Method for predicting natural frequency and loss factor of composite cylindrical shell and rectangular plate

Abstract

The present application relates to the technical field of cylindrical shell and rectangular plate vibration prediction, and particularly relates to a kind of inherent frequency and loss factor prediction method of composite cylindrical shell and rectangular plate, comprising: based on first-order shear deformation theory and sawtooth theory, introduce displacement continuous type condition sawtooth theory, establish the displacement field of each layer of structure;Penalty function is introduced to simulate the boundary conditions and coincident joint coupling conditions of structure, construct the Lagrangian energy functional equation of cylindrical shell or rectangular plate;Two-dimensional first Chebyshev polynomial is used to expand displacement component, and the Rayleigh-Ritz method is used to establish the structure free vibration control equation, to solve inherent frequency and loss factor;The present application can realize the correctness and effectiveness of the analysis of inherent frequency, loss factor and steady-state response of viscoelastic sandwich composite cylindrical shell and rectangular plate.

Description

Technical Field

[0001] This invention relates to the field of vibration prediction technology for cylindrical shells and rectangular plates, and particularly to a method for predicting the natural frequency and loss factor of composite cylindrical shells and rectangular plates. Background Technology

[0002] Composite cylindrical shells, as crucial load-bearing and protective structures in high-end equipment, possess high specific strength, lightweight, high stiffness, and excellent corrosion resistance, making them widely used in submarines, ships, and aerospace structures. These equipment often operate in complex or extreme environments, and their structural dynamic characteristics directly impact operational stability and safety. Therefore, studying the vibration characteristics of composite cylindrical shells has significant theoretical and engineering application value. Currently, research on the vibration characteristics of composite cylindrical shells mainly focuses on laminated materials and functionally graded materials (FGM) structures. With the development of computational mechanics, various numerical methods have been applied to the analysis of free and forced vibrations, including meshless methods, Fourier series methods, Galerkin methods, and wavelet discretization methods. These studies, based on first-order shear deformation theory, thin-shell theory, or three-dimensional elasticity theory, systematically reveal the vibration laws of cylindrical shells under different boundary conditions, geometric parameters, and material distributions, providing a theoretical basis for structural optimization design.

[0003] However, traditional laminated or functionally graded composite material structures have limited vibration reduction capabilities under high-noise and high-excitation environments. To meet the higher requirements of modern ships for acoustic stealth and vibration and noise reduction performance, viscoelastic damping sandwich composite cylindrical shell and rectangular plate structures have emerged. These structures generally consist of a base layer, a viscoelastic damping layer, and a constraint layer. When the cylindrical shell vibrates, the intermediate damping layer and the base layer deform together, generating alternating shear stress internally, thereby converting vibration energy into heat energy and dissipating it rapidly. Simultaneously, the constraint layer restricts the deformation of the damping layer, further enhancing the shear strain and significantly improving energy dissipation efficiency. Structures based on this mechanism are widely used in the design of ship anechoic tiles, effectively reducing structural vibration and noise radiation.

[0004] The addition of damping materials significantly enhances the multilayer coupling characteristics and viscoelastic effects of cylindrical shells, making it difficult for traditional vibration models designed for single-laminated or functionally graded structures to accurately describe their dynamic responses. Therefore, it is urgent to establish a free vibration analysis model suitable for viscoelastic sandwich composite cylindrical shells, revealing the influence of the geometric and material parameters of the base layer, damping layer, and constraint layer on vibration characteristics. This model would provide theoretical support and engineering guidance for the structural vibration reduction design of ships and other high-end equipment, ensuring the accuracy and effectiveness of the analysis of the natural frequencies, loss factors, and steady-state responses of viscoelastic sandwich composite cylindrical shells and rectangular plates. Summary of the Invention

[0005] The purpose of this invention is to provide a method for predicting the natural frequencies and loss factors of composite cylindrical shells and rectangular plates, which can ensure the accuracy and effectiveness of the analysis of the natural frequencies, loss factors, and steady-state responses of viscoelastic sandwich composite cylindrical shells and rectangular plates.

[0006] To achieve the above objectives, the present invention provides a method for predicting the natural frequency and loss factor of composite cylindrical shells and rectangular plates, comprising:

[0007] A unified layered mechanical model is established for viscoelastic sandwich composite material structures, which include cylindrical shell structures and rectangular plate structures, both of which are composed of a base layer, a constraint layer and a damping core layer located between the two.

[0008] Based on the first-order shear deformation theory, the displacement field and strain-displacement relationship at any point in each layer of the viscoelastic sandwich composite material structure are established. Based on the interlayer displacement continuity condition and sawtooth theory, the displacement field of the damping material layer is established. For the rectangular plate structure, the displacement constitutive relationship of each part of the sandwich rectangular plate is obtained by letting the radius of the cylindrical shell approach infinity. A rectangular plate mechanical model unified with the cylindrical shell is established.

