Method for calculating free vibration of thin-walled metal structures embedded with viscoelastic layers

Abstract

The application discloses an analysis method for calculating free vibration of a metal thin-walled structure embedded with a viscoelastic layer, and comprises the following steps: S1, obtaining geometric parameters, material properties and viscoelastic layer position of the metal thin-walled structure embedded with the viscoelastic layer; S2, based on the parameters obtained in S1, performing theoretical characteristic representation on the embedded viscoelastic layer, performing three-dimensional motion field modeling on a displacement field of the viscoelastic metal thin-walled sandwich structure, and constructing a kinematics control differential equation of the metal thin-walled structure embedded with a frequency-varying viscoelastic layer; S3, based on the control differential equation, establishing a frequency-varying viscoelastic dynamic stiffness model through a boundary condition and a general solution; S4, solving the frequency-varying viscoelastic dynamic stiffness model, obtaining complex eigenvalues of the metal thin-walled structure embedded with the viscoelastic layer, and then extracting complex natural frequencies and modal damping ratios of the metal thin-walled structure embedded with the viscoelastic layer, so as to complete free vibration analysis of the metal thin-walled structure embedded with the viscoelastic layer.

Description

Technical Field

[0001] This invention relates to the field of viscoelastic solid mechanics, and more specifically to an analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer. Background Technology

[0002] In aerospace, vehicle engineering, precision instruments, and high-end equipment, lightweight structures and vibration control are key challenges in improving performance, reliability, and comfort. Thin-walled metal structures (such as beams, plates, and shells) are widely used due to their high specific strength and specific stiffness; however, their inherent low damping makes them prone to severe vibrations and resonances under dynamic loads, leading to fatigue damage, excessive noise, and functional failure. To effectively suppress vibration, confined layer damping is often employed, which involves embedding or covering a viscoelastic material layer onto a metal matrix. Such composite structures can significantly improve the damping loss factor, but they also introduce complex mechanical behaviors: their dynamic characteristics are highly dependent on the frequency-dependent properties of the viscoelastic material and the coupling effects between layers. Therefore, developing an analytical method capable of accurately and efficiently calculating the free vibration characteristics of thin-walled metal structures with embedded viscoelastic layers is an indispensable theoretical foundation and technical prerequisite for optimization design, performance prediction, and engineering applications, possessing significant engineering practical value and scientific significance.

[0003] Currently, several mainstream methods have emerged for the dynamics research of such viscoelastic composite thin-walled structures. In terms of modeling theory, these methods are mainly based on laminated theory (such as classical laminated plate theory and first-order shear deformation theory) or more refined higher-order theories / three-dimensional elasticity theory, combined with constitutive models of viscoelastic materials (such as complex modulus models, fractional derivative models, and generalized Maxwell / Kelvin models) to describe their frequency / temperature-dependent storage modulus and loss modulus. Regarding solution strategies, the finite element method is the most common numerical tool for handling complex geometries and boundary conditions. The complex frequencies and loss factors of the system can be approximately obtained through direct frequency response analysis or modal strain energy methods. Furthermore, analytical / semi-analytical methods (such as the Rayleigh-Ritz method, Galerkin method, and transfer matrix method) can provide deeper theoretical insights and efficient parametric research capabilities for structures with regular geometries. Numerous studies have focused on the influence of different laminated configurations and viscoelastic material models on the modal frequencies and damping performance of the overall structure.

[0004] However, existing analytical methods still have several significant shortcomings when dealing with thin-walled metal structures with embedded viscoelastic layers, limiting their design accuracy and application scope: On the one hand, there is a dilemma in balancing modeling accuracy and computational efficiency. While a refined three-dimensional finite element model can effectively characterize the stress-strain field, simulating viscoelastic frequency-varying characteristics often requires iterative solutions in the frequency domain or inverse Laplace / Fourier transforms, resulting in high computational costs and making it unsuitable for rapid design and optimization iterations. On the other hand, simplified equivalent single-layer or layered theoretical models often lack sufficient accuracy when predicting higher-order modes or complex interlayer shear and warping deformations in locally embedded structures.

[0005] On the other hand, there are limitations in handling the strong dependence of viscoelastic materials: most studies use complex constant modulus or simple frequency-varying models, which are difficult to accurately describe the dramatic changes in modulus and damping of viscoelastic materials over a wide frequency band. More accurate generalized models often result in non-standard differential-integral forms of the governing equations, making the solution extremely complex, and there is a lack of efficient and stable algorithms suitable for free vibration eigenvalue problems.

[0006] To overcome the aforementioned shortcomings, this patent aims to propose a novel, efficient, and accurate analytical method for calculating the free vibration of thin-walled metal structures with embedded viscoelastic layers. This method is expected to provide a more refined characterization of local embedding and interface effects in modeling, employ a more realistic and tractable model in material constitutive modeling, and construct a stable and efficient nonlinear complex eigenvalue solving process in the solution algorithm. This will provide a reliable and practical analytical tool for the design of high-performance vibration-damping structures, thus promoting the advancement of related technologies. Summary of the Invention

[0007] In view of this, the present invention provides an analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer, which at least solves the technical problem in the prior art that it is difficult to balance the calculation accuracy, model universality and calculation efficiency when performing free vibration analysis on a thin-walled metal structure with an embedded viscoelastic layer.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: An analytical method for calculating the free vibration of a thin-walled metallic structure with an embedded viscoelastic layer includes the following steps: S1. Obtain the geometric parameters, material properties, and location of the viscoelastic layer of the thin-walled metal structure with an embedded viscoelastic layer; S2. Based on the parameters obtained in S1, the theoretical properties of the embedded viscoelastic layer are characterized, the displacement field of the metal thin-walled sandwich structure is modeled in three dimensions, and the control differential equations of the kinematics of the metal thin-walled structure with the embedded frequency-varying viscoelastic layer are constructed. S3. Based on the governing differential equation, a frequency-varying viscoelastic dynamic stiffness model is established through boundary conditions and general solutions; S4. Solve the frequency-varying viscoelastic dynamic stiffness model to obtain the complex eigenvalues ​​of the thin-walled metal structure with embedded viscoelastic layer, and then extract the natural frequency and modal damping ratio of the thin-walled metal structure with embedded viscoelastic layer to complete the free vibration analysis of the thin-walled metal structure with embedded viscoelastic layer.

[0009] Preferably, the specific content of S1 includes: Geometric parameters include the length, thickness, and cross-sectional shape dimensions of the thin-walled metal structure; Material properties include elastic modulus, Poisson's ratio, and material density; An embedded constrained viscoelastic damping layer is designed between thin-walled metal structures. The material frequency-varying loss factor is used as a model of the viscoelastic material properties to characterize the damping properties of the embedded viscoelastic layer.

[0010] Preferably, the specific content of S2 includes: The theoretical properties of the embedded viscoelastic layer are characterized by using the Carrera unified formula theory and Lagrange polynomial expansion to characterize the three-dimensional displacement field of the thin-walled metal structure with the embedded viscoelastic layer. A high-order accurate description is performed to achieve refined modeling of the complex cross-sectional shape of the sandwich structure. Combining linear elastic dynamics theory and constitutive relations considering frequency-varying viscoelastic materials, the governing differential equations and boundary conditions of the metal thin-walled structure with embedded viscoelastic layer are constructed by using the virtual displacement principle and combining the Carrera unified formula model. The closed analytical solution of the governing differential equation of the metal thin-walled structure unit considering frequency-varying viscoelastic materials is then solved.

[0011] Preferably, the theoretical properties of the embedded viscoelastic layer are characterized. Using the Carrera unified formula theory, a high-order accurate description of the three-dimensional displacement field of the thin-walled metal structure with the embedded viscoelastic layer is achieved through Lagrange polynomial expansion. This enables refined modeling of the complex cross-sectional shape of the viscoelastic sandwich structure. Specific details include: Based on the Carrera unified formula theory, the displacement field of a thin-walled metal structure with an embedded viscoelastic layer is analyzed. Represented as a general section function and Generalized displacement vector with axial distribution The product form: (1) In the formula, This indicates the number of terms in the function's expansion. and These represent the indexes for the number of items expanded; Using the Lagrange polynomial expansion method, the general section function is... The structure is divided into subdomains that can overlap with the physical layer, with the metallic structure and viscoelastic layer being distinct subdomains. Each subdomain employs one or more Lagrange elements, and the polynomial expansion order of each Lagrange element depends on the type of Lagrange expansion used, thus yielding a universal section function. Multinomial expressions.

