Rubber vibration reduction local installation method based on multi-order vibration of fluid conveying pipeline

Abstract

The invention provides a rubber vibration reduction local installation method based on flow conveying pipeline multi-order vibration. The rubber vibration reduction local installation method comprises the steps of simply-supported flow conveying pipeline structure building, dynamic modeling, nonlinear vibration control equation obtaining, vibration reduction effect analysis, rubber local installation position obtaining through modal weighting and calculation and verification. According to the rubber vibration reduction local installation method based on multi-order vibration of the fluid conveying pipeline, a rubber damping vibration reduction mode can still be adopted, it is guaranteed that the rubber damping vibration reduction mode gives consideration to multi-order vibration through the obtained accurate installation position of the rubber layer, the cost is reduced, meanwhile, the vibration reduction effect is guaranteed, and practicability is high.

Description

Technical Field

[0001] This invention belongs to the field of pipeline vibration reduction technology, specifically relating to a local installation method for rubber vibration damping based on multi-order vibration of a transmission pipeline. Background Technology

[0002] Pipelines, as key components for transporting fluids, are widely used in various industries and energy sectors. However, during operation, pipelines are susceptible to the influence of internal fluids and the external environment, leading to harmful vibrations. These vibrations can reduce the stability and safety of the pipeline system, causing structural fatigue and even serious problems such as pipeline rupture.

[0003] In existing technologies, vibration reduction methods typically employ spring dampers, which suffer from drawbacks such as high cost, complex installation, and difficult maintenance. Furthermore, their ability to reduce multi-order (third-order and above) vibrations in pipelines is limited. Therefore, a combined design is necessary, such as adding dampers to spring dampers. In recent years, polymer materials such as rubber have attracted considerable attention in the field of vibration reduction due to their excellent damping properties. Therefore, providing a more economical, flexible, and precisely installed vibration reduction solution for multi-order vibrations, while adhering to the premise of locally installed rubber dampers without significantly increasing or decreasing pipeline mass, is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0004] This invention provides a method for local installation of rubber vibration damping based on multi-order vibration of a transmission pipeline. It aims to solve the problems of poor practicality caused by the poor performance of existing vibration damping methods for multi-order vibration and the inability of rubber dampers to be accurately installed for multi-order vibration.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is: to provide a local installation method for rubber vibration damping based on multi-order vibration of a conveying pipeline, comprising:

[0006] Construct two simply supported conveying pipes with identical basic structures, with the distance between the two support points of each conveying pipe being L; pre-install a rubber layer on one of the conveying pipes, with the distance between the rubber layer and one of the support points being L1;

[0007] The equivalent mass m of the rubber layer is set. r and concentrated damping c r Based on the physical parameters of the conveying pipe, the parameters of the rubber layer, the set fluid flow rate, L, and L1, nonlinear vibration control equations are established for the conveying pipe without the rubber layer and for the conveying pipe with the rubber layer installed; at the same time, the simply supported boundary conditions at both ends are obtained.

[0008] The amplitude-frequency response curves of the first n stages of the flow transmission pipe with and without the rubber layer installed are compared, and the vibration reduction efficiency of each stage is obtained to determine the feasibility.

[0009] Construct the superimposed deflection curves of the first n modes of the conveying pipeline, and determine the installation position of the rubber layer that can simultaneously take into account the first n modes of vibration by using the maximum point of the superimposed deflection curves.

[0010] The vibration reduction efficiency of the rubber layer installation position, which can simultaneously take into account the first n orders of vibration, for each order of vibration in the conveying pipeline is calculated and verified.

[0011] In one possible implementation, the nonlinear vibration control equations for the transport pipeline are established using Euler-Bernoulli beam theory and the generalized Hamilton principle.

[0012] In one possible implementation, obtaining the amplitude-frequency response result curves of the first n orders of the flow transmission pipe with and without the rubber layer installed includes:

[0013] Obtain the frequency equation of the flow transmission pipe with and without the rubber layer installed, and combine the physical parameters of the flow transmission pipe and the m of the rubber layer. r and c r Substituting these values ​​into the frequency equation, the natural frequencies of the first n vibrations of the conveying pipes with and without the rubber layer are obtained respectively.

