A method of forming a filter in an acoustic metamaterial by introducing a localised resonant source

Abstract

This invention relates to the field of metamaterials technology, specifically a method for creating filtering in acoustic metamaterials by introducing localized resonant sources. The method includes: breaking symmetry in a two-dimensional five-mode classical biconical structure to form a biconical bandgap structure; establishing a two-dimensional acoustic metamaterial bandgap model with resonant sources based on the biconical bandgap structure; simplifying the dynamic behavior of the periodic structure from an infinite periodic array to a single-cell problem based on Bloch's theorem, and solving for the bandgap structure; deriving the eigenvalue problem of elastic wave propagation based on the elastic wave dispersion relation in the two-dimensional periodic structure and the bandgap structure; and adding localized resonant sources within the bandgap and embedding discrete defect modes within the bandgap to achieve narrowband-pass filtering. This invention adds localized resonant sources to the single cell, and by changing the geometric and material parameters of the localized resonant sources, the filtering range can be effectively altered, resulting in precise filtering at specific frequencies.

Description

Technical Field

[0001] This invention relates to the field of metamaterials technology, and in particular to a method for forming a filter in an acoustic metamaterial by introducing a local resonant source. Background Technology

[0002] Because traditional acoustic metamaterial filters have certain limitations, their underlying principle of achieving bandgap and introducing defect states for filtering relies on Bragg scattering. Therefore, the filter model of this type of acoustic metamaterial inevitably suffers from a series of limitations inherent in Bragg scattering crystals, such as:

[0003] (1) Size-limited (low frequency, large volume) Bragg band gap usually requires the size of the crystal unit cell (the distance between adjacent scatterers) to correspond to the wavelength of the target frequency. The lower the target frequency, the larger the lattice constant and the number of periods must be sufficient (generally, there is a significant band gap only when the number of periods is ≥8–12). This results in thick, heavy, and space-consuming devices, and it is difficult to form a band gap in the low frequency band.

[0004] (2) The finite size leads to the "shallowing of the ideal band gap". The band gap is an infinite periodic concept; the actual sample has only a finite period, which leads to boundary leakage and residual transmission in the stop band, resulting in a decrease in the suppression degree.

[0005] (3) The incident angle and polarization are sensitive, and the band gap changes significantly with the incident angle and polarization (longitudinal / transverse modes in acoustic / bullet); structural anisotropy will also result in angular leakage, which leads to poor stability of filtering in multi-angle / multi-mode environments.

[0006] Therefore, there is a need to provide a method for adding local resonant units to metamaterials with five-mode bandgap characteristics to form a filter within a certain frequency range. Summary of the Invention

[0007] To address the aforementioned technical problems, this invention provides a method for forming a filter in an acoustic metamaterial by introducing a local resonant source. The invention employs a method of first generating an acoustic bandgap in the acoustic metamaterial and then introducing defect states in the bandgap region to generate a filter. Symmetry breaking is performed on the five-mode unit to split the originally degenerate modes, forming a controllable bandgap (not dependent on pure Bragg conditions). A local resonant source is added within the bandgap, and discrete defect modes are embedded within the bandgap to achieve narrowband-pass filtering (or multi-channel filtering, through multiple tunable resonant sources).

[0008] The technical means employed in this invention are as follows: A method for forming a filter in an acoustic metamaterial by introducing a local resonant source includes: breaking symmetry in a two-dimensional five-mode classical biconical structure to form a biconical bandgap structure; establishing a two-dimensional acoustic metamaterial bandgap model with a resonant source based on the biconical bandgap structure; effectively simplifying the dynamic behavior of the periodic structure from an infinite periodic array to a single-cell problem based on Bloch's theorem, and solving for the bandgap structure; deriving the eigenvalue problem of elastic wave propagation based on the elastic wave dispersion relation in the two-dimensional periodic structure and the bandgap structure; adding a local resonant source within the bandgap and embedding discrete defect modes within the bandgap to achieve narrowband-pass filtering.

