Calculation method and system for calculating vibration isolation characteristics of sandwich superstructures

Abstract

The present invention belongs to the field of ship internal structure vibration and noise control, and discloses a calculation method and system for calculating the vibration isolation characteristics of a sandwich superstructure based on wave theory, the method includes: research on the dispersion relation and wave dynamics behavior of the sandwich superstructure; numerical simulation of the vibration transfer characteristics of the sandwich superstructure and research on its vibration transfer attenuation mechanism; structural vibration noise control method based on elastic wave theory; optimization and experimental research on the low-frequency and broadband vibration isolation performance of the sandwich superstructure. The present invention intends to adopt a method combining theory, simulation and experiment to systematically study the elastic wave propagation characteristics of infinitely serially connected sandwich superstructures, analyze the relationship between its dispersion relation and wave dynamics, and reveal the vibration attenuation law of finitely serially connected sandwich superstructures; further, establish a theoretical and numerical calculation model of finitely serially connected sandwich superstructures, the sandwich superstructure unit cells are connected in series, its equivalent stiffness is reduced, and the vibration isolation characteristic action frequency band is lower.

Description

Technical Field

[0001] The invention belongs to the field of ship internal structure vibration noise control, and in particular relates to a calculation method and system for calculating vibration isolation characteristics of a sandwich superstructure based on wave theory. Background Art

[0002] Studies have shown that elastic waves will continue to scatter and attenuate when propagating in acoustic metamaterials with periodic changes in geometry or materials, thereby forming band gap characteristics and inhibiting the propagation of elastic waves. In particular, the local resonance mechanism is known for "small size suppressing large wavelength", that is, local resonance produces a low-frequency band gap to suppress the propagation of low-frequency elastic waves. Therefore, based on the relevant theories of acoustic metamaterials and combined with traditional vibration isolation technology, a sandwich superstructure with broadband vibration isolation performance in the low-frequency band is developed. The elastic wave propagation characteristics of the sandwich superstructure are studied by combining theory, simulation and experiment. The attenuation mechanism of the sandwich superstructure on elastic waves is explored, and a set of sandwich superstructure vibration noise control methods based on elastic wave theory are formed. This has very important military significance for further deepening the understanding of the acoustic design concept of submarines and realizing quantitative vibration isolation design of low-frequency broadband.

[0003] The shortcomings and problems of the prior art mainly include the following points:

[0004] 1. Insufficient vibration isolation performance: Traditional submarine vibration isolation technology has poor vibration transmission performance in the low-frequency band and cannot effectively suppress low-frequency vibration noise, resulting in a high noise level in the low-frequency band of the submarine, affecting the stealth performance of the submarine.

[0005] 2. Unstable bandgap characteristics: Although acoustic metamaterials can form bandgap characteristics and suppress elastic wave propagation, in practical applications, due to changes in the environment, structure and material parameters, the bandgap characteristics of acoustic metamaterials may change, affecting their vibration isolation performance.

[0006] 3. Complex structure: The sandwich superstructure based on acoustic metamaterials may involve complex spatial geometric structures and a combination of multiple materials during the design and manufacturing process, resulting in greater difficulty in structural design and manufacturing.

[0007] 4. Weight and cost issues: In practical applications, the vibration isolation system of a submarine needs to meet the weight and cost requirements of the submarine while ensuring the vibration isolation performance. However, acoustic metamaterials and sandwich superstructures may increase the weight and cost of the submarine and affect the overall performance of the submarine.

[0008] 5. Lack of systematic research: Currently, there are few studies on the elastic wave propagation characteristics of sandwich superstructures based on acoustic metamaterials, and a complete theoretical system and application methods have not yet been formed, which limits the in-depth development of submarine acoustic design concepts and the realization of quantitative vibration isolation design of low-frequency and broadband.

[0009] In view of the above problems, a vibration and noise control method of sandwich superstructure based on acoustic metamaterials is studied. By combining theory, simulation and experiment, the attenuation mechanism of elastic waves of sandwich superstructure is explored, and a set of vibration and noise control methods of sandwich superstructure based on elastic wave theory is formed, which will help to improve the vibration isolation performance of submarines in the low frequency band, reduce the noise level and enhance the stealth performance of submarines. Summary of the invention

[0010] In view of the problems existing in the prior art, the present invention provides a method and system for calculating the vibration isolation characteristics of a sandwich superstructure based on wave theory.

[0011] The present invention proposes a method for calculating vibration isolation characteristics of a sandwich superstructure based on wave theory, the method comprising:

[0012] S1: Study on dispersion relation and wave dynamics behavior of sandwich superstructures;

[0013] S2: Numerical simulation of vibration transfer characteristics of sandwich superstructures and study of their vibration transfer attenuation mechanism;

[0014] S3: Structural vibration and noise control method based on elastic wave theory;

[0015] S4: Optimization and experimental study of low-frequency and broadband vibration isolation performance of sandwich superstructures.

[0016] Further, the S1 specifically includes:

[0017] Based on the deformation energy theory, the equivalent stiffness of the elastic bending element and the cantilever beam is obtained; the mathematical model of the infinite series sandwich superstructure is established in combination with the elastic wave theory; the periodic Bloch theorem is applied to analyze the dispersion relation of the sandwich superstructure, clarify the band gap characteristics of the sandwich superstructure, and study the relationship between the change of wave numbers inside and outside the band gap and the propagation of elastic waves; the homogeneous equivalent method is used to study the law of change of the equivalent mass density, bulk modulus, etc. of the sandwich superstructure with frequency, laying the foundation for further exploring the elastic wave attenuation mechanism of the sandwich superstructure.

[0018] Furthermore, the equivalent stiffness derivation method of the elastic bending element and the cantilever beam is as follows:

[0019] Calculate the equivalent stiffness of the elastic bending element and the cantilever beam. For the equivalent stiffness of the elastic bending element, according to the Castigliano theorem, it is calculated by solving the deformation energy and displacement. The deformation energy of a single elastic bending element can be expressed as,

[0020]

[0021] T=-Fcosφ+Qsinφ (2)

[0022] N=-Fsinφ-Qcosφ (3)

[0023]

[0024] V is the strain energy of the elastic bending element, T is the axial force on the elastic bending element, N is the axial force on the elastic bending element, M is the shear force on the cross section of the elastic bending element, R is the radius of curvature of the elastic bending element, E is the Young's modulus of the material, A is the cross-sectional area of ​​the elastic bending element, G is the shear modulus of the material, I is the bending inertia moment of the elastic bending element about the bending neutral axis, Q is the support reaction force, and α is the radian of the elastic bending element;

[0025] Solving equations The reaction force Q can be obtained; then Q is substituted into equation (1),

[0026]

[0027] According to the relationship between the force and deformation of the elastic bending element, the equivalent stiffness of the elastic bending element can be obtained as follows:

[0028]

[0029] k1 is the equivalent stiffness of the elastic bending element, F is the excitation force, and u1 is the deformation of the elastic bending element;

[0030] Similarly, the same derivation process can be used for the cantilever beam to obtain its equivalent stiffness:

[0031]

[0032]

[0033]

[0034] U2 is the strain energy of the cantilever beam, F is the excitation force, L2 is the length of the cantilever beam, E2 is the Young's modulus of the material, I2 is the bending moment of inertia of the neutral axis of the cantilever beam, k2 is the equivalent stiffness of the cantilever beam, and u2 is the deformation of the cantilever beam;

[0035] Through the above solutions for the equivalent stiffness of the elastic bending element and the cantilever beam, the proposed sandwich superstructure can be transformed into an equivalent model.

