Dynamic modeling method for pipeline system containing movable pipe joint

Abstract

The invention relates to the technical field of mechanical dynamics, in particular to a dynamic modeling method of a pipeline system containing a movable pipe joint, which comprises the following steps: establishing a movable pipe joint-pipeline system dynamic model based on a finite element method, in the dynamic model, establishing a model of a pipeline and a joint part according to a Tiushingeshu beam theory, and establishing a model of the pipeline and the joint part according to the Tiushingeshu beam theory; the equivalent stiffness of the movable pipe joint under the pre-compression state and the dynamic load of the sealing ring is considered; performing model verification on the established movable pipe joint-pipeline system dynamic model to obtain a verified model; and based on the verified model, simulating the influence of each excitation amplitude on the inherent frequency of the movable pipe joint-pipeline, and verifying the correctness of the movable pipe joint-pipeline system dynamic model. According to the method, the dynamic model of the aero-engine pipeline system containing the movable pipe joint is accurately constructed, and the inherent characteristics and the vibration response law of the system are accurately represented.

Description

Technical Field

[0001] This invention relates to the field of mechanical dynamics technology, and more particularly to a dynamic modeling method for a pipeline system containing movable pipe joints. Background Technology

[0002] The piping systems of aero-engines are intricate, with most series pipes connected via pipe fittings. During actual operation, piping systems containing fittings are simultaneously subjected to the combined effects of hydraulic pump fluid pulsation and casing foundation excitation. Furthermore, due to the space constraints of the casing and the unique shape of the pipes, the fit between pipes is poor, making vibration faults highly susceptible to occur at the fitting points. Currently, most scholars, both domestically and internationally, have neglected the impact of pipe fittings on the dynamic modeling of piping systems, especially regarding movable pipe fittings connected by sealing rings. The large deformation of the internal structure of movable pipe fittings makes dynamic modeling of piping systems extremely difficult; therefore, the study of dynamic modeling and inherent characteristics of piping systems containing movable fittings is of significant research importance.

[0003] While current methods for dynamic modeling of multi-pipeline systems and equivalent modeling of clamps are relatively mature, research on modeling methods for various pipe joints in aero-engine piping systems is lacking. This invention addresses the equivalent contact stiffness problem of sealing rings in movable pipe joints, deriving the relationship between translational stiffness and displacement, and proposing a calculation formula for angular stiffness and its parameter identification method. The obtained stiffness model is introduced into the finite element model of the piping system, and the results are compared with those from hammer impact tests and vibration tests, verifying the effectiveness of the established model. Summary of the Invention

[0004] To address the aforementioned technical problems in existing technologies, such as the neglect of the impact of pipe joints on the dynamic modeling of aero-engine piping systems and the difficulty in modeling movable pipe joints connected by sealing rings due to large internal deformation, this invention provides a dynamic modeling method for piping systems containing movable pipe joints. This invention considers the equivalent stiffness of the movable pipe joint under pre-compression and dynamic load conditions, establishes a model of the piping and joint components based on Timoshenko beam theory, and verifies the inherent characteristics and vibration response of the movable pipe joint-piping system. This allows for the accurate construction of a dynamic model of an aero-engine piping system containing movable pipe joints, precisely characterizing the system's inherent characteristics and vibration response patterns. This provides a theoretical basis and technical support for avoiding vibration faults at pipe joints and improving the operational stability and reliability of aero-engine piping systems.

[0005] The technical means employed in this invention are as follows: A dynamic modeling method for a piping system containing movable pipe fittings includes the following steps: Based on the finite element method, a dynamic model of the movable pipe joint-pipeline system is established. In the dynamic model, the model of the pipeline and joint is established according to the Timoshenko beam theory, and the equivalent stiffness of the movable pipe joint under pre-compression and dynamic load is considered. The established dynamic model of the active pipe joint-pipeline system was validated to obtain the validated model. Based on the validated model, the influence of each excitation amplitude on the natural frequency of the movable pipe joint-pipeline is simulated to verify the correctness of the dynamic model of the movable pipe joint-pipeline system.

[0006] Furthermore, the establishment of the dynamic model of the active pipe joint-pipeline system includes: The equivalent stiffness of the sealing ring under pre-compression state under simple harmonic action is established, and the equivalent stiffness of the sealing ring under dynamic load under simple harmonic excitation is established, thus obtaining a sealing ring model considering the equivalent stiffness. The pipeline and movable pipe joints are modeled using Timoshenko beam elements to obtain the model of the pipeline and movable pipe joints. The sealing ring model considering equivalent stiffness is combined with the models of the pipeline and the movable pipe joint to obtain the dynamic model of the movable pipe joint-pipeline system.

[0007] Furthermore, the equivalent stiffness of the sealing ring under pre-compression state under harmonic action includes: Based on the O-ring installation structure, the initial compression is expressed as:

[0008] in, This is the initial compression amount. d 0 represents the initial cross-sectional diameter of the sealing ring. d 1 represents the pre-compression section diameter of the sealing ring; Based on the initial compression and the initial cross-sectional diameter of the sealing ring, the initial compression ratio is expressed as:

[0009] in, This is the initial compression ratio; Calculate the contact width of the sealing ring under pre-compression conditions based on the medium pressure acting on the sealing ring:

[0010] in, The contact width of the sealing ring. This refers to the elastic modulus of rubber. The pressure of the medium acting on the sealing ring; Calculate the circumference of the sealing ring based on the pre-compression section diameter of the sealing ring:

[0011] in, L The circumference of the sealing ring. D This refers to the standard inner diameter of the sealing ring. Calculate the average contact stress of the sealing ring under pre-compression state based on the average contact stress of the sealing ring when there is no medium pressure and the additional contact stress caused by fluid pressure:

