A method for predicting the natural frequency of stiffened conical shells

Abstract

The present invention relates to a method for predicting the natural frequency of a stiffened conical shell, comprising the following steps: S1-S2, writing the vibration control equations of the conical shell and the concentric annular plate into a dynamic stiffness matrix related to the internal force and displacement thereof; S3, discretizing the conical shell into a plurality of sub-segments, and splicing the plurality of substructures in the stiffened conical shell by utilizing two boundary conditions of displacement continuity and force balance between the ribs and the conical shell sub-segments; S4, on the premise of obtaining the dynamic stiffness of the overall calculation model, performing a correction and substitution method on the matrix elements corresponding to each degree of freedom in the dynamic stiffness matrix; S5, obtaining a curve of the dynamic stiffness matrix determinant of the overall model varying with frequency by solving the determinant of the overall stiffness matrix, and identifying and obtaining the natural frequency of the overall structure through the maximum and minimum values in the frequency response curve, quickly evaluating the natural frequency of tail structures such as the conical shell, and improving the safety and reliability of equipment containing the conical shell structure.

Description

Technical Field

[0001] The present invention relates to the technical field of aircraft, and in particular to a method for predicting the natural frequency of a ribbed conical shell. Background Art

[0002] The conical shell structure is a typical stern structure in aircraft, missiles, torpedoes, submarines and other aerial and underwater vehicles. During flight or navigation, these equipment structures will be subject to external excitation from turbulence in the external fluid medium, engines and other equipment. When the load of the external excitation is equal to or very close to the natural frequency of the conical shell structure, it is easy to cause resonance of the stern conical shell structure. Resonance will amplify the vibration of the conical shell itself. Long-term excessive vibration displacement will cause fatigue damage to the conical shell structure, resulting in failure of the above-mentioned aircraft and other equipment, causing huge disasters. Therefore, before designing and constructing such equipment, it is necessary to evaluate and predict the natural frequency of the conical shell structure to avoid resonance between the frequency of the conical shell structure and the external excitation frequency. At the same time, in order to meet the structural strength requirements in engineering, the conical shell structure is often reinforced with ribs.

[0003] For example, Chinese patent CN 110795783 A proposes a natural frequency of a liquid-filled cylindrical shell. This invention patent mainly solves the problem of liquid-filled cylindrical shells. Liquid-filled cylindrical shells are mainly used in calculations such as fluid pipelines. This invention patent is mainly aimed at the stern structure of equipment and can be extended to ribbed structures.

[0004] For example, Chinese patent CN 11499687 A proposes a simulation calculation method for the bending vibration natural frequency of a variable-amplitude conical rod. The invention uses the Timushinko beam theory, which is applicable to beam models with a relatively large aspect ratio. The present invention uses the shell theory, which is applicable to structures with larger geometric dimensions.

[0005] The existing technology still has the following problems:

[0006] 1. Traditional conical shell natural frequency prediction methods can only be applied to simple supported, clamped and free typical boundary conditions. However, in actual engineering, conical shell structures often have a variety of installation boundaries.

[0007] 2. Traditional conical shell vibration analysis methods only study unribbed conical shells.

[0008] 3. When the existing boundary conditions are applied through the large number method, large values will be introduced into the overall dynamic stiffness matrix, causing the matrix determinant to be ill-conditioned during the natural frequency solution process.

[0009] To this end, we propose a method to predict the natural frequencies of stiffened conical shells. Summary of the Invention

[0010] In response to the shortcomings of the above-mentioned existing production technology, the applicant provides a method for predicting the natural frequency of a ribbed conical shell, which can quickly evaluate the natural frequency of tail structures such as conical shells and improve the safety and reliability of equipment containing conical shell structures.

[0011] The technical solutions adopted in the present invention are as follows:

[0012] 1. A method for predicting the natural frequency of a stiffened conical shell, comprising the following steps:

[0013] S1. Using the stress-strain relationship between the in-plane and out-of-plane forces and displacements in a conical shell, the vibration control equation of the conical shell is expressed as a dynamic stiffness matrix related to its internal forces and displacements.

[0014] S2. The ribs are simulated using concentric annular plates, and the vibration control equations of the annular plates are obtained, followed by the dynamic stiffness matrix of the annular plates.