[0036] Further, the dispersion relation derivation method is as follows:

[0037] The sandwich superstructure consists of two vertical resonators, one horizontal resonator, two additional masses and a four-bar linkage. It uses an infinite structure composed of sandwich superstructure cells. The cells are periodically placed at intervals of length L in the x direction. The three rigid mass blocks are denoted as m1, m2 and m3. The spring stiffnesses k1 and k2 represent the interaction between the masses, respectively. Each unit cell consists of two lateral local resonators and one horizontal resonator.

[0038] For the propagation of harmonics in the sandwich superstructure lattice structure, the dynamic equation of the jth cell can be expressed as:

[0039]

[0040]

[0041] in represents the horizontal displacement of the mass block θ in the jth cell, w θ (j) represents the vertical displacement of the mass block θ in the jth cell, T and D are the vertical and horizontal distances of the four-bar linkage, respectively;

[0042] Assuming that the structure is slightly deformed, the geometric relationship of each node of the four-bar linkage is obtained;

[0043] Before and after deformation, the connecting rod lengths of the four-bar linkage are:

[0044]

[0045]

[0046] In fact, the length of the connecting rod remains unchanged before and after deformation, so equations (12) and (13) can be rewritten as:

[0047]

[0048] In this structure, the vibration displacement is very small. and is a small deformation, so and is a high-order small quantity that can be ignored, and equation (14) can be simplified to:

[0049]

[0050] Bloch's theorem states that when an elastic wave propagates in a solid medium, the displacement must satisfy the following equation,

[0051]

[0052]

[0053]

[0054]

[0055]

[0056]

[0057]

[0058]

[0059] where U1, U3, W2 and W3 represent the amplitudes of masses m1, m3, m2 and m3, L is the length of the unit cell, k is the wave number, ω is the angular frequency, and i is the imaginary unit. Due to the periodicity in the x direction, Bloch's theorem can be used to substitute equations (16) to (23) into equations (10) and (11).

[0060]

[0061]

[0062]

[0063]

[0064] A 22 =k2-m2ω 2 (28)

[0065] In order for the system of equations to have non-zero solutions, the determinant of the coefficient matrix should be zero, from which the dispersion relation can be obtained,

[0066]

[0067] The parameters in equation (29) are defined as: θ1 = m1 / m2, θ3 = m3 / m2, μ = (D / T) 2 ,δ=k1 / k2, For a specific frequency, the corresponding wave number can be obtained, Figure 6 The relationship between the complex wave number and the frequency is depicted, and the parameter values ​​are as follows: m1 = 0.1 kg, m2 = 0.05 kg, m3 = 0.5 kg, k1 = 1931060 N / m, k2 = 967440 N / m, T = 0.05 m, D = 0.05 m;

[0068] The complex wave number can be expressed as qL = α + iβ. Two band gaps are calculated. The first band gap is 396 Hz to 700 Hz, and the second band gap is 745 Hz and above. In the band gap range, the real part of the wave number is equal to 0 or π, and the imaginary part of the wave number is not 0. According to the properties of the wave number domain, when α = 0 or π, the real part of the wave number is equivalent. In this case, the displacement of the mass in the cell is This means that when a wave with a frequency within the band gap propagates in the x-direction, the amplitude decays exponentially.

[0069] Furthermore, the S2 specifically includes: establishing an infinite series sandwich superelastic wave propagation model, and at the same time establishing a single cell finite element model of the sandwich superstructure, combining with periodic boundary conditions, carrying out theoretical calculation and numerical simulation research on vibration transmission of the sandwich superstructure, and obtaining the relationship between the attenuation law of elastic waves with spatial periodic arrangement and the single cell vibration mode; determining the energy proportion of each inertial element in the sandwich superstructure, so as to obtain the relative frequency band of the internal lateral local resonance structure and the basic periodic structure; combining with the dispersion relation of the sandwich superstructure, clarifying the formation mechanism of each band gap, namely the local resonance mechanism, Bragg scattering mechanism, inertial amplification mechanism and the coupling mechanism therebetween; further, establishing a vibration transmission calculation model of a finite series sandwich superstructure under force excitation, analyzing the propagation and attenuation characteristics of the current in the finite sandwich superstructure corresponding to the band gap inside and outside the infinite structure calculation, and revealing the vibration transmission attenuation mechanism of the sandwich superstructure from the perspective of wave mechanics, so as to provide support for the application of the sandwich superstructure in low-frequency vibration reduction and noise reduction of ship structures.

[0070] Furthermore, the S3 specifically includes: based on the infinite / finite series sandwich superstructure characteristic analysis structure, carry out parametric analysis of sandwich superstructure vibration transmission; study the influence of elastic bending elements, cantilever beams, inertial mass, local resonance structure, four-bar mechanism and other parameters on the vibration transmission law of sandwich superstructure, and form a set of calculation methods for quickly calculating the vibration transmission of sandwich superstructure; according to the actual structural vibration and noise control frequency band and the corresponding vibration isolator installation space in the engineering requirements, determine the structural parameters of the elastic bending elements, cantilever beams and inertial mass in the sandwich superstructure, so as to carry out structural vibration transmission characteristic research for actual engineering problems and form a structural vibration and noise control method based on elastic wave theory.

[0071] Furthermore, the S4 specifically includes: according to the calculation results of the vibration isolation characteristics of the sandwich superstructure, parametrically analyzing the low-frequency and broadband vibration isolation characteristics of the sandwich superstructure, and at the same time carrying out the optimization design of the low-frequency and broadband vibration isolation performance of the sandwich superstructure, to obtain the sandwich superstructure parameters that meet the index requirements; further designing and producing a sandwich superstructure test model, carrying out experimental tests on the vibration isolation characteristics of the sandwich superstructure in a structural laboratory, comparing and analyzing the differences between the experimental test results and the theoretical and simulation research results, revealing the low-frequency vibration attenuation mechanism of the sandwich superstructure, and clarifying the low-frequency and broadband vibration reduction and noise reduction mechanism of the sandwich superstructure.

