Full-band vibration reduction design method for metamaterial pipe based on multiple band gaps

Abstract

This invention relates to a full-frequency vibration reduction design method for metamaterial pipelines based on multiple band gaps, comprising: establishing an equivalent mechanical model of the target pipeline; periodically arranging elastic supports on the target pipeline to induce the formation of a zero-frequency band gap and a Bragg band gap, coupling the equivalent support stiffness of the elastic supports with the equivalent mechanical model of the pipeline to construct a first band gap calculation model; periodically arranging local resonant vibration absorbers on the target pipeline to form a local resonant band gap, coupling the equivalent stiffness and equivalent mass of the local resonant vibration absorbers into the first band gap calculation model to obtain a second band gap calculation model; constructing a system finite structure dynamic vibration calculation model with a second assembly as the analysis object; constructing a vibration transmissibility index, and using the system finite structure dynamic vibration calculation model as a fitness evaluation tool for a genetic algorithm to perform non-periodic optimization of the elastic support positions, solving for the optimal arrangement position of the elastic supports with the largest bandwidth of the zero-frequency band gap.

Description

Technical Field

[0001] This invention relates to the field of road system dynamics modeling and vibration reduction technology, and in particular to a full-frequency vibration reduction design method for metamaterial pipelines based on multiple band gaps. Background Technology

[0002] In critical sectors such as aerospace, shipbuilding, and energy, pipelines play a core role in fluid transport and are essential components ensuring the normal operation of equipment. However, these pipelines face extremely complex operating conditions in actual operation, such as fluid disturbances, engine excitation, and pump vibration. Under the combined effect of these complex excitations, pipelines are highly susceptible to broadband vibration responses.

[0003] Prolonged exposure to such broadband vibration can severely damage pipeline structures. Vibration can lead to fatigue cracking, compromising pipeline integrity; connections may loosen due to vibration, affecting fluid transport stability; in extreme cases, it can even cause entire system failure, resulting in serious safety accidents. For example, in the aerospace field, pipeline system failure can jeopardize flight safety; in the energy sector, it can lead to energy supply interruptions, causing huge economic losses. Therefore, effective control of pipeline vibration is crucial for ensuring the operational safety and reliability of equipment.

[0004] Currently, passive vibration reduction methods are mainly used for pipeline vibration control. These methods are widely used in practical engineering due to their simple structure and good stability. Common passive vibration reduction methods include adding damping layers and installing tuned mass vibration absorbers. Adding damping layers dissipates vibration energy by increasing the damping of the structure, thereby reducing the vibration amplitude; tuned mass vibration absorbers utilize the principle of resonance to transfer vibration energy to the absorber, thus reducing the vibration of the pipeline itself. However, traditional passive vibration reduction devices have certain limitations. With small added masses, they can usually only play an effective vibration reduction role in a narrow frequency range, and their ability to suppress low-frequency vibrations is particularly limited. When pipelines face wide-frequency excitation or changes in operating conditions, traditional passive vibration reduction devices often cannot meet the actual needs of the project and cannot effectively ensure the stable operation of the pipeline.

[0005] In recent years, mechanical metamaterials have attracted widespread attention in the field of vibration and wave control due to their unique properties. Mechanical metamaterials can generate bandgap effects through artificially designed periodic structures, meaning that elastic waves within certain frequency ranges cannot propagate in the material, thus achieving effective vibration control. Based on the Bragg scattering mechanism, the bandgap frequency is inversely related to the period length of the structure. This implies that in low-frequency applications, a large structural period scale is often required to obtain the desired bandgap frequency. However, in practical engineering, large structural scales are not conducive to engineering integration and are limited by factors such as space and weight.

[0006] To overcome this limitation of mechanical metamaterials based on Bragg scattering in low-frequency applications, researchers have proposed mechanical metamaterials based on localized resonance mechanisms. These materials, by introducing localized resonant units into their structure, can form band gaps at scales much smaller than the wavelength, thereby achieving efficient suppression of low-frequency and mid-to-low-frequency vibrations. This property gives mechanical metamaterials based on localized resonance mechanisms a significant advantage in low-frequency vibration control.

[0007] However, existing localized resonant metamaterials also face some challenges in practical applications. On the one hand, most of them rely on complex localized resonant unit designs, such as multi-degree-of-freedom oscillators and bistable structures. While these complex designs can achieve good vibration reduction effects, they also significantly increase the difficulty of design and manufacturing, raising production costs and process complexity. Furthermore, in practical engineering applications, these complex structures are prone to parameter drift due to environmental factors and long-term use, which affects the vibration reduction effect and reduces reliability. On the other hand, achieving broadband suppression at low frequencies often requires an added mass ratio exceeding 100%. However, in practical engineering scenarios such as aerospace, shipbuilding, and pipelines, there are strict weight limitations, making such a high added mass ratio impractical, which greatly limits their application scope.

[0008] On the other hand, Bragg bandgap design for pipelines has been studied to some extent in engineering. Typically, the operating frequency range of the Bragg bandgap is set at high frequencies. However, in practical applications, due to limitations such as structural dimensions and actual installation space, it is often necessary to adjust the periodicity of the support distribution, which disrupts this periodicity. Currently, there is limited research on the specific dynamic characteristics of pipelines under these disrupted periodicity conditions. A lack of in-depth understanding of these dynamic characteristics makes it difficult to accurately assess and optimize the vibration reduction performance of pipelines, further limiting the application effect of Bragg bandgap design in pipeline vibration reduction.

[0009] It is evident that existing pipeline vibration reduction methods and related materials have their own limitations in practical applications, failing to fully meet the vibration reduction requirements of critical fields such as aerospace, shipbuilding, and energy. Therefore, there is an urgent need to propose a novel pipeline vibration reduction solution that can achieve broadband vibration reduction at low frequencies, is easy to design and manufacture, has high reliability, and can adapt to practical engineering space and weight constraints. Summary of the Invention

[0010] The technical problem to be solved by the present invention is to provide a full-frequency vibration reduction design method for metamaterial pipelines based on multiple band gaps.

[0011] To achieve the above-mentioned objectives, this invention provides a full-frequency vibration reduction design method for metamaterial pipelines based on multiple band gaps, comprising the following steps:

[0012] S1. Establish an equivalent mechanical model of the pipeline based on the target pipeline; S2. Periodically arrange elastic supports on the target pipeline to induce the formation of zero-frequency bandgap and Bragg bandgap. Couple the equivalent support stiffness of the elastic support with the pipeline mechanical equivalent model to construct the first bandgap calculation model. Based on the first bandgap calculation model, identify the first key parameter in the first assembly that has a regulatory effect on the two bandgap types. S3. Periodically arrange local resonant vibration absorbers on the target pipeline to form a local resonant bandgap. Couple the equivalent stiffness and equivalent mass of the local resonant vibration absorbers into the first bandgap calculation model to obtain the second bandgap calculation model. Based on the second bandgap calculation model, identify the second key parameter in the second assembly that has a regulatory effect on the three bandgap types. S4. Taking the second assembly as the analysis object, a finite structure dynamic vibration calculation model of the system is constructed based on Euler-Bernoulli beam theory and finite element method. S5. Construct a vibration transmissibility index for quantitatively analyzing the influence of the elastic support position on the dynamic characteristics of the second assembly, and use the finite structure dynamic vibration calculation model of the system as a fitness evaluation tool for the genetic algorithm to perform non-periodic optimization of the elastic support position, and solve for the optimal arrangement position of the elastic support with the largest bandwidth of the zero frequency bandgap.