[0009] Based on the displacement field, a penalty function is introduced to simulate the boundary conditions and coupling conditions at the overlapping junctions of the structure, and the Lagrange energy functional equation of the cylindrical shell or rectangular plate is constructed, including the kinetic energy, strain potential energy, elastic potential energy and coupling potential energy of the overall structure, the elastic potential energy generated by the boundary springs.

[0010] Expand the structural displacement components using Chebyshev polynomials to construct an allowable displacement function that is continuous within the domain and independent of each other along two coordinate directions.

[0011] The established unified layered mechanical model and displacement allowable function are substituted into the Lagrange energy functional, and the extreme values ​​of the Chebyshev polynomial expansion coefficients are obtained using the Rayleigh-Ritz method. The control equation for the free vibration of the structure is then established, and the natural frequency and loss factor of the structure are solved.

[0012] The displacement field and strain-displacement relationship at any point in each layer of the viscoelastic sandwich composite structure, established based on the first-order shear deformation theory, are expressed as follows:

[0013]

[0014] in, These represent the displacements of any point on the surface of the structure along the coordinate axes. The linear displacement components in the direction, where the subscripts s, c, and v represent the constraint layer, base layer, and damping sandwich layer, respectively. Indicates the angle along the relevant plane;

[0015] The strain-displacement relationship is as follows:

[0016]

[0017] In the structure, the normal strain on the surface is expressed as: The transverse shear strain is expressed as: Bending strain is expressed as: Torsional strain is expressed as: .

[0018] The displacement field of the damping material layer, based on the interlayer displacement continuity condition and sawtooth theory, is expressed as follows:

[0019] .

[0020] Where h c h s h v These are respectively represented as the thickness of the base layer, the thickness of the constraint layer, and the thickness of the damping sandwich layer. These represent the displacements of any point on the surface of the structural damping material layer along the coordinate axes. Linear displacement components in the direction; This indicates the angle along the relevant plane, where the subscripts s, c, and v represent the constraint layer, base layer, and damping sandwich layer, respectively.

[0021] The boundary conditions and overlapping junction coupling conditions of the introduced penalty function simulation structure include: free (F), simply supported (S), and fixed (C) boundaries, which are set along... Directional stiffness of linear springs and around Shaft rotation spring stiffness The values ​​are combined to achieve the virtual spring stiffness in five directions: .

[0022] Specifically, based on the displacement field, the Lagrange energy functional equations for the cylindrical shell or rectangular plate are constructed by introducing penalty functions to simulate the boundary conditions and coupling conditions at the overlapping junctions of the structure, including:

[0023] Construct the Lagrangian energy functional expression for a cylindrical shell or rectangular plate:

[0024]

[0025] Among them T, U, U bc U cp These represent the structural kinetic energy, structural potential energy, boundary spring potential energy, and coupling spring potential energy, respectively.

[0026] The energy expression for a cylindrical shell is as follows:

[0027] The kinetic energy expression of the overall structure:

[0028]

[0029] In the formula, L represents the radial length of the cylindrical shell; θ0 represents the central angle; R i Indicates the radius of each layer of the cylindrical shell; u i Denotes the displacement vector; m i Represents the moment of inertia; the specific expression is as follows:

[0030]

[0031]

[0032] In the formula, ρ i (i=s,v,c) represents the material density of each layer; These represent the displacements of any point on the surface of the structure along the coordinate axes. Linear displacement components in the direction; Indicates the angle of rotation along the relevant plane; h i Indicates material thickness;

[0033] The unified expression for structural potential energy is as follows:

[0034]

[0035]

[0036] in, These represent the internal stress and strain matrices of the structure, respectively.

[0037] When the cylindrical shell is an open cylindrical shell, and the central angle θ0 < 2π, the potential energy of the boundary spring is:

[0038]

[0039] in, This represents the equivalent spring stiffness in five directions introduced by the artificial virtual spring; This represents the boundary constraint spring potential energy generated individually in the i-th layer of the structure. This represents the average radius of the i-th cylindrical shell layer; This represents the thickness of the i-th shell layer; This represents the displacement column vector of the i-th layer at the boundary.

[0040] When the central angle θ0 of the cylindrical shell is 2π, the potential energy of the coupled spring at the overlapping junction is:

[0041]

[0042] Let R i =∞、θ0=y / Ri The energy equation for the composite material damping sandwich rectangular plate can be obtained as follows:

[0043]

[0044] Where a and b represent the length and width of the rectangular plate, respectively.

[0045] In the step of expanding the structural displacement components using Chebyshev polynomials to construct an admissible displacement function that is continuous within the domain and independent along two coordinate directions,...

