Dynamic Modeling and Modal Analysis of Rectangular Plates with Edge Symmetric Periodic Acoustic Black Holes
Abstract
1. Introduction
2. Dynamic Model of Acoustic Black Hole Symmetric Periodic Boundary Rectangular Plate
2.1. Geometric Configuration and Mathematical Description of the Edge of a Rectangular Plate Periodic ABH
- (1)
- A rectangular plate on the edge of a semicircular annular ABH
- (2)
- A rectangular plate on the edge of a rectangular block-shaped ABH
2.2. Dynamic Model of Rectangular Plate with Periodic ABH Boundary
2.3. Model Validation
3. Modal Characteristics of Rectangular Plate with Periodic Boundary of ABHs
3.1. Influence of Power Law Exponent m of ABH on Modal Frequency of Rectangular Plate
3.2. Influence of Radius of ABH on Modal Frequency of Rectangular Plate
3.3. Influence of Combination of ABH Number and Black Hole Radius on Modal Frequency of Rectangular Plate
3.4. Influence of ABH Configuration on Modal Frequency of Rectangular Plate
4. Conclusions
- (1)
- The “remove-and-fill” substitution method proposed in this paper can effectively resolve the dynamic modeling problems of the rectangular plate with a variable cross-section in the edge ABH by constructing a local energy model of the rectangular plate in the edge ABH. Through selecting the Gaussian function as the shape function, the rapid changes in the ABH region can be captured with fewer terms, which significantly reduces the computational degrees of freedom and improves the computational efficiency and stability of the solution.
- (2)
- The edge parameters of the periodic ABHs, including the radii of the edge ABHs, the number of ABHs, and the periodic distribution of the ABHs, directly affect the overall bending stiffness of the plate structure and significantly influence its modal frequency, which is a key parameter in the regulation of the dynamic properties of the structure. In addition, the configuration of the ABH also has an important effect on the modal frequency of the plate structure, and the modal frequency of the semicircular ABHs is relatively low compared with that of the rectangular wedge-shaped ABHs. The modal frequencies of rectangular plates with four-side symmetric periodic ABHs are lower than those of two-side symmetric rectangular plates.
- (3)
- Without changing the main structure, the design and regulation of the structural modal frequency can be realized only through boundary structure design, which has significant engineering value and application prospects in the vibration and noise control of aerospace panels, automobile base plates, and body structures, etc.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Geometric Parameters | Material Parameters | ||
---|---|---|---|
Length and width of plate a × b | 0.16 m × 0.16 m | 0.40 m × 0.30 m | Steel |
Thickness of plate | 2 mm | 5 mm | ρ1 = 7800 kg/m3 |
Number of black holes, long side p, short side q | p = 1, q = 1 | p = 4, q = 2 | E1 = 210 GPa |
Power law exponent of black hole m | 2 | 2 | |
Radius of black hole | 10 mm | 20 mm | ν1 = 0.3 |
Boundary condition | Quadrilateral freedom | Quadrilateral freedom |
Modal Order | Theory ft (Hz) | Simulation fs (Hz) | Relative Error a (%) |
---|---|---|---|
(1,1) | 25.75 | 25.83 | 0.31 |
(1,2) | 79.61 | 80.27 | 0.83 |
(2,2) | 150.53 | 153.24 | 1.80 |
(1,3) | 160.36 | 161.57 | 0.75 |
(2,3) | 221.81 | 220.78 | 0.46 |
(1,4) | 259.45 | 260.46 | 0.39 |
Modal Order | Experiment ft (Hz) | Simulation fs (Hz) | Relative Error a (%) |
---|---|---|---|
(1,1) | 137.23 | 139.05 | 1.01 |
(2,1) | 164.14 | 166.46 | 1.41 |
(1,2) | 310.31 | 312.19 | 0.61 |
(3,1) | 325.88 | 328.79 | 0.89 |
(2,2) | 481.21 | 485.84 | 0.96 |
Modal Order | Power Law Exponent m | 2.5 | 3.0 |
---|---|---|---|
2 | |||
(1,1) | 84.50 | 84.42 | 84.35 |
(2,1) | 176.08 | 175.90 | 175.73 |
(1,2) | 248.93 | 248.65 | 248.48 |
(3,1) | 330.33 | 329.99 | 429.69 |
(2,2) | 339.48 | 338.97 | 338.55 |
(3,2) | 492.61 | 491.75 | 490.96 |
Geometric Parameter | Semicircular Section | Rectangular Section 1 | Rectangular Section 2 |
---|---|---|---|
m | 2.5 | 2.5 | 2.5 |
h | 2 mm | 2 mm | 2 mm |
a | 400 mm | 400 mm | 400 mm |
b | 300 mm | 300 mm | 300 mm |
0.5 mm | 0.5 mm | 0.5 mm | |
20 mm | - | - | |
- | 40 mm | 16 mm | |
- | 16 mm | 40 mm | |
4.87481/m | 4.87481/m | 4.87481/m |
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Shi, Y.; Liu, Z.; Fan, Q.; Wang, X.; Huang, Q.; Peng, J. Dynamic Modeling and Modal Analysis of Rectangular Plates with Edge Symmetric Periodic Acoustic Black Holes. Symmetry 2025, 17, 1031. https://doi.org/10.3390/sym17071031
Shi Y, Liu Z, Fan Q, Wang X, Huang Q, Peng J. Dynamic Modeling and Modal Analysis of Rectangular Plates with Edge Symmetric Periodic Acoustic Black Holes. Symmetry. 2025; 17(7):1031. https://doi.org/10.3390/sym17071031
Chicago/Turabian StyleShi, Yuanyuan, Ziyi Liu, Qiyuan Fan, Xiao Wang, Qibai Huang, and Jiangying Peng. 2025. "Dynamic Modeling and Modal Analysis of Rectangular Plates with Edge Symmetric Periodic Acoustic Black Holes" Symmetry 17, no. 7: 1031. https://doi.org/10.3390/sym17071031
APA StyleShi, Y., Liu, Z., Fan, Q., Wang, X., Huang, Q., & Peng, J. (2025). Dynamic Modeling and Modal Analysis of Rectangular Plates with Edge Symmetric Periodic Acoustic Black Holes. Symmetry, 17(7), 1031. https://doi.org/10.3390/sym17071031