A vibration prediction modeling method for ring-stiffened cone-cylinder composite shells

Abstract

The present invention relates to a vibration prediction modeling method for a ring-ribbed cone-cylinder composite shell. Starting from the basic solution of the conical shell vibration response, the dynamic stiffness matrix of the conical shell is derived. By changing the cone top angle of the conical shell, the conical shell is degenerated into a cylindrical shell and an annular plate. Then, with the conical shell, the annular plate and the cylindrical shell as the three basic substructures, multiple substructure shells are spliced ​​together by utilizing the displacement continuity and internal force balance conditions on the splicing surface of the composite shell to form the overall dynamic stiffness matrix of the composite shell in the global coordinate system. The dynamic stiffness matrix is ​​used to predict the natural frequency and vibration response of the composite shell.

Description

Technical Field

[0001] The invention relates to the technical field of vibration prediction modeling methods, in particular to a vibration prediction modeling method for a ring-ribbed cone-cylinder combined shell. Background Art

[0002] Composite hulls are typical components in submarines, aircraft, rockets, and other equipment. Due to internal equipment excitation and external environmental influences, long-term excessive vibration and noise can cause fatigue damage to the composite hull. This can lead to significant economic losses or even disasters. Therefore, to prevent resonance between the natural frequency of the composite hull and the excitation frequency of the equipment, the vibration characteristics of the composite hull must be predicted during the composite hull design phase for reference by the designers.

[0003] Common methods for calculating composite shells in the existing technology include Anasys and Abaqus. However, their vibration and noise calculations require a lot of manual experience and require remodeling for different composite shell models. This is cumbersome, time-consuming, labor-intensive, and has low accuracy. The present invention is based on a precise analytical calculation method, is easy to program, and has the advantages of parametric modeling, a simple modeling process, and high computational efficiency. Summary of the Invention

[0004] In response to the shortcomings of the above-mentioned existing production technology, the applicant provides a vibration prediction modeling method for a ring-ribbed cone-cylinder composite shell, so that rapid vibration prediction of the composite shell structure can be achieved through programming. The prediction evaluation results can provide a reference for the vibration and noise reduction design of related composite shell structures.

[0005] The technical solutions adopted in the present invention are as follows:

[0006] A vibration prediction modeling method for ring-stiffened cone-cylinder composite shells.

[0007] Confirm the governing equations for conical shell vibration:

[0008]

[0009] In the above formula, u c , v c and w c They are the axial, tangential and radial displacements of the conical shell, and each section includes four u c , v c , w c and Four displacement variables; N c , M c , T c and S c Four internal force variables;

[0010] in, and is the partial differential coefficient in the vibration control equation of the conical shell, and the specific expression is as follows:

[0011]

[0012]

[0013]

[0014]

[0015]

[0016]

[0017]

[0018]

[0019]

[0020] Where: μ is the Poisson's ratio of the conical shell material, ρ is the density of the material, h is the thickness of the conical shell, is the Laplace operator;

[0021] The dynamic model of the conical shell is based on the power series method and is solved using the power series superposition method in the literature. The displacement expressions of the conical shell in three directions can be written as:

[0022]

[0023] Among them: a m , b m , c m is the power series superposition coefficient, m is the power series superposition cutoff number, n is the circumferential mode number of the conical shell, x c is the displacement along the generatrix of the conical shell, θ is the circumferential angle, and by substituting Equation (3) into the vibration equation (1) of the conical shell, the recursive relationship between the power series coefficients can be obtained as follows:

[0024]

[0025] When the power series m>0, the coefficients in formula (4) can be expressed by the eight coefficients a0, a1, b0, b1, c0, c1, c2, and c3. Therefore, these eight coefficients are also called the displacement basis functions of the conical shell. Combining the above formulas, when the circumferential modal number n is specifically determined, the displacement function expression of the conical shell can be written as:

[0026]

[0027] The coefficient expression in the above formula is:

[0028]

[0029] Substituting equations (5) and (6) into equation (3), the final expression of the conical shell displacement is:

[0030]

