A vibration solution method for plates with arbitrary elliptical openings based on elliptical energy elements

Abstract

The present application discloses a vibration solution method for an elliptical perforated plate of arbitrary shape based on an elliptical energy element, which belongs to the field of thin-walled structure technology and includes: overall mapping of the elliptical perforated plate of arbitrary shape, establishing the elliptical equation in a dimensionless coordinate system, and implementing Boolean operation modeling on the geometry; constructing a boundary condition function to simulate the constraint; simulating the plate deformation according to the first-order shear deformation theory, and constructing a global test function in combination with Legendre polynomials and boundary condition functions; establishing a total energy functional based on the minimum potential energy principle and the Ritz method, solving the stationary value of the total energy functional, and obtaining the calculation formula of each element in the stiffness matrix and the mass matrix; constructing an elliptical energy element based on local mapping, and numerically integrating each element of the matrix based on the elliptical energy element and completing the solution to the vibration problem. In this method, the energy functional and the numerical integration formula are both constructed on the square solution domain in the overall mapping stage, so that the calculation and solution process for the elliptical perforated plate of arbitrary shape is completely standard.

Description

Technical Field

[0001] The present application relates to the technical field of thin-walled structures, and in particular to a vibration solution method for an elliptical hole plate of arbitrary shape based on an elliptical energy element. Background Art

[0002] Elliptical structures are extremely common in engineering, such as aircraft doors, ship windows, and wing rib lightening holes. At the same time, in order to meet the requirements of lightweight design and equipment accessibility and reduce the risk of stress concentration, plate and shell structures often have elliptical holes. The Ritz method based on the principle of minimum potential energy is a commonly used method to solve the vibration response of elliptical hole plates, but it requires the construction of a global trial function that meets the natural boundary conditions and high-precision numerical integration within the hole plate domain. However, during numerical integration, the description of the geometry of the elliptical hole often requires the use of a polar coordinate system, but the overall solution of the plate is usually based on a rectangular coordinate system, resulting in the geometric description and solution calculation of elliptical hole plates of arbitrary shapes needing to be converted between two coordinate systems. This poses a challenge to the standardization of formulas for solving the vibration response of elliptical hole plates using the Ritz method. Summary of the Invention

[0003] In response to the above-mentioned deficiencies in the prior art, the present application provides a vibration solution method for arbitrarily shaped elliptical hole plates based on elliptical energy elements. This method standardizes the energy functional and numerical integration formula of the Ritz method for arbitrarily shaped elliptical hole plates, and does not change with the geometric configuration of the plate and the position and number of elliptical holes. An elliptical energy element based on two-level mapping on a rectangular domain is introduced under the framework of the global trial function, and through geometric and energy Boolean operations, the geometric characterization and high-precision numerical integration of strain energy of arbitrarily shaped elliptical hole plates are realized within the standard rectangular solution domain. In the first-level overall mapping stage, the domain of the elliptical perforated plate of arbitrary shape is placed into the standard rectangular solution domain, and then the standard rectangular solution domain is mapped to the square solution domain in the dimensionless coordinate system to establish the elliptical equation in the dimensionless coordinate system; the geometric configuration of the elliptical perforated plate is simulated based on geometric Boolean operations; in the square solution domain in the dimensionless coordinate system, the global trial function is constructed by combining Legendre polynomials and boundary condition functions, and the energy functional formula of the vibration problem is derived according to the Ritz method and the first-order shear deformation theory; in the second-level local mapping stage, the elliptical domain is transformed into a square domain by mapping, and a high-precision numerical integration method based on the elliptical energy element is constructed by combining the Gauss-Legendre integral rule, and the elements in the stiffness matrix and the mass matrix are numerically integrated through energy Boolean operations; finally, the matrix eigenvalue problem is solved, the unknown coefficients in the global trial function are derived, and the solution is completed. In the vibration solution method of the elliptical perforated plate proposed in the present invention, the energy functional and the numerical integration formula are both constructed on the square solution domain in the dimensionless coordinate system, so that the calculation and solution process of the elliptical perforated plate of arbitrary shape is completely standard. In particular, when the plate shape is elliptical, the elliptical energy element can be used for numerical integration; when the plate shape is other shapes, it is necessary to use the extended interval integration to integrate it.

[0004] In order to achieve the above-mentioned invention objectives, the technical solutions adopted in this application are:

[0005] This application provides a vibration solution method for an arbitrarily shaped elliptical perforated plate based on an elliptical energy element, including:

[0006] S1: In the overall mapping stage, the elliptical plate with an arbitrary shape is overall mapped; the dimensionless geometric parameters of the ellipse in the dimensionless coordinate system are solved, and the ellipse equation is constructed based on the dimensionless geometric parameters; the elliptical plate with an arbitrary shape is geometrically modeled based on the square solution domain and geometric Boolean operations;

[0007] S2: Construct a boundary level set function describing the plate boundary and the hole boundary, assign the boundary condition index to the boundary level set function, and establish the boundary condition function;

[0008] S3: According to the first-order shear deformation theory, the deformation of the elliptical perforated plate of arbitrary shape is simulated, and a global test function is constructed based on Legendre polynomials and the boundary condition function;

[0009] S4: Based on the minimum potential energy principle and the Ritz method, a total energy functional is established in the square solution domain, and the stationary value of the total energy functional is solved to derive the calculation formulas for each element in the stiffness matrix and the mass matrix;

[0010] S5: In the local mapping stage, based on the mapping technology, the elliptic integral domain is mapped to the square integral domain via the unit circle domain; based on the Gaussian Legendre integral, the Gaussian integral points and quadrature coefficients in the square integral domain are generated, and based on the inverse mapping, the inverse solution is in the elliptic integral domain in the dimensionless coordinate system of the square solution domain in the overall mapping stage to construct the elliptic energy element;

[0011] S6: Based on the elliptical energy element, numerically integrate each element in the stiffness matrix and the mass matrix and complete the solution to the vibration problem.

[0012] Furthermore, the S1 specifically includes:

[0013] S101: The elliptical hole plate domain of the elliptical hole plate of arbitrary shape Place it in the smallest rectangular domain that can completely cover the elliptical hole plate domain, and record the smallest rectangular domain as the standard rectangular solution domain ;

[0014] S102: Use dimensionless coordinate system transformation to convert xyz Standard rectangular solution domain in the coordinate system Mapped to ζ-ζ Interval under the coordinate system The square solution domain within , where the dimensionless coordinate system is converted to:

[0015]

[0016] Where, and yes xyz Standard rectangular solution domain in coordinate system length and width;

[0017] S103: solving dimensionless geometric parameters of the ellipse in a dimensionless coordinate system, and constructing an ellipse equation based on the dimensionless geometric parameters;

[0018] S104: Based on the dimensionless coordinate system transformation, the elliptical hole plate domain Mapping to dimensionless perforated plate domain , elliptical hole Convert to dimensionless elliptical hole , the integrand In the elliptical hole plate domain The integral formula inside is transformed into:

[0019]

[0020] Where, is xyz Coordinate system conversion to ζ-ζ Jacobian value of the coordinate system;

[0021] S105: Based on geometric Boolean operations, solve the square domain The dimensionless perforated plate domain within To model the dimensionless perforated plate domain Through the elliptical plate domain without opening Subtract multiple elliptical holes get;

[0022] Accordingly, the dimensionless aperture plate domain The integral formula inside is changed to:

[0023]

[0024] Where, is an elliptical plate domain without a hole, For the An elliptical hole, is the integral within the domain of the elliptical plate without a hole, For the The integral of an elliptical hole.

