A design method for broadband curved waveguides of arbitrary shapes

Abstract

This invention discloses a design method for broadband curved waveguides of arbitrary shape, comprising the following steps: S1: determining the contour shape of the waveguide; S2: calculating the curved wave refractive index distribution of the waveguide based on quasi-conformal transformation; S3: calculating the thickness point cloud distribution of the waveguide based on the refractive index distribution; S4: optimizing the thickness point cloud distribution; S5: constructing a waveguide thickness point cloud surface based on the optimized thickness point cloud distribution; S6: establishing a three-dimensional geometric model of the waveguide; S7: embedding the waveguide into a thin plate structure. This solution has strong applicability, wide operating frequency band, and is easy to manufacture, and can improve the vibration control capability of thin plate structures under complex boundary conditions.

Description

Technical Field

[0001] This invention relates to the field of vibration control technology for thin plate structures, and specifically to a design method for broadband curved waveguides of arbitrary shapes. Background Technology

[0002] Thin plate structures, as fundamental load-bearing and functional units in the engineering field, are widely used in key areas such as aerospace, shipbuilding, precision instruments, and modern architecture. Vibration problems of thin plate structures, especially vibration modes dominated by bending waves, are directly related to the fatigue life, operational safety, noise control, and instrument accuracy of the structure. Therefore, effectively controlling the propagation path and energy of bending waves in thin plates has become a key technology for improving structural performance and intelligence.

[0003] Currently, the mainstream methods for controlling bent waves are mainly based on periodic phonon crystal structures and metamaterial and metasurface technologies. These methods generate band gaps or anomalous beam deflections in specific frequency bands by designing unit structures, thereby realizing waveguide functions. However, such methods have significant limitations: First, the waveguide performance is usually highly dependent on periodic arrangement and specific resonance, resulting in a narrow operating frequency band and difficulty in adapting to broadband vibration environments. Second, in order to achieve complex wave manipulation, the unit structure is often designed to be extremely complex, which places extremely high demands on the manufacturing process and restricts its engineering practicality. Finally, and most importantly, existing waveguide design methods are usually limited to straight lines, arcs, or simple geometric shapes, with a constant waveguide width, which cannot flexibly adapt to the complex geometric boundaries and arbitrary curved paths commonly encountered in engineering practice. This greatly limits the application potential of waveguide technology in the vibration control of complex structures.

[0004] In summary, existing thin-plate bent wave waveguide technology struggles to balance geometric adaptability, operating bandwidth, and ease of engineering implementation; phononic crystals and complex superstructures are limited by narrow bandwidth and fabrication challenges; while simple gradient refractive index waveguides are constrained by fixed geometries.

[0005] Therefore, there is an urgent need to develop a universal design method for curved waveguides that can be oriented to any shape, achieve broadband operation, and is easy to manufacture. Summary of the Invention

[0006] To address the aforementioned shortcomings of existing technologies, this invention provides a design method for broadband curved waveguides of arbitrary shapes, which solves the problems of narrow operating bandwidth, complex manufacturing process, and limited geometric shape in existing thin-plate curved waveguide technologies.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A method for designing broadband curved waveguides of arbitrary shapes is provided, which includes the following steps: S1: Determine the contour shape of the waveguide; S2: Calculate the refractive index distribution of the flexural wave of the waveguide based on quasi-conformal transformation; S3: Calculate the thickness point cloud distribution of the waveguide based on the refractive index distribution; S4: Optimize the thickness point cloud distribution; S5: Construct the waveguide thickness point cloud surface based on the optimized thickness point cloud distribution; S6: Establish the three-dimensional geometric model of the waveguide; S7: Embed the waveguide into the thin plate structure.

[0008] Furthermore, the waveguide's profile is a smooth curve, described by an analytical function or an interpolated curve.