[0014] Establish the convergence curves of the first n natural frequencies of the flow transmission pipe with or without the rubber layer installed;

[0015] Based on the convergence of the first n natural frequencies of the conveying pipe without or with the rubber layer installed, solve for the response results of the first n vibrations of the conveying pipe without or with the rubber layer installed.

[0016] In one possible implementation, the harmonic balance method is used to perform frequency sweep calculations on the response results of the first n vibrations of the conveying pipe without or with the rubber layer installed, within a frequency band covering any natural frequency of any order.

[0017] In one possible implementation, the Runge-Kutta method with adaptive variable step size is used to numerically verify the response solved by the harmonic balance method.

[0018] In one possible implementation, the superimposed deflection curves of the first n modes of the conveying pipe are constructed by adding the absolute values ​​of the modal functions of the first n vibrations of the simply supported conveying pipe.

[0019] In one possible implementation, the rubber layer is a rubber layer.

[0020] In this implementation, two simply supported conveying pipes are constructed based on the method of installing a rubber layer on the conveying pipe to simulate the installation environment of the actual conveying pipe, thereby improving accuracy. A rubber layer is pre-installed on one of the conveying pipes to form two comparable conveying pipes. Using the parameters of the rubber layer, the physical parameters of the conveying pipe, the set fluid flow rate, L, and L1, nonlinear vibration control equations are established for the conveying pipe without the rubber layer and the conveying pipe with the rubber layer installed, respectively. Simultaneously, the simply supported boundary conditions at both ends are obtained to provide a theoretical basis for subsequent calculations. Based on the obtained nonlinear vibration equations and the simply supported boundary conditions at both ends, comparison curves of the amplitude-frequency response results for each of the first n orders are constructed for the conveying pipes without and with the rubber layer installed. The vibration reduction efficiency corresponding to each natural frequency is also obtained. The feasibility of this method of calculating vibration reduction efficiency is analyzed, and a calculation basis for verifying the final rubber layer installation position is provided. By constructing the superimposed deflection curves of the first n modes of the conveying pipeline, the installation position of the rubber layer that can simultaneously accommodate the first n modes of vibration can be determined through the maximum points of the superimposed deflection curves. The local installation method for rubber damping based on multi-mode vibration of the conveying pipeline provided in this implementation still employs rubber damping for vibration reduction. Through the precise installation position of the obtained rubber layer, it ensures that the rubber damping vibration reduction method simultaneously accommodates multiple modes of vibration, reducing costs while guaranteeing vibration reduction effect, making it highly practical. Attached Figure Description

[0021] Figure 1 The diagram shows the structure of a simply supported conveying pipeline constructed according to the local installation method for rubber vibration reduction based on multi-order vibration of the conveying pipeline provided in the embodiments of the present invention (a is the conveying pipeline with a rubber layer installed, b is the conveying pipeline without a rubber layer installed);

[0022] Figure 2 A simplified diagram of the concentrated mass and damping of the rubber layer in the local installation method for rubber vibration reduction based on multi-order vibration of the conveying pipeline provided in the embodiment of the present invention;

[0023] Figure 3 The convergence curves of the first four natural frequencies of the conveying pipe with rubber layer installed are obtained in the local installation method for rubber vibration reduction based on multi-order vibration of the conveying pipe provided in the embodiment of the present invention (a is the first and second natural frequencies; b is the third and fourth natural frequencies).

[0024] Figure 4The convergence curves of the first four natural frequencies of the transmission pipeline without rubber layer (a represents the first and second natural frequencies; b represents the third and fourth natural frequencies) are obtained in the local installation method for rubber vibration reduction based on multi-order vibration of the transmission pipeline provided in the embodiments of the present invention.