[0025] Furthermore, the S103 specifically includes:

[0026] A1: Based on the dimensionless coordinate system transformation xyz The ellipse in the coordinate system is converted to ζ-ζ In the coordinate system, ζ-ζ The equation of the ellipse in the coordinate system is:

[0027]

[0028] Where, and are the lengths of the major and minor axes of the ellipse, are the coordinates of the ellipse center, It is an ellipse The angle of the shaft's counterclockwise rotation;

[0029] A2: ζ-ζ The ellipse equation in the coordinate system is rewritten as the quadratic function of the ellipse:

[0030]

[0031] Where, 、 、 、 、 and are the coefficients of the elliptic function;

[0032] A3: As a condition, the coefficient expressions of the elliptic function are obtained:

[0033]

[0034] A4: Based on the obtained coefficient expressions, calculate ζ-ζ The dimensionless semi-major axis of the ellipse in the coordinate system , semi-minor axis and the center coordinates ( , ):

[0035]

[0036] A5: Based on the expressions of each coefficient, calculate the The angle of the axis's counterclockwise rotation The value of is calculated as:

[0037]

[0038] A6: Based on ζ-ζ The dimensionless semi-major axis of the ellipse in the coordinate system , semi-minor axis 、Center coordinates( , ) and around The angle of the axis's counterclockwise rotation ,get ζ-ζ The equation of the ellipse in the coordinate system:

[0039] .

[0040] Furthermore, the S2 specifically includes:

[0041] S201: Built on ζ-ζ The plate boundary level set function that describes the plate boundary contour in the coordinate system is:

[0042]

[0043] S202: Built on ζ-ζ The hole boundary level set function that describes the boundary contour of the elliptical hole in the coordinate system is:

[0044]

[0045] S203: According to the plate boundary level set function and the hole boundary level set function, the dimensionless hole plate domain The characterization function is:

[0046]

[0047] Where, is the plate boundary level set function, For the The hole boundary level set function of an elliptical hole, is the total number of elliptical holes;

[0048] S204: Boundary condition index Assigned to the plate boundary level set function and hole boundary level set function On, export Within the boundary Boundary condition function of an edge ; Multiply the boundary condition functions of all edges within the boundary to get the boundary Boundary condition function :

[0049]

[0050] Where, It is composed of Boundaries The total number of edges, is the displacement component, For the Within the boundary The boundary condition function of the edge, is the total number of boundaries inside and outside the dimensionless perforated plate domain, including the boundaries outside the plate, is the boundary condition index, where .

[0051] Furthermore, the S3 specifically includes:

[0052] S301: According to the first-order shear deformation theory, assuming three displacement components:

[0053]

[0054] Where, and Along , The two in-plane displacement components in the direction of It is along The out-of-plane displacement component in the direction, and are the mid-plane displacements corresponding to the in-plane displacement components, and Along Axis and The rotation angle of the axis, z Represents - plane vertical z axis coordinates;

[0055] S302: Based on the Green-Lagrange strain, derive the strain components of the first-order shear deformation theory, where the strain components are:

[0056]

[0057] Where, and are the bending strain array and the shear strain array, 、 and is the bending strain term, and is the shear strain term, To find the partial derivative operator;

[0058] S303: Assume that the plate only has lateral deformation during the vibration process, that is, and As a condition, the displacement component is simplified, and the simplified displacement component is substituted into the strain component of the reduced first-order shear deformation theory. The simplified displacement component and the strain component of the reduced first-order shear deformation theory are:

[0059]

[0060]

[0061] S304: Based on the strain components of the reduced first-order shear deformation theory and Legendre polynomials, a global trial function considering boundary conditions is constructed. The global trial function is:

[0062]

[0063] Where, 、 and There are 1 displacement global test function and 2 rotation global test functions, For the boundary The boundary condition function, is the displacement component, and Along and The number of Legendre polynomials used in the direction, is the displacement component The corresponding unknown coefficients; and It is Item and Legendre polynomials;

[0064] Wherein, the Legendre polynomial It is derived through the following recursive relation:

[0065]

[0066] Where, is the first Legendre polynomial, is the second Legendre polynomial, It is Legendre polynomials, It is Legendre polynomials, It is Legendre polynomials The order of .

[0067] Furthermore, the S4 specifically includes:

[0068] S401: Construct the elastic matrix of the isotropic plate:

[0069]

[0070] Where, and is a submatrix of the elasticity matrix, is a submatrix of the elastic matrix whose elements are all 0, for The transpose of is Young's modulus, is the material Poisson's ratio, is the shear correction factor, is the stiffness coefficient, , ;

[0071] S402: Solving the square domain , calculate the dimensionless aperture plate domain The strain energy functional and kinetic energy functional of the inner elliptical hole plate are:

[0072]

[0073]

[0074] Where, is the strain energy functional, is the kinetic energy functional, is the thickness of the plate, is the Jacobian determinant, is the characteristic frequency of vibration, is the unit volume density, 、 Bending strain array and shear stress array The transpose of

[0075] S403: Based on the strain energy functional and the kinetic energy functional, a total energy functional of the elliptical perforated plate in the dimensionless perforated plate domain is constructed:

[0076]

[0077] S404: Based on the minimum potential energy principle and the Ritz method, calculate the unknown coefficients of the total energy functional with respect to the global test function The first-order partial derivative of , we get the stationary value of the total energy functional:

[0078]

[0079] S405: Rewrite the total energy functional stationary value into a matrix form:

[0080]

[0081] Where, It is a dimension About the unknown coefficient The unknown coefficient array of is the stiffness matrix, is the mass matrix, where the stiffness matrix and mass matrix They are:

[0082]

[0083]

[0084] Where, 、 、 、 、 、 、 、 and is the stiffness matrix The submatrix of represents the matrix transpose, 、 and is the mass matrix submatrix of ;

[0085] S406: Calculate the stiffness matrix and mass matrix The value of each element in is calculated as follows:

[0086]

[0087]

[0088] Where, is the intermediate calculation matrix, where , .

[0089] Furthermore, the mapping technology is based on mapping the elliptic integral domain to the square integral domain via the unit circle domain, specifically including:

[0090] B1: ζ-ζ The elliptic integral domain defined by the elliptic equation in the coordinate system is mapped to pq The unit circle domain with a radius of 1 in the coordinate system:

[0091]

[0092] B2: Inverse the solution defined in B1 p - q The unit circle domain expression with a radius of 1 in the coordinate system is x , or Expressed as about p , q A binary function of :

[0093]

[0094] Among them, ζ-ζ Coordinate system conversion to pq The Jacobian of the coordinate system is:

[0095]

[0096] Correspondingly, in the elliptic integral domain The integral formula is changed to:

[0097]

[0098] B3: pq The unit circle domain is mapped to the coordinate system as The square integration domain within :

[0099]

[0100] Among them, the integral The domain of elliptic integrals is changed to α-β In the square integral domain under the coordinate system Points:

[0101]

[0102] Where, yes pq Coordinate system conversion to α-β Jacobian of the coordinate system;

[0103] B4: Inverse solution in the square integral domain gives x , or about α , β The binary function of :

[0104] .

