Model-Based Dynamics Analysis and Design Method for Conical-Cylindrical Shells

Abstract

This paper presents a model-based dynamic analysis and design method for conical-cylindrical shells. To address the problem of low computational efficiency in structural dynamics analysis and design optimization caused by the extremely high degree of freedom in traditional finite element methods (FEM) models when dealing with conical-cylindrical shell structures of submarines, the method divides the conical-cylindrical shell structure of the submarine into two substructures and models them using the FEM. The fixed interface principal modes and constraint modes of the substructures are solved, and these are combined into a first coordinate transformation matrix condensed model, resulting in a condensed model. The two condensed models are integrated, and a second coordinate transformation matrix is ​​obtained using the interface compatibility conditions between the substructures. This second coordinate transformation matrix is ​​then used to process the integrated model, yielding a condensed model of the conical-cylindrical shell structure. The frequency response curves and time-domain response curves of the condensed model of the conical-cylindrical shell in modal space are calculated, and then transformed to physical space to obtain the frequency response curves and time-domain response curves in physical space.

Description

Technical Field

[0001] This invention relates to a dynamic analysis method, specifically a dynamic analysis method for condensation of a finite element model of a conical-cylindrical shell using the Craig-Bampton method, belonging to the field of computational mechanics. Background Technology

[0002] Currently, conical-cylindrical shell structures are widely used in various engineering fields, including mechanical, aerospace, shipbuilding, civil, and power engineering. For example, in submarine research, the midsection is typically simplified to a cylindrical shell, and the stern to a truncated conical shell, with the entire submarine being abstracted as a conical-cylindrical composite shell structure. Furthermore, in aerospace research, the forward sections of aircraft, rockets, and other flying vehicles are also mostly simplified to conical-cylindrical composite shell structures. Conical-cylindrical shell structures inevitably face harsh external environments during service, resulting in significant vibration and noise problems. Therefore, dynamic analysis and design optimization of conical-cylindrical shell structures are of great importance.

[0003] In engineering, the finite element method (FEM) is generally used for dynamic analysis and design optimization of conical-cylindrical shell structures. However, to ensure accuracy when dealing with complex structures like conical-cylindrical shells, the traditional FEM typically requires a large number of finite elements, resulting in an extremely high degree of freedom in the traditional FEM model of the conical-cylindrical shell. This significantly limits the computational efficiency of structural dynamic analysis and design optimization, which is unacceptable in the engineering field. Summary of the Invention

[0004] To address the problem that traditional finite element methods for processing conical-cylindrical shell structures of submarines typically require a large number of finite elements to ensure accuracy, resulting in an extremely high degree of freedom in the traditional finite element model of the conical-cylindrical shell structure and consequently low computational efficiency for structural dynamics analysis and design optimization, this invention proposes a model-condensed method for the dynamic analysis and design of conical-cylindrical shells.

[0005] It includes the following steps:

[0006] S1. Determine the basic dimensions and material properties of the conical-cylindrical hull structure of the submarine, and divide the conical-cylindrical hull structure into two substructures, namely, substructure one is a conical shell and substructure two is a cylindrical shell. Use the finite element method to model the two substructures respectively to obtain the finite element model of the substructure.

[0007] S2. Solve for the fixed interface principal mode and constraint mode of the substructure, and combine the fixed interface principal mode and constraint mode into the first coordinate transformation matrix. Use the first coordinate transformation matrix to condense the finite element model of the substructure to obtain the condensed substructure finite element model.

[0008] S3. Integrate the two condensation substructure finite element models to obtain the integrated finite element model. Use the interface coordination conditions between the substructures to obtain the second coordinate transformation matrix. Use the second coordinate transformation matrix to eliminate the repeated interface degrees of freedom of the integrated finite element model to obtain the condensation finite element model of the conical-cylindrical shell structure.

[0009] S4. Calculate the frequency response curve and time-domain response curve of the condensation finite element model of the conical-cylindrical shell in modal space, transform the frequency response curve and time-domain response curve in modal space to physical space, and obtain the frequency response curve and time-domain response curve in physical space to complete the dynamic analysis of the conical-cylindrical shell structure of the submarine.

[0010] Furthermore, the basic dimensions of the conical-cylindrical shell structure in S1 include the length, radius, thickness, and cone angle of the conical shell; and the length, radius, and thickness of the cylindrical shell.