[0031] According to the relationship between the internal force and displacement in the middle of a thin shell, the internal force expression on the conical shell section is:

[0032]

[0033] In the above formula:

[0034]

[0035] Substituting Equation (7) into the relationship between displacement and internal force, the internal force can also be written as an expression of basis function, and the conical shell boundary x c =-L c and x c =L c The displacement at the two nodes of the conical shell can be regarded as the dynamic stiffness matrix expression at both ends of the 8×8 conical shell:

[0036]

[0037] In the above formula [D C ] is the dynamic stiffness matrix of the conical shell, which is the most basic unit in the composite shell prediction model;

[0038] During the splicing process of the composite shell, the displacement continuity and internal force balance on the splicing surface of the two substructure shells are required.

[0039] Its further technical solution is:

[0040] During the splicing process of the conical shell and the cylindrical shell, the three substructure shells, the conical shell, the cylindrical shell, and the annular plate, can all be regarded as two-port structures. Each port edge corresponds to four displacement variables and four internal force variables. Because the conical shell and the cylindrical shell share an edge, the dynamic stiffness matrices of the two substructure shells need to be superimposed and spliced ​​on this edge.

[0041] The structural form of the combined shell is a combined shell with ribs.

[0042] It adopts circular plate ribbed structure or L-shaped ribbed structure.

[0043] In the structural form of the combined shell, the dynamic stiffness matrix of the conical shell is used as the basis for variable assembly, and eight structures are obtained by degenerating by changing the cone top angles and thicknesses of the two conical shells.

[0044] According to the substructure theory, the composite shell is divided into five substructure shells, namely:

[0045] Outer cylindrical shell;

[0046] The annular plate connecting the two cylindrical shells;

[0047] Inner cylindrical shell;

[0048] Conical shell in composite shell;

[0049] The ribs in a stiffened cylindrical shell;

[0050] Starting from the most basic conical shell foundation, the dynamic stiffness matrices of the five substructure shells are obtained by formula (10), namely D1, D2, D3, D4, and D5. Then, using the boundary conditions at the connections of the five substructure shells, they are assembled into a complex shell structure.

[0051] The beneficial effects of the present invention are as follows:

[0052] The present invention is a method for predicting the vibration of a composite shell based on the spectral element method within the frequency domain. Starting from the solution of the basic conical shell vibration response, the dynamic stiffness matrix of the conical shell is derived. By changing the cone top angle of the conical shell, the conical shell is degenerated into a cylindrical shell and an annular plate. Then, with the conical shell, the annular plate and the cylindrical shell as the three basic substructures, multiple substructure shells are spliced ​​together by utilizing the displacement continuity and internal force balance conditions on the splicing surface of the composite shell to form an overall dynamic stiffness matrix of the composite shell in the global coordinate system. The dynamic stiffness matrix is ​​used to predict the natural frequency and vibration response of the composite shell.

[0053] The present invention can realize rapid vibration prediction of the combined shell structure through programming, and the prediction evaluation result can provide a reference for the vibration reduction and noise reduction design of the related combined shell structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 It is a schematic diagram of the force and displacement of the truncated cone shell of the present invention.

[0055] Figure 2 This is a schematic diagram of the present invention when the dynamic stiffness matrices of two substructure shells are superimposed and spliced.

[0056] Figure 3 It is a structural schematic diagram of the external rib of the present invention.

[0057] Figure 4 It is a partial view of the outer rib of the present invention.

[0058] Figure 5 It is a structural schematic diagram of the inner rib of the present invention.

[0059] Figure 6 It is a partial view of the inner rib of the present invention.

[0060] Figure 7 It is a schematic diagram of the double-layer ribbed cylindrical shell structure (complex combined shell structure) of the present invention.

[0061] Figure 8 This is a physical picture of the combined shell test model of the present invention.

[0062] Figure 9 It is a structural schematic diagram of the combined shell test model of the present invention.

[0063] Figure 10 FIG1 is a comparison diagram of the method used in the present invention and the test results.

[0064] Figure 11 Figure 2 is a comparison of the method used in the present invention and the test results.