[0015] S3. Discretely divide the conical shell into multiple sub-segments, and use the two boundary conditions of displacement continuity and force balance between the ribs and the conical shell sub-segments to splice the multiple sub-structures in the ribbed conical shell using a quasi-finite element method;

[0016] S4. On the premise of obtaining the dynamic stiffness of the overall calculation model, the arbitrary boundary conditions of the stiffened conical shell structure are realized by performing the correction and substitution method on the matrix elements corresponding to each degree of freedom in the dynamic stiffness matrix;

[0017] S5. By solving the determinant of the overall stiffness matrix, a curve showing the dynamic stiffness matrix determinant of the overall model changing with frequency is obtained. The natural frequency of the overall structure can be identified and obtained through the maximum and minimum values in the frequency response curve.

[0018] Furthermore, there is no requirement for the order of S1 and S2.

[0019] Furthermore, in S1, the vibration control equation of the conical shell is:

[0020]

[0021] In the above formula, u c , v c and w c are the axial, tangential and radial displacements of the conical shell respectively;

[0022] In formula (1) and is the partial differential coefficient in the vibration control equation of the conical shell;

[0023] The impedance matrix expression at both ends of the conical shell is:

[0024]

[0025] Furthermore, in S2, the vibration equation of the annular plate is:

[0026]

[0027] in D p =Eh 3 / 12(1-μ 2 );

[0028] The total impedance matrix of the ring plate is:

[0029]

[0030] Furthermore, in S3, the conical shell and the annular plate need to meet the conditions of displacement continuity and internal force balance at the splicing connection. The conical shell and the annular plate are assembled using a splicing method similar to finite element method to simulate the overall stiffness matrix of the ribbed conical shell.

[0031] Furthermore, in S4, any boundary condition is realized by an artificial spring. The artificial spring is achieved by installing a spring k on the corresponding degree of freedom element in the overall stiffness matrix. When k is close to 0, the entire model is in a free boundary condition; when k is relatively large, such as k = 1020 N / m and above, the entire model is in a rigid fixed boundary condition; and the stiffness k can take any corresponding value to realize an elastic boundary.

[0032] Furthermore, the arbitrary boundary of the artificial spring can be applied to a specific degree of freedom, such as the axial, tangential, radial or rotational direction of the conical shell, or can be applied to several degrees of freedom at the same time.

[0033] Furthermore, the verification of the conical shell solution is also included. The free vibration natural frequency of the conical shell is compared with the existing experiments and the dimensionless frequency of the conical shell is defined as The radius of the small end of the conical shell is R1, the radius of the large end is R2, and the geometric size of the conical shell is R2 / h=100. At the same time, in the existing experiment, the apex angle of the conical shell satisfies Lsinα / R2=0.25, α is the apex angle of the cone, and the entire conical shell adopts isotropic uniform material, and the Poisson's ratio of the material is μ=0.3.

[0034] Furthermore, the displacement response of the conical shell in formula (3) requires superposition of the power series m, while in actual calculation examples, the power series m needs to be truncated. The series cannot be infinite, so the calculation cost should be reduced as much as possible while ensuring the calculation accuracy.

[0035] The beneficial effects of the present invention are as follows:

[0036] The present invention features a compact, rational, and easy-to-use structure. By expressing the vibration control equations for the conical shell and concentric annular plates as a dynamic stiffness matrix related to internal forces and displacements, and utilizing the boundary conditions of displacement continuity and force balance between the ribs and conical shell subsegments, the present invention implements arbitrary boundary conditions for the ribbed conical shell structure by applying a modified large number method to the matrix elements. Furthermore, by solving the determinant of the global stiffness matrix, the natural frequency of the overall structure is obtained. This method allows design engineers in aviation, aerospace, and shipbuilding to quickly assess the natural frequency of tail structures, such as conical shells, during the design phase, thereby improving the safety and reliability of equipment containing conical shell structures.

[0037] At the same time, the present invention also has the following advantages:

[0038] (1) The conical shell natural frequency prediction method of the present invention is applicable to any elastic boundary, including traditional simply supported, clamped and free.

[0039] (2) The present invention proposes a method for predicting the natural frequency of not only ordinary smooth conical shells but also ribbed conical shells. The calculation method proposed in the present invention is applicable to both internal and external ribs.

[0040] (3) The present invention proposes a method for correcting the ill-conditioned stiffness matrix to ensure that the data in the natural frequency calculation is true and reliable.

[0041] (4) The method for calculating the natural frequency of the conical shell proposed in the present invention is based on a theoretical analytical method, which has the advantages of fast calculation speed and high convergence. In the calculation process, only a few parameters such as the geometric dimensions and materials of the conical shell need to be input. The parameter modeling analysis in the present invention has obvious advantages. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 It is a schematic diagram of the conical shell model structure in the present invention.