[0012] Preferably, the specific content of the governing differential equations and boundary conditions for the embedded viscoelastic layer of the thin-walled metal structure constructed by the principle of virtual displacement and combined with the Carrera unified formula model includes: The motion control equations are derived using the principle of virtual displacement: (2) (3) (4) In the formula, For standard virtual variable operators, Indicates strain energy. For structural volume, In response, Indicates matrix transpose. For stress, The work done by inertial force, To consider the structural stiffness coefficient matrix of the frequency-varying viscoelastic damping model, D is the linear differential operator matrix; After integration by parts, the strain energy in the governing differential equation incorporating the viscoelastic layer frequency-varying complex elastic modulus is expressed as: (5) In the formula, The length of the thin-walled metal structure with a locally embedded viscoelastic layer. and These represent the two directions of the cross-section of the thin-walled metal structure with the embedded viscoelastic layer. and The generalized displacement vector, The linear frequency-varying differential stiffness matrix is... This is the boundary condition matrix; The imaginary component of the work done by inertial forces is expressed as: (6) In the formula, Indicates the cross section. Indicates the density of the material. and They represent and The direction is in the expansion of terms. and The cross-sectional function at time, express Regarding time The second derivative, It is the fundamental diagonal kernel of the 3×3 mass matrix; Substituting (5) and (6) into (2), we obtain the section moment parameters of the embedded viscoelastic metal material. and boundary section moment parameters The explicit form of the relevant motion control equations is: (7) (8) (9) In the formula, express corresponding and The number of terms in the directional expansion are respectively and The material section moment parameters at time, where, express middle lines and Column corresponding parameters, , =1~6, and They represent , , Any one of them; express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time The space function and the number of expansion terms are time The space function respectively on The partial derivatives; express With the number of expanded terms being time The space function and the number of expansion terms are time The space function respectively on The partial derivatives; , and They represent , and With the number of expanded terms being Time The first-order partial derivative of , where , and They represent , and Displacement in the direction; , and They represent , and exist Direction The second-order partial derivative; , and They represent , and With the number of expanded terms being Displacement at time; , and express , and Regarding time The second derivative of , i.e., acceleration; , and They represent , and With the number of expanded terms being Displacement at time; make , and The number of terms to be expanded are respectively hour , and If the generalized force vector is in the direction, then the boundary conditions are... , and for: (10) Then (7) and (10) are used as the governing differential equations and boundary equations of the metal thin-walled structure with embedded viscoelastic layer constructed by the principle of virtual displacement.

[0013] Preferably, for viscoelastic sandwich structures, the structural stiffness coefficient matrix... It is a complex matrix with a non-constant elastic modulus, specifically expressed as: (11) The specific coefficients are as follows: (12) (13) (14) In the formula, The complex elastic modulus of the viscoelastic layer; Poisson's ratio represents the material; For the constant complex model, the frequency-dependent properties of the viscoelastic sandwich are introduced by the elastic modulus, which is assumed to be a complex constant. Represented as: (15) in, The energy storage elastic modulus of the viscoelastic layer; This represents the corresponding loss factor of the frequency-varying material. These are the eigenfrequency of the structure. for ; For nonlinear frequency-dependent models Represented as: (16) in, The standard energy storage modulus representing the viscoelastic layer. The frequency correlation coefficient, For fractional-order parameters, These are the model correction coefficients.

[0014] Preferably, the specific content of obtaining the governing differential equations and boundary conditions of the Carrera unified formula model and solving the closed analytical solution of the thin-walled metal structure with embedded viscoelastic layer includes: By assuming the solution is in the form of simple harmonic motion, the time-domain motion control equations (7) and (10) are transformed into a set of three mutually coupled ordinary differential equations in the frequency domain concerning the amplitude function, and expressed in a compact matrix form, denoted as the H matrix; at the same time, the corresponding boundary conditions are also matrixized into the P matrix. The ordinary differential equation in the frequency domain is a problem about The second-order ordinary differential equation system with constant coefficients can be simplified to a first-order system through variable transformation. Therefore, the H matrix is ​​transformed into the following linear differential system S matrix: (17) in, For a system of first-order ordinary differential equations with constant coefficients, For a second-order ordinary differential equation system with constant coefficients, S denotes the S matrix; The general solution of equation (17) is as follows: (18) Where n is the total number of first-order ordinary differential equations with constant coefficients, Z n For the nth first-order ordinary differential equation with constant coefficients, It is the first of the S matrix 1 eigenvalue, It is the j-th element of the i-th eigenvector of matrix S. Let n be the nth integration constant, determined by the boundary conditions. Then equation (17) can be written in the following matrix form Z: (19) The closed-form expansion of the governing differential equation for a thin-walled metal structure with an embedded viscoelastic layer is as follows: (20) in, , and Let these represent the generalized displacement functions in the x, y, and z directions, given a single expansion term in the Carrera unified theory. , Let M be the generalized displacement function in the z-direction given the number of expansion terms M in the Carrera unified theory. M is the number of expansion terms in the Carrera unified theory; once the displacement is known, the P matrix corresponding to the boundary conditions is obtained by substituting the solution of formula (20) into the boundary conditions (10): (twenty one) in, B is an intermediate variable, and B is the boundary condition coefficient matrix. Let S be the eigenvectors of matrix S; then the boundary conditions can be written in explicit form as follows: (twenty two) In the formula, That is The value in the nth row and nth column of the matrix. , and The generalized force functions P in the x, y, and z directions given by Carrera's unified theory when the number of expansion terms is 1. It is a generalized force function P given by Carrera's unified theory with M expansion terms and z-direction.

[0015] Preferably, the specific content of S3 includes: The displacement field expression for a thin-walled metal structure with an embedded viscoelastic layer, and the boundary conditions are expressed as follows: (twenty three) In the formula, and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized displacement value at that location, When the Carrera unified theory gives an expansion of M terms, the z-direction is... The generalized displacement value at the location, the negative sign indicates that the actual direction of the displacement is opposite to the preset positive direction. and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized displacement value at that location, When the Carrera unified theory gives an expansion of M terms, the z-direction is... The generalized displacement value at the location; Through and Equation (20) is used to calculate the generalized displacement matrix relationship by applying the boundary conditions of equation (23): (twenty four) In the formula Let i = 1…n, j = 1…n-1, and be the eigenvectors of matrix S. The value in the i-th row and j-th column of the array. , … Let n be the constant value of the improper integral, and let n represent the nth one. Equation (24) can be expressed in a compact form: (25) In the formula, A is the displacement coefficient matrix, and Q is the generalized integral constant vector; the boundary conditions for the generalized nodal forces are as follows: (26) In the formula, and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized nodal force value at the location, the negative sign indicates that the actual direction of the force is opposite to the preset positive direction. When the Carrera unified theory gives an expansion of M terms, the z-direction is... The generalized nodal force value at the location, and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized nodal force value at the location, When the Carrera unified theory gives an expansion of M terms, the z-direction is... Generalized nodal force values ​​at the location; Through and By calculating equation (22) and applying the boundary conditions of equation (26), the following matrix relationship of generalized nodal forces is obtained: (27) In the formula, for The value in the nth row and nth column of the matrix; Equation (27) can be expressed in a compact form. : (28) In the formula, R is the boundary coefficient matrix and Q is the generalized integral constant vector; by associating the generalized nodal forces with the corresponding generalized displacements, the generalized integral constant vector Q in equations (25) and (28) is eliminated, thus obtaining the frequency-varying viscoelastic dynamic stiffness matrix of the thin-walled metal structure with embedded viscoelastic layer: (29) (30) In the formula, K is the viscoelastic dynamic stiffness matrix; The global viscoelastic dynamic stiffness matrix is ​​then: (31) In the formula, The global viscoelastic dynamic stiffness matrix is ​​used; the boundary conditions are applied by directly removing the rows and columns corresponding to zero degrees of freedom from the global viscoelastic dynamic stiffness matrix.

