Novel bolt flange structure analytical model and verification method thereof

Abstract

The invention discloses a novel bolt flange structure analytical model and a verification method thereof, and belongs to the technical field of numerical analysis and computational mechanics. The method comprises the following steps: discretizing and modeling a bolt flange structure; deriving a unit stiffness matrix, a unit mass matrix and a unit inertia matrix for the rotor system; calculating the equivalent stiffness of a bolt for connecting the upper flange plate and the lower flange plate, wherein the equivalent stiffness comprises the axial stiffness and the shearing stiffness of the bolt and the stiffness of the joint surface of the connected flange plates; establishing a complete system kinetic equation of the bolt flange structure; and verifying the model. The problem that an existing method excessively depends on a simplified model or numerical simulation or experience statistics is solved, and a theoretical basis is provided for forward design and performance optimization of a connection structure.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of numerical analysis and computational mechanics, and particularly relates to a new bolted flange structure analytical model and a verification method thereof. BACKGROUND

[0002] The bolted flange connection structure has important applications in aerospace, ship, pipeline transportation and the like, and the research on the dynamic characteristics thereof has been a focus of scholars. The bolted flange structure has the characteristics of simple structure and good operability, and can provide certain connection stiffness and compression strength for a system.

[0003] The flange-bolt connection structure is widely used in the rotor system of an aero-engine due to its reliable connection, strong sealing and easy maintenance. The bolt connection in the rotor structure located at the rotating part is mostly the flange-bolt connection. The mechanical characteristics of the flange connection joint are complex at high speed rotation, which can cause the vibration characteristics of the rotor system to change suddenly. In a typical flange-bolt connection rotor structure, the short bolt connects the conical drum, the rotor disc and the drum, and provides an axial compression force to make the end surfaces of the connection structure contact, and then the end surface contact stress bears part of the torque transmission and the bending load, and the stopper realizes the cylindrical centering. If the load environment of the rotor changes dramatically, the additional bending moment of the rotor increases when the rotor works, and the flange connection end surface may appear local separation phenomenon, at this time, the contact state parameters will change, causing the bending stiffness of the connection structure to be lost, and then causing the vibration of the rotor base frequency to change suddenly, which affects the stable operation of the engine. At the same time, when the relative slip occurs between the related parts of the connection interface due to movement, the interface energy dissipation will be generated, and the contact friction damping will be introduced. In addition, when the engine thrust changes dramatically during the maneuvering flight, the rotor system will bear a large axial force, which causes the interface contact stress to increase, and the interface contact stiffness to fluctuate. The changes of the above-mentioned macroscopic physical quantities of the rotor structure, such as stiffness and damping, come from the state and mechanical characteristics of the flange-bolt connection. Therefore, the fundamental scientific problem causing the above-mentioned engineering problems lies in clarifying the interaction mechanism between the interface connection characteristics and the change of the vibration response of the rotor structure.

[0004] The contact characteristics of the flange-bolt connection interface are influenced by the load, geometry, assembly and other characteristic parameters, the complex mechanism is not clear, and the function relationship between the connection characteristics and the macro physical quantities of the rotor structure is not easy to accurately characterize. The bolted flange connection structure is different from the simple bolt connection, the number of connecting bolts is more, the working load is complex, and the pre-tightening force is more dispersed, so the dynamic analysis is more complex. At the same time, when the rotor structure is working, the flange-bolt connection structure not only rotates under the action of the centrifugal load, but also produces precession under the influence of the bending load, and the load conditions of the components of the connection structure under different working conditions are complex. The geometry of the flange-bolt connection structure located at different positions of the rotor structure is also different. On the other hand, the process characteristic parameters caused by assembly and manufacturing change the stress and strain distribution of the bonding surface to a certain extent, which increases the difficulty of mechanism characterization, and makes it difficult to characterize the influence of external load and geometry on the stiffness and contact state of the bolted flange structure.

[0005] The existing research shows that regarding the bolted flange structure as a rigid or continuous structure will have a great influence on the connection stiffness, due to the discontinuity and slippability of the connection interface structure, the structure will be softened in stiffness, the friction damping energy will be consumed, and thus the rotor system will produce complex vibration characteristics, so it is necessary to model the bolted flange structure in detail. At the same time, the current model cannot comprehensively and accurately predict the static and dynamic load transmission path and vibration energy distribution of the bolted flange structure, and when the structure geometry changes, the model may fail, which brings difficulties to improve the load transmission and vibration capacity of the connection structure.

[0006] Therefore, it is urgent to propose a new analytical model of the bolted flange structure and verify its accuracy, in order to solve the above technical problems. SUMMARY

[0007] Technical problems to be solved: In order to avoid the shortcomings of the prior art, the present application provides a new analytical model of the bolted flange structure and a verification method thereof, which introduces the finite element discrete idea and the Timoshenko beam theory, decomposes the complex flange structure into variable cross-section beam elements for assembly, and strictly derives the complete set of dynamic matrices, solves the excessive dependence on simplified models, numerical simulation or empirical statistics of the existing method, and provides a theoretical basis for the forward design and performance optimization of the connection structure.

