Pyramid type gradient dot matrix sandwich panel vibration characteristic forecasting method based on FSDT

Abstract

The invention relates to an FSDT-based pyramid type gradient dot matrix sandwich panel vibration characteristic forecasting method. The method comprises the following steps: S1, calculating equivalent material attributes of a gradient dot matrix core material according to stress characteristics of pyramid type gradient dot matrix unit cells; s2, establishing a first-order shear deformation displacement field, and obtaining strain energy, elastic potential energy and kinetic energy of the pyramid-shaped gradient lattice sandwich panel in combination with a geometric equation and a constitutive relation; and S3, an allowable displacement variable meeting any boundary condition is established, the characteristic value equation is solved in combination with the Lagrange equation and the Ritz method, and the vibration frequency and mode of the pyramid type gradient dot matrix sandwich panel are determined. According to the method, a gradient dot matrix equivalent material attribute calculation method and a first-order shear deformation theory are applied to prediction analysis of the vibration characteristics of the pyramid type gradient dot matrix sandwich panel, and a more accurate and efficient vibration characteristic analysis result can be provided for the pyramid type gradient dot matrix sandwich panel.

Description

Technical Field

[0001] This invention relates to the field of predicting the dynamic characteristics of composite material structures, and more specifically, to a method for predicting the vibration characteristics of pyramid-shaped gradient lattice sandwich panels based on FSDT. Background Technology

[0002] Lattice structures, due to their high specific energy, high specific strength, and lightweight properties, have been widely used in transportation, shipbuilding, and aerospace. Furthermore, various types of lattice structures can be developed to suit different operating environments and specific needs. Among them, pyramidal lattice structures have attracted considerable attention due to their unique porous pattern, periodic layout, low density, and excellent energy absorption capacity. To meet the demands for enhanced local performance and adaptability to complex load environments in lattice sandwich panels, the concept of gradient lattice sandwich panels has emerged in recent years. These panels are characterized by a gradient change in the cross-sectional dimensions of the core material along the thickness direction.

[0003] In recent years, many researchers have proposed theoretical, finite element method, and experimental methods to study the structural characteristics and vibration response of pyramidal lattice sandwich panels. Currently, research on pyramidal lattice sandwich panels with gradient core materials remains significantly insufficient, and most theoretical methods are only applicable to lattice cores with constant cross-sectional dimensions. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a method for predicting the vibration characteristics of pyramid-shaped gradient lattice sandwich panels based on FSDT, which can more effectively provide more accurate vibration characteristic analysis results for pyramid-shaped gradient lattice sandwich panels.

[0005] The technical solution adopted by this invention to solve its technical problem is: to construct a method for predicting the vibration characteristics of a pyramid-shaped gradient lattice sandwich panel based on FSDT, including the following steps: S1. Calculate the equivalent material properties of the gradient lattice core material based on the stress characteristics of the pyramid-shaped gradient lattice unit cell. S2. Establish the first-order shear deformation displacement field, and combine the geometric equations and constitutive relations to obtain the strain energy, elastic potential energy and kinetic energy of the pyramid-shaped gradient lattice sandwich panel. S3. Establish allowable displacement variables that satisfy arbitrary boundary conditions, and solve the eigenvalue equations by combining the Lagrange equation and the Ritz method to determine the vibration frequency and modes of the pyramid-shaped gradient lattice sandwich panel.

[0006] According to the above scheme, step S1 includes the following steps: S11. Select a unit cell model with a thickness of [missing information] on the pyramid-shaped gradient lattice unit cell model. dz The infinitesimal element is assumed to have a consistent cross-sectional dimension of the core material within it; S12. Homogenize and perform core material stress analysis on the uniform lattice micro-element to obtain its equivalent material properties, expressed by the following formula:

[0007]

[0008]

[0009] in, r p , G c , E e and E c These represent relative density, shear modulus, equivalent elastic modulus, and core material elastic modulus, respectively. l c , r c and β These represent the core material length, core material radius, and the angle between the core material and the panel, respectively. S13. Based on the core material cross-sectional dimensions that vary linearly along the thickness direction, the equivalent material properties of the gradient lattice unit cell are derived as follows:

[0010] in, r T , r B and h c These represent the top cross-sectional dimensions, bottom cross-sectional dimensions, and height of the core material, respectively.

[0011] According to the above scheme, the first-order shear deformation displacement field of the pyramid-shaped gradient lattice sandwich panel established in step S2 is as follows:

[0012] in, u d , v d and w d These represent the displacement states of various points inside the pyramid-shaped gradient lattice sandwich panel. u 0d , v 0d and w 0d These represent the in-plane displacements of points at different locations on the surface in physics. i x and iy These represent the points on the surface in physics along... y , x The rotation angle of the axis.

