Inherent frequency calculation method of porous two-dimensional functionally graded material framework

Abstract

The invention discloses a method for calculating the inherent frequency of a porous two-dimensional functionally graded material framework, and belongs to the technical field of structural design, the framework is simplified into a beam model with two clamped ends, a motion control equation of a porous two-dimensional functionally graded material beam in a humid and hot environment is deduced through a Hamilton principle based on a Timoshenko beam theory, and the inherent frequency of the porous two-dimensional functionally graded material beam is calculated. Simplifying the equation by using a variable separation method; the inherent frequency of the beam is solved by adopting a differential quadrature method; a motion control equation is discretized by utilizing a differential quadrature method discretization principle, then the equation is converted into a matrix form, the problem of solving the motion equation is converted into the problem of solving a matrix eigenvalue by adopting a matrix transformation method, and finally solving is carried out by writing a corresponding program. The corresponding solving program is clear in logic and high in uniformity, the calculation efficiency and precision are greatly improved, and a theoretical reference can be provided for the design of the functional gradient skeleton in the later period.

Description

Technical Field

[0001] The invention belongs to the technical field of structural design, and in particular relates to a method for calculating the natural frequency of a porous two-dimensional functional gradient material skeleton. Background Art

[0002] The interior of an aircraft wing contains a skeleton-like structure, which often bears significant structural loads. Furthermore, as an oil reservoir, the wing's internal components must be corrosion-resistant. In traditional laminated composite structures, uniform elastic laminate beams are bonded together to achieve improved mechanical and thermal performance. The primary drawback of this assembly approach is stress concentration along the interface. Functionally graded materials (FGMs), typically composed of ceramic and metal composites, combine the mechanical properties of metal with the corrosion resistance of ceramic. Their properties vary continuously along a specific direction (or multiple directions), eliminating this stress concentration issue and making them suitable as raw materials for the skeleton. Furthermore, porous materials offer high specific strength, high specific stiffness, and low weight compared to traditional materials. These properties can be exploited to further reduce the overall weight of the structure, achieving lightweight aircraft and reaping economic benefits.

[0003] In addition, the skeleton will be affected by temperature and humidity during the assembly process. On the one hand, when the skeleton is fixed with tooling, the skeleton will generate initial axial force due to changes in temperature and humidity, which will naturally produce additional force on the bolts that fix the skeleton. If the influence of this part of the force is not considered, the stress inside the skeleton during the assembly process may be underestimated, resulting in unreasonable tooling design; on the other hand, the skeleton has fixed requirements for temperature and humidity during the assembly process. Under different temperatures and humidities, the elastic modulus and thermal expansion coefficient of the material are also different. If the temperature correlation of the material is not considered, it will affect the calculation results of the skeleton's natural frequency. The natural frequency is an important parameter for structural safety performance. Incorrect prediction will affect the design of the structure. Therefore, research on the mechanical properties of the skeleton in a hot and humid environment has very positive practical significance.

[0004] The skeleton in a wing can be viewed as a beam model. Understanding the linear mechanical behavior of functionally graded beam structures is crucial for supporting their analysis, design, and manufacturing. The natural frequencies can be determined by solving the structural equations of motion. However, because the physical properties of functionally graded materials are functions of spatial coordinates, the resulting equations of motion are variable-coefficient partial differential equations, making analytical solutions difficult to obtain.

[0005] Currently, commercial finite element software is commonly used to calculate the natural frequencies of structures. However, the modeling and meshing process is time-consuming and computationally slow. Therefore, it is imperative to utilize numerical methods to efficiently and accurately obtain numerical solutions for the natural frequencies of functionally graded structures. Summary of the Invention

[0006] To solve the above technical problems, the present invention provides a method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton. The corresponding solution program has clear logic and high uniformity, which greatly improves the calculation efficiency and accuracy, and can provide a theoretical reference for the design of functionally gradient skeletons in the future.