[0105] Furthermore, the method generates Gaussian integral points and quadrature coefficients in the square integral domain based on Gaussian Legendre integral, and solves the Gaussian integral points and quadrature coefficients in the dimensionless coordinate system of the square solution domain in the overall mapping stage based on inverse mapping to construct an elliptical energy element, which specifically includes:

[0106] C1: Based on Gauss-Legendre integral rule, α-β In the square integral domain under the coordinate system Points within Perform a numerical integration:

[0107]

[0108] Where, and For interval Gaussian integration points generated within, and is the quadrature coefficient corresponding to the Gaussian integral point, and For the Integrand ellipse domain The major and minor axes of and For the counter, and The intervals are Inside and The number of Gaussian integration points in the direction, superscript Representative Integrand ellipse domain ;

[0109] C2: α-βThe Gaussian integral points and quadrature coefficients in the square domain under the coordinate system are reversed to the original dimensionless coordinate system through inverse mapping. ζ-ζ In the domain of the elliptic integral below:

[0110]

[0111] Where, ( , )yes ζ-ζ The coordinates of the Gaussian integral point in the coordinate system, is the corresponding quadrature coefficient, For the The first Gaussian integration points, For the Elliptical domain in ζ-ζ The coordinates of the center of the ellipse in the coordinate system;

[0112] C3: The original dimensionless coordinate system obtained based on the inverse mapping ζ-ζ The Gaussian integral points and quadrature coefficients in the elliptical domain below are used to construct the elliptical energy element.

[0113] Among them, based on the elliptical energy element ζ-ζ The high-precision numerical integration in the coordinate system is:

[0114] .

[0115] Furthermore, the S6 specifically includes:

[0116] S601: The geometry of the elliptical perforated plate is simulated based on the elliptical energy element and the energy geometry Boolean operation, that is, the following formula is obtained:

[0117]

[0118] Where, and They are the dimensionless perforated plate domain with elliptical holes The stiffness matrix and mass matrix inside, and are the stiffness matrix and mass matrix of the unperforated elliptical plate domain, and Representative oval hole The stiffness matrix and mass matrix inside, is the total number of elliptical holes;

[0119] S602: Intermediate calculation matrices contained in the stiffness matrix and mass matrix Each element is calculated, and the intermediate calculation matrix The Row and Column Elements It can be calculated by the following formula:

[0120]

[0121] in,

[0122] Where, and It is Item and Legendre polynomials, and They are Item and Legendre polynomials, and is an intermediate variable;

[0123] S603: Calculation of elements based on elliptical energy elements Numerical integration of :

[0124]

[0125] Where, Represents matrix elements The integrand in ;

[0126] S604: Based on the numerical integration of each element in the obtained stiffness matrix and mass matrix, the assembly stiffness matrix and mass matrix are calculated, and the eigenvalue problem of the matrix rewritten as the stationary value of the total energy functional is solved. The corresponding eigenfrequencies and eigenvectors are derived to complete the solution to the vibration problem of the elliptical hole plate.

[0127] The beneficial effects of this application are:

[0128] This application proposes a vibration solution method for elliptical perforated plates of arbitrary shapes based on elliptical energy elements. An elliptical energy element defined in a dimensionless coordinate system is constructed, which can perform high-precision numerical integration of polynomial functions on the elliptical domain. Within the square solution domain, the geometry of an arbitrary elliptical perforated plate can be constructed based on geometric Boolean operations and elliptical energy elements, and then the energy functional formula of the elliptical perforated plate is derived under the framework of the Ritz method and the global trial function. The high-precision numerical integration of the elements in the stiffness matrix and the mass matrix needs to be implemented based on elliptical energy elements and energy geometric Boolean operations. Finally, the solution is completed by deriving the eigenfrequency and eigenvector by solving the matrix eigenproblem. The above calculation process and solution formula are completely standard for the vibration problem of elliptical plates with any number of elliptical holes. BRIEF DESCRIPTION OF THE DRAWINGS

[0129] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other embodiments can also be obtained based on these drawings.

[0130] Figure 1 A schematic flow chart of a method for solving the vibration of an elliptical hole plate of arbitrary shape based on an elliptical energy element provided in an embodiment of the present application.

[0131] Figure 2 Schematic diagram of the geometric modeling of an elliptical perforated plate provided in an embodiment of the present application.

[0132] Figure 3 The embodiment of the present application provides an elliptical domain formed by xyz Coordinate system mapped to ζ-ζ Coordinate system diagram.

[0133] Figure 4 Schematic diagram of the transformation of the elliptic integral domain provided in the embodiment of the present application, mapped to the unit circle domain and then converted to the square integral domain.

[0134] Figure 5 Schematic diagram of the fully clamped elliptical perforated plate model provided in an embodiment of the present application.

[0135] Figure 6 The first eight vibration mode diagrams of the fully clamped elliptical plate with a hole derived from the vibration problem solving method proposed in this application. DETAILED DESCRIPTION

[0136] The following will be combined with the drawings in the embodiments of this application to clearly and completely describe the technical solutions in the embodiments of this application. Obviously, the embodiments described are only part of the embodiments of this application, not all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field based on this application are within the scope of protection of this application.

[0137] The present invention provides a method for solving the vibration of an elliptical plate with an arbitrary shape based on an elliptical energy element. Figure 1 , Figure 1 The figure shows a flow chart of a vibration solution method for an elliptical plate with an arbitrary shape and an elliptical hole based on an elliptical energy element provided in an embodiment of the present application, including:

[0138] S1: Overall mapping of a plate with an elliptical hole of arbitrary shape.

[0139] like Figure 2 As shown, Figure 2The schematic diagram of geometric modeling of the elliptical hole plate provided in the embodiment of the present application, wherein (a) is the elliptical hole plate domain Insert a standard rectangular solution domain (b) is the standard rectangular solution domain Convert to ζ-ζ Square solution domain in the coordinate system (c) Modeling of elliptical plate domain with hole based on geometric Boolean operations. Consider an elliptical plate domain with hole , the outer boundary of the plate is marked with the symbol Indicates that there are several elliptical holes inside the plate , , the corresponding elliptical hole boundary is To standardize the calculation process, the elliptical hole plate domain Insert a long one that completely covers it Width The minimum rectangular domain is defined as the standard rectangular solution domain , xyz The origin of the coordinate system is at its center, such as Figure 2 As shown in (a), it is then mapped to a dimensionless coordinate system through the coordinate transformation formula ζ-ζ Down:

[0140] (1)

[0141] Based on formula (1), the standard rectangular solution domain will be converted to a range defined in The square solution domain within , dimensionless coordinate system ζ-ζ The origin of is located at the geometric center of the square solution domain. In order to standardize the solution process for elliptical holes of arbitrary shapes, all subsequent modeling and calculation solutions are based on the formula (1). ζ-ζ Dimensionless coordinate system and square solution domain Conducted within.

