Bolted flange conical shell dynamic modeling method considering amplitude dependence

Abstract

The embodiment of the invention relates to a bolted flange conical shell dynamic modeling method considering amplitude dependence. The method comprises the steps that a bolted flange boundary nonlinear mechanical model considering connection interface amplitude dependence rigidity and damping is established; the method comprises the following steps: carrying out theoretical modeling by adopting a Donnell shell theory and a displacement hypothesis of a Chebyshev polynomial, and deducing a control equation through a Lagrange equation; a modal and response test platform is established, and after frequency responses under different tightening torques are verified, the influence of bolt looseness on the nonlinear vibration response of the structure is analyzed in a mode of combining theoretical calculation and experimental verification. According to the method provided by the embodiment of the invention, the influence that the bolted connection amplitude depends on rigidity and damping is considered, and a more accurate nonlinear vibration analysis method is provided on the basis of obtaining the theoretical model of the bolted flange conical shell.

Description

Technical Field

[0001] The present invention relates to the technical field of mechanical dynamics, and in particular to a dynamic modeling method of a bolted flange conical shell considering amplitude dependence. Background Art

[0002] Conical shells are key structural elements in engineering, valued for their load-bearing capacity and geometric efficiency. Bolted flange connections are commonly used to assemble different shells to ensure structural integrity. However, complex operating environments often lead to bolt loosening, affecting the dynamic characteristics of the conical shell and even compromising structural performance. Therefore, developing a nonlinear vibration model that accurately simulates bolted flange conical shells under loose bolt conditions is crucial to ensuring structural safety and reliability, and guiding maintenance strategies and design optimization.

[0003] Based on the classical thin shell theory, Jin et al. developed an accurate modified Fourier series solution and proposed a free vibration analysis method for conical shells under elastic boundary conditions. Sofiyev and Kuruoglu proposed a theoretical method for solving the vibration problem of functionally gradient truncated conical shells under mixed classical boundary conditions. Wu and Pang combined the precise integration method with the transfer matrix method and advanced it to the study of the free vibration characteristics of conical shells. Song et al. combined Fourier series and Donnell shell theory to explore the dynamic behavior of conical shells. Li et al. used a meshless method to study the acoustic vibration response of cross-laminated composite shells under the action of acoustic media.

[0004] Due to the widespread application of bolted thin-walled structures in engineering, researchers have conducted extensive research on the vibration behavior of bolted shell structures. Based on Sanders shell theory, Tang et al. conducted research on the dynamic modeling of bolted cylindrical shells and carried out related vibration analysis. Li et al. conducted experimental research on the nonlinear vibration of thin-walled cylindrical shells under point support conditions. Li et al. studied the vibration behavior of bolted thin-walled cylindrical shells based on a semi-analytical method and analyzed the contact state changes at the connection interface. Du et al. used a discontinuous variable stiffness model to simulate the actual connection of bolts and analyzed the vibration characteristics of bolted cylindrical shells. Ma et al. used a semi-analytical method to study the nonlinear vibration behavior of bolted flange double cylindrical shells considering constrained layer damping under foundation excitation conditions. Liu et al. established a semi-analytical dynamic model of bolted flange double cylindrical shell structure under elastic boundary conditions and studied the influence of local looseness on the vibration characteristics of the structure. Liu et al. comprehensively considered the nonlinearity of carbon fiber reinforced composite materials and the nonlinearity of bolted connections and conducted research on the vibration response of bolted flange double cylindrical shells.

[0005] While extensive research has been conducted on the free vibration of conical shells, there is still a gap in the study of nonlinear dynamic modeling of bolted flange-bounded conical shells. Existing studies have largely focused on idealized boundary conditions, while the complex interactions introduced by the bolted flange remain largely unexplored. Furthermore, the specific effects of bolt loosening on the nonlinear vibration characteristics of bolted flange-bounded conical shells remain unclear. Given the impact of bolt loosening on the nonlinear vibration behavior of bolted flange-bounded conical shells and the widespread use of these structures in aeroengines, it is imperative to establish an amplitude-dependent dynamic model for bolted flange-bounded conical shells and conduct nonlinear vibration analysis.

[0006] There is no dynamic modeling method for bolted flange conical shells that takes amplitude dependence into consideration in the prior art.

[0007] The above-mentioned deficiencies are what those skilled in the art would like to overcome. Summary of the Invention

[0008] (1) Technical issues to be resolved

[0009] In order to solve the above problems in the prior art, the present invention provides a dynamic modeling method for a bolted flange conical shell considering amplitude dependence, which solves the problem that the prior art fails to effectively reveal the nonlinear vibration mechanism of the flange conical shell structure when the bolts are loose.