[0046] The structural displacement components are expanded using Chebyshev polynomials as follows:

[0047]

[0048] in, Let be the coefficient vector of the Chebyshev polynomial expansion. They represent the Chebyshev expansion along... The m-th and n-th order expansions in two directions.

[0049] The governing equation for the free vibration of the structure is expressed as:

[0050]

[0051] Where K(ω) and M represent the stiffness matrix and mass matrix of a cylindrical shell or rectangular plate.

[0052] In the step of substituting the established unified layered mechanical model and displacement allowable function into the Lagrange energy functional, using the Rayleigh-Ritz method to find the extrema of the Chebyshev polynomial expansion coefficients, establishing the structural free vibration control equation, and then solving for the natural frequency and loss factor of the structure, the calculation formulas for the structural natural frequency and loss factor are as follows:

[0053] .

[0054] This invention discloses a method for predicting the natural frequencies and loss factors of composite cylindrical shells and rectangular plates. Based on first-order shear deformation theory and sawtooth theory, it introduces displacement continuity conditional sawtooth theory to establish the displacement fields of each layer of the structure. A penalty function is introduced to simulate the boundary conditions and coupling conditions at overlapping junctions of the structure, constructing the Lagrange energy functional equation for the cylindrical shell or rectangular plate. The displacement components are expanded using two-dimensional Chebyshev polynomials of the first kind, and the Rayleigh-Ritz method is combined to establish the governing equations for the free vibration of the structure, solving for the natural frequencies and loss factors. Finally, comparison with the finite element method demonstrates that this method has good accuracy and effectiveness in analyzing the natural frequencies, loss factors, and steady-state responses of viscoelastic sandwich composite cylindrical shells and rectangular plates. Attached Figure Description

[0055] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below.

[0056] Figure 1 This is a flowchart of the method for predicting the natural frequency and loss factor of composite cylindrical shells and rectangular plates according to the present invention.

[0057] Figure 2 This is a schematic diagram of the geometric model of a cylindrical shell and a rectangular plate made of viscoelastic sandwich composite material.

[0058] Figure 3 A schematic diagram illustrating the construction of boundary conditions by introducing a penalty function for a cylindrical shell.

[0059] Figure 4 Comparison of steady-state response of composite material damping sandwich closed cylindrical shell structure.

[0060] Figure 5 Comparison of steady-state response of composite material sandwich open cylindrical shell structure.

[0061] Figure 6 A schematic diagram of constructing boundary conditions by introducing a penalty function for a rectangular plate.

[0062] Figure 7 The image shows a comparison of the steady-state response (CCCC) of a composite material damping sandwich rectangular plate. Detailed Implementation

[0063] The embodiments of the present invention are described in detail below. Examples of the embodiments are shown in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, but should not be construed as limiting the present invention.

[0064] Please see Figures 1-7 This invention provides a method for predicting the natural frequency and loss factor of composite cylindrical shells and rectangular plates, comprising:

[0065] S1 establishes a unified layered mechanical model for viscoelastic sandwich composite material structures, which include cylindrical shell structures and rectangular plate structures, both of which consist of a base layer, a constraint layer, and a damping core layer located between the two.

[0066] S2 establishes the displacement field and strain-displacement relationship at any point in each layer of the viscoelastic sandwich composite material structure based on the first-order shear deformation theory. Based on the interlayer displacement continuity condition and sawtooth theory, the displacement field of the damping material layer is established. For the rectangular plate structure, the displacement constitutive relationship of each part of the sandwich rectangular plate is obtained by letting the radius of the cylindrical shell approach infinity. A mechanical model of the rectangular plate unified with the cylindrical shell is established.

[0067] In this step, the displacement field and strain-displacement relationship at any point in each layer of the viscoelastic sandwich composite material structure, established based on the first-order shear deformation theory, are expressed as follows:

[0068]

[0069] in, These represent the displacements of any point on the surface of the structure along the coordinate axes. The linear displacement components in the direction, where the subscripts s, c, and v represent the constraint layer, base layer, and damping sandwich layer, respectively. Indicates the angle along the relevant plane;

[0070] The strain-displacement relationship is as follows:

[0071]

[0072] In the structure, the normal strain on the surface is expressed as: The transverse shear strain is expressed as: Bending strain is expressed as: Torsional strain is expressed as: .

[0073] In this step, the displacement field of the damping material layer, based on the interlayer displacement continuity condition and sawtooth theory, is expressed as follows:

[0074]

[0075] Where h c h s h v These are respectively represented as the thickness of the base layer, the thickness of the constraint layer, and the thickness of the damping sandwich layer. These represent the displacements of any point on the surface of the structural damping material layer along the coordinate axes. Linear displacement components in the direction; This indicates the angle along the relevant plane, where the subscripts s, c, and v represent the constraint layer, base layer, and damping sandwich layer, respectively.