[0009] Furthermore, the symmetry breaking in the two-dimensional five-mode classical biconical structure specifically includes: in the two-dimensional five-mode classical biconical structure, all connecting arms within the unit cell have the same length and exhibit axial symmetry in both the horizontal and vertical directions; the two connecting points of the central arm at the central position are moved in opposite directions by the same distance d in the vertical direction. e A double-cone bandgap structure was obtained.

[0010] Furthermore, in the five-module unit cell of the biconical bandgap structure, each arm structure comprises two rectangles of width d and two trapezoidal portions, with the top width of the trapezoids being d and the bottom width being D. The arms are interconnected at the intersection of the extensions of their respective rectangular portions. Three connecting arms intersect to form a connection point, with an angle of 60° between adjacent connecting arms. The length of each connecting arm is consistent and L, the unit cell length is 3L, and the width is... L.

[0011] Furthermore, the establishment of the two-dimensional acoustic metamaterial band structure specifically includes: adding solid mechanics as the background field, setting periodic boundary conditions on the two parallel sides of the rectangular unit cell, using Floquet periodic boundary conditions, and distributing Bloch vector k on the boundary of the irreducible Brillouin zone; Based on the elastic dynamics equations, the elastic wave dispersion relation in a two-dimensional periodic structure can be further written as:

[0012] in, Represents the displacement vector; Represents a position vector; denoted by Hamiltonian differential operator; λ and G are Lamé constants; ρ represents mass density; Let be the second derivative of the displacement with respect to time.

[0013] Furthermore, the solution of the band structure specifically includes: based on Bloch's theorem, when periodic boundary conditions are applied to the unit cell, the displacement fields at all discrete nodes must satisfy the phase periodicity relationship of Bloch form, that is, the displacements differ by only one phase factor related to the wave vector after crossing the period:

[0014] in, Represents the displacement vector; Represents the wave vector; Represents lattice vectors; Represents angular frequency. Represents the imaginary unit; Indicates time.

[0015] Furthermore, the matrix form of the eigenvalue problem of elastic wave propagation is expressed as follows:

[0016] in, Represents the unit vector in the x-direction; Represents the unit vector in the y-direction; a and b and are the lattice constants in the x and y directions, respectively, used to construct the reciprocal lattice vector of the first Brillouin zone. The reciprocal lattice vector of the first Brillouin zone is defined as:

[0017] in, , These represent two vectors in the first Brillouin zone of the crystal lattice.

[0018] Furthermore, the addition of a local resonant source within the bandgap specifically includes: the geometry of the filter unit cell is consistent with that of the bandgap unit cell, and a local resonant unit is added at the center of the connecting arm in the middle, wherein the local resonant unit is composed of an outer rubber layer and an inner metal layer. After adding the local resonance source, waves generated solely by the principle of local resonance can be formed within a specific frequency band gap in a finite array unit cell. By changing the geometric and material parameters of the local resonance source, the filtering range can be altered, achieving precise filtering at specific frequencies.

[0019] Compared with the prior art, the present invention has the following advantages: The present invention provides a method for forming a filter in an acoustic metamaterial by introducing a local resonant source. By adding a local resonant source to a unit cell, a wave generated by a single local resonant source can be formed within a specific frequency band gap in a finite array unit cell. This greatly reduces the size and volume of the filter (compared to models that form filters based on Bragg scattering). Furthermore, by changing the geometric and material parameters of the local resonant source, the filtering range can be effectively changed, resulting in precise filtering at specific frequencies. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0021] Figure 1 This is a diagram illustrating the construction process of the two-dimensional acoustic metamaterial with a resonant source in this invention.

[0022] Figure 2 This is a schematic diagram of the band structure of the two-dimensional acoustic metamaterial with a resonant source in this invention.

[0023] Figure 3 This is a schematic diagram of the intrinsic modes of the two-dimensional acoustic metamaterial with a resonant source in this invention.

[0024] Figure 4 This is a graph showing the variation of the bandgap and filtering range of the two-dimensional acoustic metamaterial with a resonant source in this invention as a function of parameter D.

[0025] Figure 5 This is a graph showing the variation of the bandgap and filtering range of the two-dimensional acoustic metamaterial with a resonant source in this invention as a function of parameter d.

