Novel modal shape function construction method suitable for variable cross-section flexible structure

Abstract

The invention discloses a novel modal shape function construction method suitable for a variable cross-section flexible structure, and the method comprises the following steps: firstly, building a linear kinetic equation of a flexible wing based on an Euler-Bernoulli beam theory and a Lagrange principle; secondly, geometric and physical parameters in the equation are normalized through dimensionless variable substitution, and the coupling relation between different variables is simplified; secondly, formalizing the solution into a linear combination of a first type of Bessel function and a second type of Bessel function by adopting a special function expansion method so as to ensure that boundary conditions are met and effectively reflect the influence of section change on modal characteristics; and finally, solving the inherent frequency and a modal shape function by constructing a novel modal shape function. According to the method, iteration and approximate truncation are not needed, the inherent frequency and the modal shape can be rapidly and accurately calculated, the calculation efficiency is remarkably improved, high calculation precision is guaranteed, and the method is suitable for efficient dynamic analysis of a complex structure.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of aerospace and structural dynamics, and particularly relates to a novel modal shape function construction method suitable for a variable cross-section flexible structure. BACKGROUND

[0002] In the past few decades, variable cross-section structures have been widely used in the fields of aerospace, machinery and bridges due to their significant advantages in load-carrying capacity, stiffness distribution and mass optimization. It is worth noting that the change of cross-sectional geometric parameters along the length direction of the structure not only changes the overall mechanical and dynamic characteristics, but also causes the modal shape function to present essential differences compared with the equal cross-section structure. This difference makes the spatial distribution of the structure modal function more complex, and it is difficult to accurately obtain the structure modal function by using the traditional analytical method; in the case of needing to ensure high precision and good convergence, it undoubtedly puts forward new challenges to the theoretical modeling and analytical method. SUMMARY

[0003] In view of the above defects of the prior art, the technical problem to be solved by the present application is that the existing modal shape function construction method often ignores the accurate influence of cross-section change on the modal shape and natural frequency when dealing with variable cross-section flexible structures; although the traditional method can solve some basic problems, it has the problems of insufficient calculation accuracy and low efficiency when dealing with complex geometric shapes and variable boundary conditions. The present application provides a novel modal shape function construction method suitable for a variable cross-section flexible structure, which improves the calculation accuracy and greatly improves the calculation efficiency by introducing the linear combination of the first and second Bessel functions, can accurately reflect the modal characteristics of the structure, provides reliable support for the dynamic response analysis and optimization design of the structure, and further improves the stability and performance of the flexible structure in practical application.

[0004] To achieve the above-mentioned purpose, the present application provides a novel modal shape function construction method suitable for a variable cross-section flexible structure, comprising the following steps:

[0005] Step 1: According to the geometric characteristics of the flexible wing, a dynamic model of the variable cross-section flexible structure is established based on the Euler-Bernoulli beam theory and the Lagrange principle;

[0006] Step 2: The geometric and physical parameters in the dynamic model are normalized by dimensionless variable substitution to remove the unit contrast, and a standardized equation is obtained;

[0007] Step 3: According to the standardized equation, a special function expansion method is used to express the modal function as a linear combination of Bessel functions, which describes the modal shape of the flexible structure under different conditions;

[0008] Step 4, by solving the boundary conditions, a new modal shape function suitable for variable cross-section flexible structure is constructed, which provides reliable modal data for the vibration characteristics and dynamic response analysis of flexible structure. By solving the boundary conditions, the undetermined coefficients of the newly constructed modal shape function of variable cross-section flexible structure based on Bessel function are solved , which provides reliable modal data for the vibration characteristics and dynamic response analysis of flexible structure.

[0009] Further, step 1, according to the geometric characteristics of flexible wing, based on Euler-Bernoulli beam theory and Lagrange principle, the dynamic model of variable cross-section flexible structure is established, which specifically includes defining the cross-section variation coefficient along the x-axis direction of flexible wing according to its geometric characteristics, then calculating the width distribution of a certain cross-section along the x-axis direction, then calculating the cross-section area of a certain cross-section along the x-axis direction of flexible wing according to the thickness distribution function of NACA0012 airfoil , half wing span, wing root chord length, wing tip chord length and 1 / 4 back rake angle and cross-section moment of inertia ; finally, according to Euler-Bernoulli beam theory and Galerkin method, the dynamic model of variable cross-section flexible wing is obtained.