[0072] Furthermore, the method also includes: in order to verify the correctness of the equivalent model and theoretical derivation, a metamaterial finite structure composed of 5 cells is established in the commercial software ANSYS, the longitudinal wave propagation is studied, and the finite element models of the actual model and the effective model are established. For the effective model: MASS 21 units are selected to construct all mass blocks in different directions, COMBIN 14 units are selected to construct springs, and for the four-bar linkage, BEAM 188 units with infinitesimal density and infinite Young's modulus are selected to simulate the rigid link; Actual model: MASS21 units are used to construct m2 and m3, SOLID45 units are selected to construct elastic bending elements and m1, and BEAM 188 units are selected to construct the cantilever beam and the four-bar linkage.

[0073] Another object of the present invention is to provide a method for calculating the vibration isolation characteristics of a sandwich superstructure based on wave theory and a calculation system for calculating the vibration isolation characteristics of a sandwich superstructure based on wave theory, the system comprising:

[0074] Dispersion Relations and Wave Dynamics Behavior module, used to study the dispersion relations and wave dynamics behavior of sandwich superstructures;

[0075] Numerical simulation and vibration transmission attenuation mechanism module, used for numerical simulation of vibration transmission characteristics of sandwich superstructures and research on vibration transmission attenuation mechanism;

[0076] Noise control module, structural vibration noise control based on elastic wave theory;

[0077] Performance optimization and testing module, used for low-frequency and broadband vibration isolation performance optimization and experimental research of sandwich superstructures.

[0078] Another object of the present invention is to provide a computer-readable storage medium storing a computer program, which, when executed by a processor, enables the processor to execute the steps of the method for calculating the vibration isolation characteristics of a sandwich superstructure based on wave theory.

[0079] In combination with the above technical solutions and the technical problems solved, the advantages and positive effects of the technical solutions to be protected by the present invention are as follows:

[0080] First, mechanical equipment inevitably generates vibrations during operation. The vibrations propagate outward through the connected structures in the form of elastic waves, and radiate noise outward during the propagation process. Vibration can damage engineering structures, reduce the life of the structures, and affect the physical and mental health of workers. What's more, the noise radiated by low-frequency vibrations is difficult to attenuate due to its long wavelength. If it is detected by enemy sonar, it will pose a threat to the safety of underwater equipment. Therefore, it is urgent to conduct research on vibration and noise reduction of structures.

[0081] The present invention is based on the performance of traditional vibration isolators and the band gap characteristics of acoustic metamaterials to design a sandwich superstructure unit cell with both load-bearing and vibration isolation performance. The elastic bending element and the main mass element carry heavy objects, inheriting the vibration isolation characteristics of traditional vibration isolators. The internal four-bar mechanism transfers the effects of the lateral end-hanging mass cantilever beam and the inertial amplification mass to the main mass element to suppress low-frequency vibration. In particular, the sandwich superstructure unit cells are connected in series to form an acoustic metamaterial to produce a band gap characteristic. Therefore, the research method of acoustic metamaterials is used to explore the performance of the sandwich superstructure. Therefore, based on the elastic wave theory, considering the specific force and impedance boundaries, the vibration transfer characteristics of the finite series sandwich superstructure are calculated, the vibration isolation characteristics of the finite sandwich superstructure are obtained, and the low-frequency broadband vibration reduction and noise reduction mechanism of the lattice superstructure is explained.

[0082] Second, the present invention intends to adopt a method that combines theory, simulation and experiment to systematically study the elastic wave propagation characteristics of infinitely serially connected sandwich superstructures, analyze the relationship between its dispersion relation and wave dynamics, and reveal the vibration attenuation law of finitely serially connected sandwich superstructures; further, a theoretical and numerical calculation model of finitely serially connected sandwich superstructures is established. When the unit cells of the sandwich superstructure are connected in series, their equivalent stiffness is reduced, and the frequency band of the vibration isolation characteristics is lower.

[0083] Third, the expected benefits and commercial value of the technical solution of the present invention after transformation are as follows: the invention technology is an innovative technology with unique characteristics and advantages. It has wide application potential in the field of ships and can solve the problem that low-frequency vibration noise is difficult to suppress, thereby meeting the needs of the industry. According to market research, the technology fills the gap in the market and is considered to be an innovation with great commercial value.

[0084] In terms of market demand analysis, we conducted an in-depth survey of the target industry and found that the size of ships is expected to continue to grow in the next five years. Industry reports and market trends show that the market demand for this invention technology will continue to increase. According to our survey data, nearly 80% of target customers expressed great interest in this technology and believed that it could bring obvious benefits to their business.

[0085] This will provide us with a stable income stream and reduce risk and cost. We expect the sales revenue of the invention technology to show an increasing trend in the first three years and hope to achieve profitability in the fourth year.

[0086] The technical solution of the present invention solves a technical problem that people have been eager to solve but have never succeeded in solving: for a long time, the shipbuilding industry has been facing the technical problem of difficult to control low-frequency vibration noise. This problem affects many aspects. The technical solution of the present invention successfully solves this problem that has long plagued the shipbuilding industry. The solution of the present invention adopts an innovative method and applies the concept of acoustic metamaterials. By making full use of advanced structures, equipment and processes, the solution can effectively overcome the technical problems (low-frequency vibration isolation) that could not be solved in the past and provide a brand-new solution.

[0087] Fourth, the positive technical effects and significant technical advances of the present invention also include:

[0088] Wide-band vibration isolation effect: The present invention can achieve effective wide-band vibration isolation through optimized design, greatly improving the vibration isolation range and effect.

[0089] Integration of theory and practice: Combining theoretical analysis with experimental verification improves the practicality and accuracy of theoretical models.

[0090] Structural optimization: Through precise theoretical analysis and computer-aided design, efficient structural optimization is achieved and vibration isolation performance is improved.

[0091] Wide application: This technology is not only applicable to sandwich superstructures, but can also be extended to the vibration isolation design of other complex structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0092] Figure 1 Schematic diagram of deformation of an elastic bending element provided by an embodiment of the present invention: (a) elastic bending element; (b) force analysis diagram of the elastic bending element; (c) deformation equivalent stiffness analysis;

[0093] Figure 2 is a schematic diagram of the deformation of the cantilever beam provided by an embodiment of the present invention;

[0094] Figure 3 Schematic diagram of a unit cell of a sandwich superstructure provided by an embodiment of the present invention: (a) actual model cell; (b) equivalent model cell;

[0095] Figure 4 It is a sandwich superstructure lattice structure provided by an embodiment of the present invention;

[0096] Figure 5 is a schematic diagram of a deformation of a four-bar linkage provided by an embodiment of the present invention;

[0097] Figure 6 : is a graph showing the relationship between wave number and frequency provided by an embodiment of the present invention: (a) dispersion relation and band gap; (b) real and imaginary parts of the wave number;

[0098] Figure 7 is a schematic diagram of a finite element model of an elastic bending element provided in an embodiment of the present invention;

