A distributed dynamic load identification method suitable for a vehicle wallboard structure surface

Abstract

The application discloses a distributed dynamic load identification method suitable for a vehicle wall plate structure surface, which comprises the following steps: S1, establishing a vehicle wall plate structure finite element model; S2, dividing sub-regions on the vehicle wall plate structure surface subjected to the distributed dynamic load, and performing planarization treatment on the three-dimensional structure surface to construct a transfer function matrix of the load acting on each sub-region after planarization and the strain response of a structure measuring point on the finite element model; S3, obtaining the measured strain response of the wall plate structure subjected to the distributed dynamic load at the measuring point, and solving the load values on the grid nodes of each sub-region by using the time sequence of the measured strain response and the transfer function matrix, and reconstructing the distributed dynamic load on the structure surface by using an interpolation function. The application inverses the distributed dynamic load on the whole structure surface by the strain response of the limited measuring points, provides an indirect acquisition method suitable for the distributed dynamic load of the engineering structure, and provides real and effective load information for the state monitoring and safety design of the engineering structure.

Description

Technical fields:

[0001] This invention proposes a method for identifying distributed dynamic loads on the surface of vehicle panel structures, which belongs to the field of inverse structural dynamics problems. Background technology:

[0002] Vehicle panel structures refer to the external structures of vehicles such as aircraft, ships, and vehicles. Their external load information is crucial for structural dynamics analysis and condition monitoring. However, in service, it is difficult to directly measure and obtain external dynamic load information for many vehicle panel structures by placing force sensors on the structural surface. Therefore, it is necessary to develop dynamic load identification methods to invert the external loads experienced by the structure using its response.

[0003] Traditional methods for identifying distributed dynamic loads transform the problem into estimating a small number of parameters within a distributed dynamic load identification model. Methods based on orthogonal polynomials to characterize the spatial distribution of loads generally only identify distributed dynamic loads within regular two-dimensional planar regions. Identifying continuously distributed dynamic loads on three-dimensional structural surfaces requires considering both the three-dimensional spatial distribution and time history of the load, making it significantly more challenging, and existing dynamic load identification methods have poor applicability. Therefore, a new method is needed to identify dynamic loads with complex spatial distribution characteristics on the surface of vehicle panel structures. Summary of the Invention

[0004] The purpose of this invention is to provide a method for identifying distributed dynamic loads on the surface of vehicle panel structures, solving the problem of identifying three-dimensional distributed dynamic loads with non-stationary characteristics in spatial distribution, and providing an indirect means of obtaining dynamic loads for the design and condition monitoring of engineering structures operating in three-dimensional distributed load environments.

[0005] Technical Solution: To achieve the above-mentioned objectives, this invention proposes a method for identifying distributed dynamic loads on the surface of a vehicle's wall panel structure. This method includes the following steps:

[0006] S1. Establish a finite element model of the vehicle's wall panel structure;

[0007] S2. Divide the surface of the vehicle wall panel structure under the action of distributed dynamic load into sub-regions, and planarize the three-dimensional structure surface to construct the transfer function matrix of the load acting on each sub-region and the strain response of the structural measuring points on the finite element model after planarization.

[0008] S3. Obtain the measured strain response at the measuring point of the wall panel structure under distributed dynamic load, and use the time series of the measured strain response and the transfer function matrix to solve the load values ​​on the grid nodes of each sub-region, and use the interpolation function to reconstruct the distributed dynamic load on the surface of the structure.

[0009] Furthermore, in step S2, the structural surface subjected to the distributed dynamic load is divided into sub-regions, and the three-dimensional structural surface is planarized. The transfer function matrix of the load acting on each sub-region and the structural strain response on the finite element model is constructed after planarization. The specific steps are as follows:

[0010] S21: Divide the structural surface subjected to distributed dynamic loads into sub-regions and planarize the three-dimensional structural surface. After dividing the structural surface subjected to distributed dynamic loads into sub-regions, the corresponding discretized spatial surfaces are obtained. Let the discretized spatial surface T1(x,y,z) have a corresponding unfolded plane T2. T1 and T2 have the same number of nodes N and line segments M. The coordinates of node i on T1 are {X... i ,Y i Z i}, the length of line segment j is S j The coordinates of node i on T2 are {x} i ,y i ,z i}, the length of line segment j is S j * ;

[0011] If T1 and T2 are considered as hinged rod systems, then the spatial hinged rod system corresponding to T1 can be regarded as the final structure of the planar hinged rod system corresponding to T2 after deformation due to unbalanced forces between the nodes. Here, the unbalanced force F at the i-th node... i as follows:

[0012]

[0013] In the formula, F ix F iy Let F be the unbalanced force at the i-th node. i The components in the x and y directions, M i Let l be the set of line segments that intersect at node i. j and m j Let be the cosines of line segment j along the x and y directions in the set, and let E and A be the elastic modulus and cross-sectional area of ​​the assumed line segment j, respectively.