[0076] S3, based on the displacement field, introduces a penalty function to simulate the boundary conditions and coupling conditions at the overlapping junctions of the structure, and constructs the Lagrange energy functional equation of the cylindrical shell or rectangular plate, including the kinetic energy, strain potential energy, elastic potential energy and coupling potential energy of the overall structure, and the elastic potential energy and coupling potential energy generated by the boundary springs.

[0077] In this step, the boundary conditions and coupling conditions at the coincident junctions of the introduced penalty function simulation structure include: free (F), simply supported (S), and fixed (C) boundaries, which are set along... Directional stiffness of linear springs and around Shaft rotation spring stiffness The values ​​are combined to achieve the virtual spring stiffness in five directions: .

[0078] In this step, based on the displacement field, a penalty function is introduced to simulate the boundary conditions and coupling conditions at the overlapping junctions of the structure. The Lagrange energy functional equation for the cylindrical shell or rectangular plate is constructed specifically as follows:

[0079] Construct the Lagrangian energy functional expression for a cylindrical shell or rectangular plate:

[0080]

[0081] Among them T, U, U bc U cp These represent the structural kinetic energy, structural potential energy, boundary spring potential energy, and coupling spring potential energy, respectively.

[0082] The energy expression for a cylindrical shell is as follows:

[0083] The kinetic energy expression of the overall structure:

[0084]

[0085] In the formula, L represents the radial length of the cylindrical shell; θ0 represents the central angle; R i Indicates the radius of each layer of the cylindrical shell; u i Denotes the displacement vector; m i Represents the moment of inertia; the specific expression is as follows:

[0086]

[0087]

[0088] In the formula, ρ i (i=s,v,c) represents the material density of each layer; These represent the displacements of any point on the surface of the structure along the coordinate axes. Linear displacement components in the direction; Indicates the angle of rotation along the relevant plane; h i Indicates material thickness;

[0089] The unified expression for structural potential energy is as follows:

[0090]

[0091]

[0092] in, These represent the internal stress and strain matrices of the structure, respectively.

[0093] When the cylindrical shell is an open cylindrical shell, and the central angle θ0 < 2π, the potential energy of the boundary spring is:

[0094]

[0095] in, This represents the equivalent spring stiffness in five directions introduced by the artificial virtual spring; This represents the boundary constraint spring potential energy generated individually in the i-th layer of the structure. This represents the average radius of the i-th cylindrical shell layer; This represents the thickness of the i-th shell layer; This represents the displacement column vector of the i-th layer at the boundary.

[0096] When the central angle θ0 of the cylindrical shell is 2π, the potential energy of the coupled spring at the overlapping junction is:

[0097]

[0098] Let R i =∞、θ0=y / R i The energy equation for the composite material damping sandwich rectangular plate can be obtained as follows:

[0099]

[0100] Where a and b represent the length and width of the rectangular plate, respectively.

[0101] S4 expands the structural displacement components using Chebyshev polynomials to construct an allowable displacement function that is continuous within the domain and independent of each other along two coordinate directions.

[0102] In this step, the structural displacement components are expanded using Chebyshev polynomials as follows:

[0103]

[0104] in, Let be the coefficient vector of the Chebyshev polynomial expansion. They represent the Chebyshev expansion along... The m-th and n-th order expansions in two directions.

[0105] S5 substitutes the established unified layered mechanical model and displacement allowable function into the Lagrange energy functional, uses the Rayleigh-Ritz method to find the extrema of the Chebyshev polynomial expansion coefficients, establishes the governing equations for the free vibration of the structure, and then solves for the natural frequency and loss factor of the structure. The effectiveness of this patented method can be verified by comparing the results with those obtained from references and finite element analysis.

[0106] In this step, the governing equation for the free vibration of the structure is expressed as:

[0107]

[0108] Where K(ω) and M represent the stiffness matrix and mass matrix of a cylindrical shell or rectangular plate.

[0109] In this step, the formulas for calculating the structure's natural frequency and loss factor are as follows:

[0110] .

[0111] To better understand this invention, specific embodiments are described in detail below. The embodiments of this invention aim to verify the correctness and universality of the analytical model at the mathematical and numerical levels by substituting specific physical parameters into the mathematical model described herein and comparing the final results.

[0112] Table 1 provides the panel material and core layer material parameters for the composite material damping sandwich structure, and Table 2 provides the layup parameters for the composite material damping sandwich structure. Table 3 shows the boundary conditions and related spring stiffness values ​​examined in this patent. In this patent, the boundary conditions are represented by x=0, θ=0, x=L, and θ=θ0 in sequence. For example, the boundary condition CSFS indicates that the constraints of the structure at the boundaries x=0, θ=0, x=L, and θ=θ0 are fixed (C), simply supported (S), free (F), and simply supported (S), respectively.