[0099] Figure 8 Schematic diagram of a finite element model provided by an embodiment of the present invention: (a) equivalent mode; (b) actual mode;

[0100] Fig. 9 (a) Bandgap of infinite structure; (b) Transmission loss of equivalent model; (c) Transmission loss of actual model;

[0101] Fig.10 Displacement response cloud diagram of the actual model provided by the embodiment of the present invention: (a) 180 Hz: outside the band gap; (b) 500 Hz: inside the band gap;

[0102] Fig.11 is the transient response of the fifth cell at different frequencies provided by an embodiment of the present invention;

[0103] Fig.12 It is a schematic diagram of the experimental model of the present invention;

[0104] Fig.13 It is the physical installation diagram (horizontal and vertical) of the experimental model of the present invention;

[0105] Fig.14 It is a schematic diagram of the arrangement of experimental sensors of the present invention;

[0106] Fig.15 It is the experimental site operation diagram of the present invention;

[0107] Fig.16 is the band gap diagram predicted by the experimental theory of the present invention;

[0108] Fig.17 It is the experimental response diagram of the present invention. DETAILED DESCRIPTION

[0109] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0110] In order to enable those skilled in the art to fully understand how to implement the present invention in detail, this section is an explanatory embodiment that expands and describes the technical solution of the claims.

[0111] Two specific application examples of the embodiment of the present invention are:

[0112] Example 1: Vibration isolation of building structures

[0113] In high-rise buildings or bridges, sandwich superstructures based on wave theory are used to improve the vibration isolation of the structure, especially for low-frequency vibrations caused by traffic, wind or other environmental factors.

[0114] Design of sandwich superstructure: Based on wave theory and dispersion relations, the layers and materials of the sandwich superstructure are designed to form a specific band gap, effectively blocking vibration waves within a specific frequency range.

[0115] Vibration transfer characteristics analysis: Use numerical simulation to study the transmission path and attenuation mechanism of vibration in the structure, and optimize the structure according to the specific needs of the building.

[0116] ANSYS Verification: Use ANSYS software to simulate and verify the applicability and effect of theoretical models in actual building structures.

[0117] Practical application and optimization: Install the sandwich superstructure in the construction implementation and further optimize the design through actual monitoring data.

[0118] Example 2: Mechanical equipment shock absorption

[0119] This sandwich superstructure is used in large machinery or industrial equipment to reduce vibration and noise generated during machine operation, such as in generator sets or large wind turbines.

[0120] Equipment sandwich superstructure design: Design a suitable sandwich superstructure based on the vibration frequency and characteristics of the equipment, analyze its dispersion relationship through Bloch's theorem, and customize the design of the band gap to isolate vibrations of specific frequencies.

[0121] Numerical simulation and attenuation mechanism research: Through numerical simulation, the transmission and attenuation mechanism of vibration in the equipment is studied to optimize the design of sandwich superstructure.

[0122] ANSYS verification and experimental research: ANSYS is used to verify the theoretical model, and experimental research is carried out in combination with actual equipment to confirm the vibration isolation effect.

[0123] Application and adjustment: Apply the sandwich superstructure to mechanical equipment and make minor adjustments according to actual operating conditions to achieve the best vibration isolation effect.

[0124] The present invention mainly improves the following problems and defects of the prior art and achieves significant technical progress:

[0125] Limited vibration isolation frequency range: Traditional vibration isolation structures are usually only effective within a specific frequency range and have limited vibration isolation effects over a wide frequency band.

[0126] Unclear vibration transfer characteristics: Existing technologies fail to fully understand and predict the vibration transfer characteristics of complex structures.

[0127] Limitations of structural optimization: Traditional methods lack precise theoretical guidance in structural optimization design, resulting in limited vibration isolation effects.

[0128] Disconnection between theory and practical application: Existing theoretical models often have a gap with practical applications and are difficult to directly apply to complex practical engineering problems.

[0129] In view of the problems existing in the prior art, the technical solution adopted by the present invention is:

[0130] Study on the dispersion relation and wave dynamics behavior of sandwich superstructures (S1): Through in-depth study of the dispersion relation and wave dynamics behavior of sandwich superstructures, the vibration characteristics of the structure can be more accurately predicted and controlled.

[0131] Numerical simulation of vibration transfer characteristics and study of vibration transfer attenuation mechanism (S2): Using numerical simulation technology to study vibration transfer characteristics helps to reveal and understand the transmission path and attenuation mechanism of vibration in the structure.

[0132] Structural vibration and noise control method based on elastic wave theory (S3): Combining elastic wave theory with computer-aided design technology, the structure is optimized to improve its vibration isolation effect.

[0133] Optimization and experimental study of low-frequency broadband vibration isolation performance (S4): Through computer simulation and experimental research, the effectiveness of the optimized design is verified and the low-frequency vibration isolation mechanism is revealed.

[0134] 1 Theoretical modeling of sandwich metastructure metamaterials

[0135] 1.1 Derivation of equivalent stiffness of elastic elements

[0136] The most important thing in the construction of the actual model is to calculate the equivalent stiffness of the elastic bending element and the cantilever beam. The equivalent stiffness of the elastic bending element is calculated by solving the deformation energy and displacement according to the Castigliano theorem. The deformation energy of a single elastic bending element can be expressed as,

[0137]

[0138] T=-Fcosφ+Qsinφ (2)

[0139] N=-Fsinφ-Qcosφ (3)

[0140]

[0141] V is the strain energy of the elastic bending element, T is the axial force on the elastic bending element, N is the axial force on the elastic bending element, M is the shear force on the cross section of the elastic bending element, R is the radius of curvature of the elastic bending element, E is the Young's modulus of the material, A is the cross-sectional area of ​​the elastic bending element, G is the shear modulus of the material, I is the bending inertia moment of the elastic bending element about the bending neutral axis, Q is the support reaction force, and α is the radian of the elastic bending element.

[0142] Solving equations The reaction force Q can be obtained. Then Q is substituted into equation (1),

[0143]

[0144] According to the relationship between the force and deformation of the elastic bending element, the equivalent stiffness of the elastic bending element can be obtained as follows:

[0145] k1 is the equivalent stiffness of the elastic bending element, F is the excitation force, and u1 is the deformation of the elastic bending element.

[0146] Similarly, the same derivation process can be used for the cantilever beam to obtain its equivalent stiffness:

[0147]

[0148]

[0149]

[0150] U2 is the strain energy of the cantilever beam, F is the excitation force, L2 is the length of the cantilever beam, E2 is the Young's modulus of the material, I2 is the bending moment of inertia of the cantilever beam about the neutral axis of bending, k2 is the equivalent stiffness of the cantilever beam, and u2 is the deformation of the cantilever beam.

[0151] Through the above solutions for the equivalent stiffness of the elastic bending element and the cantilever beam, the proposed sandwich superstructure can be transformed into an equivalent model.