[0011] Furthermore, the material properties of the conical-cylindrical shell structure in S1 include the material's elastic modulus, density, and Poisson's ratio.

[0012] Furthermore, in S2, the fixed interface principal modes and constraint modes of the substructure are solved, and these modes are combined into a first coordinate transformation matrix. The first coordinate transformation matrix is ​​then used to condense the substructure finite element model, resulting in a condensed substructure finite element model. The specific process is as follows:

[0013] First coordinate transformation matrix Φ c for:

[0014]

[0015] in, The fixed interface principal mode of the substructure is represented by L, which represents the lower-order components. The constraint modes of the substructure are represented by E, where E is the identity matrix;

[0016] The substructure finite element model is condensed using the first coordinate transformation matrix to obtain the condensed substructure finite element model:

[0017]

[0018]

[0019]

[0020] Where M and K are the mass matrix and stiffness matrix of the substructure finite element model, respectively, and F is the force vector of the substructure finite element model. This represents the mass matrix of the finite element model of the substructure after condensation; The stiffness matrix represents the stiffness matrix of the finite element model of the substructure after condensation. Λ represents the force vector of the finite element model of the condensed substructure; Λ represents the eigenvalue matrix of the finite element model of the condensed substructure; I and B represent the internal and interface degrees of freedom, respectively; R B This indicates the connection load between substructures.

[0021] Furthermore, in S3, the second coordinate transformation matrix is ​​obtained using the interface coordination conditions between substructures. The specific process is as follows:

[0022]

[0023] Where P represents the displacement in modal space; T C This represents the second coordinate transformation matrix.

[0024] Furthermore, the condensation finite element model of the conical-cylindrical shell structure in S3 is as follows:

[0025]

[0026] in, This represents the mass matrix of the condensation finite element model; The stiffness matrix represents the stiffness matrix of the condensation finite element model; Represents the force vector in the condensation finite element model; This represents acceleration in modal space.

[0027] Beneficial effects:

[0028] This invention first divides the conical-cylindrical hull structure of a submarine into two substructures: substructure one is a conical shell, and substructure two is a cylindrical shell. Then, the finite element method (FEM) is used to model each substructure separately, obtaining substructure finite element models. By solving the fixed interface principal modes and constraint modes of the substructures, these modes are combined into a first coordinate transformation matrix. This first coordinate transformation matrix is ​​used to condense the substructure finite element models, resulting in a condensed substructure finite element model. The two condensed substructure finite element models are integrated to obtain an integrated finite element model. The second coordinate transformation matrix is ​​then obtained using the interface compatibility conditions between the substructures. This second coordinate transformation matrix is ​​used to eliminate duplicate interface degrees of freedom in the integrated finite element model, resulting in a condensed finite element model of the conical-cylindrical hull structure. Finally, by calculating the frequency response curves and time-domain response curves of the condensation finite element model of the conical-cylindrical shell in modal space, the frequency response curves and time-domain response curves in modal space are transformed into the physical space to obtain the frequency response curves and time-domain response curves in the physical space, thus completing the dynamic analysis of the conical-cylindrical shell structure of the submarine.

[0029] This invention utilizes the Craig-Bampton method (fixed interface modal synthesis) to condense the finite element model of a conical-cylindrical shell. The model condensation accuracy is extremely high, and it can greatly reduce the number of degrees of freedom of the finite element model, significantly improving the efficiency of dynamic analysis and design optimization of conical-cylindrical shell structures. This invention is applicable to the study of conical-cylindrical shell structures of any size and material, and its theoretical framework is simple and easy to program. Attached Figure Description

[0030] Figure 1 This is a side view of a conical-cylindrical shell structure;

[0031] Figure 2 This is a top view of a conical-cylindrical shell structure;

[0032] Figure 3 This is a comparison diagram of the first-order modes of the complete finite element model and the condensed finite element model;

[0033] Figure 4 This is a comparison diagram of the second-order modes of the complete finite element model and the condensed finite element model;

[0034] Figure 5 This is a comparison diagram of the third-order modes of the complete finite element model and the condensed finite element model;

[0035] Figure 6 This is a comparison diagram of the fourth-order modes of the complete finite element model and the condensed finite element model;

[0036] Figure 7 It is a comparison chart of the frequency response curves of the complete finite element model and the condensed finite element model;

