Curvature-adaptive acoustic black hole beam structure modal identification method

Abstract

This invention discloses a curvature-adaptive modal identification method for acoustic black hole beam structures, belonging to the field of acoustic black hole structure dynamics analysis technology. This method achieves accurate and efficient solutions for the natural frequencies and mode shapes of helical acoustic black hole structures by standardizing the input data of the acoustic black hole structure, combining adaptive mesh generation, combined optimization algorithm solution, and result verification. Specifically, it includes steps such as unified input data processing, parameter fractal dimension adaptation, adaptive non-uniform mesh generation, combined optimization algorithm solution, and result verification, solving technical problems in existing technologies such as poor mesh adaptability, low accuracy in constructing the stiffness matrix of variable cross-section elements, and difficulty in balancing solution accuracy and efficiency. The mesh generated by this invention can accurately fit the variable thickness and curvature characteristics of the helical acoustic black hole beam structure, achieving high solution accuracy and significantly improved computational efficiency. The output results can provide reliable data support for the design of acoustic black hole dampers and have a wide range of applications.

Description

Technical Field

[0001] This invention relates to the field of acoustic black hole structure dynamics analysis technology, specifically to a curvature-adaptive acoustic black hole beam structure modal identification method. Background Technology

[0002] Acoustic black hole beam structures achieve elastic wave localization and energy dissipation through power-law thickness distribution, and are widely used in engineering scenarios such as vibration control and noise suppression. Their modal parameters (natural frequencies and mode shapes) are the core basis for structural design and performance optimization, and the accuracy of the solution directly determines the engineering application effect. However, existing methods for solving acoustic black hole modes still have significant shortcomings: traditional mesh generation methods struggle to match the spiral curvature and thickness power-law gradient of the acoustic black hole, leading to a surge in computation when the mesh is too dense and insufficient accuracy with errors often exceeding 1% when the mesh is too sparse; in terms of frequency solving, the commonly used finite element or ordinary differential equation solver strategies have limited accuracy in constructing the stiffness matrix of variable cross-section elements, easily resulting in non-convergence in the high-frequency region and insufficient accuracy in the low-frequency region; simultaneously, current technologies lack systematic convergence verification and error quantification mechanisms, making it impossible to establish a linkage evaluation system between mesh accuracy and solution accuracy, thus making it difficult to guarantee the reliability of modal results. Summary of the Invention

[0003] This invention aims to address the aforementioned shortcomings by proposing a high-precision and high-efficiency modal identification method applicable to acoustic black hole beam structures with variable thickness and curvature, thereby significantly improving the computational robustness and engineering applicability of modal analysis.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] A method for modal identification of acoustic black hole beam structures with curvature adaptive characteristics, comprising the following steps:

[0006] (1) Normalize and scale-consistentize the geometric input data of the acoustic black hole structure;

[0007] (2) The input parameters of the acoustic black hole structure are divided into three categories: geometric parameters, material parameters, and calculation parameters. The corresponding parameters are entered and format adapted to provide basic input for subsequent mesh generation.

[0008] (3) Based on the input parameters, complete the mesh construction of the helical acoustic black hole and generate a non-uniform mesh adapted to the variable thickness-variable curvature characteristics;

[0009] (4) The Wittrick-Williams algorithm is used to perform preliminary counting and interval localization of the natural frequencies of the helical acoustic black hole. The boundary value problem is solved using the boundary value problem solver and the exact solution is obtained to obtain the natural frequency values ​​of the acoustic black hole structure.

[0010] (5) Based on the natural frequency values ​​obtained in step (4), organize and output them; at the same time, extract and calculate the mode shape data corresponding to each frequency and display them in a visual graphic; finally, compare and verify all the obtained modal parameters with the benchmark model or experimental data, and output the final verified modal analysis results.

[0011] To optimize the above technical solution, the specific measures also include:

[0012] In step (1), the geometric input data includes the thickness function, curvature function, upper and lower surface profiles, material parameters, and structural boundary conditions.

