The bandgap characteristics of periodically arranged functionally graded stepped beams are studied based on the energy variational principle in this study. The displacement field of the beam is uniformly described using an improved Fourier series. The coupling between beam segments is achieved through artificial spring technology, which simultaneously decouples the periodic boundary conditions and the shape functions. An energy method with good convergence and accuracy has been established to solve the bandgap characteristics of the beam. Through parameter research, the influence of geometric dimensions and material properties on bandgap was analyzed. The key innovation lies in introducing the simplex method to improve the beetle optimizer, optimizing the structural parameters for the first bandgap bandwidth, effectively enhancing the local search capability of the optimizer. The first bandgap bandwidth of the optimized structure has significantly increased, which has been verified through forced vibration analysis of a 10-period beam. The bandgap characteristic exhibits good stability under different boundary conditions.
AkbaşŞD (2018) Forced vibration analysis of functionally graded porous deep beams. Composite Structures186: 293–302.
2.
AslanTANooriARTemelB (2023) An efficient approach for free vibration analysis of functionally graded sandwich beams of variable cross-section. Structures58: 105397.
3.
BelabedZBousahlaAATounsiA (2024) Vibrational and elastic stability responses of functionally graded carbon nanotube reinforced nanocomposite beams via a new quasi-3D finite element model. Computers and Concrete34: 625–648.
4.
BurlayenkoVNKouhiaRDimitrovaSD (2024) One-dimensional vs. three-dimensional models in free vibration analysis of axially functionally graded beams with non-uniform cross-sections. Mechanics of Composite Materials60: 83–102.
5.
ChenW-RChangH (2018) Vibration analysis of functionally graded Timoshenko beams. International Journal of Structural Stability and Dynamics18: 1850007.
6.
ChenYJinGZhangC, et al. (2018) Thermal vibration of FGM beams with general boundary conditions using a higher-order shear deformation theory. Composites Part B: Engineering153: 376–386.
7.
DalklintAWallinMBertoldiK, et al. (2022) Tunable phononic bandgap materials designed via topology optimization. Journal of the Mechanics and Physics of Solids163: 104849.
8.
DengJGuaschOZhengL (2020) A semi-analytical method for characterizing vibrations in circular beams with embedded acoustic black holes. Journal of Sound and Vibration476: 115307.
9.
DuYJiaDLiH, et al. (2022) A unified method to analyze free and forced vibration of stiffened plates under various edge conditions. European Journal of Mechanics - A: Solids94: 104573.
10.
DuYTangYPangF, et al. (2023) A semi-analytical method to design distributed dynamic vibration absorber of beam under general elastic edge constraints. International Journal of Structural Stability and Dynamics24: 2450181.
11.
EftekhariS (2015) A note on mathematical treatment of the Dirac-delta function in the differential quadrature bending and forced vibration analysis of beams and rectangular plates subjected to concentrated loads. Applied Mathematical Modelling39: 6223–6242.
12.
FanS-KSZaharaE (2007) A hybrid simplex search and particle swarm optimization for unconstrained optimization. European Journal of Operational Research181: 527–548.
13.
FuyadSTMAl BanAMakfidunnabiM, et al. (2024) Finite element analysis of ratcheting on beam under bending-bending loading conditions. Structural Engineering & Mechanics89: 23–31.
14.
GaoNWeiZHouH, et al. (2019) Design and experimental investigation of V-folded beams with acoustic black hole indentations. Journal of the Acoustical Society of America145: EL79–EL83.
15.
GaoCPangFLiH, et al. (2022) Forced vibration analysis of uniform and stepped circular cylindrical shells with general boundary conditions. International Journal of Structural Stability and Dynamics22: 2250126.
16.
GasparettoVEElSayedMS (2021) Multiscale optimization of specific elastic properties and microscopic frequency band-gaps of architectured microtruss lattice materials. International Journal of Mechanical Sciences197: 106320.
17.
GhannadiaslAAjirlouSK (2019) Forced vibration of multi-span cracked Euler–Bernoulli beams using dynamic green function formulation. Applied Acoustics148: 484–494.
18.
GouJLeiY-XGuoW-P, et al. (2017) A novel improved particle swarm optimization algorithm based on individual difference evolution. Applied Soft Computing57: 468–481.
19.
HajhosseiniMParranyAM (2022) A new periodic beam-like structure with special vibration-isolation characteristics. Mechanics of Advanced Materials and Structures29: 3804–3814.
20.
KushwahaMSHaleviPDobrzynskiL, et al. (1993) Acoustic band structure of periodic elastic composites. Physical Review Letters71: 2022–2025.
21.
LakhdarKSadounMAddouFY, et al. (2024) Free vibrational characteristics of various imperfect FG beam via a novel integral Timoshenko's theory. Acta Mechanica235: 6287–6304.
22.
LeeJWLeeJY (2017) Free vibration analysis of functionally graded Bernoulli-Euler beams using an exact transfer matrix expression. International Journal of Mechanical Sciences122: 1–17.
23.
LiWL (2000) Free vibrations of beams with general boundary conditions. Journal of Sound and Vibration237: 709–725.
24.
LiWLBonilhaMWXiaoJ (2012) Vibrations of two beams elastically coupled together at an arbitrary angle. Acta Mechanica Solida Sinica25: 61–72.
25.
LiHPangFChenH, et al. (2019a) Vibration analysis of functionally graded porous cylindrical shell with arbitrary boundary restraints by using a semi analytical method. Composites Part B: Engineering164: 249–264.
26.
LiHPangFMiaoX, et al. (2019b) Jacobi–Ritz method for free vibration analysis of uniform and stepped circular cylindrical shells with arbitrary boundary conditions: a unified formulation. Computers & Mathematics with Applications77: 427–440.