[0043] Figure 2 This is the displacement response diagram of the conical shell model of the present invention.

[0044] Figure 3 This is a vibration force analysis diagram of the circular ring plate in the present invention.

[0045] Figure 4 It is a flowchart of the present invention.

[0046] Figure 5 This is a schematic diagram of the ribbed conical shell structure in the form of internal ribs in the present invention.

[0047] Figure 6 This is a schematic diagram of the ribbed conical shell structure in the form of external ribs in the present invention. DETAILED DESCRIPTION

[0048] The specific embodiments of the present invention will be described below with reference to the accompanying drawings.

[0049] like Figure 4 As shown, this embodiment discloses a method for predicting the natural frequency of a ribbed conical shell. In the present invention, the stress and strain relationship between the in-plane and out-of-plane forces and displacements in the conical shell is used to write the vibration control equation of the conical shell into a dynamic stiffness matrix related to its internal forces and displacements.

[0050] The ribs are simulated by concentric circular plates, and the conical shell is discretized into multiple sub-segments. The two boundary conditions of displacement continuity and force balance between the ribs and the conical shell sub-segments are used to splice multiple substructures in the ribbed conical shell using a quasi-finite element method.

[0051] After obtaining the dynamic stiffness of the overall computational model, the method of correcting the matrix elements corresponding to each degree of freedom in the dynamic stiffness matrix is applied to achieve arbitrary boundary conditions for the stiffened conical shell structure, thereby correcting the ill-conditioned stiffness matrix. Furthermore, by solving the determinant of the overall stiffness matrix, a curve of the determinant of the overall model's dynamic stiffness matrix versus frequency is obtained. The natural frequencies of the overall structure can be identified and obtained through the maxima and minima in this frequency response curve.

[0052] At the same time, because the ribbed ribs in the present invention are simulated by annular plates, which are concentric circles with two radii R1 and R2, R2 is larger than R1. If the conical shell is spliced with the larger outer diameter R2 of the annular plate, it is an inner rib form, such as Figure 5 As shown, if the conical shell is spliced with the smaller outer diameter R1 of the annular plate, it is in the form of an external rib, such as Figure 6 The difference between the two lies in the selection of the splicing edge of the annular plate when the conical shell and the annular plate are spliced together. Therefore, the calculation method of the present invention can be applied to both ribbed conical shell structures.

[0053] like Figure 2 As shown, in the present invention, starting from the conical shell displacement vibration control equation, the displacement responses in three directions are written in the form of a power series. Through the convergence analysis of the power series, the cutoff number m and the circumferential modal cutoff number n in the displacement expressions in the three directions are determined.

[0054] According to classical shell theory, the vibration control equation of conical shell is:

[0055]

[0056] In the above formula, u c , v c and w c are the axial, tangential and radial displacements of the conical shell, respectively. Figure 2 As shown;

[0057] In formula (1) and is the partial differential coefficient in the vibration control equation of the conical shell:

[0058]

[0059] Using the power series superposition method to solve, the displacement expression of the conical shell can be written as:

[0060]

[0061] where a m , b m , c m is the power series superposition coefficient. Substituting equation (3) into the conical shell vibration equation (1), we can obtain the recursive relationship expression between the power series coefficients:

[0062]

[0063] The specific expression of the recursive coefficient in the above formula can be found in the literature. 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, 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:

[0064]

[0065] The coefficient expression in the above formula is:

[0066]

[0067] Substituting equations (5) and (6) into equation (3), the final expression of the conical shell displacement is:

[0068]

[0069] Similarly, based on the relationship between the internal force and displacement in the middle of the shell, the internal force expression on the conical shell section is:

[0070]

[0071] in:

[0072]

[0073] 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 cand x c =L c The displacement at the two nodes of the conical shell is considered. Consistent with the impedance matrix principle of the cylindrical shell, the impedance matrix expression at both ends of the conical shell can be obtained as:

[0074]

[0075] In this invention, an annular plate is used to simulate the reinforcement rib. Figure 2 As shown, the inner diameter is R1, the outer diameter is R2, and the thickness is h. p and u p is the in-plane vibration, w p and ψ p is the out-of-plane vibration.