[0008] The technical scheme of the present application is: a method for establishing and verifying an analytical model of a bolted flange structure, comprising the following steps: S1, structure discretization modeling: Based on the finite element discretization concept, the flange in the bolted flange structure is uniformly divided into several sector regions in the circumferential direction, and the flange body structure of each sector region is modeled as one or more variable cross-section beam elements. The variable cross-section beam elements are constructed based on Timoshenko beam theory and are used to accurately simulate the elastic deformation of the flange under the combined action of tension, compression, bending, shear and torsion. S2. Derivation of the unit dynamics matrix: For each variable cross-section beam element established in step S1, derive its element stiffness matrix, element mass matrix, and element inertia matrix for the rotor system. S3. Establish the bolt connection model: Calculate the equivalent stiffness of the bolts connecting the upper and lower flanges. The equivalent stiffness includes the axial stiffness and shear stiffness of the bolts themselves, as well as the stiffness of the mating surfaces of the flanges being connected. S4. System dynamics equations assembly: The dynamic matrices of all the variable cross-section beam elements in step 2 are assembled into the overall dynamic matrix of the disk. The overall dynamic matrix includes the overall mass matrix, overall damping matrix, overall inertia matrix, and overall stiffness matrix. Then, the equivalent stiffness of the bolts is used as an external excitation to combine the stiffness matrices of the upper and lower disks. The system damping matrix is ​​introduced to establish the complete system dynamic equations of the bolt flange structure:

[0009] in, These are the overall mass matrix, overall damping matrix, overall inertia matrix, overall stiffness matrix, and external excitation matrix of the bolted flange structure, respectively. These are the structure's acceleration vector, velocity vector, and displacement vector, respectively. The rotational speed of the structure; S5. Model Validation: The structural natural frequencies calculated by this analytical model are compared with the simulation and experimental results through finite element simulation and / or experimental modal testing to verify the accuracy and effectiveness of the analytical model. A further technical solution of the present invention is: in step S1, the flange is divided into multiple fan-shaped regions in the circumferential direction, and each region is discretized into a first variable cross-section beam element and a second variable cross-section beam element in the circumferential direction. The second variable cross-section beam element is set to avoid deformation divergence of the first variable cross-section beam element at the boundary. The number of sector regions is a multiple of the number of bolts in the bolted flange structure; the first variable cross-section beam is the main body of the sector region, and its axial length is the difference between the outer diameter and the inner diameter of the solid flange; the central angle of the second variable cross-section beam is smaller than that of the first variable cross-section beam, and its axial length satisfies the aspect ratio of the Timoshenko beam.

[0010] A further technical solution of the present invention is: in step S2, when deriving the unit mass matrix and the unit inertia matrix, the mass distribution and rotational inertia distribution per unit length of the variable cross-section beam unit are considered, and obtained by integration through the shape function matrix.

[0011] A further technical solution of the present invention is: in step S2, the element stiffness matrix for:

[0012] in, This is the axial tensile / compressive stiffness element of the beam element; the axial direction of the beam element is... direction; This is the axial torsional stiffness element for the beam element. These are non-axial stiffness matrix elements. ; Unit mass matrix for:

[0013] in, For the axial mass matrix elements of the beam element, These are non-axial mass matrix elements. For non-axial inertial matrix elements, ; Element inertia matrix for rotor systems for:

[0014] in, These are non-axial gyroscope matrix elements. .

[0015] A further technical solution of the present invention is: based on the element stiffness matrix and element mass matrix, using the Rayleigh damping model, the element damping matrix is ​​obtained. The expression is as follows:

[0016] in, is the damping coefficient of Ruili.

[0017] A further technical solution of the present invention is: in step S3, the axial stiffness of the bolt itself The calculation formula is as follows:

[0018] in, The equivalent length of the bolt shank, This is the equivalent length of the bolt head. This is the equivalent length of the screw. This is the equivalent length of the threaded portion. The cross-sectional area of ​​the bolt's smooth rod. Let be the equivalent cross-section of the bolt. The elastic modulus of the bolt material; The formula for calculating the shear stiffness is as follows:

[0019] in, Where is the nominal diameter of the bolt. This is the length of the screw.

[0020] The formula for calculating the stiffness of the mating surfaces of the connected flanges is as follows:

[0021] in, The outer diameter of the nut cross-section. Where is the nominal diameter of the bolt. The elastic modulus of the flange material. For the thickness of the flange, The included angle of the cone-shaped model.

[0022] A further technical solution of the present invention is: in step S4, when assembling the overall stiffness matrix and the overall mass matrix of the system, a transformation matrix is ​​introduced to transform and assemble the dynamic matrix of each variable cross-section beam element in the local coordinate system into the global coordinate system. A further technical solution of the present invention is: in step S5, the experimental modal test adopts the hammer impact method, and obtains the frequency response function of the overall structure of the bolt flange through multi-point excitation and multi-point vibration pickup, and identifies the experimental modal parameters accordingly.