[0065] Figure 12 This is a comparison chart of the calculation of the complex composite shell of the present invention using this method and finite element software.

[0066] Figure 13 This is a comparative analysis diagram of the complex composite shell of the present invention using this method and finite element software calculation.

[0067] Figure 14 The present invention changes the angles of the two cone apexes and the thickness of the cone shells, degenerating into eight typical structural diagrams. DETAILED DESCRIPTION

[0068] The specific embodiments of the present invention will be described below with reference to the accompanying drawings.

[0069] like Figures 1-13 As shown, a vibration prediction modeling method for a ring-ribbed cone-cylinder composite shell in this embodiment is provided. Cone-cylinder, cone-cone, cone-cylinder-sphere and other composite shells are typical structural components in aviation and underwater vehicles. Long-term vibration will cause fatigue damage to the structure. Therefore, it is necessary to establish a dynamic model of this composite shell to carry out dynamic characteristic analysis.

[0070] The invention is based on the conical shell vibration control equation, which is:

[0071]

[0072] In the above formula, u c , v c and w cThey are the axial, tangential and radial displacements of the conical shell respectively. The force and displacement of the truncated conical shell are shown as follows Figure 1 As shown in the figure, each section includes four u c , v c , w c and Four displacement variables; N c , M c , T c and S c Four internal force variables.

[0073] In formula (1) and is the partial differential coefficient in the vibration control equation of the conical shell, and the specific expression is as follows:

[0074]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080]

[0081]

[0082]

[0083] In the above formula, μ is the Poisson's ratio of the conical shell material, ρ is the density of the material, and h is the thickness of the conical shell. is the Laplace operator. The dynamic model of the conical shell is based on the power series method and is solved using the power series superposition method in the literature. The displacement expressions of the conical shell in three directions can be written as:

[0084]

[0085] where a m , b m , c m is the power series superposition coefficient, m is the power series superposition cutoff number, n is the circumferential mode number of the conical shell, x c is the displacement along the generatrix of the conical shell, and θ is the circumferential angle. Substituting Equation (3) into the vibration equation (1) of the conical shell, the recursive relationship between the power series coefficients can be obtained as follows:

[0086]

[0087] The specific expression of the recursive coefficient in the above formula can be found in the literature [Caresta]. According to the above recursive relationship, when the power series m>0, the coefficients in formula (4) can be expressed by the eight coefficients a0, a1, b0, b1, c0, c1, c2, and c3. Therefore, these eight coefficients are also called the displacement basis functions of the conical shell. Combining the above formulas, when the circumferential mode number n is specifically determined, the displacement function expression of the conical shell can be written as:

[0088]

[0089] The coefficient expression in the above formula is:

[0090]

[0091] Substituting equations (5) and (6) into equation (3), the final expression of the conical shell displacement is:

[0092]

[0093] According to the relationship between the internal force and displacement in the middle of a thin shell, the internal force expression on the conical shell section is:

[0094]

[0095] In the above formula:

[0096]

[0097] Substituting Equation (7) into the relationship between displacement and internal force, the internal force can also be written as an expression of basis function. c =-L c and x c =L c The displacement at the two nodes of the conical shell is considered. The dynamic stiffness matrix expression of the two ends of the 8×8 conical shell can be obtained as follows:

[0098]

[0099] In the above formula [D C ] is the dynamic stiffness matrix of the conical shell, which is the most basic unit in the combined shell prediction model of the present invention. During the splicing process of the combined shell, the displacement continuity and internal force balance on the splicing surface of the two substructure shells are required. Figure 2The typical conical shell and cylindrical shell splicing process shown in the figure, the three sub-structure shells of the conical shell, cylindrical shell and annular plate in the present invention can be regarded as two-port structures, and each port edge corresponds to 4 displacement variables and 4 internal force variables. Because the conical shell and the cylindrical shell share one edge, it is necessary to superimpose the dynamic stiffness matrices of the two sub-structure shells on this edge. Figure 2 As shown in .

