Inherent frequency characteristic analysis method for thin film structure in any shape

Abstract

The invention discloses an inherent frequency characteristic analysis method of a thin film structure in any shape, which can directly establish a semi-analytical expression of an energy functional based on a contour curve equation by dividing any film domain into curved edge trapezoid sub-domains and combining an orthogonal polynomial trial function and a boundary penalty factor method. According to the method, the dependence of a traditional finite element on a dense grid is avoided, and the calculation efficiency and the convergence speed under complex geometry are remarkably improved; more importantly, the obtained inherent frequency result is presented in a series form, and an explicit mathematical relationship between structural geometric parameters and vibration characteristics is established, so that efficient and high-precision analysis of the inherent frequency of the film structure in any shape is realized, and a theoretical basis and a calculation tool are provided for film structure shape optimization based on vibration performance; the method is suitable for membrane structures with complex shapes and different boundary conditions.

Description

Technical Field

[0001] This invention belongs to the field of engineering mechanics technology, specifically relating to a method for analyzing the inherent frequency characteristics of thin film structures of arbitrary shapes. Background Technology

[0002] Thin-film structures, with their high tensile strength, excellent flexibility, high folding ratio, lightweight, and low cost, are increasingly widely used in aerospace and civil engineering, especially in space structures, where they have become an indispensable form, such as thin-film antennas and solar sails. However, due to the complexity of the working environment, especially the space environment, and the low stiffness of membrane materials, the vibration characteristics of the membrane become a key factor affecting the normal operation, safety, and durability of the structure. A deep understanding and precise grasp of its inherent frequency characteristics are of paramount importance for the design and application of thin-film structures.

[0003] Given the varying requirements for the geometry of thin-film structures under different working conditions, the existence of complex profile curves significantly increases the difficulty of modeling and solving the vibration problems of thin-film structures. Currently, most analytical methods for the vibration characteristics of complex-shaped thin films are limited to numerical approximation methods, such as the finite element method, and no method has been reported that can reveal the analytical relationship between the thin-film profile curve and its natural frequencies. Therefore, establishing an analytical method based on the profile curve equation that better suits the actual needs of engineering design for arbitrarily shaped thin-film structures under different boundary conditions has significant theoretical and practical value. Summary of the Invention

[0004] The purpose of this invention is to provide a method for analyzing the intrinsic frequency characteristics of thin film structures of arbitrary shapes, so as to solve the problems mentioned in the background art.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: a method for analyzing the intrinsic frequency characteristics of a thin film structure of arbitrary shape, comprising the following steps: Step 1: Divide the membrane integration domain into several curvilinear trapezoidal domains, and correspondingly divide the boundary into several segments; Step 2: Based on the segmented curved trapezoidal domains, the double integral forms of the deformation energy and kinetic energy of the membrane vibration are rewritten as the sum of the second integrals over each curved trapezoidal domain; Step 3: For the boundary conditions of the membrane, a translational penalty factor and a rotational penalty factor are introduced to limit the boundary displacement, and the boundary energy of each segment is calculated based on the boundary segmentation results in Step 1, and then the total boundary energy is obtained. Step 4: Select any set of orthogonal polynomials as the displacement trial function and construct an approximate expression for the lateral displacement; Step 5: Substitute the polynomial trial function selected in Step 4 into the deformation energy and kinetic energy expressions in Step 2, transform its quadratic integral into a definite integral with respect to a single variable, and solve the definite integral analytically or obtain a semi-analytical result by using a numerical integration method according to the specific form of the contour curve equation defined in Step 1. Step Six: Based on the semi-analytical results of deformation energy, kinetic energy and boundary energy obtained in Step Five, construct the Lagrangian function, minimize the function and apply the variational method to solve the generalized eigenvalue problem, thereby obtaining the intrinsic frequency characteristics of the membrane structure.