[0025] Figure 5 The above is a comparison chart of the amplitude-frequency response curves of the first four orders of the transmission pipeline with and without the rubber layer, obtained in the local installation method for rubber vibration reduction based on multi-order vibration of the transmission pipeline provided in the embodiments of the present invention (a, b, c, and d represent the first, second, third, and fourth orders, respectively).

[0026] Figure 6 The superimposed deflection curves of the first four modes of the conveying pipeline constructed in the local installation method for rubber vibration reduction based on multi-order vibration of the conveying pipeline provided in the embodiments of the present invention;

[0027] Figure 7 This is a bar chart showing the vibration order versus vibration reduction efficiency obtained from the specific installation location of the rubber layer in the local installation method for rubber vibration reduction based on multi-order vibration of a conveying pipeline provided in an embodiment of the present invention.

[0028] Figure 8 A flowchart of a local installation method for rubber vibration damping based on multi-order vibration of a conveying pipeline, provided in an embodiment of the present invention;

[0029] Figure 9 This is a flowchart illustrating the comparison of amplitude-frequency response curves for each of the first n orders of the transmission pipeline with and without the rubber layer, in the local installation method for rubber vibration damping based on multi-order vibration of the transmission pipeline provided in this embodiment of the invention. Detailed Implementation

[0030] To make the technical problems, solutions, and beneficial effects of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0031] Please refer to the following: Figures 1 to 9 The present invention will now describe the local installation method for rubber vibration damping based on multi-order vibration of a conveying pipeline. The local installation method for rubber vibration damping based on multi-order vibration of a conveying pipeline includes:

[0032] S101 Simply Supported Transmission Pipeline Structure Construction: Construct two simply supported transmission pipelines with identical basic structures. The distance between the two support points of each transmission pipeline is L. A rubber layer is pre-installed on one of the transmission pipelines, with the distance between the rubber layer and one of the support points being L1.

[0033] S102 Dynamic Modeling and Obtaining Nonlinear Vibration Control Equations: Equivalently Setting the Concentrated Mass m of the Rubber Layer r and concentrated damping c r Based on the physical parameters of the conveying pipe, the parameters of the rubber layer, the set fluid flow rate, L, and L1, nonlinear vibration control equations are established for both the conveying pipe without a rubber layer and the conveying pipe with a rubber layer. Simultaneously, the simply supported boundary conditions at both ends are obtained.

[0034] S103 vibration reduction effect analysis: Obtain the amplitude-frequency response curves of each of the first n stages of the transmission pipeline with and without rubber layer, and obtain the vibration reduction efficiency of each stage to determine the feasibility.

[0035] S104 Modal weighting to obtain the local installation position of rubber: Construct the superimposed deflection curve of the first n modes of the conveying pipeline, and determine the installation position of the rubber layer that can simultaneously take into account the first n vibrations by using the maximum point of the superimposed deflection curve.

[0036] S105 Calculation and Verification: Calculate and verify the vibration reduction efficiency of the rubber layer installation position in the conveying pipeline for each order of vibration, which can simultaneously take into account the first n orders of vibration.

[0037] In this embodiment, regarding the vibration reduction effect analysis step S103, firstly, based on the pre-installation position of the rubber layer, the amplitude-frequency response curves of each of the first n orders of the transmission pipe with and without the rubber layer are obtained. The vibration reduction efficiency of each order is obtained by comparing the response results of the transmission pipe with and without the rubber layer corresponding to each resonance frequency, and the feasibility of this installation position is determined. In the calculation and verification step S105, after the rubber layer is installed at the calculated installation position, the results obtained from the simple-supported transmission pipe structure construction step S101, the dynamic modeling and nonlinear vibration control equation acquisition step S102, and the vibration reduction effect analysis step S103 are used for verification.