[0113] Table 1 Material parameters of composite material damping sandwich structure panel

[0114]

[0115] Table 2 Different boundary conditions and corresponding spring stiffness

[0116]

[0117] Example 1: Verification of the natural frequency and steady-state response of a viscoelastic sandwich composite cylindrical shell

[0118] Geometric model of viscoelastic sandwich composite cylindrical shell as follows Figure 2 As shown, the cutoff numbers along the x and θ directions are M and N, respectively.

[0119] S1 constructs the displacement field at any point in each layer of the structure.

[0120] Substituting the parameters of the viscoelastic sandwich composite cylindrical shell into the theoretical model, the displacement field at any point in each layer of the structure is constructed according to the first-order shear deformation theory (FSDT):

[0121]

[0122] S2 establishes the displacement field of the damping material layer.

[0123] Based on the inter-layer displacement continuity condition and sawtooth theory, the displacement field of the damping material layer is established:

[0124]

[0125] S3 introduces a penalty function to simulate the boundary conditions and coupling conditions at overlapping junctions of the structure.

[0126] like Figure 3 As shown, introduce direction , around Shaft rotation spring stiffness Virtual spring stiffness in 5 directions:

[0127]

[0128] Lagrange energy functional expression for a cylindrical shell with structure S4

[0129]

[0130] The energy expression for the cylindrical shell is as follows:

[0131] Substitute the kinetic energy expression of the entire structure:

[0132]

[0133] Substituting the unified expression for structural potential energy:

[0134]

[0135]

[0136] When the cylindrical shell is an open cylindrical shell, and the central angle θ0 < 2π, the potential energy of the boundary spring is:

[0137]

[0138] The displacement components of the S5 structure are expanded using Chebyshev polynomials:

[0139]

[0140] Substituting S6 into the governing equation for free vibration of the structure, we can solve for the vibration characteristics of the structure:

[0141]

[0142]

[0143] The following section will verify the calculation accuracy and reliability of the viscoelastic sandwich composite cylindrical shell by comparing the numerical results of the method presented in this paper with those of existing literature.

[0144] Table 3 shows a comparison of the accuracy of the natural frequencies of the composite damped sandwich cylindrical shell structure with FEM results under different central angles. The cylindrical shell length L = 6 m; radius R = 2 m; central angle θ0 is taken as 360°, 270°, 180°, and 90° respectively; the boundary condition for the closed cylindrical shell is set to CC, and the rest are CCCC; h c =h s =h v =0.01 m; the base layer and restraint layer materials are selected as M1, the sandwich layer material is selected as M2, and the laying angle is 0° for all; cut-off number: M=N=20.

[0145] As shown in Table 3, the natural frequencies calculated by the method of this invention generally agree well with the FEM results. Further analysis of the data in Table 3 reveals that, under the same modal order, the natural frequencies of the cylindrical shell structure generally increase as the opening angle gradually decreases from 360° to 90°. The relative deviation of the first 10 natural frequencies does not exceed 5%, and the trends of frequency values ​​are basically consistent. This indicates that the method proposed in this invention can accurately predict the natural frequencies of composite material damped sandwich cylindrical shells, demonstrating high calculation accuracy and good reliability. For each natural frequency under different opening angles, the results obtained by the method of this invention maintain a high degree of consistency with the FEM calculation results, indicating that this method is not only applicable to closed cylindrical shell structures but also can be used well for predicting the natural frequencies of composite material damped sandwich cylindrical shell structures with different opening angles.

[0146] Table 3 Comparison of the correctness of the natural frequency (Hz) of composite material damping sandwich cylindrical shell structures

[0147]

[0148] To further verify the applicability of the dynamic analysis model established in this invention in the steady-state response analysis of composite material damped sandwich closed cylindrical shells, relevant typical examples were selected and compared with finite element results.

[0149] like Figure 4 As shown. Structural geometric parameters: cylindrical shell length L = 6 m; radius R = 1 m; central angle θ0 is 360°; boundary conditions are set as fixed supports CC at both ends; h c =h s =h v=0.01 m; the material for the first and second layers and the constraint layer is M1, and the material for the sandwich layer is M2. The layup angle for both layers is 0°; the cutoff number is M=N=20. The point excitation is applied to the center of the outer wall of the structure, and its coordinates are (x t ,θ t = (3, 0°). Two structural measuring points were selected for response comparison. The geometric coordinates of measuring point 1# and measuring point 2# are (x, y, y) respectively. m1 ,θ m1 ) = (1.5, 90°) and (x m2 ,θ m2 ) = (3, 180°). From Figure 4 The comparison results show that the steady-state response curve obtained by the method of the present invention is highly consistent with the result obtained by the finite element method, indicating that the theoretical analysis model established by the present invention has good versatility and reliability in the steady-state response analysis of composite sandwich closed cylindrical shells.