[0010] Further, according to the dynamic model of variable cross-section flexible wing, the i-th order displacement of flexible wing is calculated, and the variable separation is obtained, then the dimensionless variable is substituted into, according to the variable separation and dimensionless variable substitution dynamic equation, considering the cantilever beam boundary condition, finally the standard modal function is obtained.

[0011] Further, the variable separation is to separate the spatial variable and time variable in the dynamic equation, and the spatial-time coupling equation suitable for variable cross-section flexible wing is obtained.

[0012] Further, by variable substitution, the dynamic equation is transformed into the standard form of Bessel equation, so that the modal function is expressed as the linear combination of Bessel function, which can describe the modal shape of flexible structure under different conditions.

[0013] Further, the modal shape of variable cross-section flexible structure is expressed as the linear combination of first and second type Bessel function.

[0014] Further, in the process of constructing modal shape, the boundary conditions are fully considered, and through the processing of boundary conditions, the boundary behavior of the mode is limited, so as to obtain the modal solution conforming to the actual situation.

[0015] Further, based on the constructed modal shape function, the is used to calculate the natural frequency of variable cross-section flexible structure, which provides basic data for subsequent dynamic response analysis.

[0016] Further, the modal characteristics of the modal shape include the natural frequency and the shape function under different modes.

[0017] Technical effects

[0018] The application provides a modal shape function construction method suitable for a variable cross-section flexible structure. The method can accurately describe the modal characteristics of the flexible structure under different states by introducing a linear combination of the first and second Bessel functions. Through the method, the natural frequency and the modal shape of the flexible structure can be accurately calculated, and the calculation accuracy and efficiency are significantly improved. When processing the variable cross-section structure, the method can effectively consider the influence of the cross-section change on the modal characteristics, and does not need to rely on complex iterative calculation and approximate truncation. In this way, reliable support can be provided for the dynamic response analysis and optimization design of the structure, and the stability and performance of the structure in actual application are improved. The method not only improves the calculation efficiency, but also provides a new solution for the dynamic analysis of complex structures, and can adapt to more complex structures and boundary conditions.

[0019] The concept, specific structure and generated technical effects of the application will be further described below with reference to the drawings, so as to fully understand the purpose, features and effects of the application. BRIEF DESCRIPTION OF DRAWINGS

[0020] Figure 1 is a flexible wing schematic diagram of a modal shape function construction method suitable for a variable cross-section flexible structure according to a preferred embodiment of the application;

[0021] Figure 2 is a comparison diagram of the first three modal shapes of ANSYS, AM-STD and AM-BES when pb=0.6, 0.7 and 0.8, which is a modal shape function construction method suitable for a variable cross-section flexible structure according to a preferred embodiment of the application;

[0022] Figure 3 is a modal guarantee criterion value diagram between the two groups of modal shapes of ANSYS and AM-BES, which is a modal shape function construction method suitable for a variable cross-section flexible structure according to a preferred embodiment of the application;

[0023] Figure 4 is a flowchart of a modal shape function construction method suitable for a variable cross-section flexible structure according to a preferred embodiment of the application. DETAILED DESCRIPTION

[0024] In order to make the technical problems, technical solutions and beneficial effects of the application clearer and more apparent, the application will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the application, and are not used to limit the application.

[0025] In the following description, for purposes of explanation and not limitation, specific details are set forth such as particular procedures, techniques, etc. in order to provide a thorough understanding of the embodiments of the present application. However, it will be apparent to those skilled in the art that the present application can be practiced in other embodiments that depart from these specific details. In other instances, detailed descriptions of well-known systems, devices, circuits, and methods are omitted so as not to obscure the description of the present application with unnecessary detail.

[0026] As shown in Figure 4 , the present application provides a new modal shape function construction method suitable for variable cross-section flexible structure, including the following steps:

[0027] Step 1, according to the geometric characteristics of flexible wing, based on Euler-Bernoulli beam theory and Lagrange principle, the dynamic model of variable cross-section flexible structure is established; specifically including according to the geometric characteristics of flexible wing, the cross-section variation coefficient along the x-axis direction is defined as , then the cross-section area along the x-axis direction is calculated, then according to the thickness distribution function of NACA0012 airfoil , half wing span, wing root chord length, wing tip chord length and 1 / 4 back rake angle, the cross-section area of flexible wing along the x-axis direction is calculated and cross-section moment of inertia ; finally, according to Euler-Bernoulli beam theory and Galerkin method, the dynamic model of variable cross-section flexible wing is obtained.