[0013] According to one aspect of the present invention, in step S1, in the step of establishing a pipeline mechanical equivalent model based on the target pipeline, a slender pipeline with elastic deformation capability is selected as the basic model of the target pipeline, wherein the pipeline mechanical equivalent model is expressed as:

[0014] in, The quality matrix of the pipeline, Here is the damping matrix of the pipeline. Here is the stiffness matrix of the pipeline. Let u be the load vector and u be the displacement vector.

[0015] According to one aspect of the present invention, in step S2, the first key parameter that has a regulatory effect on the two band gaps in the first assembly based on the first band gap calculation model includes: the equivalent support stiffness of the elastic support and the lattice constant that divides the first assembly. The first key parameter is obtained by applying Bloch boundary conditions based on the first band gap calculation model, using the equivalent support stiffness of the elastic support and the lattice constant that divides the first assembly as analysis variables, and employing a processing method of parametric simulation, solving dispersion relations, and constructing a quantitative relationship between parameters and band gap characteristics to obtain the modulation law of the structural parameters in the first assembly on the zero-frequency band gap and the Bragg band gap.

[0016] According to one aspect of the present invention, in the step of obtaining the modulation law of the structural parameters in the first assembly on the zero-frequency bandgap and the Bragg bandgap, the modulation law for the zero-frequency bandgap is: based on the equivalent support stiffness of the elastic support to raise the lowest order bending mode of the first assembly, thereby realizing the control of the zero-frequency bandgap. The modulation law of the Bragg bandgap is as follows: the range of the Bragg bandgap is controlled by the equivalent support stiffness of the elastic support, and the bandwidth of the Bragg bandgap is controlled by the lattice constant. Specifically, Bloch boundary conditions are applied to the first bandgap calculation model to calculate the band characteristics of the Bragg bandgap and extract the bandgap features to quantify the modulation law of the range of the Bragg bandgap by the elastic support structure parameters. The bandgap features include: start and end frequencies, bandwidth, and vibration attenuation.

[0017] According to one aspect of the invention, the Bloch boundary condition is expressed as:

[0018] in, The field quantity at the boundary of the periodic unit outputs. The field quantity at the boundary of the periodic unit. The imaginary unit, , For wave vector, It is a periodic vector.

[0019] According to one aspect of the invention, in the step of applying Bloch boundary conditions to a first bandgap calculation model and calculating the band characteristics of the Bragg bandgap, the number of calculation cycles used is greater than or equal to 6.

[0020] According to one aspect of the present invention, step S2, which couples the equivalent support stiffness of the elastic support with the equivalent model of pipeline mechanics to construct a first bandgap calculation model, includes: Based on the finite element method, the equivalent mechanical model of the pipeline is discretized into a global matrix equation, and then the global matrix equation is decoupled according to the nodal degrees of freedom to extract the target pipeline. Equilibrium equations for transverse vibration of a finite element node; Get installed in the The equivalent support stiffness of the elastic support at each finite element node is directly superimposed onto the first finite element node. In the stiffness term of the lateral vibration equilibrium equation of each finite element node, the equivalent support stiffness is coupled with the equivalent model of pipeline mechanics. After traversing all finite element nodes with elastic supports to complete the stiffness superposition of the lateral vibration equilibrium equations, the overall stiffness matrix of the assembly is reassembled, thus completing the construction of the first bandgap calculation model; wherein, the first bandgap calculation model is expressed as:

[0021] in, This represents the concentrated mass of a finite element node on a combination of rigid supports and target pipelines. and These represent the linear stiffness and damping of the rigid support, respectively. Indicates the first Concentrated mass at each finite element node Lateral displacement, Indicates the first The interaction forces generated inside the target pipeline at each finite element node.

[0022] According to one aspect of the present invention, step S3, which involves periodically arranging local resonant vibration absorbers on the target pipeline to form a local resonant bandgap, and coupling the equivalent stiffness and equivalent mass of the local resonant vibration absorbers into the first bandgap calculation model to obtain the second bandgap calculation model, includes: S31. Determine the operating frequency of the local resonant vibration absorber on the target pipeline; S32. Obtain the first correlation between the equivalent stiffness and equivalent mass of the vibration absorber based on the operating frequency of the local resonant vibration absorber; wherein, the first correlation is expressed as:

[0023] in, This represents the equivalent linear stiffness of the vibration absorber. This indicates the operating frequency of the vibration absorber on the target pipeline. This represents the equivalent mass of a single vibration absorber; S33. Introduce the obtained first correlation into the first bandgap calculation model to obtain the second bandgap calculation model for the second assembly.

[0024] According to one aspect of the present invention, step S33, in which the obtained first correlation relationship is introduced into the first bandgap calculation model to obtain a second bandgap calculation model for the second assembly, includes: Based on the first bandgap calculation model, the target pipeline is extracted. Equilibrium equations for transverse vibration of a finite element node; Get installed in the The equivalent stiffness and equivalent mass of the local resonant vibration absorber at each finite element node are superimposed on the equivalent stiffness at the first node. In the stiffness term of the lateral vibration equilibrium equation for the nth finite element node, the equivalent mass is superimposed onto the nth node. In the mass term of the transverse vibration equilibrium equation of each finite element node, the local coupling between the dynamic characteristics of the local resonant vibration absorber and the first bandgap calculation model is realized. After traversing all finite element nodes with local resonant vibration absorbers to complete the superposition of equivalent stiffness and equivalent mass in the lateral vibration equilibrium equation, the overall stiffness matrix and overall mass matrix of the assembly are updated. Based on the first correlation relationship, the first bandgap calculation model coupling the dynamic characteristics of the local resonant vibration absorbers is transformed into a second bandgap calculation model; wherein, the second bandgap calculation model is expressed as:

[0025] in, This represents the concentrated mass of a finite element node on a combination of rigid supports and target pipelines. This represents the equivalent mass of the local resonant vibration absorber. and These represent the linear stiffness and damping of the local resonant vibration absorber, respectively. Indicates the target pipeline number Concentrated mass at each finite element node Lateral displacement, Indicates the target pipeline number Lateral displacement of the local resonant vibration absorber at each finite element node. Indicates the first The interaction forces generated inside the target pipeline at each finite element node.

[0026] According to one aspect of the present invention, in step S5, the step of constructing a vibration transmissibility index for quantitatively analyzing the influence of the elastic support position on the dynamic characteristics of the second assembly, the vibration transmissibility index is expressed as:

[0027] in, Indicates vibration transmissibility. Indicates the displacement of the response point. Indicates the displacement of the excitation. Indicates decibels; In step S5, the genetic algorithm optimization function used is: The genetic algorithm optimizes the elastic support positions aperiodically to find the optimal arrangement of elastic supports with the largest bandwidth at zero frequency gap, based on the finite structure dynamic vibration calculation model of the system as the fitness evaluation tool.