[0013] According to the above scheme, in step S2, the strain at each point of the pyramid-shaped gradient lattice sandwich panel is calculated using the following formula:

[0014]

[0015] .

[0016] According to the above scheme, the stress-strain relationship of the pyramid-shaped gradient lattice sandwich panel established in step S2 is as follows:

[0017] in, , These are the stress vector and strain vector, respectively, and the stiffness coefficient. Q ij Obtained from the following formula:

[0018]

[0019] in, E ( z ) is the elastic modulus. v It is Poisson's ratio.

[0020] According to the above scheme, in step S2, the in-plane forces, in-plane bending moments, and shear forces are calculated using the following formulas:

[0021]

[0022] Where the shear correction coefficient α =5 / 6, The extension, bending-extension coupling, and bending stiffness coefficients are respectively calculated by the following formula: .

[0023] According to the above scheme, in step S2, strain energy, elastic potential energy and kinetic energy are calculated using the following formulas; Calculate the strain energy of a pyramid-shaped gradient lattice sandwich panel. U ss :

[0024] Calculate the elastic potential energy of a pyramid-shaped gradient lattice sandwich panel.U sp :

[0025]

[0026] Calculate the kinetic energy of a pyramid-shaped gradient lattice sandwich panel. T :

[0027] Among them, subscript t Represents the derivative with respect to time. r ( z ) is density, ( H 0, H 1, H 2) is the inertial term, which can be calculated by the following formula.

[0028] in, z Represents the thickness coordinate.

[0029] According to the above scheme, the method for establishing the allowable displacement variable that satisfies arbitrary boundary conditions in step S3 is as follows:

[0030] in, This is the mode shape coefficient.

[0031]

[0032] .

[0033] According to the above scheme, the method of solving the eigenvalue equation by combining the Lagrange equation and the Ritz method in step S3 includes the following steps: Lagrange equations L :

[0034] Strain energy, elastic potential energy, and kinetic energy are substituted into the Lagrange equation and calculated using the Ritz method.

[0035] Solve the eigenvalue equations:

[0036] in, , These are the stiffness matrix and the mass matrix, respectively. For frequency, { ψ 1, ψ 2, ψ 3, ψ 4, ψ 5} T It is the Fourier coefficient vector; by substituting the Fourier coefficient vector into the displacement variables that satisfy any boundary conditions, the mode can be obtained.

[0037] The present invention also provides a vibration characteristic prediction device for a pyramid-shaped gradient lattice sandwich panel based on FSDT, comprising: The equivalent material property acquisition module is used to calculate the equivalent material properties of the gradient lattice core material based on the stress characteristics of the pyramid-shaped gradient lattice unit cell. The sandwich panel energy harvesting module establishes a first-order shear deformation displacement field and, in conjunction with geometric equations and constitutive relations, obtains the strain energy, elastic potential energy, and kinetic energy of the pyramid-shaped gradient lattice sandwich panel. The vibration frequency and mode determination module is used to establish allowable displacement variables that satisfy arbitrary boundary conditions, and to solve the eigenvalue equations by combining the Lagrange equation and the Ritz method to determine the vibration frequency and modes of the pyramid-shaped gradient lattice sandwich panel.

[0038] The vibration characteristic prediction method for pyramid-shaped gradient lattice sandwich panels based on FSDT of the present invention has the following beneficial effects: This invention calculates the equivalent gradient material properties based on the micro-element configuration of a pyramid-shaped gradient lattice unit cell. It models a pyramid-shaped gradient lattice sandwich panel based on first-order shear deformation theory and energy principles. Using this model and energy principles, the Lagrange equation is obtained, and the eigenvalue equations are solved using the Ritz method to obtain the natural frequencies of the pyramid-shaped gradient lattice panel. The corresponding Fourier coefficient vectors are then substituted into the displacement variables to obtain the vibration modes, enabling the prediction of the vibration characteristics of the pyramid-shaped gradient lattice sandwich panel. Furthermore, the first-order model in this invention achieves progressively higher prediction accuracy as the micro-element configuration is optimized, allowing it to be extended to other types of gradient lattice sandwich panels, providing strong support for the prediction of vibration characteristics of different types of gradient lattice sandwich panels. Attached Figure Description

