique theoretically describes the vibration isolation effect of a finite group of pile barriers on cylindrical waves, addressing the lack of consideration of the impact of incident wave curvature in previous theoretical analyses. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 This is a scattering analysis model of cylindrical waves by periodically distributed pile groups in an embodiment of the present invention;

[0034] Figure 2 It is the form of the interior domain Graf addition formula between any two coordinate systems;

[0035] Figure 3 The displacement contours for different incident wave curvatures under cylindrical P waves; (a) y0 = 10; (b) y0 = 20; (c) y0 = 50;

[0036] Figure 4 Displacement contours for different incident wave curvatures under cylindrical SV waves; where (a) y0 = 10; (b) y0 = 20; (c) y0 = 50. DETAILED DESCRIPTION

[0037] To facilitate understanding of the present invention, the present invention will be described in more detail below with reference to the accompanying drawings and specific embodiments. However, the present invention can be implemented in many different forms and is not limited to the embodiments described in the specification. On the contrary, the purpose of providing these embodiments is to make the understanding of the present invention more thorough and comprehensive.

[0038] The analytical calculation method for cylindrical wave scattering by a periodic barrier structure provided by the present invention comprises the following steps:

[0039] 1. Construct a pile vibration isolation model with cylindrical waves as the incident wave field, such as Figure 1 As shown in the figure, the piles are located in an isotropic homogeneous elastic soil and are numbered in rows and columns as “(1,1)”, “(1,2)”, “(1,3)”, etc., where the horizontal axis represents the number of rows and the vertical axis represents the number of columns, with a total of p rows. Figure 1 The model diagram takes a plum blossom arrangement as an example, but the arrangement form to which the method of the present invention can be applied is not limited to this. Any arrangement of piles can be solved using the theory proposed by the present invention by simply marking the position.

[0040] by Figure 1 The center of the pile body in the 1st row and 0th column is taken as the origin of coordinates to establish the overall plane rectangular coordinate system (x, y). At the same time, any pile body is taken as the research object, that is, the center of the pile body in the cth row and jth column (any pile body) is taken as the origin of coordinates to establish the local plane rectangular coordinate system (x c,j ,y c,j ). Among them, the polar coordinate system corresponding to the global rectangular coordinate system (x, y) is (r, θ), and the local rectangular coordinate system (x c,j ,y c,j ) corresponds to the polar coordinate system (r c,j ,θ c,j ).

[0041] The material properties of the soil are determined by its Lame constants λ1 and μ1 and mass density ρ1. All piles in the pile group barrier are identical and their material properties are determined by their Lame constants λ2 and μ2 and mass density ρ2. The pile cross-section radius is a, and the piles are distributed equiperiodically along the x-axis with a period of b. The row spacing along the y-axis is h. The incident waves are cylindrical SV and P waves, and the angles between the incident directions and the x-axis are θ and θ, respectively. SV and θ P , the incident circular frequency is ω, and the corresponding wave velocities in the soil are c β1 and c α1 , the corresponding wave velocity in the pile is c β2 and c α2 .

[0042] like Figure 1As shown in the figure, when cylindrical P and SV waves are incident, the wave function expansion method is used to represent the cylindrical wave source. Then, the Graf addition theorem applicable to the transformation between arbitrary coordinate systems is used to represent the incident wave field in an arbitrary scattered wave field. The total wave field after the incident wave field and the scattered wave field are superimposed to satisfy the boundary conditions of each pile body, and the equation group is constructed and solved to solve all the scattering coefficients to obtain the solution of the entire wave field.

[0043] 2. Representation of Cylindrical Wave Free Field

[0044] The key to solving this problem is how to represent the cylindrical wave free field in each scattered wave field. Therefore, the cylindrical wave expansion form is first derived and discussed. Here, the cylindrical P wave is used as an example for a detailed derivation. The derivation of the cylindrical SV wave only requires the transformation of the free field.

[0045] In the local coordinate system (r0, θ0) of the cylindrical wave source, the incident cylindrical P wave can be expressed as:

[0046]

[0047] in, is the amplitude of the incident cylindrical P wave, is the wave field representation of the incident cylindrical P wave in the local coordinate system (r0, θ0), α1 is the wave number of the P wave in the soil, the positive form i represents the imaginary unit, and the superscript (i) represents the incident wave field; To simplify the derivation, the time factor exp(-iωt) is omitted.

[0048] The incident cylindrical wave is expressed in the local coordinate system corresponding to each pile body. Here we use the Graf addition theorem applicable to the transformation between arbitrary coordinate systems, that is:

[0049]

[0050] in:

[0051]