[0150] like Figure 5 As shown, to further verify the applicability of the method of the present invention to open cylindrical shell structures, a steady-state response comparison verification will be carried out on a composite material sandwich open cylindrical shell as the analysis object. Structural geometry and material parameters: Cylindrical shell length L = 6 m; radius R = 1 m; h c =h s =h v =0.01 m; central angle θ0=300°; base and constraint layer materials are selected as M1, core layer material is selected as M2, and layup angle is 0° for both; boundary condition is CCCC, cutoff number M=N=28. The coordinates of the excitation load application location are (x t ,θ t = (3, 180°), the positions of measuring point 1# and measuring point 2# are respectively (x m1 ,θ m1 ) = (1.5, 90°) and (x m2 ,θ m2 = (1.5, 270°).

[0151] like Figure 5 As shown, the analytical model of this invention maintains good consistency with the steady-state response curve of the composite sandwich open cylindrical shell obtained by the finite element method, indicating that the theoretical analytical model established by this invention also has good prediction accuracy for open cylindrical shell structures, thus further verifying the correctness of the method of this invention.

[0152] Example 2: Verification of the natural frequency loss factor and steady-state response of a viscoelastic sandwich composite rectangular plate

[0153] Geometric model of a viscoelastic sandwich composite rectangular plate as follows Figure 2 As shown, the number of cutoffs along the x and y directions are M and N, respectively.

[0154] S1 Model Parameter Substitution and Mathematical Object Definition

[0155] Substituting the parameters of the viscoelastic sandwich composite rectangular plate into the theoretical model, the displacement field at any point in each layer of the structure is constructed according to the first-order shear deformation theory (FSDT):

[0156]

[0157] S2 establishes the displacement field of the damping material layer.

[0158] Based on the inter-layer displacement continuity condition and sawtooth theory, the displacement field of the damping material layer is established:

[0159]

[0160] S3 introduces a penalty function to simulate the boundary conditions of the structure.

[0161] like Figure 6 As shown, introduce direction , around Shaft rotation spring stiffness Virtual spring stiffness in 5 directions:

[0162]

[0163] Lagrange energy functional expression for the S4-constructed rectangular plate

[0164]

[0165] The energy expression for the rectangular plate is as follows:

[0166] Substitute the kinetic energy expression of the entire structure:

[0167]

[0168] Substituting the unified expression for structural potential energy:

[0169]

[0170]

[0171] Where a and b represent the length and width of the rectangular plate, respectively.

[0172]

[0173] The displacement components of the S5 structure are expanded using Chebyshev polynomials:

[0174]

[0175] Substituting S6 into the governing equation for free vibration of the structure, we can solve for the vibration characteristics of the structure:

[0176]

[0177]

[0178] The following section will verify the calculation accuracy and reliability of the viscoelastic sandwich composite rectangular plate by comparing the numerical results of the method presented in this paper with those of existing literature.

[0179] To verify the effectiveness of the method of this invention in the dynamic analysis of composite damped sandwich rectangular plates, Table 4 presents the comparison results of the first 10 frequency parameters and loss factors between the method of this invention and FEM under (CCCC) boundary conditions. The base layer and constraint layer materials were selected as M1, the core layer material as M2, and the layup angle was 0°. The thicknesses of the constraint layer and base layer were 0.01 m, and the core layer thickness was 0.01 m; the cutoff number M=N=20. The comparison results show that the frequency parameters obtained by the method of this invention are highly consistent with the finite element results, indicating that this invention can effectively predict the inherent vibration characteristics of composite damped sandwich rectangular plates, and has high accuracy and good applicability.

[0180] Table 4 Comparison of the first 10 natural frequencies (Hz) and loss factors of composite material damping sandwich rectangular plates

[0181]

[0182] To further verify the effectiveness and accuracy of the dynamic analysis model established in this invention in the steady-state response analysis of composite material damped sandwich rectangular plates, a comparative study was conducted using typical rectangular plate structures. Figure 7 The steady-state response of the composite damped sandwich rectangular plate calculated using the method of this invention is compared with the FEM results. The structural parameters are a=2 m, b=1 m. The damping coefficient of the damping sandwich layer material is adjusted to ηᵥ=0.9683, and the structural damping is set to 0. The remaining parameters are consistent with those in Table 6. A point excitation is applied to the structural constraint layer, and its position in the xy-plane coordinate system is (x...). t ,y t =(1.5,0.5); Measuring points 1# and 2# are arranged on the plane of the structural base layer, and their coordinates are (x, y, y) respectively. m1 ,y m1 ) = (1.7, 0.6) and (x m2 ,y m2 = (0.8, 0.3).

[0183] like Figure 7As shown, when the core layer of the composite sandwich rectangular plate uses a damping material, the peak value of the steady-state response and its trend with frequency obtained by the method of the present invention and the finite element method are highly similar. This result indicates that even in the presence of energy dissipation, the dynamic analysis model of the composite damped sandwich rectangular plate established in this invention can still accurately characterize the steady-state response characteristics of the structure, thus further verifying the correctness and applicability of the method of the present invention.