[0012] in, The average contact stress under pre-compression conditions. The average contact stress when there is no medium pressure. The average contact stress when there is no medium pressure. Poisson's ratio; Calculate the average contact area of ​​the sealing ring based on the contact width of the sealing ring under pre-compression conditions:

[0013] in, This represents the average contact area of ​​the sealing ring; Calculate the radial contact force based on the average contact area of ​​the sealing ring and the average contact stress under pre-compression conditions:

[0014] in, Radial contact force, This refers to the standard inner diameter of the sealing ring. Under bolt preload, the sealing ring undergoes a small displacement. Based on the radial contact force and the small displacement, the contact pressure vector sum of the sealing ring in the x and y directions is calculated:

[0015]

[0016] in, The first in the x-direction i The contact pressure vector sum, The first in the x-direction i A tiny displacement, The first in the x-direction i +1 basic pressure vector sum, The first in the x-direction i +1 tiny displacement, The first in the y-direction i The contact pressure vector sum, The first in the y-direction i A tiny displacement, The first in the y-direction i +1 basic pressure vector sum, The first in the y-direction i +1 tiny displacement; Based on the contact pressure vector in the x-direction and the small displacement in the x-direction, the translational equivalent stiffness under pre-compression in the x-direction is calculated. Similarly, based on the contact pressure vector in the y-direction and the small displacement in the x-direction, the translational equivalent stiffness under pre-compression in the x-direction is also calculated.

[0017]

[0018] in, Let be the translational equivalent stiffness under pre-compression in the x-direction. The translational equivalent stiffness in the pre-compression state in the y direction; Using fractal contact theory, the relationship between the minute displacement in the z-direction and the contact stress vector sum in the z-direction is obtained. The formula for calculating the minute displacement in the z-direction is as follows:

[0019] in, For the first i A small displacement in the z-direction, The Poisson's ratio of the sealing ring. The equivalent shear modulus of the contact surface. The coefficient of friction of the contact surface. The first in the z-direction i The sum of the basic pressure vectors, For the first i +1 small displacement in the z direction, The first in the z-direction i +1 basic pressure vector sum; Based on the contact pressure vector in the z-direction and the small displacement in the z-direction, the translational equivalent stiffness under pre-compression state in the z-direction is calculated:

[0020] in, The translational equivalent stiffness under pre-compression in the z-direction; Based on the translational equivalent stiffness in the pre-compression state in the x-direction and the translational equivalent stiffness in the pre-compression state in the z-direction, the equivalent stiffness of the sealing ring in the pre-compression state under simple harmonic action is constructed.

[0021] Furthermore, the formula for calculating the elastic modulus of the rubber is:

[0022] in, and These are the coefficients of the first model and the coefficients of the second model, respectively. Hardness The first in the x direction i The formula for calculating a small displacement is:

[0023] The formula for calculating the equivalent shear modulus of the contact surface is:

[0024] in, For elastic modulus, It is calculated using the following formula:

[0025] in, For the pipe fitting Poisson's ratio, The elastic modulus of the sealing ring. This is the elastic modulus of the pipe fitting.

[0026] Furthermore, the equivalent stiffness of the sealing ring under dynamic load during harmonic excitation includes: Based on the shape characteristics of the sealing ring itself, calculate the contact force of the sealing ring at a certain angle:

[0027] in, For the x-th i One sealing ring in α Contact force at angle This is the time-dependent displacement. This is the radial deformation. For the x-th i +1 sealing ring α Contact force at an angle; Based on the contact force of the sealing ring at a certain angle, the linear stiffness of the sealing ring in the x-direction and the linear stiffness of the sealing ring in the y-direction are calculated:

[0028]

[0029] in, Let be the linear stiffness of the sealing ring in the x-direction. For deformation Contact force at angle Before deformation Contact force at angle Let be the linear stiffness of the sealing ring in the y-direction. For the y-direction i One sealing ring in β Contact force at angle For the y-direction i +1 sealing ring β Contact force at angle For deformation β Contact force at angle Before deformation β Contact force at an angle; Calculate the linear stiffness of the sealing ring in the z-direction based on the axial displacement of the sealing ring:

[0030] in, Let be the linear stiffness of the sealing ring in the z-direction. As an intermediate variable, For the axial displacement of the sealing ring, The coefficient of friction of the sealing ring material; Based on the variation law of translational stiffness, assuming that the angular stiffness of the sealing ring in each direction is a quadratic function, the formula for calculating the angular stiffness is:

[0031] in, , For angular stiffness, This represents the equivalent angular stiffness under pre-compression conditions. This is the equivalent angular stiffness coefficient under pre-compression conditions. Let be the angular displacement at any time t. This is the first coefficient value. This is the second coefficient value; Based on the linear stiffness of the sealing ring in the x-direction, the linear stiffness of the sealing ring in the y-direction, the linear stiffness of the sealing ring in the z-direction, and the angular stiffness, the stiffness matrix of the sealing ring under dynamic load under harmonic excitation is constructed:

[0032]

[0033] in, This is the stiffness matrix of the sealing ring under dynamic loads during harmonic excitation. For the self-compiled spring element stiffness matrix; Based on the stiffness matrix of the sealing ring under dynamic load under harmonic excitation, the equivalent stiffness of the sealing ring under dynamic load under harmonic excitation is obtained.

[0034] Furthermore, the formula for calculating the time-related displacement is as follows:

[0035] in, Let x represent the relative displacement of each sealing ring's equivalent spring node in the x-direction at any time t. The formula for calculating the radial deformation is: , The formula for calculating the intermediate variable is as follows: .