[0006] Preferably, dividing the membrane integration domain into several curved trapezoidal domains, and correspondingly dividing the boundary into several segments, includes: The contour of the membrane structure of arbitrary shape in the plane is divided into arbitrary piecewise smooth curves, and each smooth curve is represented by an equation. Description; among them, there are a total of smooth curve edges. n +1 segment, number of intersections is n +1; The arbitrary-shaped membrane domain, which serves as the integration domain for energy integration, passes through the intersection points of the membrane profile. x The perpendicular line to the axis is divided into n A curved trapezoidal domain, simultaneously dividing the membrane boundary into 2 n A curved segment, using or The equation is written as follows: or , p =1,2,,…, n ; Intersection at the origin of the coordinate system x Coordinates are x 0 = 0, the other intersection points x Coordinate marking x p ; Points on the membrane profile curve y The maximum and minimum values ​​of the coordinates are denoted as follows: y max and y min .

[0007] Preferably, the segmented curvilinear trapezoidal domain rewrites the double integral form of the deformation energy and kinetic energy of the membrane vibration as the sum of the second integrals over each curvilinear trapezoidal domain, including: The deformation energy of the thin film during free vibration is: (1); where, S t This represents the uniform tension per unit length of the thin film; W ( x ,y ) is the lateral displacement; A Let be the integration domain of the plane containing the thin film; Based on the segmentation of the membrane integral domain in step one, the deformation energy is rewritten as the sum of quadratic integrals over each curvilinear trapezoidal domain: (2); The kinetic energy of the membrane during free vibration is: (3); among which, ρ It is the mass of the film per unit area. ω It is angular frequency; The kinetic energy integral can be rewritten as the sum of quadratic integrals over the curvilinear trapezoidal domain: (4).

[0008] Preferably, for the boundary conditions of the membrane, translational penalty factors and rotational penalty factors are introduced to limit boundary displacement, and the boundary energy of each segment is calculated based on the boundary segmentation results in step one, thereby obtaining the total boundary energy, including: Two boundary penalty factors are introduced at the membrane boundary to restrict the lateral translational and rotational displacements of the boundary, respectively. k t and k r express; The boundary conditions for free vibration of the membrane are achieved by changing the value of the penalty factor. When a certain boundary constraint is rigid, the corresponding penalty factor value is infinite, with 10 as an example. 10 Alternative; when there are no boundary constraints, the corresponding penalty factor is 0; The boundary energy of the membrane's free vibration can be calculated based on the piecewise results: (5); in k t and k r These are the translational penalty factor and the rotational penalty factor at the boundary, respectively; and For the two endpoints of the boundary x Coordinates; if it is a free boundary condition k t =0 and k r =10 10 If it is a fixed boundary condition k t =10 10 and k r =0.

[0009] The total boundary energy of the membrane is the sum of the boundary energies of each segment: (6).

[0010] Preferably, the step of selecting an arbitrary set of orthogonal polynomials as the displacement trial function and constructing an approximate expression for the lateral displacement includes: The displacement form is: (7); among which For trial functions, C j , C u,j , C v,j For expansion coefficients, j= 1,2, …… , n .

[0011] Choosing orthogonal Jacobi polynomials as trial functions; Jacobi polynomials Orthogonal on the interval [-1, 1], where, , i =1,2,…; That is, the trial function is: (8); in, n =0,1,2,3…, N ; m =0,1,2,3…, M , N , M This represents the highest degree in the Jacobi polynomial. Normalize the coordinates: (9); among which x max , x min Points on the membrane profile curve x The maximum and minimum values ​​of the coordinates; The displacement function is: (10).

[0012] Preferably, the step of substituting the polynomial trial function selected in step four into the deformation energy and kinetic energy expressions in step two, transforming its quadratic integral into a definite integral with respect to a single variable, and then analytically solving the definite integral or obtaining a semi-analytical result using a numerical quadrature method according to the specific form of the contour curve equation defined in step one, includes: Substituting the polynomial displacement function (10) into the deformation energy equation (2) and kinetic energy equation (4) obtained in step two, respectively, the integrand in both equations becomes about x and y The bivariate polynomials, combined, are expressed as: (11); among which, This represents deformation energy or kinetic energy; c The constant coefficients in both equations; (12), for the purpose of discussing x and y A bivariate polynomial, s =0,1,2,…, S , t =0,1,2,…, T , S and T Coordinate variables x and y The highest number of times it can be retrieved; For coefficients; polynomial about y Antiderivative with analyticity : (13); Equation (11) can be rewritten as: (14); Based on the contour curve and The specific form determines the solution method of equation (14), including: If the equation of the contour curve allows the integral to be calculated analytically, then the energy... An analytical solution can be obtained; otherwise... The solution is obtained by numerical integration, such as the Gauss quadrature formula; the calculation method of the boundary energy definite integral (5) is the same.