[0016] Preferably, the specific content of S4 includes: (i) Select a trial frequency To calculate the global inviscid elastic dynamic stiffness matrix of the structure ; (ii) Using Gaussian elimination method to... Converting to upper triangle form, we get The algorithm counts the number of negative values ​​on its main diagonal; this number is called the sign count of the algorithm. ; (iii) Structure at the test frequency The following modal frequencies modulus for: (32) in, This indicates that when the node boundaries of a thin-walled metal structure with an embedded viscoelastic layer are fully fixed, due to... and The number of natural frequencies between them; Corresponding to time Number of modes; Indicates to Transform the matrix into upper triangular form and count the number of negative values ​​on its main diagonal. s(...) represents the transformation of the matrix into upper triangular form. Solve using an indirect method : when The half-wave number in the direction is At that time, the mode count of a thin-walled metallic structural unit with an embedded viscoelastic layer having all simply supported conditions. The method for obtaining it is as follows: (33) (34) Among them, when The half-wave number in the direction is hour, for ; For simply supported thin-walled metal structural elements with embedded viscoelastic layers, the frequency-varying dynamic stiffness matrix is ​​given. In obtaining And through statistics The number of negative terms was obtained Under the premise of this, the arbitrary natural frequency is limited to the experimental frequency by the dichotomy method. Within the upper and lower bounds, and to achieve the required accuracy; Obtaining the global inviscid elastic dynamic stiffness matrix After obtaining the undamped eigenvalues, the corresponding eigenvectors are calculated, and the eigenvalue problem of the inviscid elastic dynamic stiffness model is expressed as: (35) in, The global inviscid elastic dynamic stiffness matrix. Represents real eigenvalues. Let be a vector of real parameters. It is a real eigenvector; and Determined using the Wittrick-Williams algorithm; If frequency-varying viscoelastic properties are considered in the dynamic stiffness model, then the formula for frequency-varying viscoelastic dynamic stiffness is expressed as: (36) Compared with the inviscid dynamic stiffness formula (35), the eigenvalues and eigenvectors perturbation momentum and All are damping-related parameter vector perturbations Nonlinear functions; Therefore, the homotopy method is used to solve for complex eigenvalues ​​and eigenvectors, and its specific form is as follows: (37) In homotopy analysis, let h be the convergence control parameter, including the loss factor or characteristic time constant; where Indicates an undamped state. The final target damping coefficient, and These represent the final complex eigenvalues ​​and complex eigenvectors of the viscoelastic dynamic stiffness matrix, respectively. The growth method of parameter h is as follows: (38) (39) (40) (41) in Let k be the total homotopy, and k represent the current iteration step. ; , and These represent the convergence control parameters, complex eigenvalues, and complex eigenvectors obtained in each iteration, respectively. based on , and The iterative process yields the complex eigenvalues ​​and complex eigenvectors at the (k+1)th step; finally, based on the adopted frequency-varying viscoelastic damping model, the global frequency-varying viscoelastic dynamic stiffness matrix is ​​calculated. The complex eigenvalues ​​and complex eigenvectors are the complex natural frequencies and complex mode shapes of the thin-walled metal structure with embedded viscoelastic layers.

[0017] As can be seen from the above technical solution, compared with the prior art, the present invention discloses an analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer, which has the following beneficial effects: 1. This invention, through the use of layering theory and component methods, has developed a high-precision CUF(LE)-dynamic stiffness formula for defining the displacement field on the cross-section of a thin-walled metal structure with an embedded viscoelastic layer. This method enables the convenient assembly of arbitrary sub-components or layups with different material loss factors, which is crucial for characterizing sandwich structures with non-uniform damping properties.

[0018] 2. The shape functions used in the dynamic stiffness method of this invention are derived from the frequency-domain governing differential equations of structural vibration. This allows the method to efficiently and directly characterize viscoelastic damping, which can be frequency-dependent or frequency-independent, linear or nonlinear, and possess integer or fractional order viscoelastic properties. The frequency-varying dynamic stiffness formula eliminates the need for separate remodeling based on different damping models, providing a general and unified approach.

[0019] 3. The high-order CUF dynamic stiffness model established in this invention can accurately identify various classical and non-classical vibration modes, including torsion, shear and coupled modes, and exhibits high accuracy and high stability for various beam and plate structures. It provides a powerful and reliable analysis tool for the parametric research and optimization design of thin-walled metal structures with embedded viscoelastic layers. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0021] Figure 1 A flowchart of the analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer provided by the present invention; Figure 2 The coordinate system provided for the general thin-walled metal structure with embedded viscoelastic layer in the embodiments of the present invention; Figure 3The present invention provides an analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer, which includes the assembly process of the frequency-varying stiffness matrix corresponding to the frequency-varying stiffness model of the thin-walled metal structure with an embedded viscoelastic layer. Figure 4 The viscoelastic sandwich beam structure dimensions and cross-sectional node arrangement provided in the embodiments of the present invention; Figure 5 This invention provides a schematic diagram comparing the modal shapes and finite element analysis of a viscoelastic sandwich beam structure obtained using this invention, as provided in this embodiment. Figure 5 (a) Figure 5 (c) Figure 5 (e) and Figure 5 (g) These are the first four mode shapes calculated using this method. Figure 5 (b) Figure 5 (d) Figure 5 (f) and Figure 5 (h) represent the first four mode shapes obtained using finite element simulation; Figure 6 The present invention provides a viscoelastic metal thin-walled stiffened ring structure with specific dimensions, shape, and cross-sectional node arrangement; wherein... Figure 6 (a) shows the structural outline and dimensions; Figure 6 (b) is a side view of section 1, arranged with 52 L9 elements. Figure 6 (c) is a side view of section 2 and an arrangement of 40 L9 elements; Figure 7 The present invention provides a schematic diagram comparing the modal shapes and finite element analysis of the viscoelastic thin-walled stiffened circular ring structure obtained by the present invention in an embodiment of the invention; wherein... Figure 7 (a) Figure 7 (c) Figure 7 (e) and Figure 7 (g) These are the first four mode shapes calculated using this method. Figure 7 (b) Figure 7 (d) Figure 7 (f) and Figure 7 (h) represent the first four mode shapes obtained by finite element simulation. Detailed Implementation

[0022] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0023] This invention provides an analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer, such as... Figure 1 As shown, it includes the following steps: S1. Obtain the geometric parameters, material properties, and location of the viscoelastic layer of the thin-walled metal structure with an embedded viscoelastic layer; S2. Based on the parameters obtained in S1, the theoretical properties of the embedded viscoelastic layer are characterized, the displacement field of the metal thin-walled sandwich structure is modeled in three dimensions, and the control differential equations of the kinematics of the metal thin-walled structure with the embedded frequency-varying viscoelastic layer are constructed. S3. Based on the governing differential equation, a frequency-varying viscoelastic dynamic stiffness model is established through boundary conditions and general solutions; S4. Solve the frequency-varying viscoelastic dynamic stiffness model to obtain the complex eigenvalues ​​of the thin-walled metal structure with embedded viscoelastic layer, and then extract the natural frequency and modal damping ratio of the thin-walled metal structure with embedded viscoelastic layer to complete the free vibration analysis of the thin-walled metal structure with embedded viscoelastic layer.

[0024] It should be noted that: In this embodiment, the coordinate system of the general higher-order beam model of the thin-walled metal structure with embedded viscoelastic material layer is as follows: Figure 2 As shown.

[0025] To further implement the above technical solution, the specific content of S1 includes: Geometric parameters include the length, thickness, and cross-sectional shape dimensions of the thin-walled metal structure; Material properties include elastic modulus, Poisson's ratio, and material density; An embedded constrained viscoelastic damping layer is designed between thin-walled metal structures. The material frequency-varying loss factor is used as a model of the viscoelastic material properties to characterize the damping properties of the embedded viscoelastic layer.