[0036] Furthermore, the piping and movable pipe joint sections are modeled using Timoshenko beam elements, including: The nodal displacement vector of the element is expressed as:

[0037] in, Let be the nodal displacement vector of the element. for i Translational displacement of the node in the z-direction. for i Translational displacement of the node in the y-direction. for i Translational displacement of the node in the x-direction. for i Angular displacement of the node in the z-direction for i Angular displacement of the node in the y-direction for i Angular displacement of the node in the x-direction. for j Displacement of node in the z-direction for j Displacement of the node in the y-direction. for j Node displacement in the x-direction for j Angular displacement of the node in the z-direction for j Angular displacement of the node in the x-direction. for j Angular displacement of the node in the x-direction. The kinetic energy of the unit is expressed as:

[0038] in, The kinetic energy of the unit. For pipeline density, The cross-sectional area of ​​the pipe. , , , , , Differentiate the time in the six directions of the element. Let ox be the moment of inertia of the cross section. Let Oy be the moment of inertia of the cross section. Let the moment of inertia of the cross section be oz. For the first k The length and cross-sectional area of ​​each unit With functionals u, v, w, θ, and As independent variables, the potential energy of the unit is expressed as:

[0039] in, The kinetic energy of the unit. The elastic modulus of the pipe fitting. , , , , , Differentiate the spatial displacements of the element in six directions. The y-axis shear coefficient. For unit shear modulus, The z-axis shear coefficient is... The moment of inertia is the torsional section. Furthermore, the sealing ring model considering equivalent stiffness is combined with the models of the pipeline and the movable pipe joint to obtain a dynamic model of the movable pipe joint-pipeline system, including: Grouping the element matrices yields the motion differential equations of the active pipe fitting-piping system:

[0040] in, The quality matrix of the pipeline, For the generalized displacement vector of the pipeline system, The first derivative of the generalized displacement vector is... The second derivative of the generalized displacement vector is... Here is the damping matrix of the pipeline. Here is the stiffness matrix of the pipeline. Here is the time-varying stiffness matrix of the pipe joint. Let the basic external excitation vector be the vector acting on the pipeline system. The formula for calculating the basic external excitation vector of the pipeline system is as follows:

[0041] in, Let the basic external excitation vector be the vector acting on the pipeline system. For displacement indication vector, To determine the level of incentive, For the excitation frequency, To incentivize time.

[0042] Furthermore, based on the validated model, the influence of each excitation amplitude on the natural frequency of the movable pipe joint is simulated to verify the correctness of the dynamic model of the movable pipe joint, including: The natural frequency of the movable pipe joint-pipeline system was obtained based on multiple hammer impact tests. A frequency sweep test was conducted on the dynamic model of the movable pipe joint-pipe system to identify the nonlinear stiffness. The identified nonlinear stiffness coefficients were substituted into the calculation formula of the angular stiffness in each direction of the movable pipe joint. The obtained dynamic stiffness and modal damping ratio were introduced into the system dynamic model to obtain the natural frequency of the dynamic model. The correctness of the model was verified by comparing the natural frequencies of the active pipe fitting-pipeline system with the natural frequencies of the dynamic model.

[0043] Compared with the prior art, the present invention has the following advantages: Force analysis was performed on the contact state of the O-ring seal in the movable joint. The translational stiffness in each direction was obtained through the relationship between force and deformation. Based on this, a dynamic model of the pipeline system containing the movable joint was established using Timoshenko beam elements. Modal tests were conducted on the pipeline system, and the results were compared with simulations to verify the accuracy of the model. The dynamic model of the pipeline system containing the movable joint provided by this invention enables faster and more intelligent establishment of arbitrary pipeline system models through appropriate dynamic modeling methods, greatly improving efficiency.

[0044] Based on the above reasons, this invention can be widely applied in fields such as mechanical dynamics. Attached Figure Description

[0045] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0046] Figure 1 This is a stress analysis diagram of the movable pipe joint containing a sealing ring according to the present invention.

[0047] Figure 2 This is a schematic diagram of preload deformation in various directions according to the present invention; (a) x direction, (b) z direction.

[0048] Figure 3 The present invention provides a dynamic deformation of the seal and its mechanical model; (a) dynamic deformation of the sealing ring, (b) deformation cross section, and (c) mechanical model of the movable pipe joint based on the spring unit.

[0049] Figure 4 This is a schematic diagram of the Timoshenko beam element used in an embodiment of the present invention.

[0050] Figure 5 This invention provides a comparison between experimental and simulated frequency response functions.

[0051] Figure 6 The results of experiments and simulations under different excitation amplitudes of the present invention are shown below: (a) 0.5g, (b) 1.0g, (c) 1.5g, (d) 2.0g, (e) 5.5g. Detailed Implementation

[0052] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0053] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0054] This invention provides a dynamic modeling method for a piping system containing movable pipe fittings, comprising the following steps: S1. Based on the finite element method, a dynamic model of the movable pipe joint-pipeline system is established. In the dynamic model, the model of the pipeline and joint parts is established according to the Timoshenko beam theory, and the equivalent stiffness of the movable pipe joint under pre-compression and dynamic load is considered.

[0055] The equivalent stiffness of the movable pipe joint is calculated according to the formula, and the equivalent stiffness is input into the dynamic model.

[0056] Figure 1 middle, Figure 1 (a) is a cross-sectional view of the O-ring seal. Figure 1 (b) is a stress analysis diagram of the movable pipe joint. The movable pipe joint mainly achieves self-tightening sealing by deforming the sealing ring to make the contact compressive force greater than the internal pressure of the sealing medium. In the figure, d0 is the initial cross-sectional diameter of the sealing ring, d1 is the pre-compression cross-sectional diameter of the sealing ring, D is the standard inner diameter of the sealing ring, D1 is the inner diameter of the pipe, D2 is the outer diameter of the pipe, D3 is the small end diameter of the pipe joint, and D4 is the large end diameter of the pipe joint.