[0010] (2) Technical solution

[0011] In order to achieve the above objectives, the main technical solutions adopted by the present invention include:

[0012] An embodiment of the present invention provides a method for dynamic modeling of a bolted flange conical shell considering amplitude dependence, which includes:

[0013] A nonlinear mechanical model of bolt flange boundary considering the amplitude-dependent stiffness and damping of the connection interface was established.

[0014] Theoretical modeling was carried out using Donnell shell theory and Chebyshev polynomial displacement assumptions, and the governing equations were derived using the Lagrange equations.

[0015] A modal and response test platform was built, and after verifying the frequency response under different tightening torques, the influence of bolt loosening on the nonlinear vibration response of the structure was analyzed by combining theoretical calculation with experimental verification.

[0016] In one embodiment of the present invention, the flange conical shell is formed by N sThe bolts are connected to the base, and four groups of artificial springs are used to simulate the bolt constraints. It is assumed that the connection interface always maintains a contact state and undergoes radial stick-slip motion, and the influence of the axial constraint force on the normal pressure is not considered. A nonlinear mechanical model of the bolt flange boundary is established considering the amplitude-dependent stiffness and damping of the connection interface.

[0017] In one embodiment of the present invention, the strain energy U of the conical shell is ε It can be expressed as:

[0018]

[0019] In one embodiment of the present invention, the kinetic energy T of the conical shell can be expressed as:

[0020]

[0021] In one embodiment of the present invention, taking into account the translational and rotational kinetic energy of the flange, the kinetic energy of the flange can be calculated as:

[0022]

[0023] In one embodiment of the present invention, the strain energy of the flange can be expressed as:

[0024]

[0025] In one embodiment of the present invention, the bolt connection interface is under the preload force F pre The contact is maintained under the action of , and the spring stiffness remains unchanged in the axial, circumferential and angular directions. The constraint force in these directions can be expressed as:

[0026] f u =K bu u s ,f v =K bv v s ,

[0027] Among them, K bu ,K bv ,K bθ Represents the linear connection stiffness in the above three directions, u s ,v s , Represent the displacements in the three directions mentioned above.

[0028] In one embodiment of the present invention, the radial connection stiffness K of the bolt is bw Depending on the actual vibration amplitude, the radial constraint force of the bolt point during the entire vibration cycle in the sticking, positive slip, and negative slip contact states can be expressed as:

[0029]

[0030] Where w0 is the initial radial displacement, f w0 It represents the radial friction force at the beginning of the sticking state, and μ is the friction coefficient, which is usually 0.3 between the steel and iron interfaces.

[0031] In one embodiment of the present invention, the tightening torque T of the bolt t and nominal diameter d determine the preload force F pre , tightening torque T t The calculation formula is:

[0032] T t =KF pre d

[0033] Where K represents the torque coefficient, its reference range is 0.1-0.2, and the final value is 0.15.

[0034] In one embodiment of the present invention, the radial displacement, velocity and acceleration of the bolt point are respectively expressed as:

[0035] w s =γcos(τ),

[0036] Where τ = ωt is the dimensionless time, and γ represents the radial displacement amplitude of the bolt point.

[0037] In one embodiment of the present invention, the motion state of the bolt point can be divided into four stages within one cycle: (1) positive slip-viscosity transition stage, i.e. (2) Viscous-negative slip transition stage, i.e. (3) Negative slip-viscosity transition stage, i.e. (4) Viscous-positive slip transition stage, i.e.

[0038] The equations for each stage can be listed as follows:

[0039]

[0040] By solving the equation we get:

[0041]

[0042] In one embodiment of the present invention, the radial restraint force of the bolt point during a complete vibration cycle can be expressed as:

[0043]

[0044] According to the first-order Fourier series expansion, the constraint force can be reformulated as:

[0045]

[0046] where K eq represents the equivalent stiffness (ES), C eq stands for equivalent damping (ED):

[0047]

[0048] In one embodiment of the present invention, the potential energy generated in the bolt connection area can be expressed as:

[0049]

[0050] In one embodiment of the present invention, the theoretical modeling is performed using the Donnell shell theory and the displacement assumption of Chebyshev polynomials, and the control equation is derived using the Lagrange equation, including:

[0051] Applying lateral foundation vibration excitation to the bolted flange conical shell, the excitation forces in three directions can be expressed as:

[0052]

[0053] Among them, ω e is the excitation frequency, F u ,F v and F w Represent the excitation force coefficients in different directions respectively.

