A method for modeling dynamics of a piping system considering soft nonlinearity of a clamp

Abstract

The application provides a pipeline system dynamics modeling method considering the soft nonlinearity of a clamp, comprising: establishing a nonlinear clamp-pipeline system dynamics model based on a finite element method; verifying the established nonlinear clamp-pipeline system model and identifying nonlinear parameters; further verifying the nonlinear clamp-pipeline system model under different boundary conditions and designing vibration response characteristics by using the verified model and the identified nonlinear parameters. Based on a genetic algorithm and a finite element method, the application proposes a multi-degree-of-freedom nonlinear clamp-pipeline system model, identifies nonlinear parameters by taking frequency errors and response amplitude errors between tests and simulations as objective functions, and verifies the established model through modal and vibration response tests under different boundary conditions. The application considers the soft nonlinearity of a metal rubber clamp when predicting vibration characteristics of a pipeline system, and has higher accuracy.

Description

Technical Field

[0001] This invention relates to the field of mechanical dynamics technology, and in particular to a dynamic modeling method for pipeline systems that considers the soft nonlinearity of clamps. Background Technology

[0002] Piping, as a crucial component connecting aero-engine lubrication, fuel, regulation, and starting systems with other accessories, is typically fixed to the engine casing or interconnected via clamps, forming a complex piping system. Various excitation modes from the aero-engine are transmitted to the piping system through these clamps; therefore, establishing a dynamic model of the clamps and understanding the vibration characteristics of the clamp-piping system has become a current research hotspot. Although some scholars have studied the dynamic characteristics of clamps, most studies have neglected their nonlinear effects during modeling, and the dynamic analysis of piping systems incorporating clamp nonlinearities is still insufficient.

[0003] One of the functions of clamps is to reduce the vibration of pipeline systems. In the study of the vibration characteristics of pipeline systems, rigid constraints or flexible springs are usually used to model the clamps equivalently. However, clamps are segmented structures, and the gaskets are made of metal-rubber materials. This determines that the clamps have soft nonlinearity. When subjected to external excitation, friction occurs between the metal wires in the metal-rubber, thereby generating nonlinear forces. Therefore, the linear model of the clamp will affect the accurate prediction of the pipeline vibration response. Establishing a nonlinear model of the metal-rubber clamp and introducing it into the pipeline model to study the dynamic characteristics of the nonlinear clamp-pipeline system is of great significance.

[0004] In the process of nonlinear modeling of clamps, determining the nonlinear coefficients is crucial. The back-inference identification method is a technique based on experimentation and genetic algorithms to identify unknown parameters. It uses the natural frequency error and response amplitude error obtained from frequency sweeping as the objective function, and the nonlinear coefficients as the optimization variables. By combining the frequency response curves of the pipeline system obtained from experiments, the optimization variables corresponding to minimizing the objective function are the identified clamp nonlinear coefficients. Introducing the identified nonlinear coefficients into the dynamic model yields an accurate nonlinear clamp-pipeline system dynamic model, which can support subsequent vibration analysis. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention provides a dynamic modeling method for pipeline systems considering the flexible nonlinearity of clamps. This method offers high solution efficiency, fills a gap in research on flexible nonlinear clamp-pipeline systems based on the finite element method, and provides support for subsequent pipeline vibration analysis.

[0006] The technical solution adopted in this invention is as follows:

[0007] The present invention proposes a dynamic modeling method for pipeline systems considering the nonlinearity of clamps, comprising:

[0008] A dynamic model of the nonlinear clamp-pipeline system was established based on the finite element method.

[0009] The established nonlinear clamp-pipeline system model was validated and its nonlinear parameters were identified.

[0010] Using the validated model and identified nonlinear parameters, the nonlinear clamp-pipe system model under different boundary conditions is further validated, and vibration response characteristics are designed.

[0011] Furthermore, the establishment of a nonlinear clamp-pipeline system dynamic model based on the finite element method specifically includes: establishing a pipeline model;

[0012] Introducing nonlinear clamps;

[0013] Identify the nonlinear coefficients.

[0014] Furthermore, the establishment of the pipeline model specifically includes:

[0015] According to the finite element method, the entire pipeline is discretized using Timoshenko beam elements. Each beam element has two nodes, and assuming each node has four degrees of freedom, the element node displacement vector is defined as follows:

[0016] q e =[v i ,w i ,θ yi ,θ zi ,v j ,w j ,θ yj ,θ zj ] T

[0017] The displacement vector of any node within the element can be expressed as:

[0018]

[0019] Where N(x) is the unit shape function, and its expression is as follows:

[0020]

[0021] Based on Hamilton's principle and the principle of minimum potential energy, the partial differential equations of the pipeline are obtained. Substituting the element displacement vectors into the equations, the stiffness matrix and mass matrix of the element in the local coordinate system are obtained. The damping matrix adopts the Rayleigh damping form.