[0057] Specifically, the movable pipe fitting-piping system includes: S11. Establish the equivalent stiffness of the sealing ring under pre-compression state under simple harmonic action, and establish the equivalent stiffness of the sealing ring under dynamic load under simple harmonic excitation to obtain a sealing ring model considering the equivalent stiffness.

[0058] The steps for establishing the equivalent stiffness of the sealing ring under pre-compression state under simple harmonic action are as follows: Step 1: Based on the O-ring installation structure, the initial compression is expressed as:

[0059] in, This is the initial compression amount. d 0 represents the initial cross-sectional diameter of the sealing ring. d 1 represents the pre-compression cross-sectional diameter of the sealing ring. Based on the initial compression amount and the initial cross-sectional diameter of the sealing ring, the initial compression ratio is expressed as:

[0060] in, The initial compression ratio is given. Based on the medium pressure acting on the sealing ring, the contact width of the sealing ring under pre-compression is calculated:

[0061] in, The contact width of the sealing ring. This refers to the elastic modulus of rubber. The pressure of the medium acting on the sealing ring. When "When" indicates the contact state under conditions of no medium pressure.

[0062] Step 2: Calculate the circumference of the sealing ring based on the pre-compression section diameter of the sealing ring.

[0063] in,L The circumference of the sealing ring. D This is the standard inner diameter of the sealing ring. Based on the average contact stress of the sealing ring under no-medium pressure and the additional contact stress caused by fluid pressure, calculate the average contact stress of the sealing ring under pre-compression conditions:

[0064] in, The average contact stress under pre-compression conditions. The average contact stress when there is no medium pressure. The average contact stress when there is no medium pressure. It is Poisson's ratio.

[0065] Considering that the sealing ring material is a nonlinear polymer such as nitrile rubber, fluororubber, and silicone rubber, its constitutive relation adopts the Mooney-Rivlin model, and the elastic modulus of the rubber can be expressed as:

[0066] Considering that the sealing ring material is a nonlinear polymer such as nitrile rubber, fluororubber, and silicone rubber, its constitutive relation adopts the Mooney-Rivlin model. and The coefficients are those of the Mooney-Rivlin model. Hardness.

[0067] Step 3: Calculate the average contact area of ​​the sealing ring based on the contact width of the sealing ring under pre-compression conditions.

[0068] in, This represents the average contact area of ​​the sealing ring.

[0069] Step 4: Calculate the radial contact force based on the average contact area of ​​the sealing ring and the average contact stress under pre-compression conditions.

[0070] in, Radial contact force, This refers to the standard inner diameter of the sealing ring.

[0071] A small displacement δ is applied in the x, y, and z directions. ai and δ a(i+1) (a=x,y,z). Assuming the system is radially symmetric in structure, then k x =k y The deformation is illustrated as follows: Figure 2 As shown, where F is the contact pressure at a certain angle.t It is a tangential contact force.

[0072] To obtain the stiffness in each direction, a small displacement δ is applied in the x, y, and z directions under the simulated bolt preload condition. ai and δ a(i+1) (a=x,y,z). Since the movable pipe joint with sealing ring is radially symmetrical in the structure under pre-compression, the stiffness in the x and y directions is considered to be the same. Therefore, subsequent radial analyses will take the x direction as an example. Figure 2 This diagram illustrates the deformation before and after pre-compression and displacement in the x and z directions. Here, x and y represent radial directions, and z represents the tangential direction.

[0073] Step 5: Considering that the compressibility varies with the circumferential angle after displacement, and that the force component in the x-direction is small at other axial angles due to the small applied displacement, only the contact pressure at the maximum displacement in the x-direction, i.e., α=0° and α=180°, is summed. The resulting force vector sum in the x-direction is the loading force at that displacement. Based on the radial contact force and the small displacement, the contact pressure vector sum of the sealing ring in the x and y directions is calculated:

[0074]

[0075] in, The first in the x-direction i The contact pressure vector sum, The first in the x-direction i A tiny displacement, The first in the x-direction i +1 basic pressure vector sum, The first in the x-direction i +1 tiny displacement, The first in the y-direction i The contact pressure vector sum, The first in the y-direction i A tiny displacement, The first in the y-direction i +1 basic pressure vector sum, The first in the y-direction i +1 tiny displacement.

[0076] The first in the x-direction i The formula for calculating a small displacement is:

[0077] Step 6: Based on the contact pressure vector in the x-direction and the small displacement in the x-direction, calculate the translational equivalent stiffness in the pre-compression state in the x-direction. Based on the contact pressure vector in the y-direction and the small displacement in the x-direction, calculate the translational equivalent stiffness in the pre-compression state in the x-direction.

[0078]

[0079] in, Let be the translational equivalent stiffness under pre-compression in the x-direction. Let y be the translational equivalent stiffness under pre-compression in the y-direction.

[0080] Step 7: Using fractal contact theory, the relationship between the infinitesimal displacement in the z-direction and the contact stress vector sum in the z-direction is obtained. The formula for calculating the infinitesimal displacement in the z-direction is:

[0081] in, For the first i A small displacement in the z-direction, The Poisson's ratio of the sealing ring. The equivalent shear modulus of the contact surface. The coefficient of friction of the contact surface. The first in the z-direction i The sum of the basic pressure vectors, For the first i +1 small displacement in the z direction, The first in the z-direction i +1 basic pressure vector sum.

[0082] The formula for calculating the equivalent shear modulus of the contact surface is:

[0083] in, It is the elastic modulus.

[0084] It is calculated using the following formula:

[0085] in, For the pipe fitting Poisson's ratio, The elastic modulus of the sealing ring. This is the elastic modulus of the pipe fitting.