[0142] Therefore, you need to first xyz The ellipse in the coordinate system is converted to ζ-ζ In the dimensionless coordinate system, Figure 3 As shown, Figure 3 The embodiment of the present application provides an elliptical domain formed by xyz Coordinate system mapped to ζ-ζ Schematic diagram of the coordinate system, where (a) is xyz The ellipse in the coordinate system, (b) is ζ-ζ The domain of elliptic integral in the coordinate system is x - y The equation of the ellipse in the coordinate system is:

[0143] (2)

[0144] Where, and are the lengths of the major and minor axes of the ellipse, respectively. Figure 3 In (a), the coordinates of the ellipse center are , the ellipse around The axis rotates counterclockwise by an angle Substituting formula (1) into formula (2), we can obtain ζ-ζ The equation of the ellipse in the coordinate system:

[0145] (3)

[0146] Expand the above formula and rewrite it into the form of an elliptical quadratic function:

[0147] (4)

[0148] Where, 、 、 、 、 and are the coefficients of the elliptic function;

[0149] Assume that there is , then the expressions of the coefficients in the equation can be derived:

[0150] (5)

[0151] Accordingly, ζ-ζ The dimensionless major semi-axis of the ellipse (see 3(b) in the figure) in the coordinate system , semi-minor axis and the center coordinates ( , )for:

[0152] (6)

[0153] Around The angle of the axis's counterclockwise rotation It is determined by the following formula:

[0154] (7)

[0155] Combining equations (6) and (7), the dimensionless geometric parameters of the ellipse ( , , , , ) is substituted into formula (2), and we can derive ζ-ζThe equation of the ellipse in the coordinate system:

[0156] (8)

[0157] At the same time, after the coordinate system transformation is completed, the elliptical hole plate domain Mapped into a dimensionless hole plate domain Ξ, the elliptical hole Then it is converted into a dimensionless elliptical hole ,like Figure 2 As shown in (b) in . Therefore, let the integrand be The integral in the elliptical hole plate domain Ω is:

[0158] (9)

[0159] Then based on the transformation of formula (1), the integral formula (9) will be converted into the following form:

[0160] (10)

[0161] In the formula, the Jacobian determinant is .for Figure 2 The dimensionless perforated plate domain Ξ in (b) can be modeled based on geometric Boolean operations. The plate domain is considered as a plate domain derived from an elliptical plate domain. There are 3 oval holes in the middle ( ), see Figure 2 (c) in , so the integral formula (10) is changed to:

[0162] (11)

[0163] Where, is the integral within the domain of the elliptical plate without a hole, For the The integral of an elliptical hole.

[0164] S2: Construct a boundary level set function that describes the plate boundary and the hole boundary, assign the boundary condition index to the boundary level set function, and establish the boundary condition function.

[0165] To simulate the boundary conditions of the elliptical hole plate, it is necessary to define ζ-ζ The plate boundary level set function that describes the plate boundary contour in the coordinate system , its general form is:

[0166] (12)

[0167] Similarly, the hole boundary level set function describing the boundary contour of the elliptical hole is It is defined as:

[0168] (13)

[0169] In formulas (12) and (13), and The corresponding dimensionless geometric parameters of the ellipse ( , , , , ) is substituted into Equation (11) to obtain. Furthermore, the dimensionless perforated plate domain Ξ can be expressed based on the above boundary level set function:

[0170] (14)

[0171] In the formula L is the total number of elliptical holes.

[0172] For the vibration problem of the elliptical hole plate, different boundary conditions can be applied to the plate and hole boundaries respectively. ( ) on different edges within .

[0173] Therefore, by giving Within the boundary Edges with respect to displacement components Boundary condition index To the boundary level set function On the above, we can derive the characterization of Within the boundary Boundary condition function for each boundary condition ; Afterwards, The boundary condition functions of all edges within the boundary are multiplied together to form the boundary Γ. i Boundary condition function :

[0174] (15)

[0175] Where, It is composed of Boundaries The total number of edges, is the displacement component, For the Within the boundary The boundary condition function of the edge, is the total number of boundaries inside and outside the dimensionless perforated plate domain, including the external boundaries of the plate, and the boundary condition index It needs to be determined according to the following formula:

[0176] (16)

[0177] Combining equations (15) and (16), the boundary condition function of the elliptical hole plate with arbitrary shape can be derived according to the actual boundary conditions applied.

[0178] S3: According to the first-order shear deformation theory, the deformation of the elliptical perforated plate of arbitrary shape is simulated, and a global test function is constructed based on the Legendre polynomials and the boundary condition function.

[0179] S301: To accurately simulate the vibration deformation of an elliptical plate with a hole, according to the first-order shear deformation theory, it is necessary to assume two in-plane displacement components and one out-of-plane displacement component:

[0180] (17)

[0181] In the formula and Along , The two in-plane displacement components in the direction of It is along The out-of-plane displacement component in the direction, and are the mid-plane displacements corresponding to the in-plane displacement components, and Along Axis and The rotation angle of the axis, z Represents - plane vertical z Axis coordinates.

[0182] S302: Based on the Green-Lagrange strain, the strain component expression of the first-order shear deformation theory can be given:

[0183] (18)

[0184] in,

[0185] (19)

[0186] Where, and are the bending strain array and the shear strain array, 、 and is the bending strain term, and is the shear strain term, To find the partial derivative operator.

[0187] S303: Assume that the plate only has lateral deformation during the vibration process, that is, and , then the displacement component in equation (17) can be simplified as:

[0188] (20)

[0189] Accordingly, the strain components can be reduced to the following form:

[0190] (twenty one)

[0191] S304: Based on the first-order shear deformation theory (21) and Legendre polynomials, a global trial function considering boundary conditions is constructed. It is necessary to assume a displacement global trial function and 2 corner global test functions 、 :

[0192] (twenty two)

[0193] Where, 、 and There are 1 displacement global test function and 2 rotation global test functions, For the boundary The boundary condition function, is the displacement component, and Along and The number of Legendre polynomials used in the direction, is the displacement component The corresponding unknown coefficients; and It is Item and The Legendre polynomials are defined by the following recursive relation:

[0194] (twenty three)

[0195] In the formula is the first Legendre polynomial, is the second Legendre polynomial, It is Legendre polynomials, It is Legendre polynomials, It is Legendre polynomials The order of .

[0196] S4: Based on the minimum potential energy principle and the Ritz method, the total energy functional in the square solution domain is established, and the stationary value of the total energy functional is solved to derive the calculation formulas for each element in the stiffness matrix and the mass matrix.

[0197] S401: For an isotropic plate, its elastic matrix can be defined as:

[0198] (twenty four)

[0199] Where, and is a submatrix of the elasticity matrix, is a submatrix of the elastic matrix whose elements are all 0, for The transpose of is Young's modulus, is the material Poisson's ratio, is the shear correction factor, is the stiffness coefficient, , .

[0200] S402: To unify the energy functional and numerical integration formula, in Equations (17) to (21) x - y The coordinate system needs to be converted to a dimensionless coordinate system through formula (1) x - or Then, calculate again. Based on the square solution domain established in step S1 , the dimensionless strain energy functional calculation formula in the perforated plate domain Ξ can be derived:

[0201] (25)

[0202] Where, is the thickness of the plate, is the Jacobian determinant, 、 Bending strain array and shear stress array The transpose of .