[0026] To further implement the above technical solution, the specific content of S2 includes: The theoretical properties of the embedded viscoelastic layer are characterized by using the Carrera unified formula theory and Lagrange polynomial expansion to characterize the three-dimensional displacement field of the thin-walled metal structure with the embedded viscoelastic layer. A high-order accurate description is performed to achieve refined modeling of the complex cross-sectional shape of the sandwich structure. Combining linear elastic dynamics theory and constitutive relations considering frequency-varying viscoelastic materials, the governing differential equations and boundary conditions of the metal thin-walled structure with embedded viscoelastic layer are constructed by using the virtual displacement principle and combining the Carrera unified formula model. The closed analytical solution of the governing differential equation of the metal thin-walled structure unit considering frequency-varying viscoelastic materials is then solved.

[0027] To further implement the above technical solution, the theoretical properties of the embedded viscoelastic layer are characterized. Using the Carrera unified formula theory, a high-order accurate description of the three-dimensional displacement field of the thin-walled metal structure with the embedded viscoelastic layer is achieved through Lagrange polynomial expansion. Specific details of this refined modeling of the complex cross-sectional shape of the sandwich structure include: Based on the Carrera unified formula theory, the displacement field of a thin-walled metal structure with an embedded viscoelastic layer is analyzed. Represented as a general section function and Generalized displacement vector with axial distribution The product form: (1) In the formula, This indicates the number of terms in the function's expansion. and These represent the indexes for the number of items expanded; Using the Lagrange polynomial expansion method, the general section function is... The structure is divided into subdomains that can overlap with the physical layer, with the metallic structure and viscoelastic layer being distinct subdomains. Each subdomain employs one or more Lagrange elements, and the polynomial expansion order of each Lagrange element depends on the type of Lagrange expansion used, thus yielding a universal section function. Multinomial expressions.

[0028] It should be noted that: This invention employs the Lagrange expansion function as the section function. Using the Lagrange expansion method, the section of the viscoelastic sandwich structure element is divided into subdomains that coincide with the physical layer, namely the metal layer and the viscoelastic layer. Each subdomain can use one or more Lagrange elements. The order of the polynomial expansion for each element depends on the type of Lagrange expansion used; specifically, four-point (L4) bilinear, nine-point (L9) cubic, and sixteen-point (L16) fourth-order polynomials can be used to construct different beam structure theories. The order of the beam structure model is directly related to the choice of the section polynomial. Here, the L9 polynomial expression is used as an example for illustration: ; in, and This indicates the position of a vertex in the natural coordinate system.

[0029] To further implement the above technical solution, the specific content of constructing the governing differential equations and boundary conditions for the metal thin-walled structure with an embedded viscoelastic layer based on the virtual displacement principle and combined with the Carrera unified formula model includes: The motion control equations are derived using the principle of virtual displacement: (2) (3) (4) In the formula, For standard virtual variable operators, Indicates strain energy. For structural volume, In response, Indicates matrix transpose. For stress, The work done by inertial force, To consider the structural stiffness coefficient matrix of the frequency-varying viscoelastic damping model, D is the linear differential operator matrix; After integration by parts, the strain energy in the governing differential equation incorporating the viscoelastic layer frequency-varying complex elastic modulus is expressed as: (5) In the formula, The length of the thin-walled metal structure with a locally embedded viscoelastic layer. and These represent the two directions of the cross-section of the thin-walled metal structure with the embedded viscoelastic layer. and The generalized displacement vector, whose formal mathematical expression depends neither on the order of beam theory nor on the section function. Selection; The linear frequency-varying differential stiffness matrix is... This is the boundary condition matrix; Figure 3 This diagram illustrates the assembly process of the frequency-varying viscoelastic dynamic stiffness matrix corresponding to the frequency-varying viscoelastic dynamic stiffness model of a thin-walled metal structure containing a viscoelastic layer. It demonstrates the assembly process of a linear differential stiffness matrix for a three-layer cross-section with a frequency-varying viscoelastic layer and a metal layer using this method. The Lagrangian element model used in this invention can be refined in two ways: firstly, by employing higher-order polynomials at the global level; and secondly, by locally combining polynomials within each subdomain of the viscoelastic and metal layer cross-sections.

[0030] The imaginary component of the work done by inertial forces is expressed as: (6) In the formula, Indicates the cross section. Indicates the density of the material. and They represent and The direction is in the expansion of terms. and The cross-sectional function at time, express Regarding time The second derivative, It is the fundamental diagonal kernel of the 3×3 mass matrix; Substituting (5) and (6) into (2), we obtain the section moment parameters of the embedded viscoelastic metal material. and boundary section moment parameters The explicit form of the relevant motion control equations is: (7) (8) (9) In the formula, express corresponding and The number of terms in the directional expansion are respectively and The material section moment parameters at time, where, express middle lines and Column corresponding parameters, , =1~6, and They represent , , Any one of the following; general item It is a generalized section moment parameter for thin-walled metal structures containing viscoelastic material layers, used to describe the equivalent mechanical properties of viscoelastic sandwich structures.

[0031] express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time The space function and the number of expansion terms are time The space function respectively on The partial derivatives; express With the number of expanded terms being time The space function and the number of expansion terms are time The space function respectively on The partial derivatives; , and They represent , and With the number of expanded terms being Time The first-order partial derivative of , where , and They represent , and Displacement in the direction; , and They represent , and exist Direction The second-order partial derivative; , and They represent , and With the number of expanded terms being Displacement at time; , and express , and Regarding time The second derivative of , i.e., acceleration; , and They represent , and With the number of expanded terms being Displacement at time; make , and The number of terms to be expanded are respectively hour , and If the generalized force vector is in the direction, then the boundary conditions are... , and for: (10) Then (7) and (10) are used as the governing differential equations and boundary equations of the metal thin-walled structure with embedded viscoelastic layer constructed by the principle of virtual displacement.

[0032] To further implement the above technical solution, for viscoelastic sandwich structures, the structural stiffness coefficient matrix... It is a complex matrix with a non-constant elastic modulus, specifically expressed as: (11) The specific coefficients are as follows: (12) (13) (14) In the formula, The complex elastic modulus of the viscoelastic layer; Poisson's ratio represents the material; Two models are used to theoretically characterize the viscoelastic modulus properties. The first is a constant complex model, where the real and imaginary parts are constants. The second model is a nonlinear frequency-dependent model. For the constant complex model, the frequency-dependent properties of the viscoelastic sandwich are introduced by the elastic modulus, which is assumed to be a complex constant. Represented as: (15) in, The energy storage elastic modulus of the viscoelastic layer; This represents the corresponding loss factor of the frequency-varying material. These are the eigenfrequency of the structure. for For this model, the loss factor of the viscoelastic layer affects the vibration behavior of the structure. For the nonlinear frequency-dependent model, polyvinyl butyral material is selected as the frequency-variable viscoelastic interlayer in this embodiment. By fitting the master curve at 20°C, its frequency-dependent elastic modulus can be found in the general Maxwell model. Represented as: (16) in, The standard energy storage modulus representing the viscoelastic layer. The frequency correlation coefficient, For fractional-order parameters, These are the model correction coefficients.

[0033] To further implement the above technical solution, obtain the governing differential equations and boundary conditions of the Carrera unified formula model, and solve the closed analytical solution of the thin-walled metal structure with embedded viscoelastic layer, the specific content includes: By assuming the solution is in the form of simple harmonic motion, the time-domain motion control equations (7) and (10) are transformed into a set of three mutually coupled ordinary differential equations concerning the amplitude function in the frequency domain, and expressed in a compact matrix form, denoted as the H matrix; at the same time, the corresponding boundary conditions are also matrixized into the P matrix; for each layer of the viscoelastic sandwich structure, the interlayer theoretical model is constructed by applying multiple Lagrange expansions on the cross-sectional domains of the viscoelastic layer and the metal layer. Utilizing this characteristic of the unified beam model of Lagrange expansion, the high-level interlayer theoretical model can be implemented in a simple and direct manner.