[0022]

[0023]

[0024]

[0025] In the formula, ρ p l k and A p Let I represent the density, length, and cross-sectional area of ​​the k-th pipe element, respectively; y and I z Let θ represent the moments of inertia of the cross sections about oy and oz, respectively; v and w represent the translational displacements of any cross section along the y and z axes, respectively; y and θ z Let E and G represent the angular displacement of any cross section about the y and z directions, respectively, and let κ represent Young's modulus and shear modulus, respectively. y and κ z represents the shear coefficients about the y and z axes, respectively, with a value of 0.5 for thin-walled cylindrical parts; the superscript of the variable with an apostrophe (′) indicates that the first derivative is taken with respect to the x coordinate, f1 and f2 are the first two natural frequencies, and ξ1=ξ2=0.02 are the first two modal damping ratios;

[0026] Furthermore, the introduction of the nonlinear clamp specifically includes:

[0027] Considering the influence of the clamp width, the metal felt single clamp is equivalently discretized along the axial direction into two linear springs and two nonlinear springs. The measured stiffness of the clamp introduced by the linear springs was obtained through multiple experimental measurements: k cy =1×10 7 N / m, k cz =8×10 6 N / m, k cθy =70 Nm / rad, k cθz =30Nm / rad, and the stiffness of each linear spring is 1 / 2 of the measured stiffness in that direction;

[0028] When an external excitation is present, a nonlinear restoring force is generated between the metal wires. Assume that the nonlinear restoring force f ni With displacement x i The relationship is:

[0029] f ni =k ni x i 3 i = v, w, θ y ,θ z

[0030] In the formula, k ni For the unknown nonlinear stiffness coefficient of the clamp, x i Let k be the displacement of the clamp in this direction. niWhen the stiffness is 0, the support stiffness degenerates from a nonlinear spring to a linear spring. To solve the problem of difficulty in solving the nonlinear term in the stiffness matrix, the nonlinear stiffness term is shifted to the right into the force vector. Therefore, the dynamic equation of the nonlinear clamp-pipe system is obtained as follows:

[0031]

[0032] In the formula, M p C p and K p These are the mass matrix, damping matrix, and stiffness matrix of the pipeline, respectively; C c K is the damping matrix of the clamp; c q is the linear stiffness matrix of the clamp; q is the generalized displacement vector of the system; F is the external force vector; F n For nonlinear force vectors:

[0033]

[0034] To simplify the calculation, we assume that the nonlinear stiffness coefficients k in the four directions are... ni If they are equal, then k needs to be adjusted based on vibration response tests. ni Perform identification.

[0035] The equivalent viscous damping coefficient of the metal-rubber clamp was determined by analyzing the hysteresis loop, and the energy dissipated by the clamp was expressed as:

[0036] ΔW=πPa=πxT

[0037] maxnmaxn

[0038] In the formula, a n and T n P represents the intercepts of the elliptic loop on the coordinate axes. max The restoring force at the point of maximum deformation is expressed as the elastic potential energy of the system:

[0039]

[0040] Therefore, the damping loss factor can be expressed as:

[0041]

[0042] The equation of the ellipse can be expressed as:

[0043]

[0044] For a viscous damped system, let c c Let be the equivalent viscous damping coefficient, then the restoring force in the dry friction system is:

[0045]

[0046] When x = 0, the restoring force of the system is:

[0047] P(x) x＝0 =c c x max ω

[0048] According to the equivalence principle, the restoring force in a dry friction hysteresis system is equal to the equivalent viscous damping force in an equivalent viscous system, therefore:

[0049]

[0050] therefore:

[0051]

[0052] In the formula, k is the measured stiffness of the clamp, η is the damping loss factor, and ω is the excitation frequency. Due to the nonlinearity of the metal rubber material, the damping changes with the increase of the excitation amplitude. Therefore, in order to correct the equivalent viscous damping coefficient c... c Introducing a modified nonlinear damping coefficient c n The corrected nonlinear damping coefficient is c n c c ;

[0053] Furthermore, the identification of nonlinear coefficients specifically includes:

[0054] To determine the unknown nonlinear stiffness coefficient k in the dynamic equations of the nonlinear clamp-system. n and nonlinear damping coefficient c n Parameter identification is performed based on the input excitation and output response results of the experiment;

[0055] An objective function is constructed using the frequency error and response amplitude error of the experiment, and identified by reverse engineering using a genetic algorithm; two frequency points ω are selected on both sides of the resonance frequency ω0. m and ω n And ω m >ω n The response amplitudes corresponding to these two points are equal, i.e., A. mop =A nop Calculate the experimental and simulation values ​​at ω respectively. n The response amplitude error at ω0 was calculated, and the genetic algorithm and Newmark-β method were used for iterative solution to finally obtain the identified nonlinear coefficients.