[0023] A system for establishing and verifying an analytical model of a bolted flange structure, characterized by comprising: The structural discretization modeling module is used to divide the flange in the bolted flange structure into several sector regions in the circumferential direction based on the finite element discretization concept, and to model the flange body structure of each sector region as one or more variable cross-section beam elements based on Timoshenko beam theory to simulate the elastic deformation of the flange under combined loads. The element dynamics matrix derivation module is communicatively connected to the structure discretization modeling module and is used to derive the element stiffness matrix, element mass matrix, and element inertia matrix for the rotor system for each of the established variable cross-section beam elements. The bolt connection model calculation module is used to calculate the equivalent stiffness of the bolts connecting the upper and lower flanges. The equivalent stiffness integrates the axial stiffness and shear stiffness of the bolt itself, as well as the compressive stiffness of the mating surface of the connected flanges. The system dynamics equation assembly module is communicatively connected to the element dynamics matrix derivation module and the bolt connection model calculation module, respectively. It is used to assemble the dynamics matrices of all variable cross-section beam elements, combine the stiffness matrices of the upper and lower plates by taking the equivalent stiffness of the bolts as an external excitation, and introduce the system damping matrix to establish the complete system dynamics equations of the bolt flange structure. The model verification module communicates with the system dynamics equation building module and is used to compare the calculation results of this analytical model with the external finite element simulation results and / or experimental modal test data to verify the accuracy and effectiveness of the analytical model.

[0024] An electronic device, characterized in that it includes at least one processor and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program executable by the at least one processor, the computer program being executed by the at least one processor to enable the at least one processor to perform the method for establishing and verifying the analytical model of the bolt flange structure.

[0025] Beneficial effects The beneficial effects of this invention are as follows: This invention is the first to combine Timoshenko beam theory with the finite element discretization concept for dynamic modeling of bolted flange structures. By establishing variable cross-section beam elements, the model can accurately simulate the complex deformation of the flange under combined tension, compression, bending, shear, and torsion. Its analytically derived stiffness matrix, mass matrix, etc., have clear physical meanings. The effectiveness and accuracy of the model have been verified through experiments and the finite element method, fundamentally overcoming the shortcomings of traditional spring models that oversimplify the connection interface and fail to reflect the deformation coordination relationship of the continuum.

[0026] This invention establishes a complete system dynamics equation, capable of simultaneously predicting the deformation and stress distribution of a structure under static loads, as well as its natural frequencies, mode shapes, and unbalanced responses under dynamic loads. This makes the model particularly suitable for the design and analysis of high-end equipment such as aero-engine rotor systems, which have stringent requirements for vibration characteristics.

[0027] This invention establishes the correlation between external loads and geometry on the stiffness and contact state of bolted flange structures, enabling comprehensive and accurate prediction of static and dynamic load transmission paths and vibration energy distribution, thus facilitating the introduction of friction and slippage. Furthermore, this invention lays the foundation for the optimized design of bolted flange structures (such as the number, size, and geometric distribution of bolts). Attached Figure Description Figure 1 This is a schematic diagram of the bolt flange structure discretized into a tapered variable cross-section beam in an embodiment of the present invention; Figure 2 These are the first six modes in this embodiment of the invention; (a) the initial disk; (b) the discretized disk. Figure 3 This is a cross-sectional view of the beam in the YOX plane in an embodiment of the present invention; Figure 4 This is a single-disk assembly model in an embodiment of the present invention; Figure 5 This is the single-disk assembly process in an embodiment of the present invention; Figure 6 This is the bolt stiffness model in the embodiments of the present invention; Figure 7 This is the flange mating surface stiffness model in an embodiment of the present invention; Figure 8 This is a variable cross-section cantilever beam model in an embodiment of the present invention; Figure 9 This is the modal testing device in the embodiments of the present invention; Figure 10 These are the experimental modal data in the embodiments of the present invention; Figure 11 This is the finite element model of the bolt flange structure in the embodiment of the present invention; Figure 12 These are the first six modes in the embodiments of the present invention. Detailed Implementation The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the invention, and should not be construed as limiting the invention.

[0028] Currently, the analysis and evaluation methods for bolted flange structures are mainly divided into the following categories, but all of them have significant limitations: 1. Methods based on traditional engineering algorithms and strength assessment (such as CN104156498A): These methods perform stress analysis by establishing a finite element model of the bolt, which can quickly assess the bolt's strength limit under complex loads. However, their focus is on static strength and yield criterion, and they cannot reflect the overall dynamic behavior of the connection structure under dynamic loads (such as high-speed rotation and vibration), load transfer path, and vibration energy distribution, let alone predict the critical speed and mode shape of the rotor system.

[0029] 2. Methods based on spring models and load distribution calculations (e.g., CN106295024B): These methods simplify the connection structure into a network of spring elements, efficiently calculating the load distribution of multi-bolted connections and considering nonlinear factors such as clearance and friction. However, their models are highly simplified, treating the connection interface as discrete springs, and cannot characterize the deformation coordination relationship of the flange as a continuous body. Therefore, it is difficult to accurately predict the overall stiffness matrix, mass matrix, and dynamic response of the structure, and it is especially unsuitable for rotor system analysis with strict requirements on vibration characteristics.

[0030] 3. Methods based on assembly process and interaction optimization (e.g., CN110610057A): These methods optimize the initial preload of bolts through an elastic interaction coefficient matrix, aiming to make the final preload distribution more uniform and improve sealing performance. The core is to solve the mechanical coupling problem during assembly; however, the model itself is static and quasi-linear, and does not involve the dynamic equations of the structure under working conditions (such as under centrifugal force or gyroscopic effects), nor can it analyze key dynamic parameters such as the structure's natural frequencies and modes.