[0034] The ordinary differential equation in the frequency domain is a problem about The second-order ordinary differential equation system with constant coefficients can be simplified to a first-order system through variable transformation. Therefore, the H matrix is ​​transformed into the following linear differential system S matrix: (17) in, For a system of first-order ordinary differential equations with constant coefficients, For a second-order ordinary differential equation system with constant coefficients, S denotes the S matrix; The general solution of equation (17) is as follows: (18) Where n is the total number of first-order ordinary differential equations with constant coefficients, Z n For the nth first-order ordinary differential equation with constant coefficients, It is the first of the S matrix 1 eigenvalue, It is the j-th element of the i-th eigenvector of matrix S. Let n be the nth integration constant, determined by the boundary conditions. Then equation (17) can be written in the following matrix form Z: (19) The vector Z contains not only displacements but also their first derivatives. If only displacements are needed, we only need to consider rows 1, 3, 5, ..., n−1, thus obtaining a solution of the following form: the expansion of the closed-form solution of the governing differential equation of the thin-walled metal structure with embedded viscoelastic layers is as follows: The closed-form expansion of the governing differential equation for a thin-walled metal structure with an embedded viscoelastic layer is as follows: (20) in, , and Let these represent the generalized displacement functions in the x, y, and z directions, given a single expansion term in the Carrera unified theory. , Let M be the generalized displacement function in the z-direction given the number of expansion terms M in the Carrera unified theory. M is the number of expansion terms in the Carrera unified theory, and n = 6×M is the dimension of the unknown vector, which is also the number of differential equations; once the displacement is known, by substituting the solution of formula (20) into the boundary condition (10), the P matrix corresponding to the boundary condition is obtained: (twenty one) in, B is an intermediate variable, and B is the boundary condition coefficient matrix. Let S be the eigenvectors of matrix S; then the boundary conditions can be written in explicit form as follows: (twenty two) In the formula, That is The value in the nth row and nth column of the matrix. , and The generalized force functions P in the x, y, and z directions given by Carrera's unified theory when the number of expansion terms is 1. It is a generalized force function P given by Carrera's unified theory with M expansion terms and z-direction.

[0035] To further implement the above technical solution, the specific content of S3 includes: The displacement field expression for a thin-walled metal structure with an embedded viscoelastic layer, and the boundary conditions are expressed as follows: (twenty three) In the formula, and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized displacement value at that location, When the Carrera unified theory gives an expansion of M terms, the z-direction is... The generalized displacement value at the location, the negative sign indicates that the actual direction of the displacement is opposite to the preset positive direction. and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized displacement value at that location, When the Carrera unified theory gives an expansion of M terms, the z-direction is... The generalized displacement value at the location; Through and Equation (20) is used to calculate the generalized displacement matrix relationship by applying the boundary conditions of equation (23): (twenty four) In the formula Let i = 1…n, j = 1…n-1, and be the eigenvectors of matrix S. The value in the i-th row and j-th column of the array. , … Let n be the constant value of the improper integral, and let n represent the nth one. Equation (24) can be expressed in a compact form: (25) In the formula, A is the displacement coefficient matrix, and Q is the generalized integral constant vector; the boundary conditions for the generalized nodal forces are as follows: (26) In the formula, and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized nodal force value at the location, the negative sign indicates that the actual direction of the force is opposite to the preset positive direction. When the Carrera unified theory gives an expansion of M terms, the z-direction is... The generalized nodal force value at the location, and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized nodal force value at the location, When the Carrera unified theory gives an expansion of M terms, the z-direction is... Generalized nodal force values ​​at the location; Through and By calculating equation (22) and applying the boundary conditions of equation (26), the following matrix relationship of generalized nodal forces is obtained: (27) In the formula, for The value in the nth row and nth column of the matrix; Equation (27) can be expressed in a compact form. : (28) In the formula, R is the boundary coefficient matrix and Q is the generalized integral constant vector; by associating the generalized nodal forces with the corresponding generalized displacements, the generalized integral constant vector Q in equations (25) and (28) is eliminated, thus obtaining the frequency-varying viscoelastic dynamic stiffness matrix of the thin-walled metal structure with embedded viscoelastic layer: (29) (30) In the formula, K is the frequency-varying viscoelastic dynamic stiffness matrix; The frequency-varying viscoelastic dynamic stiffness matrix is ​​the fundamental unit for calculating the accurate complex modal frequencies of high-order thin-walled metallic structures with embedded viscoelastic layers. The global frequency-varying viscoelastic dynamic stiffness matrix can be obtained by assembling element matrices using the classical finite element method. Specifically, for complex structures composed of stiffened structures and thin-walled elements, the overall frequency-varying viscoelastic dynamic stiffness matrix can be constructed by assembling the element frequency-varying viscoelastic dynamic stiffness matrices. This can be achieved by introducing corresponding Lagrangian elements to the entire cross-section, or by dividing the cross-section of the thin-walled metallic structure with embedded viscoelastic layers into several subdomains. This method, because it models the displacement variables of each structural component using Lagrangian elements at the cross-sectional level, is called the component method. Figure 3 A component-based model of a viscoelastic sandwich beam structure is presented, where each component is modeled using a one-dimensional Lagrangian element. These Lagrangian elements are then assembled to obtain a global frequency-varying viscoelastic dynamic stiffness matrix. The illustration also shows the assembly methods of the frequency-varying viscoelastic dynamic stiffness matrix for different cross-sectional shapes.

[0036] The global viscoelastic dynamic stiffness matrix is ​​then: (31) In the formula, The global viscoelastic dynamic stiffness matrix is ​​used; the boundary conditions are applied by directly removing the rows and columns corresponding to zero degrees of freedom from the global viscoelastic dynamic stiffness matrix.

[0037] To further implement the above technical solution, the specific content of S4 includes: The homotopy perturbation method is applied to the damped dynamic stiffness formula of thin-walled metallic structures with embedded viscoelastic layers to obtain the desired complex frequencies. This method uses the undamped eigenvalues ​​and eigenvectors calculated by the enhanced Wittrick-Williams (WW) algorithm as initial solutions, and then solves for the complex eigenvalues ​​through inverse iteration combined with homotopy perturbation. Obtaining the undamped eigenvalue solutions is a crucial component of this method. The basic working principle of the algorithm can be briefly summarized in the following steps: (i) Select a trial frequency To calculate the global inviscid elastic dynamic stiffness matrix of the structure ; (ii) Using Gaussian elimination method to... Converting to upper triangle form, we get The algorithm counts the number of negative values ​​on its main diagonal; this number is called the sign count of the algorithm. ; (iii) Structure at the test frequency The following modal frequencies modulus for: (32) in, This indicates that when the node boundaries of a thin-walled metal structure with an embedded viscoelastic layer are fully fixed, due to... and The number of natural frequencies between them; Corresponding to time Number of modes; it should be noted that, The introduction of this is because when all nodes of the structure are fully fixed, the damping dynamic stiffness matrix can still consider the infinite number of natural frequencies generated by the vibration of individual or multiple elements between structural nodes. This is solved using an indirect method. Enhancements to the Wittrick-Williams (WW) algorithm: when The half-wave number in the direction is At that time, the mode count of a thin-walled metallic structural unit with an embedded viscoelastic layer having all simply supported conditions. The method for obtaining it is as follows: (33) (34) Among them, when The half-wave number in the direction is hour, for ; For simply supported thin-walled metal structural elements with embedded viscoelastic layers, the frequency-varying dynamic stiffness matrix is ​​given. In obtaining And through statistics The number of negative terms was obtained Under the premise of this, the arbitrary natural frequency is limited to the experimental frequency by the dichotomy method. Within the upper and lower bounds, and to achieve the required accuracy; Obtaining the global inviscid elastic dynamic stiffness matrix After obtaining the undamped eigenvalues, the corresponding eigenvectors are calculated, and the eigenvalue problem of the inviscid elastic dynamic stiffness model is expressed as: (35) in, The global inviscid elastic dynamic stiffness matrix. Represents real eigenvalues. Let be a vector of real parameters. It is a real eigenvector; and Determined using the Wittrick-Williams algorithm; If frequency-varying viscoelastic properties are considered in the dynamic stiffness model, then the formula for frequency-varying viscoelastic dynamic stiffness is expressed as: (36) Compared with the inviscid dynamic stiffness formula (35), the eigenvalues and eigenvectors perturbation momentum and All are damping-related parameter vector perturbations Nonlinear functions; Therefore, the homotopy method is used to solve for complex eigenvalues ​​and eigenvectors, and its specific form is as follows: (37) In homotopy analysis, let h be the convergence control parameter, including the loss factor or characteristic time constant; where Indicates an undamped state. The final target damping coefficient, and These represent the final complex eigenvalues ​​and complex eigenvectors of the viscoelastic dynamic stiffness matrix, respectively.