[0056] Furthermore, the establishment of the nonlinear clamp-pipe system model is validated and nonlinear parameters are identified, specifically including modal validation and vibration response validation of the clamp-pipe system by hammer impact test.

[0057] Furthermore, the testing instruments required for the modal verification of the clamp-pipe system hammer test include a triaxial accelerometer, a force hammer, and a 12-channel LMS system. To ensure the accuracy of the test results, the natural frequencies of the clamp-pipe system are obtained through multiple hammer tests. The finite element method is used for dynamic modeling, and the natural frequencies and mode shapes of the pipe are obtained by solving for eigenvalues ​​and eigenvectors. The simulation and experimental frequency and mode shape results are compared to verify the model.

[0058] Furthermore, the vibration response verification of the clamp-pipe system under hammer impact test specifically includes: using an electromagnetic vibration table to subject the clamp-pipe system to simple harmonic excitation at different excitation amplitudes of 0.5g, 1g, 2g, and 3g; setting the sweep frequency bandwidth based on the hammer impact test results; the duration of each sweep frequency test being 2.5 minutes; the acceleration signal at the target location being processed by the LMS SCADAS mobile front end and transmitted to the LMS mobile workstation; using a genetic algorithm and the test frequency response curve to identify the nonlinear coefficient of the clamp; after bringing it back to the model, identifying the natural frequency of the pipe by the frequency corresponding to the resonance peak in the frequency response under sweep frequency excitation; and verifying the model by comparing the frequency and response amplitude of the simulation and the test under sweep frequency excitation.

[0059] Furthermore, by utilizing the validated model and identified nonlinear parameters, the nonlinear clamp-pipe system model under different boundary conditions is further validated, and the vibration response characteristics are designed. The different boundary conditions specifically include:

[0060] Genetic algorithms and experimental frequency response curves were used to identify the nonlinear coefficients of the clamps. The identified nonlinear coefficients were then brought back to the clamp-pipeline system at the fixed-elastic boundary and the clamp-pipeline system at the elastic-elastic boundary, respectively. The natural frequency of the pipeline was identified by the frequency corresponding to the resonant peak in the frequency response under frequency sweep excitation. Finally, the model was further verified by comparing the frequency and response amplitude of the simulation and the experiment.

[0061] Compared with the prior art, the present invention has the following advantages:

[0062] 1. The present invention provides a pipeline system dynamics modeling method that considers the soft nonlinearity of clamps. This model takes into account the soft nonlinearity of metal rubber clamps, thereby improving the calculation accuracy of the model.

[0063] 2. The present invention provides a dynamic modeling method for pipeline systems considering the nonlinearity of clamps. First, based on Hamilton's principle and the principle of minimum potential energy, a dynamic model of the pipeline system including clamps is derived, incorporating the measured linear and nonlinear stiffness of the clamps, and verifying the accuracy of the model through a pipe impact test. Second, the nonlinear coefficients are identified through frequency response curves obtained by frequency sweep tests, and experimental verification is performed. Finally, the identified nonlinear coefficients are substituted into nonlinear clamp-pipeline models under different boundary conditions, and experimental verification is conducted for each.

[0064] 3. The pipeline system dynamic modeling method considering the flexible nonlinearity of clamps provided by this invention fills the gap in pipeline system dynamic modeling considering the flexible nonlinearity of clamps, and provides support for subsequent pipeline system vibration response analysis. Attached Figure Description

[0065] 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.

[0066] Figure 1 A flowchart illustrating the pipeline dynamics modeling method considering the flexible nonlinearity of clamps in this invention;

[0067] Figure 2 A schematic diagram of a pipeline system model considering the nonlinearity of clamps in this invention;

[0068] Figure 3 This is a flowchart illustrating the optimization and iteration process of the nonlinear coefficient of the clamp in this invention.

[0069] Figure 4 This is a comparison of simulation and experimental results of the frequency response function of the nonlinear clamp-pipeline system of this invention;

[0070] Figure 5 This is a comparison diagram of the vibration mode simulation and experiment of the nonlinear clamp-pipeline system of this invention;

[0071] Figure 6 The figure shows the test results of the pipeline system under different excitation amplitudes according to the present invention;

[0072] Figure 7 This is a diagram illustrating the optimization and iteration process of the nonlinear coefficient of the clamp in this invention.

[0073] Figure 8 The figure shows the comparison between simulation and experimental results of the vibration response of the nonlinear clamp-pipe system of this invention;

[0074] Figure 9 The figure shows a comparison of the simulation and experimental results of the frequency response function of the nonlinear clamp-pipe system under two boundary conditions according to the present invention.