[0014] Let the set of all unbalanced forces at all nodes be a vector {F}. u}, use the Newton-Raphson method to iteratively solve {F} u The structural geometric displacement under the action of {F} is gradually reduced and eventually eliminated. u Finally, the unfolded plane of the spatial surface is obtained; by performing planarization processing on the surface, the discretized sub-region spatial surface (x,y,z) is unfolded into a discrete sub-region plane.

[0015] S22: When the structural surface subjected to distributed dynamic loads is divided into sub-regions, the load distribution within each sub-region is determined by the load values ​​on the shared nodes of its boundaries. The load magnitude within a sub-region is obtained by multiplying the node load value by the interpolation function corresponding to the node in that grid. Thus, the distributed dynamic load within a single sub-region is expressed as follows:

[0016]

[0017] In the formula, m N This indicates the number of nodes in the sub-region. p represents the interpolation function corresponding to the i-th node in this sub-region. i (t) represents the load value at the node that changes over time. After dividing the load sub-regions, the interpolation function corresponding to each node is selected. Furthermore, the shape function of the finite element corresponding to the shape of the sub-region is directly selected as the interpolation function. The interpolation function determines the distribution form of the load in each sub-region. The load value at the node controls the coefficient of the distribution function. Solving for the load values ​​at all sub-region nodes reconstructs the distributed dynamic load of the entire structural surface. The distributed dynamic load of the structural surface is shown in the following formula:

[0018]

[0019] In the formula, m P m represents the number of nodes contained in the subregion. e This represents the number of sub-regions connected to the j-th node. Let represent the interpolation function within the i-th subregion to which the j-th node belongs;

[0020] S23: To obtain the strain response at the measuring point, the distributed dynamic load is transformed into nodal loads on the structural finite element model using the finite element method. The element nodal loads are obtained by integration within the structural finite element model.

[0021]

[0022] In the formula, The shape function of an element in a structural finite element model. p represents the interpolation function corresponding to the i-th node in this sub-region. i (t) represents the nodal load value as a function of time;

[0023] The nodal loads on the structural finite element model, which are equivalent to the distributed dynamic loads on the sub-region, are applied to the finite element model, and the strain response at the model measurement points is calculated.

[0024] S24: Based on the Green's function of a linear system, the transfer function relationship between the distributed dynamic load and the strain response at the measuring point on the sub-region is obtained as follows:

[0025]

[0026] In the formula, ε i Let p be the strain response vector at the i-th measuring point. i Let m be the distributed dynamic load vector on the i-th sub-region node. S m represents the number of strain measurement points. P G represents the number of nodes in the load sub-region. i,j This represents the transfer matrix between the distributed load corresponding to the node in the j-th sub-region and the strain response at the i-th measuring point;

[0027] Equation (5) can be simplified to the following:

[0028] {ε}=[G]{P} (6)

[0029] In the formula, {ε} is the strain response vector, which contains the time-domain response sequence at each strain measurement point, [G] is the system transfer matrix, and {P} is the load vector, which contains the load time sequence at each sub-region grid node;

[0030] For each node, a unit distributed pulse load in the form of a shape function is applied to the sub-region connected to that node. By acquiring the transient strain response at the measuring point, the transfer relationship matrix G between the nodal load values ​​and the structural strain response in the sub-region is solved. ij Repeat the operation on all nodes to solve for the system transfer matrix G.

[0031] Furthermore, in step S2, when the sub-region is a quadrilateral, the load magnitude within each sub-region is obtained by multiplying the load values ​​at the four nodes of that region by the corresponding interpolation function and then summing them. Also, when the sub-region is a quadrilateral, the specific expression for the interpolation function of a single node in the four adjacent sub-regions is:

[0032]

[0033] In the formula, L x ,L y Each sub-region represents the following: The length of the direction, D1 to D4, represents the four sub-regions connected by the node.