[0038] This embodiment provides a local installation method for rubber vibration damping based on multi-order vibration of a conveying pipeline. Compared with existing technologies, this method constructs two simply supported conveying pipelines by installing a rubber layer on the pipeline to simulate the actual installation environment of a conveying pipeline, thus improving accuracy. A rubber layer is pre-installed on one of the conveying pipelines to form two comparable pipelines. Using the parameters of the rubber layer, the physical parameters of the conveying pipeline, the set fluid flow rate, L, and L1, nonlinear vibration control equations are established for both the pipeline without and with the rubber layer. Simultaneously, the simply supported boundary conditions at both ends are obtained to provide a theoretical basis for subsequent calculations. Based on the obtained nonlinear vibration equations and simply supported boundary conditions, comparison curves of the amplitude-frequency response results for each of the first n orders of the conveying pipelines with and without the rubber layer are constructed. The vibration damping efficiency corresponding to each resonance frequency is also obtained. The feasibility of this installation position is analyzed, and a calculation basis for verifying the final rubber layer installation position is provided. By constructing the superimposed deflection curves of the first n modes of the conveying pipeline, the installation position of the rubber layer that can simultaneously accommodate the first n modes of vibration can be determined through the maximum points of the superimposed deflection curves. The local installation method of rubber damping based on multi-mode vibration of the conveying pipeline provided in this embodiment can still adopt the vibration reduction method of rubber damping. By obtaining the precise installation position of the rubber layer, it can ensure that the rubber damping vibration reduction method can simultaneously accommodate multi-mode vibration, reducing costs while ensuring vibration reduction effect, and has strong practicality.

[0039] Of course, in this embodiment, the calculation of the installation position of the rubber layer in different transmission pipelines (with different parameters) may be involved. For the first type of transmission pipeline, the feasibility can be determined by sequentially performing the following steps: S101 simply supported transmission pipeline structure construction, S102 dynamic modeling and obtaining the nonlinear vibration control equation, and S103 vibration reduction effect analysis. For other transmission pipelines, the calculation of the rubber layer installation position can directly proceed to the S105 modal weighting step to obtain the local rubber installation position, followed by sequential analysis and verification through the S101 simply supported transmission pipeline structure construction, S102 dynamic modeling and obtaining the nonlinear vibration control equation, and S103 vibration reduction effect analysis steps.

[0040] In some embodiments, see Figure 1 and Figure 2 The nonlinear vibration control equations for the conveying pipeline were established using Euler-Bernoulli beam theory and the generalized Hamilton principle.

[0041] Specifically, one possible implementation is that the rubber layer, under the influence of mass and viscosity, can generate relative vibration with the conveying pipe, effectively dissipating vibration energy. Using a concentrated mass m... r and concentrated damping cr Representing this effect equivalently is a widely used simplification method in vibration system modeling. Therefore, the concentrated effect of the material is modeled as a mass-damped system acting at a single point. The absolute displacement of this system is represented by v1(t). See also... Figure 2 Ignoring shear strain and directly setting the position of the pre-installed rubber layer as a = L / 2, the calculation method for the nonlinear vibration control equation of the conveying pipeline can be obtained by applying Euler-Bernoulli beam theory and the generalized Hamilton principle to obtain the control equation for the vibration of the conveying pipeline:

[0042]

[0043] Among them, I p It is the moment of inertia of the cross-section of the conveying pipe about the neutral layer. E p and ρ p Let P0 represent the Young's modulus and density of the conveying pipe, respectively. P0 is the initial axial force acting on the conveying pipe, and μ is the viscous damping coefficient of the conveying pipe. a is the installation position of the rubber layer on the conveying pipe. R1(t) is the interaction force between the conveying pipe and the rubber layer, and δ is the Dirac function.