[0075] Figure 10 This is a comparison of simulation and experimental results of the vibration response of the nonlinear clamp-pipe system under the fixed-elastic boundary of the present invention.

[0076] Figure 11 This is a comparison of the vibration response simulation and experimental results of the nonlinear clamp-pipeline system under the spring-support-spring-support boundary of the present invention. Detailed Implementation

[0077] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0078] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the present invention or its application or use. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0079] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of exemplary embodiments according to the invention. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0080] Unless otherwise specifically stated, the relative arrangement, numerical expressions, and values ​​of the components and steps described in these embodiments do not limit the scope of the invention. It should also be understood that, for ease of description, the dimensions of the various parts shown in the drawings are not drawn to actual scale. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it need not be further discussed in subsequent figures.

[0081] In the description of this invention, it should be understood that the orientation or positional relationship indicated by directional terms such as "front, back, up, down, left, right", "horizontal, vertical, horizontal" and "top, bottom" is generally based on the orientation or positional relationship shown in the accompanying drawings, and is only for the convenience of describing this invention and simplifying the description. Unless otherwise stated, these directional terms do not indicate or imply that the device or element referred to must have a specific orientation or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation on the scope of protection of this invention. The directional terms "inner" and "outer" refer to the inner and outer contours relative to the outline of each component itself.

[0082] For ease of description, spatial relative terms such as "above," "over," "on the upper surface of," "above," etc., are used herein to describe the spatial positional relationship of a device or feature as shown in the figures to other devices or features. It should be understood that spatial relative terms are intended to encompass different orientations in use or operation besides the orientation of the device as described in the figures. For example, if the device in the figures is inverted, a device described as "above" or "above" other devices or structures would subsequently be positioned as "below" or "under" other devices or structures. Thus, the exemplary term "above" can include both "above" and "below." The device may also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatial relative descriptions used herein will be interpreted accordingly.

[0083] Furthermore, it should be noted that the use of terms such as "first" and "second" to define components is merely for the purpose of distinguishing the corresponding components. Unless otherwise stated, the above terms have no special meaning and therefore should not be construed as limiting the scope of protection of this invention.

[0084] like Figure 1 ,2 As shown, this invention provides a dynamic modeling method for pipeline systems considering the nonlinearity of clamps, specifically including the following steps:

[0085] S1. Based on the finite element method, establish a dynamic model of the nonlinear clamp-pipeline system;

[0086] S2. Perform model verification and nonlinear parameter identification on the established nonlinear clamp-pipeline system model;

[0087] S3. Using the validated model and identified nonlinear parameters, further validate the nonlinear clamp-pipeline system model under different boundary conditions, and design vibration response characteristics.

[0088] In a specific implementation, as a preferred embodiment of the present invention, the specific process of step S1 is as follows:

[0089] S11. Establish a pipeline model;

[0090] The entire pipeline is discretized using Timoshenko beam elements. Each beam element has two nodes, and assuming each node has four degrees of freedom, the element nodal displacement vector can be defined as follows:

[0091] q e =[v i ,w i ,θ yi ,θ zi ,v j ,w j ,θ yj ,θ zj ] T (1)

[0092] The displacement vector of any node within an element can be expressed as:

[0093]

[0094] Where N(x) is the unit shape function, and its expression is as follows:

[0095]

[0096] Based on Hamilton's principle and the principle of minimum potential energy, the partial differential equations of the pipeline can be obtained. Substituting the element displacement vectors into the equations, the stiffness matrix and mass matrix of the element in the local coordinate system can be obtained. The damping matrix adopts the Rayleigh damping form.

[0097]

[0098]

[0099]

[0100] In the formula, ρ p l k and A p These refer to the density, length, and cross-sectional area of ​​the k-th pipe unit, respectively. y and I z Let represent the moments of inertia of the cross sections about oy and oz, respectively; v and w are the translational displacements of any cross section along the y and z axes, respectively; θ y and θ z Let E and G represent the angular displacement of any cross section about the y and z directions, respectively, and let κ represent Young's modulus and shear modulus, respectively. y and κ z These represent the shear coefficients about the y and z axes, respectively, with a value of 0.5 for thin-walled cylindrical parts. The superscript of the variable with an apostrophe (′) indicates that the first derivative is taken with respect to the x-axis. f1 and f2 are the first two natural frequencies, and ξ1=ξ2=0.02 are the first two modal damping ratios.