[0026] Figure 6 This is a graph showing the variation of the bandgap and filtering range of the two-dimensional acoustic metamaterial with a resonant source as a function of the width of the cladding in this invention.

[0027] Figure 7 This is a graph showing the variation of the bandgap and filtering range of the two-dimensional acoustic metamaterial with a resonant source in this invention with the density of the cladding.

[0028] Figure 8 This is a graph showing the variation of the bandgap and filtering range of the two-dimensional acoustic metamaterial with a resonant source in this invention with the Young's modulus of the cladding.

[0029] Figure 9 This is a graph showing the variation of the bandgap and filtering range of the two-dimensional acoustic metamaterial with a resonant source as a function of the radius of the resonant source in this invention.

[0030] Figure 10 This is a graph showing the variation of the bandgap and filtering range of the two-dimensional acoustic metamaterial with a resonant source as a function of the resonant source density in this invention.

[0031] Figure 11 This is a graph showing the variation of the bandgap and filtering range of the two-dimensional acoustic metamaterial with a resonant source in this invention with the Young's modulus of the resonant source.

[0032] Figure 12 This is a simulation diagram of the vibration isolation of the two-dimensional acoustic metamaterial with a resonant source in this invention.

[0033] Figure 13 This is a stress curve analysis diagram of the two-dimensional acoustic metamaterial with a resonance source in this invention. Detailed Implementation

[0034] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0035] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the present invention or its application or use. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0036] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of exemplary embodiments according to the invention. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0037] Unless otherwise specifically stated, the relative arrangement, numerical expressions, and values ​​of the components and steps described in these embodiments do not limit the scope of the invention. It should also be understood that, for ease of description, the dimensions of the various parts shown in the drawings are not drawn to actual scale. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it need not be further discussed in subsequent figures.

[0038] This invention provides a method for forming a filter in an acoustic metamaterial by introducing a local resonant source, comprising: breaking symmetry in a two-dimensional five-mode classical biconical structure to form a biconical bandgap structure; in a specific implementation, as a preferred embodiment of this invention, Figure 1 In the two-dimensional five-mode classical biconical structure in (a), all connecting arms within the unit cell have the same length and exhibit axisymmetric structure in both the horizontal and vertical directions. Due to the scalability of acoustic metamaterials, the dimensions are normalized to L, where L is 100 mm. The specific geometric parameters are listed in Table 1. The two connecting points of the central arm at the center position are moved by the same distance d in opposite directions in the vertical direction. e get Figure 1 (b) shows the biconical bandgap structure. In this case, the lengths of all connecting arms within the unit cell are no longer the same, and their values ​​change with the distance traveled.

[0039] Table 1. Geometric parameters of two-dimensional acoustic metamaterials with resonant sources

[0040] In a preferred embodiment of the invention, in the five-module unit cell of the double-cone bandgap structure, each arm structure comprises two rectangles of width d and two trapezoidal portions, with the top width of the trapezoids being d and the bottom width being D. The arms are interconnected at the intersection of the extensions of their respective rectangular portions. Three connecting arms intersect to form a connection point, with an angle of 60° between adjacent connecting arms. The length of each connecting arm is the same and is L, the unit cell length is 3L, and the width is... L.

[0041] A localized resonance source of radius r, enclosed by a cladding layer of width t, is introduced at the geometric center of the central arm, thereby obtaining... Figure 1 (c) Two-dimensional acoustic metamaterial with a resonant source. The matrix material was set to aluminum, the cladding material was initially set to rubber, and the local resonant source material was set to copper. The material parameters of the cladding and the local resonant unit are listed in Table 2, where D is the major diameter of the connecting arm, d is the minor diameter of the connecting arm, r is the radius of the local resonant source, and t is the thickness of the cladding.