[0028] in, Let represent the objective function for optimization, and be the vibration transmissibility. Below the threshold Continuous frequency bandwidth at that time The upper boundary of the stopband can be abbreviated as: , Let represent the set of locations of elastic supports, which are design variables for optimization, and be expressed as . , These represent the position coordinates of each elastic support. Indicates the minimum position coordinates. This represents the maximum position coordinates.

[0029] The technical effects of this invention are as follows: According to one aspect of the present invention, a collaborative vibration reduction framework guided by three mechanisms—zero-frequency bandgap induced by elastic support, Bragg bandgap, and local resonance bandgap induced by vibration absorber—achieves full-frequency vibration reduction of pipelines. By adjusting the configuration of structural parameters and the non-periodic arrangement of support conditions, a suppression frequency band from low to high frequencies is constructed, providing a basis for the design and optimization of broadband vibration control schemes for pipelines.

[0030] According to one aspect of the present invention, this approach induces a zero-bandgap (0 Hz—) in the system by arranging elastic supports along the pipeline. f (1Hz). The elastic support introduces additional elastic constraint stiffness into the originally nearly free or weakly constrained pipeline structure, reconstructing the boundary conditions and overall stiffness matrix of the structure. Due to the introduction of the support stiffness, the lowest-order bending mode of the pipeline system no longer starts from zero frequency, but is instead raised to a certain non-zero cutoff frequency. This prevents the originally freely propagating ultra-low-frequency bending waves from forming effective propagation modes below this cutoff frequency. Based on this, the wave dispersion characteristics of the target pipeline are effectively altered, causing its lowest-order dispersion curve to shift upwards, constructing an elastic wave bandgap region starting from zero frequency, and achieving effective interception of low-frequency vibrations and quasi-rigid body modes approaching 0Hz.

[0031] According to one aspect of the present invention, the zero-bandgap constructed in this aspect does not rely on the addition of large mass, but on the reasonable configuration of elastic support stiffness, which has the advantages of simple structure, limited weight increase and strong engineering feasibility.

[0032] According to one aspect of the present invention, the impedance mismatch between the support stiffness of the elastic support and the bending stiffness of the target pipeline can generate wave reflection and transmission phenomena at the support. The scattered waves generated by multiple support points coherently superimpose in space. When the incident wavelength satisfies the Bragg condition related to the period length, a strong interference cancellation effect will be formed, preventing the elastic wave from propagating continuously within a specific frequency range, thereby forming a Bragg band gap (…). f 2Hz~ f (3Hz). By covering the high-frequency range with the Bragg bandgap, it is possible to achieve the lifting of the ultra-low frequency mode at zero frequency bandgap, while also inducing the formation of the Bragg scattering bandgap to effectively suppress high-frequency vibrations. Thus, without increasing the complexity of the structure, it can meet the control requirements of both ultra-low frequency and high frequency, and significantly improve the overall vibration reduction efficiency.

[0033] According to one aspect of the present invention, this approach constructs a localized resonant bandgap by arranging vibration absorbers on the target pipeline. f 3Hz~ f (4Hz), where the vibration absorber is a resonant oscillator system coupled to the target pipeline. When the external excitation frequency is close to the natural frequency of the resonant oscillator, the local mass of the vibration absorber generates a strong resonant response, causing abnormal changes in the equivalent dynamic parameters of the system near that frequency, thereby forming a local resonant bandgap in the frequency domain. This achieves efficient interception of mid-to-high frequency vibrations and fills the frequency domain gap between the zero-frequency bandgap and the Bragg bandgap, making the three bandgap mechanisms form a continuous connection in the frequency domain.

[0034] According to one aspect of the present invention, this solution can flexibly match the mass and stiffness parameters of the vibration absorber based on the characteristic frequencies of common power sources in the target pipeline, such as the pump blade passing frequency, motor speed frequency and its harmonic components, and in combination with the frequency range not yet fully covered between the zero-frequency bandgap and the Bragg bandgap, so that the local resonant frequency is distributed within the target suppression range. Without changing the main structure of the pipeline, efficient interception of mid-to-high frequency vibrations is achieved through modular installation and adjustable parameters, filling the frequency domain gap between the zero-frequency bandgap and the Bragg bandgap, thus creating a continuous connection between the three bandgap mechanisms in the frequency domain.

[0035] According to one aspect of the present invention, this scheme constructs an approximately continuous elastic wave bandgap curve in the frequency domain through the coordinated design of zero-frequency bandgap, Bragg bandgap, and local resonant bandgap. This enables the pipeline system to achieve effective attenuation when facing complex excitations with a wide frequency range, regardless of whether the excitation originates from ultra-low frequency overall vibration, mid-frequency structural fluctuations, or periodic excitations at specific frequencies. The three bandgap mechanisms are uniformly coupled in the dynamic model, and the continuity of frequency band coverage and the maximization of bandwidth are achieved through parameter coordination. Thus, while ensuring controlled added mass and feasible structural modularity, the goal of suppressing vibration across the entire frequency band of the pipeline is achieved, significantly improving the stability and engineering reliability of the system.

[0036] According to one aspect of the present invention, by synergistically utilizing the zero-frequency bandgap, Bragg bandgap, and local resonant bandgap of a metamaterial structure, this invention overcomes the limitations of traditional vibration reduction technologies, achieving broadband vibration suppression of pipelines from ultra-low frequencies to mid-high frequencies. This effectively resolves the contradiction between mass and vibration reduction bandwidth that a single vibration reduction mechanism cannot balance. It does not rely on an extremely large added mass ratio; through lightweight design, vibration reduction efficiency is improved. It adapts to actual engineering installation constraints, and the vibration reduction bandwidth can be expanded by optimizing the support layout (including non-periodic arrangements), balancing installation convenience and vibration reduction effect, thus lowering the threshold for practical application. The design concept has strong universality and is applicable to various types of pipelines, providing a general solution for vibration control in multiple fields such as machinery, aviation, and shipbuilding.

[0037] According to one aspect of the present invention, the excitation of both the zero-frequency bandgap and the Bragg bandgap can be achieved simultaneously by using periodic elastic supports, without the need for additional high-frequency damping structures. This simplifies the structure while taking into account the damping requirements of both ultra-low and high frequencies, thereby improving the overall damping efficiency. Attached Figure Description

[0038] Figure 1 This is a flowchart illustrating the pipeline vibration reduction design method based on metamaterial structures of the present invention. Figure 2 This is a flowchart of the pipeline vibration reduction design method based on metamaterial structures of the present invention; Figure 3 This is a structural diagram of the target pipeline, elastic support, and local resonant vibration absorber assembly of the present invention, wherein, Figure 3 (a) is a structural model diagram of the target pipeline, rigid support, and vibration absorber assembly. Figure 3 (b) is Figure 3 (a) Enlarged view of the target pipeline cross-section at position α. Figure 3 (c) is Figure 3 (a) Equivalent principle diagram of the cell at position β; Figure 4 A physical diagram of the target pipeline, rigid support, and vibration absorber assembly structure; Figure 5This is a diagram illustrating the coordinated influence of the Bragg bandgap and local resonant bandgap in this invention, wherein... Figure 5 (a) Influence diagram of support stiffness; Figure 5 (b) Effect diagram of additional mass ratio; Figure 5 (c) Graph showing the variation of lattice constant; Figure 6 This is a diagram illustrating the impact of the non-periodic support of this invention on system dynamics; Figure 7 This is a diagram showing the experimental results from Embodiment 2 of the present invention. Figure 7 (a) shows the effect of lattice constant on bandgap characteristics; Figure 7 (b) shows the coupling characteristics of the Bragg bandgap and the local resonant bandgap; Figure 7 (c) shows a comparison of the time domain frequency sweep from 400Hz to 630Hz; Figure 7 (d) shows a structural schematic diagram of periodic and non-periodic arrangement of elastic supports; Figure 7 (e) shows a comparison of the characteristics of periodic and non-periodic arrangements of elastic supports. Detailed Implementation

[0039] To more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the embodiments will be described in detail below.

