Nanoparticle filling axial gradient material structure optimization method

Abstract

The invention discloses a nano-particle filling axial gradient material structure optimization method, which comprises the following steps of: constructing equivalent material parameters according to agglomeration quantity parameters of a nano-particle filling axial gradient material; a perturbation amount is set according to the agglomeration amount parameter, and a vibration control equation of the axial gradient beam is constructed according to the equivalent material parameter and the perturbation amount; perturbation expansion is carried out on the vibration control equation; solving according to the perturbation expanded vibration control equation to obtain a mapping relation between perturbation amount and frequency; and acquiring a design frequency, and calculating the design frequency according to a mapping relation between perturbation and frequency to obtain corresponding agglomeration parameter data of the nano-particle filled axial gradient material, namely a structure optimization result. According to the method, the mathematical corresponding relation between the agglomeration amount parameter and the vibration frequency can be established, so that quantitative analysis, reverse design and structural optimization of the structure of the nano-particle filling axial gradient material are facilitated.

Description

Technical Field

[0001] This invention belongs to the field of composite material structure optimization technology, and particularly relates to a method for optimizing the structure of an axial gradient material filled with nanoparticles. Background Technology

[0002] Nanoparticle-filled gradient materials are a novel type of composite material that combines the advantages of gradient materials and filler particles. Specific physical and mechanical effects are achieved by adding nanoparticles to a matrix gradient, where the material gradient can vary along the axial, radial, or thickness directions as needed. For example, adding graphene or carbon nanotubes can significantly improve the strength and stiffness of the matrix and optimize its thermal conductivity and coefficient of thermal expansion. This material exhibits outstanding designability and has broad application prospects in various fields such as aerospace, automotive, and sensors. However, in reality, due to van der Waals forces between the nanoparticles and the matrix, it is difficult for the filler particles to be uniformly or distributed according to the designed gradient within the matrix, instead resulting in agglomeration. This causes deviations in the material's properties and mechanical characteristics from those of uniformly distributed or ideally gradient-distributed particles, and can even lead to calculation errors. Therefore, agglomeration cannot be ignored in theoretical calculations. Currently, some theories describe the agglomeration phenomenon and equivalent material parameters, and can be used to calculate the natural frequencies of beams, plates, and shell structures. However, the analysis of the impact of agglomeration on vibration characteristics is limited and mostly based on qualitative analysis, failing to provide intuitive and quantitative analytical expressions and hindering reverse design. Existing technologies can calculate frequencies based on aggregation effects, but they cannot directly obtain the required aggregation parameters for the desired frequencies. This necessitates extensive calculations and parameter analysis to determine parameters such as the matrix material, nanoparticle filler material, and their proportions. Therefore, existing technologies provide a more intuitive and quantitative design for nanoparticle-filled gradient materials. Summary of the Invention

[0003] To address the aforementioned technical problems, this invention proposes a method for optimizing the structure of nanoparticle-filled axial gradient materials, thereby resolving the issues present in the prior art.

[0004] To achieve the above objectives, the present invention provides a method for optimizing the structure of an axially gradient material filled with nanoparticles, comprising:

[0005] Based on the aggregation parameters of nanoparticle-filled axial gradient materials, equivalent material parameters are constructed.

[0006] The perturbation amount is set according to the agglomeration parameter, and the vibration control equation of the axial gradient beam is constructed according to the equivalent material parameter and the perturbation amount; the vibration control equation is then perturbated and expanded.

[0007] The vibration control equations derived from the perturbation are solved to obtain the mapping relationship between the perturbation quantity and the frequency.

[0008] The design frequency is obtained, and the design frequency is calculated through the mapping relationship between perturbation and frequency. The corresponding agglomeration parameter data of the nanoparticle-filled axial gradient material is obtained, which is the structural optimization result.

[0009] Optionally, the aggregation parameter includes the volume fraction of aggregated microclusters and the volume fraction of nanoparticles within the aggregated microclusters.

[0010] Optionally, the equivalent material parameters include the equivalent elastic modulus E and Poisson's ratio v of an isotropic axial gradient beam in a nanoparticle-filled axial gradient material.