[0101] S12, Introduce nonlinear clamps;

[0102] Considering the influence of the clamp width, the metal felt single clamp is equivalently discretized along the axial direction into two linear springs and two nonlinear springs. The linear springs can be introduced into the clamp's measured stiffness, which is obtained through multiple experimental measurements: k cy =1×10 7 N / m, k cz =8×10 6 N / m, k cθy =70 Nm / rad, k cθz =30Nm / rad, and the stiffness of each linear spring is 1 / 2 of the measured stiffness in that direction.

[0103] In previous experiments, the natural frequency of the clamp-pipe system often exhibited a dependence on the excitation amplitude. This is mainly due to the strong soft nonlinearity of the elastic restoring force of the metal-rubber material, which is therefore typically characterized using a high-order nonlinear model. The amplitude-frequency characteristics of the dry friction system composed of metal-rubber are primarily determined by a third-order nonlinear displacement factor. Therefore, when an external excitation is present, a nonlinear restoring force will be generated between the metal wires. Assuming the nonlinear restoring force f... ni With displacement x i The relationship is:

[0104] f ni =k ni x i 3 i = v, w, θ y ,θ z (7)

[0105] In the formula, k niFor the unknown nonlinear stiffness coefficient of the clamp, x i Let k be the displacement of the clamp in this direction. ni When the stiffness is 0, the support stiffness degenerates from a nonlinear spring to a linear spring. To address the difficulty of solving for the nonlinear term in the stiffness matrix, the nonlinear stiffness term is shifted to the right into the force vector. Therefore, the dynamic equation of the nonlinear clamp-pipe system can be obtained as follows:

[0106]

[0107] In the formula, M p C p and K p These are the mass matrix, damping matrix, and stiffness matrix of the pipeline, respectively; C c K is the damping matrix of the clamp; c q is the linear stiffness matrix of the clamp; q is the generalized displacement vector of the system; F is the external force vector; F n It is a nonlinear force vector.

[0108]

[0109] To simplify the calculation, we assume that the nonlinear stiffness coefficients k in the four directions are... ni If they are equal, then k needs to be adjusted based on vibration response tests. ni Perform identification.

[0110] Since metal-rubber damping elements exhibit both structural damping and viscous damping, the concept of an equivalent viscous damping coefficient is introduced to replace the complex damping mechanism. This equivalent viscous damping coefficient is determined through analysis of the hysteresis loop of the metal-rubber clamp. The energy dissipated by the clamp can be expressed as:

[0111] ΔW=πP max a n =πx max T n (10)

[0112] In the formula, a n and T n P represents the intercepts of the elliptic loop on the coordinate axes. max Let be the restoring force at the point of maximum deformation. The elastic potential energy of the system can be expressed as:

[0113]

[0114] Therefore, the damping loss factor can be expressed as:

[0115]

[0116] The equation of an ellipse can be expressed as:

[0117]

[0118] For a viscous damped system, let c c Let be the equivalent viscous damping coefficient, then the restoring force in the dry friction system is:

[0119]

[0120] When x = 0, the restoring force of the system is:

[0121] P(x) x＝0 =c c x max ω (15)

[0122] According to the equivalence principle, the restoring force in a dry friction hysteresis system is equal to the equivalent viscous damping force in an equivalent viscous system, therefore:

[0123]

[0124] therefore:

[0125]

[0126] In the formula, k is the measured stiffness of the clamp, η is the damping loss factor, and ω is the excitation frequency. Due to the nonlinearity of the metal rubber material, the damping changes with the increase of the excitation amplitude. Therefore, in order to correct the equivalent viscous damping coefficient c... c Introducing a modified nonlinear damping coefficient c n The corrected nonlinear damping coefficient is c n c c .

[0127] S13. Identify the nonlinear coefficients;

[0128] To determine the unknown nonlinear stiffness coefficient k in the dynamic equations of the nonlinear clamp-system. n and nonlinear damping coefficient c n Parameter identification is required based on the input (excitation) and output (response) results of the experiment.

[0129] An objective function is constructed using the frequency error and response amplitude error of the experiment, and then identified using a genetic algorithm. Figure 3 As shown. The error in the response amplitude is mainly obtained based on the improved half-power bandwidth method, which takes two frequency points ω on both sides of the resonant frequency ω0. m and ω n (ω m >ω n The response amplitudes corresponding to these two points are equal (A). mop =A nop), calculate the experimental and simulation values ​​at ω respectively. n The response amplitude error at ω0 is calculated, and the genetic algorithm and Newmark-β method are used for iterative solution to finally obtain the identified nonlinear coefficients.

[0130] In a specific implementation, as a preferred embodiment of the present invention, step S2 involves model verification and nonlinear parameter identification of the established nonlinear clamp-pipeline system model, specifically including modal verification and vibration response verification of the clamp-pipeline system by hammer impact test.