[0037] Figure 8 This is a comparison chart of the time-domain response curves of the complete finite element model and the condensed finite element model;

[0038] Figure 9 This is a comparison diagram showing the natural frequencies of the complete finite element model and the condensed finite element model as a function of θ. Detailed Implementation

[0039] Specific implementation method one: Combining Figures 1-9 This embodiment describes a model-based condensation-based dynamic analysis and design method for conical-cylindrical shells, which includes the following steps:

[0040] S1. Based on actual requirements, determine the basic dimensions and material properties of the submarine's conical-cylindrical hull structure. Divide the conical-cylindrical hull structure into two substructures: substructure one is a conical shell, and substructure two is a cylindrical shell. Use the finite element method to model the two substructures separately, obtaining the finite element models of the substructures, namely the conical shell finite element model and the cylindrical shell finite element model, as follows: Figure 1 and 2 As shown, the basic dimensions include: a conical shell with length L1, radius R1, thickness h, and cone angle θ; and a cylindrical shell with length L2, radius R2, and thickness h. Material properties include the elastic modulus E, density ρ, and Poisson's ratio μ of the conical-cylindrical shell structure material.

[0041] S2. Solve for the fixed interface principal mode and constraint mode of the substructure, and combine the fixed interface principal mode and constraint mode into the first coordinate transformation matrix. Use the first coordinate transformation matrix to condense the finite element model of the substructure to obtain the condensed finite element model of the substructure.

[0042] Based on the finite element model of the substructure, the finite element dynamic equation of the substructure is obtained as follows:

[0043]

[0044] Where M and K are the mass matrix and stiffness matrix of the substructure finite element model, respectively, F is the force vector of the substructure finite element model, and the subscripts "I" and "B" represent the internal and interface degrees of freedom, respectively. B The symbol represents the connection load between substructures, and q represents the coordinates in physical space. It represents acceleration in physical space.

[0045] The first coordinate transformation matrix Φ is:

[0046] c

[0047]

[0048] in, For the fixed interface main mode of the substructure, the superscript "L" indicates a low-order component. Let E be the constraint mode of the substructure, and E be the identity matrix. The fixed interface principal mode refers to the normalized mode of the substructure when the interface is fixed; the constraint mode refers to the static equilibrium relationship between the interface degrees of freedom and the internal degrees of freedom of the substructure. It is obtained in the following way, let

[0049]

[0050] Therefore, the finite element equation of motion for the fixed interface of the substructure is:

[0051]

[0052] Solving the above equation yields the main mode of the fixed interface. Then select the lower-order components. This can be obtained using Guyan's method, a static condensation method proposed by Guyan in the 1960s. It is used to solve for the static equilibrium relationship between the interface degrees of freedom and the internal degrees of freedom of a substructure. Therefore, we can obtain...

[0053]

[0054] The substructure finite element model is condensed using the first coordinate transformation matrix. The operation is as follows:

[0055]

[0056]

[0057]

[0058] in, This represents the mass matrix of the finite element model of the substructure after condensation; The stiffness matrix represents the stiffness matrix of the finite element model of the substructure after condensation. Λ represents the force vector of the finite element model of the substructure after condensation; Λ represents the eigenvalue matrix of the finite element model of the substructure after condensation; thus, the condensation of the finite element model of the substructure is completed.

[0059] S3. Integrate the two condensation substructure finite element models to obtain the integrated finite element model. Use the interface coordination conditions between the substructures to obtain the second coordinate transformation matrix. Use the second coordinate transformation matrix to eliminate the repeated interface degrees of freedom of the integrated finite element model to obtain the condensation finite element model of the conical-cylindrical shell structure.

[0060] The two finite element models of the condensation substructures are integrated, and the integrated finite element equations of motion are:

[0061]

[0062] Where P represents the displacement in modal space; This represents acceleration in modal space; the subscripts "I" and "II" represent substructure one and substructure two, respectively. Furthermore:

[0063]

[0064] Utilizing the interface compatibility conditions (displacement compatibility conditions) P between substructures B,I =P B,II The following relationship is obtained:

[0065]

[0066] Among them, T CThis is the second coordinate transformation matrix. The second coordinate transformation matrix is ​​used to eliminate duplicate interface degrees of freedom in the integrated finite element model.