[0031] 4. Methods based on experimental statistics and condition monitoring (e.g., CN116542072A): These methods infer slippage and deformation at the connection interface by monitoring changes in bolt assembly torque, thereby assessing the health status of the rotor system. While this method provides an effective engineering monitoring tool, it is essentially a "black box" or "grey box" empirical statistical method, lacking analytical models that can explain the intrinsic relationship between connection stiffness, damping, external loads, and geometric parameters from a physical mechanism perspective. Therefore, it is difficult to use in early-stage optimization design.

[0032] In summary, existing technologies share a common core deficiency: they either focus on static strength or rely on simplified models and empirical statistics, failing to establish an analytical model that accurately describes the overall dynamic characteristics of bolted flange connection structures based on their physical essence. Therefore, this invention proposes a method for establishing and verifying an analytical model of bolted flange structures, comprising the following steps: S1. Structural Discretization Modeling: Based on the finite element discretization concept, the flange in the bolted flange structure is uniformly divided into several sector regions in the circumferential direction, and the flange body structure of each sector region is modeled as one or more variable cross-section beam elements. The variable cross-section beam elements are constructed based on Timoshenko beam theory and are used to accurately simulate the elastic deformation of the flange under the combined action of tension, compression, bending, shear and torsion. S2. Derivation of the unit dynamics matrix: For each variable cross-section beam element established in step S1, derive its element stiffness matrix, element mass matrix, and element inertia matrix for the rotor system. S3. Establish the bolt connection model: Calculate the equivalent stiffness of the bolts connecting the upper and lower flanges. The equivalent stiffness includes the axial stiffness and shear stiffness of the bolts themselves, as well as the stiffness of the mating surfaces of the flanges being connected. S4. System dynamics equations assembly: The dynamic matrices of all the variable cross-section beam elements in step 2 are assembled into the overall dynamic matrix of the disk. The overall dynamic matrix includes the overall mass matrix, overall damping matrix, overall inertia matrix, and overall stiffness matrix. Then, the equivalent stiffness of the bolts is used as an external excitation to combine the stiffness matrices of the upper and lower disks. The system damping matrix is ​​introduced to establish the complete system dynamic equations of the bolt flange structure:

[0033] in, These are the overall mass matrix, overall damping matrix, overall inertia matrix, overall stiffness matrix, and external excitation matrix of the bolted flange structure, respectively. These are the structure's acceleration vector, velocity vector, and displacement vector, respectively. The rotational speed of the structure; S5. Model Validation: The structural natural frequencies calculated by this analytical model are compared with the simulation and experimental results through finite element simulation and / or experimental modal testing to verify the accuracy and effectiveness of the analytical model.

[0034] This invention also proposes a system for establishing and verifying an analytical model of a bolted flange structure, comprising: The structural discretization modeling module is used to divide the flange in the bolted flange structure into several sector regions in the circumferential direction based on the finite element discretization concept, and to model the flange body structure of each sector region as one or more variable cross-section beam elements based on Timoshenko beam theory to simulate the elastic deformation of the flange under combined loads. The element dynamics matrix derivation module is communicatively connected to the structure discretization modeling module and is used to derive the element stiffness matrix, element mass matrix, and element inertia matrix for the rotor system for each of the established variable cross-section beam elements. The bolt connection model calculation module is used to calculate the equivalent stiffness of the bolts connecting the upper and lower flanges. The equivalent stiffness integrates the axial stiffness and shear stiffness of the bolt itself, as well as the compressive stiffness of the mating surface of the connected flanges. The system dynamics equation assembly module is communicatively connected to the element dynamics matrix derivation module and the bolt connection model calculation module, respectively. It is used to assemble the dynamics matrices of all variable cross-section beam elements, combine the stiffness matrices of the upper and lower plates by taking the equivalent stiffness of the bolts as an external excitation, and introduce the system damping matrix to establish the complete system dynamics equations of the bolt flange structure. The model verification module communicates with the system dynamics equation building module and is used to compare the calculation results of this analytical model with the external finite element simulation results and / or experimental modal test data to verify the accuracy and effectiveness of the analytical model.

[0035] The present invention also proposes an electronic device, including at least one processor and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program executable by the at least one processor, the computer program being executed by the at least one processor to enable the at least one processor to execute the method for establishing and verifying the analytical model of the bolt flange structure.

[0036] The present invention also proposes a computer-readable storage medium storing computer instructions, which enable a processor to implement the method for establishing and verifying the analytical model of the bolt flange structure when executed.

[0037] The above technical solution will be further explained below with examples: In one embodiment, the method for establishing and verifying the analytical model of a bolted flange structure includes: Step 1: Bolted Flange Structure Modeling In traditional rotor system modeling, the transfer matrix method and the finite element method are commonly used. The finite element method (FEM) is a commonly used method for numerically solving partial differential equations in engineering and mathematical modeling, widely applied in structural analysis, heat transfer, and fluid flow. It discretizes complex problems into simple finite elements, which are then combined into a global system of equations for solution. Based on this idea, the bolted flange structure can be discretized into a tapered variable cross-section beam for solution, with the two discs connected by a bolt-linear spring model, such as... Figure 1 .