[0009] Furthermore, the refractive index distribution in step S2 is calculated using numerical methods, specifically including: S21: applying first and second type boundary conditions based on Cauchy-Riemann conditions on the waveguide boundary in physical space; S22: Solve the Laplace equation under boundary conditions using numerical methods to obtain the distribution of the transformation function; S23: Calculate the inverse Jacobian matrix of the coordinate transformation from physical space to virtual space based on the transformation function and its conjugate harmonic function; S24: Calculate the refractive index distribution based on the inverse Jacobian matrix.

[0010] Furthermore, when a negative value appears in the calculated refractive index distribution, the waveguide is trimmed, and the exit cutoff of the trimmed waveguide is parallel to the contour lines of the dependent variable in the Laplace equation.

[0011] Furthermore, the formula for calculating the thickness distribution in step S3 is as follows:

[0012] in, h 0 represents the substrate thickness. n z This represents the refractive index distribution.

[0013] Furthermore, the optimization process in step S4 includes: smoothing the thickness point cloud using natural neighborhood interpolation and Gaussian filtering to eliminate edge burrs and surface roughness.

[0014] Furthermore, step S5 specifically includes: constructing a waveguide thickness point cloud surface using a parametric surface modeling method.

[0015] Furthermore, step S6 specifically includes: establishing a three-dimensional geometric model of the waveguide based on the waveguide thickness point cloud surface and through geometric Boolean operations.

[0016] Furthermore, after the waveguide is embedded in the thin plate structure in step S7, its profile thickness is thicker than, equal to, or thinner than that of the thin plate structure.

[0017] Furthermore, a Blu-Tack absorbing material is provided in the region where the waveguide profile is thicker than the substrate to reduce edge scattering caused by thickness discontinuity; a piezoelectric material for vibration energy harvesting is provided at the thinnest part of the waveguide.

[0018] The beneficial effects of this invention are as follows: 1. This scheme is based on the quasi-conformal transformation theory, which maps waveguides with arbitrary smooth curve profiles in physical space to regular rectangles in virtual space. By solving the boundary value problem of the Laplace equation corresponding to this mapping relationship, the complex geometry is directly transformed into a computable mathematical and physical model. It breaks through the geometric constraints of traditional waveguide design, which is limited by straight lines, arcs or fixed-width channels. The waveguide path can be freely defined according to the complex boundary of the actual engineering structure. It has strong geometric adaptability and realizes the bypassing, redirection or convergence of vibration energy, which greatly expands the application scenarios of waveguide technology in real complex structures.

[0019] 2. This scheme uses thickness gradient to control the refractive index distribution, enabling the waveguide to maintain good wave guiding performance over a wide frequency band, making it suitable for broadband vibration environments.

[0020] 3. The waveguide structure in this scheme achieves its function through thickness variation, without the need for complex periodic units, and can be fabricated using conventional processes such as CNC machining, reducing manufacturing costs and difficulty.

[0021] 4. This solution supports multiple functions such as vibration isolation, steering, and energy capture by flexibly defining input and output boundary curves and sidewall profiles, effectively guiding the propagation path of bending waves and improving the vibration control capability of thin plate structures.

[0022] 5. This solution eliminates edge burrs and surface roughness caused by numerical discretization through post-processing such as trimming and optimization of negative refractive index regions and smoothing filtering of thickness point clouds, thus ensuring the smoothness and performance stability of the processing model. Attached Figure Description

[0023] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly described below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. The above and other objects, features, and advantages of the present invention will become clearer through the accompanying drawings. The same reference numerals indicate the same parts in all the drawings. The drawings are not intentionally drawn to scale to actual dimensions; the focus is on illustrating the main points of the invention.

[0024] Figure 1 A flowchart of a method for designing broadband curved waveguides of arbitrary shapes.

[0025] Figure 2 A flowchart for calculating the refractive index of broadband curved waveguides of arbitrary shapes.

[0026] Figure 3 (a) is a schematic diagram of the quasi-conformal transformation of the vibration isolation waveguide. Figure 3 (b) is a schematic diagram of the refractive index distribution of the vibration isolation waveguide.