[0100] The side ② at the bottom of the conical shell overlaps the side ③ at the top of the cylindrical shell. Therefore, the displacement continuity and internal force continuity conditions of the side ② and the side ③ of the cylindrical shell must be met. The dynamic stiffness matrix of the conical shell is written as a two-port structure. The dynamic stiffness matrix of the conical shell and the cylindrical shell in the figure above can be written as:

[0101]

[0102] In the above formula, D ij is a 4×4 matrix. Since edges ② and ③ overlap, U2=U3=U * 、F2=F3=F * ,U * is the displacement matrix on the merged splicing edge, F * is the force matrix on the merged edge. The two equations in the above formula (11) are written together as follows:

[0103]

[0104] The same applies to the splicing of other multiple substructure shells.

[0105] (1) When the truncation angle is not 0°, it is a conical shell structure;

[0106] (2) When the cone apex angle of the conical shell is 0 degrees, the dynamic model of the conical shell can be degenerated into the dynamic model of the cylindrical shell. (3) When the cone apex angle of the conical shell is 90 degrees, the dynamic model of the conical shell can be degenerated into the dynamic model of the annular plate.

[0107] The structural forms of the combined shells in the present invention can be obtained by varying the dynamic stiffness matrix of the conical shells. The present invention mainly uses the cone-cone combined shell structure. By changing the cone top angles and the thickness of the two conical shells, it can be degenerated into 8 typical structures. Figure 14 .

[0108] In the present invention, the annular plate is used as a ribbed component and is spliced ​​with the conical shell, cylindrical shell or a combined shell thereof. Figure 3-Figure 7 shown.

[0109] Among them, the complex double-layer composite shell structure is first divided into 5 substructure shells according to the substructure theory:

[0110] ① is the outer cylindrical shell;

[0111] ② is the annular plate connecting the two cylindrical shells;

[0112] ③ is the inner cylindrical shell;

[0113] ④ is the conical shell in the combined shell;

[0114] ⑤ is the rib part in the ribbed cylindrical shell.

[0115] Although there are five different shell structures, starting from the most basic conical shell foundation, the dynamic stiffness matrices of the five substructure shells can be obtained by simply using Equation (10): D1, D2, D3, D4, and D5. Using the boundary conditions at the connections of the five substructure shells, a complex shell structure can be assembled.

[0116] In addition to the ordinary circular plate ribbed structure, it can also be expanded to an L-shaped ribbed structure.

[0117] It should be noted that the above-mentioned dynamic stiffness matrices of conical shells are all derived in their local coordinate systems. For example, when splicing multiple combined shell structures, each substructure conical shell structure needs to be converted to the global coordinate system for splicing.

[0118] The vibration prediction method of the combined shell in the patent of this invention was used to compare with the classical finite element and experimental results, which further proved the correctness of the combined shell calculation method in the present invention.

[0119] The calculation method described in the present invention has the advantages of high calculation effect, fast convergence, easy programming implementation, and the ability to realize parametric law analysis. Compared with traditional commercial finite element software, it does not require repeated modeling and meshing, and only requires the input of a few parameters to realize parametric modeling. In traditional finite element calculations, it is necessary to divide the combined shell into grids. The size of the grid determines the accuracy of the calculation frequency range. The smaller the grid, the higher the calculation accuracy. However, this requires a lot of time and computing resources. The calculation method proposed in the patent of this invention has a displacement shape function that strictly satisfies the control equation. An entire conical shell structure only needs to be divided into a whole, and the calculation frequency range can reach the high frequency range.

[0120] This method, based on the dynamic stiffness matrix of a conical shell, degenerates it into an annular plate or cylindrical shell by changing its apex angle, eliminating the need to derive the vibration control equations for the annular plate or cylindrical shell. This significantly reduces the number of modeling parameters and the theoretical derivation process for cylindrical shells and annular plates. The overall dynamic stiffness matrix of the combined shells is then spliced ​​using the equilibrium and continuity conditions at the connecting sections of the combined shells, encompassing a total of eight combined shell structures.

[0121] The present invention simulates the rib-plus-rib structure, and is applicable to an outer rib form, an inner rib form, and a double-layer shell structure of a combined shell structure.