[0038] The growth method of parameter h is as follows: (38) (39) (40) (41) in Let k be the total homotopy, and k represent the current iteration step. ; , and These represent the convergence control parameters, complex eigenvalues, and complex eigenvectors obtained in each iteration; based on , and Through iterative processes, the complex eigenvalues ​​and complex eigenvectors at the (k+1)th step are obtained; finally, based on the adopted frequency-varying damping model, the global frequency-varying viscoelastic dynamic stiffness matrix is ​​calculated. The complex eigenvalues ​​and complex eigenvectors are the complex natural frequencies and mode shapes of the thin-walled metal structure with embedded viscoelastic layers.

[0039] The value of directly affects the convergence of the final complex eigenvalues ​​and complex eigenvectors. For different types of viscoelastic sandwich structures, appropriate values ​​need to be selected. The value is adjusted to improve convergence performance. It is worth noting that this method is not limited by the magnitude of the frequency-varying damping parameter; it only requires setting an appropriate value. The value can be used to solve the vibration system under high damping conditions in segments.

[0040] The invention will be further illustrated by the following experiments: Two simple numerical examples illustrate the application of the proposed invention in the free vibration analysis of thin-walled metal structures with embedded viscoelastic layers. The dimensions and cross-sectional node arrangement of the viscoelastic sandwich beam structure are shown below. Figure 4 As shown, the thickness of the glass layer is The thickness of the viscoelastic layer is The beam's width and length are 100mm and 1m, respectively. The glass's elastic modulus is 64.5 GPa. Its Poisson's ratio is 0.22. The material's density is 2737 kg / m³. 3 The elastic modulus of the nonlinear frequency-dependent viscoelastic layer is 0.00134 GPa. The Poisson's ratio is 0.4. The material density is 999 kg / m³. 3 . The standard storage modulus of the viscoelastic layer is 0.235 GPa, and the frequency correlation coefficient is... The value is 0.3979, a fractional-order parameter. The model correction coefficient is 0.46. The total homology of the mezzanine beam structure is 0.1946. It is 20.

[0041] The dimensions, shape, and cross-sectional node arrangement of the viscoelastic metal thin-walled stiffened circular ring structure are as follows: Figure 6 As shown; where Figure 6 (a) shows the structural outline and dimensions; Figure 6(b) is a side view of section 1, arranged with 52 L9 elements. Figure 6 (c) shows the side view of section 2 and the arrangement of 40 L9 elements. The damping ring of this structure is composed of three materials: the main body is aluminum, the middle layer is a constant viscoelastic layer, and the two end faces are steel sections. The centroid radii of the innermost, middle, and outermost layers are respectively, and their corresponding thicknesses are respectively... and The ring also has a thickness of [missing information]. The rectangular reinforcing ribs, while the side length of the end section of the hollow square steel is... Thickness is The total length of the ring is The elastic modulus of aluminum is 69 GPa. Its Poisson's ratio is 0.3. The material density is 2766 kg / m³. 3 The elastic modulus of the steel is 206 GPa. Poisson's ratio is 0.25. The material density is 7900 kg / m³. 3 The elastic modulus of the constant viscoelastic layer is 0.001794 GPa. Poisson's ratio is 0.3. The material density is 958.1 kg / m³. 3 The material loss factor is 0.95. The total homology of the thin-walled, stiffened circular metal ring structure is... It is 20.

[0042] Table 1 shows the results of comparing the first four modal frequencies and their modal damping ratios of the viscoelastic sandwich beam structure obtained by various algorithms with those obtained by the finite element method. Modal mode comparisons are also provided. Figure 5 As shown.

[0043] Table 2 shows the results of comparing the first six modal frequencies and modal damping ratios of the viscoelastic thin-walled stiffened circular ring structure obtained by various algorithms with those obtained by the finite element method. Modal mode comparisons are also provided. Figure 7 As shown.

[0044] Table 1. First four modal frequencies and modal damping ratios of the edge-fixed viscoelastic sandwich beam structure. ; Table 2. First six modal frequencies and modal damping ratios of simply supported viscoelastic thin-walled stiffened circular ring structures. ; The first four natural frequencies and damping of the viscoelastic sandwich beam structure under fixed-fixed boundary conditions are shown in Table 1. The results of this invention deviate from those of other methods by less than 1%, demonstrating excellent consistency. This confirms that even when analyzing damped structures dominated by complex frequency-dependent viscoelastic models, the modeling strategy and corresponding complex eigenvalue solution process proposed in this invention remain highly reliable. Even with the 9L9 Lagrange section discretization model, the method of this invention requires only 1.8 seconds of computation time, while the finite element method requires 19.4 seconds with a coarse mesh and 95 seconds with a fine mesh. Therefore, the computational efficiency of the method of this invention is one to two orders of magnitude higher than that of commercial finite element solvers. It is worth noting that commercial finite element software packages typically require significant mesh refinement to handle nonlinear frequency-dependent damping models; otherwise, the solution cannot converge, leading to a significant increase in the computational cost of complex modal analysis. Furthermore, even with mesh refinement and extended computation time, the finite element solution may still be difficult to converge. Figure 5 The first four mode shapes obtained by the finite element method and this invention are shown, where (a), (c), (e), and (g) are the first four mode shapes calculated by this method, and (b), (d), (f), and (h) are the first four mode shapes obtained by finite element simulation. The two methods show good agreement. The fourth mode corresponds to torsional deformation, while the other modes are mainly dominated by bending deformation. The comparison shows a high degree of agreement, verifying the accuracy of this method.

[0045] The geometric configuration of a viscoelastic thin-walled stiffened circular ring structure and the corresponding CUF-Lagrange expansion model are as follows: Figure 6 As shown. Figure 6 The key focus is on the nodal discretization strategy employed at two typical cross-sections (denoted as Cross-section 1 and Cross-section 2) to account for the geometric changes introduced by the stiffeners. Cross-section 1 employs... Figure 6 (b) shows the discrete method, while section 2 uses... Figure 6 (c) shows the discretization method. The labels "40L9 & 52L9" indicate that section 1 was discretized using 40 L9 elements, and section 2 used 52 L9 elements, reflecting the differences in local geometric features at the two sections. The discretized sections were then assembled through shared node interfaces to ensure motion coordination and consistent transmission of generalized displacement along the longitudinal direction.

[0046] Table 2 lists the first six natural frequencies (Hz) and modal damping ratios of the viscoelastic thin-walled stiffened circular ring structure under simply supported boundary conditions. The predicted results of this invention were compared with reference three-dimensional finite element results obtained using commercial finite element software. In the commercial finite element software, C3D20 solid elements were used; the coarse mesh model contained 59,796 degrees of freedom, and the fine mesh model contained 108,628 degrees of freedom. The comparison results show that the method of this invention exhibits high accuracy at all frequencies and damping ratios, and the deviation relative to the fine mesh finite element results remains within a small range for each considered mode. Among the four discrete models of this invention, the 52L16 & 64L16 configurations agree best with the three-dimensional finite element results. This high accuracy stems from the richer cross-sectional kinematics described by the L16 unfolding. Compared with low-order discretization, L16 assigns more interpolation points on the annular thin-walled profile, thereby introducing more generalized displacement variables. This allows for a more detailed characterization of local bending, circumferential deformation, and shell-like deformation effects related to enhanced geometric features, thus confirming that the method is applicable to the vibration analysis of complex thin-walled viscoelastic structures.