[0076] In this embodiment, the vibration equation of the annular plate is:

[0077]

[0078] in D p =Eh 3 / 12(1-μ 2 ). The first two equations in equation (11) are in-plane vibration equations, and the third one is out-of-plane vibration. The displacement solution of the annular plate can be written as:

[0079]

[0080] The displacement of the annular plate can be written as the standard form of Bessel function, and its displacement in three directions can be written as follows:

[0081]

[0082] where k PB =(ρhω 2 / D p ) 1 / 4 , are the in-plane and out-of-plane vibration wave numbers, and the rotation angle J n and Y n are the first and second kind Bessel functions, I n and K n are the modified Bessel functions of the first and second kind respectively. 1m , A 2m , A 3m , A 4m , B 1m , B 2m , B 3m , B 4m is the coefficient of the Bessel function. The relationship between the internal force and displacement of the annular plate is as follows:

[0083]

[0084] Because the in-plane vibration and lateral vibration of the annular plate are decoupled from each other, they can be solved and analyzed separately. According to equations (13) and (14), the in-plane displacement and internal force matrix of the annular plate can be written as:

[0085]

[0086] Combining the above two equations, we can eliminate the unknown quantity [B 1n B 2n B 3n B 4n ] T , the in-plane impedance matrix of the circular ring plate is obtained as:

[0087]

[0088] Similarly, the out-of-plane impedance matrix of the circular ring plate can be obtained: Therefore, the total impedance matrix of the ring plate is:

[0089]

[0090] Splicing of ribbed structures:

[0091] In equations (10) and (18), the conical shell and the annular plate must satisfy displacement continuity and internal force equilibrium conditions at the splice joint. A finite element-like splicing method is used. The conical shell and the annular plate are assembled to simulate the overall stiffness matrix of the stiffened conical shell.

[0092] When the annular plate and the conical shell are spliced together, the impedance matrix of the conical shell is shown in Equation 10, and the impedance matrix of the annular plate is shown in Equation 18.

[0093] The conical shell and the annular plate divide the above two impedance matrices into four parts according to their ends. If an inner rib shape is used, the inner diameter of the conical shell is equal to the outer diameter of the annular plate. Therefore, the following conditions must be met: Therefore, the conical shell area ④ and the annular plate area ④ are used for splicing. If it is an external rib, the outer diameter of the conical shell is equal to the smaller outer diameter R1 of the annular plate, so it is necessary to meet the following requirements: Therefore, the conical shell area ④ and the annular plate area ① are spliced during the splicing process.

[0094] Description of boundary conditions:

[0095] Elastic boundary conditions can be achieved by artificial springs. Artificial springs can be achieved by installing springs k on the corresponding degree of freedom elements in the overall stiffness matrix. When k is close to 0, the entire model is in a free boundary condition; when k is relatively large, such as k = 10 20When the spring is at or above 0.001N / m, the entire model is in a rigidly clamped boundary condition; the stiffness k can take any corresponding value to achieve an elastic boundary. Artificial springs can achieve arbitrary boundaries by applying them to a specific degree of freedom, such as the axial, tangential, radial, or rotational directions of a conical shell. They can also be applied simultaneously to several degrees of freedom, addressing difficult areas that are difficult to address with classical boundaries in traditional shells.

[0096] Typical example table:

[0097] In order to verify the correctness of the conical shell solution, the free vibration natural frequency of the conical shell is compared with the literature, as shown in Table 1 and Table 2. First, the dimensionless frequency of the conical shell is defined as The radius of the small end of the conical shell is R1, and the radius of the large end is R2. The conical shell geometry is R2 / h = 100. According to the literature, the cone apex angle satisfies Lsinα / R2 = 0.25, where α is the cone apex angle. The entire conical shell is made of isotropic homogeneous material with a Poisson's ratio of μ = 0.3. Because the above uses dimensionless natural frequencies, there is no need to set a specific material density value.

[0098]

[0099] Table 1 Verification of natural frequency of conical shell simply supported at both ends

[0100]

[0101] Table 2 Verification of natural frequency of conical shell with simple support at small end and clamped support at large end

[0102] As can be seen from the two tables above, the calculated dimensionless natural frequency of the conical shell in this invention is essentially consistent with the results in the literature, proving the correctness of the calculation method of this invention. Because it is a dimensionless natural frequency, the dimensionless natural frequency can be calculated by giving the above parameters. No specific geometric parameters of the conical shell are required; any conical shell that meets the above geometric parameter ratios can meet the calculation requirements. The displacement response of the conical shell in formula (3) requires the superposition of the power series m. However, in actual calculation examples, the power series m must be truncated. This series cannot be infinite, so the calculation cost should be minimized while ensuring the calculation accuracy.