[0044] The equation of motion for the rubber layer is:

[0045]

[0046] The interaction forces can be expressed as:

[0047]

[0048] The displacement and velocity of the attachment point can be expressed as:

[0049] v a1 (t)=v(a,t),

[0050]

[0051] Substituting the interaction forces and the displacement and velocity of the attachment points into the governing equations of the pipeline vibration and the motion equations of the rubber layer, we can obtain:

[0052]

[0053] Consider a uniformly distributed harmonic external excitation F sin (Ω t The force acts vertically along the pipe, where F and Ω represent the force amplitude and frequency, respectively. Therefore, the final nonlinear vibration control equation for the conveying pipe is determined as follows:

[0054]

[0055] Simply supported boundary conditions at both ends (derived from vibration dynamics):

[0056] v(0,t)=0, v(L,t=0,

[0057] v,xx(0,t)=0,v,xx(L,t=0

[0058] It should be noted that x and t represent the horizontal axis and time. v(x,t) represents the lateral displacement of the pipe, and L is the pipe length (the comma before x or t indicates the partial derivative with respect to x or t). E p and ρ p Let P0 represent the Young's modulus and density of the conveying pipe, respectively. P0 is the initial axial force acting on the conveying pipe, and μ is the viscous damping coefficient of the conveying pipe. The fluid inside the conveying pipe flows at velocity Γ, and the fluid density is ρ. f The density of the rubber layer is ρ r A p Let A be the cross-sectional area of ​​the pipeline. f A is the cross-sectional area of ​​the fluid. r Let I be the cross-sectional area of ​​the rubber layer. p This is the moment of inertia of the pipeline cross-section about the neutral layer. The mass-damped system is laid at position a (here a = L / 2), with a lateral displacement of v1(t), and the concentrated mass and concentrated damping of the rubber layer are m and m, respectively. r and c r The parameters are determined based on the constitutive model of the rubber material, including its diameter d. r density ρ r The parameters of the rubber layer, such as width b, can be obtained directly through measurement and will not be elaborated here.

[0059] It should also be noted that for calculations of pipelines without a rubber layer, the parameters related to the rubber layer are directly set to 0.

[0060] In some embodiments, see Figure 1 , Figure 3 , Figure 4 as well as Figure 5 Comparison charts of amplitude-frequency response results for each of the first n orders of the transmission pipeline with and without rubber layer were obtained, including:

[0061] Obtaining the natural frequency of Y101: Obtain the frequency equation of the transmission pipe with and without rubber layer installed, and combine the physical parameters of the transmission pipe and the m of the rubber layer. r and c r Substituting these values ​​into the frequency equation, we obtain the natural frequencies of the first n vibrations of the conveying pipes with and without rubber layers.

[0062] Y102 Natural Frequency Truncation Convergence Acquisition: Establish the first n natural frequency convergence curves of the transmission pipe with / without rubber layer installed.

[0063] Y103 Response Results: Based on the convergence of the first n natural frequencies of the conveying pipe without / with rubber layer installed, the response results of the first n vibrations of the conveying pipe without / with rubber layer installed are obtained.

[0064] In this embodiment, the frequency equation of the conveying pipe is first obtained. By inputting the physical parameters of the conveying pipe and the concentrated mass and concentrated damping of the rubber layer into this frequency equation, the only unknown remains the natural frequency. Therefore, the nth natural frequency of the conveying pipe with and without the rubber layer can be obtained. The nth natural frequencies are sorted from smallest to largest to obtain the natural frequencies of the first n vibrations. Then, convergence curves of the first n natural frequencies of the conveying pipe with and without the rubber layer are established (the horizontal axis represents fluid velocity, and the vertical axis represents natural frequency). These convergence curves allow the acquisition of the natural frequencies of each order at the same flow velocity. Finally, frequency sweep calculations are performed within the resonant frequency range of each order to obtain the response result of that vibration. (See also...) Figure 5 The obtained amplitude-frequency response curves of the transmission pipes with and without rubber layers are compared. The difference rate between the curves of the transmission pipes with and without rubber layers at the highest point can be defined as the vibration reduction efficiency.