[0011] The process of obtaining the equivalent elastic modulus E and Poisson's ratio v is as follows:

[0012]

[0013] Wherein, K represents the equivalent bulk modulus and G represents the equivalent shear modulus, wherein the equivalent bulk modulus and equivalent shear modulus are calculated based on the aggregation parameter.

[0014] Optionally, the process of setting the perturbation includes:

[0015] One or more parameters are selected from the aggregation parameters as perturbation factors, wherein the volume fraction of nanoparticles within the aggregated microclusters is selected as the perturbation factor.

[0016] Optionally, the vibration control equation is:

[0017]

[0018] in, To represent the state vector,

[0019]

[0020] ω is the natural frequency. This represents the dimensionless displacement of the neutral layer of the beam along the axial direction. These represent the mode shapes of the beam along the z-direction, x-direction, and rotation about the y-axis, respectively. There is a mapping relationship between the mode shapes and the natural frequencies. This is the coefficient matrix in the equation;

[0021] The boundary conditions corresponding to the vibration control equation are:

[0022] M(ω)η(0,ω)+N(ω)η(1,ω)=0

[0023] In the formula, M(ω) and N(ω) represent the left end of the beam, respectively. Time and right end The boundary condition selection matrix at that time.

[0024] Optionally, the mapping relationship between perturbation and frequency is as follows:

[0025] ω(ε)=ω p0 +εω p1

[0026] Where ω(ε) is the frequency, ω p0 ω is the zeroth natural frequency. p1 ε is the first-order natural frequency, and ε is the perturbation.

[0027] Optionally, the zero-order natural frequency ω p0 The calculation is performed using the following formula:

[0028]

[0029] Where det[] represents the det function, Φ0 represents the coefficient matrix of the vibration control equation, and M0 and N0 represent the zero-order boundary condition selection matrices for the left and right ends of the beam, respectively. Φ0, M0, and N0 are all related to ω. p0 There is a computational relationship between them.

[0030] Optionally, the calculation process for the first-order natural frequency is as follows:

[0031]

[0032] Where D0 and D1 are the zeroth-order coefficients and the first-order coefficients:

[0033]

[0034] Where V0 represents When established, the matrix The corresponding eigenvectors, where P represents the matrix The corresponding eigenvector matrix, ζ represents any integration variable between 0 and 1, M1 and N1 represent the first-order boundary condition selection matrices for the left and right ends of the beam, Φ 10 M 10 N 10 The result of expanding Φ1, M1, and N1 without ω p1 Φ 11 M 11 N 11 To expand Φ1, M1, and N1 to include ω p1 Items.

[0035] Optionally, the process of obtaining aggregation quantity parameter data includes:

[0036] The volume fraction of nanoparticles within the aggregated microclusters was selected as the perturbation factor.

[0037] When the volume fraction of aggregated microclusters is obtained through measurement, the design frequency is directly calculated using the mapping relationship between perturbation and frequency, thus obtaining the volume fraction data of nanoparticles within the aggregated microclusters.

[0038] When the volume fraction of aggregated microclusters cannot be obtained by measurement, different sampling points are set for the volume fraction of aggregated microclusters and the volume fraction of nanoparticles within the aggregated microclusters. Based on the numerical relationship surface between the sampling point fitting frequency and the aggregation amount parameter, the aggregation amount parameter corresponding to the design frequency is obtained from the relationship surface, and the aggregation amount parameter data set is obtained.

[0039] On the other hand, the present invention also provides a nanoparticle-filled axial gradient material structure optimization system for performing the above-described method.

[0040] Compared with the prior art, the present invention has the following advantages and technical effects:

[0041] The method proposed in this invention can provide a precise correspondence between the material parameters and corresponding frequencies of nanoparticle-filled axial gradient materials. Compared to the method proposed in this invention, traditional methods require frequency calculations through fitting and interpolation of material parameters for most data points other than the sampling points, resulting in lower accuracy and reliability. Compared to traditional methods, this invention significantly reduces the number of calculations, improves computational efficiency, and lowers computational costs; it can rapidly and quantitatively design the material parameters of nanocrystals while balancing accuracy and computational cost. Attached Figure Description

[0042] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings:

[0043] Figure 1 This is a schematic diagram of a beam and any representative volume element of an embodiment of the present invention;

[0044] Figure 2 This is a schematic diagram illustrating the volume fraction of aggregated microclusters and the relationship between the volume fraction of nanoparticles within aggregated microclusters and frequency, according to an embodiment of the present invention.