[0038] The more beams there are, the more accurate the approximation, but the computational load will also increase. This invention divides the flange into 24 first variable cross-section beam elements and 24 second variable cross-section beam elements at an angle of 15 degrees. The second variable cross-section beam elements are used to avoid the boundary divergence of the first variable cross-section beam elements.

[0039] The number of sector regions is determined by the number of bolts. For example, in this example, 12 bolts are used, so the sector regions should be divided at 360 / 12 = 30 degrees. However, to ensure the accuracy of the model simulation, the angle of division should be as small as possible. Therefore, this paper divides the region a second time based on 30 degrees, with an angle of 15 degrees, dividing the disc into 24 sector regions. The first variable cross-section beam is the main body of the sector region, with an angle of 14°. The length of the beam is determined by the difference between the outer and inner diameters of the solid flange. The second variable cross-section beam is used to prevent the deformation divergence of the first variable cross-section beam element at the boundary. It is located between two adjacent first variable cross-section beam elements, with an angle of 1°. Its length should be as small as possible, but it should meet the aspect ratio of a short and thick beam like the Timoshenko beam, which is more in line with the theoretical model.

[0040] To verify the feasibility of the method, finite element software was used to compare its first six mode shapes and natural frequencies, and the results are as follows: Figure 2 As shown in Table 1.

[0041] Table 1. First six natural frequencies

[0042] As can be seen, the first six modes of vibration are the same for both the disk model and the discrete model, and the maximum error in the natural frequency is 9.52%, demonstrating the reliability of the discrete model. The next step is to derive its analytical model.

[0043] Step 2: Flange Model Establishment After obtaining the discrete model, its dynamic equations are derived. The key step lies in deriving the element matrix of the variable cross-section beam. In the finite element method, the variable cross-section beam is often discretized using 2-node elements. Let the point axis element be a Timoshenko beam, with each node having 6 degrees of freedom in three directions: displacement in three directions and rotation in three directions. Therefore, one axis element has 12 degrees of freedom, and its generalized coordinates are... (1) in, This represents the displacement of the two ends of the axis element in the axial direction (X direction); This represents the displacement of the two ends of the axis element in the Y direction; This represents the displacement of the axis element in the Z direction; The rotation angle of the two ends of the axis element around the X direction; The angle of rotation around the Y direction at both ends of the axis element; Let be the rotation angle of the two ends of the axis element around the Z direction. The degree of freedom in the X direction is caused by the tensile and compressive deformation of the axis segment, while the degrees of freedom in the Y and Z directions are caused by the bending and torsional deformation.

[0044] First, derive the relevant matrices for bending and torsional deformation, starting with the kinetic and potential energy of the infinitesimal segment. Displacement... With corner The relationship can be approximated as: (2) Displacement of any point within the axis element Generalized coordinates of the endpoints can be used to represent ,Right now (3) in, For shape function matrix (4) Similarly, the angle of rotation at any point within the axis element. It can also be represented by the generalized coordinates of the endpoints. ,Right now (5) in, For shape function matrix (6) In the YOX plane, as shown in Figure x, the equilibrium condition of the infinitesimal element is: (7) Based on the Timoshenko beam assumption, we have (8) In the formula, For bending moment, The elastic modulus of the material, Let the moment of inertia of the cross section be... For shear force, For shear modulus, The effective shear area of ​​a rectangular beam is expressed as follows: (9) in, and Let be the width and height of the rectangular cross-section, respectively. Then its moment of inertia... .

[0045] From equations (7) and (8), we can obtain (10) Assume the solution to equation (10) is (11) Combining equation (9-11), we can obtain (12) in, Let be the coordinates of any point on the x-axis. These are undetermined coefficients. Substitute them into the boundary conditions. (13) It can be obtained Substituting into equations (11) and (12) yields the following results. and The expression for generalized coordinates can be simplified to obtain... (14) Similarly, following the above process, it can also be determined in the ZOX plane. and Expressions for generalized coordinates (15) Then the elastic bending deformation energy of this shaft segment and kinetic energy They are respectively (16) (17) In the formula, For shear deformation in the YOX plane; For shear deformation of the ZOX plane; The mass per unit length; The moment of inertia per unit length is denoted as , where , ; The polar moment of inertia per unit length; This is the rotor's rotational speed.

[0046] After substituting and integrating over the total length of the elements, we get (18) In the formula, (19) From the Lagrange equation (20) The equations of motion for the axis elements are obtained as follows: (twenty one) In the formula, Let be the element inertia matrix in the rotor system; The element stiffness matrix; For external excitation on the shaft element, Here is the element damping matrix.

[0047] Among them, each matrix is (twenty two) in, These are non-axial mass matrix elements. For non-axial inertial matrix elements, These are non-axial stiffness matrix elements. These are non-axial gyroscope matrix elements.

[0048] Next, the axial tensile and compressive stiffness of the shaft element is derived. By the law of conservation of energy, the strain energy of the system is equal to the work done by the external force. Assume the external force acting on it is... Then the work done by the external force is (twenty three) And strain energy is (twenty four) in, ,again Therefore (25) From strain energy Equal to external work ,get Therefore, tensile stiffness (26) The torsional stiffness of a shaft element can be calculated using the following formula: (27) According to mechanics of materials, This is called torsional stiffness, where and These are the height and width of the cross-section (the longer side is the height, and the shorter side is the width). For the ratio The relevant parameters are shown in Table 2.