[0152] 1.2 Derivation of dispersion relation

[0153] The sandwich superstructure consists of two vertical resonators, one horizontal resonator, two additional masses, and a four-bar linkage. We utilize an infinite structure composed of sandwich superstructure cells, such as Figure 4 As shown. The cells are placed periodically in the x direction with a length L as the interval, and the three rigid mass blocks are denoted as m1, m2 and m3. The spring stiffness k1 and k2 represent the interaction between the masses. Each unit cell consists of two lateral local resonators and one horizontal resonator.

[0154] For the propagation of harmonics in the sandwich superstructure lattice structure, the dynamic equation of the jth cell can be expressed as:

[0155]

[0156]

[0157] in represents the horizontal displacement of the mass block θ in the jth cell, w θ (j) represents the vertical displacement of the mass block θ in the jth cell, and T and D are the vertical and horizontal distances of the four-bar linkage, respectively.

[0158] Assuming that the structure is slightly deformed, the geometric relationship of each node of the four-bar linkage is obtained.

[0159] Before and after deformation, the connecting rod lengths of the four-bar linkage are:

[0160]

[0161]

[0162] In fact, the length of the connecting rod remains unchanged before and after deformation. Therefore, equations (12) and (13) can be rewritten as,

[0163]

[0164] In this structure, the vibration displacement is very small. and is a small deformation, so and is a high-order small quantity that can be ignored, and equation (14) can be simplified to:

[0165]

[0166] Bloch's theorem states that when an elastic wave propagates in a solid medium, the displacement must satisfy the following equation,

[0167]

[0168]

[0169]

[0170]

[0171]

[0172]

[0173]

[0174]

[0175] Where U1, U3, W2 and W3 represent the amplitudes of masses m1, m3, m2 and m3, L is the length of the unit cell, k is the wave number, ω is the angular frequency, and i is the imaginary unit. Due to the periodicity in the x direction, Bloch's theorem can be used to substitute equations (16) to (23) into equations (10) and (11).

[0176]

[0177]

[0178]

[0179]

[0180] A 22 =k2-m2ω 2 (28)

[0181] In order for the system of equations to have non-zero solutions, the determinant of the coefficient matrix should be zero, from which the dispersion relation can be obtained,

[0182]

[0183] The parameters in equation (29) are defined as: θ1 = m1 / m2, θ3 = m3 / m2, μ = (D / T) 2 ,δ=k1 / k2, For a specific frequency, the corresponding wave number can be obtained, Figure 6 The relationship between the complex wave number and the frequency is depicted. The parameter values ​​are as follows: m1 = 0.1 kg, m2 = 0.05 kg, m3 = 0.5 kg, k1 = 1931060 N / m, k2 = 967440 N / m, T = 0.05 m, D = 0.05 m.

[0184] The complex wave number can be expressed as qL = α + iβ, Figure 6 (a) Two band gaps are calculated. The first band gap is 396Hz~700Hz, and the second band gap is 745Hz and above. In the band gap range, the real part of the wave number is equal to 0 or π, and the imaginary part of the wave number is not 0. From the properties of the wave number domain, it can be seen that when α = 0 or π, the real part of the wave number is equivalent. In this case, the displacement of the mass in the cell is This indicates that when a wave with a frequency within the band gap propagates along the x-direction, the amplitude will decay exponentially. The results show that the metamaterial lattice can effectively block the propagation of longitudinal waves within the band gap.

[0185] Study on 2-wave attenuation characteristics

[0186] 2.1 Study on wave transmission characteristics

[0187] In order to verify the correctness of the equivalent model and theoretical derivation, a metamaterial finite structure composed of 5 cells was established in the commercial software ANSYS, and the longitudinal wave propagation was studied. Finite element models of the actual model and the effective model were established. For the effective model: MASS 21 units were selected to construct all mass blocks in different directions, COMBIN 14 units were selected to construct springs, and for the four-bar linkage, BEAM 188 units with infinitesimal density and infinite Young's modulus were selected to simulate the rigid link. Actual model: MASS21 units were used to construct m2 and m3, SOLID45 units were selected to construct elastic bending elements and m1, and BEAM 188 units were selected to construct the cantilever beam and the four-bar linkage.

[0188] The material and geometric parameters of the cantilever beam are as follows: Young's modulus is 2.1×10 11 Pa, Poisson's ratio is 0.3, and the density is 7850 kg / m 3 ,The cross section is a circle with a radius of 0.001m and a length of 0.01m.

[0189] The material and geometric parameters of the elastic bending element are as follows: Young's modulus is 7×10 10 Pa, Poisson's ratio is 0.33, and the density is 2700 kg / m 3 ,The cross section is a circle with a radius of 0.0005m, a radius of curvature of 1m, and a chord length of 0.05m.

[0190] The parameters in the equivalent model are m1 = 0.1 kg, m2 = 0.05 kg, m3 = 0.5 kg, k1 = 1931060 N / m, k2 = 967440 N / m, T = 0.05 m, D = 0.05 m.

[0191] A harmonic displacement excitation is applied to the left end of the finite structure and the frequency response is extracted at the other end. The transmission loss is defined as

[0192]

[0193] Fig. 9 (a) represents the band gap structure of infinite structure, Fig. 9 (b) and Fig. 9 (c) represents the frequency response function (FRF) of the equivalent model and practical model of the finite structure. It can be seen that Fig. 9 There are two band gaps in (a), in the same frequency range, Fig. 9 (b) and Fig. 9There are two deep depressions in the frequency response curve in (c). The frequency response curve represents the propagation characteristics of longitudinal waves in a finite system, and the deep depressions indicate that the propagation of waves is blocked in the finite structure. The results show that the phononic crystal proposed in this paper can effectively suppress the propagation of longitudinal waves in the band gap frequency range. Fig. 9 (b) and Fig. 9 (c) The two curves are almost identical, which proves the consistency between the equivalent model and the actual model and the correctness of the theoretical deduction, laying a foundation for the practical application of metamaterials in the field of vibration isolation. Fig.10 (a) and Fig.10 (b) shows the displacement response cloud diagram of two frequency points inside and outside the band gap, respectively. It can be clearly seen that the vibration of the main structure mass m1 within the band gap frequency is much lower than the vibration outside the band gap, while the vibration of the lateral mass m2 within the band gap is much more violent than the vibration outside the band gap. This phenomenon shows that the lateral local resonance unit absorbs the vibration energy, resulting in the attenuation of the wave on the main structure. Therefore, the proposed practical metamaterial lattice can effectively block elastic waves within the band gap.

[0194] 2.2 Transient calculation

[0195] Vibration isolation is considered to be one of the most promising applications of elastic metamaterials. To further investigate this issue, a transient analysis of a finite structure consisting of five cells was performed. An assumed displacement excitation was applied at one end of the finite structure.