[0054] For the bolted flange conical shell, the virtual work of external excitation can be expressed as:

[0055]

[0056] The axial displacement of the conical shell is approximated by a linear combination of the first kind Chebyshev polynomials. The displacement of the conical shell is expressed as:

[0057]

[0058] Where N and G are the number of truncated terms, a ng ,b ng and c ng is the unknown coefficient, T g (η) represents the g-th Chebyshev polynomial, n is the circumferential wave number, ω represents the vibration angular frequency, U, V and W represent the modal vectors, q u ,q v and q w represents a generalized coordinate vector.

[0059] Substituting the strain energy of the conical shell, the kinetic energy of the conical shell, the kinetic energy of the flange, the strain energy of the flange, the potential energy generated in the bolted connection area, and the virtual work of the external excitation into the Lagrange equation, the dynamic equation of the bolted flange conical shell is obtained:

[0060]

[0061] Among them, M,M f ,C,K,K f and K b They represent the conical shell mass matrix, flange mass matrix, Rayleigh damping matrix, structural stiffness matrix, flange stiffness matrix, and boundary stiffness matrix, respectively. q represents the generalized displacement, and F represents the excitation force vector.

[0062] In one embodiment of the present invention, the dynamic equation of the bolted flange conical shell is solved using the Newmark numerical method.

[0063] In one embodiment of the present invention, the influence of the bolt loosening on the nonlinear vibration response is analyzed by combining theoretical calculation with experimental verification.

[0064] In one embodiment of the present invention, the hammer test of the bolted flange conical shell specimen is performed by building a modal test platform to verify the accuracy of the dynamic model and obtain the artificial spring stiffness.

[0065] In one embodiment of the present invention, the stiffness of the bolt connection can be determined by using a back-stepping identification method after obtaining the natural frequency of each constraint type from experimental tests. The fitness function is:

[0066]

[0067] In one embodiment of the present invention, the response experiment of the bolted flange conical shell specimen is performed by building a response test platform to verify the dynamic response under foundation excitation.

[0068] In one embodiment of the present invention, the nonlinear vibration mechanism of the bolted flange conical shell can be described as follows:

[0069] The dynamic analysis of the bolted flange boundary conical shell structure requires consideration of the displacement-dependent characteristics of the connection interface. The results show that the response of the bolt point increases with increasing excitation level, leading to nonlinear viscous slip motion at the bolted flange connection interface, causing fluctuations in equivalent stiffness and equivalent damping. Ultimately, this leads to nonlinear vibration phenomena in the bolted flange conical shell, including a leftward shift in the resonant frequency and attenuation of the resonant response. By comparing the vibration responses under different bolt loosening boundary conditions, we gain a deeper understanding of the impact of friction slip and stiffness degradation on the nonlinear vibration behavior of the structure, providing a basis for dynamic prediction and design of bolted connection structures in engineering applications.

[0070] (3) Beneficial effects

[0071] The beneficial effects of the present invention are as follows: the method provided by the embodiment of the present invention takes into account the influence of amplitude dependence, and proposes a more accurate nonlinear vibration analysis method based on the theoretical model of the bolted flange conical shell. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figure 1 A flowchart of a method for dynamic modeling of a bolted flange conical shell considering amplitude dependence provided by one embodiment of the present invention;

[0073] Figure 2 Schematic diagram of a bolted flange conical shell in one embodiment of the present invention;

[0074] Figure 3 Schematic diagram of the geometric shape of the flange on the conical shell in one embodiment of the present invention;

[0075] Figure 4 Schematic diagram of a mechanical model of a bolt flange boundary in one embodiment of the present invention;

[0076] Figure 5 Schematic diagram of a modal testing system for a bolted flange conical shell according to an embodiment of the present invention;

[0077] Figure 6 Schematic diagram of specific distribution positions of different numbers of bolts in one embodiment of the present invention;

[0078] Figure 7 Schematic diagram of an iterative process for optimizing spring stiffness based on a genetic algorithm in one embodiment of the present invention;

[0079] Figure 8 1 is a comparison diagram of experimental and theoretical frequency responses of constraint type A in one embodiment of the present invention;

[0080] Figure 9 1 is a comparison diagram of experimental and theoretical frequency responses of constraint type B in one embodiment of the present invention;

[0081] Figure 101 is a comparison diagram of experimental and theoretical frequency responses of constraint type C in one embodiment of the present invention;

[0082] Figure 11 1 is a comparison diagram of experimental and theoretical frequency responses of constraint type D in one embodiment of the present invention;

[0083] Figure 12 Schematic diagram of a response test system for a bolted flange conical shell according to an embodiment of the present invention;

[0084] Figure 13 Graph showing experimental and theoretical amplitude-frequency curves for constraint type A in one embodiment of the present invention;