[0131] For the model established in this invention, modal verification was first performed. The parameters of the piping system are shown in Table 1. The testing instruments required for the modal experiment included a triaxial accelerometer (PCB 356A01), a force hammer (PCB 086C01), and a 12-channel LMS system (SC-XS12-A). The clamp consists of two straps and a metal-rubber assembly, with a tightening torque of 8 N·m. The stiffness of the clamp, k, was obtained through multiple measurements using a self-designed fixture. cy =1×10 7 N / m, k cz =8×10 6 N / m, k cθy =70 Nm / rad, k cθz = 30 Nm / rad.

[0132] To ensure the accuracy of the experimental results, multiple hammer tests were conducted to obtain the frequency response function and mode shapes of the pipeline system under constrained modes. The natural frequencies of the pipeline were identified by the peak values ​​of the frequency response function, as shown in Table 2. A comparison of the frequency response functions from simulation and experiment is shown in the figure below. Figure 4 As shown, the mode shapes are as follows: Figure 5 As shown, the experiment and simulation agree well, which preliminarily verifies the effectiveness of the clamp-pipeline model.

[0133] Table 1 Parameters of the Piping System

[0134]

[0135] Table 2 Comparison of natural frequencies of pipeline systems under constraints

[0136]

[0137] Based on the above dynamic model, an electromagnetic vibration table was used to simulate foundation excitation. The sweep frequency range was set to 360Hz-400Hz according to the hammer impact test. Sweep frequency tests were conducted with excitation amplitudes of 0.5g, 1g, 2g, and 3g to obtain the system's vibration response results. Identifying the peak values ​​of the responses allowed for accurate determination of the natural frequencies corresponding to different sweep frequency excitation amplitudes. Transitivity is an important indicator for evaluating clamp vibration isolation. Dynamic transitivity can be defined as:

[0138]

[0139] Among them, a out For the system output acceleration amplitude, a in The system is input with acceleration amplitude, and the dynamic transfer curves under different excitation amplitudes are as follows: Figure 6 As shown, with the increase of excitation amplitude, the natural frequency of the system decreases, and the transmissibility of the clamp gradually decreases, indicating that the clamp has soft nonlinearity and can more effectively suppress severe vibration when the excitation amplitude increases, with a significant vibration reduction effect.

[0140] The nonlinear coefficients of the clamp under different excitation amplitudes are identified by back-calculation using the Newmark-β method and genetic algorithm. The optimization iteration results are shown in the figure below. Figure 7 As shown in Table 3, after obtaining the nonlinear coefficients, they were substituted into the nonlinear clamp-pipeline system, and the final results of the simulation and experimental frequency response curves were obtained. Figure 8 As shown, the curve trends remain largely consistent, and the errors are within the acceptable range for engineering applications, verifying the correctness of the established model and the effectiveness of parameter identification. With increasing excitation, the first-order modal frequency of the system shifts to the left, which is caused by the soft nonlinear characteristics of the clamp, further validating the model's effectiveness.

[0141] Table 3 Test results under different excitation amplitudes

[0142]

[0143] In a specific implementation, as a preferred embodiment of the present invention, in step S3, the nonlinear clamp-pipeline system model under different boundary conditions is further verified using the model verified in step S2 and the identified nonlinear parameters, and its vibration response characteristics are studied.

[0144] To further verify the established nonlinear clamp-pipe model, dynamic characteristic tests of the pipe system under two boundary conditions were conducted: a pipe system with a fixed left end and a clamp support at the right end, and a pipe system with clamp supports at both ends. The pipes used in the tests were all 400 mm long and 8 mm in diameter. Hammer impact tests were performed to obtain their natural frequencies and frequency response functions. Corresponding finite element models were established for each, and the comparison results of the natural frequencies obtained from the simulation and experiments are shown in Table 4. The frequency response function comparisons are as follows: Figure 9 As shown, the basic trends of the curves exhibit good consistency, indicating that the established model is relatively accurate.

[0145] Table 4. Comparison of simulation and experimental results of natural frequencies under different boundary conditions.

[0146]

[0147] Vibration response tests were conducted on a nonlinear clamp-pipeline system under fixed-elastic boundary conditions. Based on the results of hammer impact tests, a sweep frequency range of 300-320Hz was set, and sweep frequency tests were performed at different excitation amplitudes (0.5g, 1g, 2g, 3g) to obtain the system's vibration response results. Identifying the response peak values ​​allowed for accurate determination of the natural frequencies corresponding to different sweep frequency excitation amplitudes. Substituting the identified nonlinear coefficients back into the pipeline model yielded a comparison of the simulated and experimental vibration responses. Figure 10 As shown in Table 5, the natural frequency and acceleration response amplitude are consistent. It can be seen that the curves show a generally consistent trend, and the errors in both the natural frequency and acceleration vibration response amplitude are small, controllable within 6%. This indicates that the identification results have universality, and the nonlinear clamp-pipeline system model has high accuracy.