[0086] Step 8: Based on the contact pressure vector in the z-direction and the small displacement in the z-direction, calculate the translational equivalent stiffness in the pre-compression state in the z-direction:

[0087] in, It represents the translational equivalent stiffness under pre-compression in the z-direction.

[0088] Step 9: Based on the translational equivalent stiffness in the pre-compression state in the x-direction and the translational equivalent stiffness in the pre-compression state in the z-direction, construct the equivalent stiffness of the sealing ring in the pre-compression state under simple harmonic action.

[0089] The steps to establish the equivalent stiffness of the sealing ring under dynamic loads under harmonic excitation are as follows: First, we will analyze the degradation patterns of the clamps. Figure 3 This refers to the contact deformation state of the sealing ring of the movable pipe joint during vibration, including static contact and compression in the x and y directions. , These represent the relative displacements of the sealing ring in the x and y directions, respectively. When the system exhibits static stiffness. When the time is right, the stiffness changes with deformation. The stiffness can be expressed as:

[0090] in, Let be the stiffness of the sealing ring in the x-direction. The contact pressure varies in the x-direction. A small displacement applied in the x-direction.

[0091] Step 1: Calculate the contact force of the sealing ring at a certain angle based on its shape characteristics.

[0092] in, For the x-th i One sealing ring in α Contact force at angle This is the time-dependent displacement. This is the radial deformation. For the x-th i +1 sealing ring α Contact force at an angle.

[0093] The formula for calculating time-dependent displacement is:

[0094] in, Let x represent the relative displacement of each sealing ring's equivalent spring node in the x-direction at any time t.

[0095] The formula for calculating radial deformation is: , Radial displacement Δ xt It is related to the magnitude of the initial compression δ0, and the total radial displacement is the maximum when α=0°, which is Δδ+δ0, and the contact pressure is also the maximum at this point.

[0096] The second step is to calculate the linear stiffness of the sealing ring in the x-direction and the linear stiffness of the sealing ring in the y-direction based on the contact force of the sealing ring at a certain angle.

[0097]

[0098] in, Let be the linear stiffness of the sealing ring in the x-direction. For deformation Contact force at angle Before deformation Contact force at angle Let be the linear stiffness of the sealing ring in the y-direction. For the y-direction i One sealing ring in β Contact force at angle For the y-direction i +1 sealing ring β Contact force at angle For deformation β Contact force at angle Before deformation β Contact force at an angle.

[0099] Thirdly, under the influence of the basic excitation, the axial stiffness of the movable pipe joint is affected by the axial displacement of the sealing ring. The axial translational stiffness is then processed to obtain the relationship between the axial displacement Δ... zt The relationship between stiffness and linear stiffness. Based on the axial displacement of the sealing ring, calculate the linear stiffness of the sealing ring in the z-direction:

[0100] in, Let be the linear stiffness of the sealing ring in the z-direction. As an intermediate variable, For the axial displacement of the sealing ring, denoted as the coefficient of friction of the sealing ring material.

[0101] The formula for calculating intermediate variables is: .

[0102] Fourth step: From the translational stiffness formulas above, it can be seen that the change in stiffness with increasing displacement is non-linear. Since radial displacement directly affects the angular stiffness in each direction, referring to the variation law of translational stiffness, let the angular stiffness of the sealing ring in each direction be a quadratic function y=c+ax+bx 2 The formula for calculating angular stiffness is:

[0103] in, , For angular stiffness, This represents the equivalent angular stiffness under pre-compression conditions. This is the equivalent angular stiffness coefficient under pre-compression conditions. Let be the angular displacement at any time t. This is the first coefficient value. This is the second coefficient value.

[0104] Step 5: Based on the linear stiffness of the sealing ring in the x-direction, the linear stiffness of the sealing ring in the y-direction, the linear stiffness of the sealing ring in the z-direction, and the angular stiffness, construct the stiffness matrix of the sealing ring under dynamic loads under harmonic excitation:

[0105]

[0106] in, This is the stiffness matrix of the sealing ring under dynamic loads during harmonic excitation. This is a custom-designed stiffness matrix for spring elements.

[0107] Step 6: Based on the stiffness matrix of the sealing ring under dynamic load under simple harmonic excitation, obtain the equivalent stiffness of the sealing ring under dynamic load under simple harmonic excitation.

[0108] S12. Model the pipeline and movable pipe joint using Timoshenko beam elements to obtain the model of the pipeline and movable pipe joint.

[0109] Specifically, the steps include: Step 1, Timoshenko beam unit, as follows Figure 4 As shown, the diagram includes two nodes, i and j, and displays the six degrees of freedom of the two nodes at both ends of the pipe fitting. The node displacement vector can then be expressed as:

[0110] in, Let be the nodal displacement vector of the element. for i Translational displacement of the node in the z-direction. for iTranslational displacement of the node in the y-direction. for i Translational displacement of the node in the x-direction. for i Angular displacement of the node in the z-direction for i Angular displacement of the node in the y-direction for i Angular displacement of the node in the x-direction. for j Displacement of node in the z-direction for j Displacement of the node in the y-direction. for j Node displacement in the x-direction for j Angular displacement of the node in the z-direction for j Angular displacement of the node in the x-direction. for j Angular displacement of the node in the x-direction.

[0111] The second step is to express the kinetic energy of the unit as:

[0112] in, The kinetic energy of the unit. For pipeline density, The cross-sectional area of ​​the pipe. , , , , , Differentiate the time in the six directions of the element. Let ox be the moment of inertia of the cross section. Let Oy be the moment of inertia of the cross section. Let the moment of inertia of the cross section be oz. For the first k The length and cross-sectional area of ​​each unit The third step involves using functionals u, v, w, θ. and As independent variables, the potential energy of the unit is expressed as:

[0113] in, The kinetic energy of the unit. The elastic modulus of the pipe fitting. , , , , , Differentiate the spatial displacements of the element in six directions. The y-axis shear coefficient. For unit shear modulus, The z-axis shear coefficient is... The moment of inertia is the torsional section.