[0184] This invention discloses a method for predicting the natural frequencies and loss factors of composite cylindrical shells and rectangular plates. Based on FSDT theory, this invention establishes the displacement field components of the base layer, constraint layer, and core layer of viscoelastic sandwich composite cylindrical shells and rectangular plates. According to the interlayer displacement continuity condition and sawtooth theory, the displacement coupling relationship between the layers is derived. Then, virtual spring technology is introduced to establish the coupling potential energy and structural boundary potential energy at the central angles θ=0 and θ=2π of the closed cylindrical shell. The structural displacement field is expanded using Chebyshev polynomials of the first kind. Based on this, the vibration characteristics and loss factor are solved using the Rayleigh-Ritz method. This not only enriches the research results in this field but also provides theoretical reference for the design of related engineering practices.

[0185] Compared with the prior art, the present invention has the following significant advantages:

[0186] 1. High Accuracy: This invention combines first-order shear deformation theory and sawtooth theory, enabling precise capture of the complex deformation modes of viscoelastic sandwich composite cylindrical shells, particularly the shear effect of the core layer and the discontinuity of interlayer displacement. Comparison with results from the finite element method (FEM) and other literature demonstrates high consistency and excellent accuracy in calculating natural frequencies and loss factors.

[0187] 2. High versatility: This invention employs virtual spring technology to apply boundary conditions and internal coupling conditions. By simply changing the spring stiffness value, it can simulate various situations from open to closed, and from classical boundaries to elastic boundaries, forming a unified analysis framework. This avoids the tedious process of re-deriving formulas for different boundary conditions, exhibiting extremely high versatility and flexibility.

[0188] 3. High computational efficiency: As a semi-analytical method, this invention uses global Chebyshev polynomials to approximate the displacement field. Compared with the finite element method, which requires dense meshing, this method can obtain high-precision solutions with fewer degrees of freedom (expansion coefficients), thereby significantly improving computational efficiency. It is particularly suitable for a large number of parametric analyses and optimization designs.

[0189] 4. This invention can not only predict the vibration characteristics of structures, but also reveal the influence of material parameters (such as ply angle and stiffness ratio) and geometric parameters (such as thickness) of the base layer and constraint layer on the natural frequency and loss factor of the structure through parameter analysis. It can provide direct and profound theoretical guidance for the low-noise and high-performance design of viscoelastic sandwich composite cylindrical shells and rectangular plates.

[0190] The above-disclosed embodiments are merely one or more preferred embodiments of this application and should not be construed as limiting the scope of this application. Those skilled in the art can understand that implementing all or part of the above embodiments and making equivalent changes in accordance with the claims of this application still fall within the scope of this application.

Claims

1. A method for predicting the natural frequency and loss factor of a composite cylindrical shell and rectangular plate, characterized in that, include: A unified layered mechanical model is established for viscoelastic sandwich composite material structures, which include cylindrical shell structures and rectangular plate structures, both of which are composed of a base layer, a constraint layer and a damping core layer located between the two. Based on the first-order shear deformation theory, the displacement field and strain-displacement relationship at any point in each layer of the viscoelastic sandwich composite material structure are established. Based on the interlayer displacement continuity condition and sawtooth theory, the displacement field of the damping material layer is established. For the rectangular plate structure, the displacement constitutive relationship of each part of the sandwich rectangular plate is obtained by letting the radius of the cylindrical shell approach infinity. A rectangular plate mechanical model unified with the cylindrical shell is established. Based on the displacement field, a penalty function is introduced to simulate the boundary conditions and coupling conditions at the overlapping junctions of the structure, and the Lagrange energy functional equation of the cylindrical shell or rectangular plate is constructed, including the kinetic energy, strain potential energy, elastic potential energy and coupling potential energy of the overall structure, the elastic potential energy generated by the boundary springs. Expand the structural displacement components using Chebyshev polynomials to construct an allowable displacement function that is continuous within the domain and independent of each other along two coordinate directions. The established unified layered mechanical model and displacement allowable function are substituted into the Lagrange energy functional, and the extreme values ​​of the Chebyshev polynomial expansion coefficients are obtained using the Rayleigh-Ritz method. The control equation for the free vibration of the structure is then established, and the natural frequency and loss factor of the structure are solved.

2. The method for predicting the natural frequency and loss factor of composite cylindrical shells and rectangular plates as described in claim 1, characterized in that, The displacement field and strain-displacement relationship at any point in each layer of the viscoelastic sandwich composite structure, based on the first-order shear deformation theory, are expressed as follows: in, These represent the displacements of any point on the surface of the structure along the coordinate axes. The linear displacement components in the direction, where the subscripts s, c, and v represent the constraint layer, base layer, and damping sandwich layer, respectively. Indicates the angle along the relevant plane; The strain-displacement relationship is as follows: In the structure, the normal strain on the surface is expressed as: The transverse shear strain is expressed as: Bending strain is expressed as: Torsional strain is expressed as: .