[0042] Table 2 Material parameters of two-dimensional acoustic metamaterials with resonant sources

[0043] Modeling and simulation were performed using COMSOL Multiphysics. A two-dimensional acoustic metamaterial band structure with a resonant source was established based on the biconical bandgap structure. In a preferred embodiment of the invention, solid mechanics was added as the background field. Periodic boundary conditions were set on the two parallel sides of the rectangular unit cell. Floquet periodic boundary conditions were used, and Bloch vector k was distributed on the boundary of the irreducible Brillouin zone. Based on the elastic dynamics equations, the elastic wave dispersion relation in a two-dimensional periodic structure can be further written as:

[0044] in, Represents the displacement vector; Represents a position vector; denoted by Hamiltonian differential operator; λ and G are Lamé constants; ρ represents mass density; Let be the second derivative of the displacement with respect to time.

[0045] Based on Bloch's theorem, the dynamic behavior of periodic structures is effectively simplified from an infinite periodic array to a single-cell problem, and the band structure is solved. In a preferred embodiment of this invention, based on Bloch's theorem, when periodic boundary conditions are applied to the single cell, the displacement fields at all discrete nodes must satisfy the Bloch-form phase periodicity relationship, that is, the displacements differ by only one phase factor related to the wave vector after crossing a period.

[0046] in, Represents the displacement vector; Represents the wave vector; Represents lattice vectors; Represents angular frequency. Represents the imaginary unit; Indicates time.

[0047] Based on the elastic wave dispersion relation and band structure in a two-dimensional periodic structure, the eigenvalue problem of elastic wave propagation is derived. In a preferred embodiment of this invention, the eigenvalue problem of elastic wave propagation is expressed in matrix form as follows:

[0048] in, Represents the unit vector in the x-direction; Represents the unit vector in the y-direction; a and b These are the lattice constants in the x and y directions, respectively, used to construct the reciprocal lattice vector of the first Brillouin zone. The reciprocal lattice vector of the first Brillouin zone is defined as:

[0049] in, , These represent two vectors in the first Brillouin zone of the crystal lattice.

[0050] By adding a local resonant source within the bandgap and embedding discrete defect modes within the bandgap, narrow-bandpass filtering can be achieved. In a preferred embodiment of this invention, the geometry of the filter unit cell is identical to that of the bandgap unit cell. A local resonant unit is added at the center of the connecting arm, consisting of an outer rubber layer and an inner metal layer. After adding the local resonant source, waves generated solely based on the principle of local resonance can be formed within a specific frequency bandgap within the finite array unit cell. By changing the geometric and material parameters of the local resonant source, the filtering range can be altered, achieving precise filtering at specific frequencies.

[0051] exist Figure 2 Figures (d) to (f) illustrate the band structure of acoustic metamaterials. The band diagrams of each acoustic metamaterial are obtained by scanning the wave vector k in the range (0,3) and presented in the form of a two-dimensional plotting set. The horizontal axis represents the high symmetry point of the irreducible Brillouin zone of the wave vector k. The paths traversed correspond to k values ​​of 0, 1, 2, and 3, respectively. The vertical axis represents the eigenfrequency obtained through eigenvalue calculation, corresponding to the vibration frequency of different propagation modes. Each curve in the figure represents the trajectory of the eigenfrequency of a certain propagation mode (eigenmode) as a function of the wave vector. In the band structure diagram, each curve characterizes the dispersion relationship of different eigenmodes in the structure, and their frequencies vary with the wave vector. k The changes reflect the propagation characteristics of elastic waves in periodic structures. When the intrinsic mode curves are continuously distributed and overlap within a certain frequency range, it indicates that the elastic wave can propagate effectively within that frequency band, corresponding to the passband characteristics of the structure; conversely, when frequency intervals without mode distribution appear between the curves, a region of impeded propagation, i.e., a band gap, is formed.

[0052] The filtering effect in this invention is mainly manifested in the introduction of local resonant modes within the bandgap or in its adjacent frequency bands. These modes typically exhibit approximately flat dispersion branches with group velocities close to zero, indicating that wave energy is locally confined within the resonant units, thereby achieving selective suppression or transmission of specific frequency components. Therefore, the bandgap primarily serves as the frequency boundary of the filtering structure, while the actual filtering behavior is dominated by local resonance.