[0045] Figure 3 This is a flowchart illustrating the optimization strategy for the nanoparticle-filled axial gradient material structure in an embodiment of the present invention. Detailed Implementation

[0046] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0047] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0048] The closest prior art to this patent application is the patent with patent number 202310846767.5, entitled "A Method and Algorithm for Obtaining the Natural Frequency of a Bidirectional Functionally Graded Beam in a Thermal Environment." That patent discloses a method for obtaining the natural frequency of a bidirectional functionally graded beam based on an improved perturbation method. While this invention is also based on the perturbation concept, it differs from the existing patent. The main differences are as follows: First, the existing patent addresses the vibration problem of functionally graded beams, providing an expression for the frequency based on physical characteristics; this research object is relatively common. However, this invention addresses a new problem, focusing on the influence of aggregation effects on frequency. Second, the existing patent focuses on improving the calculation steps of the traditional perturbation method, increasing computational efficiency and accuracy. This invention aims to construct perturbation parameters and their physical meanings for a new physical scenario, enabling them to express the physical characteristics of aggregation and derive an expression for the influence of aggregation effects on frequency.

[0049] Existing technologies can calculate frequencies based on aggregation effects, but they cannot directly obtain the required aggregation parameters for the desired frequencies. This necessitates extensive calculations and parameter analysis to determine parameters such as the matrix material, nanoparticle filler material, and their proportions. In summary, this invention, targeting new physical scenarios and application backgrounds, assigns new physical meaning to perturbations, improving upon traditional perturbation methods. It enables the direct expression of the relationship between aggregation effects and frequencies, allowing for a more intuitive, effective, and rapid design of nanoparticle-filled gradient materials through a reverse design approach.

[0050] Existing research on the influence of agglomeration characteristics on the vibration characteristics of axially gradient beam structures is mostly based on qualitative analysis, lacking analytical quantitative correspondence. This invention addresses this problem by proposing a structural optimization method for nanoparticle-filled axially gradient materials. This method establishes a mathematical correspondence between the degree of agglomeration and vibration frequency and mode shape, facilitating quantitative analysis, reverse design, and structural optimization.

[0051] To address the aforementioned technical objectives, this invention provides a method for optimizing the structure of an axially gradient material filled with nanoparticles, comprising the following steps:

[0052] Step 1: Based on the micromechanical model, establish the equivalent material parameters of the material after considering agglomeration;

[0053] Step 2: Based on the aggregation parameters, design the perturbation parameters and establish the vibration control equations;

[0054] Step 3: Based on the perturbation concept, expand the vibration control equations using perturbation.

[0055] Step 4: Solve the expanded vibration control equations to obtain analytical expressions for the frequency and mode shape;

[0056] Step 5: Based on the analytical expressions of the frequencies and mode shapes obtained above, obtain the material parameters of the nanoparticle-filled gradient material through reverse design.

[0057] like Figure 3 As shown, the relevant steps of the above technical solution are described in detail:

[0058] Step 1: Based on the aggregated micromechanical model, establish the equivalent material parameters of the axial gradient beam.

[0059] Taking a nanoparticle-filled gradient material with carbon nanotube (CNT) particles in a matrix as an example, the specific implementation steps are detailed. The CNT particles are gradually distributed along the axial direction of the beam, forming an axial gradient. The dimensions and coordinates of the beam are defined as follows: Figure 1 As shown. Taking any representative volume element on the beam, its microstructure is refined. It is assumed that some CNTs are distributed in the matrix, and the remainder aggregate into CNT microclusters due to the aggregation effect, such as... Figure 1 As shown. Among them, in Figure 1 In this context, z represents the z-axis direction, x represents the x-axis direction, and L represents the length of the axial gradient beam.

[0060] Volume fraction f of CNTs in beam r Represented as:

[0061] f r =x p f rmax (1)

[0062] Among them, f rmax The maximum value of the CNT volume fraction is represented by x, which is the axial coordinate of the beam, and p is the gradient coefficient.