[0085] Figure 14 Graph showing experimental and theoretical amplitude-frequency curves for constraint type B in one embodiment of the present invention;

[0086] Figure 15 Graph showing experimental and theoretical amplitude-frequency curves for constraint type C in one embodiment of the present invention;

[0087] Figure 16 Graph showing experimental and theoretical amplitude-frequency curves of constraint type D in one embodiment of the present invention;

[0088] Figure 17 Graph showing the resonance amplitude of bolt points under different constraint types in one embodiment of the present invention;

[0089] Figure 18 Comparison of the equivalent stiffness of different bolts under different excitation amplitudes in one embodiment of the present invention (Type C): (a) Bolt 1; (b) Bolt 2; (c) Bolt 3; (d) Bolt 5;

[0090] Figure 19 Comparison of equivalent stiffness of different bolts under different excitation amplitudes in one embodiment of the present invention (D type): (a) Bolt 2; (b) Bolt 3;

[0091] Figure 20 Comparison of equivalent damping of different bolts under different excitation amplitudes in one embodiment of the present invention (Type C): (a) Bolt 1; (b) Bolt 2; (c) Bolt 3; (d) Bolt 5;

[0092] Figure 21 Comparison of equivalent damping of different bolts under different excitation amplitudes in one embodiment of the present invention (D type): (a) Bolt 2; (b) Bolt 3. DETAILED DESCRIPTION

[0093] In order to better explain the present invention and facilitate understanding, the present invention is described in detail below through specific implementation methods in conjunction with the accompanying drawings.

[0094] All technical and scientific terms used herein have the same meanings as those commonly understood by those skilled in the art to which the present invention pertains. The terms used herein in the specification of the present invention are for the purpose of describing specific embodiments only and are not intended to limit the present invention. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.

[0095] The method provided in the embodiment of the present invention takes into account the influence of amplitude dependence. On the basis of obtaining the theoretical model of the bolted flange conical shell, a more accurate nonlinear vibration analysis method is proposed, and the effectiveness of the bolted flange conical shell theoretical model and the nonlinear vibration analysis method are verified through modal and response experiments.

[0096] Figure 1 A flowchart of a method for dynamic modeling of a bolted flange conical shell considering amplitude dependence is provided in one embodiment of the present invention, such as Figure 1 As shown, the method includes the following steps:

[0097] like Figure 1 As shown, in step S110, a nonlinear mechanical model of the bolt flange boundary is established considering the amplitude-dependent stiffness and damping of the connection interface;

[0098] like Figure 1 As shown, in step S120, theoretical modeling is performed using the Donnell shell theory and the displacement assumption of Chebyshev polynomials, and the control equation is derived using the Lagrange equation;

[0099] like Figure 1 As shown, in step S130, a modal and response test platform is built, and after verifying the frequency response under different tightening torques, the influence of bolt loosening on the nonlinear vibration response of the structure is analyzed by combining theoretical calculation with experimental verification.

[0100] exist Figure 1 The technical solution provided in the embodiment of the present invention provides a dynamic modeling method for a bolted flange conical shell considering amplitude dependence, and proposes a more accurate nonlinear vibration analysis method by considering the amplitude dependence characteristics of the bolt point connection stiffness and damping.

[0101] The following Figure 1 The specific implementation of each step of the embodiment shown is described in detail:

[0102] In step S110 , a nonlinear mechanical model of the bolt flange boundary is established considering the amplitude-dependent stiffness and damping of the connection interface.

[0103] Figure 2 This is a schematic diagram of a bolted flange conical shell in an embodiment of the present invention. The shell is connected to the Ns The cone angle of the conical shell is α, the generatrix length of the conical shell is L, the radius of any point on the conical shell is r, and the upper and lower end radii of the conical shell are R0 and R1 respectively.

[0104] Figure 2 Taking a point on the middle surface of the upper end of the conical shell as the origin, an orthogonal coordinate system (x, θ, z) is defined on the shell. The displacements along the three coordinate axes are represented by u, v, and w, respectively. In order to simplify subsequent analysis, the dimensionless parameter η = x / L is introduced.