[0052] Among them, C n (r1) represents the coefficient of the fluctuation at a specific position; cosnθ1 and sinnθ1 represent the angular distribution of the fluctuation; cosmθ0 and sinmθ0 represent the cosine and sine components of the fluctuation at angle θ0, respectively, and are used in the Fourier series expansion to describe the contribution of the fluctuation at different angles; specifically, cosnθ1 and sinnθ1 describe the angular distribution of the fluctuation in the local coordinate system (r1,θ1), while cosmθ0 and sinmθ0 describe the angular distribution of the fluctuation in the initial coordinate system (r0,θ0); in Figure 2In the figure, we can see the relationship between the two coordinate systems. By using Graf's addition theorem, these angular distributions can be converted between the two coordinate systems. m (r0) is the first kind m-order Bessel function, which is used to describe the radial fluctuation distribution; and is a function describing the relationship between two coordinate systems, where and denote the radial wave components propagating in the forward and reverse directions, respectively, and It represents the angular wave components of forward and reverse propagation; e is the distance between the two coordinate systems, δ is the angle between the two coordinate systems. The relative position relationship between the two coordinate systems can be uniquely determined by a set of e and δ. The relationship between the two coordinate systems is as follows: Figure 2 As shown; ε m is a sign factor, which takes the value of 1 when m=0 and takes the value of 2 when m≠0; C n+m (e) and C n+m (e) represents the coefficient function related to the coordinate system transformation, which is used to describe the fluctuation contribution at different positions; n and m are the harmonic order of the angular distribution and the order of the Bessel function, respectively, indicating the contribution of the fluctuation at different orders.

[0053] Substituting the expression of the incident cylindrical P wave into equation (2), we can obtain the representation of the free field in any coordinate system:

[0054]

[0055] in, is the incident cylindrical P wave in the local coordinate system (r c,j ,θ c,j ) in the wave field representation, is the amplitude of the incident wave cylinder P wave, and α1 is the wave number of the P wave in the soil. K m (e) is the combination of Hankel functions, is the first kind m-order Hankel function.

[0056] When the incident wave is a cylindrical SV wave, only the incident field is different. Let ψ0 be the amplitude of the cylindrical SV wave, and the incident wave field can be expressed as:

[0057]

[0058] in, is the wave field representation of the incident cylindrical SV wave in the local coordinate system (r0, θ0), ψ0 is the amplitude of the incident cylindrical SV wave, and β1 is the wave number of the SV wave in the soil.

[0059] 3. Solving the Scattering Wave Field of Pile Groups

[0060] The presence of piles in the soil will generate scattered wave fields under the action of the incident wave field, including P waves and SV waves. Therefore, it is necessary to express the scattered wave fields in the local coordinate system corresponding to each pile. Figure 1 The local polar coordinate system (r c,j ,θ c,j ), the Fourier-Bessel series expression of the pile scattered wave field is:

[0061]

[0062] in, They represent the P wave and SV wave in the local coordinate system (r c,j ,θ c,j ) in the scattered wave field, the superscript (r) represents the scattered wave field; represents the nth-order Hankel function of the first kind. and is the scattering coefficient of the pile (c, j) in the soil domain. The subscript n represents the number of terms in the Fourier series expansion, which describes the contributions of different orders of the wave motion. Specifically, each term in the Fourier series corresponds to a specific harmonic component of the wave motion, where n is the order of that harmonic. By summing all the harmonic components, the complete wave motion field can be constructed.

[0063] At this time, according to the relative positions (e, δ) between the local coordinate systems of different piles, the Graf addition formula between arbitrary coordinate systems is applied again to transform the scattered wave fields of all piles into the polar coordinate systems of all the piles to be solved. Here, the pile numbered (p, q) is taken as an example, and the scattered wave fields of other arbitrary piles (c, j) are transformed into the polar coordinate systems of the local coordinate system (r p,q ,θ p,q ) is expressed as follows:

[0064]

[0065] 4. Then, the incident wave field and all scattered wave fields are superimposed to obtain the total wave field in the soil:

[0066]

[0067] The scattered wave field in the pile body can be expressed as:

[0068]

[0069] in, and is the scattering coefficient to be determined in the pile domain, α2 and β2 are the wave numbers of P wave and SV wave in the pile, respectively.

[0070] 5. Express the total wave field in the soil and pile domains as stress σ rr , σ rθ and displacement u r 、u θ , using the stress and displacement continuity conditions at the pile-soil interface and the zero stress condition at the inner diameter of the pile:

[0071]

[0072] The total wave fields (8a) and (8b) in the soil and the scattered wave field (10) in the pile are substituted into the boundary condition (11). Based on the linear independence of trigonometric functions of different orders, the solution can be obtained quickly.

[0073] 6. Solve the equations to obtain all the unknown coefficients. Substitute them into equation (7) to obtain the scattered fields of all piles. Then superimpose the incident field equation (1) to obtain the total wave field expression in the soil, and then all the stresses and displacements can be obtained.

[0074] The method of the present invention is applied to the analytical calculation analysis of the scattering of cylindrical waves by periodic barrier structures. Taking PCC pipe piles (i.e., cast-in-place thin-walled concrete hollow pipe piles) as an example, the Poisson's ratios of soil and pipe piles are μ1=0.33 and μ2=0.25, respectively, the pile-soil shear wave velocity ratio is μ2 / μ1=1000, and the pile-soil density ratio is ρ2 / ρ1=1.3. The vibration isolation law of pile rows under cylindrical wave sources is analyzed using the method of the present invention, and the displacement contour map after the pile rows are incident with cylindrical P waves and SV waves is drawn as shown below. Figure 3 and Figure 4 shown.