[0063] Considering the aggregation effect of CNTs, the aggregation effect is described by the volume fraction μ of aggregated microclusters and the volume fraction η of CNTs within aggregated microclusters. The volume fraction μ of aggregated microclusters is defined as:

[0064]

[0065] Where V is the volume of a representative volume unit, V clusrer It represents the total volume of aggregated microclusters in the unit. μ represents the volume percentage of aggregated microclusters within the unit volume. When μ = 1, the aggregation effect disappears, and CNTs are uniformly distributed throughout the entire volume unit. As μ decreases, the degree of aggregation becomes more significant.

[0066] η is defined as:

[0067]

[0068] in, V represents the total volume of CNTs in aggregated microclusters. r The total volume of CNTs in a representative volumetric unit includes both aggregated and unaggregated CNTs uniformly distributed in the matrix. η represents the ratio of the volume of CNTs in aggregated microclusters to the total volume of all CNTs in the unit, representing the degree of aggregation. When η = 1, all CNTs are in aggregated microclusters. As η decreases, CNTs gradually become more uniformly distributed in the matrix.

[0069] Equivalent bulk modulus K within aggregated microclusters in and shear modulus G in It can be written as:

[0070]

[0071] Where the subscript m represents the matrix parameter, K m G represents the equivalent bulk modulus of the matrix. m Indicates the shear modulus of the matrix;

[0072] δ r α r β r η r The CNT correlation coefficient is defined as follows:

[0073]

[0074] Where, k r , l r m r n r and p r The Hill modulus of elasticity represents CNTs, while the other two represent the Hill plane strain bulk modulus, transverse modulus, transverse shear modulus, axial modulus, and axial shear modulus normalized to the CNT fiber direction.

[0075] The equivalent bulk modulus K outside the aggregated micro-clusters out and shear modulus G out for:

[0076]

[0077] Therefore, by the Mori-Tanaka method, the equivalent bulk modulus K and shear modulus G of the beam are obtained as follows:

[0078]

[0079] Among them, K out and G outLet α and β represent the equivalent bulk modulus and shear modulus outside the aggregate micro-groups, respectively, and let α and β represent coefficients related to Poisson's ratio, defined as:

[0080]

[0081] Where, ν out This represents the Poisson's ratio of aggregated micro-macromaterials.

[0082] When the CNT direction is randomly distributed, the equivalent elastic modulus E and Poisson's ratio v of the isotropic axial gradient beam are:

[0083]

[0084] Where K represents the equivalent bulk modulus and G represents the equivalent shear modulus.

[0085] Step 2: Design aggregation perturbation parameters and establish vibration control equations.

[0086] Based on the concept of perturbation, the volume fraction μ of aggregated microclusters and the volume fraction η of CNTs within the aggregated microclusters can be selected as perturbation quantities; here, η is used as an example. Let:

[0087] ε=η (9)

[0088] Where ε is a small perturbation quantity, when ε = 0, the agglomeration effect disappears, and CNTs are dispersed in the matrix. Therefore, the elastic modulus and Poisson's ratio of the material can both be expressed as functions of ε, i.e., E(x,ε) and G(x,ε). Correspondingly, according to Timoshenko beam theory, the vibration governing equation of the beam can be written as:

[0089]

[0090] in, Let represent the dimensionless displacement of the neutral layer of the beam along the axial direction (x direction), represent the rotation angle of any point on the neutral layer about the y-axis, and L be the length of the beam. Let I0, I1, and I2 represent the mode shapes of the beam along the z-direction, x-direction, and y-axis, respectively, where ω is the natural frequency. I0, I1, and I2 are coefficients related to the moment of inertia of the cross-section mass, and A... xx B xx C xz D xx A1, A2, and A3 are coefficients related to the cross-sectional stiffness, and are equation simplification coefficients. Each coefficient is defined as follows:

[0091]

[0092] Where Ks represents the cross-sectional shape shear correction factor, h represents the cross-sectional height of the beam, ρ represents the density, G represents the shear modulus, and E represents the elastic modulus.