[0065] As a specific calculation method for obtaining the inherent frequency in this embodiment:

[0066] Ignoring the effects of external force terms and nonlinear terms, we can obtain a linear derived system:

[0067]

[0068] Based on the simply supported boundary conditions, assume the solution of the linear derived system is:

[0069]

[0070] Where, q i (t) is a generalized coordinate, φ i (x) is a modal function. Therefore:

[0071]

[0072] Substitute the hypothetical solution into the equation of the linearly derived system, and multiply the equation of the linearly derived system by ζ. j (x) = sin(jπx / L), and integrating over [0, L] yields:

[0073]

[0074] After the M-order truncation, due to the orthogonality of the modal functions, the original partial differential equation will be projected into the modal space, becoming an equation relating to the modal coordinates q. i A set of ordinary differential equations for v1(t) and v2(t):

[0075]

[0076] Where M is the mass matrix, G is the gyroscope matrix, C is the pipe damping matrix, and C0 is the pipe damping matrix. ra and C rb Let K be the damping matrix of the rubber layer, and K be the stiffness matrix. The specific expressions for the matrix elements are:

[0077]

[0078] cra′ is the damping element at the pipe attachment point, and its expression is:

[0079]

[0080] In addition, q i v1(t) and v2(t) are respectively:

[0081]

[0082] Among them, Q ai and Q b1 The coefficients are constants. Substituting the above equation into q... i A set of ordinary differential equations and linear derived systems of v1(t) and v2(t) yields a series of algebraic equations:

[0083]

[0084] The condition for the existence of a non-zero solution in the above equation is that the determinant of the coefficient matrix of the system of equations is zero. Therefore, by extracting the Jacobian matrix of the coefficients in the above equation and setting its determinant to zero, we can obtain the following frequency equation:

[0085]

[0086] In some embodiments, see Figure 5 The harmonic balance method is used to calculate the response of the first n orders of vibration of the transmission pipeline with or without rubber layer by sweeping the frequency within a frequency band covering any natural frequency.

[0087] The harmonic balance method (HBM) is an approximate analytical method for analyzing nonlinear vibration systems. This method assumes that the response of a nonlinear system undergoing periodic motion can be represented by a Fourier series expansion. The core idea of ​​HBM is to substitute the assumed periodic solution into the differential equations describing the nonlinear vibration system. By utilizing the orthogonality of trigonometric functions, an integral is performed over one period. This rearranges the differential equations into a system of algebraic equations about the Fourier coefficients. Solving this system yields an approximate solution to the system response. Furthermore, this method features low memory consumption, high computational speed, and the ability to accurately determine the system response at arbitrary points. In practical applications, it not only reduces costs but also maintains high accuracy.

[0088] As one specific implementation method of calculation in this embodiment:

[0089] Based on the boundary conditions, the solution to the previously derived nonlinear vibration control equation for the conveying pipeline is assumed to be in the following form:

[0090]

[0091] Where M is the order of the Galerkin cutoff (here, the order can be set to 4). Substituting the solution into the previously derived nonlinear vibration control equation for the flow-carrying pipeline, we get:

[0092]

[0093] Multiply both sides of the above equation by... Then integrate over the interval from 0 to L to obtain the modal space with respect to q. i M truncated ordinary differential equations (ODEs) of (t). Simplifying, we get:

[0094]

[0095] in,

[0096]

[0097] Based on the selectivity of the Dirac function:

[0098]

[0099] Therefore, ordinary differential equations (ODEs) can be simplified using the above properties, and the final equation of the system is:

[0100]

[0101] Based on HBM, assuming the solution q i (t) and v i(t) is formed by the superposition of harmonics, and its specific form is as follows:

[0102]

[0103] Where m is the harmonic order. The response of the transmission pipeline at any time and any location can be written as the following expression:

[0104]

[0105] After HBM processing, the algebraic equation for the nonlinear coupled harmonic coefficients is obtained, namely:

[0106] R(z) = 0

[0107] Where z = (a0, a, b), a0 = (a 10 , ..., a i0 r 10 ) T , a = (a 11 , ..., a im r m ) T b = (b 11 , ..., b im b m ) T After deriving the convergent discrete system, the forced vibrations of the system are studied.

[0108] In some embodiments, the Runge-Kutta method with adaptive variable step size is used to numerically verify the response solved by the harmonic balance method.