[0040] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. The embodiments cannot be described in detail here, but the embodiments of the present invention are not limited to the following embodiments.

[0041] Combination Figure 1 and Figure 2 As shown, according to one embodiment of the present invention, a full-frequency vibration reduction design method for metamaterial pipelines based on multiple band gaps includes the following steps: S1. Establish an equivalent mechanical model of the pipeline based on the target pipeline; S2. Periodically arrange elastic supports on the target pipeline to induce the formation of zero-frequency bandgap and Bragg bandgap. Couple the equivalent support stiffness of the elastic support with the pipeline mechanical equivalent model to construct the first bandgap calculation model. Based on the first bandgap calculation model, identify the first key parameter in the first assembly that has a regulatory effect on the two bandgap types. S3. Periodically arrange local resonant vibration absorbers on the target pipeline to form a local resonant bandgap. Couple the equivalent stiffness and equivalent mass of the local resonant vibration absorbers into the first bandgap calculation model to obtain the second bandgap calculation model. Based on the second bandgap calculation model, identify the second key parameter in the second assembly that has a regulatory effect on the three bandgap types. S4. Taking the second assembly as the analysis object, a finite structure dynamic vibration calculation model of the system is constructed based on Euler-Bernoulli beam theory and finite element method; S5. Construct a vibration transmissibility index for quantitatively analyzing the influence of the position of elastic supports on the dynamic characteristics of the second assembly, and use the finite structure dynamic vibration calculation model of the system as a fitness evaluation tool for the genetic algorithm to perform non-periodic optimization of the position of elastic supports, and solve for the optimal arrangement position of elastic supports with the largest bandwidth of zero frequency band gap.

[0042] According to one embodiment of the present invention, in step S1, in the step of establishing a pipeline mechanical equivalent model based on the target pipeline, a slender pipeline with elastic deformation capability is selected as the basic model of the target pipeline. In this embodiment, based on the Euler-Bernoulli beam theory, the displacement control equation of the pipeline structure is established, which is expressed as:

[0043] in, This represents the Young's modulus of the target pipeline. This represents the cross-sectional moment of the target pipeline. Indicates the density of the target pipeline. Represents the area of ​​the cross section. This indicates lateral displacement.

[0044] Furthermore, the constructed displacement control equations are spatially discretized, transforming the continuous partial differential equations into a system of ordinary differential equations for a finite-degree-of-freedom system, thus constructing an equivalent pipeline mechanical model, which is expressed in matrix form as follows:

[0045] in, The quality matrix of the pipeline, Here is the damping matrix of the pipeline. Here is the stiffness matrix of the pipeline. Let u be the load vector and u be the displacement vector.

[0046] In this embodiment, the target pipeline is a metal circular pipe, which serves as the main path for vibration propagation. The material can be stainless steel, carbon steel, or aluminum alloy to meet the strength and corrosion resistance requirements of different engineering scenarios.

[0047] like Figure 3 As shown in (a), according to one embodiment of the present invention, in step S2, in the step of periodically arranging elastic supports on the target pipeline to induce the formation of zero-frequency bandgap and Bragg bandgap, the elastic supports are installed on the target pipeline in a periodic arrangement along the extension direction of the target pipeline. The elastic supports are detachably fixed to the outer wall of the pipeline via pipe clamps to ensure reliable connection. In this embodiment, the spacing of the elastic supports along the axial direction of the pipeline is 'a'.

[0048] In this embodiment, without the addition of elastic supports, the target pipeline approximates a free-free boundary condition, with its lowest-order bending vibration mode originating near 0Hz, and ultra-low-frequency vibrations and quasi-rigid body modes propagating freely along the pipeline. Furthermore, by adding periodic elastic supports, additional radial and bending constraint stiffness is introduced into the target pipeline body, reconstructing the boundary conditions and overall stiffness matrix of the target pipeline. This causes the lowest-order bending mode of the pipeline system (i.e., the first assembly consisting of the target pipeline and the elastic supports) to be raised to a non-zero cutoff frequency. f 1 (The specific value can be adjusted by regulating the stiffness of the elastic support), thus preventing the normally freely propagating ultra-low frequency bending waves from forming effective propagation modes below this cutoff frequency range. This design alters the wave dispersion characteristics of the pipeline structure, shifting its lowest-order dispersion curve upwards, constructing an elastic wave bandgap region starting from zero frequency, and achieving effective interception of low-frequency vibrations and quasi-rigid body modes approaching 0Hz. Therefore, at 0Hz— f Within the 1Hz frequency band, the bending waves of the target pipeline cannot form an effective propagation mode, thus achieving effective interception of ultra-low frequency vibrations and quasi-rigid body modes. The construction of this zero-frequency bandgap does not require the addition of a large mass block; it is achieved only through the reasonable configuration of the equivalent support stiffness of the elastic support. The structure is simple, the increase in weight is small, and the engineering feasibility is strong. Thus, the induced formation of the zero-frequency bandgap can be realized.

[0049] Furthermore, each elastic support is considered as a periodically distributed point impedance unit. There is a significant impedance mismatch between the stiffness of the elastic support unit and the bending stiffness of the target pipeline. When the elastic wave propagates along the axial direction of the pipeline to the support unit, reflection and transmission occur at the support. Consequently, the scattered waves generated by multiple periodically arranged elastic supports coherently superimpose. When the wavelength of the incident elastic wave satisfies the Bragg condition, the scattered waves exhibit a strong interference cancellation effect, preventing the elastic wave from propagating continuously in a specific high-frequency range, thus forming a Bragg bandgap. The frequency range of this Bragg bandgap is... f 2 to f 3. This allows for the induced formation of the Bragg bandgap. Specifically, the Bragg bandgap primarily covers the high-frequency range, and its frequency range is determined by the elastic support spacing, support stiffness, and target pipeline parameters. Through the rational design of the periodic parameters, the position of the Bragg bandgap and the elastic support spacing can be precisely controlled. The key advantage of this design is that, using the same set of periodic elastic support systems, while achieving the lifting of the ultra-low-frequency mode from the zero-frequency bandgap, it can also induce the formation of a Bragg scattering bandgap, effectively suppressing high-frequency vibrations. Thus, without increasing the complexity of the structure, it addresses both ultra-low-frequency and high-frequency control needs, significantly improving overall vibration reduction efficiency.