[0093] Define the state vector:

[0094]

[0095] Substituting into the equation, the governing equation can be written as:

[0096]

[0097] In the formula, is the coefficient matrix in the equation.

[0098] The corresponding boundary conditions are:

[0099] M(ω)η(0,ω)+N(ω)η(1,ω)=0 (14)

[0100] In the formula, M(ω) and N(ω) represent the left end of the beam, respectively. and the right end The boundary condition selection matrix.

[0101] Step 3: Develop the vibration control equations based on the perturbation concept.

[0102] According to the perturbation concept, the governing equation containing the perturbation amount ε can be perturbed into a reference equation and a first-order perturbation equation. When ε = 0, the aggregation effect disappears, and CNTs are dispersed in the matrix, corresponding to the reference equation. When ε > 0, it corresponds to the first-order perturbation equation; as the perturbation amount ε increases, the degree of aggregation intensifies, and ε can directly indicate the degree of aggregation.

[0103] Using the perturbation method, a first-order expansion of the parameters containing the perturbation is given by:

[0104] ω=ω p0 +εω p1 (15)

[0105]

[0106] Where, ω p0 and ω p1 These represent the zeroth and first natural frequencies, respectively. A subscript of 0 represents a zeroth-order parameter, and a subscript of 1 represents a first-order parameter. η represents the state vector, and Φ represents the coefficient matrix of the governing equations.

[0107] And the reference equation is obtained:

[0108]

[0109] M0(ω p0 )η0(0,ω p0 )+N0(ω p0 )η0(1,ω p0 )=0 (21)

[0110] And the first-order perturbation equation:

[0111]

[0112] Step 4: Solve the vibration control equations to obtain the correspondence between aggregation, frequency, and mode shape.

[0113] The zeroth-order perturbation equation is a partial differential equation with definite coefficients. Expressions for the frequency and mode shape, such as the natural frequency ω, can be obtained through the transfer function method or other analytical methods. p0 Calculated using the following formula:

[0114]

[0115] The natural mode shape is:

[0116]

[0117] Where V0 is ω p0 The corresponding eigenvector.

[0118] The coefficient matrix Φ1 and boundary condition matrices M1 and N1 are calculated according to the expansion formulas (16)-(19):

[0119]

[0120] Where, Φ mn M mn N mn Let represent the coefficients in the m-th row and n-th column of the coefficient matrix Φ, the boundary condition selection matrix M, and N, respectively. 10 M 10 N 10 To obtain the result without ω after expanding Φ1, M1, and N1. p1 The corresponding item, Φ 11 M 11 N 11 For containing ω p1 Items.

[0121] Therefore, solving equations (22)-(23) yields...

[0122]

[0123] The coefficients are:

[0124]

[0125] Where P represents a matrix The corresponding eigenvector matrix, where ζ represents any integral variable between 0 and 1, and V0 represents... When established, the matrix The corresponding feature vector, V10 Indicates that it does not contain ω p1 The first-order eigenvector coefficients, V 11 Indicates containing ω p1 The first-order eigenvector coefficients, D0 represents the coefficients without ω. p1 The zeroth-order auxiliary coefficient, D1 represents the coefficient containing ω. p1 The first-order auxiliary coefficient.

[0126] The frequency is:

[0127] ω(ε)=ω p0 +εω p1 (29)

[0128] The corresponding mode shape is:

[0129]

[0130] Step 5: Based on the analytical expressions of the frequencies and mode shapes obtained above, the material parameters of the nanoparticle-filled gradient material are obtained through reverse design.

[0131] Through the above derivation, given the volume fraction μ of a certain aggregate, we obtain the corresponding expression for the frequency ω(ε) and the perturbation ε, i.e., the volume fraction η of CNTs within the aggregate, which can be simplified as follows:

[0132] ω(η)=f1(η) (31)

[0133] Next, the analytical expression derived above will be applied to reverse engineering. In the reverse engineering problem, given the natural frequency ω(η,μ) of the required beam structure, the undetermined parameters are the volume fraction η of CNTs within the aggregated microclusters and the volume fraction μ of a certain aggregated microcluster.

[0134] Case 1: The volume fraction μ of a given agglomerate can be predicted by observation or material properties.