[0105] In one embodiment of the present invention, the strain energy U of the conical shell is ε , which can be expressed as:

[0106]

[0107] In one embodiment of the present invention, the kinetic energy T of the conical shell can be expressed as:

[0108]

[0109] Figure 2 A schematic diagram of the flange geometry on the conical shell in an embodiment of the present invention, b f is the width of the flange, h f is the thickness of the flange, R f is the distance from the center of mass of the flange to the axis of rotation of the conical shell, e f is the eccentricity of the flange, which can be expressed as:

[0110]

[0111] Taking into account the translational and rotational kinetic energy of the flange, the kinetic energy of the flange can be calculated as:

[0112]

[0113] The strain energy of the flange can be expressed as:

[0114]

[0115] Figure 4 Schematic diagram of the mechanical model of the bolt flange boundary. The bolt flange conical shell passes through N s The bolts are connected to the base, and four groups of artificial springs are used to simulate the bolt constraints. It is assumed that the connection interface always maintains a contact state and undergoes radial stick-slip motion, and the influence of the axial constraint force on the normal pressure is not considered. A nonlinear mechanical model of the bolt flange boundary is established considering the amplitude-dependent stiffness and damping of the connection interface.

[0116] In one embodiment of the present invention, the bolt connection interface is under the preload force Fpre The contact is maintained under the action of , and the spring stiffness remains unchanged in the axial, circumferential and angular directions. The constraint force in these directions can be expressed as:

[0117] f u =K bu u s ,f v =K bv v s ,

[0118] Among them, K bu , K bv , K bθ Represents the linear connection stiffness in the above three directions, u s 、v s 、 Represent the displacements in the three directions mentioned above.

[0119] In one embodiment of the present invention, the radial connection stiffness K of the bolt bw Depending on the actual vibration amplitude, the radial constraint force of the bolt point in the entire vibration cycle under sticking, positive slip, and negative slip contact states can be expressed as:

[0120]

[0121] Where w0 is the initial radial displacement, f w0 It represents the radial friction force at the beginning of the sticking state, and μ is the friction coefficient, which is usually 0.3 between the steel and iron interfaces.

[0122] In one embodiment of the present invention, the tightening torque T of the bolt is t and nominal diameter d determine the preload force F pre , tightening torque T t The calculation formula is:

[0123] T t =KF pre d

[0124] Where K represents the torque coefficient, its reference range is 0.1-0.2, and the final value is 0.15.

[0125] In one embodiment of the present invention, the radial displacement, velocity and acceleration of the bolt point are respectively expressed as:

[0126] w s =γcos(τ),

[0127] Where τ = ωt is the dimensionless time, and γ represents the radial displacement amplitude of the bolt point.

[0128] In one embodiment of the present invention, the motion state of the bolt point can be divided into four stages within one cycle: (1) positive slip-viscosity transition stage, i.e. (2) Viscous-negative slip transition stage, i.e. (3) Negative slip-viscosity transition stage, i.e. (4) Viscous-positive slip transition stage, i.e.

[0129] The equations for each stage can be listed as follows:

[0130]

[0131] By solving the equation we get:

[0132]

[0133] In one embodiment of the present invention, the radial restraint force of the bolt point during a complete vibration cycle can be expressed as:

[0134]

[0135] According to the first-order Fourier series expansion, the constraint force can be reformulated as:

[0136]

[0137] where K eq represents the equivalent stiffness (ES), C eq stands for equivalent damping (ED):

[0138]

[0139] In one embodiment of the present invention, the potential energy generated in the bolt connection area can be expressed as:

[0140]

[0141] In step S120 , theoretical modeling is performed using the Donnell shell theory and the displacement assumption of Chebyshev polynomials, and the control equation is derived using the Lagrange equation.

[0142] Applying lateral foundation vibration excitation to the bolted flange conical shell, the excitation forces in three directions can be expressed as:

[0143]

[0144] Among them, ω e is the excitation frequency, F u ,F v and Fw Represent the excitation force coefficients in different directions respectively.

[0145] For the bolted flange conical shell, the virtual work of external excitation can be expressed as:

[0146]

[0147] The axial displacement of the conical shell is approximated by a linear combination of the first kind Chebyshev polynomials. The displacement of the conical shell is expressed as:

[0148]

[0149] Where N and G are the number of truncated terms, a ng ,b ng and c ng is the unknown coefficient, T g (η) represents the g-th Chebyshev polynomial, n is the circumferential wave number, ω represents the vibration angular frequency, U, V and W represent the modal vectors, q u ,q v and q w represents a generalized coordinate vector.

[0150] Substituting the strain energy of the conical shell, the kinetic energy of the conical shell, the kinetic energy of the flange, the strain energy of the flange, the potential energy generated in the bolted connection area, and the virtual work of the external excitation into the Lagrange equation, the dynamic equation of the bolted flange conical shell is obtained:

[0151]

[0152] Among them, M,M f ,C,K,K f and K b They represent the conical shell mass matrix, flange mass matrix, Rayleigh damping matrix, structural stiffness matrix, flange stiffness matrix and boundary stiffness matrix respectively. q represents the generalized displacement, and F represents the excitation force vector.