[0047] Figure 7 The first four modes of a viscoelastic thin-walled stiffened circular ring structure under simply supported boundary conditions are presented. (a), (c), (e), and (g) are the mode shapes calculated using this method, while (b), (d), (f), and (h) are the mode shapes obtained from finite element simulation. Comparison of the results from this method and the three-dimensional finite element method shows that the deformation modes of each mode obtained by the two methods are almost identical. Specifically, the first and second modes mainly exhibit overall bending deformation of the ring; the third and fourth modes show a mixed bending-distortion deformation behavior related to the thin-walled geometry; and higher-order modes, such as the fifth and sixth modes, exhibit more complex deformation characteristics and localized shell-like modes. The accuracy of the results obtained by this method is comparable to that of the three-dimensional finite element model, but the degrees of freedom used are only 3.4% of the latter.

[0048] It is evident that the present invention possesses the following beneficial effects: 1. This invention, through the use of layering theory and component methods, has developed a high-precision CUF(LE)-dynamic stiffness formula for defining the displacement field on the cross-section of a thin-walled metal structure with an embedded viscoelastic layer. This method enables the convenient assembly of arbitrary sub-components or layups with different material loss factors, which is crucial for characterizing sandwich structures with non-uniform damping properties.

[0049] 2. The shape functions used in the CUF(LE)-dynamic stiffness method employed in this invention are derived from the frequency-domain governing differential equations of structural vibration. This allows the method to efficiently and directly characterize viscoelastic damping, which can be frequency-dependent or frequency-independent, linear or nonlinear, and possess integer-order or fractional-order viscoelastic properties. The frequency-varying dynamic stiffness formula eliminates the need for separate remodeling based on different damping models, providing a general and unified approach.

[0050] 3. The high-order CUF dynamic stiffness model established in this invention can accurately identify various classical and non-classical vibration modes, including torsion, shear and coupled modes, and exhibits high accuracy and high stability for various beam and plate structures. It provides a powerful and reliable analysis tool for the parametric research and optimization design of thin-walled metal structures with embedded viscoelastic layers.

[0051] The above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application, and should all be included within the protection scope of this application.

Claims

1. An analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer, characterized in that, Includes the following steps: S1. Obtain the geometric parameters, material properties, and location of the viscoelastic layer of the thin-walled metal structure with an embedded viscoelastic layer; S2. Based on the parameters obtained in S1, the theoretical properties of the embedded viscoelastic layer are characterized, the displacement field of the metal thin-walled sandwich structure is modeled in three dimensions, and the control differential equations of the kinematics of the metal thin-walled structure with the embedded frequency-varying viscoelastic layer are constructed. S3. Based on the governing differential equation, a frequency-varying viscoelastic dynamic stiffness model is established through boundary conditions and general solutions; S4. Solve the frequency-varying viscoelastic dynamic stiffness model to obtain the complex eigenvalues ​​of the thin-walled metal structure with embedded viscoelastic layer, and then extract the complex natural frequency and modal damping ratio of the thin-walled metal structure with embedded viscoelastic layer to complete the free vibration analysis of the thin-walled metal structure with embedded viscoelastic layer.

2. The analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer according to claim 1, characterized in that, The specific content of S1 includes: Geometric parameters include the length, thickness, and cross-sectional shape dimensions of the thin-walled metal structure; Material properties include elastic modulus, Poisson's ratio, and material density; An embedded constrained viscoelastic damping layer is designed between thin-walled metal structures. The material frequency-varying loss factor is used as a model of the viscoelastic material properties to characterize the damping properties of the embedded viscoelastic layer.

3. The analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer according to claim 1, characterized in that, The specific content of S2 includes: The theoretical properties of the embedded viscoelastic layer are characterized by using the Carrera unified formula theory and Lagrange polynomial expansion to characterize the three-dimensional displacement field of the thin-walled metal structure with the embedded viscoelastic layer. A high-order accurate description is performed to achieve refined modeling of the complex cross-sectional shape of the sandwich structure. Combining linear elastic dynamics theory and constitutive relations considering frequency-varying viscoelastic materials, the governing differential equations and boundary conditions of the metal thin-walled structure with embedded viscoelastic layer are constructed by using the virtual displacement principle and combining the Carrera unified formula model. The closed analytical solution of the governing differential equation of the metal thin-walled structure unit considering frequency-varying viscoelastic materials is then solved.

4. The analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer according to claim 3, characterized in that, Using the Carrera unified formula theory, the three-dimensional displacement field of a thin-walled metallic structure with an embedded viscoelastic layer is investigated through Lagrange polynomial expansion. The specific content of performing high-order accurate descriptions and achieving refined modeling of the complex cross-sectional shapes of sandwich structures includes: Based on the Carrera unified formula theory, the displacement field of a thin-walled metal structure with an embedded viscoelastic layer is analyzed. Represented as a general section function and Generalized displacement vector with axial distribution The product form: （1） In the formula, This indicates the number of terms in the function's expansion. and These represent the indexes for the number of items expanded; Using the Lagrange polynomial expansion method, the general section function is... The structure is divided into subdomains that can overlap with the physical layer, with the metallic structure and viscoelastic layer being distinct subdomains. Each subdomain employs one or more Lagrange elements, and the polynomial expansion order of each Lagrange element depends on the type of Lagrange expansion used, thus yielding a universal section function. Multinomial expressions.

5. The analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer according to claim 3, characterized in that, The specific content of constructing the governing differential equations and boundary conditions for a thin-walled metallic structure with an embedded viscoelastic layer based on the principle of virtual displacement and combined with the Carrera unified formula model includes: The motion control equations are derived using the principle of virtual displacement: （2） （3） （4） In the formula, For standard virtual variable operators, Indicates strain energy. For structural volume, In response, Indicates matrix transpose. For stress, The work done by inertial force, To consider the structural stiffness coefficient matrix of the frequency-varying viscoelastic damping model, D is the linear differential operator matrix; After integration by parts, the strain energy in the governing differential equation incorporating the viscoelastic layer frequency-varying complex elastic modulus is expressed as: （5） In the formula, The length of the thin-walled metal structure with a locally embedded viscoelastic layer. and These represent the two directions of the cross-section of the thin-walled metal structure with the embedded viscoelastic layer. and The generalized displacement vector, The linear differential stiffness matrix is... This is the boundary condition matrix; The imaginary component of the work done by inertial forces is expressed as: （6） In the formula, Indicates the cross section. Indicates the density of the material. and They represent and The direction is in the expansion of terms. and The cross-sectional function at time, express Regarding time The second derivative, It is the fundamental diagonal kernel of the 3×3 mass matrix; Substituting (5) and (6) into (2), we obtain the section moment parameters of the embedded viscoelastic metal material. and boundary section moment parameters The explicit form of the relevant motion control equations is: （7） （8） （9） In the formula, express corresponding and The number of terms in the directional expansion are respectively and The material section moment parameters at time, where, express middle lines and Column corresponding parameters, , =1~6, and They represent , , Any one of them; express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time Space function pairs The partial derivatives; express With the number of expanded terms being time The space function and the number of expansion terms are time The space function respectively on The partial derivatives; express With the number of expanded terms being time The space function and the number of expansion terms are time The space function respectively on The partial derivatives; , and They represent , and With the number of expanded terms being Time The first-order partial derivative of , where , and They represent , and Displacement in the direction; , and They represent , and exist Direction The second-order partial derivative; , and They represent , and With the number of expanded terms being Displacement at time; , and express , and Regarding time The second derivative of , i.e., acceleration; , and They represent , and With the number of expanded terms being Displacement at time; make , and The number of terms to be expanded are respectively hour , and If the generalized force vector is in the direction, then the boundary conditions are... , and for: （10） Then (7) and (10) are used as the governing differential equations and boundary equations of the metal thin-walled structure with embedded viscoelastic layer constructed by the principle of virtual displacement.

6. The analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer according to claim 5, characterized in that, For viscoelastic sandwich structures, the structural stiffness coefficient matrix It is a complex matrix with a non-constant elastic modulus, specifically expressed as: （11） The specific coefficients are as follows: （12） （13） （14） In the formula, The complex elastic modulus of the viscoelastic layer; Poisson's ratio represents the material; For the constant complex model, the frequency-dependent properties of the viscoelastic sandwich are introduced by the elastic modulus, which is assumed to be a complex constant. Represented as: （15） in, The energy storage elastic modulus of the viscoelastic layer; This represents the corresponding loss factor of the frequency-varying material. These are the eigenfrequency of the structure. for ; For nonlinear frequency-dependent models Represented as: （16） in, The standard energy storage modulus representing the viscoelastic layer. The frequency correlation coefficient, For fractional-order parameters, These are the model correction coefficients.