[0050] Based on this, the band structure characteristics of the Bragg bandgap can be calculated using the Bloch boundary condition. In this embodiment, the Bloch boundary condition is a specific boundary condition for wave propagation analysis in periodic elastic structures. Its core function is to describe the field relationships of elastic waves (vibration waves) at the boundaries of periodic elements: when the smallest periodic element of the periodic structure is taken as the object of analysis, after the wave passes through the element, its displacement, stress / strain, and other field quantities do not change abruptly; only a phase shift related to the wave vector and period length occurs, i.e., the field quantity at the element's outgoing edge = the field quantity at the incoming edge × the phase factor. This condition is the core basis for solving the dispersion relation of periodic structures and determining the frequency range of the Bragg bandgap, and it is also the theoretical foundation for the design of periodic bandgap.

[0051] Therefore, in step S2, the first key parameter that controls the two band gaps in the first assembly is determined based on the first band gap calculation model. The first key parameter includes: the equivalent support stiffness of the elastic support and the lattice constant that divides the first assembly. Specifically, based on the first band gap calculation model, Bloch boundary conditions are applied, and the equivalent support stiffness of the elastic support and the lattice constant that divides the first assembly are used as analysis variables. The processing method of parametric simulation, solving the dispersion relation, and constructing the quantitative relationship between parameters and band gap characteristics is adopted to obtain the modulation law of the structural parameters in the first assembly on the zero-frequency band gap and the Bragg band gap, thus realizing the confirmation of the first key parameter.

[0052] In this embodiment, the step of obtaining the modulation law of the structural parameters in the first assembly on the zero-frequency bandgap and the Bragg bandgap is as follows: the modulation law for the zero-frequency bandgap is: based on the equivalent support stiffness of the elastic support, the lowest order bending mode of the first assembly is raised to achieve the control of the zero-frequency bandgap; wherein, the equivalent support stiffness of the elastic support can be adjusted by the design of the structural parameters itself, and will not be elaborated here. The modulation law for the Bragg bandgap is: based on the equivalent support stiffness of the elastic support, the range of the Bragg bandgap is controlled, and based on the lattice constant, the bandwidth of the Bragg bandgap is controlled; wherein, Bloch boundary conditions are applied to the first bandgap calculation model, the band characteristics of the Bragg bandgap are calculated, and the bandgap features of the Bragg bandgap are extracted to quantify the modulation law of the elastic support structural parameters on the range of the Bragg bandgap, wherein the bandgap features include: start and end frequencies, bandwidth, and vibration attenuation. In this embodiment, the range of the Bragg bandgap is a key factor providing theoretical basis and design benchmark for coordinated matching with other bandgap (i.e., the zero-frequency bandgap and subsequent local resonance bandgap) and non-periodic optimization of the elastic support position. Therefore, the modulation law of the Bragg bandgap range is mainly analyzed by combining Bloch boundary conditions. The Bloch boundary conditions used are expressed as follows:

[0053] in, The field quantity at the boundary of the periodic unit outputs. The field quantity at the boundary of the periodic unit. The imaginary unit, , For wave vector, It is a periodic vector.

[0054] In this embodiment, in the step of applying Bloch boundary conditions to the first bandgap calculation model and calculating the band structure characteristics of the Bragg bandgap, the number of calculation cycles used is greater than or equal to 6. Based on this set number of calculation cycles, the dynamic characteristics of the finite-size structure calculation are essentially the same as those of the infinite-size structure calculation, effectively ensuring calculation accuracy while reducing the consumption of computational resources.

[0055] According to one embodiment of the present invention, step S2, which couples the equivalent support stiffness of the elastic support with the equivalent model of pipeline mechanics to construct the first bandgap calculation model, includes: Based on the finite element method, the equivalent mechanical model of the pipeline is discretized into a global matrix equation, and then the global matrix equation is decoupled according to the nodal degrees of freedom to extract the target pipeline. Equilibrium equations for transverse vibration of a finite element node; Get installed in the The equivalent support stiffness of the elastic support at each finite element node is directly superimposed onto the first finite element node. In the stiffness term of the lateral vibration equilibrium equation of each finite element node, the equivalent support stiffness is coupled with the equivalent model of pipeline mechanics. After traversing all finite element nodes with elastic supports to complete the stiffness superposition of the lateral vibration equilibrium equations, the overall stiffness matrix of the assembly is reassembled, thus completing the construction of the first bandgap calculation model; wherein, the first bandgap calculation model is expressed as:

[0056] in, This represents the concentrated mass of a finite element node on a combination of rigid supports and target pipelines. and These represent the linear stiffness and damping of the rigid support, respectively. Indicates the first Concentrated mass at each finite element node Lateral displacement, Indicates the first The interaction forces generated inside the target pipeline at each finite element node.

[0057] Combination Figure 3 (a) and Figure 3As shown in (c), according to an embodiment of the present invention, in step S3, the step of periodically arranging local resonant vibration absorbers on the target pipeline to form a local resonant bandgap, and coupling the equivalent stiffness and equivalent mass of the local resonant vibration absorbers into the first bandgap calculation model to obtain the second bandgap calculation model includes: S31. Determine the operating frequency of the local resonance vibration absorber on the target pipeline; In this embodiment, since the local resonance vibration absorber is used for vibration reduction control of the target pipeline, the operating frequency of the absorber can be directly determined based on the working scenario and state of the target pipeline. In this embodiment, the metamaterial structure in the local resonance vibration absorber is implemented using a resonator structure of "elastic element + mass block". Based on the typical power source characteristic frequencies (such as pump blade passing frequency, motor speed frequency and its harmonic components) of the pipeline system (i.e., the second assembly including the target pipeline, elastic support and vibration absorber), combined with the uncovered frequency band between the zero-frequency bandgap and the Bragg bandgap, the natural frequency of the local resonance unit is designed so that the natural frequency of the absorber is distributed within this uncovered frequency band, thereby forming a local resonance bandgap. Thus, the operating frequency of the absorber is determined based on the frequency range of the local resonance bandgap. f 3 to f 4.

[0058] S32. Obtain the first correlation between the equivalent stiffness and equivalent mass of the vibration absorber based on the operating frequency of the local resonant vibration absorber; wherein, the first correlation is expressed as:

[0059] in, This represents the equivalent linear stiffness of the vibration absorber. This indicates the operating frequency of the vibration absorber on the target pipeline. This represents the equivalent mass of a single vibration absorber; S33. Introduce the obtained first correlation relationship into the first bandgap calculation model to obtain a second bandgap calculation model for the second assembly; in this embodiment, it includes: Based on the first bandgap calculation model, the target pipeline is extracted. Equilibrium equations for transverse vibration of a finite element node; Get installed in the The equivalent stiffness and equivalent mass of the local resonant vibration absorber at each finite element node are superimposed on the equivalent stiffness at the first node. In the stiffness term of the lateral vibration equilibrium equation for the nth finite element node, the equivalent mass is superimposed onto the nth node. In the mass term of the transverse vibration equilibrium equation of each finite element node, the local coupling between the dynamic characteristics of the local resonant vibration absorber and the first bandgap calculation model is realized. After traversing all finite element nodes with local resonant vibration absorbers to complete the superposition of equivalent stiffness and equivalent mass in the lateral vibration equilibrium equation, the overall stiffness matrix and overall mass matrix of the assembly are updated. Based on the first correlation relationship, the first bandgap calculation model coupling the dynamic characteristics of the local resonant vibration absorbers is transformed into a second bandgap calculation model; wherein, the second bandgap calculation model is expressed as:

[0060] in, This represents the concentrated mass of a finite element node on a combination of rigid supports and target pipelines. This represents the equivalent mass of the local resonant vibration absorber. and These represent the linear stiffness and damping of the local resonant vibration absorber, respectively. Indicates the target pipeline number Concentrated mass at each finite element node Lateral displacement, Indicates the target pipeline number Lateral displacement of the local resonant vibration absorber at each finite element node. Indicates the first The interaction forces generated inside the target pipeline at each finite element node.