27.
LiWMengFChenY, et al. (2019c) Topology optimization of photonic and phononic crystals and metamaterials: a review. Advanced Theory and Simulations2: 1900017.
28.
LiHPangFGaoC, et al. (2020) A Jacobi-Ritz method for dynamic analysis of laminated composite shallow shells with general elastic restraints. Composite Structures242: 112091.
29.
LiuZZhangXMaoY, et al. (2000) Locally resonant sonic materials. Science (New York, N.Y.)289: 1734–1736.
30.
LuQLiuCWangP (2022) Band gap enhancement and vibration reduction of functionally graded sandwich metastructure beam. Composite Structures292: 115650.
31.
MaoZXYangZLuoD, et al. (2025) A multi-strategy enhanced dung beetle algorithm for solving real-world engineering problems. Artificial Intelligence Review58: 253.
32.
NetoAG (2017) Simulation of mechanisms modeled by geometrically-exact beams using Rodrigues rotation parameters. Computational Mechanics59: 459–481.
33.
PanahiEHosseinkhaniAFrangiA, et al. (2022) A novel low-frequency multi-bandgaps metaplate: genetic algorithm based optimization and experimental validation. Mechanical Systems and Signal Processing181: 109495.
34.
PangFTangYQinY, et al. (2023) Analysis of acoustic radiation characteristic of laminated paraboloidal shell based on Jacobi-Ritz-spectral BEM. Ocean Engineering280: 114686.
35.
RongJYeW (2019) Topology optimization design scheme for broadband non-resonant hyperbolic elastic metamaterials. Computer Methods in Applied Mechanics and Engineering344: 819–836.
36.
SarkarKGanguliR (2014) Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed–fixed boundary condition. Composites Part B: Engineering58: 361–370.
37.
ShiDWangQShiX, et al. (2015) An accurate solution method for the vibration analysis of Timoshenko beams with general elastic supports. Proceedings of the Institution of Mechanical Engineers - Part C: Journal of Mechanical Engineering Science229: 2327–2340.
38.
ŞimşekM (2010) Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nuclear Engineering and Design240: 697–705.
39.
SuZJinGYeT (2018) Vibration analysis of multiple-stepped functionally graded beams with general boundary conditions. Composite Structures186: 315–323.
40.
SuginoCLeadenhamSRuzzeneM, et al. (2016) On the mechanism of bandgap formation in locally resonant finite elastic metamaterials. Journal of Applied Physics120: 134501.
41.
TangYWeiJHQinYX, et al. (2024a) Sound insulation characteristics of sandwich thin-plate acoustic metamaterial: analysis and optimization. International Journal of Computational Methods.
42.
TangYZhaoZQinY, et al. (2024b) Experimental and numerical investigation on vibro-acoustic performance of a submerged stiffened cylindrical shell under multiple excitations. Thin-Walled Structures197: 111569.
43.
TaoPLiuYDuJ, et al. (2023) Vibration and bandgap characteristics analysis of multiple beams with arbitrary connection angles. Structures58: 105534, Elsevier.
44.
TounsiABelabedZBounouaraF, et al. (2024) A finite element approach for forced dynamical responses of porous FG nanocomposite beams resting on viscoelastic foundations. International Journal of Structural Stability and Dynamics.
45.
WuXLiYZuoS (2020) The study of a locally resonant beam with aperiodic mass distribution. Applied Acoustics165: 107306.
46.
XiaoYWangSXLiYQ, et al. (2021) Closed-form bandgap design formulas for beam-type metastructures. Mechanical Systems and Signal Processing159: 107777.
47.
XieKWangYFanX, et al. (2020) Nonlinear free vibration analysis of functionally graded beams by using different shear deformation theories. Applied Mathematical Modelling77: 1860–1880.
48.
YanLShiDZhangY, et al. (2023) Vibration band gap characteristics of two-dimensional functionally graded grids using the spectral element method. Journal of Vibration and Control29: 4948–4958.
49.
YaylaciMYaziciogluAYaylaciEU, et al. (2025) Evaluation of the contact problem of two layers one of functionally graded, loaded by circular rigid block and resting on a Pasternak foundation by analytical and numerical (FEM and MLP) methods. Archive of Applied Mechanics95(4): 1–23.
50.
YuDLLiuYZWangG, et al. (2006) Flexural vibration band gaps in Timoshenko beams with locally resonant structures. Journal of Applied Physics100: 124901.
51.
YukselOYilmazC (2015) Shape optimization of phononic band gap structures incorporating inertial amplification mechanisms. Journal of Sound and Vibration355: 232–245.
52.
ZhangZJiGHanX (2019) Optimization design of a novel zigzag lattice phononic crystal with holes. International Journal of Modern Physics B33: 1950124.
53.
ZhangZQianDZouP (2023) Study on band gap characteristics of phononic crystal double-layer beam structure attached by multilayer cylinder. Crystals13: 638.
54.
ZhangDMWangZJZhaoYQ, et al. (2024) Multi-strategy fusion improved dung beetle optimization algorithm and engineering design application. IEEE Access12: 97771–97786.
55.
ZhaoJFLiYLiuWK (2015) Predicting band structure of 3D mechanical metamaterials with complex geometry via XFEM. Computational Mechanics55: 659–672.
56.
ZhouJWangKXuD, et al. (2017) Local resonator with high-static-low-dynamic stiffness for lowering band gaps of flexural wave in beams. Journal of Applied Physics121: 044902.
57.
ZhouJDouLWangK, et al. (2019) A nonlinear resonator with inertial amplification for very low-frequency flexural wave attenuations in beams. Nonlinear Dynamics96: 647–665.