[0053] like Figure 2 As shown in (d)–(f), in Figure 2 In (d), the yellow shaded area indicates the frequency range corresponding to the five-mode structure, and the difference between its upper and lower boundary frequencies is defined as the five-mode bandwidth. Within this frequency band, only the propagation mode dominated by compression waves exists, while the shear wave-related modes are completely suppressed, indicating that the structure in this frequency range only supports the propagation of a single type of wave, demonstrating obvious mode selectivity. Figure 2(e) illustrates the band structure's band characteristics, where the blue shaded area corresponds to a full bandgap, and the difference between its upper and lower boundary frequencies is defined as the bandgap width. Within this frequency range, no continuous intrinsic mode distribution was observed, indicating that elastic waves cannot propagate effectively in this interval, and both compressional and shear waves are significantly suppressed. Figure 2 As shown in (f), the red shaded area corresponds to the effective operating frequency band of the filtering model. By introducing local resonant units into the bandgap structure, a controllable local resonant mode is formed within the original bandgap, thereby achieving selective transmission of specific frequency components. Further adjustment of the structural parameters of the local resonant units allows for continuous control of the filter center frequency and its response characteristics, enabling the structure to maintain bandgap suppression capability while possessing good designability and tunability.

[0054] In the band structure of the five-mode structure, an isolated dispersion branch can be observed in the frequency range of 115.42–1413.5 Hz. This branch corresponds to a propagation mode dominated solely by the compression wave. This mode exhibits significant direction selectivity, with its vibrations primarily along... x The direction occurs, and yThe negligible directional displacement indicates that the structure supports only longitudinal wave propagation in a single direction within this frequency band. This highly directional modal characteristic stems from the geometric constraints and modal coupling mechanism of the unit structure, effectively suppressing the transverse degree of freedom and thus forming a propagation channel with single-mode characteristics. By further adjusting the distance between the connection points in the unit structure, the energy band diagram of the structure changes significantly. Under the new parameter configuration, a distinct bandgap region appears in the frequency range of 261.75–822.13 Hz in the energy band diagram. No continuous intrinsic mode distribution was observed in this frequency band, indicating that elastic waves cannot propagate in the structure within this frequency range. Further modal analysis shows that neither compressional nor shear waves form effective propagation modes in this frequency band, indicating that the bandgap simultaneously suppresses the propagation behavior of both longitudinal and transverse wave components. In the band structure diagram, a local resonant mode with a group velocity close to zero has a characteristic frequency within the band gap of 256.02-774.15 Hz. Within this band gap, only one eigenmode dominated by local resonance exists, while all other propagation modes are completely suppressed, indicating that only a single mode is allowed to participate in energy transfer within this frequency band. These results demonstrate that the structure can form a single propagation mode dominated by the local resonance mechanism within a specific frequency range. This mode originates from the coupling between the local resonant unit and the overall structure, allowing only one eigenmode to exist within this frequency band, while all other modes are effectively suppressed. Consequently, the structure exhibits significant single-mode propagation characteristics within the corresponding frequency range, thus achieving selective transmission of elastic waves, i.e., forming a filtering effect with a defined operating frequency band. These results show that by rationally designing the structural parameters of the local resonant unit, precise control of the filtering frequency band can be achieved while maintaining the overall band gap characteristics.