[0075] from Figure 3 and Figure 4 It can be seen from the figure that for cylindrical P waves, as the curvature of the incident wave decreases (i.e., the distance between the wave source and the pile row increases), the effective vibration isolation range increases and the optimal vibration isolation area moves backward; for cylindrical SV waves, as the curvature of the incident wave decreases, the effective vibration isolation range increases and the weaker displacement attenuation area decreases, resulting in a more obvious vibration isolation effect.

Claims

1. Analytical calculation method for cylindrical wave scattering by periodic barrier structure, characterized by: The following steps are involved: (1) Construct a pile row vibration isolation model with cylindrical waves as the incident wave field. The piles are located in isotropic homogeneous elastic soil and numbered by row and column. (2) The cylindrical wave source is represented by the wave function expansion method, and the Graf addition theorem is used to obtain the representation of the incident wave field in the arbitrary scattered wave field. In the local coordinate system (r0, θ0) of the cylindrical wave source, the incident cylindrical wave is expressed as: in, is the amplitude of the incident cylindrical P wave, α1 is the wave number of the P wave in the soil, and i represents the imaginary unit; is the incident cylindrical SV wave in the local coordinate system (r c,j ,θ c,j ) is represented by the wave field, ψ0 is the amplitude of the incident wave cylinder SV wave, β1 is the wave number of the SV wave in the soil; represents the first kind of zero-order Hankel function; The incident cylindrical wave is expressed in the local coordinate system corresponding to each pile body. Here we use the Graf addition theorem applicable to the transformation between arbitrary coordinate systems, that is: in: Among them, C n (r1) represents the coefficient of the fluctuation at a preset position; cos nθ1 and sin nθ1 represent the angular distribution of the fluctuation; J m (r0) is the first-order m-order Bessel function, which is used to describe the radial fluctuation distribution; F1 ± (e,δ) and It is a function describing the relationship between two coordinate systems, where F1 + (e, δ) and F1 - (e, δ) represent the radial wave components propagating in the forward and reverse directions, respectively. and It represents the angular wave components of forward and reverse propagation; e is the distance between the two coordinate systems, δ is the angle between the two coordinate systems, and the relative position relationship between the two coordinate systems can be uniquely determined by a set of e and δ; ε m is a sign factor, which takes the value of 1 when m=0 and takes the value of 2 when m≠0; C n+m (e) and C n+m (e) represents the coefficient function related to the coordinate system transformation, which is used to describe the contribution of fluctuations at different positions; n and m are the harmonic order of the angular distribution and the order of the Bessel function, respectively, indicating the contribution of fluctuations at different orders; Substituting the expression of the incident cylindrical wave into the Graf addition theorem expression, we can obtain the representation of the free field in any coordinate system; (3) Determine the scattered wave field of the pile group; superimpose the incident wave field and all scattered wave fields to obtain the total wave field in the soil; (4) The total wave field expression in the soil and pile is expressed as stress σ rr , σ rθ and displacement u r 、u θ ,; Bring the total wave field in the soil and the scattered wave field in the pile into the boundary condition formula; (5) Solve the equations to obtain all the unknown coefficients, substitute these coefficients into the expression of the scattered wave field of any pile in the local coordinate system of the pile to be determined, and obtain the scattered field of all piles; (6) The scattered fields of all piles are superimposed on the incident field formula to obtain the total wave field expression in the soil, and then all the stresses and displacements are obtained.

2. The analytical calculation method for cylindrical wave scattering by a periodic barrier structure according to claim 1, characterized in that: In the step (1), the pile vibration isolation model takes the center of the pile body in the first row and the 0th column as the coordinate origin to establish the overall plane rectangular coordinate system (x, y); take any one of the pile bodies as the research object, take its center as the coordinate origin, and establish the local plane rectangular coordinate system (x c,j ,y c,j ); Among them, the polar coordinate system corresponding to the global plane rectangular coordinate system (x, y) is (r, θ), and the local plane rectangular coordinate system (x c,j ,y c,j ) corresponds to the polar coordinate system (r c,j ,θ c,j ).

3. The analytical calculation method for cylindrical wave scattering by a periodic barrier structure according to claim 2, characterized in that: In step (3), the solution process of the pile group scattered wave field is: Solve the scattered wave field of each pile in its corresponding local coordinate system: in, and is the scattering coefficient of the pile (c, j) in the soil domain; According to the relative positions (e, δ) between the local coordinate systems of different piles, the Graf addition formula between arbitrary coordinate systems is applied to transform the scattered wave fields of all piles into the polar coordinate systems of all the piles to be determined: Among them, (p,q) is the pile to be determined; (r p,q ,θ p,q ) is the local polar coordinate system of the pile to be determined; (c, j) is any other pile.

4. The analytical calculation method for cylindrical wave scattering by a periodic barrier structure according to claim 3, characterized in that: In step (3), the total wave field in the soil is expressed as:

5. The analytical calculation method for cylindrical wave scattering by a periodic barrier structure according to claim 1, characterized in that: In the step (4), the boundary conditions are that the stress and displacement at the pile-soil interface are continuous and the stress at the inner diameter of the pipe pile is zero.

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