[0203] Similarly, for the dimensionless perforated plate domain Ξ, the calculation formula of the kinetic energy functional within the domain is defined as follows:

[0204] (26)

[0205] Where, is the characteristic frequency of vibration, is the unit volume density.

[0206] S403: According to equations (25) and (26), the total energy functional formula within the dimensionless perforated plate domain Ξ can be expressed as:

[0207] (27)

[0208] S404: Find the total energy functional based on the minimum potential energy principle and the Ritz method P Regarding the unknown coefficients { , , The first-order partial derivative of} can be used to obtain the stationary value of the total energy functional:

[0209] (28)

[0210] S405: The above formula is a set of unknown coefficients { , , }, rewrite it into matrix form:

[0211] (29)

[0212] In the formula It is a dimension About the unknown coefficients { , , } unknown coefficient array, is the stiffness matrix, is the mass matrix, corresponding to the eigenvector of a certain order eigenfrequency, the dimension of the stiffness matrix K and the mass matrix M is 3 MN ×3 MN :

[0213] (30)

[0214] Where, 、 、 、 、 、 、 、 and is the stiffness matrix The sub-matrix of 、 and is the mass matrix submatrix of .

[0215] S406: The calculation formulas for each element in the stiffness matrix K and mass matrix M in the above formula are:

[0216] (31)

[0217] as well as

[0218] (32)

[0219] It can be seen that equations (31) and (32) contain the calculation of the intermediate matrix ( , ), and each element needs to be numerically integrated within the dimensionless perforated plate domain Ξ containing the elliptical hole.

[0220] S5: In the local mapping stage, based on the mapping technology, the elliptic integral domain is mapped to the square integral domain through the unit circle domain; based on the Gaussian Legendre integral, the Gaussian integral points and integration coefficients in the square integral domain are generated, and based on the inverse mapping, the inverse solution is performed to the elliptic integral domain in the dimensionless coordinate system of the square solution domain in the overall mapping stage to construct the elliptic energy element.

[0221] S501: Using mapping technology to transform the elliptical domain into a square integral domain.

[0222] In one embodiment of the present application, the intermediate calculation matrix in equations (31) and (32) is In order to perform high-precision numerical integration on the elements of , and avoid the conversion of the elliptic coordinate system during integration, and to standardize the numerical integration formula, it is necessary to use mapping technology to transform the elliptic integral domain. First, the elliptic integral domain is mapped to a unit circle domain, and then the unit circle domain is mapped to a square integral domain. Then, Gaussian integral points are generated in the square integral domain for numerical integration, such as Figure 4 As shown, Figure 4 Schematic diagram of the transformation from the elliptic integral domain to the unit circle domain and then to the square integral domain provided in the embodiment of the present application, where (a) is ζ-ζ The elliptic integral domain in the coordinate system; (b) is pq Unit circle domain in the coordinate system; (c) is α-β The square integral domain in the coordinate system. Therefore, in formula (8) ζ-ζ Elliptic integral domain in coordinate system (see Figure 4 (a)) in will first be mapped to a pq The unit circle with a radius of 1 in the coordinate system (see Figure 4 (b) in the following example:

[0223] (33)

[0224] To facilitate calculation, reverse equation (33) and write x , or Expressed as about p , q A binary function of :

[0225] (34)

[0226] Therefore ζ-ζ Coordinate system conversion to pq The Jacobian of the coordinate system is:

[0227] (35)

[0228] Accordingly, ( ) is also expressed by the integral formula (11) ζ-ζ Coordinate system conversion to pq Coordinate system:

[0229] (36)

[0230] It can be seen that the variables in the above formula q The upper and lower limits of the integral are still about p It is not convenient to use Gauss-Legendre integral rule to integrate the function, so it is necessary to further use mapping technology to convert pq The unit circle domain is mapped to the coordinate system as The square integration domain within :

[0231] (37)

[0232] Therefore, the integral of formula (36) is Finally α-β In the square integral domain under the coordinate system Perform integration (see Figure 4 (c) in the following example:

[0233] (38)

[0234] In the formula yes pq Coordinate system conversion to α-β Jacobian of the coordinate system:

[0235] (39)

[0236] Substituting equation (37) into equation (34), we can obtain x , or about α , β The binary function of :

[0237] (40)

[0238] It can be seen that for any elliptical domain, the integral is ultimately in the square integral domain Conducted within.

[0239] S502: Based on the inverse mapping, the Gaussian integral points in the square integral domain are inversely solved to a dimensionless coordinate system to construct an elliptical energy element.

[0240] In one embodiment of the present application, equation (38) is numerically integrated based on the Gauss-Legendre integral rule:

[0241] (41)

[0242] In the formula and For interval Gaussian integration points generated within, and is the quadrature coefficient corresponding to the Gaussian integral point, and are the major and minor axes of the ellipse domain, and For the counter, and The intervals are Inside x and or The number of Gaussian integration points in the direction, superscript Representative Integrand ellipse domain To facilitate numerical integration calculation, the Gaussian integral points and quadrature coefficients in the square domain of Equation (41) are inversely mapped to the dimensionless coordinate system of the original square domain in the overall mapping stage. ζ-ζ In the domain of the elliptic integral below:

[0243] (42)

[0244] Where ( , )yes ζ-ζ The coordinates of the Gaussian integral point in the coordinate system, is the corresponding quadrature coefficient, For the The first Gaussian integration points, For the Elliptical domain in ζ-ζ Coordinates of the ellipse center in the coordinate system; Substitute equation (42) into equation (41), so that the numerical integration of the ellipse domain of arbitrary shape is still ζ-ζ Coordinate system and interval Conducted within:

[0245] (43)

[0246] When the function FWhen representing the energy functional integrand in Equations (25) to (27), Equation (43) defines ζ-ζ The elliptical energy element in the coordinate system is an elliptical energy integration unit that characterizes the energy on the elliptical domain and can perform high-precision numerical integration of the energy in the integrated elliptical domain. x - y Any elliptic integral domain in the coordinate system is converted into an interval based on mapping technology After the square integration domain is inside, the five geometric parameters ( , , , , ) Determine the i Elliptical energy element Ξ i Size and position (see equations (6) to (8)), and then calculate ζ-ζ The corresponding Gaussian integration points and quadrature coefficients in the coordinate system are then calculated. Finally, based on the energy geometry Boolean operation (i.e., assigning positive and negative attributes to each elliptical energy element based on the geometric Boolean operation of the elliptical energy element that forms the elliptical hole plate, see Equation (11)), a high-precision numerical integral can be solved within the domain Ω of an elliptical hole plate of any shape. The numerical integration formula based on the local mapping elliptical energy element is completely standard and applicable to any elliptical plate with any number of elliptical holes.

[0247] S6: Based on the elliptical energy element, numerically integrate each element in the stiffness matrix and the mass matrix and complete the solution to the vibration problem.