[0007] The object of the present invention is achieved through the following technical solutions: A method for calculating the natural frequency of a porous two-dimensional functional gradient material skeleton is proposed. The skeleton is simplified into a beam model with two ends fixed. Based on Timoshenko beam theory and Hamilton's principle, the motion control equations of the porous two-dimensional functional gradient material beam in a humid and hot environment are derived, and the equations are simplified using the separation of variables method. The natural frequency of the beam is solved using the differential quadrature method. The motion control equations are discretized using the discretization principle of the differential quadrature method, and then the equations are converted into matrix form. The problem of solving the motion equations is transformed into the problem of solving the matrix eigenvalues ​​using the matrix transformation method, and finally the solution is achieved by writing a corresponding program.

[0008] The following steps are involved: Step 1: Based on Timoshenko beam theory, the motion control equation of the porous two-dimensional functional gradient beam is established by Hamilton principle; Step 2: Use the separation of variables method to process the equation; Step 3: Discretize the motion equation using the differential quadrature method; Step 4: Convert the motion control equation into matrix form; Step 5: Write the corresponding program to solve the problem.

[0009] Preferably, in step 1, the motion control equation of the porous two-dimensional functional gradient beam is: (1); (2); (3); in, is the axial displacement of any point in the beam; is the axial displacement of any point in the beam; is the density of ceramic / metal; is the elastic modulus of ceramic / metal; is the thermal / humidity expansion coefficient; is the thermal / moisture axial force; is the natural frequency of the beam; is the shear correction factor; is the rotation angle of the cross section about the vertical direction, all is the inertia term.

[0010] Preferably, in the step 2: (4); in, is the natural frequency of the beam; is the rotation angle of the cross section about the vertical direction, x is the spatial coordinate, and t is the time variable.

[0011] Preferably, in step three: (5); (6); (7); in: is the weight coefficient in the differential quadrature method; Preferably, the cross section of the beam in step one is rectangular.

[0012] Preferably, the motion control equation established in step 1 is based on classical continuum theory.

[0013] Preferably, the beam is made of a composite of ceramic and metal, and the physical parameters of the beam vary continuously along the axial direction and thickness direction of the beam, and are expressed as a function of coordinates x and z: (8); (9); (10); (11); At the same time, the physical properties of the material change with temperature: (12); (13); (14); (15); in, is the axial displacement of any point in the beam; is the axial displacement of any point in the beam; is the density of ceramic / metal; is the elastic modulus of ceramic / metal; is the thermal / humidity expansion coefficient; is the thermal / moisture axial force; is the natural frequency of the beam; （i =−1, 0, 1, 2, 3) is the material temperature correlation coefficient; For ceramic / metal; is the functional gradient index; is the porosity; is the rotation angle of the cross section about the vertical direction; is the shear correction factor; Temperature / humidity varies uniformly across the thickness of the beam: (16); (17); in, T 0=300 K and C 0=0% indicates the initial temperature and humidity.

[0014] Preferably, the stiffness components and inertia terms in equations (1) to (3) are: (18); (19); (20).

[0015] Preferably, when the differential quadrature method is used to discretize the continuous system in step 2, the nodes are selected as follows: (twenty one); Where N is the total number of discrete nodes and L is the length of the beam.

[0016] Preferably, in step 4, the motion control equation is converted into a matrix form: (twenty two); in, K 1 、M are the stiffness matrix and mass matrix, X is the displacement component.

[0017] Preferably, mathematical processing is performed on formula (22) (the specific processing process is as follows Figure 2 ): First, the matrix form of the motion control equation is standardized, then the motion control equation is converted into a block matrix form, and finally matrix operations are performed to obtain Equation (23). In this way, the problem of solving the motion equation is transformed into the problem of solving the matrix eigenvalue (the minimum eigenvalue of the matrix corresponds to the natural frequency of the beam): ;matrix S The eigenvalue of is the natural frequency of the beam.