[0034] Furthermore, in step S3, the measured strain response of the wall panel structure under distributed dynamic load at the measuring point is obtained, and the load values ​​on the grid nodes of each sub-region are solved using the time series of the measured strain response and the transfer function matrix. The distributed dynamic load on the structural surface is reconstructed using the interpolation function. The specific steps are as follows:

[0035] S31. Using the least squares method, from equation (6), we can obtain,

[0036] {P}=[G]+ {ε} (8)

[0037] In the formula, [G] + The generalized inverse matrix of the system transfer function matrix is ​​used to solve for the time history of the load values ​​at the grid nodes in the sub-region.

[0038] S32. After solving for the load values ​​at the nodes of each sub-region using equation (8), the distributed dynamic load on the entire structural surface is expressed as:

[0039]

[0040] In the formula, m P m represents the number of nodes contained in the sub-region. N N represents the number of sub-regions connected to the j-th node. ji This represents the interpolation function for the j-th node within the i-th sub-region it is connected to.

[0041] Beneficial effects: Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

[0042] (1) Existing distributed dynamic load identification technologies are generally applicable to the identification of two-dimensional distributed loads. There is a lack of effective identification methods for three-dimensional distributed dynamic loads. The method for identifying distributed dynamic loads using strain response information from finite measuring points provided in this invention can accurately identify distributed dynamic loads on the surface of three-dimensional structures and has certain advantages.

[0043] (2) Using shape functions to represent the spatial distribution of loads has certain advantages over the spatial representation of distributed dynamic loads based on orthogonal polynomials. Attached Figure Description

[0044] Figure 1 This is a flowchart illustrating the logic of the method of the present invention.

[0045] Figure 2 This is a schematic diagram of a curved panel subjected to distributed dynamic loads.

[0046] Figure 3 This is a schematic diagram of the load sub-region.

[0047] Figure 4 a represents the time history distribution identification result of the distributed dynamic load acting on the curved panel structure, b represents the time history distribution error of the distributed dynamic load acting on the curved panel structure, and c represents the spatial distribution reference value and identification result of the three-dimensional distributed dynamic load. Detailed Implementation

[0048] The technical solution of the present invention will be described in detail below through embodiments. However, the embodiments are only preferred embodiments of the present invention. It should be noted that for those skilled in the art, several improvements and equivalent substitutions can be made to the structure and dynamic load form without departing from the principle of the present invention. These improved and equivalent substitutions of the technical solutions of the claims of the present invention all fall within the protection scope of the present invention.

[0049] Example, for such Figure 2 The curved panel structure shown is used to identify the distributed dynamic load acting on its surface. It is 0.2m long, 0.15m wide, and 0.002m thick, with a maximum curvature of 0.0014 along its two wide sides. The boundary condition is that all four sides are fixed. The material is steel with an elastic modulus of 2.1 × 10⁻⁶. 11 Pa, Poisson's ratio is 0.3, and density is 7850 kg·m³. -3 The modal damping ratios of the structure are ξ. i =0.02. The time history distribution of the reference distributed dynamic load adopts a sinusoidal distribution, and the spatial distribution is a quadratic surface. The specific expression is as follows:

[0050] F(x,y,t)=(2x 2 +3y 2 +5)×(sin(2πt)+0.3sin(3πt)+0.5sin(4πt)) (1)

[0051] The technique of this invention for identifying distributed dynamic loads from measured structural strain response data specifically includes the following steps:

[0052] S1: Establish the finite element model of the vehicle's panel structure. In this example, the structure is a simple curved panel structure, including the following steps:

[0053] S11: Establish the initial finite element model of the actual structure;

[0054] S12: Obtain modal data of the actual structure based on modal testing;

[0055] S13: Based on the modal data from the modal tests, the initial finite element model is modified to obtain the modified finite element model.