[0135] Once the volume fraction μ of the aggregated micro-clusters is determined, only the volume fraction η of CNTs within the aggregated micro-clusters mentioned above is used as the independent variable. At this time, the value of η can be directly given by equation (31). The required number of operations is 1.

[0136] Traditional methods can only calculate ω given an η. This requires taking multiple points (e.g., 11 points) within the range η = 0-1, calculating the corresponding ω, plotting the corresponding curve, and finally interpolating and reverse-engineering from this curve to obtain the η value corresponding to a given ω. This requires 11 calculations. Its disadvantage is that the curve is derived from differences; if the number of interpolation points is small, the curve accuracy is low; conversely, the computational load is large. For the design of material parameters for gradient-filled nanoparticle materials, it is impossible to balance accuracy and computational cost.

[0137] Case 2: The matrix material is to be determined, therefore its corresponding η and μ are also to be determined.

[0138] First, by changing μ within the range of 0-1 (for example, taking 11 points), the relationship curve obtained from equation (31) is extended to a three-dimensional space containing μ, resulting in 11 corresponding relationship curves between ω and η. Then, the perturbation is changed to μ, and 11 η points are taken to obtain 11 corresponding relationship curves between ω and μ. The two intertwine to form a three-dimensional surface showing the variation of ω(η,μ) with the independent variables η and μ, as shown in the example. Figure 2 As shown in (a) in the figure.

[0139] Then, let the required frequency be ω, where ω is a plane parallel to the η-μ plane. The intersection of this plane with the aforementioned curved surface is the set of (η, μ) that meets the requirements. This allows us to obtain the specific values ​​of the material parameters for the corresponding nanoparticle-filled gradient material. Based on these values, we can then select appropriate matrix materials, reinforcing particle materials, and processing techniques according to the degree of agglomeration.

[0140] Traditional methods can only calculate ω given μ and η, therefore the frequency surface needs to be determined using a point-plotting method, such as... Figure 2 (b) In this case, 11 μ points and 11 η points are selected, and their corresponding ω is calculated one by one, requiring 11 × 11 calculations. Its obvious disadvantage is the dramatic increase in computational complexity.

[0141] In summary, the advantages of this invention are mainly twofold. First, the method proposed in this invention can provide a precise correspondence between η and ω. Compared to the method proposed in this invention, traditional methods require interpolation to obtain ω for most values ​​other than the selected point (e.g., η = 0.15), resulting in lower accuracy and reliability. Second, compared to traditional methods, this invention significantly reduces the number of calculations, improves computational efficiency, and lowers computational costs. It can achieve a balance between accuracy and computational cost for the rapid quantitative design of material parameters for nanocrystals.

[0142] This invention proposes a theoretical analytical method for the vibration characteristics of nanoparticle-filled axial gradient beams considering agglomeration effects. Based on the traditional perturbation method, this method assigns special physical meaning to the perturbation quantity according to the application scenario, enabling it to express the agglomeration effect. Furthermore, through the improvement of this method, the corresponding analytical expressions for frequency and mode shape and agglomeration amount are directly obtained, solving the problem that existing methods can only perform forward calculations and cannot derive design values ​​inversely from frequency. That is, based on the natural frequency of the required nanoparticle beam structure, the corresponding agglomeration amount parameter can be quickly obtained. Based on the agglomeration amount, the nanoparticles can be designed (e.g., selecting suitable matrix materials, reinforcing particle materials, and selecting processing technology according to the degree of agglomeration), effectively generating nanoparticle beams that meet the above requirements. This provides support for the optimization and reverse design of nanoparticle-filled gradient beams. Through this analytical expression, reverse design can be achieved, that is, based on the required frequency, the volume fraction of a specific agglomerated microcluster and the volume fraction of CNTs within the agglomerated microcluster can be calculated, improving calculation accuracy and significantly reducing the computational load.