[0055] like Figure 3As shown, when the wave vector k reaches point M, points e, f, p, and q in the band structure diagram correspond to the modes of the acoustic metamaterial. Points e and p are located at the upper and lower limits of the band gap where the filter is located, point f is the characteristic frequency of the filter, and point q is a point on any passband. At point e, the overall mode shape exhibits a clear "rotation-stretch coupling" characteristic. This mode is not a simple longitudinal wave compression, but rather involves relative rotation between elements and bending of the connecting arms. Its physical essence can be attributed to a mixed mode generated by local resonance and lattice coupling, representing a critical state transitioning from the propagation state to the band gap state. At point p, the connecting arms do not exhibit significant rotation or deformation; instead, the local resonant elements show unidirectional movement, indicating that this mode is a critical mode dominated by local resonance. At the passband of point q, the overall structure undergoes slight torsion, while the local resonant source does not deform or move. At the filtering point f, it can be observed that the overall structure does not undergo any movement or torsion. The most significant deformation is the overall stretching of the local resonant unit, which is a typical local resonant mode. This indicates the inherent vibration state of the local resonant unit in the structure, exhibiting a strong response at this characteristic frequency. Its dynamic behavior is mainly dominated by local vibration, while the propagation effect of the overall structure is significantly suppressed. Furthermore, only a single mode dominated by local resonance exists, indicating that the system exhibits significant single-mode response characteristics at this frequency. Therefore, by adjusting the material parameters of the local resonant source and its cladding layer, the location of the filtering characteristic frequency can be effectively changed, while changing the geometric parameters of the structure has little effect on its filtering range.

[0056] Based on the mode characteristics of the filtering location, it has been determined that changing the geometric parameters of the unit cell will not significantly affect the filtering range. Therefore, the geometric configuration can be changed by altering the major diameter D and minor diameter d of the biconical structure. Figure 4 and Figure 5 The diagrams show the unit cell geometry and band structure as the major and minor diameters (D and d) of the biconical structure change, with normalized dimensions. The x-axis represents the ratio of each variable to L, and the y-axis represents the band gap and filtering frequency. The blue line represents the upper and lower limits of the band gap, and the red line represents the filtering position. The diagrams show that changing the major and minor diameters of the biconical structure does not significantly alter the filtering range; only the band gap changes due to the structural influence. This fully verifies that the filtering generated by the localized resonant mode is not affected by changes in geometric parameters.

[0057] like Figure 6As shown, while keeping the radius of the local resonant unit constant, the positional transformation of the filtering and bandgap is verified by increasing the width of the cladding. The figure shows that as the width increases, the filtering range approaches the lower limit of the bandgap, while the overall position of the bandgap shifts downwards with increasing width. This is because the rubber cladding layer essentially acts as a spring connecting the local resonant core and the rigid outer shell. Increasing the width is equivalent to lengthening the spring, thus leading to an increase in equivalent stiffness. Reduced. Since the formula for local resonant frequency is approximately: The decrease in stiffness leads to a decrease in the resonant frequency, which in turn causes the filter passband to shift to lower frequencies. Furthermore, the low-frequency band is absorbed by the outer bandgap, resulting in a reduction in the overall usable filtering range.

[0058] The material parameters of the coating have a direct impact on the filtering range, such as Figure 7 As shown, since the mass of the cladding layer is much smaller than the mass of the local resonant core (in high-density materials), the mass of the cladding layer is significantly smaller than the equivalent mass. The impact is negligible. The resonant frequency is mainly determined by the core mass and equivalent stiffness, which means that changes in the cladding density have little effect on the filtering.

[0059] Since the cladding completely encapsulates the local resonant unit, the rubber cladding layer essentially acts as a spring connecting the local resonant core to the rigid outer shell, which results in the cladding layer's stiffness... Directly replacing the overall Young's modulus (Young's modulus directly determines the stiffness of the coating layer), such as Figure 8 As shown, Increasing the frequency leads to a higher resonant frequency, extending the filter band to higher frequencies and thus increasing the filtering range. From... Figure 7 and Figure 8 It can be seen that the change in the coating density has no effect on the band gap or the filtering position, while the increase in Young's modulus directly leads to an upward shift in the filtering range, which fully verifies the above theory.

[0060] Figure 9 This diagram illustrates how the bandgap and filtering range vary with the radius of the local resonant unit while keeping the cladding width constant. Figure 9 As shown in (b), increasing the radius of the local resonant unit does not significantly change the upper and lower limits of the bandgap, but it does produce a relatively obvious downward trend in the filtering range. This is because the increase in the core radius of the local resonant unit leads to an increase in the core mass m. core Increase, due to , m core Increasing the frequency will reduce the filtering frequency, thereby limiting the filtering range to the lower edge of the bandgap and reducing the usable filtering range.