[0039] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings: Figure 1 This is a flowchart of the vibration characteristic prediction method for pyramid-shaped gradient lattice sandwich panels based on FSDT according to the present invention; Figure 2 Different embodiments of the present invention L b / L a The effect on vibration frequency is shown in the diagram. Figure 3 Different embodiments of the present invention h c / h f The effect on vibration frequency is shown in the diagram. Figure 4 This is a schematic diagram showing the frequency and modal relationship of a gradient pyramid lattice sandwich panel according to an embodiment of the present invention. Detailed Implementation

[0040] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0041] like Figure 1 As shown, the vibration characteristic prediction method for pyramid-shaped gradient lattice sandwich panels based on FSDT of the present invention includes the following steps: S1. Calculate the equivalent gradient material properties based on the micro-element configuration of the gradient pyramid lattice unit cell; S2. Establish the first-order shear deformation displacement field, and combine the geometric equations and constitutive relations to establish the strain energy, elastic potential energy and kinetic energy of the pyramid-shaped gradient lattice sandwich panel. S3. Establish displacement variables that satisfy arbitrary boundary conditions, and calculate the eigenvalue equations by combining the Lagrange equation and the Rayleigh-Ritz method. Finally, determine the vibration frequency and modes of the pyramid-shaped gradient lattice sandwich panel.

[0042] Furthermore, step S1, which involves calculating the equivalent gradient material properties based on the micro-element configuration of the gradient pyramid lattice unit cell, further includes: S11. Select a unit cell model with a thickness of [missing information] on the pyramid-shaped gradient lattice unit cell model. dz The infinitesimal element is assumed to have a consistent cross-sectional dimension of the core material within it.

[0043] S12. Homogenize the uniform lattice micro-element and perform core material stress analysis to obtain the equivalent material properties of the uniform lattice micro-element, as shown below: In this embodiment of the invention, the equivalent material properties described in step S12 are specifically shown in Table 1.

[0044] Table 1 Equivalent elastic modulus, shear modulus, and relative density

[0045] in, r p , G c , E e and E e These represent the equivalent density, shear modulus, equivalent elastic modulus, and core material elastic modulus, respectively. l c , rc and β These represent the core material length, core material radius, and the angle between the core material and the panel, respectively.

[0046] S13. Based on the core material cross-sectional dimensions that vary linearly along the thickness direction, derive the equivalent material properties of the gradient lattice unit cell.

[0047]

[0048] in, r T , r B and h c These represent the top cross-sectional dimensions, bottom cross-sectional dimensions, and height of the core material, respectively.

[0049] Furthermore, the step S2, which involves establishing a first-order shear deformation displacement field and, in conjunction with geometric equations and constitutive relations, establishing the strain energy, elastic potential energy, and kinetic energy of the pyramid-shaped gradient lattice sandwich panel, further includes: S21. Establish the first-order shear deformation displacement field of the pyramid-shaped gradient lattice sandwich panel:

[0050] in, u d , v d and w d These represent the displacement states of various points inside the pyramid-shaped gradient lattice sandwich panel. u 0d , v 0d and w 0d These represent the in-plane displacements of points at different locations on the surface in physics. i x and i y These represent the points on the surface in physics along... y , x The rotation angle of the axis.

[0051] S22. Calculate the strain at each point of the pyramid-shaped gradient lattice sandwich panel:

[0052]

[0053]

[0054] S23. Establish the stress-strain relationship of the pyramid-shaped gradient lattice sandwich panel:

[0055] in, , These are stress components and strain components, and elastic constants, respectively. Q ij Obtained from the following formula:

[0056]

[0057] in, E ( z ) is the elastic modulus. v It is Poisson's ratio.

[0058] S24. Calculate in-plane forces, in-plane bending moments, and shear forces:

[0059]

[0060] Where the shear correction coefficient α =5 / 6, The stiffness coefficient is calculated using the following formula:

[0061] S25. Calculate strain energy, elastic potential energy, and kinetic energy: Calculate the strain energy of a pyramid-shaped gradient lattice sandwich panel. U ss :

[0062] Calculate the elastic potential energy of a pyramid-shaped gradient lattice sandwich panel. U sp :

[0063]

[0064] Calculate the kinetic energy of a pyramid-shaped gradient lattice sandwich panel. T :

[0065] Among them, subscript t Represents the derivative with respect to time, i.e. ( S ,t )= ( S ) / t , r ( z) is mass density, ( H 0, H 1, H 2) is the inertial term, which can be calculated by the following formula.

[0066] in, z Represents the thickness coordinate.

[0067] In this embodiment of the invention, the stiffness coefficient of the pyramid-shaped gradient lattice sandwich panel in step S24 is shown in Table 2, and the inertia coefficient of the pyramid-shaped gradient lattice sandwich panel in step S24 is shown in Table 3.