[0114] Step 4: Obtain the partial differential equations for the six directions of the element based on Hamilton's principle:

[0115] in, , , , , , The second derivatives are the six directions of the element.

[0116] Based on the second derivatives of the six directions of the element, the shape functions of the six directions of the element are calculated:

[0117] in, These are the shape functions for the six directions of the element. These are the components of the shape function in the first direction in the two directions, respectively. These are the components of the shape function in the second direction in the four directions. These are the components of the shape function in the third direction in the four directions. These are the components of the shape function in the fourth direction in two directions, respectively. These are the components of the shape function in the fifth direction in the four directions. , These are the components of the shape function in the sixth direction in the four directions.

[0118] Step 5: Express the mass matrix of the element as follows:

[0119] in, This is the quality matrix.

[0120] The stiffness matrix of the element is expressed as:

[0121] in, Here is the stiffness matrix. Let x be the derivative of the axial displacement shape function vector with respect to x. For pipeline density, The y-axis shear coefficient. Shear modulus Let x be the derivative of the shape function vector of the lateral displacement in the y-direction with respect to x. Let be a angular function vector about the y-axis. The z-axis shear coefficient is... Let x be the derivative of the shape function vector of the lateral displacement in the z-direction with respect to x. Let be the bending angle function vector about the z-axis. For the torsional moment of inertia of the cross section, Let x be the derivative of the twisted angular function vector about the x-axis with respect to x. Let x be the derivative of the angular vector of the bending function about the z-axis. Let x be the derivative of the angular vector of the function about the y-axis with respect to x. Let be the shape function vector of the lateral displacement in the z-direction. Let be the bending angle function vector about the z-axis. Let be a angular function vector about the y-axis.

[0122] S13. Combine the sealing ring model considering equivalent stiffness with the models of the pipeline and the movable pipe joint to obtain the dynamic model of the movable pipe joint-pipeline system.

[0123] The specific steps for combining the sealing ring model considering equivalent stiffness with the models of the pipeline and the movable pipe joint to obtain the dynamic model of the movable pipe joint-pipeline system are as follows: Grouping the element matrices yields the motion differential equations of the active pipe fitting-piping system:

[0124] in, The quality matrix of the pipeline, For the generalized displacement vector of the pipeline system, The first derivative of the generalized displacement vector is... The second derivative of the generalized displacement vector is... Here is the damping matrix of the pipeline. Here is the stiffness matrix of the pipeline. Here is the time-varying stiffness matrix of the pipe joint. This is the basic external excitation vector experienced by the pipeline system.

[0125] The formula for calculating the basic external excitation vector of the piping system is:

[0126] in, Let the basic external excitation vector be the vector acting on the pipeline system. This is the displacement indicator vector, with a value of 1 in the excitation direction and 0 in all other directions. In the simulation, an inertial force is applied to each node of the pipeline system along the force direction, effectively converting the excitation force into a distributed force. To determine the level of incentive, For the excitation frequency, To incentivize time.

[0127] S2. The established dynamic model of the active pipe joint-pipeline system is validated to obtain the validated model.

[0128] S3. Based on the validated model, simulate the influence of each excitation amplitude on the natural frequency of the movable pipe joint-pipeline system to verify the correctness of the dynamic model of the movable pipe joint-pipeline system.

[0129] The specific steps are as follows: The first step is to obtain the natural frequency of the movable pipe joint-pipeline system based on multiple hammer tests.

[0130] In contrast to simulation, Figure 5 For the comparison of field and frequency response functions of the fixed-supported modal test, the comparison between test and simulation results is shown in Table 1. The errors of the first four natural frequencies are all within 5%, and the trends of the frequency response function curves are basically consistent, indicating that the stiffness model of the movable pipe joint meets the accuracy requirements.

[0131] Table 1 Comparison of experimental and simulation modal results under fixed boundary conditions.

[0132] The second step is to conduct a frequency sweep test on the dynamic model of the movable pipe joint-pipe system to identify the nonlinear stiffness. The identified nonlinear stiffness coefficients are then substituted into the calculation formula for the angular stiffness of the movable pipe joint in each direction. The obtained dynamic stiffness and modal damping ratio are then introduced into the system dynamic model to obtain the natural frequency of the dynamic model.

[0133] The translational nonlinear stiffness of the movable pipe joint is:

[0134]

[0135] The nonlinear stiffness coefficient k was identified through multiple iterations. nx =k ny =73.52 N·m / rad, k nz =45.60 N·m / rad. The final formula for calculating the angular stiffness of the movable pipe joint in each direction is:

[0136]

[0137] The third step is to verify the correctness of the model by comparing the natural frequency of the active pipe fitting-pipeline system with the natural frequency of the dynamic model.

[0138] The aforementioned dynamic stiffness and modal damping ratio were incorporated into the system dynamics model, and the response under various excitation amplitudes was simulated and calculated. The results were then compared with experimental results, as shown below. Figure 6 As shown. Figure 6 The comparison results are quantified in Table 2. The maximum frequency error is only 0.28%, and the maximum amplitude error is 3.88%, indicating that the established nonlinear model has high accuracy and applicability.

[0139] Table 2 Comparison of simulation results for different excitation amplitudes

[0140] In summary, this invention addresses the equivalent contact stiffness problem of the sealing ring in a movable pipe joint. It utilizes the force-deformation relationship to obtain the translational stiffness in each direction under pre-compression. By introducing the vibration deformation of the sealing ring, the relationship between translational stiffness and displacement is derived, and a calculation formula for angular stiffness and its parameter identification method are proposed. The obtained stiffness model is introduced into the finite element model of the pipeline system, and the results are compared with those from hammer impact tests and vibration tests, verifying the effectiveness of the established model.