[0061] Therefore, by analyzing the modulation law of the structural parameters in the second assembly on the zero-frequency bandgap, Bragg bandgap, and local resonant bandgap based on the second bandgap calculation model, it can be seen that when the external excitation frequency of the target pipeline is close to the natural frequency of the vibration absorber, the resonant oscillator structure generates a strong resonant response, concentrating the vibration energy of the target pipeline in the vibration absorber and dissipating it through the damping effect of the elastic element in the vibration absorber. This causes abnormal changes in the equivalent dynamic parameters (equivalent stiffness, equivalent mass) of the system near this frequency band, thereby forming a local resonant bandgap, filling the frequency domain gap between the zero-frequency bandgap and the Bragg bandgap, and realizing the continuous connection of the three bandgap types.

[0062] In this embodiment, based on the obtained second bandgap calculation model, the added mass ratio of the local resonant absorber, the equivalent support stiffness of the elastic support, and the lattice constant are used as analysis variables. Through parametric simulation, solving for dispersion relations, and constructing a quantitative relationship between parameters and bandgap characteristics, the coupling modulation law of structural parameters in the second assembly on the zero-frequency bandgap, Bragg bandgap, and local resonant bandgap is analyzed, clarifying the parameter design criteria for the coordinated and continuous connection of the three bandgap components. Therefore, the second key parameters include: the added mass ratio of the local resonant absorber, the equivalent support stiffness of the elastic support, and the lattice constant.

[0063] In this embodiment, the localized resonance vibration absorber is installed on the outer wall of the target pipeline, between two adjacent elastic supports. The localized resonance vibration absorber includes a mass block, an elastic element, and a mounting base. The mounting base is fixed to the outer wall of the target pipeline with bolts, without damaging the main structure of the target pipeline. The number and position of the absorbers can be flexibly adjusted according to vibration reduction requirements. In this embodiment, the mounting base has a semi-circular hollow structure. The hollow part of the mounting base is semi-circular, and the mass block can be set as a block with a fan-shaped cross-section to fit the hollow part of the mounting base. Furthermore, one axial end of the mass block is fixedly connected to the wall of the hollow part using an elastic element. The elastic element can be a metal rod, and its elasticity is set according to actual requirements, which will not be elaborated here. In this embodiment, the outer side of the mass block can be covered with a flexible rubber layer. Furthermore, two mass blocks can be symmetrically arranged inside the mounting base, and there are gaps between adjacent mass blocks, between the covering layer of the mass block and the wall of the hollow part of the mounting base. Therefore, when the vibration is small, the restoring force is provided solely by the elastic element, while when the vibration is small, the restoring force is provided by the combined action of the elastic element and the coating layer, so as to achieve the vibration absorption capability under different vibration states.

[0064] According to one embodiment of the present invention, in step S4, taking the second assembly as the analysis object, the step of constructing a system finite structure dynamic vibration calculation model based on Euler-Bernoulli beam theory and the finite element method is used to verify the correctness of the second bandgap calculation model and to provide a simulation tool for quantitative evaluation of vibration reduction performance for subsequent genetic algorithm non-periodic optimization of elastic support positions. In this embodiment, the mathematical form of the system finite structure dynamic vibration calculation model is consistent with the equivalent model of pipeline mechanics, and will not be described again here.

[0065] According to one embodiment of the present invention, in step S5, the step of constructing a vibration transmissibility index for quantitatively analyzing the influence of the elastic support position on the dynamic characteristics of the second assembly, the vibration transmissibility index is expressed as:

[0066] in, Indicates vibration transmissibility. Indicates the displacement of the response point. Indicates the displacement of the excitation. It represents decibels.

[0067] According to one embodiment of the present invention, in step S5, where the system finite structure dynamic vibration calculation model is used as the fitness evaluation tool of the genetic algorithm to perform aperiodic optimization of the elastic support positions and solve for the optimal arrangement of elastic supports with the largest bandwidth of the zero-frequency bandgap, the elastic support positions are optimized aperiodically based on the system finite structure dynamic vibration calculation model through a genetic algorithm process of model iteration, population fitness calculation, and selection crossover mutation to solve for the optimal arrangement of elastic supports with the largest bandwidth of the zero-frequency bandgap. The genetic algorithm optimization function used is:

[0068] in, Let represent the objective function for optimization, and be the vibration transmissibility. Below the threshold Continuous frequency bandwidth at that time The upper boundary of the stopband can be abbreviated as: , Let represent the set of locations of elastic supports, which are design variables for optimization, and be expressed as . , These represent the position coordinates of each elastic support. Indicates the minimum position coordinates. This represents the maximum position coordinates.

[0069] In this embodiment, based on support position constraints This effectively ensures that the rigid supports are located within the feasible region of the structure; based on the position monotonicity constraint Effectively ensures that the support position monotonically increases with the design sequence number; based on boundary distance constraints: This ensures that the distance between the rigid supports at both ends and the end of the target pipeline is not less than [amount missing]. This effectively avoids the impact of boundary effects on structural integrity.

[0070] According to this invention, this solution constructs a multi-bandgap vibration reduction system that synergistically utilizes zero-frequency bandgap, Bragg bandgap, and local resonant bandgap by arranging elastic support structures and local resonant vibration absorption units on the target pipeline, achieving continuous vibration suppression from ultra-low frequencies to mid-high frequencies. Specifically, by setting elastic supports with equivalent stiffness to alter the pipeline boundary constraint characteristics, a zero-frequency bandgap is formed to control ultra-low frequency vibration; a periodic support structure is arranged along the pipeline direction to form a Bragg bandgap for high-frequency vibration suppression; and vibration absorbers are installed at key locations to form a local resonant bandgap, achieving precise control of the target frequency band. A genetic algorithm is used for aperiodic optimization design of the support positions to maximize the widening of the stopband. This invention achieves full-frequency vibration reduction of pipelines with relatively low added mass conditions, has a simple structure, is easy to integrate into engineering, and has good potential for widespread application.

[0071] To further illustrate this plan, an example will be provided.

[0072] Example 1 Combination Figure 3 (a) Figure 3 (b) and Figure 3 As shown in (c), in this embodiment, the outer diameter of the target pipeline... , inner diameter ,length Young's modulus Poisson's ratio ,density The equivalent stiffness of the support is lattice constant The operating frequency of the local resonant vibration absorber The total added mass ratio is .exist Figure 3 In (a), This represents the instantaneous displacement excitation function at the end opposite to the response point. Indicates the displacement of the excitation. The angular frequency representing the excitation displacement. Indicates time. Figure 3 In (c), This indicates the displacement of the left boundary of the cell; The force representing the left boundary of the cell; This indicates the right boundary displacement of the cell; The force representing the right boundary of the cell; like Figure 5 As shown in (a), by analyzing the influence of support stiffness, it can be seen that increasing the equivalent support stiffness of elastic support will expand the range of the three Bragg band gaps and enhance low-frequency attenuation, while shrinking the local resonant band gap and raising its peak frequency.