[0153] In one embodiment of the present invention, the dynamic equations of the bolted flange conical shell are solved using the Newmark numerical method.

[0154] In step S130, the influence of the bolt loosening on the nonlinear vibration response is analyzed by combining theoretical calculation with experimental verification.

[0155] Figure 5 This is a schematic diagram of a modal testing system for a bolted flange conical shell in one embodiment of the present invention. The modal testing platform is used to perform hammer tests to verify the accuracy of the dynamic model and obtain the artificial spring stiffness and natural frequency.

[0156] Table 1 provides the material and dimensional parameters of the bolted flange conical shell.

[0157]

[0158] In one embodiment of the present invention, 80 measuring points are arranged on the surface of the specimen, wherein an accelerometer is placed at the 65th measuring point to capture the vibration signal. A hammer is used to strike each measuring point in turn, and a data acquisition system records the experimental data.

[0159] Table 2 lists the types of constraints with different bolt numbers and tightening torques.

[0160] Constraint Type A B C D <![CDATA[Number of bolts / N s > 16 16 12 8 Tightening torque / Nm 20 10 5 5

[0161] Figure 6 Schematic diagram of the specific distribution positions of different bolt quantities in one embodiment of the present invention

[0162] In one embodiment of the present invention, the stiffness of the bolt connection can be determined using an inverse identification method after obtaining the natural frequency of each constraint type from experimental tests, and the fitness function is:

[0163]

[0164] Figure 7 The schematic diagram of the iterative process of spring stiffness optimization based on genetic algorithm. After 35 iterations, the fitness function tends to converge, where the torque T t It is 20N m.

[0165] Table 3 shows the spring stiffness identified based on modal testing under different constraint types.

[0166] type <![CDATA[T t (Nm)]]> <![CDATA[K bu (N / m)]]> <![CDATA[K bv (N / m)]]> <![CDATA[K bw (N / m)]]> <![CDATA[K bθ (N·m / rad)]]> A 20 <![CDATA[8.54×10 8 ]]> <![CDATA[8.25×10 8 ]]> <![CDATA[6.25×10 8 ]]> <![CDATA[6.29×10 9 ]]> B 10 <![CDATA[2.24×10 8 ]]> <![CDATA[9.35×10 7 ]]> <![CDATA[1.36×10 8 ]]> <![CDATA[3.27×10 8 ]]> C 5 <![CDATA[4.69×10 7 ]]> <![CDATA[5.30×10 7 ]]> <![CDATA[8.50×10 7 ]]> <![CDATA[5.40×10 7 ]]> D 5 <![CDATA[4.69×10 7 ]]> <![CDATA[5.30×10 7 ]]> <![CDATA[8.50×10 7 ]]> <![CDATA[5.40×10 7 ]]>

[0167] Figure 8 、 9 , 10, and 11 are the comparison diagrams of the experimental and theoretical frequency responses of constraint types A, B, C, and D, respectively. The hammer point is located at point 33, and the theoretical and experimental frequency responses show good consistency.

[0168] Table 4 provides a detailed comparison of the experimental and theoretical frequencies, showing that the maximum error is within 1.05%.

[0169]

[0170] Table 5 provides a comparison of the theoretical and experimental mode shapes for restraint type A.

[0171]

[0172] In one embodiment of the present invention, the comparison results confirm the effectiveness of the established kinetic model.

[0173] In one embodiment of the present invention, the response experiment of the bolted flange conical shell specimen is performed by building a response test platform to verify the dynamic response under foundation excitation.

[0174] Figure 12 Schematic diagram of the response test system for bolted flange conical shell. A frequency sweep experiment near the fundamental frequency was performed on the shell specimen.

[0175] In one embodiment of the present invention, five sets of frequency sweep experiments were conducted on a bolted flange conical shell under four types of constraints, with excitation amplitudes of 1g, 2g, 3g, 4g, and 5g, respectively.

[0176] Figure 13 、 14 The experimental and theoretical amplitude-frequency curves for constraint types A and B show that the theoretical and experimental resonance responses increase linearly with increasing excitation amplitude, while the resonant frequency remains constant, with the maximum error in the response amplitude not exceeding 7.20%. When transitioning from constraint A to constraint B (reducing tightening torque), the resonance amplitude increases slightly and the resonant frequency decreases slightly. This indicates that when the bolt tightening torque reaches 10 Nm, the bolt flange boundary is almost fully constrained, and at an excitation amplitude of 5 g, the connection interface remains in a viscous state.