[0248] In one embodiment of the present application, in order to calculate the high-precision numerical integral of each element in the stiffness matrix K and the mass matrix M defined in the dimensionless perforated plate domain Ξ containing an elliptical hole (see Equations (31) and (32)), it is necessary to simulate the geometry of the plate based on the elliptical energy element and the energy geometry Boolean operation, that is, the following formula is obtained:

[0249] (44)

[0250] Where K0 and M0 are the stiffness matrix and mass matrix of the elliptical plate domain without holes (see Figure 2 The Ξ0 in (c) is regarded as a large-scale elliptical energy element Ξ0), and Representative oval hole The stiffness matrix and mass matrix inside, is the total number of elliptical holes (elliptical holesΞ i See Figure 2 (c) in the figure is considered as several smaller elliptical energy elements Ξ i ). Except for the 5 geometric parameters ( , , , , ), the numerical integration formulas of these elliptical energy elements are exactly the same. Based on formula (44), the intermediate calculation matrix contained in formulas (31) and (32) can be ( , ) is calculated for each element in the matrix. r Row and s Column Elements It can be calculated by the following formula:

[0251]

[0252] In the formula, the intermediate variable and Calculated by the following formula:

[0253] (45)

[0254] In the formula and , and Representing the k Item and l Item, Item and The Legendre polynomials are shown in equation (23). Finally, the numerical integral equation (43) in the elliptic energy element is introduced to complete the element Numerical integration of :

[0255] (46)

[0256] In the formula F Represents matrix elements The integrand in , , , are the Gaussian integration points and their quadrature coefficients within the elliptical energy element, respectively, see Equation (42). After completing the numerical integration and assembly of each element in the stiffness matrix K and the mass matrix M, the corresponding eigenfrequency and eigenvector can be derived by solving the eigenvalue problem of Equation (29), completing the solution to the vibration problem of the elliptical perforated plate.

[0257] In one embodiment of the present application, the vibration problem solving method described in this application is applied to the vibration problem of an elliptical hole plate with both inner and outer boundaries clamped. Clamped boundary conditions are applied to the outer boundary of the plate and the inner boundary of the elliptical hole. The first eight vibration characteristic frequencies of the plate are solved and the corresponding vibration mode diagram is drawn. The material constants of the elliptical hole plate are: E=200GPa, n =0.3, r =7800kg / m3; geometric parameters are: a 0 / b 0= a 1 / b 1=2, b 1 / b 0=0.25, h / b 0=0.005, b 0=1m, the rotation angle of the elliptical hole in the counterclockwise direction c 1=π / 4, the rotation angle of the elliptical hole itself i 1=π / 4, e 1 is the distance from the center of the elliptical hole to the origin of the coordinate system O The distance, parameter = e 1 / ( e 1) max =0.9 represents the distance from the center of the elliptical hole to O The ratio of the distance of the point to the distance to the plate boundary along the path. The geometric model and discrete model of the elliptical hole plate are as follows: Figure 5 As shown, Figure 5 Schematic diagram of the fully-fixed elliptical perforated plate model provided in an embodiment of the present application, where (a) is its geometric model; (b) is the elliptical energy element discrete model.

[0258] The specific implementation steps of this embodiment are as follows:

[0259] Step 1: Overall mapping of the elliptical hole plate of arbitrary shape. First, geometric modeling of the elliptical hole plate is performed, such as Figure 5 The gray area in (a) is the elliptical hole plate domain Ω, which has an elliptical hole Ω1 inside. It is its length a Width b The standard rectangular solution domain, with the origin located at The center of . Using the dimensionless coordinate transformation formula (1) x - y Standard rectangular solution domain in coordinate system Mapped to ζ-ζ Interval under the coordinate system The square solution domain is inside . xyz The unperforated elliptical plate domain and the elliptical hole domain in the coordinate system are transformed into ζ-ζ coordinate system, and derive from equations (2) to (8): ζ-ζ The equation of the ellipse in the coordinate system. This elliptical plate domain is modeled based on geometric Boolean operations. The geometric configuration of the plate is obtained by drilling an elliptical hole in the unperforated elliptical plate domain.

[0260] Step 2: Since the inner and outer boundaries of the elliptical hole plate are both ellipses, the boundary level set function is the ellipse equation, see Equation (8). According to the applied fully clamped boundary conditions, the boundary condition function of the plate is:

[0261] (47)

[0262] Where the rotation angle It can be obtained by formula (7).

[0263] Step 3: Deformation simulation of the plate with an elliptical hole of arbitrary shape and construction of global trial functions based on the first-order shear deformation theory. According to the first-order shear deformation theory, while ignoring the lateral deformation of the plate during vibration, the reduced strain component formula (21) is derived. Based on the Legendre polynomials and the boundary condition function formula (47) in step 1, a displacement global trial function and two rotation global trial functions are constructed.

[0264] Step 4: Based on the minimum potential energy principle and the Ritz method, the energy functional in the square solution domain is established and solved. First, the elastic matrix of the isotropic plate (24) is derived and expressed in the dimensionless coordinate system. ζ-ζ In the square solution domain below, the strain energy and kinetic energy functional formulas of the elliptical hole plate are derived, and finally the total energy functional formula of the plate is written. Then, based on the minimum potential energy principle and the Ritz method, the total energy functional formula is obtained. P About the unknown coefficients in the global test function { , , The first-order partial derivative of} obtains the stationary value of the functional, resulting in a set of linear equations, which are further rewritten into matrix form to derive the expressions of each element in the stiffness matrix K and the mass matrix M, as shown in Equations (31) and (32).

[0265] Step 5: Based on the local mapping, construct the elliptical energy element for high-precision numerical integration. ζ-ζ In the coordinate system, the unperforated elliptical plate domain and the elliptical hole domain are mapped as pq The unit circle domain with a radius of 1 in the coordinate system is then further mapped to α-β Square integration domain in the coordinate system , see formula (33) to formula (41). To standardize the numerical integral calculation formula, α-β The Gaussian integral points and quadrature coefficients in the square integral domain under the coordinate system are inversely mapped back to the dimensionless coordinate system of the square solution domain in the original overall mapping stage. ζ-ζ In the elliptic integral domain under the equation (42), the energy concept and Gauss-Legendre integral rule are combined to establish ζ-ζElliptical energy element in the coordinate system. For the elliptical perforated plate in this embodiment, two elliptical energy elements are used for numerical calculation, and 43×43=1849 Gaussian integral points are arranged in each elliptical energy element. Figure 5 (b) in the.

[0266] Step 6: Introduce elliptical energy elements to perform numerical integration of elements in stiffness matrix and mass matrix and complete the solution of vibration problem. Calculate matrix based on elliptical energy elements and energy Boolean operation The numerical integration of each element is shown in Equations (44) and (46). After that, the stiffness matrix K and the mass matrix M are calculated and assembled. By solving the eigenvalue problem of Equation (29), the first eight eigenfrequencies and eigenvectors can be derived, and the first eight vibration mode diagrams can be drawn, completing the solution to the vibration problem of the elliptical hole plate.

[0267] The vibration problem of the fully fixed elliptical perforated plate with inner and outer boundaries was analyzed by proposing a vibration solution method for arbitrarily shaped elliptical perforated plates based on elliptical energy elements. The first 8 order eigenfrequencies were solved, the corresponding vibration modal diagrams were drawn, and compared with the existing results. It can be seen that the results of the proposed vibration problem solution method based on elliptical energy elements are very close to the existing technology results. Table 1 shows the results of the proposed vibration problem solution method in terms of the number of Legendre polynomials. M×N The numerical results when taking 33×33, and the comparison with the existing technology results, Figure 6 The first eight vibration mode diagrams of the fully clamped elliptical plate with a hole derived from the vibration problem solving method proposed in this application.