[0018] The beneficial effects of this technical solution are as follows: 1. The present invention provides a method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton. The corresponding solution program has clear logic and high uniformity (applicable to beams of different cross-sections and different functionally gradient material distributions). It greatly improves the calculation efficiency (compared with the finite element method, it does not require modeling and meshing, and the calculation time is only 20% of the finite element method) and accuracy (the error is within 1%). Later, in-depth research can be conducted on the influence of various parameters on the dynamic characteristics of the beam.

[0019] 2. The present invention provides a method for calculating the natural frequency of a porous two-dimensional functional gradient material skeleton. The calculation model has universal applicability: the previous calculation model was for a simple unidirectional functional gradient material beam and did not consider the effect of voids on the dynamic characteristics of the beam. This method can calculate the natural frequency of more complex porous two-dimensional functional gradient material beams. Significantly improve the calculation efficiency: This method requires matrix transformation of the stiffness matrix before calculation (see Figure 2 ), which significantly shortens calculation time while maintaining accuracy (the program execution time corresponding to this method is only 3.64% of the traditional method of directly solving the matrix determinant). The solution program based on this principle is highly unified: the natural frequency of the beam under different working conditions can be calculated by directly modifying the program parameters. The program is simple, easy to understand, and convenient to operate, allowing for further research on the impact of various parameters on the dynamic characteristics of the beam. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 It is a porous two-dimensional functionally graded material beam model; Figure 2 The specific process of processing the motion control equations; Figure 3 Take screenshots of some programs; Figure 4 Comparison of the natural frequencies of beams calculated using this method and the finite element method. DETAILED DESCRIPTION

[0021] The present invention will be further described in detail below with reference to the examples, but the embodiments of the present invention are not limited thereto.

[0022] Example 1 The present invention proposes a numerical calculation method for the natural frequency of a porous two-dimensional functional gradient material skeleton in a hot and humid environment, wherein the skeleton is considered to be in the form of a clamped support at both ends. The skeleton is composed of a composite of ceramic (Al2O3) and metal (SUS304), and the physical properties of the ceramic and metal vary continuously along the thickness and axial direction of the skeleton (e.g. Figure 1 As shown in Figure 3 ), considering the temperature dependence of the material physical properties, the temperature and humidity change uniformly along the thickness direction of the skeleton, and a numerical calculation model of the natural frequency of the porous two-dimensional functional gradient material skeleton is established.

[0023] To solve for the skeleton frequency, the skeleton is simplified into a beam model with two clamped ends. Based on Timoshenko beam theory and Hamilton's principle, the governing equations for the motion of a porous two-dimensional functionally graded material beam in a humid and hot environment are derived and simplified using the separation of variables method. The beam's natural frequency is solved using the differential quadrature method. The governing equations are discretized using the discretization principle of the differential quadrature method and then converted into a matrix form. Using matrix transformation, the problem of solving the equation of motion is transformed into the problem of solving the matrix eigenvalues. Finally, the solution is achieved by writing a corresponding program.

[0024] Example 2 The present invention proposes a numerical calculation method for the natural frequency of a porous two-dimensional functional gradient material skeleton in a hot and humid environment, wherein the skeleton is considered to be in the form of a clamped support at both ends. The skeleton is composed of a composite of ceramic (Al2O3) and metal (SUS304), and the physical properties of the ceramic and metal vary continuously along the thickness and axial direction of the skeleton (e.g. Figure 1 As shown in Figure 3 ), considering the temperature dependence of the material physical properties, the temperature and humidity change uniformly along the thickness direction of the skeleton, and a numerical calculation model of the natural frequency of the porous two-dimensional functional gradient material skeleton is established.

[0025] To solve for the skeleton frequency, the skeleton is simplified into a beam model with two clamped ends. Based on Timoshenko beam theory and Hamilton's principle, the governing equations for the motion of a porous two-dimensional functionally graded material beam in a humid and hot environment are derived and simplified using the separation of variables method. The beam's natural frequency is solved using the differential quadrature method. The governing equations are discretized using the discretization principle of the differential quadrature method and then converted into a matrix form. Using matrix transformation, the problem of solving the equation of motion is transformed into the problem of solving the matrix eigenvalues. Finally, the solution is achieved by writing a corresponding program.