[0049] Table 2 Torsional coefficient of rectangular section

[0050] It is worth mentioning that when At this time, the cross-section becomes a long and narrow rectangle. Furthermore, since the beam element in this paper is a variable cross-section beam, this ratio is constantly changing, and the coordinates of any point on the axis on the X-axis are also constantly changing. When a certain value is reached, the height and width of the cross-section will interchange, that is, when , Sometimes, ;when ,have Therefore, the method of taking the median value in segments is adopted.

[0051] In the first case, when hour, ,So ;when hour, ,So This can be divided into 10 segments in total, until... , This can be used to calculate Similarly, the torsional stiffness of each segment can be calculated for the second case. In summary, the axial torsional stiffness of the shaft segment can be calculated as follows: .

[0052] In summary, the overall element stiffness matrix of the shaft element can be obtained as follows: (28) The overall mass matrix of the shaft element is: (29) In the formula, , , , .

[0053] This yields the stiffness and mass matrices of the axial elements of the first variable cross-section beam element. The derivation process for the second variable cross-section beam element is the same as that for the first, the difference being the integration region. After obtaining the stiffness and mass matrices of the axial elements of the first and second variable cross-section beam elements, matrix assembly can be performed to obtain the overall stiffness and mass matrix of the flange. Before assembly, the matrix is ​​multiplied by the transpose matrix, where the transpose matrices of the first and second variable cross-section beam elements are respectively... (30) (31) in, and Representing the first The first variable cross-section beam element and the first The transpose matrix of the second variable cross-section beam element assembled into the first beam element. After multiplying by the transpose matrix, the assembly process is as follows, as shown in the figure; this yields the mass matrix and stiffness matrix of the flange, and its damping matrix adopts the Rayleigh damping form, i.e. (32) in, is the damping coefficient of Ruili.

[0054] Step 3: Bolt Model The upper and lower flanges are connected by bolts, and the connection rigidity is... It can be divided into the stiffness of the bolt itself. Stiffness of the joint surface According to bolt calculation method VDI2230, bolt stiffness calculation can be simplified to a stepped shaft form. The cross-sectional area of ​​the bolt's smooth rod. Let be the equivalent cross-section of the bolt. , , , These are the equivalent lengths of the bolt head, bolt shank, bolt rod, and threaded portion, respectively.

[0055] like Where is the nominal diameter of the bolt. Let be the pitch, then we have: (33) The bolt stiffness is then: (34) The equivalent length of the bolt rod in the above formula Equivalent length of screw The actual dimensions of the bolted connection are easily obtained; the equivalent length of the bolt head of a typical hexagonal head bolt is... The equivalent length of the bolt head of the internal hexagonal head bolt. Therefore, the key issue lies in determining the equivalent length of the threaded portion. .

[0056] In the threaded section, under the action of axial external force, in addition to the deformation of the bolt shaft, the mating threads also undergo elastic deformation. Therefore, the method for calculating the stiffness of the threaded section differs from that of the bolt shaft. The formula for calculating the equivalent length of the threaded section is as follows: (35) In the formula: (36) (37) (38) In the formula, , The elastic modulus of bolts and nuts; , Poisson's ratio for bolts and nuts; The thread pitch diameter; It is a tooth-shaped half-angle; This is the equivalent cross-sectional area of ​​the nut; For thread helix angle; The outer diameter of the nut's cross-section; This refers to the length of the threaded portion (usually the thickness of the nut).

[0057] Meanwhile, the shear stiffness of the bolt (39) in, Where is the nominal diameter of the bolt. This is the length of the screw.

[0058] Regarding the stiffness of the connected surfaces, the internal normal stress distribution of the connected surfaces during bolted connections follows a conical distribution, such as... Figure 7 As shown.

[0059] Its stiffness calculation formula is: (40) in, The outer diameter of the nut cross-section. Where is the nominal diameter of the bolt. The elastic modulus of the flange material. For the thickness of the flange, The included angle of the cone-shaped model.

[0060] (41) Therefore, the equation of motion for the bolted flange structure can be established, as shown in the following equation. (42) in, These are the overall mass matrix, overall damping matrix, overall gyro matrix, and stiffness matrix of the integral bolted flange structure. The integral bolted flange structure includes upper and lower flanges and bolts. The matrix forms for the lower and upper flanges are identical. Regarding the overall stiffness... Distributed along the diagonal in the middle, External excitations for the model include bolt forces and external forces for fixing constraints.

[0061] Step 4: Model Validation Verification of element stiffness matrix To verify the correctness of the above expression for the element stiffness matrix, a variable cross-section cantilever beam as shown in the figure is used as the research object, and its finite element model is established using finite element software. Beam length The starting point of the beam's integral. The endpoint of the points ,thickness included angle elastic modulus ,density shear modulus A cantilever beam is free at one end and fixed at the other, with a load applied to the free end. .