[0027] Figure 3 (c) is a schematic diagram of the distribution of the thickness of the vibration isolation waveguide.

[0028] Figure 4 (a) is a schematic diagram of the refractive index distribution of the vibration-directing waveguide.

[0029] Figure 4 (b) is a schematic diagram of the refractive index distribution of the vibration-directing waveguide after cutting.

[0030] Figure 4 (c) is a schematic diagram of the distribution of the refractive index of the waveguide trapped by vibrational energy.

[0031] Figure 5 The coordinate graph of the interpolation function for the vibration-directing waveguide profile.

[0032] Figure 6 Displacement and absolute displacement field diagrams obtained from experiments of vibration isolation waveguide, vibration steering waveguide and vibration energy trapping waveguide.

[0033] Figure 7 Dispersion curves of bending wave and Lamb wave A0 modes for a 2mm aluminum alloy sheet.

[0034] Figure 8 (a) shows the insertion loss of the vibration isolation waveguide in the range of 100kHz-120kHz.

[0035] Figure 8 (b) is the open-circuit output voltage diagram of the vibration energy trapping waveguide.

[0036] Figure 8 (c) is a diagram of the load output power of the vibration energy trapping waveguide. Detailed Implementation

[0037] 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 components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0038] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0039] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0040] Furthermore, the terms "first," "second," etc., are used only to distinguish descriptions and should not be interpreted as indicating or implying relative importance.

[0041] like Figure 1 As shown, the design method for arbitrary-shaped broadband curved waveguides in this scheme includes the following steps: S1: Determine the contour shape of the waveguide; S2: Calculate the curved wave refractive index distribution of the waveguide based on quasi-conformal transformation; S3: Calculate the thickness point cloud distribution of the waveguide based on the refractive index distribution; S4: Optimize the thickness point cloud distribution; S5: Construct the waveguide thickness point cloud surface based on the optimized thickness point cloud distribution; S6: Establish a three-dimensional geometric model of the waveguide; S7: Embed the waveguide into a thin plate structure. This scheme, based on quasi-conformal transformation, designs arbitrary-shaped broadband curved waveguides with specific functions by adjusting the thickness. It designs three types of arbitrary-shaped curved waveguides: vibration isolation, vibration redirection, and vibration energy trapping.

[0042] like Figure 2 As shown, the steps for calculating the refractive index distribution of a flexural wave in a waveguide include: S21: Applying first and second type boundary conditions based on Cauchy-Riemann conditions to the waveguide boundary in physical space; S22: Solving the Laplace equation under the boundary conditions using numerical methods to obtain the distribution of the transformation function; S23: Calculating the inverse Jacobian matrix of the coordinate transformation from physical space to virtual space based on the transformation function and its conjugate harmonic function; S24: Calculating the refractive index distribution based on the inverse Jacobian matrix.

[0043] like Figure 3 As shown in (a), the virtual space The refractive index within the rectangle ABCDEFGHIJ is denoted as . physical space The refractive index within waveguides A'B'C'D'E'F'G'H'I'J' in the diagram is denoted as .

[0044] According to the theory of conformal transformation, the refractive index of virtual space and physical space refractive index satisfy:

[0045] In the formula The inverse Jacobian matrix:

[0046] The inverse Jacobian matrix is ​​crucial for realizing quasi-conformal transformation. The inverse Jacobian matrix of a vibrating isolating waveguide can be obtained by solving the Laplace equation under specific boundary conditions. In quasi-conformal transformation theory, the transformation... The Cauchy-Riemann boundary conditions must be satisfied:

[0047] The Laplace equation can be derived from the Cauchy-Riemann boundary conditions:

[0048] Based on the shape of the vibration isolation waveguide, the line segments in the transformed virtual space and Curves mapped to physical space and The equation of the curve is Where a and b are 20mm and 50mm respectively, the first and second type boundary conditions are applied at the waveguide boundary:

[0049] The inverse Jacobian matrix can be solved by combining the two equations. Physical space refractive index For a substrate of uniform thickness with a refractive index, therefore The value is 1, and thus we can calculate... .