[0067]

[0068]

[0069]

[0070] Therefore, a condensation finite element model of the conical-cylindrical shell structure can be obtained:

[0071]

[0072] Highly efficient kinetic analysis and design optimization can be performed based on the condensation finite element model.

[0073] S4. Calculate the frequency response curve and time-domain response curve of the condensation finite element model of the conical-cylindrical shell in modal space, transform the frequency response curve and time-domain response curve in modal space to physical space, and obtain the frequency response curve and time-domain response curve in physical space to complete the dynamic analysis of the conical-cylindrical shell structure of the submarine.

[0074] The external stimulus is a simple harmonic stimulus. Where P0 and ω0 are the amplitude and frequency of the external excitation, and t is time; therefore, the solution to the equation can be assumed to be...

[0075] P = A0 sinω0t + B0 cosω0t

[0076] A0 and B0 are the inherent coefficients of the formula.

[0077] Substituting the solution back into the equations of the condensed finite element model and simplifying the solution form yields the following result.

[0078] B0=0

[0079] Therefore, the displacement in modal space can be obtained

[0080]

[0081] The above describes the displacement in modal space. Further conversion to physical space is required, which can be achieved through the following methods.

[0082]

[0083] Where u is the displacement in physical space, the dynamic analysis of the structure can be completed through u, including but not limited to frequency response curve analysis.

[0084] Example

[0085] I. Based on the specific application requirements of the conical-cylindrical shell structure in the actual working conditions of small underwater detectors, the specific dimensions of the conical-cylindrical shell structure are designed, and the material is selected. The conical-cylindrical shell structure is divided into conical shell structure and cylindrical shell structure, such as... Figure 1 and 2 As shown, the structural and material parameters in this embodiment include: elastic modulus E = 200 GPa, density ρ = 7850 kg / m³. 3 Poisson's ratio μ = 0.3, the conical shell structure has a length L1 = 0.05m, a radius R1 = 0.04113m, a thickness h = 0.001m, and a cone angle θ = 30°, and the cylindrical shell structure has a length L2 = 0.2m, a radius R2 = 0.07m, and a thickness h = 0.001m.

[0086] II. After determining the structural and material parameters of the conical-cylindrical shell, the conical and cylindrical shell structures are first modeled using the finite element method (FEM) to obtain the finite element dynamic equations. Next, the principal and constraint modes of the fixed interfaces of the substructures are solved, and these are combined into the first coordinate transformation matrix. This first coordinate transformation matrix is ​​used to condense the finite element models of the substructures. Then, the two substructure finite element models are integrated. Finally, the second coordinate transformation matrix is ​​obtained using the interface compatibility conditions between the substructures. This second coordinate transformation matrix is ​​used to eliminate the redundant interface degrees of freedom in the integrated finite element model, thus obtaining the condensed finite element model of the conical-cylindrical shell.

[0087] Third, after completing the second step, to verify the calculation accuracy of the method of the present invention, the natural frequencies and modes were calculated and compared using both the traditional finite element model of the conical-cylindrical shell and the condensation finite element model of the conical-cylindrical shell. The comparison results of the natural frequencies are given in the table below.

[0088]

[0089] As can be seen from the table, the natural frequency calculation accuracy of the condensation finite element model is extremely high. A comparison of the first-order modes of the complete finite element model and the condensation finite element model is shown below. Figure 3 As shown, the second-order mode comparison diagram is as follows: Figure 4 As shown, the comparison diagram of the third-order modes is as follows: Figure 5 As shown, the fourth-order mode comparison diagram is as follows: Figure 6 As shown in the figure, the modal calculation accuracy of the condensation finite element model is also extremely high. Furthermore, the degree of freedom of the finite element model using the method of this invention is only 3.9% of that of the traditional finite element model. This demonstrates the extremely high condensation accuracy of the model achieved by the method of this invention.