[0268] Table 1 The first eight normalized eigenfrequencies of the fully clamped elliptical plate with an elliptical hole

[0269]

[0270] The present application proposes a vibration solution method for an elliptical hole plate of arbitrary shape based on an elliptical energy element. In the first-level overall mapping stage, the domain of the elliptical hole plate of arbitrary shape is placed in a standard rectangular solution domain, and then the standard rectangular solution domain is mapped to a square solution domain in a dimensionless coordinate system to establish an elliptical equation in the dimensionless coordinate system; the geometric configuration of the elliptical hole plate is simulated based on geometric Boolean operations; in the square solution domain in the dimensionless coordinate system, a global trial function is constructed by combining Legendre polynomials and boundary condition functions, and the energy functional formula of the vibration problem is derived according to the Ritz method and the first-order shear deformation theory; in the second-level local mapping stage, the elliptical domain is transformed into a square domain through mapping, and the elliptical energy element is constructed by combining the Gauss-Legendre integral rule, and the elements in the stiffness matrix and the mass matrix are numerically integrated with high precision through energy Boolean operations; finally, the matrix eigenvalue problem is solved, the assumed unknown coefficients are derived, and the solution is completed. In the vibration solution method for elliptical hole plates proposed in this application, the energy functional and numerical integration formula are constructed on the square solution domain under the dimensionless coordinate system in the overall mapping stage, making the calculation and solution process for elliptical hole plates of arbitrary shapes completely standard.

[0271] It should be noted that those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of this application, and it should be understood that the scope of protection of this application is not limited to such specific descriptions and embodiments. Those skilled in the art can make various other specific variations and combinations based on the technical teachings disclosed in this application without departing from the essence of this application, and such variations and combinations are still within the scope of protection of this application.

Claims

1. A vibration solution method for an elliptical hole plate of arbitrary shape based on elliptical energy element, characterized in that: include: S1: In the overall mapping stage, the elliptical plate with an arbitrary shape is overall mapped; the dimensionless geometric parameters of the ellipse in the dimensionless coordinate system are solved, and the ellipse equation is constructed based on the dimensionless geometric parameters; the elliptical plate with an arbitrary shape is geometrically modeled based on the square solution domain and geometric Boolean operations; S2: Construct a boundary level set function describing the plate boundary and the hole boundary, assign the boundary condition index to the boundary level set function, and establish the boundary condition function; S3: According to the first-order shear deformation theory, the deformation of the elliptical perforated plate of arbitrary shape is simulated, and a global test function is constructed based on Legendre polynomials and the boundary condition function; S4: Based on the minimum potential energy principle and the Ritz method, a total energy functional is established in the square solution domain, and the stationary value of the total energy functional is solved to derive the calculation formulas for each element in the stiffness matrix and the mass matrix; S5: In the local mapping stage, based on the mapping technology, the elliptic integral domain is mapped to the square integral domain via the unit circle domain; based on the Gaussian Legendre integral, the Gaussian integral points and quadrature coefficients in the square integral domain are generated, and based on the inverse mapping, the inverse solution is in the elliptic integral domain in the dimensionless coordinate system of the square solution domain in the overall mapping stage to construct the elliptic energy element; S6: Based on the elliptical energy element, numerically integrate each element in the stiffness matrix and the mass matrix and complete the solution to the vibration problem.

2. The vibration solution method of an elliptical plate with an arbitrary shape based on an elliptical energy element according to claim 1 is characterized in that: Said S1 specifically includes: S101: The elliptical hole plate domain of the elliptical hole plate of arbitrary shape Place it in the smallest rectangular domain that can completely cover the elliptical hole plate domain, and record the smallest rectangular domain as the standard rectangular solution domain ; S102: Use dimensionless coordinate system transformation to convert xy Standard rectangular solution domain in the coordinate system Mapped to ξ-η Interval under the coordinate system The square solution domain , where the dimensionless coordinate system is converted to: Where, and yes xy Standard rectangular solution domain in the coordinate system length and width; S103: solving dimensionless geometric parameters of the ellipse in a dimensionless coordinate system, and constructing an ellipse equation based on the dimensionless geometric parameters; S104: Based on the dimensionless coordinate system transformation, the elliptical hole plate domain Mapping to dimensionless perforated plate domain , elliptical hole Convert to dimensionless elliptical hole , the integrand In the elliptical hole plate domain The integral formula inside is transformed into: Where, is xy Coordinate system conversion to ξ-η Jacobian value of the coordinate system; S105: Based on geometric Boolean operations, solve the square domain The dimensionless perforated plate domain within To model the dimensionless perforated plate domain Through the elliptical plate domain without opening Subtract multiple elliptical holes get; Accordingly, the dimensionless aperture plate domain The integral formula inside is changed to: Where, is an elliptical plate domain without a hole, For the An elliptical hole, is the integral within the domain of the elliptical plate without a hole, For the The integral of an elliptical hole.

3. The vibration solution method of an elliptical plate with an arbitrary shape based on an elliptical energy element according to claim 2 is characterized in that: The S103 specifically includes: A1: Based on the dimensionless coordinate system transformation xy The ellipse in the coordinate system is converted to ξ-η In the coordinate system, ξ-η The equation of the ellipse in the coordinate system is: Where, and are the lengths of the major and minor axes of the ellipse, are the coordinates of the ellipse center, It is an ellipse The angle of the shaft's counterclockwise rotation; A2: ξ-η The ellipse equation in the coordinate system is rewritten as the quadratic function of the ellipse: Where, 、 、 、 、 and are the coefficients of the elliptic function; A3: As a condition, the coefficient expressions of the elliptic function are obtained: A4: Based on the obtained coefficient expressions, calculate ξ-η The dimensionless semi-major axis of the ellipse in the coordinate system , semi-minor axis and the center coordinates ( , ): A5: Based on the expressions of each coefficient, calculate the The angle of the axis's counterclockwise rotation The value of is calculated as: A6: Based on ξ-η The dimensionless semi-major axis of the ellipse in the coordinate system , semi-minor axis 、Center coordinates( , ) and around The angle of the axis's counterclockwise rotation ,get ξ-η The equation of the ellipse in the coordinate system: 。 4. The vibration solution method of an elliptical plate with an arbitrary shape based on an elliptical energy element according to claim 3 is characterized in that: The S2 specifically includes: S201: Built on ξ-η The plate boundary level set function that describes the plate boundary contour in the coordinate system is: S202: Built on ξ-η The hole boundary level set function that describes the boundary contour of the elliptical hole in the coordinate system is: S203: According to the plate boundary level set function and the hole boundary level set function, the dimensionless hole plate domain The characterization function is: Where, is the plate boundary level set function, For the The hole boundary level set function of an elliptical hole, is the total number of elliptical holes; S204: Boundary condition index Assigned to the plate boundary level set function and hole boundary level set function On, export Within the boundary Boundary condition function of an edge ; Multiply the boundary condition functions of all edges within the boundary to get the boundary Boundary condition function : Where, It is composed of Boundaries The total number of edges, is the displacement component, For the Within the boundary The boundary condition function of the edge, is the total number of boundaries inside and outside the dimensionless perforated plate domain, including the boundaries outside the plate, is the boundary condition index, where .