[0050] This embodiment solves the Laplace equation under boundary conditions using COMSOL's PDE module, and calculates... like Figure 3 As shown in (b), the maximum refractive index of the vibrating isolation waveguide is 1.37, located at the center of the lower boundary of the vibrating isolation waveguide, and the minimum refractive index is 0.55, located at the center of the upper boundary of the vibrating isolation waveguide. The distribution of the refractive index at the upper boundary satisfies the rule that the more concave the boundary, the smaller the refractive index value, while the distribution of the refractive index at the lower boundary is the opposite.

[0051] The phase velocity of bending wave propagation in a thin plate structure is:

[0052] According to Snell's law, the refractive index of physical space for:

[0053] In the formula h 0 represents the thickness of the plate of uniform thickness. and The bending wave phase velocities in regions of uniform and variable thickness are respectively used to deduce the thickness distribution in physical space. for:

[0054] like Figure 3 As shown in (c), the maximum thickness is 6.66 mm and the minimum thickness is 1.06 mm, which occur at the minimum and maximum refractive indices, respectively. The thickness distribution pattern is opposite to the refractive index distribution pattern. After the bent wave enters the waveguide from the left, it moves along the trajectory of the red line in the figure due to the influence of the thickness change, thus achieving the function of vibration isolation. The vibration isolation area is the triangular area enclosed by the red dashed line in the figure.

[0055] The refractive index distribution of the vibrating isolating waveguide is a solution to the Laplace equation under specified boundary conditions. Since the numerical method for solving the Laplace equation is the finite element method, the refractive index distribution of the vibrating isolating waveguide is discrete, and therefore the corresponding thickness distribution is point cloud data. The PDE module in COMSOL is used to set the boundary conditions and solve the Laplace equation. Subsequently, the thickness point cloud data is calculated using MATLAB, and the data undergoes double smoothing processing using natural neighbor interpolation and Gaussian filtering. Modeling with unfiltered point cloud data results in edge jaggedness, while modeling with Gaussian-filtered point cloud data produces a model with neat edges and a smoother surface. Finally, the smoothed thickness point cloud... Data was imported into COMSOL's "Parameterized Surface" function, and then geometric Boolean operations were used to complete the model construction. Subsequently, the vibration isolation waveguide was processed and tested. The experimental test included two test samples: a vibration isolation waveguide plate and a plate of uniform thickness. Both plates were made of 6061 aluminum alloy. The waveguide plate was manufactured by CNC machining, with uniform plate dimensions of 250mm×100mm×2mm. To reduce the influence of boundary reflections of the vibration isolation waveguide on the experimental results, Blu-Tack was applied to the edges of the plates to absorb boundary reflections. In addition, Blu-Tack was applied to the edges where the waveguide section thickness exceeded 2mm to reduce boundary scattering caused by abrupt changes in waveguide boundary thickness.

[0056] This embodiment uses the same method to design and fabricate vibration steering and vibration energy trapping waveguides, whose respective refractive indices are as follows: Figure 4 As shown in (a) and (c), the upper and lower profile curves of the vibration-directing waveguide are plotted using an interpolation function, and the coordinates of the interpolation function are as follows: Figure 5 As shown; the upper and lower profile curve functions of the vibration energy trapping waveguide are as follows: , where a and b are 22.5 mm and 85 mm respectively. Figure 4In (a), the refractive index calculated for the vibration-directing waveguide has a negative value, therefore the waveguide needs to be trimmed. The refractive index distribution after trimming is as follows: Figure 4 As shown in (b), the cut-off line Y'Q' is parallel to the contour lines of the dependent variable u in the Laplace equation to ensure that the flexural wave maintains its planar wave characteristics at the waveguide outlet.