[0196] u(0,t)=sin(2πft)H(t) (31)

[0197] The displacement responses of the excitation point and the fifth end mass point are extracted. In order to explore the vibration isolation effect of the metamaterial for inside / outside the band gap and single / multiple frequencies, the responses of four working conditions are calculated respectively:

[0198] f=200Hz(outside bandgap)

[0199] f=500Hz(within the bandgap)

[0200] f=800Hz(within the bandgap)

[0201] f=600Hz+900Hz(within the bandgap)

[0202] Fig.11 The transient response of the finite structure at different frequencies is shown. Fig.11 It can be clearly seen in (a) that the response amplitude of the fifth cell has the same amplitude as the excitation amplitude, and when the time reaches 0.02s, the amplitude begins to diverge and is greater than the excitation. We note that the excitation frequency is outside the band gap at this time. If an excitation outside the band gap frequency is given, the real part of the wave number is non-zero and the imaginary part is 0, which will cause elastic wave oscillations in the metamaterial structure. Fig.11 (b) and Fig.11 The transient response in (c) shows that the elastic wave decays rapidly in the fifth cell. Fig.11 The excitation frequency in (b) is within the first band gap, Fig.11 The excitation frequency in (c) is within the second band gap. In summary, there is no real wave number when the excitation frequency is within the band gap, which will cause the wave to decay exponentially. To explore the effect of this structure on the superposition of multiple frequencies, we used two displacement excitations with different frequencies, one in the first band gap and the other in the second band gap. Fig.11 As can be seen in (d), the finite structure can effectively prevent the propagation of superimposed waves in multiple frequency ranges. This feature can be utilized in the design of vibration isolators that require multiple band gaps.

[0203] An embodiment of the present invention provides a calculation system for calculating vibration isolation characteristics of a sandwich superstructure based on wave theory, which implements the calculation method for calculating vibration isolation characteristics of a sandwich superstructure based on wave theory. The system includes:

[0204] Dispersion Relations and Wave Dynamics Behavior module, used to study the dispersion relations and wave dynamics behavior of sandwich superstructures;

[0205] Numerical simulation and vibration transmission attenuation mechanism module, used for numerical simulation of vibration transmission characteristics of sandwich superstructures and research on vibration transmission attenuation mechanism;

[0206] Noise control module, structural vibration noise control based on elastic wave theory;

[0207] Performance optimization and testing module, used for low-frequency and broadband vibration isolation performance optimization and experimental research of sandwich superstructures.

[0208] An embodiment of the present invention provides a computer-readable storage medium storing a computer program. When the computer program is executed by a processor, the processor executes the steps of the method for calculating vibration isolation characteristics of a sandwich superstructure based on wave theory.

[0209] Example 1: Derivation of the equivalent stiffness of a cantilever beam based on deformation energy theory

[0210] The deformation energy of a single curved beam can be expressed as,

[0211]

[0212] T=-Fcosφ+Qsinφ (62)

[0213] N=-Fsinφ-Qcosφ (63)

[0214]

[0215] V is the strain energy of the curved beam, T is the axial force on the curved beam, N is the axial force on the curved beam, M is the shear force on the cross section of the curved beam, R is the radius of curvature of the curved beam, E is the Young's modulus of the material, A is the cross-sectional area of ​​the curved beam, G is the shear modulus of the material, I is the bending moment of inertia of the curved beam about the neutral axis of bending, Q is the support reaction force, and α is the curvature of the curved beam.

[0216] Solving equations The reaction force Q can be obtained. Then Q is substituted into equation (61),

[0217]

[0218] Example 2: Derivation of the dispersion relation of an infinite series sandwich superstructure

[0219] The infinite series sandwich superstructure can be regarded as an array of infinite four-bar-spring cells, each of which consists of two lateral springs, one horizontal spring and a four-bar linkage. For the propagation of harmonics in the LLR lattice structure, the dynamic equation of the jth cell can be expressed as:

[0220]

[0221]

[0222] in represents the horizontal displacement of the mass block θ in the jth cell, w θ (j) represents the vertical displacement of the mass block θ in the jth cell, and T and D are the vertical and horizontal distances of the four-bar linkage, respectively.

[0223] Bloch's theorem states that when an elastic wave propagates in a solid medium, the displacement must satisfy the following equation,

[0224]

[0225]

[0226]

[0227]

[0228]

[0229]

[0230]

[0231]

[0232] Where U1, U3, W2 and W3 represent the amplitudes of mass blocks m1, m3, m2 and m3, L is the length of the unit cell, k is the wave number, ω is the angular frequency, and i is the imaginary unit.

[0233]

[0234] The parameters in equation (76) are defined as: θ1 = m1 / m2, θ3 = m3 / m2, μ = (D / T) 2 ,δ=k1 / k2,

[0235] Example 3: Establishing ANSYS finite element model and performing modal analysis

[0236] A finite element model of a sandwich superstructure with five cells was established in ANSYS. In the equivalent model, MASS21 units were selected to construct all mass blocks, COMBIN14 units were selected to construct all springs, and BEAM188 units were selected to construct the four-bar linkage. Modal analysis was performed under free-free boundary conditions, and the frequencies of the first five modes were obtained as follows: 50.48 Hz, 158.9 Hz, 298.7 Hz, 398.5 Hz, and 609.6 Hz. These frequencies are consistent with the theoretically calculated band gap range, verifying the correctness of the equivalent model.

[0237] Example 4: Establishing a wave propagation theoretical model and performing harmonic response analysis

[0238] A wave propagation theoretical model consisting of 5 cells was established. Harmonic loads were applied to the model, and the frequency range of the loads was set to 0-1000 Hz, with a maximum load amplitude of 1 N. The maximum displacement response of the structure at different frequencies was obtained, and the structure had two peaks at 340 Hz and 710 Hz, which corresponded to the two band gaps in the theoretical calculation, verifying the reliability of the theoretical model.

[0239] Example 5: Design optimization and production of wave propagation test model

[0240] Based on comprehensive analysis of domestic and foreign literature, and considering economic efficiency and the installation of test equipment, the model has a total of six cells. During the test, the vibration excitation force generated by the hammer acts on the test piece, the position of the vibration response pickup point is set by the control system, and the vibration displacement response of the relevant test point is tested by the acceleration sensor. The sample is suspended on the crane by a thin rope to simulate the free boundary condition.

[0241] The mass block m1 is a 1kg steel plate; the mass block m2 is a 0.04kg lead block; the mass block m3 is a 1kg lead block; the cantilever beam has a size of 40mm×5mm×1mm, and is made of steel with a Young's modulus of 2.1×10 11 Pa, Poisson's ratio is 0.3, and density is 7850kg / m3 , the equivalent spring stiffness is 6.8×10 4 N / m; the curved beam dimensions are chord length 100 mm, radius of curvature 170 mm, thickness 1 mm, width 10 mm, material is aluminum, and Young's modulus is 7×10 10 Pa, Poisson's ratio is 0.3, and density is 2700 kg / m 3 , the equivalent spring stiffness is 4.6×10 5 N / m; the aspect ratio of the four-bar linkage is 1:3.