[0109] To evaluate the accuracy of the analytical results obtained through HBM, the Runge-Kutta method (RK) was used to numerically verify the system response. When using RK to solve the numerical solution, the previously obtained higher-order ODEs are first discretized into a first-order system. Then, the displacement and velocity of the system are calculated at each time step to progressively approximate the true solution. In the standard RK method, the step size is fixed, meaning that the same time interval is used for each calculation step. Although the fixed-step-size method is simple and easy to use, it may lead to insufficient accuracy or low computational efficiency when dealing with some nonlinear problems. Therefore, the Adaptive Step-size Runge-Kutta method is adopted. It can dynamically adjust the step size based on the current local error, thereby improving the accuracy and efficiency of the numerical solution. This can be achieved using the ODE45 solver in MATLAB, which itself implements the adaptive step-size RK method and can therefore be used to handle the control equations of this model.

[0110] As one specific implementation method of calculation in this embodiment:

[0111] First, reduce the order of the final system equations obtained after the Galerkin truncation by letting...

[0112]

[0113] Substituting this into the truncated system of ordinary differential equations, we get:

[0114]

[0115] Inverse solution to each and The expression is denoted as:

[0116]

[0117] The results are further written in the following form:

[0118]

[0119] Using the ODE45 solver, given initial conditions, the steady-state response of the transmission pipeline under different external excitation conditions can be obtained after multiple iterations.

[0120] In this embodiment, please refer to Figure 5 The amplitude-frequency response of the first four orders of forced vibration in two types of conveying pipes was studied and numerically verified using the RK method. Under the same external excitation F = 1 N / m, the rubber layer effectively reduced the amplitude of the first three orders of the conveying pipe's response. However, for the first-order vibration, its damping efficiency was as high as 98.26%, while it had almost no damping effect on the fourth-order vibration. Furthermore, the rubber layer pre-installed at the midpoint of the conveying pipe had a greater effect on odd-order vibrations than on even-order vibrations. This is consistent with the conclusions obtained using the harmonic balance method.

[0121] In some embodiments, see Figure 6 The superimposed deflection curves of the first n modes of the conveying pipeline are constructed by adding the absolute values ​​of the modal functions of the first n vibrations of the simply supported conveying pipeline.

[0122] The optimal placement of the rubber layer, which simultaneously suppresses the first n modes of vibration, can effectively ensure that the rubber damping vibration reduction method can simultaneously accommodate multiple vibrations, reducing costs while guaranteeing the vibration reduction effect.

[0123] As one specific implementation method of calculation in this embodiment:

[0124] A simply supported transmission pipe can essentially be viewed as a simply supported beam, and its vibration mode function φ i for:

[0125]

[0126] Where i = 1, 2, 3, 4 corresponds to the first four modes. The absolute values ​​of the first four modes are added together to construct the superimposed deflection curve D(x):

[0127]

[0128] Based on the extreme value condition of the deflection curve, the maximum point is determined by solving D'(x) = 0. See also... Figure 6 Since D(x) is symmetrical about the midpoint, the first two maximum points a = 0.165L and 0.4L are selected as key analysis locations.

[0129] In some embodiments, see Figure 7 The vibration reduction efficiency of the rubber layer installation position in the conveying pipeline, which can simultaneously take into account the first n orders of vibration, is calculated to verify the obtained rubber layer installation position.

[0130] Specifically, the vibration reduction efficiencies of a = 0.165L and 0.4L obtained earlier can be calculated directly using the following steps: S101 Simple Supported Pipeline Structure Construction, S102 Dynamic Modeling and Obtaining Nonlinear Vibration Control Equations, and S103 Vibration Reduction Effect Analysis. See also... Figure 7 By optimizing the placement of the rubber layer using modal weighting, the vibration suppression effect of even-order modes can be improved while maintaining the vibration reduction efficiency of odd-order modes. Furthermore, both installation positions exhibit superior vibration control effects for the first three modes of the pipeline. For the fourth mode of the conveying pipeline, the installation position at 0.165L is closer to the minimum value of the modal amplitude. Therefore, this position significantly reduces the fourth-order vibration reduction efficiency. In contrast, installing the rubber layer at 0.4L is closer to the maximum value of the modal amplitude, thus achieving a vibration reduction efficiency higher than 17.07%. Furthermore, considering the order of magnitude of the amplitudes of the first four modes, the vibration reduction efficiency of the rubber layer decreases with increasing resonance order. In particular, the vibration reduction efficiency for the first-order vibration can reach over 90%.