[0056] The distributed dynamic load time-domain identification method, in step S2, divides the structural surface where the distributed dynamic load acts into sub-regions, and performs planarization processing on the three-dimensional structural surface to construct the transfer function matrix between the load acting on each sub-region and the strain response of the structural measuring points on the finite element model. The specific method includes the following steps:

[0057] S21: Divide the surface of the vehicle's wall panel structure under distributed dynamic loads into sub-regions and planarize the three-dimensional surface. After dividing the surface into sub-regions, the corresponding discretized spatial surfaces are obtained. Let the discretized spatial surface T1(x,y,z) have a corresponding unfolded plane T2. T1 and T2 have the same number of nodes N and line segments M. The coordinates of node i on T1 are {X... i ,Y i Z i}, the length of line segment j is S j The coordinates of node i on T2 are {x} i ,y i ,z i}, the length of line segment j is S j * ,

[0058] If T1 and T2 are considered as hinged rod systems, then the spatial hinged rod system corresponding to T1 can be regarded as the final structure after deformation due to unbalanced forces between the nodes of the planar hinged rod system corresponding to T2, where the unbalanced force F at the i-th node is... i writing:

[0059]

[0060] In the formula, F ix F iy Let F be the unbalanced force at the i-th node. i The components in the x and y directions, M i Let l be the set of line segments that intersect at node i. j m j Let be the cosines of line segment j along the x and y directions in the set, and let E and A be the assumed elastic modulus and cross-sectional area of ​​line segment j, respectively.

[0061] Let the set of all unbalanced forces at all nodes be a vector {F}. u}, use the Newton-Raphson method to iteratively solve {F} u The structural geometric displacement under the action of {F} is gradually reduced and eventually eliminated. u Finally, the unfolded plane of the spatial surface can be obtained.

[0062] By performing planarization on the surface, the discretized sub-region spatial surface (x,y,z) is equivalent to a discrete sub-region plane.

[0063] S22. Sub-region division of the load-bearing surface as follows: Figure 3As shown, the system comprises 36 rectangular regions and 49 shared nodes. The load magnitude within each sub-region is obtained by multiplying the load values ​​at the four nodes of that region by their corresponding interpolation functions and then summing the results. The specific expression for the interpolation function of a single node in its four adjacent sub-regions is as follows:

[0064]

[0065] In the formula, L x ,L y Each sub-region represents the following: The length of the direction, D1 to D4, represents the four sub-regions connected by the node.

[0066] Therefore, the distributed dynamic load within a single sub-region can be expressed as:

[0067]

[0068] In the formula, m N This indicates the number of nodes in the sub-region. p represents the interpolation function corresponding to the i-th node in this sub-region. i (t) represents the load value at the center node of the sub-region that varies with time. After dividing the load sub-regions, the interpolation function corresponding to each node is selected. Here, the shape function of the finite element corresponding to the sub-region can be directly selected as the interpolation function. The interpolation function determines the distribution of the load in each sub-region, and the load value at the center node controls the coefficient of the distribution function. By solving for the load values ​​at all nodes, the distributed dynamic load on the entire structural surface can be reconstructed. The distributed dynamic load on the structural surface is shown in the following equation:

[0069]

[0070] In the formula, m P m represents the number of nodes contained in the subregion. e This represents the number of sub-regions connected to the j-th node. This represents the interpolation function for the j-th node within the i-th sub-region it is connected to.

[0071] S23. To obtain the strain response at the measuring point, the distributed dynamic load is first transformed into nodal loads on the structural finite element model using the finite element method. The nodal loads are obtained by integrating within the structural finite element model:

[0072]

[0073] In the formula, The shape function of an element in a structural finite element model. p represents the interpolation function corresponding to the i-th node in this sub-region. i(t) represents the nodal load value that varies with time.

[0074] S24: Based on the Green's function of a linear system, the transfer function relationship between the distributed load and the strain response at the measuring point is obtained, specifically in the following form:

[0075]

[0076] In the formula, {ε i},{P j Let} represent the response column vector at the i-th measuring point and the distributed dynamic load P caused by the j-th grid node, respectively. j The column vector consisting of the values ​​at each time point; m s ,m L These represent the number of measuring points and the number of nodes in the sub-region, respectively. In this embodiment, 20 strain measuring points are selected in three columns along the length of the plate, i.e., m s =60 (Take the three-directional strain at 20 measuring points, ε) x ,ε y and ε xy ), m L =49. G ij This represents the impulse response function g obtained at the i-th measuring point under the distributed dynamic load corresponding to the j-th sub-region node of the structure. ij The impulse response matrix composed of (t).