[0013] The specific method for step (1) is as follows:

[0014] First, the thickness function of the acoustic black hole beam... With curvature function Scale normalization is performed, where The spatial coordinates representing the direction of the beam axis, with a range of values ​​of [value range missing]. , Given the total length of the structure, the thickness function is normalized as follows:

[0015] ,

[0016] in This represents the maximum value of the thickness function.

[0017] The curvature function is normalized as follows:

[0018] ,

[0019] in For the beam profile at position curvature, To achieve the maximum curvature, this stage employs uniformity and normalization to keep material parameters within the same scale range, thereby avoiding ill-conditioned matrices when solving higher-order differential equations. Furthermore, the boundary conditions of the structure are initialized, and the cantilever beam structure satisfies:

[0020] ,

[0021] in For displacement function, For coordinates The first derivative of the input data is used in this stage to ensure that the input data structure is uniform and the values ​​are stable, providing reliable initial information for subsequent steps.

[0022] The specific method for step (2) is as follows:

[0023] Geometric parameters include beam length ,width Thickness function curvature function And the spiral radius of a spiral acoustic black hole Pitch Power index ,in This represents the radial distance from the center of the helix to the centerline of the beam. This represents the axial displacement of the helical surface during one rotation. Controlling the power-law decay of the thickness of an acoustic black hole involves material parameters including elastic modulus. ,density Damping ratio All parameters are allowed to vary according to spatial location; calculation parameters include the minimum mesh size. Maximum size Target search frequency zone Frequency convergence threshold With stiffness convergence threshold ,in This indicates the operating frequency; the completed parameters will serve as the core input for subsequent adaptive mesh generation algorithms.

[0024] The specific method for step (3) is as follows:

[0025] The initialization process involves inputting the helical parameters of the helical acoustic black hole, including the helical radius, pitch, and exponent. The total arc length of the helical curve is calculated to determine the mesh generation range, and the upper boundary of the element size is set. and lower boundary As a constraint on mesh density, the initial angle of the helix is ​​set to... To determine the starting position of the grid division,

[0026] Traverse the spiral and adaptively control the control unit size to determine the current angle. Is it less than the maximum angle of the spiral? ,like Then calculate the current Helical curvature at corresponding position , combined and Determine the element size at this location. After recording the size and position information of the current unit, the angle is updated. , Use the angle variable and repeat this step; if Then it proceeds to the subsequent grid processing stage.

[0027] A non-uniform mesh is generated and element parameters are calculated. Based on the element size obtained by adaptive control, a non-uniform mesh for the helical acoustic black hole is generated. For each mesh element, its core geometric properties, including element area, are calculated. Moment of inertia Polar moment of inertia ,

[0028] Verify grid convergence and iteratively adjust it using the frequency convergence criterion. With stiffness convergence criterion Determine if the mesh meets the accuracy requirements. If not, adjust the mesh parameters and return to the initialization process to re-execute element size control. If it meets the requirements, output the final adaptive mesh. These are the modal frequencies calculated in the nth iteration. These are the modal frequencies calculated in the (n+1)th iteration. It is the stiffness matrix of the structure. It is a displacement vector, and ||·|| is the magnitude of the vector.

[0029] The boundary value problem solver is BVP4C.

[0030] The specific method for step (4) is as follows:

[0031] Initialization process: Input the geometric parameters, material parameters, and operating frequency of the unit. , element stiffness matrix Initialize to a zero matrix, i.e. This provides an empty carrier for the subsequent assembly of column vectors, where the geometric parameters of the element include the cross-sectional area. and moment of inertia Material parameters include elastic modulus ,

[0032] Iteratively solve for the column vectors of the element stiffness matrix, setting the loop variable. ,against Construct boundary condition vectors for each degree of freedom. ,satisfy Only the middle One component is a unit constraint, and the rest are zero. An initial guessed solution to the ODE boundary value problem is generated based on the element's geometric properties. Configure the accuracy threshold of the boundary value problem solver. Then, it is used to solve the boundary value problem of the dynamic governing differential equations of the variable cross-section beam:

[0033] ,

[0034] In the formula, The elasticity matrix of the element. It is the cross-sectional area function. Let the moment of inertia of the cross section be... Represents a diagonal matrix. The mass matrix of the unit, For material density, For coordinates The second derivative,

[0035] The solution results are evaluated and the column vectors are assembled. The boundary value problem solver is then checked for convergence. If the solution is successful, the internal forces at both ends of the element are extracted. ,in, The j-th internal force consists of six components: axial force, shear force, bending moment, nodal x-direction force, nodal y-direction force, and nodal bending moment. These components are considered as... The Column, i.e. If the solution fails, the predefined static stiffness matrix is ​​called. List Perform assembly, and then determine the result. Check if the condition is met. If it is, return to process the next degree of freedom; otherwise, proceed to the result output stage.

[0036] Output the element stiffness matrix. After assembling the column vectors corresponding to all 6 degrees of freedom, output the final element stiffness matrix. .

[0037] The beneficial effects of this invention are:

[0038] (1) Adaptive Mesh Accurately Matches Structural Features. Existing technologies for generating meshes for helical acoustic black holes mostly use uniform meshes or empirically derived non-uniform meshes, which are difficult to adapt to the changes in helical curvature and thickness gradient distribution (power-law thickness function). This results in a surge in computation when the mesh is too dense and insufficient accuracy when it is too sparse. This invention controls the size of the control unit through curvature adaptive control, based on the local curvature of the helix. Dynamically adjust unit size Combined with frequency convergence criterion With stiffness convergence criterion Iterative optimization generates a non-uniform mesh that precisely matches the geometric characteristics of a helical acoustic black hole—with denser elements in regions of high curvature and drastic thickness changes, and sparser elements in regions of gentle curvature, while ensuring the accuracy of modal solving (error controlled within a certain range). While increasing the scale (by a factor of 1), it significantly reduces the number of grids and computational costs.

[0039] (2) Combinatorial optimization algorithm ensures accuracy and convergence. Existing technologies for solving acoustic black hole modes often employ the finite element method for direct discretization or a single algorithm to handle boundary value problems, resulting in low accuracy in constructing the stiffness matrix of the variable cross-section element and non-convergence in solving boundary value problems. This invention innovatively adopts a combined optimization algorithm scheme: the Wittrick-Williams algorithm is used to achieve accurate counting and interval positioning of natural frequencies; the BVP4C solver is used to solve the dynamic control equations of the variable cross-section beam. By combining a column-by-column assembly strategy for the element stiffness matrix (solving iteratively across 6 degrees of freedom) and introducing a static stiffness matrix mechanism, the stiffness matrix of the variable cross-section element is guaranteed. It improves the accuracy of construction and solves the convergence problem of boundary value problem solving. Attached Figure Description

[0040] Figure 1 The flowchart shows the overall process of a curvature-adaptive acoustic black hole beam structure modal identification method.

[0041] Figure 2 A flowchart for an adaptive mesh and cell size control method;

[0042] Figure 3 A diagram illustrating the method for constructing the element stiffness matrix;

[0043] Figure 4 This is an example diagram of a spiral acoustic black hole structure. Detailed Implementation

[0044] The invention will now be described in further detail with reference to the accompanying drawings.

[0045] It should be noted that the terms such as "upper", "lower", "left", "right", "front", and "back" used in the invention are only for clarity of description and are not intended to limit the scope of the invention. Changes or adjustments to their relative relationships, without substantially altering the technical content, should also be considered within the scope of the invention.

[0046] The present invention provides a curvature-adaptive acoustic black hole beam structure modal recognition method, the overall process of which is as follows: Figure 1 As shown, it includes the following steps:

[0047] Initial Stage: The initial stage includes unified preprocessing of the geometric parameters, material parameters, and boundary conditions of the acoustic black hole beam to ensure the numerical stability of subsequent calculations. This method first preprocesses the thickness function of the acoustic black hole beam. With curvature function Scale normalization is performed, where The spatial coordinates representing the direction of the beam axis, with values ​​ranging from [value range missing]. , Let be the total length of the structure. The thickness function is normalized as follows:

[0048] ,

[0049] in This represents the maximum value of the thickness function.