[0131] In some embodiments, see Figure 1 The rubber layer is a rubber layer that is sleeved on the transmission pipeline.

[0132] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for local installation of rubber vibration damping based on multi-order vibration of a transmission pipeline, characterized in that, include: Construct two simply supported conveying pipes with identical basic structures, with the distance between the two support points of each conveying pipe being L; pre-install a rubber layer on one of the conveying pipes, with the distance between the rubber layer and one of the support points being L1; The equivalent mass m of the rubber layer is set. r and concentrated damping c r Based on the physical parameters of the conveying pipe, the parameters of the rubber layer, the set fluid flow rate, L, and L1, nonlinear vibration control equations are established for the conveying pipe without the rubber layer and for the conveying pipe with the rubber layer installed; at the same time, the simply supported boundary conditions at both ends are obtained. The amplitude-frequency response curves of the first n stages of the flow transmission pipe with and without the rubber layer installed are compared, and the vibration reduction efficiency of each stage is obtained to determine the feasibility. Construct the superimposed deflection curves of the first n modes of the conveying pipeline, and determine the installation position of the rubber layer that can simultaneously take into account the first n modes of vibration by using the maximum point of the superimposed deflection curves. The vibration reduction efficiency of the rubber layer installation position, which can simultaneously take into account the first n orders of vibration, for each order of vibration in the conveying pipeline is calculated and verified.

2. The method for local installation of rubber vibration damping based on multi-order vibration of a conveying pipeline as described in claim 1, characterized in that, The nonlinear vibration control equations for the transport pipeline are established using Euler-Bernoulli beam theory and the generalized Hamilton principle.

3. The method for local installation of rubber vibration damping based on multi-order vibration of a conveying pipeline as described in any one of claims 1-2, characterized in that, The step of obtaining comparative graphs of amplitude-frequency response results for each of the first n orders in the flow transmission pipes with and without the rubber layer includes: Obtain the frequency equation of the flow transmission pipe with and without the rubber layer installed, and combine the physical parameters of the flow transmission pipe and the m of the rubber layer. r and c r Substituting these values ​​into the frequency equation, the natural frequencies of the first n vibrations of the conveying pipes with and without the rubber layer are obtained respectively. Establish the convergence curves of the first n natural frequencies of the flow transmission pipe with or without the rubber layer installed; Based on the convergence of the first n natural frequencies of the conveying pipe without or with the rubber layer installed, solve for the response results of the first n vibrations of the conveying pipe without or with the rubber layer installed.

4. The method for local installation of rubber vibration damping based on multi-order vibration of a conveying pipeline as described in claim 3, characterized in that, The harmonic balance method is used to perform frequency sweep calculations on the response results of the first n vibrations of the transmission pipe without or with the rubber layer installed, within a frequency band covering any natural frequency of any order.

5. The method for local installation of rubber vibration damping based on multi-order vibration of a conveying pipeline as described in claim 4, characterized in that, The response obtained by the harmonic balance method was numerically verified using the adaptive variable step size Runge-Kutta method.

6. The method for local installation of rubber vibration damping based on multi-order vibration of a conveying pipeline as described in claim 3, characterized in that, The superimposed deflection curves of the first n modes of the conveying pipeline are constructed by adding the absolute values ​​of the modal functions of the first n vibrations of the simply supported conveying pipeline.

7. The method for local installation of rubber vibration damping based on multi-order vibration of a conveying pipeline as described in claim 1, characterized in that, The rubber layer is a rubber layer.

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