[0073] like Figure 5 As shown in (b), by analyzing the effect of the added mass ratio, increasing the added mass ratio will reduce the bandwidth of the quasi-Bragg bandgap and widen the other two Bragg bandgap and the local resonance bandgap.

[0074] like Figure 5 As shown in (c), by analyzing the effect of the lattice constant, the increase of the lattice constant will reduce the bandwidth of the low-frequency quasi-Bragg bandgap and shift the entire local resonant bandgap to higher frequencies.

[0075] like Figure 6 As shown, by analyzing the influence of aperiodic supports, the aperiodic arrangement of supports exhibits a strong low-frequency vibration attenuation effect, while maintaining a significant mid-to-low frequency bandgap, and at the operating frequency... Significant suppression effect is observed in the vicinity. Effective vibration suppression is defined as a transmissibility below -10dB (allowing one frequency to exceed this threshold). Comparative analysis reveals that the quasi-Bragg bandgap caused by the periodic group with periodic elastic support is in the range of 0-145Hz, while the elastic support with a non-periodic configuration achieves wider bandgap suppression in the range of 0-233Hz (non-periodic group 1) and 0-250Hz (non-periodic group 2).

[0076] Example 2 In this embodiment, the vibration of the vibration absorber is induced by an exciter, and the velocity signal of the locally resonant vibration absorber is measured using a laser vibrometer, thereby analyzing its natural vibration frequency. .

[0077] A metamaterial pipeline model is constructed on a finite-sized target pipeline by periodically arranging the aforementioned elastic supports and localized resonant vibration absorbers. Taking a one-dimensional linear metamaterial pipeline model with elastic supports as an example, such as... Figure 4 As shown, the average mass of a single oscillator of the additional localized resonant vibration absorber Target pipe outer diameter , inner diameter Made of stainless steel, with a length of The distance between any two elastic supports is defined as the lattice constant.

[0078] Furthermore, vibration tests were conducted to examine the vibration transmission characteristics of a one-dimensional linear metamaterial pipeline model with elastic supports and local resonant vibration absorbers under swept-frequency excitation, as well as its time-domain attenuation characteristics at the target frequency. This verified the effectiveness of the synergistic vibration reduction design utilizing a low-frequency quasi-Bragg bandgap and a mid-to-high-frequency local resonant bandgap. Sweeped-frequency excitation was applied to one side of the right end of the linear metamaterial pipeline model, with the excitation amplitude controlled by a power amplifier. A laser vibrometer was used to test the vibration response near the excitation point and at the leftmost end of the pipeline.

[0079] By changing only the lattice constant (363 mm, 275 mm, 236 mm), comparative tests were conducted to explore the regulation of the lattice constant on the band gap characteristics. The test results are as follows: Figure 7 As shown in (a), the results show that for the quasi-Bragg bandgap, decreasing the lattice constant can broaden the bandwidth from 205 Hz to 230 Hz; the second Bragg bandgap shifts to higher frequencies as the lattice constant decreases, and the bandwidth increases synchronously, with the vibration attenuation efficiency gradually improving, demonstrating the precise control effect of the lattice constant on the Bragg bandgap, and providing a parameter basis for bandgap coupling.

[0080] To further investigate the impact of ultra-wideband coupling between the Bragg bandgap and the local resonant bandgap on vibration reduction performance, a lattice constant of 363 mm was selected for comparative testing to explore the effects of the two bandgap coupling methods. The test results are as follows: Figure 7As shown in (b). The results show that 25% added mass can produce a strong bandgap coupling effect, thus forming a continuous ultra-wide bandgap (370Hz-640Hz). Under a sweep frequency excitation from 400Hz to 630Hz, the model with the local resonant damper has a significantly better vibration suppression effect compared to the model without the damper, such as... Figure 7 As shown in (c), quantitative analysis showed that the root mean square velocity decreased from 0.004456 m / s to 0.000097 m / s, a decrease of 97.83%; the absolute peak velocity decreased from 0.024141 m / s to 0.000708 m / s, an attenuation rate of 97.07%.

[0081] To further investigate the impact of aperiodic installation of elastic supports on vibration reduction performance, while keeping the number of elastic supports constant, two aperiodic models (i.e., aperiodic model 1 and aperiodic model 2) were constructed by systematically adjusting the spacing of the elastic supports. The specific arrangement is as follows: Figure 7 As shown in (d), a comparative test was conducted with the periodic model, and the test results are as follows. Figure 7 As shown in (e), the low-frequency vibration reduction bandwidth of the periodic model is 220Hz; the bandwidth of the non-periodic model 1 increases to 243Hz, which is 10.45% higher than that of the periodic structure; the bandwidth of the non-periodic model 2 is further broadened to 265Hz, an increase of 20.45%. This result confirms that the non-periodic support arrangement can effectively broaden the low-frequency vibration reduction bandwidth, solve the problem of spatial constraints on periodic arrangements in engineering, and at the same time ensure the low-frequency vibration reduction performance.

[0082] The above description is merely an example of a specific solution of the present invention. For any devices and structures not described in detail herein, it should be understood that they are implemented using common devices and methods already available in the art.

[0083] The above description is merely one embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the invention by those skilled in the art. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A full-frequency vibration reduction design method for metamaterial pipelines based on multiple band gaps, characterized in that, Includes the following steps: S1. Establish an equivalent mechanical model of the pipeline based on the target pipeline; S2. Periodically arrange elastic supports on the target pipeline to induce the formation of zero-frequency bandgap and Bragg bandgap. Couple the equivalent support stiffness of the elastic support with the pipeline mechanical equivalent model to construct the first bandgap calculation model. Based on the first bandgap calculation model, identify the first key parameter in the first assembly that has a regulatory effect on the two bandgap types. S3. Periodically arrange local resonant vibration absorbers on the target pipeline to form a local resonant bandgap. Couple the equivalent stiffness and equivalent mass of the local resonant vibration absorbers into the first bandgap calculation model to obtain the second bandgap calculation model. Based on the second bandgap calculation model, identify the second key parameter in the second assembly that has a regulatory effect on the three bandgap types. S4. Taking the second assembly as the analysis object, a finite structure dynamic vibration calculation model of the system is constructed based on Euler-Bernoulli beam theory and finite element method. S5. Construct a vibration transmissibility index for quantitatively analyzing the influence of the elastic support position on the dynamic characteristics of the second assembly, and use the finite structure dynamic vibration calculation model of the system as a fitness evaluation tool for the genetic algorithm to perform non-periodic optimization of the elastic support position, and solve for the optimal arrangement position of the elastic support with the largest bandwidth of the zero frequency bandgap.

2. The pipeline vibration reduction design method based on metamaterial structures according to claim 1, characterized in that, In step S1, the step of establishing a pipeline mechanical equivalent model based on the target pipeline involves selecting a slender pipeline with elastic deformation capability as the basic model of the target pipeline. The pipeline mechanical equivalent model is expressed as follows: in, The quality matrix of the pipeline. Here is the damping matrix of the pipeline. Here is the stiffness matrix of the pipeline. Let u be the load vector and u be the displacement vector.