[0141] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A dynamic modeling method for a piping system containing movable pipe fittings, characterized in that, Includes the following steps: Based on the finite element method, a dynamic model of the movable pipe joint-pipeline system is established. In the dynamic model, the model of the pipeline and joint is established according to the Timoshenko beam theory, and the equivalent stiffness of the movable pipe joint under pre-compression and dynamic load is considered. The established dynamic model of the active pipe joint-pipeline system was validated to obtain the validated model. Based on the validated model, the influence of each excitation amplitude on the natural frequency of the movable pipe joint-pipeline is simulated to verify the correctness of the dynamic model of the movable pipe joint-pipeline system.

2. The dynamic modeling method for a piping system containing movable pipe joints according to claim 1, characterized in that, The establishment of the dynamic model of the active pipe joint-pipeline system includes: The equivalent stiffness of the sealing ring under pre-compression state under simple harmonic action is established, and the equivalent stiffness of the sealing ring under dynamic load under simple harmonic excitation is established, thus obtaining a sealing ring model considering the equivalent stiffness. The pipeline and movable pipe joints are modeled using Timoshenko beam elements to obtain the model of the pipeline and movable pipe joints. The sealing ring model considering equivalent stiffness is combined with the models of the pipeline and the movable pipe joint to obtain the dynamic model of the movable pipe joint-pipeline system.

3. The dynamic modeling method for a piping system containing movable pipe joints according to claim 2, characterized in that, The equivalent stiffness of the sealing ring under pre-compression state under simple harmonic action includes: Based on the O-ring installation structure, the initial compression is expressed as: in, This is the initial compression amount. d 0 represents the initial cross-sectional diameter of the sealing ring. d 1 represents the pre-compression section diameter of the sealing ring; Based on the initial compression and the initial cross-sectional diameter of the sealing ring, the initial compression ratio is expressed as: in, This is the initial compression ratio; Calculate the contact width of the sealing ring under pre-compression conditions based on the medium pressure acting on the sealing ring: in, The contact width of the sealing ring. This refers to the elastic modulus of rubber. The pressure of the medium acting on the sealing ring; Calculate the circumference of the sealing ring based on the pre-compression section diameter of the sealing ring: in, L The circumference of the sealing ring. D This refers to the standard inner diameter of the sealing ring. Calculate the average contact stress of the sealing ring under pre-compression state based on the average contact stress of the sealing ring when there is no medium pressure and the additional contact stress caused by fluid pressure: in, The average contact stress under pre-compression conditions. The average contact stress when there is no medium pressure. The average contact stress when there is no medium pressure. Poisson's ratio; Calculate the average contact area of ​​the sealing ring based on the contact width of the sealing ring under pre-compression conditions: in, This represents the average contact area of ​​the sealing ring; Calculate the radial contact force based on the average contact area of ​​the sealing ring and the average contact stress under pre-compression conditions: in, Radial contact force, This refers to the standard inner diameter of the sealing ring. Under bolt preload, the sealing ring undergoes a small displacement. Based on the radial contact force and the small displacement, the contact pressure vector sum of the sealing ring in the x and y directions is calculated: in, The first in the x-direction i The contact pressure vector sum, The first in the x-direction i A tiny displacement, The first in the x-direction i +1 basic pressure vector sum, The first in the x-direction i +1 tiny displacement, The first in the y-direction i The contact pressure vector sum, The first in the y-direction i A tiny displacement, The first in the y-direction i +1 basic pressure vector sum, The first in the y-direction i +1 tiny displacement; Based on the contact pressure vector in the x-direction and the small displacement in the x-direction, the translational equivalent stiffness under pre-compression in the x-direction is calculated. Similarly, based on the contact pressure vector in the y-direction and the small displacement in the x-direction, the translational equivalent stiffness under pre-compression in the x-direction is also calculated. in, Let be the translational equivalent stiffness under pre-compression in the x-direction. The translational equivalent stiffness in the pre-compression state in the y direction; Using fractal contact theory, the relationship between the minute displacement in the z-direction and the contact stress vector sum in the z-direction is obtained. The formula for calculating the minute displacement in the z-direction is as follows: in, For the first i A small displacement in the z-direction, The Poisson's ratio of the sealing ring. The equivalent shear modulus of the contact surface. The coefficient of friction of the contact surface. The first in the z-direction i The sum of the basic pressure vectors, For the first i +1 small displacement in the z direction, The first in the z-direction i +1 basic pressure vector sum; Based on the contact pressure vector in the z-direction and the small displacement in the z-direction, the translational equivalent stiffness under pre-compression state in the z-direction is calculated: in, The translational equivalent stiffness under pre-compression in the z-direction; Based on the translational equivalent stiffness in the pre-compression state in the x-direction and the translational equivalent stiffness in the pre-compression state in the z-direction, the equivalent stiffness of the sealing ring in the pre-compression state under simple harmonic action is constructed.

4. The dynamic modeling method for a piping system containing a movable pipe joint according to claim 3, characterized in that, The formula for calculating the elastic modulus of the rubber is: in, and These are the coefficients of the first model and the coefficients of the second model, respectively. Hardness The first in the x direction i The formula for calculating a small displacement is: The formula for calculating the equivalent shear modulus of the contact surface is: in, For elastic modulus, It is calculated using the following formula: in, For the pipe fitting Poisson's ratio, The elastic modulus of the sealing ring. This is the elastic modulus of the pipe fitting.