[0062] Based on the above derivation process, the element stiffness matrix is ​​obtained. The calculation process is shown in equations (22), (27), and (28), combined with the load array. Solve the static equilibrium equations The displacement of the free end of the cantilever beam is obtained. The results were compared with those from the finite element software, and the results are shown in Table 3.

[0063] Table 3 Comparison of free end displacements of variable cross-section cantilever beams

[0064] As can be seen, the displacement along the beam axis and displacement perpendicular to the flange face The small error verifies the correctness of the stiffness matrix expressions for the variable cross-section beam elements in these two directions. The displacement error along the beam tangent, i.e., the flange tangent, is larger, but since the flange is unlikely to deform in the tangential direction, the displacement in this direction is not the focus of this paper. Furthermore, subsequent research shows that there is no corresponding modal vibration in this direction.

[0065] Model Modal Validation To further verify the correctness of the model, natural frequency analysis was performed on the overall structure of the bolted flange. Experimental verification was conducted using the bolted flange structure as the research object. The bolted flange structure mainly consists of two flange discs, upper and lower, 12 M4 bolts, and short, thick shafts on both sides for connection. The disc thickness... elastic modulus ,density shear modulus .

[0066] Modal testing of the bolted flange structure was performed using the impact test method. The testing system mainly included: a signal acquisition system, a force hammer, an accelerometer, and a computer. Figure 9 The natural frequencies of the structure were obtained using a multi-point excitation and multi-point vibration pickup method. Points 1-14 were the excitation points for the force hammer, distributed on the wheel, to obtain the corresponding vibration response. Points 2-14 were the mounting points for the accelerometer to pick up the response signal. The experimental results are shown in the figure. Due to the limitation of the force hammer, the input energy attenuates significantly after 2500Hz, so only data between 0 and 2500Hz can be measured.

[0067] Furthermore, to fully verify the effectiveness and accuracy of the proposed method, a finite element model of the overall bolted flange structure was established using finite element software, such as... Figure 11 As shown, the connection between the root of the short, thick shaft and the disk surface is rigid, and the shaft end is fixedly supported and constrained. Table 4 shows the first six natural frequencies of the bolted flange structure obtained through experiments, finite element analysis, and the method presented in this paper.

[0068] Table 4. First Six Natural Frequencies

[0069] Note: Error A in the table represents the error analysis between the method presented in this paper and the finite element model; Error B represents the error analysis between the method presented in this paper and the experimental results.

[0070] As can be seen, due to the limitations of the hammer, the input energy attenuates significantly after 2500Hz, so only data between 0 and 2500Hz can be measured. However, within this range, the first three modes are basically consistent. Nevertheless, natural frequencies such as 1665Hz, 1721Hz, and 2100Hz, which are not found in theoretical methods or finite element models, still appear. This is because a base was added to the bolt flange structure to ensure the boundary conditions of the fixed support during the experiment. Due to cost considerations, the weight of the base was not negligible, resulting in additional modes. Furthermore, welding errors of the shaft and bolts on the disk surface during processing also contribute to the error. As shown in the table, the maximum error A is 9%. This error originates from the applied boundary conditions. The method used in this paper applies high stiffness to the center node of the disk to simulate fixed support. In the finite element model, fixed support is applied to the short, thick section of the shaft, which leads to differences in the predicted natural frequencies. The maximum error B is 15.08%, caused by factors including the weight of the base, inaccurate shaft welding, alignment of bolts on the disk surface during manufacturing, and the influence of sensor weight. Analysis of the first six vibration modes in Figure 12 shows that there are no vibration modes along the flange tangent, thus confirming the previous conclusion.

[0071] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A method for establishing and verifying an analytical model of a bolted flange structure, characterized in that, Includes the following steps: S1. Structural Discretization Modeling: Based on the finite element discretization concept, the flange in the bolted flange structure is uniformly divided into several sector regions in the circumferential direction, and the flange body structure of each sector region is modeled as one or more variable cross-section beam elements. The variable cross-section beam elements are constructed based on Timoshenko beam theory and are used to accurately simulate the elastic deformation of the flange under the combined action of tension, compression, bending, shear and torsion. S2. Derivation of the unit dynamics matrix: For each variable cross-section beam element established in step S1, derive its element stiffness matrix, element mass matrix, and element inertia matrix for the rotor system. S3. Establish the bolt connection model: Calculate the equivalent stiffness of the bolts connecting the upper and lower flanges. The equivalent stiffness includes the axial stiffness and shear stiffness of the bolts themselves, as well as the stiffness of the mating surfaces of the flanges being connected. S4. System dynamics equations assembly: The dynamic matrices of all the variable cross-section beam elements in step 2 are assembled into the overall dynamic matrix of the disk. The overall dynamic matrix includes the overall mass matrix, overall damping matrix, overall inertia matrix, and overall stiffness matrix. Then, the equivalent stiffness of the bolts is used as an external excitation to combine the stiffness matrices of the upper and lower disks. The system damping matrix is ​​introduced to establish the complete system dynamic equations of the bolt flange structure: in, These are the overall mass matrix, overall damping matrix, overall inertia matrix, overall stiffness matrix, and external excitation matrix of the bolted flange structure, respectively. These are the structure's acceleration vector, velocity vector, and displacement vector, respectively. The rotational speed of the structure; S5. Model Validation: The structural natural frequencies calculated by this analytical model are compared with the simulation and experimental results through finite element simulation and / or experimental modal testing to verify the accuracy and effectiveness of the analytical model.