[0148] Table 5. Simulation and experimental comparison results of nonlinear clamp-pipeline under fixed-elastic support.

[0149]

[0150] Vibration response tests were conducted on a nonlinear clamp-pipeline system under the condition of a spring-loaded boundary. Based on the results of the hammer impact test, a sweep frequency range of 260-280Hz was set, and sweep frequency tests were performed at different excitation amplitudes (0.5g, 1g, 2g, 3g) to obtain the vibration response results of the system. Identifying the peak response allows for accurate determination of the natural frequencies corresponding to different sweep frequency excitation amplitudes. Substituting the identified nonlinear coefficients back into the pipeline model yields a comparison of the simulated and experimental vibration responses. Figure 11As shown in Table 6, the natural frequency and acceleration response amplitude are also shown. The information from the tables indicates that the frequency response curves of the simulation and the experiment generally follow the same trend, and the errors in the natural frequency and acceleration vibration response amplitude are small, controllable within 2%. This demonstrates the universality of the identification results and the high accuracy of the established nonlinear clamp-pipeline system model.

[0151] Table 6. Comparison of Simulation and Experiment Results of Nonlinear Clamp-Pipeline under Spring Support-Spring Support

[0152]

[0153]

[0154] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for dynamic modeling of pipeline systems considering the nonlinearity of clamps, characterized in that, include: A dynamic model of a nonlinear clamp-pipeline system is established based on the finite element method. In the dynamic model of the nonlinear clamp-pipeline system, the nonlinear restoring force of the clamp is characterized by a cubic nonlinear stiffness model, and the nonlinear damping is characterized by a modified nonlinear damping coefficient obtained based on hysteresis loop analysis. The established nonlinear clamp-pipeline system model was validated and its nonlinear parameters were identified. The nonlinear parameter identification employs a genetic algorithm combined with Newmark- The method uses the frequency error and response amplitude error of the experiment and simulation as dual-index constrained objective functions to perform global optimization identification of the cubic nonlinear stiffness coefficient and the modified nonlinear damping coefficient. Using the validated model and identified nonlinear parameters, the nonlinear clamp-pipe system model under different boundary conditions is further validated, and vibration response characteristics are designed. The different boundary conditions specifically include: using a genetic algorithm and experimental frequency response curves to identify the nonlinear coefficients of the clamps; substituting the identified nonlinear coefficients back into the clamp-pipe system with fixed-elastic boundary and the clamp-pipe system with elastic-elastic boundary, respectively; identifying the natural frequency of the pipe by the frequency corresponding to the resonance peak in the frequency response under frequency sweep excitation; and finally, further validating the model by comparing the frequency and response amplitude of the simulation and experiment.

2. The pipeline system dynamics modeling method considering the flexible nonlinearity of clamps as described in claim 1, characterized in that, The establishment of a nonlinear clamp-pipeline system dynamic model based on the finite element method specifically includes: Establish a pipeline model; Introducing nonlinear clamps; Identify the nonlinear coefficients.

3. The pipeline system dynamics modeling method considering the flexible nonlinearity of clamps according to claim 2, characterized in that, The establishment of the pipeline model specifically includes: According to the finite element method, the entire pipeline is discretized using Timoshenko beam elements. Each beam element has two nodes, and assuming each node has four degrees of freedom, the element node displacement vector is defined as follows: The displacement vector of any node within the element can be expressed as: in, For a unit shape function, its expression is as follows: Based on Hamilton's principle and the principle of minimum potential energy, the partial differential equations of the pipeline are obtained. Substituting the element displacement vectors into the equations, the stiffness matrix and mass matrix of the element in the local coordinate system are obtained. The damping matrix adopts the Rayleigh damping form. In the formula, , and A p They represent the first k The density, length, and cross-sectional area of ​​each piping unit; and They represent about oy and oz The moment of inertia of the cross section; and Indicates any cross section along y and z Translational displacement of the axis; and Represents any cross section around y, z Angular displacement in direction, E and G These represent Young's modulus and shear modulus, respectively. and They represent about y and z The shear coefficient of the shaft is 0.5 for thin-walled cylindrical parts; the superscript of the variable with an apostrophe indicates that it is about the coordinate. x Take the first derivative, , These are the first two natural frequencies. This represents the damping ratio of the first two modes.