[0028] Step 2, the geometric and physical parameters in the dynamic model are normalized by dimensionless variable substitution, and the unit contrast is removed to obtain the standardized equation form; according to the dynamic model of variable cross-section flexible wing, the i-th order displacement of flexible wing is calculated to obtain variable separation, then the dimensionless variable is substituted, according to the variable separation and dimensionless change substitution dynamic equation, considering the cantilever beam boundary condition, finally the standard modal function is obtained. Variable separation is to separate the spatial variable and time variable in the dynamic equation to obtain a spatial-time coupled equation suitable for variable cross-section flexible wing. In the present application, the geometric and physical parameters in the dynamic equation are normalized by dimensionless variable substitution, which simplifies the coupling relationship in the equation and obtains the standardized equation form, thereby reducing the calculation complexity and improving the solving efficiency.

[0029] Step 3, according to the standardized equation, the modal function is expressed as a linear combination of Bessel functions by using special function expansion method, which describes the modal shape of flexible structure under different conditions;

[0030] Step 4, the undetermined coefficients of the modal shape function of the new variable cross-section flexible structure based on Bessel function are solved by solving the boundary conditions and standardization equation , and reliable modal data are provided for the vibration characteristics and dynamic response analysis of the flexible structure. In the construction process of the modal shape, the boundary conditions are fully considered, and the boundary behavior of the mode is limited through the processing of the boundary conditions, so that the modal solution conforming to the actual situation is obtained. Based on the constructed modal shape function, the natural frequency of the variable cross-section flexible structure is calculated, which provides basic data for subsequent dynamic response analysis. The modal characteristics of the modal shape include the natural frequency and the mode shape function under different modes, which can provide detailed data for the dynamic response analysis of the flexible structure.

[0031] Based on the obtained modal shape function, it can be widely applied to the dynamic analysis of variable cross-section structures in the fields of aerospace, bridge and machinery, and provide theoretical support for efficient structure design and optimization.

[0032] The specific steps of a new modal shape function construction method suitable for variable cross-section flexible structure of the application will be specifically explained below.

[0033] As shown in Figure 1 , it is a variable cross-section flexible wing, wherein the geometric parameters are: the wing length, the wing root chord length, the wing tip chord length, and the 1 / 4 back rake angle is 0°. The method comprises the following steps:

[0034] Step 101: first, according to the geometric characteristics of the flexible wing, the cross-section variation coefficient of the flexible wing along the x-axis direction is defined as:

[0035]

[0036] According to the definition, the cross-section width distribution along the z-axis direction can be expressed as:

[0037]

[0038] The cross-section area of the flexible wing along the x-axis direction and the cross-section moment of inertia can be expressed as:

[0039]

[0040]

[0041] The thickness distribution function of the standard NACA airfoil is defined as:

[0042] ​​

[0043] where, is the ratio of maximum thickness to chord length at the airfoil.

[0044] Step 102: According to the characteristics of the flexible wing structure, the Euler-Bernoulli beam theory and the Galerkin method, the dynamic model of the variable cross-section flexible wing is obtained:

[0045]

[0046] where, is the mass matrix, is the Young's modulus, is the linear damping term.

[0047] Let the i-th order displacement of the flexible wing be expressed as:

[0048]

[0049] where, is the undetermined modal space distribution function, is a time function. Subsequently, the dimensionless variable is substituted to achieve consistency with the corresponding physical scale transformation:

[0050]

[0051] According to the above variable separation and dimensionless transformation, substitute into the dynamic equation to obtain:

[0052]

[0053] Considering the cantilever beam boundary conditions, we can obtain:

[0054]

[0055] It is well known that for a homogeneous cantilever beam with constant cross-section, the modal function expression is as follows:

[0056]

[0057] This set of basis functions is called the standard modal function, and the corresponding analytical method is simply referred to as AM-STD.

[0058] According to the transformed dynamic equation, and ignoring the damping part, by performing variable separation in the spatial domain and the time domain, we can obtain:

[0059]

[0060]

[0061] Step 103: To obtain the exact analytical solution of the spatial domain after variable separation, the modal function is expressed as a linear combination of the first and second class Bessel functions in this study. Both of them have good asymptotic properties and orthogonality, so they are very suitable for characterizing complex modal shapes, which are as follows:

[0062]

[0063] Here, The subscript i (i = 1, 2, 3) represents the modal order.

[0064] The spatial domain dynamics equation is transformed into the equation expressed in Bessel functions:

[0065]

[0066] where, .