[0077] For each node, an impact load is applied to the four sub-regions connected to that node. By acquiring the transient strain response at the measuring points, the transfer relationship matrix G between the nodal load values ​​and the structural strain response in the sub-regions is solved. ij Repeat the operation on all nodes to solve for the system transfer matrix G.

[0078] S3. Obtain the measured strain response at the measuring point of the wall panel structure under distributed dynamic load, and use the time series of the measured strain response and the transfer function matrix to solve for the load values ​​on the grid nodes of each sub-region. Reconstruct the distributed dynamic load on the structural surface using the interpolation function. The specific steps include:

[0079] S31. Using the least squares method, solve for the numerical results of the load mesh node values ​​changing with time, and transform equation (7) into:

[0080] {P}=[G] + {ε} (8)

[0081] In the formula, {P} is the column vector of sub-region nodal loads that varies with time; {ε} is the column vector of strain response of the wall panel structure under distributed dynamic load in three directions at the measuring point; [G] is the system transfer function matrix.

[0082] S42. After solving for the load values ​​at each node in the sub-region, the distributed dynamic load on the entire structural surface is expressed as:

[0083]

[0084] In the formula, m P m represents the number of nodes contained in the entire sub-region of the structure's surface. N N represents the number of sub-regions connected to the j-th node. ji This represents the interpolation function for the j-th node within the i-th sub-region it is connected to.

[0085] Figure 4 In the figure, Figures a and b show the identification results and errors of the dynamic load time history distribution, respectively. Figure 4 Figure c shows the actual spatial distribution and identification results of the three-dimensional distributed dynamic load. It can be seen that the identification method in this invention can accurately identify the spatially distributed dynamic load that changes over time through the strain sequence at the measuring point. Compared with existing two-dimensional distributed dynamic load identification methods, the method in this invention can more accurately describe the local characteristics of the dynamic load on a three-dimensional structure. In summary, this invention has certain advantages.