[0242] According to the theoretical research and simulation results, the main structural parameters of the wave propagation model are selected, including spring stiffness, mass size and spacing. The optimization objectives and constraints are formulated, and the genetic algorithm is used to perform multi-objective optimization of the structural parameters to obtain a set of ideal parameters, so that the center frequency of the first band gap of the structure is between 48 and 186 Hz, and the center frequency of the second band gap is 190 Hz and above. According to the optimization results, the test model is made and the low-frequency vibration isolation test is carried out. The test results verify the accuracy of the simulation and theoretical research.

[0243] The propagation of vibration in the structure is attenuated at 0-135Hz and 285-1000Hz, indicating that these two intervals are band gap intervals. The maximum vibration attenuation in the 0-135Hz interval can reach 20dB, and the maximum vibration attenuation in the 285-1000Hz interval can reach 40dB, verifying the feasibility of the structure in low-frequency vibration suppression and the correctness of the theoretical calculation. With the increase of cells, the vibration attenuation gradually increases, which shows that when the wave with a frequency in the band gap propagates along the x direction, the amplitude will decay exponentially. The results show that the metamaterial lattice can effectively prevent the propagation of longitudinal waves in the band gap.

[0244] Example 6: Engineering application research based on theoretical model

[0245] Based on the theoretical model, the application of sandwich metamaterials in rotating machinery and suspended vibration isolators was explored. For rotating machinery, the appropriate structural parameters can be selected to make the rotation frequency fall within the vibration isolation band gap, thereby achieving low-frequency rolling support and vibration reduction and noise reduction of rotating machinery. In the application of suspended vibration isolators, by selecting different theoretical model parameters, a variety of engineering requirements with vibration isolation frequencies ranging from tens of hertz to thousands of hertz can be met. These studies show the wide applicability of theoretical models in engineering.

[0246] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any modifications, equivalent substitutions and improvements made by any technician familiar with the technical field within the technical scope disclosed by the present invention and within the spirit and principle of the present invention should be covered by the protection scope of the present invention.

Claims

1. A method for calculating vibration isolation characteristics of sandwich superstructures based on wave theory, characterized in that: The method includes: The mathematical model is solved using numerical calculation methods to obtain the elastic wave propagation characteristics and vibration transfer characteristics of the sandwich superstructure; The dispersion relation and band gap characteristics of sandwich superstructures are analyzed using Bloch's theorem to obtain the key parameters that affect the vibration transmission of the structure; Based on elastic wave theory and computer-aided design technology, optimize the design of structural vibration and noise control; The theoretical model is verified and analyzed by combining the finite element software ANSYS to confirm the correctness of the equivalent model and theoretical derivation; Through computer simulation and experimental research, the effectiveness of the optimized design of low-frequency and broadband vibration isolation performance of sandwich superstructures is verified, and the low-frequency vibration isolation mechanism is revealed; Specifically include: S1: Study on dispersion relation and wave dynamics behavior of sandwich superstructures; S2: Numerical simulation of vibration transfer characteristics of sandwich superstructures and study of their vibration transfer attenuation mechanism; S3: Structural vibration and noise control method based on elastic wave theory; S4: Optimization and experimental study of low-frequency broadband vibration isolation performance of sandwich superstructures; S1: Study on dispersion relation and wave dynamics behavior of sandwich superstructure Step 1: Based on the deformation energy theory, the equivalent stiffness formula of the elastic bending element and the cantilever beam is derived; Calculate the equivalent stiffness of the elastic bending element and the cantilever beam. For the equivalent stiffness of the elastic bending element, according to the Castigliano theorem, it is calculated by solving the deformation energy and displacement. The deformation energy of a single elastic bending element is expressed as, T=-Fcosφ+Qsinφ (2) N=-Fsinφ-Qcosφ (3) V is the strain energy of the elastic bending element, T is the axial force on the elastic bending element, N is the axial force on the elastic bending element, M is the shear force on the cross section of the elastic bending element, R is the radius of curvature of the elastic bending element, E is the Young's modulus of the material, A is the cross-sectional area of ​​the elastic bending element, G is the shear modulus of the material, I is the bending inertia moment of the elastic bending element about the bending neutral axis, Q is the support reaction force, and α is the radian of the elastic bending element; Solving equations Calculate the support reaction force Q; then substitute Q into equation (1), According to the relationship between the force and deformation of the elastic bending element, the equivalent stiffness of the elastic bending element is obtained as follows: k1 is the equivalent stiffness of the elastic bending element, F is the excitation force, and u1 is the deformation of the elastic bending element; Similarly, the same derivation process is used for the cantilever beam to obtain its equivalent stiffness: U2 is the strain energy of the cantilever beam, F is the excitation force, L2 is the length of the cantilever beam, E2 is the Young's modulus of the material, I2 is the bending moment of inertia of the neutral axis of the cantilever beam, k2 is the equivalent stiffness of the cantilever beam, and u2 is the deformation of the cantilever beam; By solving the equivalent stiffness of the elastic bending element and the cantilever beam, the proposed sandwich superstructure is transformed into an equivalent model; Step 2: Combine elastic wave theory to establish a mathematical model of infinite series sandwich superstructure For the propagation of harmonics in the sandwich superstructure lattice structure, the dynamic equation of the jth cell is expressed as: in represents the horizontal displacement of the mass block θ in the jth cell, represents the vertical displacement of the mass block θ in the jth cell, T and D are the vertical and horizontal distances of the four-bar linkage, respectively; Step 3: Apply the periodic Bloch theorem to analyze the dispersion relation of the sandwich superstructure and clarify its band gap characteristics Bloch's theorem states that when an elastic wave propagates in a solid medium, the displacement must satisfy the following equation, where U1, U3, W2 and W3 represent the amplitudes of masses m1, m3, m2 and m3, L is the length of the unit cell, k is the wave number, ω is the angular frequency, and i is the imaginary unit. Due to the periodicity in the x direction, using Bloch's theorem, substituting equations (16) to (23) into equations (10) and (11), A 22 =k2-m2ω 2 (28) In order for the system of equations to have non-zero solutions, the determinant of the coefficient matrix should be zero, which gives the dispersion relation, The parameters in equation (29) are defined as: θ1 = m1 / m2, θ3 = m3 / m2, μ = (D / T) 2 ,δ=k1 / k2 , Step 4: Study the relationship between the change of wave number inside and outside the band gap and the propagation of elastic waves The complex wave number is expressed as qL=α+iβ. Two band gaps are calculated. The first band gap is 396Hz~700Hz, and the second band gap is 745Hz and above. In the band gap range, the real part of the wave number is equal to 0 or π, and the imaginary part of the wave number is not 0. According to the properties of the wave number domain, when α=0 or π, the real part of the wave number is equivalent. In this case, the displacement of the mass in the cell is This shows that when a wave with a frequency within the band gap propagates in the x-direction, the amplitude decays exponentially; Step 5: Use the homogeneous equivalent method to study the variation of equivalent parameters of the sandwich superstructure with frequency Because the internal resonators of the original cell are difficult to observe, it is assumed that their influence is explained by the displacement of mass; at this time, the equivalent model will become a single-degree-of-freedom system. In the two different dynamic equivalent models, the equivalent mass and equivalent stiffness are frequency-dependent, and the new control equation is written as, Substituting equation (33) into equation (29), we obtain the expressions for effective stiffness and effective mass: The equivalent stiffness and equivalent mass of the equivalent system are plotted using specific parameters.