[0013] Preferably, based on the semi-analytical results of deformation energy, kinetic energy, and boundary energy obtained in step five, a Lagrangian function is constructed. By minimizing this function and applying the variational method, the generalized eigenvalue problem is solved, thereby obtaining the intrinsic frequency characteristics of the membrane structure, including: The Lagrangian function of an arbitrary-shaped membrane structure is expressed as: (15); Make the Lagrange function L minimize (16); Where C is a vector composed of expansion coefficients, the intrinsic frequency characteristics of the membrane are obtained by using the variational method.

[0014] Technical effects and advantages of the present invention: The method for analyzing the intrinsic frequency characteristics of thin film structures of arbitrary shapes proposed in this invention has the following advantages compared with the prior art: This invention, by dividing an arbitrary membrane domain into curvilinear trapezoidal subdomains and combining orthogonal polynomial trial functions with the boundary penalty factor method, can directly establish a semi-analytical expression of the energy functional based on the contour curve equation. This method avoids the dependence of traditional finite element methods on dense meshes, significantly improving computational efficiency and convergence speed under complex geometries. More importantly, the obtained natural frequency results are presented in series form, establishing an explicit mathematical relationship between structural geometric parameters and vibration characteristics. This enables efficient and high-precision analysis of the natural frequencies of arbitrary-shaped thin-film structures and provides a theoretical basis and computational tools for membrane structure shape optimization based on vibration performance. It is applicable to membrane structures with complex shapes and different boundary conditions. Attached Figure Description

[0015] Figure 1 A schematic diagram showing the arbitrary-shaped thin film structure of the present invention divided into curved trapezoidal regions; Figure 2 This is a geometric schematic diagram of the semi-elliptical-triangular thin film structure of the present invention. Detailed Implementation

[0016] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The specific embodiments described herein are merely used to explain the present invention and are not intended to limit the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0017] This invention provides a method for analyzing the intrinsic frequency characteristics of thin film structures of arbitrary shapes, comprising the following steps: Step 1: Divide the thin-film integration domain into several curved trapezoidal domains, and correspondingly divide the boundary into several segments, such as... Figure 1 As shown.

[0018] Membrane structures can be of arbitrary shapes within a plane, their outlines bounded by arbitrary piecewise smooth curves, each of which can be derived from a shape such as... The equation describes (or approximates) the smooth curve edges. n +1 segment, the number of their intersection points is also... n +1. The arbitrary-shaped membrane domain, which serves as the integration domain for energy integration, passes through the intersection points of the membrane profile. x The perpendicular line to the axis is divided into n Each region comprises a curvilinear trapezoidal domain (including a curvilinear triangular domain), and the membrane boundary is divided into 2... n A curved segment, using and The equation is written as follows: or ,p =1,2,,…, n The intersection point at the origin of the coordinate system x Coordinates are x 0 = 0, the other intersection points x Coordinate marking x p Points on the membrane profile curve y The maximum and minimum values ​​of the coordinates are denoted as follows: y max and y min .

[0019] Step 2: Rewrite the double integral form of the deformation energy and kinetic energy of the membrane vibration as the sum of the double integrals over all curvilinear trapezoidal domains.

[0020] The deformation energy of the thin film during free vibration is: (1) in, S t This represents the uniform tension per unit length of the thin film; W ( x , y ) is the lateral displacement; A Let be the integration domain of the plane containing the thin film.

[0021] Based on the segmentation of the membrane integral domain in step one, the deformation energy is rewritten as the sum of quadratic integrals over each curvilinear trapezoidal domain: (2) The kinetic energy of the membrane during free vibration is: (3) in, ρ It is the mass of the film per unit area. ω It is angular frequency.

[0022] Similar to deformation energy, the kinetic energy integral can be rewritten as the sum of second integrals over the curvilinear trapezoidal domain: (4) Step 3: For the boundary conditions of the membrane, introduce a boundary penalty factor to restrict the boundary displacement, and calculate the boundary energy according to the piecewise results in Step 1.