[0177] Figure 15 、 16 The experimental and theoretical amplitude-frequency curves for constraint types C and D show that, as the excitation amplitude increases, both the theoretical and experimental amplitude-frequency curves exhibit dynamic softening, with the resonant frequency gradually decreasing. The maximum error in the response amplitude does not exceed 2.46%. When transitioning from constraint B to constraint C (reducing the tightening torque and number of bolts), the response amplitude increases nearly threefold. However, when transitioning from constraint C to constraint D (reducing the number of bolts), the resonance amplitude decreases by nearly half.

[0178] Figure 14 、 15 , 16 It can be seen that due to the combined influence of the bolt boundary stiffness and the interface contact state, the response amplitude in constraint types B, C, and D first increases and then decreases. When the constraint type changes from B to C, the bolt boundary stiffness decreases significantly and the bolt slips. At this time, the increase in amplitude due to the decrease in the bolt boundary stiffness is more significant. When the constraint type changes from C to D, the reduction in the number of bolts leads to a decrease in vibration transmission efficiency. At this time, energy dissipation is mainly due to friction slip at the connection interface, resulting in an increase in the amplitude attenuation effect. In addition, for constraint type D, although the slip effect is more significant, the frequency shift to the left is not enhanced due to the small number of constraint bolts.

[0179] Figure 17 The following plots the resonance amplitudes of the bolt points under different constraint types. It can be seen that the resonance amplitudes at each bolt point increase with increasing excitation level. The displacement trends of the bolt points under different constraint types are similar to those of the shell structure. Since the fundamental frequency has a circumferential wave number of six, the resonance amplitude curves exhibit six peaks and six troughs, with the troughs representing the nodal lines of vibration.

[0180] In one embodiment of the present invention, once the resonance amplitude of the bolt point is determined, the ES (equivalent stiffness K eq ) and ED (equivalent damping C eq ) can be calculated using Eq.

[0181] Figure 18 、 19 The following graphs compare the equivalent stiffness under different excitation amplitudes for constraint types C and D, respectively. When the excitation amplitude is low, the equivalent stiffness at the bolt point remains unchanged. As the excitation frequency approaches the resonant frequency, the structural vibration response increases, and the equivalent stiffness gradually decreases, reaching its minimum value at the resonant frequency. As the excitation amplitude increases, the decrease in equivalent stiffness becomes more significant, and the frequency range over which stiffness attenuation occurs widens. This gradual reduction in constraint stiffness weakens the capacity of the bolted connection, ultimately leading to a decrease in the resonant frequency.

[0182] Figure 20 、 21 The following are comparisons of equivalent damping under different excitation amplitudes for constraint types C and D. When the excitation amplitude is small, the bolt point remains in a viscous state, and the equivalent damping is zero. As the excitation amplitude increases, the excitation frequency approaches the resonance region, and the equivalent damping gradually increases. At large excitation amplitudes, the equivalent damping reaches a local minimum at the resonance frequency.

[0183] A dynamic analysis of the bolt flange boundary conical shell structure was performed. Considering the displacement-dependent characteristics of the connection interface, it was found that the response of the bolt point increases with the increase of the excitation level, resulting in nonlinear stick-slip motion at the bolt flange connection interface, causing fluctuations in the equivalent stiffness and equivalent damping, and ultimately leading to nonlinear vibration phenomena in the bolt flange conical shell, including the left shift of the resonant frequency and the attenuation of the resonant response.

[0184] In one embodiment of the present invention, the frequency range in which the equivalent stiffness decreases is consistent with the frequency range in which the equivalent damping increases. Within this range, the bolt flange boundary interface exhibits viscous slip motion, which is called the sensitive frequency range. As the excitation amplitude increases, this range expands and the nonlinear characteristics of the bolt connection become more obvious.

[0185] In this embodiment, on the one hand, the nonlinear vibration characteristics of the bolted flange conical shell under the condition of bolt loosening are analyzed through theoretical modeling and experimental research, and a nonlinear dynamic model considering the stiffness and damping of the bolt connection amplitude dependence is established; on the other hand, the influence of bolt loosening on the dynamic response is studied, and the results are verified through modal and response tests, which can provide a certain reference for actual engineering design.