[0057] like Figure 6 (a)- Figure 6 As shown in (c), the vibration displacement field test results of the vibration isolation waveguide at frequencies of 100kHz, 110kHz, and 120kHz are respectively. It can be seen that the plane wave excited by the line source propagates along the preset waveguide path. During the propagation process, the wavefront remains perpendicular to the upper and lower boundaries of the waveguide. The flexural wave, guided by the waveguide, bypasses the designated vibration isolation area. The displacement amplitude in this area is significantly lower than that inside the waveguide, thus verifying that the vibration isolation waveguide has a good vibration isolation effect. In addition, the wavefront is completely emitted as a plane wave at the outlet of the vibration isolation waveguide, indicating that the vibration isolation waveguide has good broadband wave guiding performance.

[0058] It should be noted that the bending wave is a low-frequency approximation of the Lamb wave A0 mode in thin-plate structures. In a 2mm aluminum alloy sheet, the bending wave within 120kHz can be approximately equivalent to the Lamb wave A0 mode, such as... Figure 7 As shown, the A0 mode dispersion curves of the bending wave and the Lamb wave at 100kHz almost overlap, while the deviation gradually increases for frequencies above 180kHz. Therefore, the applicable frequency band of this method needs to be determined based on the dispersion curves of the bending wave and the A0 Lamb wave of a plate structure with specific materials and thicknesses.

[0059] like Figure 6 (d)- Figure 6 As shown in (f), these are the measured displacement field diagrams of the vibration-directing waveguide at frequencies of 65kHz, 75kHz, and 85kHz. It can be seen that the plane wave excited by the line source undergoes compression and expansion of the wavefront during propagation, and finally achieves vibration deflection at the waveguide outlet. The flexural wavefronts at the three frequencies still maintain the plane wave characteristics upon exiting, indicating that the vibration-directing waveguide has good broadband vibration deflection performance.

[0060] like Figure 6 (g)- Figure 6 As shown in (i), these are the measured absolute displacement field diagrams of the vibration energy trapping waveguide at frequencies of 100kHz, 110kHz and 120kHz. It can be found that the plane wave excited by the left-side line effectively converges in the central region of the waveguide after entering the vibration energy trapping waveguide, and the wavefront at the waveguide exit still maintains the characteristics of a plane wavefront. This indicates that the vibration energy trapping waveguide successfully realizes broadband guided wave propagation while effectively concentrating the energy of the bending wave.

[0061] This embodiment uses insertion loss to quantitatively reflect the vibration isolation performance of the vibration-isolated waveguide, and the calculation area is... Figure 3 (c) shows the triangular region and the triangular region corresponding to the plate of uniform thickness; such as Figure 8 As shown in (a), the insertion loss of the vibration isolation waveguide is positive at all test frequencies, confirming the effective vibration isolation at all test frequencies. The maximum insertion loss is 12.3 dB at 107.5 kHz, the minimum is 1.6 dB at 117.5 kHz, and the average insertion loss in the frequency band is 6.2 dB. This stable positive insertion loss fully demonstrates that the vibration isolation waveguide has good vibration isolation performance.

[0062] In this embodiment, the energy harvesting performance of the vibration energy harvesting waveguide is evaluated using open-circuit output voltage and load output power; such as Figure 8 As shown in (b), the red and black lines represent the open-circuit voltages measured under no-load conditions for the vibration energy trapping waveguide and the plate of equal thickness, respectively. The vibration energy trapping waveguide generated a peak-to-peak voltage of 7.76V, which is 2.2 times that of the plate of equal thickness. Figure 8 As shown in (c), the output power of the vibration energy trapping waveguide and the plate of equal thickness is calculated under a load of 200Ω. The vibration energy trapping waveguide has a maximum power of 6.52mW and an average power of 0.47mW, which are 2.7 times and 4.7 times that of the plate of equal thickness (maximum 2.43mW, average 0.10mW), respectively.

