Study on Vibration Band Gap Characteristics of a Branched Shape Periodic Structure Using the GDQR

Subject Areas : Mechanics of Solids

M Hajhosseini ¹ , A Abshahi ²

1 - Department of Mechanical Engineering, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran
2 - Department of Mechanical Engineering, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

Received: 2021-03-26 Accepted : 2021-05-27 Published : 2021-09-30

Keywords: Close band gaps, GDQR method, periodic structure, Transverse vibration, ANSYS,

Abstract :

In this study, a new periodic structure with special vibration band gap properties is introduced. This structure consists of a main beam and several cantilever beam elements connected to this main beam in the branched shape. Two models with different number of beam elements and geometrical parameters are considered for this periodic structure. The transverse vibrations of beams are solved using the generalized differential quadrature rule (GDQR) method to calculate the first four band gaps of each model. Investigating the influences of geometrical parameters on the band gaps shows that some bands are close to each other for specific ranges of geometrical parameters values. Furthermore, as the number of beam elements increases, the number of close band gaps increases. Having more than two close band gaps means that this periodic structure has a relatively wide band gap in total. Furthermore, this wide band can move to low frequency ranges by changing the geometrical parameters. Absorbing vibrations over a wide band gap at low frequency ranges makes this periodic structure a good vibration absorber. Verification of the analytical method using ANSYS software shows that the GDQR method can be used for vibration analysis of beam-like structures with high accuracy.

References:

[1] Hajhosseini M., Rafeeyan M., 2016, Modeling and analysis of piezoelectric beam with periodically variable cross-sections for vibration energy harvesting, Applied Mathematics and Mechanics 37(8): 1053-1066.
[2] Wen J.H., Wang G., Yu D.L., Zhao H.G., Liu Y.Z., 2008, Study on the vibration band gap and vibration attenuation property of phononic crystals, Science in China Series E-Technological Sciences 51(1): 85-99.
[3] Olhoff N., Niu B., Cheng G., 2012, Optimum design of band-gap beam structures, International Journal of Solids and Structures 49(22): 3158-3169.
[4] Kamotski L.V., Smyshlyaev V.P., 2019, Band gaps in two-dimensional high-contrast periodic elastic beam lattice materials, Journal of the Mechanics and Physics of Solids 123: 292-304.
[5] Xiang H.J., Cheng Z.B., Shi Z.F., Yu X.Y., 2014, In-plane Band Gaps in a Periodic Plate with Piezoelectric Patches, Journal of Solid Mechanics 6(2): 194-207.
[6] Zouari S., Brocail J., Génevaux J.M., 2018, Flexural wave band gaps in metamaterial plates: A numerical and experimental study from infinite to finite models, Journal of Sound and Vibration 435: 246-263.
[7] Hajhosseini M., Rafeeyan M., Ebrahimi S., 2017, Vibration band gap analysis of a new periodic beam model using GDQR method, Mechanics Research Communications 79: 43-50.
[8] Hajhosseini M., Mahdian Parrany A., 2019, Vibration band gap properties of a periodic beam-like structure using the combination of GDQ and GDQR methods, Waves in Random and Complex Media 2019: 1-17.
[9] Zhu Z.W., Deng Z.C., Huang B., Du J.K., 2019, Elastic wave propagation in triangular chiral lattices: Geometric frustration behavior of standing wave modes, International Journal of Solids and Structures 158: 40-51.
[10] Wu Z.J., Li F.M., Zhang C., 2014, Vibration properties of piezoelectric square lattice structures, Mechanics Research Communications 62: 123-131.
[11] Hajhosseini M., Ebrahimi S., 2019, Analysis of vibration band gaps in an Euler–Bernoulli beam with periodic arrays of meander-shaped beams, Journal of Vibration and Control 25(1): 41-51.
[12] Guo X., Liu H., Zhang K., Duan H., 2018, Dispersion relations of elastic waves in two-dimensional tessellated piezoelectric phononic crystals, Applied Mathematical Modelling 56: 65-82.
[13] Wang G., Wen J., Liu Y., Wen X., 2004, Lumped-mass method for the study of band structure in two-dimensional phononic crystals, Physical Review B - Condensed Matter and Materials Physics 69: 184302.
[14] Liang X., Wang T., Jiang X., Liu Z., Ruan Y., Deng Y., 2019, A numerical method for flexural vibration band gaps in a phononic crystal beam with locally resonant oscillators, Crystals 9(6): 293.
[15] Chang I.L., Liang Z.X., Kao H.W., Chang S.H., Yang C.Y., 2018, The wave attenuation mechanism of the periodic local resonant metamaterial, Journal of Sound and Vibration 412: 349-359.
[16] Zhou X., Xu Y., Liu Y., Lv L., Peng F., Wang L., 2018, Extending and lowering band gaps by multilayered locally resonant phononic crystals, Applied Acoustics 133: 97-106.
[17] Leissa A.W., Qatu M.S., 2011, Vibration of Continuous Systems, McGraw-Hill Professional.
[18] Kittel C., 2005, Introduction to Solid State Physics, John Wiley & Sons, New York.
[19] Bellman R., Casti J., 1971, Differential quadrature and long term integration, Journal of Mathematical Analysis and Applications 34(2): 235-238.
[20] Liang X., Zha X., Jiang X., Cao Z., Wang Y., Leng J., 2019, A semi-analytical method for the dynamic analysis of cylindrical shells with arbitrary boundaries, Ocean Engineering 178: 145-155.
[21] Liang X., Deng Y., Jiang X., Cao Z., Ruan Y., Leng J., Wang T., Zha X., 2019, Three-dimensional semi-analytical solutions for the transient response of functionally graded material cylindrical panels with various boundary conditions, Journal of Low Frequency Noise, Vibration and Active Control 39: 1002-1023.
[22] Wu T.Y., Liu G.R., 1999, A differential quadrature as a numerical method to solve differential equations, Computational Mechanics 24(3): 197-205.
[23] Hajhosseini M., 2020, Analysis of complete vibration bandgaps in a new periodic lattice model using the differential quadrature method, Journal of Vibration and Control 26: 1708-1720.
[24] Shu C., 2000, Differential Quadrature and its Applications in Engineering, Springer-Verlag, London.