7. The analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer according to claim 6, characterized in that, The specific details of obtaining the governing differential equations and boundary conditions of the Carrera unified formula model, and solving the closed analytical solution of the thin-walled metal structure with embedded viscoelastic layer include: By assuming the solution is in the form of simple harmonic motion, the time-domain motion control equations (7) and (10) are transformed into a set of three mutually coupled ordinary differential equations in the frequency domain concerning the amplitude function, and expressed in a compact matrix form, denoted as the H matrix; at the same time, the corresponding boundary conditions are also matrixized into the P matrix. The ordinary differential equation in the frequency domain is a problem about The second-order ordinary differential equation system with constant coefficients can be simplified to a first-order system through variable transformation. Therefore, the H matrix is ​​transformed into the following linear differential system S matrix: （17） in, For a system of first-order ordinary differential equations with constant coefficients, For a second-order ordinary differential equation system with constant coefficients, S denotes the S matrix; The general solution of equation (17) is as follows: （18） Where n is the total number of first-order ordinary differential equations with constant coefficients, Z n For the nth first-order ordinary differential equation with constant coefficients, It is the first of the S matrix 1 eigenvalue, It is the S matrix of the first generation. The th eigenvector of the th feature vector One element, Let n be the nth integration constant, determined by the boundary conditions. Then equation (17) can be written in the following matrix form Z: （19） The closed-form expansion of the governing differential equation for a thin-walled metal structure with an embedded viscoelastic layer is as follows: （20） in, , and Let these represent the generalized displacement functions in the x, y, and z directions, given a single expansion term in the Carrera unified theory. , Let M be the generalized displacement function in the z-direction given the number of expansion terms M in the Carrera unified theory. M is the number of expansion terms in the Carrera unified theory; once the displacement is known, the P matrix corresponding to the boundary conditions is obtained by substituting the solution of formula (20) into the boundary conditions (10): （21） in, B is an intermediate variable, and B is the boundary condition coefficient matrix. Let S be the eigenvectors of matrix S; then the boundary conditions can be written in explicit form as follows: （22） In the formula, That is The value in the nth row and nth column of the matrix. , and The generalized force functions P in the x, y, and z directions given by Carrera's unified theory when the number of expansion terms is 1. It is a generalized force function P given by Carrera's unified theory with M expansion terms and z-direction.

8. The analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer according to claim 7, characterized in that, The specific content of S3 includes: The displacement field expression for a thin-walled metal structure with an embedded viscoelastic layer, and the boundary conditions are expressed as follows: （23） In the formula, and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized displacement value at that location, When the Carrera unified theory gives an expansion of M terms, the z-direction is... The generalized displacement value at the location, the negative sign indicates that the actual direction of the displacement is opposite to the preset positive direction. and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized displacement value at that location, When the Carrera unified theory gives an expansion of M terms, the z-direction is... The generalized displacement value at the location; Through and Equation (20) is used to calculate the generalized displacement matrix relationship by applying the boundary conditions of equation (23): （24） In the formula Let i = 1…n, j = 1…n-1, and be the eigenvectors of matrix S. The value in the i-th row and j-th column of the array. , … Let n be the constant value of the improper integral, and let n represent the nth one. Equation (24) can be expressed in a compact form: （25） In the formula, A is the displacement coefficient matrix, and Q is the generalized integral constant vector; the boundary conditions for the generalized nodal forces are as follows: （26） In the formula, and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized nodal force value at the location, the negative sign indicates that the actual direction of the force is opposite to the preset positive direction. When the Carrera unified theory gives an expansion of M terms, the z-direction is... The generalized nodal force value at the location, and The Carrera unified theory gives the x, y, and z directions when the number of expansion terms is 1. The generalized nodal force value at the location, When the Carrera unified theory gives an expansion of M terms, the z-direction is... Generalized nodal force values ​​at the location; Through and By calculating equation (22) and applying the boundary conditions of equation (26), the following matrix relationship of generalized nodal forces is obtained: （27） In the formula, for The value in the nth row and nth column of the matrix; Equation (27) can be expressed in a compact form. : （28） In the formula, R is the boundary coefficient matrix, and Q is the generalized integral constant vector; by associating the generalized nodal forces with the corresponding generalized displacements, the generalized integral constant vector Q in equations (25) and (28) is eliminated, thus obtaining the frequency-varying viscoelastic dynamic stiffness matrix of the metal thin-walled structure with embedded viscoelastic layer: （29） （30） In the formula, K is the viscoelastic dynamic stiffness matrix; The global viscoelastic dynamic stiffness matrix is ​​then: （31） In the formula, The global viscoelastic dynamic stiffness matrix is ​​used; the boundary conditions are applied by directly removing the rows and columns corresponding to zero degrees of freedom from the global viscoelastic dynamic stiffness matrix.

9. The analytical method for calculating the free vibration of a thin-walled metal structure with an embedded viscoelastic layer according to claim 8, characterized in that, The specific content of S4 includes: (i) Select a trial frequency To calculate the global inviscid elastic dynamic stiffness matrix of the structure ; (ii) Using Gaussian elimination method to... Converting to upper triangle form, we get The algorithm counts the number of negative values ​​on its main diagonal; this number is called the sign count of the algorithm. ; (iii) Structure at the test frequency The following modal frequencies modulus for: （32） in, This indicates that when the node boundaries of a thin-walled metal structure with an embedded viscoelastic layer are fully fixed, due to... and The number of natural frequencies between them; Corresponding to time Number of modes; Indicates to Transform the matrix into upper triangular form and count the number of negative values ​​on its main diagonal. s(...) represents the transformation of the matrix into upper triangular form. Solve using an indirect method : when The half-wave number in the direction is At that time, the mode count of a thin-walled metallic structural unit with an embedded viscoelastic layer having all simply supported conditions. The method for obtaining it is as follows: （33） （34） Among them, when The half-wave number in the direction is hour, for ; For simply supported thin-walled metal structural elements with embedded viscoelastic layers, the frequency-varying dynamic stiffness matrix is ​​given. In obtaining And through statistics The number of negative terms was obtained Under the premise of this, the arbitrary natural frequency is limited to the experimental frequency by the dichotomy method. Within the upper and lower bounds, and to achieve the required accuracy; Obtaining the global inviscid elastic dynamic stiffness matrix After obtaining the undamped eigenvalues, the corresponding eigenvectors are calculated, and the eigenvalue problem of the inviscid elastic dynamic stiffness model is expressed as: （35） in, The global inviscid elastic dynamic stiffness matrix. Represents real eigenvalues. Let be a vector of real parameters. It is a real eigenvector; and Determined using the Wittrick-Williams algorithm; If frequency-varying viscoelastic properties are considered in the dynamic stiffness model, then the formula for frequency-varying viscoelastic dynamic stiffness is expressed as: （36） Compared with the inviscid dynamic stiffness formula (35), the eigenvalues and eigenvectors perturbation momentum and All are damping-related parameter vector perturbations Nonlinear functions; Therefore, the homotopy method is used to solve for complex eigenvalues ​​and eigenvectors, and its specific form is as follows: （37） In homotopy analysis, let h be the convergence control parameter, including the loss factor or characteristic time constant; where Indicates an undamped state. The final target damping coefficient, and These represent the final complex eigenvalues ​​and complex eigenvectors of the viscoelastic dynamic stiffness matrix, respectively. The growth method of parameter h is as follows: （38） （39） （40） （41） in Let k be the total homotopy, and k represent the current iteration step. ; , and These represent the convergence control parameters, complex eigenvalues, and complex eigenvectors obtained in each iteration, respectively. based on , and The iterative process yields the complex eigenvalues ​​and complex eigenvectors at the (k+1)th step; finally, based on the adopted frequency-varying viscoelastic damping model, the global frequency-varying viscoelastic dynamic stiffness matrix is ​​calculated. The complex eigenvalues ​​and complex eigenvectors are the complex natural frequencies and complex mode shapes of the thin-walled metal structure with embedded viscoelastic layers.

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