[0063] In summary, this scheme employs quasi-conformal transformation, enabling waveguide design with arbitrary smooth curve profiles, adapting to complex engineering boundary conditions, and breaking through the shape limitations of traditional waveguides. By controlling the refractive index distribution through thickness gradient, the waveguide maintains good waveguide performance over a wide frequency band, making it suitable for broadband vibration environments. The waveguide structure achieves its function through thickness variation, eliminating the need for complex periodic units and allowing for fabrication using conventional processes such as CNC machining, reducing manufacturing costs and complexity. This scheme supports multiple functions such as vibration isolation, deflection, and energy harvesting, effectively guiding the propagation path of flexural waves and enhancing the vibration control capability of thin-plate structures. Furthermore, this scheme can be combined with absorbing materials and piezoelectric materials to further optimize boundary scattering problems and achieve vibration energy harvesting and utilization, expanding the functional application range of waveguides.

[0064] Although the specific embodiments of the invention have been described in detail with reference to the accompanying drawings, this should not be construed as limiting the scope of protection of this patent; various modifications and variations that can be made by a person skilled in the art without inventive effort within the scope described in the claims are still within the scope of protection of this patent.

Claims

1. A design method for broadband curved waveguides of arbitrary shape, characterized in that, Includes the following steps: S1: Determine the contour shape of the waveguide; S2: Calculate the flexural refractive index distribution of the waveguide based on the quasi-conformal transformation; S3: Calculate the thickness point cloud distribution of the waveguide based on the refractive index distribution; S4: Optimize the thickness point cloud distribution; S5: Construct a waveguide thickness point cloud surface based on the optimized thickness point cloud distribution; S6: Establish a three-dimensional geometric model of the waveguide; S7: Embed the waveguide into the thin plate structure.

2. The design method for broadband curved waveguides of arbitrary shape according to claim 1, characterized in that, The waveguide has a smooth curve profile, described by an analytical function or an interpolated curve.

3. The design method for broadband curved waveguides of arbitrary shape according to claim 1, characterized in that, The refractive index distribution mentioned in step S2 is calculated using numerical methods, specifically including: S21: Apply first and second type boundary conditions based on Cauchy-Riemann conditions to the waveguide boundary in physical space; S22: Solve the Laplace equation under the given boundary conditions using numerical methods to obtain the distribution of the transformation function; S23: Calculate the inverse Jacobian matrix of the coordinate transformation from physical space to virtual space based on the transformation function and its conjugate harmonic function; S24: Calculate the refractive index distribution based on the inverse Jacobian matrix.

4. The design method for broadband curved waveguides of arbitrary shape according to claim 3, characterized in that, When a negative value appears in the calculated refractive index distribution, the waveguide is trimmed, and the exit cutoff of the trimmed waveguide is parallel to the contour lines of the dependent variable in the Laplace equation.

5. The design method for broadband curved waveguides of arbitrary shape according to claim 1, characterized in that, The formula for calculating the thickness distribution in step S3 is: in, h 0 represents the substrate thickness. n z This represents the refractive index distribution.

6. The design method for broadband curved waveguides of arbitrary shape according to claim 1, characterized in that, The optimization process in step S4 includes: using natural neighborhood interpolation and Gaussian filtering to smooth the thickness point cloud in order to eliminate edge burrs and surface roughness.

7. The design method for broadband curved waveguides of arbitrary shape according to claim 1, characterized in that, Step S5 specifically includes: constructing a waveguide thickness point cloud surface using a parametric surface modeling method.

8. The design method for broadband curved waveguides of arbitrary shape according to claim 1, characterized in that, Step S6 specifically includes: establishing a three-dimensional geometric model of the waveguide based on the waveguide thickness point cloud surface and through geometric Boolean operations.

9. The design method for broadband curved waveguides of arbitrary shape according to claim 1, characterized in that, After the waveguide is embedded in the thin plate structure in step S7, its profile thickness is greater than, equal to or less than that of the thin plate structure.

10. The design method for broadband curved waveguides of arbitrary shape according to claim 9, characterized in that, Blu-Tack absorber is provided in the region where the waveguide profile is thicker than the substrate to reduce boundary scattering caused by thickness discontinuity; piezoelectric material for vibration energy harvesting is provided at the thinnest part of the waveguide.

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