[0122] The above description is an explanation of the present invention, not a limitation of the present invention. The scope of the present invention is defined in the claims. Any modifications may be made within the scope of protection of the present invention.

Claims

1. A vibration prediction modeling method for a ring-stiffened cone-cylinder composite shell, characterized by: Confirm the governing equations for conical shell vibration: In the above formula, u c , v c and w c They are the axial, tangential and radial displacements of the conical shell, and each section includes four u c , v c , w c and Four displacement variables; N c , M c , T c and S c Four internal force variables; in, and is the partial differential coefficient in the vibration control equation of the conical shell, and the specific expression is as follows: Where: μ is the Poisson's ratio of the conical shell material, ρ is the density of the material, h is the thickness of the conical shell, is the Laplace operator; The dynamic model of the conical shell is based on the power series method and is solved using the power series superposition method in the literature. The displacement expressions of the conical shell in three directions can be written as: Among them: a m , b m , c m is the power series superposition coefficient, m is the power series superposition cutoff number, n is the circumferential mode number of the conical shell, x c is the displacement along the generatrix of the conical shell, θ is the circumferential angle, and by substituting Equation (3) into the vibration equation (1) of the conical shell, the recursive relationship between the power series coefficients can be obtained as follows: When the power series m>0, the coefficients in formula (4) can be expressed by the eight coefficients a0, a1, b0, b1, c0, c1, c2, and c3. Therefore, these eight coefficients are also called the displacement basis functions of the conical shell. Combining the above formulas, when the circumferential modal number n is specifically determined, the displacement function expression of the conical shell can be written as: The coefficient expression in the above formula is: Substituting equations (5) and (6) into equation (3), the final expression of the conical shell displacement is: According to the relationship between the internal force and displacement in the middle of a thin shell, the internal force expression on the conical shell section is: In the above formula: Substituting Equation (7) into the relationship between displacement and internal force, the internal force can also be written as an expression of basis function, and the conical shell boundary x c =-L c and x c =L c The displacement at the two nodes of the conical shell can be regarded as the dynamic stiffness matrix expression at both ends of the 8×8 conical shell: In the above formula [D C ] is the dynamic stiffness matrix of the conical shell, which is the most basic unit in the composite shell prediction model; During the splicing process of the composite shell, the displacement continuity and internal force balance on the splicing surface of the two substructure shells are required.

2. The vibration prediction modeling method for a ring-stiffened cone-cylinder composite shell according to claim 1, characterized in that: During the splicing process of the conical shell and the cylindrical shell, the three substructure shells, the conical shell, the cylindrical shell, and the annular plate, can all be regarded as two-port structures. Each port edge corresponds to four displacement variables and four internal force variables. Because the conical shell and the cylindrical shell share an edge, the dynamic stiffness matrices of the two substructure shells need to be superimposed and spliced ​​on this edge.

3. The vibration prediction modeling method for a ring-stiffened cone-cylinder composite shell according to claim 1, characterized in that: The structural form of the combined shell is a combined shell with ribs.

4. The vibration prediction modeling method for a ring-stiffened cone-cylinder composite shell according to claim 3, characterized in that: It adopts circular plate ribbed structure or L-shaped ribbed structure.

5. The vibration prediction modeling method for a ring-stiffened cone-cylinder composite shell according to claim 1, characterized in that: In the structural form of the combined shell, the dynamic stiffness matrix of the conical shell is used as the basis for variable assembly, and eight structures are obtained by degenerating by changing the cone top angles and thicknesses of the two conical shells.

6. The vibration prediction modeling method for a ring-stiffened cone-cylinder composite shell according to claim 5, characterized in that: According to the substructure theory, the composite shell is divided into five substructure shells, namely: Outer cylindrical shell; The annular plate connecting the two cylindrical shells; Inner cylindrical shell; Conical shell in composite shell; The ribs in a stiffened cylindrical shell; Starting from the most basic conical shell foundation, the dynamic stiffness matrices of the five substructure shells are obtained by formula (10), namely D1, D2, D3, D4, and D5. Then, using the boundary conditions at the connections of the five substructure shells, they are assembled into a complex shell structure.

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