[0186] In summary, the present invention takes the bolt flange conical shell in the aircraft engine as the research object, and establishes a nonlinear mechanical model of the bolt flange boundary that takes into account the amplitude-dependent stiffness and damping of the connection interface. Using the Donnell shell theory and the displacement assumption of the Chebyshev polynomial, the control equation is derived through the Lagrange equation. A modal and response test platform is constructed, and after verifying the frequency response under different tightening torques, the influence of bolt loosening on the nonlinear vibration response of the structure is analyzed by combining theoretical calculation with experimental verification. The amplitude-frequency analysis reveals the trend of the resonance response caused by bolt loosening to first rise and then fall, as well as the phenomenon of the resonance frequency shifting to the left. This phenomenon can be attributed to the combined influence of the contact state at the bolt flange boundary and the connection stiffness. By comparing the vibration responses under different bolt loosening conditions, the present invention provides an in-depth understanding of the influence of friction slip and stiffness degradation on the nonlinear vibration behavior of the structure, and provides a theoretical basis for the dynamic prediction and design of bolt structures in engineering applications.

[0187] It should be noted that, although several modules or units of the device for action execution are mentioned in the above detailed description, this division is not mandatory. In fact, according to an embodiment of the present invention, the features and functions of two or more modules or units described above can be concretized in one module or unit. Conversely, the features and functions of one module or unit described above can be further divided into multiple modules or units to be concretized.

[0188] Through the description of the above embodiments, it is easy for those skilled in the art to understand that the example embodiments described herein can be implemented by software or by combining software with necessary hardware. Therefore, the technical solution according to the embodiments of the present invention can be embodied in the form of a software product, which can be stored in a non-volatile storage medium (which can be a CD-ROM, a USB flash drive, a mobile hard disk, etc.) or on a network, and includes a number of instructions to enable a computing device (which can be a personal computer, a server, a touch terminal, or a network device, etc.) to execute the method according to the embodiments of the present invention.

[0189] Other embodiments of the present invention will readily occur to those skilled in the art after considering the specification and practicing the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the present invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein. The description and examples are to be considered as exemplary only, with the true scope and spirit of the invention being indicated by the following claims.

[0190] It will be appreciated that the present invention is not limited to the precise construction described above and shown in the accompanying drawings and that various modifications may be made without departing from its scope, which is limited only by the appended claims.

Claims

1. A dynamic modeling method for bolted flange conical shell considering amplitude dependence, characterized in that: It includes: A nonlinear mechanical model of bolt flange boundary considering the amplitude-dependent stiffness and damping of the connection interface is established. Theoretical modeling was carried out using Donnell shell theory and Chebyshev polynomial displacement assumptions, and the system dynamics equations were derived using the Lagrange equation. A modal and response test platform was built, and after verifying the frequency response under different tightening torques, the influence of bolt loosening on the nonlinear vibration response of the structure was analyzed by combining theoretical calculation with experimental verification.

2. The method for dynamic modeling of bolted flange conical shell considering amplitude dependence according to claim 1 is characterized in that: The conical flange shell passes N s The bolts are connected to the base, and four groups of artificial springs are used to simulate the bolt constraints. The connection interface is set to always maintain a contact state and experience radial stick-slip motion. The influence of the axial constraint force on the normal pressure is not considered, and a nonlinear mechanical model of the bolt flange boundary is established considering the amplitude-dependent stiffness and damping of the connection interface.

3. The method for dynamic modeling of bolted flange conical shell considering amplitude dependence according to claim 1 is characterized in that: The Donnell shell theory and the displacement assumption of Chebyshev polynomials were used to carry out theoretical modeling and the dynamic equation of the bolted flange boundary conical shell was derived through the Lagrange equation:

4. The method for dynamic modeling of bolted flange conical shell considering amplitude dependence according to claim 1 is characterized in that: The hammer test of the bolted flange conical shell specimen is carried out by building a modal test platform to verify the accuracy of the dynamic model and use the back-stepping identification method to determine the artificial spring stiffness.

5. The method for dynamic modeling of bolted flange conical shell considering amplitude dependence according to claim 1 is characterized in that: The effect of bolt loosening on the nonlinear vibration response of the structure is revealed by combining theoretical calculations with shaking table foundation excitation response experiments.

6. The method for dynamic modeling of bolted flange conical shell considering amplitude dependence according to claim 1 is characterized in that: The nonlinear vibration mechanism of the bolted flange conical shell can be described as follows: When performing dynamic analysis on the bolt flange boundary conical shell structure, the displacement dependence characteristics of the connection interface need to be considered. The response of the bolt point increases with the increase of the excitation level, resulting in nonlinear viscous slip motion at the bolt flange connection interface, causing fluctuations in equivalent stiffness and equivalent damping, and ultimately leading to nonlinear vibration phenomena in the bolt flange conical shell, including left shift of the resonant frequency and attenuation of the resonant response. By comparing the vibration responses under different bolt loosening boundary conditions, we can deeply understand the influence of friction slip and stiffness degradation on the nonlinear vibration behavior of the structure, and provide a basis for the dynamic prediction and design of bolted connection structures in engineering applications.

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