[0061] exist Figure 10The figure shows the changes in bandgap and filtering range with resonant source density. It can be seen from the figure that increasing the density of local resonant units significantly reduces the filtering range, while the upper and lower limits of the bandgap remain essentially unchanged. This is consistent with... Figure 7 The dotted-line graph of the filter changing with density is clearly formed. This is because the density of the local resonant material is much greater than that of the cladding material. This makes the mass of the local resonant unit the dominant core mass. The increase in density leads to the increase in core mass, which in turn leads to the decrease in resonant frequency and further narrows the filtering range.

[0062] exist Figure 11 The figure shows the variation of bandgap and filtering range with Young's modulus of the resonant source. The reason why the filtering range does not change in the figure is that since the cladding layer completely encloses the local resonant unit, the elastic effect of the cladding layer and the local resonant unit is mainly provided by the cladding layer. Therefore, the change of Young's modulus of the local resonant unit will not significantly change the equivalent stiffness of the local resonance. So the resonant frequency is almost unaffected by the core Young's modulus, making the filtering range basically unchanged.

[0063] The configured geometric parameters in Table 3 are applied to a two-dimensional acoustic metamaterial with a resonant source, and then arrayed along a two-dimensional plane to obtain the arrayed two-dimensional acoustic metamaterial as shown below. Figure 12 As shown in (a), a pressure load of 1 N is applied to the left boundary of the model, perpendicular to the left incident plane. A fixed constraint is applied to the right boundary, such that the displacement u=0 to ensure that the pressure on the left can be transmitted to the right, while providing the necessary constraint conditions. Periodic boundary conditions are applied to the upper and lower boundaries, and Floquet periodic boundary conditions are selected to simulate the structure of the unit cell after infinite arraying in the x and y directions. Figure 12 (b) shows the band structure of the unit cell under these geometric parameters. It can be seen that the filtering range exists between 256.1 and 804.28 Hz within the band gap, and a single wave caused by local resonance is generated near 549 Hz. The propagation of the elastic wave in the two-dimensional metamaterial behind the array is observed at 500 Hz and 549 Hz, respectively. Figure 12 As shown in (c) and (d), in Figure 12 As shown in (c), when the frequency is within the bandgap range, the pressure wave at the left boundary attenuates after passing through the left baffle to the second metamaterial layer, and attenuates to 0 at the third metamaterial layer. This verifies that the model can significantly suppress elastic wave propagation within this frequency range; Figure 12 As can be seen in (d), when the pressure wave propagates from left to right, the metamaterial vibrates with different amplitudes, indicating that the elastic wave propagates effectively within this range, thus verifying the filtering characteristics within this range.

[0064] Table 3 Geometric parameters of the two-dimensional acoustic metamaterial after arraying

[0065] To more intuitively demonstrate the filtering characteristics of metamaterials within the bandgap range, a 4×4 element of the aforementioned array metamaterial was selected as the analysis object, such as... Figure 13 As shown in (a), a boundary load is applied to the left end of the 4×4 array metamaterial as the incident surface, and a fixed constraint is set at the right end as the exit surface. Stress analysis is performed on the exit and incident surfaces using Comsol Multiphysics software, and the results are as follows: Figure 13 (b) shows the stress curve, where the x-axis represents stress, the y-axis represents the corresponding frequency, and the yellow area represents the filtering range within the theoretical unit cell bandgap. As can be seen in the figure, when the frequency is within the bandgap (256.1-804.28 Hz), the stress is very small or almost zero, with only a few points showing relatively high outgoing stress, resulting in a slightly higher stress than normal. However, when the frequency falls within the filtering range (around 549 Hz), the stress shows a significant surge, then decays after an increment of approximately 10 Hz, followed by a second stress peak at 780 Hz. This is because this range exceeds the filtering range and reaches the upper limit of the bandgap, allowing elastic waves to propagate. This is consistent with theoretical predictions, thus fully verifying the filtering characteristics of the two-dimensional acoustic metamaterial.