5. The dynamic modeling method for a piping system containing movable pipe joints according to claim 2, characterized in that, The equivalent stiffness of the sealing ring under dynamic load under harmonic excitation includes: Based on the shape characteristics of the sealing ring itself, calculate the contact force of the sealing ring at a certain angle: in, For the x-th i One sealing ring in α Contact force at angle This is the time-dependent displacement. This is the radial deformation. For the x-th i +1 sealing ring α Contact force at an angle; Based on the contact force of the sealing ring at a certain angle, the linear stiffness of the sealing ring in the x-direction and the linear stiffness of the sealing ring in the y-direction are calculated: in, Let be the linear stiffness of the sealing ring in the x-direction. For deformation Contact force at angle Before deformation Contact force at angle Let be the linear stiffness of the sealing ring in the y-direction. For the y-direction i One sealing ring in β Contact force at angle For the y-direction i +1 sealing ring β Contact force at angle For deformation β Contact force at angle Before deformation β Contact force at an angle; Calculate the linear stiffness of the sealing ring in the z-direction based on the axial displacement of the sealing ring: in, Let be the linear stiffness of the sealing ring in the z-direction. As an intermediate variable, For the axial displacement of the sealing ring, The coefficient of friction of the sealing ring material; Based on the variation law of translational stiffness, assuming that the angular stiffness of the sealing ring in each direction is a quadratic function, the formula for calculating the angular stiffness is: in, , For angular stiffness, This represents the equivalent angular stiffness under pre-compression conditions. This is the equivalent angular stiffness coefficient under pre-compression conditions. Let be the angular displacement at any time t. This is the first coefficient value. This is the second coefficient value; Based on the linear stiffness of the sealing ring in the x-direction, the linear stiffness of the sealing ring in the y-direction, the linear stiffness of the sealing ring in the z-direction, and the angular stiffness, the stiffness matrix of the sealing ring under dynamic load under harmonic excitation is constructed: in, This is the stiffness matrix of the sealing ring under dynamic loads during harmonic excitation. For the self-compiled spring element stiffness matrix; Based on the stiffness matrix of the sealing ring under dynamic load under harmonic excitation, the equivalent stiffness of the sealing ring under dynamic load under harmonic excitation is obtained.

6. The dynamic modeling method for a piping system containing a movable pipe joint according to claim 5, characterized in that, The formula for calculating the time-related displacement is: in, Let x represent the relative displacement of each sealing ring's equivalent spring node in the x-direction at any time t. The formula for calculating the radial deformation is: ， The formula for calculating the intermediate variable is as follows: 。 7. The dynamic modeling method for a piping system containing movable pipe joints according to claim 2, characterized in that, The piping and movable pipe joints are modeled using Timoshenko beam elements, including: The nodal displacement vector of the element is expressed as: in, Let be the nodal displacement vector of the element. for i Translational displacement of the node in the z-direction. for i Translational displacement of the node in the y-direction. for i Translational displacement of the node in the x-direction. for i Angular displacement of the node in the z-direction for i Angular displacement of the node in the y-direction for i Angular displacement of the node in the x-direction. for j Displacement of node in the z-direction for j Displacement of the node in the y-direction. for j Node displacement in the x-direction for j Angular displacement of the node in the z-direction for j Angular displacement of the node in the x-direction. for j Angular displacement of the node in the x-direction. The kinetic energy of the unit is expressed as: in, The kinetic energy of the unit. For pipeline density, The cross-sectional area of ​​the pipe. , , , , , Differentiate the time in the six directions of the element. Let ox be the moment of inertia of the cross section. Let Oy be the moment of inertia of the cross section. Let the moment of inertia of the cross section be oz. For the first k The length and cross-sectional area of ​​each unit With functionals u, v, w, θ, and As independent variables, the potential energy of the unit is expressed as: in, The kinetic energy of the unit. The elastic modulus of the pipe fitting. , , , , , Differentiate the spatial displacements of the element in six directions. The y-axis shear coefficient. For unit shear modulus, The z-axis shear coefficient is... The moment of inertia is the torsional section.

8. The dynamic modeling method for a piping system containing a movable pipe joint according to claim 2, characterized in that, The sealing ring model, which considers equivalent stiffness, is combined with the models of the pipeline and the movable pipe joint to obtain a dynamic model of the movable pipe joint-pipeline system, including: Grouping the element matrices yields the motion differential equations of the active pipe fitting-piping system: in, The quality matrix of the pipeline. For the generalized displacement vector of the pipeline system, The first derivative of the generalized displacement vector is... The second derivative of the generalized displacement vector is... Here is the damping matrix of the pipeline. Here is the stiffness matrix of the pipeline. Here is the time-varying stiffness matrix of the pipe joint. Let the basic external excitation vector be the vector acting on the pipeline system. The formula for calculating the basic external excitation vector of the pipeline system is as follows: in, Let the basic external excitation vector be the vector acting on the pipeline system. For displacement indication vector, To determine the level of incentive, For the excitation frequency, To incentivize time.

9. The dynamic modeling method for a piping system containing a movable pipe joint according to claim 1, characterized in that, Based on the validated model, the influence of various excitation amplitudes on the natural frequency of the movable pipe joint is simulated to verify the correctness of the dynamic model of the movable pipe joint, including: The natural frequency of the movable pipe joint-pipeline system was obtained based on multiple hammer impact tests. A frequency sweep test was conducted on the dynamic model of the movable pipe joint-pipe system to identify the nonlinear stiffness. The identified nonlinear stiffness coefficients were substituted into the calculation formula of the angular stiffness in each direction of the movable pipe joint. The obtained dynamic stiffness and modal damping ratio were introduced into the system dynamic model to obtain the natural frequency of the dynamic model. The correctness of the model was verified by comparing the natural frequencies of the active pipe fitting-pipeline system with the natural frequencies of the dynamic model.

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