[0067] Finally, the following four base functions are obtained, which are all in the form of the first and second class Bessel functions:

[0068]

[0069] The first class Bessel function, the first class modified Bessel function, the second class Bessel function, and the second class modified Bessel function are defined as follows:

[0070]

[0071]

[0072]

[0073]

[0074] Step 2: To solve the unknown coefficients and , the corresponding coefficient matrix is constructed according to the dimensionless boundary conditions, and the solution is obtained accordingly:

[0075]

[0076] To obtain the non-zero solution of the above equation, the determinant of the corresponding coefficient matrix must be set to 0, from which the unknown parameters can be determined. By fixing any one coefficient in as a non-zero real number, the remaining three coefficients can be determined accordingly, so the modal shape function of the flexible structure is constructed by the linear combination of the first and second class Bessel functions, and this analytical method is simply called AM-BES. At the same time, to facilitate subsequent calculations, the four constants The normalized condition is determined as follows:

[0077]

[0078] The calculation formula of the dimensionless natural frequency of the flexible structure under different section influence factors is as follows:

[0079]

[0080] wherein, is a dimensionless mass matrix, is a dimensionless stiffness matrix.

[0081] By applying the formula the dimensionless natural frequency obtained from the above formula can be converted into the theoretical value of the actual natural frequency.

[0082] The following gives an example of using the above new modal shape function construction method.

[0083] The geometric size of the flexible wing model selected in the embodiment is as follows: the wing half-span length is 500 mm, the wing root chord length is 80 mm, the wing tip chord length changes with the section influence factor , and the 1 / 4 back rake angle is 0°. The material parameters of the flexible wing are as follows: the density =1130 kg / m 3 , the Young's modulus Yw= Pa, and the Poisson's ratio .

[0084] According to the natural frequency related formula, the present application takes p b =0.6, 0.7, 0.8 as examples, and table 1 gives the natural frequencies of the flexible wing calculated by the finite element (FE) method, the AM-STD method and the AM-BES method. The results show that the results obtained by the method proposed in the present application are highly consistent with the finite element method, and the natural frequency deviation is less than 0.5%.

[0085] Table 1:

[0086]

[0087] Note:

[0088]

[0089] By normalizing the length and amplitude of the flexible wing and comparing with the first three order modal shapes of ANSYS and AM-STD respectively, as shown in the following table, it can be observed that, with the increase of the section influence factor Figure 2 p b ​As the amplitude increases, the difference between the mode shape function obtained by the AM-STD method using the standard mode shape function and the result obtained by this method becomes increasingly significant. This phenomenon is... p b The increasing trend of the calculated natural frequency difference provides a more intuitive physical explanation. Furthermore, the good coincidence between AM-BES and AM0STD demonstrates the reliability of their models. Subsequently, the modal guarantee criterion values ​​between the two sets of modal modes of ANSYS and AM-BES were further calculated and evaluated, with results as follows: Figure 3 As shown. With the influence factor of the cross section p b As the cross-sectional influence factor gradually increases, the values ​​on the main diagonal remain close to 1. This indicates that the proposed AM-BES method is highly consistent with the finite element method, thus verifying that the method maintains a strong and robust predictive ability for the main modal characteristics. p b The increasing values ​​of off-diagonal elements indicate a gradual upward trend, suggesting modal coupling to some extent. This reflects a weakening of the theoretical model's ability to achieve modal decoupling under significant cross-sectional changes. Nevertheless, the model maintains high robustness across the entire parameter space. In summary, the proposed AM-BES method exhibits high consistency with the finite element method, demonstrating its strong and robust predictive ability for key modal characteristics.

[0090] To more comprehensively verify the theoretical accuracy of the AM-BES method, this invention compares the results obtained using existing methods with relevant research results in the literature to fully evaluate its reliability. The comparison results are shown in Table 2. As can be seen from Table 2, the maximum relative error between the AM-BES method proposed in this study and the results reported in the literature is only 0.028%, which fully demonstrates the high consistency between the two. Furthermore, it can be observed from the table that the AM-BES method significantly outperforms the AM-STD method in the calculation of higher-order modes of beams with variable cross-sections, thus making it more suitable for the dynamic analysis of complex structural geometries.

[0091] Table 2:

[0092]

[0093] Note:

[0094]

[0095]

[0096] To further verify the advantage of the proposed theoretical method in calculation efficiency, all models and programs are constructed by ANSYS and MATLAB platforms under the same calculation hardware configuration, and then the finite element (FE) method, the method in the related literature and the AM-BES method are compared in detail, and the performance of the two methods in calculation efficiency is analyzed in depth, and the specific comparison results are shown in Tables 3, 4 and 5. K

[0097] Table 3

[0098]

[0099] Table 4

[0100]

[0101] Table 5

[0102]

[0103] The embodiment fully verifies the effectiveness of the method in the modal analysis of the variable cross-section flexible structure. By introducing high-precision modal shape functions and efficient calculation methods, the design parameters are systematically optimized, and the modal characteristics solution is better than that of the traditional method, which provides a scientific and accurate solution for the dynamic analysis and optimization design of complex structures.