[0050] The curvature function is normalized as follows:

[0051] ,

[0052] in For the beam profile at position curvature, To maximize the curvature, this stage employs uniformity and normalization to keep material parameters within the same scale range, thus avoiding ill-conditioned matrices when solving higher-order differential equations. Furthermore, the boundary conditions of the structure are initialized; for example, a cantilever beam structure satisfies:

[0053] ,

[0054] in It is a displacement function. For coordinates The first derivative. This stage of processing ensures that the input data structure is uniform and the values ​​are stable, providing reliable initial information for subsequent steps.

[0055] The next step involves parameter input and fractal processing to classify, organize, and adapt geometric, material, and computational parameters. Geometric parameters include beam length. ,width Thickness function curvature function And the spiral radius required for a spiral acoustic black hole. Pitch Power index Parameters, among which This represents the radial distance from the center of the helix to the centerline of the beam. This represents the axial displacement of the helical surface during one rotation. Controlling the power-law decay law of the acoustic black hole thickness. Material parameters include elastic modulus. ,density Damping ratio All parameters are allowed to vary according to spatial location. Calculation parameters include the minimum mesh size. Maximum size Target search frequency zone Frequency convergence threshold With stiffness convergence threshold ,in This represents the operating frequency. The completed parameters will serve as the core input for the subsequent adaptive mesh generation algorithm.

[0056] like Figure 2 As shown, the next step is to generate a non-uniform adaptive mesh that adapts to the local curvature and thickness variations of the acoustic black hole. Before mesh construction, the radius of the helical curve is input based on the geometry of the helical acoustic black hole. Pitch and angular parameters The spatial path of the beam is represented as a function of angle. A changing spiral, and calculate its arc length:

[0057] ,

[0058] in Indicates the distance from the starting point to the angle. The length of the spiral curve. The mesh cell size is determined by the curvature control function, through... The system automatically densifies cells in regions with high curvature (i.e., acoustic black hole regions) and sparses cells in regions with low curvature. In the formula... Indicates the current angle The unit size below, This represents the local curvature at that location. (This is achieved by traversing...) Achieve non-uniform mesh construction across the entire beam domain.

[0059] After the mesh is generated, the area of ​​each cell is calculated:

[0060] ,

[0061] in The width of the beam. Unit size;

[0062] Moment of inertia:

[0063] ,

[0064] Polar moment of inertia is defined based on element size and geometric relationship:

[0065] ,

[0066] After the adaptive mesh is constructed, the frequency convergence criterion is used:

[0067] ,

[0068] And the stiffness convergence criterion:

[0069] ,

[0070] Verify mesh adaptability, where and They represent the first Next and first The modal frequencies obtained in the second iteration This is the frequency-dependent stiffness matrix. It is a displacement vector. This represents the magnitude of the vector. When the criterion is not met, the mesh parameters are automatically adjusted and the mesh is regenerated until the accuracy requirements are met.

[0071] The next step is to search for the intrinsic frequency and numerically solve the boundary value problem. This invention employs the Wittrick-Williams algorithm to count frequencies within the target frequency range and calculates the frequency indication function. Determine the interval containing each natural frequency. In the formula... For frequency counting functions, This indicates the effect of structural boundaries on frequency counting. The number of frequency oscillations contributed to each element. This method can quickly narrow down the search range for each natural frequency.