Claims

1. A method for identifying distributed dynamic loads on the surface of a vehicle's wall panel structure, characterized in that, The method includes the following steps: S1. Establish a finite element model of the vehicle's wall panel structure; S2. Divide the surface of the vehicle wall panel structure under the action of distributed dynamic load into sub-regions, and planarize the three-dimensional structure surface to construct the transfer function matrix of the load acting on each sub-region and the strain response of the structural measuring points on the finite element model after planarization. S3. Obtain the measured strain response at the measuring point of the wall panel structure under distributed dynamic load, and use the time series of the measured strain response and the transfer function matrix to solve the load values ​​on the grid nodes of each sub-region, and use the interpolation function to reconstruct the distributed dynamic load on the surface of the structure. In step S2, the structural surface subjected to distributed dynamic loads is divided into sub-regions, and the three-dimensional structural surface is planarized. The transfer function matrix of the loads acting on each sub-region and the strain response of the structure on the finite element model is constructed after planarization. The specific steps are as follows: S21: Divide the structural surface subjected to distributed dynamic loads into sub-regions and planarize the three-dimensional structural surface. After dividing the structural surface subjected to distributed dynamic loads into sub-regions, the corresponding discretized spatial surfaces are obtained. Let the discretized spatial surface T1(x,y,z) have a corresponding unfolded plane T2. T1 and T2 have the same number of nodes N and line segments M. The coordinates of node i on T1 are {X... i ,Y i Z i }, the length of line segment j is S j The coordinates of node i on T2 are {x} i ,y i ,z i }, the length of line segment j is S j * ; If T1 and T2 are considered as hinged rod systems, then the spatial hinged rod system corresponding to T1 is considered as the final structure after deformation due to nodal unbalanced forces between the nodes of the planar hinged rod system corresponding to T2. Here, the nodal unbalanced force F at the i-th node... i as follows: (1) In the formula, F ix F iy Let F be the unbalanced force at the i-th node. i The components in the x and y directions, M i Let l be the set of line segments that intersect at node i. j and m j Let be the cosines of line segment j along the x and y directions in the set, and let E and A be the elastic modulus and cross-sectional area of ​​the assumed line segment j, respectively. Let the set of all unbalanced forces at all nodes be a vector {F}. u }, use the Newton-Raphson method to iteratively solve {F} u The structural geometric displacement under the action of} is gradually reduced and eventually eliminated. u Finally, the unfolded plane of the spatial surface is obtained; by performing planarization processing on the surface, the discretized sub-region spatial surface (x,y,z) is unfolded into a discrete sub-region plane. ; S22: When the structural surface subjected to distributed dynamic loads is divided into sub-regions, the load distribution within each sub-region is determined by the load values ​​on the shared nodes of its boundaries. The load magnitude within a sub-region is obtained by multiplying the node load value by the interpolation function corresponding to the node in that grid. Thus, the distributed dynamic load within a single sub-region is expressed as follows: (2) In the formula, m N N represents the number of nodes in this sub-region. i p represents the interpolation function corresponding to the i-th node in this sub-region. i (t) represents the load value at the node that changes over time. After dividing the load sub-regions, the interpolation function corresponding to each node is selected. Furthermore, the shape function of the finite element corresponding to the shape of the sub-region is directly selected as the interpolation function. The interpolation function determines the distribution form of the load in each sub-region. The load value at the node controls the coefficient of the distribution function. Solving for the load values ​​at all sub-region nodes reconstructs the distributed dynamic load of the entire structural surface. The distributed dynamic load of the structural surface is shown in the following formula: (3) In the formula, m P m represents the number of nodes contained in the subregion. e This represents the number of sub-regions connected to the j-th node. Let represent the interpolation function within the i-th subregion to which the j-th node belongs; S23: To obtain the strain response at the measuring point, the distributed dynamic load is transformed into nodal loads on the structural finite element model using the finite element method. The element nodal loads are obtained by integration within the structural finite element model. (4) In the formula, N represents the shape function of an element in a structural finite element model. i p represents the interpolation function corresponding to the i-th node in this sub-region. i (t) represents the nodal load value as a function of time; The nodal loads on the structural finite element model, which are equivalent to the distributed dynamic loads on the sub-region, are applied to the finite element model, and the strain response at the model measurement points is calculated. S24: Based on the Green's function of a linear system, the transfer function relationship between the distributed dynamic load and the strain response at the measuring point on the sub-region is obtained as follows: (5) In the formula, i Let be the strain response vector at the i-th measuring point. i Let m be the distributed dynamic load vector on the i-th sub-region node. S m represents the number of strain measurement points. P G represents the number of nodes in the load sub-region. i,j This represents the transfer matrix between the distributed load corresponding to the node in the j-th sub-region and the strain response at the i-th measuring point; Equation (5) can be simplified to the following: (6) In the formula, {G} is the strain response vector, containing the time-domain response sequence at each strain measurement point; {P} is the system transfer matrix; and {P} is the load vector, containing the load time sequence at each sub-region grid node. For each node, a unit distributed pulse load in the form of a shape function is applied to the sub-region connected to that node. By acquiring the transient strain response at the measuring point, the transfer relationship matrix G between the nodal load values ​​and the structural strain response in the sub-region is solved. ij Repeat the operation on all nodes to solve for the system transfer matrix G.

2. The method for identifying distributed dynamic loads on the surface of a vehicle wall panel structure according to claim 1, characterized in that, In step S2, when the sub-region is a quadrilateral, the load magnitude within each sub-region is obtained by multiplying the load values ​​at the four nodes of that region by the corresponding interpolation function and then summing them. Furthermore, when the sub-region is a quadrilateral, the specific expression for the interpolation function of a single node in its four adjacent sub-regions is as follows: (7) In the formula, L x ,L y Each sub-region represents the following: The length of the direction, D1~D4 represents the four sub-regions connected by the node.

3. A method for identifying distributed dynamic loads on the surface of a vehicle wall panel structure according to claim 2 or 1, characterized in that, In step S3, the measured strain response of the wall panel structure under distributed dynamic load at the measuring point is obtained, and the load values ​​on the grid nodes of each sub-region are solved using the time series of the measured strain response and the transfer function matrix. The distributed dynamic load on the structural surface is reconstructed using the interpolation function. The specific steps are as follows: S31. Using the least squares method, from equation (6), we can obtain, (8) In the formula, [G] + The generalized inverse matrix of the system transfer function matrix is ​​used to solve for the time history of the load values ​​at the grid nodes in the sub-region. S32. After solving for the load values ​​at the nodes of each sub-region using equation (8), the distributed dynamic load on the entire structural surface is expressed as: (9) In the formula, m P m represents the number of nodes contained in the sub-region. N N represents the number of sub-regions connected to the j-th node. ji This represents the interpolation function for the j-th node within the i-th subregion it is connected to.

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