[0068] Table 2 Stiffness Coefficients of Pyramid-Shaped Gradient Lattice Sandwich Panels

[0069] Table 3. Inertia Coefficients of Pyramid-Shaped Gradient Lattice Sandwich Panels

[0070] Furthermore, based on the establishment of displacement variables satisfying arbitrary boundary conditions as described in step S3, and by combining the Lagrange equation and the Ritz method to solve the eigenvalue equations, the vibration frequencies and modes of the pyramid-shaped gradient lattice sandwich panel are finally determined, further including: S31. Establish displacement variables that satisfy the boundary conditions:

[0071] in, This is the mode shape coefficient.

[0072]

[0073]

[0074] S32. Solve the eigenvalue equations by combining the Lagrange equation and the Ritz method: The Lagrange equation is L :

[0075] Substitute the strain energy, elastic potential energy, and kinetic energy obtained from S25 into the Lagrange equation and perform further calculations using the Rayleigh-Ritz method.

[0076]

[0077] Solve the eigenvalue equations:

[0078] in, , These are the stiffness matrix and the mass matrix, respectively. For frequency, { ψ 1, ψ 2, ψ 3, ψ 4, ψ 5} T It is the Fourier coefficient vector. Substituting the Fourier coefficient vector into the displacement variables that satisfy arbitrary boundary conditions yields the mode.

[0079] In this embodiment of the invention, the plate width b =300mm, plate length a =300mm, sandwich panel thickness h =26mm, core material thickness h c =20mm. The panel where the four core materials intersect is defined as the top panel, and the other panel is defined as the bottom panel. The core material structure uses the cross-sectional dimensions of the top panel. r T =3.75mm, base plate cross-sectional dimensions r B =1.5mm. The vibration frequency and modes of the pyramid-shaped gradient lattice sandwich panel can be obtained through step S33, such as... Figure 4 As shown. From Figure 2 It can be seen that, with L b / L a The increase is significant; compared to lower-order frequencies, higher-order frequencies exhibit a higher growth rate. Figure 3 Showing different thickness ratios h c / h f The frequencies of the pyramid-shaped gradient lattice sandwich panel show that the fourth and fifth order frequencies are the same, and the frequencies of each order vary with... h c / h f The higher the frequency, the more sensitive it becomes to changes in the thickness ratio. Compared to lower frequencies, higher frequencies are more sensitive to changes in the thickness ratio.

[0080] The present invention also provides a vibration characteristic prediction device for a pyramid-shaped gradient lattice sandwich panel based on FSDT, comprising: The equivalent material property acquisition module is used to calculate the equivalent material properties of the gradient lattice core material based on the stress characteristics of the pyramid-shaped gradient lattice unit cell. The sandwich panel energy harvesting module establishes a first-order shear deformation displacement field and, in conjunction with geometric equations and constitutive relations, obtains the strain energy, elastic potential energy, and kinetic energy of the pyramid-shaped gradient lattice sandwich panel. The vibration frequency and mode determination module is used to establish allowable displacement variables that satisfy arbitrary boundary conditions, and to solve the eigenvalue equations by combining the Lagrange equation and the Ritz method to determine the vibration frequency and modes of the pyramid-shaped gradient lattice sandwich panel.

[0081] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A method for predicting the vibration characteristics of a pyramid type gradient dot lattice sandwich panel based on FSDT, characterized in that, It comprises the following steps: S1, calculating the equivalent material properties of the gradient lattice core material according to the stress characteristics of the pyramid gradient lattice unit cell; S2, establishing a first-order shear deformation displacement field, and combining geometric equations and constitutive relations to obtain the strain energy, elastic potential energy and kinetic energy of the pyramid gradient lattice sandwich plate; S3, establishing an allowable displacement variable that meets arbitrary boundary conditions, combining Lagrange equations and Ritz method to solve the eigenvalue equation, and determining the vibration frequency and mode of the pyramid gradient lattice sandwich plate.

2. The FSDT-based pyramid-type gradient lattice sandwich panel vibration characteristics prediction method according to claim 1, characterized in that, The step S1 comprises the following steps: S11, a micro-element with a thickness of dz is selected on the pyramid gradient lattice unit cell model, and it is assumed that the core material cross-sectional size remains consistent; S12, homogenizing the uniform lattice microelement and analyzing the core stress to obtain the equivalent material properties of the uniform lattice microelement, which is expressed by the following formula: wherein, ρ p , G c , E e and E c represent the relative density, the shear modulus, the equivalent elastic coefficient and the core material elastic coefficient, respectively, l c , r c and β represent the core material length, the core material radius and the angle between the core material and the panel, respectively. S13, according to the linear change of the core section size along the thickness direction, the equivalent material properties of the gradient lattice unit cell are derived as follows: wherein, r T , r B and h c represent the top cross-sectional dimension, the bottom cross-sectional dimension and the height of the core material, respectively.