3. The method for predicting the natural frequency and loss factor of composite cylindrical shells and rectangular plates as described in claim 2, characterized in that... Based on the interlayer displacement continuity condition and sawtooth theory, the displacement field of the damping material layer is expressed as follows: Where h c h s h v These are respectively represented as the base layer thickness, the constraint layer thickness, and the damping sandwich layer thickness; These represent the displacements of any point on the surface of the structural damping material layer along the coordinate axes. Linear displacement components in the direction; This indicates the angle along the relevant plane, where the subscripts s, c, and v represent the constraint layer, base layer, and damping sandwich layer, respectively.

4. The method for predicting the natural frequency and loss factor of composite cylindrical shells and rectangular plates as described in claim 3, characterized in that, The boundary conditions and coupling conditions at the coincident junctions of the introduced penalty function simulation structure include: free (F), simply supported (S), and fixed (C) boundaries, which are set along... Directional stiffness of linear springs and around Shaft rotation spring stiffness The values ​​are combined to achieve the virtual spring stiffness in five directions: .

5. The method for predicting the natural frequency and loss factor of composite cylindrical shells and rectangular plates as described in claim 4, characterized in that, Based on the displacement field, a penalty function is introduced to simulate the boundary conditions and coupling conditions at the overlapping junctions of the structure. The Lagrange energy functional equations for the cylindrical shell or rectangular plate are constructed as follows: Construct the Lagrangian energy functional expression for a cylindrical shell or rectangular plate: Among them T, U, U bc U cp These represent the structural kinetic energy, structural potential energy, boundary spring potential energy, and coupling spring potential energy, respectively. The energy expression for a cylindrical shell is as follows: The kinetic energy expression of the overall structure: In the formula, L represents the radial length of the cylindrical shell; θ0 represents the central angle; R i Indicates the radius of each layer of the cylindrical shell; u i m represents the displacement vector; i Represents the moment of inertia; the specific expression is as follows: In the formula, ρ i (i=s,v,c) represents the material density of each layer; These represent the displacements of any point on the surface of the structure along the coordinate axes. Linear displacement components in the direction; Indicates the angle of rotation along the relevant plane; h i Indicates material thickness; The unified expression for structural potential energy is as follows: in, These represent the internal stress and strain matrices of the structure, respectively. When the cylindrical shell is an open cylindrical shell, and the central angle θ0 < 2π, the potential energy of the boundary spring is: in, This represents the equivalent spring stiffness in five directions introduced by the artificial virtual spring; This represents the boundary constraint spring potential energy generated individually in the i-th layer of the structure. This represents the average radius of the i-th cylindrical shell layer; This represents the thickness of the i-th shell layer; This represents the displacement column vector of the i-th layer at the boundary; When the central angle θ0 of the cylindrical shell is 2π, the potential energy of the coupled spring at the overlapping junction is: Let R i =∞、θ0=y / R i The energy equation for the composite material damping sandwich rectangular plate can be obtained as follows: Where a and b represent the length and width of the rectangular plate, respectively.

6. The method for predicting the natural frequency and loss factor of composite cylindrical shells and rectangular plates as described in claim 5, characterized in that, In the step of expanding the structural displacement components using Chebyshev polynomials to construct admissible displacement functions that are continuous within the domain and independent of each other along two coordinate directions... The structural displacement components are expanded using Chebyshev polynomials as follows: in, Let be the coefficient vector of the Chebyshev polynomial expansion; They represent the Chebyshev expansion along... The m-th and n-th order expansions in two directions.

7. The method for predicting the natural frequency and loss factor of composite cylindrical shells and rectangular plates as described in claim 6, characterized in that, The governing equation for the free vibration of the structure is expressed as: Where K(ω) and M represent the stiffness matrix and mass matrix of a cylindrical shell or rectangular plate.

8. The method for predicting the natural frequency and loss factor of composite cylindrical shells and rectangular plates as described in claim 1, characterized in that, In the process of substituting the established unified layered mechanical model and allowable displacement function into the Lagrange energy functional, using the Rayleigh-Ritz method to find the extrema of the Chebyshev polynomial expansion coefficients, establishing the structural free vibration control equations, and then solving for the natural frequencies and loss factors of the structure, the formulas for calculating the structural natural frequencies and loss factors are as follows: in, This represents taking the real part of the eigenvalues ​​of a complex number; This represents taking the imaginary part of the eigenvalues ​​of a complex number; Represents the system's inherent frequency; This represents the modal loss factor of the system.

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