[0023] The boundary conditions of the membrane can be fixed or free. Two boundary penalty factors are introduced at the membrane boundary to restrict the lateral translational and rotational displacements of the boundary, respectively. k t and k rThis indicates that the boundary conditions for the free vibration of the membrane are achieved by changing the values ​​of these penalty factors. When a certain boundary constraint is rigid, the corresponding penalty factor value takes an infinite value, with 10... 10 Alternative; when there are no boundary constraints, the corresponding penalty factor is 0.

[0024] The boundary of the membrane's free vibration can be calculated based on the piecewise results. (Based on the edge...) For example, its boundary energy is (5) in k t and k r These are the translational penalty factor and the rotational penalty factor at the boundary, respectively. and For the two ends of the boundary x Coordinates. If it is a free boundary condition, k t =0; and k r =10 10 If the boundary conditions are fixed, k t =10 10 ;and k r =0.

[0025] The total boundary energy of the membrane is the sum of the boundary energies of each segment: (6) Step 4: Select any set of orthogonal polynomials as the displacement trial function.

[0026] Assume the displacement has the following form: (7) in For trial functions, C j , C u,j , C v,j For expansion coefficients, j= 1,2, …… , n .

[0027] Classical orthogonal polynomials such as Jacobi polynomials can generally be chosen as trial functions. Orthogonal on the interval [-1, 1] i Jacobi polynomial recurrence formula for (8a) (8b) (8c) in, , i =1,2,…

[0028] Jacobi polynomials The orthogonality condition for weighted functions on [-1,1] is: (9) Wherein, the weight function , It is the gamma function. yes δ function.

[0029] That is, the trial function is (10) in, n =0,1,2,3…, N ; m =0,1,2,3…, M , N , M This represents the highest degree in the Jacobi polynomial.

[0030] Since the Jacobi polynomial is defined in the interval [-1, 1], the coordinates need to be normalized: (11) in x max , x min Points on the membrane profile curve x The maximum and minimum values ​​of the coordinates.

[0031] Therefore, the displacement function (7) has the following form: (12) Step 5: Use polynomial trial functions to transform the second integrals of deformation energy and kinetic energy into definite integrals, and then analytically calculate or use numerical integration methods to obtain semi-analytical results based on the form of the profile curve equation.

[0032] Substituting the polynomial displacement function (12) into the deformation energy (2) and kinetic energy (4) obtained in step two, respectively, the integrands in the two equations become about x and y The bivariate polynomials, combined, are expressed as: (13) This expression still has the form of a quadratic integral. Among them, This represents deformation energy or kinetic energy; c The constant coefficients in both equations; (14), For about x and y A bivariate polynomial, s =0,1,2,…, s =0,1,2,…, S , t =0,1,2,…, T , S and T Coordinate variables x and y The highest number of times it can be retrieved; is a coefficient.

[0033] polynomial about y primitives with analytic properties number: (15) This expression is still a polynomial, therefore equation (13) can be rewritten as follows: (16) At this point, according to the contour curve and The specific form determines how equation (16) is solved. If the contour curve equation allows the integral to be calculated analytically, then the energy... An analytical solution can be obtained; otherwise... The solution is obtained using numerical integration methods such as the Gauss quadrature formula. The boundary energy definite integral (5) is calculated in the same way.

[0034] Step 6: Obtain the Lagrangian function from the energy integral, minimize it, and use the variational method to obtain the intrinsic frequency characteristics of the membrane.

[0035] The Lagrangian function of an arbitrary-shaped membrane structure can be expressed as: (17) Make the Lagrange function L Minimize, have (18) Where C is a vector composed of expansion coefficients. The natural frequency characteristics of the membrane are obtained by solving this equation using the variational method.

[0036] The above equation can be rewritten in matrix form: (19) Where K and M are the stiffness matrix and mass matrix, respectively. By solving this equation, the natural frequencies and mode shapes of the membrane structure can be obtained.