3. The FSDT-based pyramid-type gradient lattice sandwich panel vibration characteristics prediction method of claim 1, wherein The first-order shear deformation displacement field of the pyramid gradient lattice sandwich plate established in the step S2 is as follows: wherein, u d , v d and w d respectively represent the displacement state of each position point inside the pyramid type gradient lattice sandwich plate, u 0d , v 0d and w 0d respectively represent the in-plane displacement of each position point on the physical mid-surface, θ x and θ y respectively represent the rotation angle of each position point on the physical mid-surface along the y , x axis.

4. The FSDT-based pyramid-type gradient lattice sandwich panel vibration characteristics prediction method of claim 1, wherein The strain of each point of the pyramid gradient lattice sandwich plate is calculated in the step S2 by the following formula: 。 5. The FSDT-based pyramid-type gradient lattice sandwich panel vibration characteristics prediction method of claim 1, wherein The stress-strain relationship of the pyramid gradient lattice sandwich plate established in the step S2 is as follows: wherein , are respectively the stress and strain vectors, the stiffness coefficient Q ij is obtained from the following equation: wherein E z is the elastic modulus, v is the Poisson's ratio.​ 6. The FSDT-based pyramid-type gradient lattice sandwich panel vibration characteristics prediction method of claim 1, wherein The in-plane force, in-plane bending moment and shear force are calculated in the step S2 by the following formula: where the shear correction factor α = 5 / 6, are the extension, flexure-extension coupling and flexural stiffness coefficients, respectively, calculated from the following equations: 。 7. The FSDT-based pyramid-type gradient lattice sandwich panel vibration characteristics prediction method of claim 1, wherein, The strain energy, elastic potential energy and kinetic energy are calculated in the step S2 by the following formula: Strain energy of pyramid type gradient lattice sandwich panel U ss : Elastic potential energy of pyramid type gradient dot matrix sandwich panel U sp : Kinetic energy of pyramid type gradient dot matrix sandwich panel T : where the subscript t represents differentiation with respect to time, The method for establishing the allowable displacement variable that meets arbitrary boundary conditions in the step S3 is as follows: ( z ) is the density, ( H 0, H 1, H 2) is the inertial term, which can be calculated from the formula wherein z represents the thickness coordinate.

8. The FSDT-based pyramid-type gradient lattice sandwich panel vibration characteristics prediction method of claim 1, wherein, The method for solving the eigenvalue equation by combining Lagrange equations and Ritz method in the step S3 comprises the following steps: wherein is the mode shape coefficient. 。 9. The FSDT-based pyramid-type gradient lattice sandwich panel vibration characteristics prediction method of claim 1, wherein, The strain energy, elastic potential energy and kinetic energy are substituted into the Lagrange equation, and Ritz method is used for calculation; Lagrange equation L : Solving the eigenvalue equation: The method for solving the eigenvalue equation by combining Lagrange equations and Ritz method in the step S3 comprises the following steps: wherein, , are the stiffness matrix and the mass matrix, respectively; is the frequency, The strain energy, elastic potential energy and kinetic energy are substituted into the Lagrange equation, and Ritz method is used for calculation; 1, Solving the eigenvalue equation: 2, The method for solving the eigenvalue equation by combining Lagrange equations and Ritz method in the step S3 comprises the following steps: 3, The strain energy, elastic potential energy and kinetic energy are substituted into the Lagrange equation, and Ritz method is used for calculation; 4, Solving the eigenvalue equation: 5} T is the Fourier coefficient vector; the modal shapes are obtained by bringing the Fourier coefficient vector into the displacement variable that satisfies arbitrary boundary conditions.

10. A device for predicting the vibration characteristics of a pyramid type gradient dot lattice sandwich panel based on FSDT, characterized in that, It comprises: An equivalent material property acquisition module for calculating the equivalent material properties of the gradient lattice core material according to the stress characteristics of the pyramid gradient lattice unit cell; A sandwich plate energy acquisition module for establishing a first-order shear deformation displacement field, and combining geometric equations and constitutive relations to obtain the strain energy, elastic potential energy and kinetic energy of the pyramid gradient lattice sandwich plate; A vibration frequency and mode determination module for establishing an allowable displacement variable that meets arbitrary boundary conditions, combining Lagrange equations and Ritz method to solve the eigenvalue equation, and determining the vibration frequency and mode of the pyramid gradient lattice sandwich plate.

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