[0072] like Figure 3 As shown, after determining the frequency range, the differential equations of the acoustic black hole beam dynamics are solved using the BVP4C solver:

[0073] ,

[0074] in The location-dependent elasticity matrix, For the quality matrix, For operating frequency, This represents the displacement function. The element stiffness matrix is ​​constructed through column-by-column assembly. For a six-DOF element, the degrees of freedom are numbered as follows: Set boundary condition vectors for each column. Only the first one One component is 1, and the rest are 0. Solving the above differential equation using BVP4C yields the internal force vectors at both ends of the element:

[0075] ,

[0076] And as the stiffness matrix of the first Column, that is:

[0077] ,

[0078] in, These are the six components of the j-th internal force: axial force, shear force, bending moment, nodal x-direction force, nodal y-direction force, and nodal bending moment.

[0079] If the solution for a certain column fails to converge, the corresponding column of the static stiffness matrix will be automatically used to replace it, so as to ensure that the element stiffness matrix can always be constructed.

[0080] Finally, this method performs overall assembly of the element stiffness matrix obtained under the adaptive mesh, based on the characteristic equation:

[0081] ,

[0082] Solve for the natural frequencies and mode shapes of the structure, where For mode shape vectors, This represents the global stiffness matrix. The obtained modal results include the natural frequency sequence and the corresponding mode shape distribution. Mode shape visualization can display the energy focusing characteristics of the acoustic black hole region (e.g., ...). Figure 4 (Example of a spiral acoustic black hole structure shown). Finally, the results of this method are compared with finite element models or experimental data. If the frequency error is controlled within... If the magnitude of the vibration mode and the cross-correlation between the vibration modes are greater than 0.98, then the solution obtained by this method is considered reliable.

[0083] Through the above process, this invention successfully achieves high-precision and high-efficiency acoustic black hole beam mode solving and identification.

[0084] The above are merely preferred embodiments of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principles of the present invention should be considered within the scope of protection of the present invention.

Claims

1. A method for modal recognition of an acoustic black hole beam structure with curvature adaptive characteristics, characterized in that, Includes the following steps: (1) Normalize and scale-consistentize the geometric input data of the acoustic black hole structure; (2) The input parameters of the acoustic black hole structure are divided into three categories: geometric parameters, material parameters, and calculation parameters. The corresponding parameters are entered and format adapted to provide basic input for subsequent mesh generation. (3) Based on the input parameters, complete the mesh construction of the helical acoustic black hole and generate a non-uniform mesh adapted to the variable thickness-variable curvature characteristics; (4) The Wittrick-Williams algorithm is used to perform preliminary counting and interval localization of the natural frequencies of the helical acoustic black hole. The boundary value problem is solved using the boundary value problem solver and the exact solution is obtained to obtain the natural frequency values ​​of the acoustic black hole structure. (5) Based on the natural frequency values ​​obtained in step (4), organize and output them; at the same time, extract and calculate the mode shape data corresponding to each frequency and display them in a visual graphic; finally, compare and verify all the obtained modal parameters with the benchmark model or experimental data, and output the final verified modal analysis results.

2. The method for modal recognition of a curvature-adaptive acoustic black hole beam structure according to claim 1, characterized in that, In step (1), the geometric input data includes the thickness function, curvature function, upper and lower surface profiles, material parameters, and structural boundary conditions.

3. The method for modal recognition of an acoustic black hole beam structure with curvature adaptive according to claim 2, characterized in that, The specific method for step (1) is as follows: First, the thickness function of the acoustic black hole beam... With curvature function Scale normalization is performed, where The spatial coordinates representing the direction of the beam axis, with a range of values ​​of [value range missing]. , Given the total length of the structure, the thickness function is normalized as follows: ， in This represents the maximum value of the thickness function. The curvature function is normalized as follows: ， in For the beam profile at position curvature, To determine the maximum curvature, the boundary conditions of the structure are also initialized. The cantilever beam structure satisfies: ， in For displacement function, For coordinates The first derivative.