[0026] The following steps are involved: Step 1: Based on Timoshenko beam theory, the motion control equation of the porous two-dimensional functional gradient beam is established by Hamilton principle; Step 2: Use the separation of variables method to process the equation; Step 3: Discretize the motion equation using the differential quadrature method; Step 4: Convert the motion control equation into matrix form; Step 5: Write the corresponding program to solve the problem.

[0027] Wherein, in step 1, the motion control equation of the porous two-dimensional functional gradient beam is: (1); (2); (3); in, is the axial displacement of any point in the beam; is the axial displacement of any point in the beam; is the density of ceramic / metal; is the elastic modulus of ceramic / metal; is the thermal / humidity expansion coefficient; is the thermal / moisture axial force; is the natural frequency of the beam; is the shear correction factor; is the rotation angle of the cross section about the vertical direction, all is the inertia term.

[0028] Wherein, in the step 2: (4); in, is the natural frequency of the beam; is the rotation angle of the cross section about the vertical direction, x is the spatial coordinate, and t is the time variable.

[0029] Wherein, in said step three: (5); (6); (7); in: is the weight coefficient in the differential quadrature method; Wherein, the cross section of the beam in step 1 is rectangular.

[0030] The motion control equation established in step 1 is based on classical continuum theory.

[0031] The beam is made of a composite of ceramic and metal. The physical parameters of the beam vary continuously along the axial direction and thickness direction of the beam, and are expressed as functions of coordinates x and z: (8); (9); (10); (11); At the same time, the physical properties of the material change with temperature: (12); (13); (14); (15); in, is the axial displacement of any point in the beam; is the axial displacement of any point in the beam; is the density of ceramic / metal; is the elastic modulus of ceramic / metal; is the thermal / humidity expansion coefficient; is the thermal / moisture axial force; is the natural frequency of the beam; （i =−1, 0, 1, 2, 3) is the material temperature correlation coefficient; For ceramic / metal; is the functional gradient index; is the porosity; is the rotation angle of the cross section about the vertical direction; is the shear correction factor; Temperature / humidity varies uniformly across the thickness of the beam: (16); (17); in, T 0=300 K and C 0=0% indicates the initial temperature and humidity.

[0032] Among them, the stiffness components and inertia terms in equations (1) to (3) are: (18); (19); (20).

[0033] Preferably, when the differential quadrature method is used to discretize the continuous system in step 2, the nodes are selected as follows: (twenty one); Where N is the total number of discrete nodes and L is the length of the beam.

[0034] Preferably, in step 4, the motion control equation is converted into a matrix form: (twenty two); in, K 1 、M are the stiffness matrix and mass matrix, X is the displacement component.

[0035] Perform mathematical processing on Equation (22) (the specific processing process is as follows Figure 2): First, the matrix form of the motion control equation is standardized, then the motion control equation is converted into a block matrix form, and finally matrix operations are performed to obtain Equation (23). In this way, the problem of solving the motion equation is transformed into the problem of solving the matrix eigenvalue (the minimum eigenvalue of the matrix corresponds to the natural frequency of the beam): ;matrix S The eigenvalue of is the natural frequency of the beam.

[0036] in, Figure 3 This is a screenshot of some of the procedures in this application; Figure 4 As shown in FIG, the natural frequency comparison diagram of the beam calculated by this application and the finite element method. Figure 4 It can be seen that the accuracy of this application is high (the error is within 1%).

[0037] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any form. Any simple modification or equivalent change made to the above embodiment based on the technical essence of the present invention shall fall within the scope of protection of the present invention.