2. The method for establishing and verifying an analytical model of a bolted flange structure according to claim 1, characterized in that: In step S1, the flange is divided into multiple sector regions in the circumferential direction. Each region is discretized into a first variable cross-section beam element and a second variable cross-section beam element in the circumferential direction. The second variable cross-section beam element is set to avoid deformation divergence of the first variable cross-section beam element at the boundary. The number of sector regions is a multiple of the number of bolts in the bolted flange structure; the first variable cross-section beam is the main body of the sector region, and its axial length is the difference between the outer diameter and the inner diameter of the solid flange; the central angle of the second variable cross-section beam is smaller than that of the first variable cross-section beam, and its axial length satisfies the aspect ratio of the Timoshenko beam.

3. The method for establishing and verifying an analytical model of a bolted flange structure according to claim 2, characterized in that: In step S2, when deriving the element mass matrix and element inertia matrix, the mass distribution and moment of inertia distribution per unit length of the variable cross-section beam element are considered, and obtained by integration using the shape function matrix.

4. The method for establishing and verifying an analytical model of a bolted flange structure according to claim 3, characterized in that: In step S2, the element stiffness matrix for: in, This is the axial tensile / compressive stiffness element of the beam element; the axial direction of the beam element is... direction; This is the axial torsional stiffness element for the beam element. These are non-axial stiffness matrix elements. ; Unit mass matrix for: in, For the axial mass matrix elements of the beam element, These are non-axial mass matrix elements. For non-axial inertial matrix elements, ; Element inertia matrix for rotor systems for: in, These are non-axial gyroscope matrix elements. .

5. The method for establishing and verifying an analytical model of a bolted flange structure according to claim 4, characterized in that: Based on the element stiffness matrix and element mass matrix, the element damping matrix is ​​obtained using the Rayleigh damping model. The expression is as follows: in, is the damping coefficient of Ruili.

6. The method for establishing and verifying an analytical model of a bolted flange structure according to claim 5, characterized in that: In step S3, the axial stiffness of the bolt itself The calculation formula is as follows: in, The equivalent length of the bolt shank, This is the equivalent length of the bolt head. This is the equivalent length of the screw. This is the equivalent length of the threaded portion. The cross-sectional area of ​​the bolt's smooth rod. Let be the equivalent cross-section of the bolt. The elastic modulus of the bolt material; The formula for calculating the shear stiffness is as follows: in, Where is the nominal diameter of the bolt. The length of the screw; The formula for calculating the stiffness of the mating surfaces of the connected flanges is as follows: in, The outer diameter of the nut cross-section. Where is the nominal diameter of the bolt. The elastic modulus of the flange material. For the thickness of the flange, The included angle of the cone-shaped model.

7. The method for establishing and verifying an analytical model of a bolted flange structure according to claim 6, characterized in that: In step S4, when assembling the overall stiffness matrix and overall mass matrix of the system, a transformation matrix is ​​introduced to transform and assemble the dynamic matrices of each variable cross-section beam element in the local coordinate system into the global coordinate system.

8. The method for establishing and verifying an analytical model of a bolted flange structure according to claim 7, characterized in that: In step S5, the experimental modal test adopts the hammer impact method, and obtains the frequency response function of the overall structure of the bolt flange through multi-point excitation and multi-point vibration pickup, and identifies the experimental modal parameters accordingly.

9. A system for establishing and verifying an analytical model of a bolted flange structure, used to execute the method for establishing and verifying an analytical model of a bolted flange structure as described in any one of claims 1-8; characterized in that... include: The structural discretization modeling module is used to divide the flange in the bolted flange structure into several sector regions in the circumferential direction based on the finite element discretization concept, and to model the flange body structure of each sector region as one or more variable cross-section beam elements based on Timoshenko beam theory to simulate the elastic deformation of the flange under combined loads. The element dynamics matrix derivation module is communicatively connected to the structure discretization modeling module and is used to derive the element stiffness matrix, element mass matrix, and element inertia matrix for the rotor system for each of the established variable cross-section beam elements. The bolt connection model calculation module is used to calculate the equivalent stiffness of the bolts connecting the upper and lower flanges. The equivalent stiffness integrates the axial stiffness and shear stiffness of the bolt itself, as well as the compressive stiffness of the mating surface of the connected flanges. The system dynamics equation assembly module is communicatively connected to the element dynamics matrix derivation module and the bolt connection model calculation module, respectively. It is used to assemble the dynamics matrices of all variable cross-section beam elements, combine the stiffness matrices of the upper and lower plates by taking the equivalent stiffness of the bolts as an external excitation, and introduce the system damping matrix to establish the complete system dynamics equations of the bolt flange structure. The model verification module communicates with the system dynamics equation building module and is used to compare the calculation results of this analytical model with the external finite element simulation results and / or experimental modal test data to verify the accuracy and effectiveness of the analytical model.

10. An electronic device, characterized in that: It includes at least one processor and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program executable by the at least one processor, the computer program being executed by the at least one processor to enable the at least one processor to perform the method for establishing and verifying the analytical model of the bolt flange structure according to any one of claims 1-8.

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