[0104] In summary, the application verifies the effectiveness of the application in solving the problems of dynamic analysis and design optimization of complex structures through the modal analysis of the variable cross-section flexible wing, and has important engineering application value and broad popularization prospect.

[0105] The preferred embodiments of the application are described in detail above. It should be understood that those skilled in the art can make many modifications and changes without creative labor based on the concept of the application. Therefore, any technical solution obtained by logical analysis, reasoning or limited experiment based on the existing technology according to the concept of the application shall be within the protection scope determined by the claims.​

Claims

1. A novel method for constructing modal mode functions applicable to flexible structures with variable cross-sections, characterized in that, Includes the following steps: Step 1: Based on the geometric characteristics of the flexible wing, and using the Euler-Bernoulli beam theory and Lagrange's principle, establish a dynamic model of the variable cross-section flexible structure. Step 2: Normalize the geometric and physical parameters in the dynamic model by dimensionless variable substitution, remove unit comparison, and obtain standardized equations; Step 3: Based on the standardized equations, a special function expansion method is used to express the modal functions as a linear combination of Bessel functions, describing the mode shapes of the flexible structure under different conditions; Step 4: By solving the boundary conditions and standardized equations, the undetermined coefficients are obtained for the modal vibration functions of the novel variable cross-section flexible structure based on the Bessel function. This provides reliable modal data for the vibration characteristics and dynamic response analysis of flexible structures.

2. The novel modal function construction method applicable to flexible structures with variable cross-sections as described in claim 1, characterized in that, Step 1: Based on the geometric characteristics of the flexible wing, and using the Euler-Bernoulli beam theory and Lagrange's principle, establish a dynamic model of the variable cross-section flexible structure. Specifically, this includes defining the coefficient of cross-sectional variation along the x-axis based on the geometric characteristics of the flexible wing, then calculating the thickness distribution of a certain cross-section along the z-axis, followed by calculating the area and moment of inertia of a certain cross-section along the x-axis of the flexible wing; finally, based on the half-wing span, wing section, root chord length, tip chord length, 1 / 4 sweep angle, Euler-Bernoulli beam theory, and Galerkin method, obtain the dynamic model of the variable cross-section flexible wing.

3. The novel modal function construction method applicable to flexible structures with variable cross-sections as described in claim 1, characterized in that, Based on the dynamic model of the variable cross-section flexible airfoil, the i-th order displacement of the flexible airfoil is calculated to obtain variable separation. Then, the dimensionless variables are substituted into the dynamic equation based on the variable separation and dimensionless changes. At the same time, the cantilever beam boundary conditions are considered to finally obtain the standard modal function.

4. The novel modal function construction method for flexible structures with variable cross-sections as described in claim 3, characterized in that, Variable separation specifically involves separating the spatial and temporal variables in the dynamic equations to obtain a space-time coupled equation applicable to flexible wings with variable cross-sections.

5. The novel modal function construction method applicable to flexible structures with variable cross-sections as described in claim 1, characterized in that, By substituting variables, the dynamic equations are transformed into the standard form of the Bessel equations, thereby representing the modal functions as a linear combination of Bessel functions, which describes the mode shapes of the flexible structure under different conditions.

6. The novel modal function construction method applicable to flexible structures with variable cross-sections as described in claim 5, characterized in that, The modal vibration modes of the variable cross-section flexible structure are represented as a linear combination of the first and second type Bessel functions.

7. The novel modal function construction method for flexible structures with variable cross-sections as described in claim 5, characterized in that, In the process of constructing mode shapes, boundary conditions are fully considered. By processing the boundary conditions, the boundary behavior of the mode shapes is limited, thereby obtaining a mode solution that conforms to the actual situation.

8. The novel modal function construction method for flexible structures with variable cross-sections as described in claim 7, characterized in that, Based on the constructed modal function, using The natural frequencies of the variable cross-section flexible structure were calculated, providing fundamental data for subsequent dynamic response analysis.

9. A novel modal function construction method for flexible structures with variable cross-sections as described in claim 5, characterized in that, The modal characteristics of a mode shape include its natural frequencies and mode shape functions for different modes.

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