[0037] Following the steps above, the following calculations were performed: Figure 2 The vibration characteristics of the semi-elliptical-triangular thin-film structure are shown. The structure's shape is a combination of an equilateral triangle and a semi-ellipse. The geometric dimensions are: side length of the equilateral triangle... l =1m, the semi-major axis of the semi-ellipse r a =1m, semi-minor axis r b =0.5m, film thickness is h =0.001m. Mass density per unit area of ​​the thin film. ρ =7.805kg / m 2 Tension per unit length S t =23000N. Thin film structures under both fixed and free boundary conditions were analyzed, and the results are shown in Table 1: Table 1. Results of the first 8 natural frequencies (Hz) of the semi-elliptical-triangular membrane structure.

[0038] In summary, this invention provides analytical or semi-analytical results for the vibration characteristics of membrane structures of arbitrary shapes. Compared to commercial finite element software, it reveals the analytical mathematical relationship between the geometry of the membrane structure and the vibration process. This facilitates the rapid design and optimization of the membrane structure's geometry in engineering design by modifying the membrane structure's profile curve equation based on vibration characteristics. The natural frequency results are exact solutions in series form, and the calculation accuracy can be adjusted as needed, with rapid convergence. It is applicable to membrane structures with complex shapes and different boundary conditions.

[0039] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for analyzing the intrinsic frequency characteristics of a thin film structure of arbitrary shape, characterized in that, Includes the following steps: Step 1: Divide the membrane integration domain into several curvilinear trapezoidal domains, and correspondingly divide the boundary into several segments; Step 2: Based on the segmented curved trapezoidal domains, the double integral forms of the deformation energy and kinetic energy of the membrane vibration are rewritten as the sum of the second integrals over each curved trapezoidal domain; Step 3: For the boundary conditions of the membrane, a translational penalty factor and a rotational penalty factor are introduced to limit the boundary displacement, and the boundary energy of each segment is calculated based on the boundary segmentation results in Step 1, and then the total boundary energy is obtained. Step 4: Select any set of orthogonal polynomials as the displacement trial function and construct an approximate expression for the lateral displacement; Step 5: Substitute the polynomial trial function selected in Step 4 into the deformation energy and kinetic energy expressions in Step 2, transform its quadratic integral into a definite integral with respect to a single variable, and solve the definite integral analytically or obtain a semi-analytical result by using a numerical integration method according to the specific form of the contour curve equation defined in Step 1. Step Six: Based on the semi-analytical results of deformation energy, kinetic energy and boundary energy obtained in Step Five, construct the Lagrangian function, minimize the function and apply the variational method to solve the generalized eigenvalue problem, thereby obtaining the intrinsic frequency characteristics of the membrane structure.

2. The method for analyzing the intrinsic frequency characteristics of an arbitrary-shaped thin-film structure according to claim 1, characterized in that, The process of dividing the membrane integral domain into several curvilinear trapezoidal domains, and correspondingly dividing the boundary into several segments, includes: The contour of the membrane structure of arbitrary shape in the plane is divided into arbitrary piecewise smooth curves, and each smooth curve is represented by an equation. Description; among them, there are a total of smooth curve edges. n +1 segment, number of intersections is n +1; The arbitrary-shaped membrane domain, which serves as the integration domain for energy integration, passes through the intersection points of the membrane profile. x The perpendicular line to the axis divides the membrane into n curvilinear trapezoidal domains, and simultaneously divides the membrane boundary into 2... n A curved segment, using or The equation is written as follows: or , p =1,2,,…, n ; Intersection at the origin of the coordinate system x Coordinates are x 0=0, the other intersection points x Coordinate marking x p ; Points on the membrane profile curve y The maximum and minimum values ​​of the coordinates are denoted as follows: y max and y min .