5. The vibration solution method of an elliptical plate with an arbitrary shape based on an elliptical energy element according to claim 4 is characterized in that: The S3 specifically includes: S301: According to the first-order shear deformation theory, assuming three displacement components: Where, and Along , The two in-plane displacement components in the direction of It is along The out-of-plane displacement component in the direction, and are the mid-plane displacements corresponding to the in-plane displacement components, and Along Axis and The rotation angle of the axis, z Represents - plane vertical z axis coordinates; S302: Based on the Green-Lagrange strain, derive the strain components of the first-order shear deformation theory, where the strain components are: Where, and are the bending strain array and the shear strain array, 、 and is the bending strain term, and is the shear strain term, To find the partial derivative operator; S303: Assume that the plate only has lateral deformation during the vibration process, that is, and As a condition, the displacement component is simplified, and the simplified displacement component is substituted into the strain component of the reduced first-order shear deformation theory. The simplified displacement component and the strain component of the reduced first-order shear deformation theory are: S304: Based on the strain components of the reduced first-order shear deformation theory and Legendre polynomials, a global trial function considering boundary conditions is constructed. The global trial function is: Where, 、 and There are 1 displacement global test function and 2 rotation global test functions, For the boundary The boundary condition function, is the displacement component, and Along and The number of Legendre polynomials used in the direction, is the displacement component The corresponding unknown coefficients; and It is Item and Legendre polynomials; Wherein, the Legendre polynomial It is derived through the following recursive relation: Where, is the first Legendre polynomial, is the second Legendre polynomial, It is Legendre polynomials, It is Legendre polynomials, It is Legendre polynomials The order of .

6. The vibration solution method for an elliptical plate with an arbitrary shape based on an elliptical energy element according to claim 5 is characterized in that: The S4 specifically includes: S401: Construct the elastic matrix of the isotropic plate: Where, and is a submatrix of the elasticity matrix, is a submatrix of the elastic matrix whose elements are all 0, for The transpose of is Young's modulus, is the material Poisson's ratio, is the shear correction factor, is the stiffness coefficient, , ; S402: Solving the square domain , calculate the dimensionless aperture plate domain The strain energy functional and kinetic energy functional of the inner elliptical hole plate are: Where, is the strain energy functional, is the kinetic energy functional, is the thickness of the plate, is the Jacobian determinant, is the characteristic frequency of vibration, is the unit volume density, 、 Bending strain array and shear stress array The transpose of S403: Based on the strain energy functional and the kinetic energy functional, a total energy functional of the elliptical perforated plate in the dimensionless perforated plate domain is constructed: S404: Based on the minimum potential energy principle and the Ritz method, calculate the unknown coefficients of the total energy functional with respect to the global test function The first-order partial derivative of , we get the stationary value of the total energy functional: S405: Rewrite the total energy functional stationary value into a matrix form: Where, It is a dimension About the unknown coefficient The unknown coefficient array of is the stiffness matrix, is the mass matrix, where the stiffness matrix and mass matrix They are: Where, 、 、 、 、 、 、 、 and is the stiffness matrix The submatrix of represents the matrix transpose, 、 and is the mass matrix submatrix of ; S406: Calculate the stiffness matrix and mass matrix The value of each element in is calculated as follows: Where, is the intermediate calculation matrix, where , .

7. The vibration solution method of an elliptical plate with an arbitrary shape based on an elliptical energy element according to claim 6 is characterized in that: The mapping technology is based on mapping the elliptic integral domain to the square integral domain via the unit circle domain, specifically including: B1: ξ-η The elliptic integral domain defined by the elliptic equation in the coordinate system is mapped to pq The unit circle domain with a radius of 1 in the coordinate system: B2: Inverse the solution defined in B1 p - q The unit circle domain expression with a radius of 1 in the coordinate system is ξ , η Expressed as about p , q A binary function of : Among them, ξ-η Coordinate system conversion to pq The Jacobian of the coordinate system is: Correspondingly, within the elliptic integral domain The integral formula is changed to: B3: pq The unit circle domain is mapped to the coordinate system as The square integration domain within : Among them, the integral The domain of elliptic integrals is changed to α-β In the square integral domain under the coordinate system Points: Where, yes pq Coordinate system conversion to α-β Jacobian of the coordinate system; B4: Inverse solution in the square integral domain gives , about , The binary function of : 。 8. The vibration solution method for an elliptical plate with an arbitrary shape and an elliptical hole based on an elliptical energy element according to claim 7 is characterized in that: The method of generating Gaussian integral points and quadrature coefficients in the square integral domain based on Gaussian Legendre integral, and inversely solving the Gaussian integral points and quadrature coefficients in the dimensionless coordinate system of the square solution domain in the overall mapping stage based on inverse mapping to construct an elliptical energy element specifically includes: C1: Based on Gauss-Legendre integral rule, α-β In the square integral domain under the coordinate system Points within Perform a numerical integration: Where, and For interval Gaussian integration points generated within, and is the quadrature coefficient corresponding to the Gaussian integral point, and For the Integrand ellipse domain The major and minor axes of and For the counter, and The intervals are Inside and The number of Gaussian integration points in the direction, superscript Representative Integrand ellipse domain ; C2: α-β The Gaussian integral points and quadrature coefficients in the square domain under the coordinate system are reversed to the original dimensionless coordinate system through inverse mapping. ξ-η In the domain of the elliptic integral below: Where, ( , )yes ξ - η The coordinates of the Gaussian integral point in the coordinate system, is the corresponding quadrature coefficient, For the The first Gaussian integration points, For the Elliptical domain in ξ - η The coordinates of the center of the ellipse in the coordinate system; C3: The original dimensionless coordinate system obtained based on the inverse mapping ξ-η The Gaussian integral points and quadrature coefficients in the elliptical domain below are used to construct the elliptical energy element. Among them, based on the elliptical energy element ξ-η The high-precision numerical integration in the coordinate system is: 。 9. The vibration solution method for an elliptical plate with an arbitrary shape based on an elliptical energy element according to claim 8, characterized in that: The S6 specifically includes: S601: The geometry of the elliptical perforated plate is simulated based on the elliptical energy element and the energy geometry Boolean operation, that is, the following formula is obtained: Where, and They are the dimensionless perforated plate domain with elliptical holes The stiffness matrix and mass matrix inside, and are the stiffness matrix and mass matrix of the unperforated elliptical plate domain, and Representative oval hole The stiffness matrix and mass matrix inside, is the total number of elliptical holes; S602: Intermediate calculation matrices contained in the stiffness matrix and mass matrix Each element is calculated, and the intermediate calculation matrix The Row and Column Elements It can be calculated by the following formula: in, Where, and It is Item and Legendre polynomials, and They are Item and Legendre polynomials, and is an intermediate variable; S603: Calculation of elements based on elliptical energy elements Numerical integration of : Where, Represents matrix elements The integrand in ; S604: Based on the numerical integration of each element in the obtained stiffness matrix and mass matrix, the assembly stiffness matrix and mass matrix are calculated, and the eigenvalue problem of the matrix rewritten as the stationary value of the total energy functional is solved. The corresponding eigenfrequencies and eigenvectors are derived to complete the solution to the vibration problem of the elliptical hole plate.

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