3. The pipeline vibration reduction design method based on metamaterial structures according to claim 2, characterized in that, In step S2, the first key parameter that controls the two band gaps in the first assembly is determined based on the first band gap calculation model. The first key parameter includes: the equivalent support stiffness of the elastic support and the lattice constant that divides the first assembly. Based on the first band gap calculation model, Bloch boundary conditions are applied, and the equivalent support stiffness of the elastic support and the lattice constant that divides the first assembly are used as analysis variables. The processing method of parametric simulation, solving the dispersion relation, and constructing the quantitative relationship between parameters and band gap characteristics is adopted to obtain the modulation law of the structural parameters in the first assembly on the zero-frequency band gap and the Bragg band gap.

4. The pipeline vibration reduction design method based on metamaterial structures according to claim 3, characterized in that, In the step of obtaining the modulation law of the structural parameters in the first assembly on the zero-frequency bandgap and the Bragg bandgap, the modulation law for the zero-frequency bandgap is as follows: based on the equivalent support stiffness of the elastic support, the lowest order bending mode of the first assembly is raised to achieve the control of the zero-frequency bandgap. The modulation law of the Bragg bandgap is as follows: the range of the Bragg bandgap is controlled by the equivalent support stiffness of the elastic support, and the bandwidth of the Bragg bandgap is controlled by the lattice constant. Specifically, Bloch boundary conditions are applied to the first bandgap calculation model to calculate the band characteristics of the Bragg bandgap and extract the bandgap features to quantify the modulation law of the range of the Bragg bandgap by the elastic support structure parameters. The bandgap features include: start and end frequencies, bandwidth, and vibration attenuation.

5. The pipeline vibration reduction design method based on metamaterial structures according to claim 4, characterized in that, Bloch boundary conditions are expressed as follows: in, The field quantity at the boundary of the periodic unit outputs. The field quantity at the boundary of the periodic unit. The imaginary unit, , For wave vector, It is a periodic vector.

6. The pipeline vibration reduction design method based on metamaterial structures according to claim 5, characterized in that, In the step of applying Bloch boundary conditions to the first bandgap calculation model and calculating the band characteristics of the Bragg bandgap, the number of calculation cycles used is greater than or equal to 6.

7. The pipeline vibration reduction design method based on metamaterial structures according to any one of claims 2 to 6, characterized in that, Step S2, which couples the equivalent support stiffness of the elastic support with the equivalent model of pipeline mechanics to construct the first bandgap calculation model, includes: Based on the finite element method, the equivalent mechanical model of the pipeline is discretized into a global matrix equation, and then the global matrix equation is decoupled according to the nodal degrees of freedom to extract the target pipeline. Equilibrium equations for transverse vibration of a finite element node; Get installed in the The equivalent support stiffness of the elastic support at each finite element node is directly superimposed onto the first finite element node. In the stiffness term of the lateral vibration equilibrium equation of each finite element node, the equivalent support stiffness is coupled with the equivalent model of pipeline mechanics. After traversing all finite element nodes with elastic supports to complete the stiffness superposition of the lateral vibration equilibrium equations, the overall stiffness matrix of the assembly is reassembled, thus completing the construction of the first bandgap calculation model; wherein, the first bandgap calculation model is expressed as: in, This represents the concentrated mass of a finite element node on a combination of rigid supports and target pipelines. and These represent the linear stiffness and damping of the rigid support, respectively. Indicates the first Concentrated mass at each finite element node Lateral displacement, Indicates the first The interaction forces generated inside the target pipeline at each finite element node.

8. The pipeline vibration reduction design method based on metamaterial structures according to claim 7, characterized in that, In step S3, the step of periodically arranging local resonant vibration absorbers on the target pipeline to form a local resonant bandgap, and coupling the equivalent stiffness and equivalent mass of the local resonant vibration absorbers into the first bandgap calculation model to obtain the second bandgap calculation model includes: S31. Determine the operating frequency of the local resonant vibration absorber on the target pipeline; S32. Obtain the first correlation between the equivalent stiffness and equivalent mass of the vibration absorber based on the operating frequency of the local resonant vibration absorber; wherein, the first correlation is expressed as: in, This represents the equivalent linear stiffness of the vibration absorber. This indicates the operating frequency of the vibration absorber on the target pipeline. This represents the equivalent mass of a single vibration absorber; S33. Introduce the obtained first correlation into the first bandgap calculation model to obtain the second bandgap calculation model for the second assembly.

9. The pipeline vibration reduction design method based on metamaterial structures according to claim 8, characterized in that, Step S33, which involves introducing the obtained first correlation into the first bandgap calculation model to obtain the second bandgap calculation model for the second assembly, includes: Based on the first bandgap calculation model, the target pipeline is extracted. Equilibrium equations for transverse vibration of a finite element node; Get installed in the The equivalent stiffness and equivalent mass of the local resonant vibration absorber at each finite element node are superimposed on the equivalent stiffness at the first node. In the stiffness term of the lateral vibration equilibrium equation for the nth finite element node, the equivalent mass is superimposed onto the nth node. In the mass term of the transverse vibration equilibrium equation of each finite element node, the local coupling between the dynamic characteristics of the local resonant vibration absorber and the first bandgap calculation model is realized. After traversing all finite element nodes with local resonant vibration absorbers to complete the superposition of equivalent stiffness and equivalent mass in the lateral vibration equilibrium equation, the overall stiffness matrix and overall mass matrix of the assembly are updated. Based on the first correlation relationship, the first bandgap calculation model coupling the dynamic characteristics of the local resonant vibration absorbers is transformed into a second bandgap calculation model; wherein, the second bandgap calculation model is expressed as: in, This represents the concentrated mass of a finite element node on a combination of rigid supports and target pipelines. This represents the equivalent mass of the local resonant vibration absorber. and These represent the linear stiffness and damping of the local resonant vibration absorber, respectively. Indicates the target pipeline number Concentrated mass at each finite element node Lateral displacement, Indicates the target pipeline number Lateral displacement of the local resonant vibration absorber at each finite element node. Indicates the first The interaction forces generated inside the target pipeline at each finite element node.

10. The pipeline vibration reduction design method based on metamaterial structures according to claim 9, characterized in that, In step S5, the vibration transmissibility index, used to quantitatively analyze the influence of the elastic support position on the dynamic characteristics of the second assembly, is expressed as follows: in, Indicates vibration transmissibility. Indicates the displacement of the response point. Indicates the displacement of the excitation. Indicates decibels; In step S5, the genetic algorithm optimization function used is: The genetic algorithm optimizes the elastic support positions aperiodically to find the optimal arrangement of elastic supports with the largest bandwidth at zero frequency gap, based on the finite structure dynamic vibration calculation model of the system as the fitness evaluation tool. in, Let represent the objective function for optimization, and be the vibration transmissibility. Below the threshold Continuous frequency bandwidth at that time The upper boundary of the stopband can be abbreviated as: , Let represent the set of locations of elastic supports, which are design variables for optimization, and be expressed as . , These represent the position coordinates of each elastic support. Indicates the minimum position coordinates. This represents the maximum position coordinates.

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