[0090] Fourth, the efficiency verification of the finite element model dynamic analysis and design optimization of the method of this invention will be carried out next. First, the frequency response curves and time-domain response curves of the traditional finite element model of the conical-cylindrical shell and the condensation finite element model of the conical-cylindrical shell were calculated respectively. For example... Figure 7 and 8 As shown in the figure, the frequency response curve and the time-domain response curve are very close, while the calculation time of the frequency response curve of the finite element model of the present invention is only 7.1% of that of the traditional finite element model. This proves that the finite element model of the present invention has extremely high efficiency in dynamic analysis. In addition, when the cone angle θ changes from 15° to 45° in a step of 1°, the first six natural frequencies of the complete finite element model and the condensed finite element model were calculated, and the calculation results are as follows. Figure 9 As shown in the figure, the calculation results are very close, but the calculation time of the finite element model using the method of this invention is only 27.8% of that of the traditional finite element model. This proves that the finite element model design optimization method of this invention is extremely efficient.

[0091] As can be seen from this embodiment, the finite element model of the method of the present invention has extremely high accuracy, and the efficiency of dynamic analysis and design optimization is extremely high.

Claims

1. A method of dynamic analysis and design of conic-cylindrical shells based on model condensation, characterized in that: It includes the following steps: S1. Determine the basic dimensions and material properties of the conical-cylindrical hull structure of the submarine, and divide the conical-cylindrical hull structure into two substructures, namely, substructure one is a conical shell and substructure two is a cylindrical shell. Use the finite element method to model the two substructures respectively to obtain the finite element model of the substructure. S2. Solve for the fixed interface principal mode and constraint mode of the substructure, and combine the fixed interface principal mode and constraint mode into the first coordinate transformation matrix. Use the first coordinate transformation matrix to condense the finite element model of the substructure to obtain the condensed substructure finite element model. S3. Integrate the two condensation substructure finite element models to obtain the integrated finite element model. Use the interface coordination conditions between the substructures to obtain the second coordinate transformation matrix. Use the second coordinate transformation matrix to eliminate the repeated interface degrees of freedom of the integrated finite element model to obtain the condensation finite element model of the conical-cylindrical shell structure. The condensation finite element model of the conical-cylindrical shell structure is as follows: wherein, denotes the mass matrix of the condensed finite element model; denotes the stiffness matrix of the condensed finite element model; denotes the force vector of the condensed finite element model; denotes the acceleration in modal space; S4. Calculate the frequency response curve and time-domain response curve of the condensation finite element model of the conical-cylindrical shell in modal space, transform the frequency response curve and time-domain response curve in modal space to physical space, and obtain the frequency response curve and time-domain response curve in physical space to complete the dynamic analysis of the conical-cylindrical shell structure of the submarine.

2. The model-based agglomerated conic-cylindrical shell dynamics analysis and design method recited in claim 1, wherein: The basic dimensions of the conical-cylindrical shell structure in S1 include the length, radius, thickness, and cone angle of the conical shell; and the length, radius, and thickness of the cylindrical shell.

3. The model-based condensation-based dynamic analysis and design method for conical-cylindrical shells as described in claim 2, characterized in that: The material properties of the conical-cylindrical shell structure in S1 include the elastic modulus, density, and Poisson's ratio.

4. The model-based condensation-based dynamic analysis and design method for conical-cylindrical shells according to claim 3, characterized in that: In S2, the fixed interface principal modes and constraint modes of the substructure are solved, and the fixed interface principal modes and constraint modes are combined into the first coordinate transformation matrix. The first coordinate transformation matrix is ​​then used to condense the finite element model of the substructure to obtain the condensed finite element model of the substructure. The specific process is as follows: First coordinate transformation matrix for: in, The fixed interface principal mode of the substructure is represented by L, which represents the lower-order components. The constraint modes of the substructure are represented by E, where E is the identity matrix; The substructure finite element model is condensed using the first coordinate transformation matrix to obtain the condensed substructure finite element model: Where M and K are the mass matrix and stiffness matrix of the substructure finite element model, respectively, and F is the force vector of the substructure finite element model. This represents the mass matrix of the finite element model of the substructure after condensation; The stiffness matrix represents the stiffness matrix of the finite element model of the substructure after condensation. Represents the force vector in the finite element model of the substructure after condensation; This represents the eigenvalue matrix of the finite element model of the condensed substructure; I and B represent the interior and interface degrees of freedom, respectively. This indicates the connection load between substructures.

5. The model-based condensation-based dynamic analysis and design method for conical-cylindrical shells according to claim 4, characterized in that: In S3, the second coordinate transformation matrix is ​​obtained by utilizing the interface coordination conditions between substructures. The specific process is as follows: in, T represents displacement in modal space. C This represents the second coordinate transformation matrix.

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