4. The method for modal recognition of a curvature-adaptive acoustic black hole beam structure according to claim 3, characterized in that, The specific method for step (2) is as follows: Geometric parameters include beam length ,width Thickness function curvature function And the spiral radius of a spiral acoustic black hole Pitch Power index ,in This represents the radial distance from the center of the helix to the centerline of the beam. This represents the axial displacement of the helical surface during one rotation. Controlling the power-law decay of the thickness of an acoustic black hole involves material parameters including elastic modulus. ,density Damping ratio All parameters are allowed to vary according to spatial location; calculation parameters include the minimum mesh size. Maximum size Target search frequency zone Frequency convergence threshold With stiffness convergence threshold ,in This indicates the operating frequency; the completed parameters will serve as the core input for subsequent adaptive mesh generation algorithms.

5. The method for modal recognition of a curvature-adaptive acoustic black hole beam structure according to claim 4, characterized in that, The specific method for step (3) is as follows: The initialization process involves inputting the helical parameters of the helical acoustic black hole, including the helical radius, pitch, and exponent. The total arc length of the helical curve is calculated to determine the mesh generation range, and the upper boundary of the element size is set. and lower boundary As a constraint on mesh density, the initial angle of the helix is ​​set to... To determine the starting position of the grid division, Traverse the spiral and adaptively control the control unit size to determine the current angle. Is it less than the maximum angle of the spiral? ,like Then calculate the current Helical curvature at corresponding position , combined and Determine the element size at this location. After recording the size and position information of the current unit, the angle is updated. , Use the angle variable and repeat this step; if Then it proceeds to the subsequent grid processing stage. A non-uniform mesh is generated and element parameters are calculated. Based on the element size obtained by adaptive control, a non-uniform mesh for the helical acoustic black hole is generated. For each mesh element, its core geometric properties, including element area, are calculated. Moment of inertia Polar moment of inertia , Verify grid convergence and iteratively adjust it using the frequency convergence criterion. With stiffness convergence criterion Determine if the mesh meets the accuracy requirements. If not, adjust the mesh parameters and return to the initialization process to re-execute element size control. If it meets the requirements, output the final adaptive mesh. These are the modal frequencies calculated in the nth iteration. These are the modal frequencies calculated in the (n+1)th iteration. It is the stiffness matrix of the structure. It is a displacement vector, and ||·|| is the magnitude of the vector.

6. The method for modal recognition of a curvature-adaptive acoustic black hole beam structure according to claim 5, characterized in that, The boundary value problem solver is BVP4C.

7. The method for modal recognition of an acoustic black hole beam structure with curvature adaptive according to claim 5, characterized in that, The specific method for step (4) is as follows: Initialization process: Input the geometric parameters, material parameters, and operating frequency of the unit. , element stiffness matrix Initialize to a zero matrix, i.e. This provides an empty carrier for the subsequent assembly of column vectors, where the geometric parameters of the element include the cross-sectional area. and moment of inertia Material parameters include elastic modulus , Iteratively solve for the column vectors of the element stiffness matrix, setting the loop variable. ,against Construct boundary condition vectors for each degree of freedom. ,satisfy Only the middle One component is a unit constraint, and the rest are zero. An initial conjectured solution to the ODE boundary value problem is generated based on the element's geometric properties. Configure the accuracy threshold of the boundary value problem solver. Then, it is used to solve the boundary value problem of the dynamic governing differential equations of the variable cross-section beam: ， In the formula, The elasticity matrix of the element. It is the cross-sectional area function. Let the moment of inertia of the cross section be... Represents a diagonal matrix. The mass matrix of the unit, For material density, For coordinates The second derivative, The solution results are evaluated and the column vectors are assembled. The boundary value problem solver is then checked for convergence. If the solution is successful, the internal forces at both ends of the element are extracted. ,in, The j-th internal force consists of six components: axial force, shear force, bending moment, nodal x-direction force, nodal y-direction force, and nodal bending moment. These components are considered as... The Column, i.e. If the solution fails, the predefined static stiffness matrix is ​​called. List Perform assembly, and then determine the result. Check if the condition is met. If it is, return to process the next degree of freedom; otherwise, proceed to the result output stage. Output the element stiffness matrix. After assembling the column vectors corresponding to all 6 degrees of freedom, output the final element stiffness matrix. .

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