[0143] The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A method for optimizing the structure of an axially gradient material filled with nanoparticles, characterized in that, include: Based on the aggregation parameters of nanoparticle-filled axial gradient materials, equivalent material parameters are constructed. The perturbation amount is set according to the agglomeration parameter, and the vibration control equation of the axial gradient beam is constructed according to the equivalent material parameter and the perturbation amount; the vibration control equation is then perturbated and expanded. The vibration control equations derived from the perturbation are solved to obtain the mapping relationship between the perturbation quantity and the frequency. The design frequency is obtained, and the design frequency is calculated through the mapping relationship between perturbation and frequency. The corresponding agglomeration parameter data of the nanoparticle-filled axial gradient material is obtained, which is the structural optimization result.

2. The method according to claim 1, characterized in that, The aggregation parameter includes the volume fraction of aggregated microclusters and the volume fraction of nanoparticles within the aggregated microclusters.

3. The method according to claim 1, characterized in that, The equivalent material parameters include the equivalent elastic modulus E and Poisson's ratio v of an isotropic axial gradient beam in a nanoparticle-filled axial gradient material. The process of obtaining the equivalent elastic modulus E and Poisson's ratio v is as follows: Wherein, K represents the equivalent bulk modulus and G represents the equivalent shear modulus, wherein the equivalent bulk modulus and equivalent shear modulus are calculated based on the aggregation parameter.

4. The method according to claim 2, characterized in that, The process of setting the perturbation includes: One or more parameters are selected from the aggregation parameters as perturbation factors, wherein the volume fraction of nanoparticles within the aggregated microclusters is selected as the perturbation factor.

5. The method according to claim 1, characterized in that, The vibration control equation is: in, To represent the state vector, ω is the natural frequency. This represents the dimensionless displacement of the neutral layer of the beam along the axial direction. These represent the mode shapes of the beam along the z-direction, x-direction, and rotation about the y-axis, respectively. There is a mapping relationship between the mode shapes and the natural frequencies. This is the coefficient matrix in the equation; The boundary conditions corresponding to the vibration control equation are: M(ω)η(0,ω)+N(ω)η(1,ω)=0 In the formula, M(ω) and N(ω) represent the left end of the beam, respectively. Time and right end The boundary condition selection matrix at that time.

6. The method according to claim 1, characterized in that, The mapping relationship between perturbation and frequency is as follows: ω(e)=ω p0 +oh p1 Where ω(ε) is the frequency, ω p0 ω is the zeroth natural frequency. p1 ε is the first-order natural frequency, and ε is the perturbation.

7. The method according to claim 6, characterized in that, The zero-order natural frequency ω p0 The calculation is performed using the following formula: Where det[] represents the det function, Φ0 represents the coefficient matrix of the vibration control equation, and M0 and N0 represent the zero-order boundary condition selection matrices for the left and right ends of the beam, respectively. Φ0, M0, and N0 are all related to ω. p0 There is a computational relationship between them. When the above formula is satisfied, the matrix... The eigenvector is V0.

8. The method according to claim 7, characterized in that, The calculation process for the first-order natural frequency is as follows: Where D0 and D1 are the zeroth-order coefficients and the first-order coefficients: Where P represents a matrix The eigenvector matrix is ​​denoted by ζ, which represents any integration variable between 0 and 1, and M1 and N1 represent the first-order boundary condition selection matrices for the left and right ends of the beam, respectively. 10 M 10 N 10 The result of expanding Φ1, M1, and N1 without ω p1 Φ 11 M 11 N 11 To expand Φ1, M1, and N1 to include ω p1 Items.

9. The method according to claim 1, characterized in that, The process of obtaining aggregation parameter data includes: The volume fraction of nanoparticles within the aggregated microclusters was selected as the perturbation factor. When the volume fraction of aggregated microclusters is obtained through measurement, the design frequency is directly calculated using the mapping relationship between perturbation and frequency, thus obtaining the volume fraction data of nanoparticles within the aggregated microclusters. When the volume fraction of aggregated microclusters cannot be obtained by measurement, different sampling points are set for the volume fraction of aggregated microclusters and the volume fraction of nanoparticles within the aggregated microclusters. Based on the numerical relationship surface between the sampling point fitting frequency and the aggregation amount parameter, the aggregation amount parameter corresponding to the design frequency is obtained from the relationship surface, and the aggregation amount parameter data set is obtained.

10. A nanoparticle-filled axial gradient material structure optimization system, characterized in that, Used to perform the method according to any one of claims 1-9.

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