4. The pipeline system dynamics modeling method considering the flexible nonlinearity of clamps according to claim 2, characterized in that, The introduction of the nonlinear clamp specifically includes: Considering the influence of the clamp width, the metal felt single clamp is equivalently discretized along the axial direction into two linear springs and two nonlinear springs. The measured stiffness of the clamp introduced by the linear springs was obtained through multiple experimental measurements. , , , The stiffness of each linear spring is half of the measured stiffness in that direction; When an external excitation is present, a nonlinear restoring force is generated between the metal wires. Assuming this nonlinear restoring force... With displacement The relationship is: In the formula, For the unknown nonlinear stiffness coefficient of the clamp, For the displacement of the clamp in this direction, when At this time, the support stiffness degenerates from a nonlinear spring to a linear spring; in order to solve the problem of the difficulty in solving the nonlinear term in the stiffness matrix, the nonlinear stiffness term is shifted to the right into the force vector, thus obtaining the dynamic equation of the nonlinear clamp-pipe system as follows: In the formula, , and These are the mass matrix, damping matrix, and stiffness matrix of the pipeline, respectively. The damping matrix of the clamp; Here is the linear stiffness matrix of the clamp; The generalized displacement vector of the system; This is the vector of external forces; For nonlinear force vectors: To simplify the calculation, we assume nonlinear stiffness coefficients in four directions. k ni If they are equal, then it is necessary to determine the vibration response test results. Perform identification; The equivalent viscous damping coefficient of the metal-rubber clamp was determined by analyzing the hysteresis loop, and the energy dissipated by the clamp was expressed as: In the formula, and These are the intercepts of the elliptic loop on the coordinate axes, respectively. The restoring force at the point of maximum deformation is expressed as the elastic potential energy of the system as: Therefore, the damping loss factor can be expressed as: The equation of the ellipse can be expressed as: For a viscous damped system, let... Let be the equivalent viscous damping coefficient, then the restoring force in the dry friction system is: when At that time, the system's resilience is: According to the equivalence principle, the restoring force in a dry friction hysteresis system is equal to the equivalent viscous damping force in an equivalent viscous system, therefore: therefore: In the formula, For the actual measured stiffness of the clamp, The damping loss factor, Given the excitation frequency, and due to the nonlinearity of the metal rubber material, the damping changes with the increase of the excitation amplitude. Therefore, in order to correct the equivalent viscous damping coefficient... Introducing a modified nonlinear damping coefficient The corrected nonlinear damping coefficient is .

5. The pipeline system dynamics modeling method considering the flexible nonlinearity of clamps according to claim 2, characterized in that, The identification of nonlinear coefficients specifically includes: To determine the unknown nonlinear stiffness coefficients in the dynamic equations of the nonlinear clamp-system. k n and nonlinear damping coefficient Parameter identification is performed based on the input excitation and output response results of the experiment; The objective function is constructed using the frequency error and response amplitude error of the experiment, and identified by reverse engineering using a genetic algorithm; at the resonant frequency... Two frequency points are taken on both sides. and and The response amplitudes corresponding to these two points are equal, that is... Calculate the experimental and simulation results respectively. and The response amplitude error at the location was determined using a genetic algorithm and Newmark- The method iteratively solves the problem and finally obtains the identified nonlinear coefficients.

6. The pipeline system dynamics modeling method considering the flexible nonlinearity of clamps according to claim 1, characterized in that, The establishment of the nonlinear clamp-pipe system model is verified and nonlinear parameters are identified, specifically including modal verification and vibration response verification of the clamp-pipe system by hammer impact test.

7. The pipeline system dynamics modeling method considering the flexible nonlinearity of clamps according to claim 6, characterized in that: The testing instruments required for modal verification of the clamp-pipe system hammer test include a triaxial accelerometer, a force hammer, and a 12-channel LMS system. To ensure the accuracy of the test results, the natural frequencies of the clamp-pipe system are obtained through multiple hammer tests. The finite element method is used for dynamic modeling, and the natural frequencies and mode shapes of the pipe are obtained by solving for eigenvalues ​​and eigenvectors. The simulation and experimental frequency and mode shape results are compared to verify the model.

8. The method for dynamic modeling of pipeline systems considering the flexible nonlinearity of clamps according to claim 6, characterized in that: The vibration response verification of the clamp-pipeline system under hammer impact test specifically includes: using an electromagnetic vibration table to subject the clamp-pipeline system to simple harmonic excitation at different excitation amplitudes of 0.5g, 1g, 2g, and 3g; setting the sweep bandwidth based on the hammer impact test results; the duration of each sweep test being 2.5 min; the acceleration signal at the target location being processed by the LMS SCADAS mobile front end and transmitted to the LMS mobile workstation; using a genetic algorithm and the test frequency response curve to identify the nonlinear coefficient of the clamp; after bringing it back to the model, identifying the natural frequency of the pipeline by the frequency corresponding to the resonance peak in the frequency response under sweep excitation; and verifying the model by comparing the frequency and response amplitude of the simulation and the test under sweep excitation.

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