[0066] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for forming a filter in an acoustic metamaterial by introducing a local resonant source, characterized in that, include: Symmetry breaking is performed on the two-dimensional five-mode classical biconical structure to form a biconical bandgap structure; A two-dimensional acoustic metamaterial bandgap model with a resonant source is established based on the described double-cone bandgap structure; Based on Bloch's theorem, the dynamic behavior of periodic structures is effectively simplified from an infinite periodic array to a single-cell problem, and the band structure is solved. Based on the elastic wave dispersion relation and band structure in two-dimensional periodic structures, we can deduce the problem of eigenvalues ​​for elastic wave propagation. By adding a local resonant source within the band gap and embedding discrete defect modes within the band gap, narrow bandpass filtering can be achieved.

2. The method for forming a filter in an acoustic metamaterial by introducing a local resonant source according to claim 1, characterized in that, The aforementioned symmetry breaking in the two-dimensional five-mode classical biconical structure specifically includes: In the aforementioned two-dimensional five-mode classical biconical structure, all connecting arms within the unit cell are of the same length and exhibit axial symmetry in both the horizontal and vertical directions. Moving the two connecting points of the central arm at the center position in opposite directions by the same distance d in the vertical direction... e A double-cone bandgap structure was obtained.

3. The method for forming a filter in an acoustic metamaterial by introducing a local resonant source according to claim 1, characterized in that, In the five-module unit cell of the biconical bandgap structure, each arm structure comprises two rectangles of width d and two trapezoidal portions, with the top width of the trapezoids being d and the bottom width being D. The arms are interconnected via their respective rectangular portions at the intersection of the trapezoidal extensions. Three connecting arms intersect to form a connection point, with an angle of 60° between adjacent connecting arms. All connecting arms have the same length, L, and the unit cell length is 3L with a width of [missing value]. L.

4. The method for forming a filter in an acoustic metamaterial by introducing a local resonant source according to claim 1, characterized in that, The establishment of the two-dimensional acoustic metamaterial band structure specifically includes: Add solid mechanics as the background field, set periodic boundary conditions on the two parallel sides of the rectangular unit cell, use Floquet periodic boundary conditions, and distribute Bloch vector k on the boundary of the irreducible Brillouin zone; Based on the elastic dynamics equations, the elastic wave dispersion relation in a two-dimensional periodic structure can be further written as: in, Represents the displacement vector; Represents a position vector; denoted by Hamiltonian differential operator; λ and G are Lamé constants; ρ represents mass density; Let be the second derivative of the displacement with respect to time.

5. The method for forming a filter in an acoustic metamaterial by introducing a local resonant source according to claim 1, characterized in that, The solution to the band structure specifically includes: Based on Bloch's theorem, when periodic boundary conditions are applied to a unit cell, the displacement fields at all discrete nodes must satisfy the Bloch-form phase periodicity relation, meaning that the displacements differ by only one phase factor related to the wave vector after crossing a period: in, Represents the displacement vector; Represents the wave vector; Represents lattice vectors; Represents angular frequency. Represents the imaginary unit. Indicates time.

6. The method for forming a filter in an acoustic metamaterial by introducing a local resonant source according to claim 1, characterized in that, The matrix form of the eigenvalue problem of elastic wave propagation is as follows: in, Represents the unit vector in the x-direction; Represents the unit vector in the y-direction; a and b and are the lattice constants in the x and y directions, respectively, used to construct the reciprocal lattice vector of the first Brillouin zone. The reciprocal lattice vector of the first Brillouin zone is defined as: in, , These represent two vectors in the first Brillouin zone of the crystal lattice.

7. The method for forming a filter in an acoustic metamaterial by introducing a local resonant source according to claim 1, characterized in that, The addition of a localized resonant source within the bandgap specifically includes: The geometry of the filter unit cell is consistent with that of the bandgap unit cell. A local resonant unit is added at the center of the connecting arm in the middle. The local resonant unit consists of an outer rubber layer and an inner metal layer. After adding the local resonance source, waves generated solely by the principle of local resonance can be formed within a specific frequency band gap in a finite array unit cell. By changing the geometric and material parameters of the local resonance source, the filtering range can be altered, achieving precise filtering at specific frequencies.

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