[0103] The method for predicting the natural frequency of a ribbed conical shell proposed by the present invention can help design engineers in aviation, aerospace, and shipbuilding to quickly evaluate the natural frequency of tail structures such as conical shells during the design stage of conical shell structures. The geometric parameters of the conical shell can be optimized based on the evaluation calculation results, thereby avoiding resonance between the natural frequency of the conical shell and the frequency of the external excitation load, and improving the safety and reliability of equipment containing conical shell structures.

[0104] References:

[0105] [1]Tong L.Free vibration of composite laminated conical shells[J].International Journal of Mechanical Science,1993.Vol.35,No.1:47-61;[2]T.Irie,G Yamada,Y Kaneko.Natural frequencies of truncated conicalshells[J].Journal of Sound and Vibration,1984,92,447.

[0106] 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 method for predicting the natural frequency of a stiffened conical shell, characterized in that: The steps include: S1. Using the stress-strain relationship between the in-plane and out-of-plane forces and displacements in a conical shell, the vibration control equation of the conical shell is expressed as a dynamic stiffness matrix related to its internal forces and displacements. S2. The ribs are simulated using concentric annular plates, and the vibration control equations of the annular plates are obtained, followed by the dynamic stiffness matrix of the annular plates. S3. Discretely divide the conical shell into multiple sub-segments, and use the two boundary conditions of displacement continuity and force balance between the ribs and the conical shell sub-segments to splice the multiple sub-structures in the ribbed conical shell using a quasi-finite element method; S4. On the premise of obtaining the dynamic stiffness of the overall calculation model, the arbitrary boundary conditions of the stiffened conical shell structure are realized by performing the correction and substitution method on the matrix elements corresponding to each degree of freedom in the dynamic stiffness matrix; S5. By solving the determinant of the overall stiffness matrix, a curve showing the dynamic stiffness matrix of the overall model changing with frequency is obtained. The natural frequency of the overall structure can be identified and obtained through the maximum and minimum values in the frequency response curve. In S1, the vibration control equation of the conical shell is: In the above formula, u c , v c and w c are the axial, tangential and radial displacements of the conical shell respectively; In formula (1) and is the partial differential coefficient in the vibration control equation of the conical shell; the impedance matrix expression at both ends of the conical shell is: In S2, the vibration equation of the annular plate is: in D p =Eh 3 / 12(1-μ 2 ); The total impedance matrix of the ring plate is:

2. The method for predicting the natural frequency of a stiffened conical shell according to claim 1, wherein: There is no requirement for the order of S1 and S2.

3. The method for predicting the natural frequency of a stiffened conical shell according to claim 1, wherein: In S3, the conical shell and the annular plate need to meet the conditions of displacement continuity and internal force balance at the splicing connection. The conical shell and the annular plate are assembled using a splicing method similar to that in finite element method to simulate the overall stiffness matrix of the ribbed conical shell.

4. A method for predicting the natural frequency of a stiffened conical shell according to claim 3, characterized in that: In S4, any boundary condition is achieved by an artificial spring. The artificial spring is achieved by installing a spring k on the corresponding degree of freedom element in the overall stiffness matrix. When k is close to 0, the entire model is in a free boundary condition. When k=10 20 When N / m or above, the entire model is in a rigidly fixed boundary condition; and the stiffness k takes any corresponding value to achieve an elastic boundary.

5. The method for predicting the natural frequency of a stiffened conical shell according to claim 4, wherein: The arbitrary boundary of the artificial spring is applied to a specific degree of freedom, such as the axial, tangential, radial or rotational direction of the conical shell, or is applied to several degrees of freedom at the same time.

6. The method for predicting the natural frequency of a stiffened conical shell according to claim 1, wherein: It also includes the verification of the solution results of the conical shell. The natural frequency of the free vibration of the conical shell will be compared with the existing experiments. The dimensionless frequency of the conical shell is defined as The radius of the small end of the conical shell is R1, the radius of the large end is R2, and the geometric size of the conical shell is R2 / h=100. At the same time, in the existing experiment, the apex angle of the conical shell satisfies Lsinα / R2=0.25, α is the apex angle of the cone, and the entire conical shell adopts isotropic uniform material, and the Poisson's ratio of the material is μ=0.

3.

7. The method for predicting the natural frequency of a stiffened conical shell according to claim 1, wherein: The displacement response of the conical shell in formula (3) requires superposition of the power series m, but in actual calculation examples, the power series m needs to be truncated. The series cannot be infinite, so the calculation cost should be reduced as much as possible while ensuring the calculation accuracy.

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