3. The method for analyzing the intrinsic frequency characteristics of an arbitrary-shaped thin-film structure according to claim 2, characterized in that, The segmented curvilinear trapezoidal domain rewrites the double integral form of the deformation energy and kinetic energy of membrane vibration as the sum of second integrals over each curvilinear trapezoidal domain, including: The deformation energy of the thin film during free vibration is: (1); where, S t This represents the uniform tension per unit length of the thin film; W ( x , y ) is the lateral displacement; A Let be the integration domain of the plane containing the thin film; Based on the segmentation of the membrane integral domain in step one, the deformation energy is rewritten as the sum of quadratic integrals over each curvilinear trapezoidal domain: （2）； The kinetic energy of the membrane during free vibration is: (3); among which, ρ It is the mass of the film per unit area. ω It is angular frequency; The kinetic energy integral can be rewritten as the sum of quadratic integrals over the curvilinear trapezoidal domain: （4）。 4. The method for analyzing the intrinsic frequency characteristics of an arbitrary-shaped thin-film structure according to claim 3, characterized in that, For the boundary conditions of the membrane, translational and rotational penalty factors are introduced to limit boundary displacement, and the boundary energy of each segment is calculated based on the boundary segmentation results in step one, thereby obtaining the total boundary energy, including: Two boundary penalty factors are introduced at the membrane boundary to restrict the lateral translational and rotational displacements of the boundary, respectively. k t and k r express; The boundary conditions for free vibration of the membrane are achieved by changing the value of the penalty factor. When a certain boundary constraint is rigid, the corresponding penalty factor value is infinite, with 10 as an example. 10 Alternative; when there are no boundary constraints, the corresponding penalty factor is 0; The boundary energy of the membrane's free vibration can be calculated based on the piecewise results: （5）； in k t and k r These are the translational penalty factor and the rotational penalty factor at the boundary, respectively; and For the two endpoints of the boundary x Coordinates; if it is a free boundary condition k t =0 and k r =10 10 If it is a fixed boundary condition k t =10 10 and k r =0; The total boundary energy of the membrane is the sum of the boundary energies of each segment: (6).

5. The method for analyzing the intrinsic frequency characteristics of an arbitrary-shaped thin-film structure according to claim 4, characterized in that, The step of selecting an arbitrary set of orthogonal polynomials as the displacement trial function and constructing an approximate expression for the lateral displacement includes: The displacement form is: (7); among which For trial functions, C j , C u,j , C v,j For expansion coefficients, j= 1,2, …… , n ; Choosing orthogonal Jacobi polynomials as trial functions; Jacobi polynomials Orthogonal on the interval [-1, 1], where, ，i =1,2,…; That is, the trial function is: (8); in, n =0,1,2,3…, N ; m =0,1,2,3…, M , N , M This represents the highest degree in the Jacobi polynomial. Normalize the coordinates: (9); among which x max , x min Points on the membrane profile curve x The maximum and minimum values ​​of the coordinates; The displacement function is: (10).

6. The method for analyzing the intrinsic frequency characteristics of an arbitrary-shaped thin-film structure according to claim 5, characterized in that, The step of substituting the polynomial trial function selected in step four into the expressions for deformation energy and kinetic energy in step two, transforming its quadratic integral into a definite integral with respect to a single variable, and then, based on the specific form of the contour curve equation defined in step one, analytically solving the definite integral or obtaining a semi-analytical result using a numerical quadrature method, includes: Substituting the polynomial displacement function (10) into the deformation energy equation (2) and kinetic energy equation (4) obtained in step two, respectively, the integrand in both equations becomes about x and y The bivariate polynomials, combined, are expressed as: (11); among which, This represents deformation energy or kinetic energy; c The constant coefficients in both equations; (12), for the purpose of discussing x and y A bivariate polynomial, s =0,1,2,…, S , t =0,1,2,…, T , S and T Coordinate variables x and y The highest number of times it can be retrieved; For coefficients; polynomial about y Antiderivative with analyticity : （13）； Equation (11) can be rewritten as: (14)； According to the contour curve and The specific form determines the solution method of equation (14), including: If the equation of the contour curve allows the integral to be calculated analytically, then the energy... An analytical solution can be obtained; otherwise... The solution is obtained by numerical integration, such as the Gauss quadrature formula; the calculation method of the boundary energy definite integral (5) is the same.

7. The method for analyzing the intrinsic frequency characteristics of an arbitrary-shaped thin-film structure according to claim 6, characterized in that, Based on the semi-analytical results of deformation energy, kinetic energy, and boundary energy obtained in step five, a Lagrangian function is constructed. By minimizing this function and applying the variational method, the generalized eigenvalue problem is solved, thereby obtaining the intrinsic frequency characteristics of the membrane structure, including: The Lagrangian function of an arbitrary-shaped membrane structure is expressed as: (15); Make the Lagrange function L minimize (16); Where C is a vector composed of expansion coefficients, the intrinsic frequency characteristics of the membrane are obtained by using the variational method.

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