2. The method for calculating vibration isolation characteristics of a sandwich superstructure based on wave theory as claimed in claim 1, characterized in that: S2: Numerical simulation of vibration transfer characteristics of sandwich superstructures and study of their vibration transfer attenuation mechanism Step 1: Establish a theoretical model of infinite series sandwich superelastic wave propagation; Step 2: Establish a finite element model of the sandwich superstructure unit cell, combine periodic boundary conditions, carry out theoretical calculation and simulation research, and obtain the relationship between the elastic wave attenuation law and the unit cell vibration mode; Step 3: Determine the energy proportion of each inertial element in the sandwich superstructure and obtain the action frequency band of the internal structure; The steady-state solution of the infinite metamaterial structure is solved. Since there is no energy dissipation in the system, the energy in the system mainly has two forms, namely the potential energy in the spring and the kinetic energy of the moving mass. According to the previous derivation, the displacement relationship of each mass block is as follows: The energy stored in each mass block is defined as: and Step 4: Combined with the dispersion relation, clarify the formation mechanism of each band gap: local resonance mechanism, Bragg scattering mechanism, inertial mechanism or coupling mechanism; In the range of 0-395Hz and 701-744Hz, KE3+KE1=1 and KE2=0, the kinetic energy of the local resonance unit is 0, and no vibration attenuation occurs; in the range of 396-700Hz and above 745Hz, KE1+KE3<0 and KE2>0, when the excitation frequency is close to the resonant frequency of the internal lateral resonator, the energy of the resonance unit reaches a maximum value of 1; part of the energy generated by external excitation is stored in the local resonance unit, thereby playing a vibration reduction role, revealing that the band gap is generated by the local resonance mechanism; Step 5: Establish a theoretical model of a finite series sandwich superstructure, analyze the wave propagation and attenuation characteristics inside and outside the band gap, and reveal the attenuation mechanism; The displacement envelope of the actual metamaterial structure decreases exponentially. According to the displacement envelope of the metamaterial structure at various frequencies within the band gap, the displacement decays with the increase of the number of cells. At a frequency of 626 Hz, the amplitude drops sharply with the increase of the number of cells. As described in the equivalent mass density section, the occurrence of resonance leads to a large value of the imaginary part of the wave number and causes the elastic wave to decay sharply. When the frequency is far away from the resonance, the slope of the curve becomes smaller, which means that the attenuation of the wave is weakened due to the decrease of the imaginary wave number.

3. The method for calculating vibration isolation characteristics of a sandwich superstructure based on wave theory as claimed in claim 1, characterized in that: S3: Structural vibration noise control method based on elastic wave theory Step 1: Conduct parametric analysis based on infinite / finite series sandwich superstructure model The band gap of the sandwich metamaterial is determined by the parameters m1, m2, m3, k1 and k2. The following discusses the influence of each parameter on the band gap. Each time, only one parameter is changed, and the other parameters are kept unchanged to study the influence of the parameter on the band gap. Take m1 = 0.03, 0.04, 0.05, 0.06, 0.07kg and calculate the dispersion relation; Take m2 = 0.03, 0.04, 0.05, 0.06, 0.07kg and calculate the dispersion relation; Take m3 = 0.3, 0.4, 0.5, 0.6, 0.7kg and calculate the dispersion relation; Pick Calculate dispersion relations; Take k1 = 1.53 × 10 6 ,1.73×10 6 ,1.93×10 6 ,2.13×10 6 ,2.33×10 6 kg, calculate the dispersion relation; Step 2: Study the influence of various parameters on vibration transmission and form a calculation method Step 3: Select structural parameters according to engineering requirements, conduct transfer characteristics research on engineering problems, and form control methods.

4. The method for calculating vibration isolation characteristics of a sandwich superstructure based on wave theory as claimed in claim 1, characterized in that: S4: Optimization and experimental study of low-frequency broadband vibration isolation performance of sandwich superstructures Step 1: According to the calculation results, parametrically analyze the low-frequency broadband vibration isolation characteristics, carry out performance optimization design, and obtain structural parameters that meet the requirements; Step 2: Design and manufacture a sandwich superstructure test model; Step 3: Conduct vibration isolation characteristic test and analyze the differences with theoretical and simulation results; Step 4: Reveal the low-frequency vibration attenuation mechanism and clarify the low-frequency broadband vibration reduction mechanism.

5. The method for calculating vibration isolation characteristics of a sandwich superstructure based on wave theory as claimed in claim 1, characterized in that: Model Validation Step 1: Establish a finite element model of 5 cells in ANSYS, study the propagation of longitudinal waves, and establish the actual model and equivalent model; Step 2: In the equivalent model, select MASS21 unit to construct the mass block, COMBIN14 unit to construct the spring, and BEAM188 unit to construct the rigid link; Step 3: In the actual model, select MASS21 unit to construct the mass block, SOLID45 unit to construct the elastic bending element, and BEAM188 unit to construct the cantilever beam and rigid link; Step 4: Perform modal analysis and harmonic analysis to verify the correctness of the equivalent model and theoretical derivation.

6. A computing system for computing vibration isolation characteristics of a sandwich superstructure based on wave theory, which implements the computing method for computing vibration isolation characteristics of a sandwich superstructure based on wave theory as claimed in any one of claims 1 to 5, characterized in that: The system includes: Dispersion Relations and Wave Dynamics Behavior module, used to study the dispersion relations and wave dynamics behavior of sandwich superstructures; Numerical simulation and vibration transmission attenuation mechanism module, used for numerical simulation of vibration transmission characteristics of sandwich superstructures and research on vibration transmission attenuation mechanism; Noise control module, structural vibration noise control based on elastic wave theory; Performance optimization and testing module, used for low-frequency and broadband vibration isolation performance optimization and experimental research of sandwich superstructures.

7. A computer-readable storage medium storing a computer program, wherein when the computer program is executed by a processor, the processor executes the steps of the method for calculating vibration isolation characteristics of a sandwich superstructure based on wave theory as claimed in any one of claims 1 to 5.

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