Claims

1. A method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton, characterized in that: The skeleton is simplified into a beam model with two clamped ends. Based on Timoshenko beam theory and Hamilton's principle, the motion governing equations of a porous two-dimensional functionally graded material beam in a humid and hot environment are derived. The equations are simplified using the separation of variables method. The natural frequency of the beam is solved using the differential quadrature method. The motion control equations are discretized using the discretization principle of differential quadrature method, and then the equations are transformed into matrix form. The matrix transformation method is used to transform the problem of solving the motion equations into the problem of solving the matrix eigenvalues, and finally the solution is achieved by writing the corresponding program.

2. The method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton according to claim 1, characterized in that: The following steps are involved: Step 1: Based on Timoshenko beam theory, the motion control equation of the porous two-dimensional functional gradient beam is established by Hamilton principle; Step 2: Use the separation of variables method to process the equation; Step 3: Discretize the equation of motion using the differential quadrature method; Step 4: Convert the motion control equation into matrix form; Step 5: Write the corresponding program to solve the problem.

3. The method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton according to claim 2, characterized in that: In step 1, the motion control equation of the porous two-dimensional functional gradient beam is: （1）； （2）； （3）； in, is the axial displacement of any point in the beam; is the axial displacement of any point in the beam; is the density of ceramic / metal; is the elastic modulus of ceramic / metal; is the thermal / humidity expansion coefficient; is the thermal / moisture axial force; is the natural frequency of the beam; is the shear correction factor; is the rotation angle of the cross section about the vertical direction, all is the inertia term.

4. The method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton according to claim 3, characterized in that: In the step 2: （4）； in, is the natural frequency of the beam; is the rotation angle of the cross section about the vertical direction, x is the spatial coordinate, and t is the time variable.

5. The method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton according to claim 4, characterized in that: In the step three: （5）； （6）； （7）； in: are the weighted coefficients in the differential quadrature method.

6. The method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton according to claim 5, characterized in that: The cross section of the beam in step 1 is rectangular.

7. The method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton according to claim 6, characterized in that: The motion control equation established in step 1 is based on classical continuum theory.

8. The method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton according to claim 7, characterized in that: The beam is made of a composite of ceramic and metal. The physical parameters of the beam vary continuously along the axial direction and thickness direction of the beam, and are expressed as functions of coordinates x and z: （8）； （9）； （10）； （11）； At the same time, the physical properties of the material change with temperature: （12）； （13）； （14）； （15）； in, is the axial displacement of any point in the beam; is the axial displacement of any point in the beam; is the density of ceramic / metal; is the elastic modulus of ceramic / metal; is the thermal / humidity expansion coefficient; is the thermal / moisture axial force; is the natural frequency of the beam; （i =−1, 0, 1, 2, 3) is the material temperature correlation coefficient; For ceramic / metal; is the functional gradient index; is the porosity; is the rotation angle of the cross section about the vertical direction; is the shear correction factor; Temperature / humidity varies uniformly across the thickness of the beam: （16）； （17）； in, T 0=300 K and C 0=0% indicates the initial temperature and humidity.

9. The method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton according to claim 8, characterized in that: The stiffness components and inertia terms in equations (1) to (3) are: （18）； （19）； （20）。 10. The method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton according to claim 9, characterized in that: When the differential quadrature method is used to discretize the continuous system in step 2, the node selection is: （21）； Where N is the total number of discrete nodes and L is the length of the beam.

11. The method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton according to claim 10, characterized in that: In step 4, the motion control equation is converted into a matrix form: （22）； in, K 1 、M are the stiffness matrix and mass matrix, X is the displacement component.

12. The method for calculating the natural frequency of a porous two-dimensional functionally gradient material skeleton according to claim 11, characterized in that: Mathematical processing is performed on Equation (22): first, the matrix form of the motion control equation is standardized, then the motion control equation is converted into the form of a block matrix, and finally matrix operations are performed to obtain Equation (23), so that the problem of solving the motion equation is converted into the problem of solving the matrix eigenvalue: ;matrix S The eigenvalue of is the natural frequency of the beam.

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