Abstract
Oscillations and waves in multisection periodic systems with symmetric subsystems are investigated. Many engineering systems, building structures, as well as models of acoustic metamaterials have a similar structure. The structure of the natural frequency spectrum for such a class of systems is found, taking into account the elastic properties of the constituent subsystems. It is shown that such systems have band gaps of a harmonic signal and the dispersion curve consists of n branches according to the number of degrees of freedom in the subsystem. The boundaries of the harmonic signal band gaps have been found in the analytical form. Vibration modes have been obtained in different frequency ranges. Modulated waves are shown to arise in the system due to modulation by lower frequencies that correspond to system oscillations without taking into account the elasticity of the constituent subsystems. In the analysis of symmetric structures, the theory of groups is used in combination with wave dispersion equations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Banakh LY, Barmina OV, Volokhovskaya OA (2021) Oscillations and waves in multi-section rotor systems. Journal of Machinery Manufacture and Reliability 50(5):396–404
Zingoni A (2005) On the symmetries and vibration modes of layered space grids. Engineering Structures 27(4):629–638
Mohan SJ, Pratap R (2004) A natural classification of vibration modes of polygonal ducts based on group theoretic analysis. Journal of Sound and Vibration 269(3):745–764
Dyachenko MI, Pavlov AM, Temnov AN (2015) Longitudinal elastic vibrations of multistage liquid-propellant launch vehicle body (in Russ.). Vestn Mosk Gos Tekh Univ im NE Baumana (5):14–24
Banakh LY, Grinkevitch VK, Ovchinnikova NF, Fedoseev YN (1988) Analysis of the dynamics of mechanisms using symmetry groups. Mashinovedenie (2):67–70
Zingoni A (2014) Group-theoretic insights on the vibration of symmetric structures in engineering. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372(2008):20120,037
Ching FDK, Onouye BS, Zuberbuhler D (2014) Building Structures Illustrated: Patterns, Systems, and Design, 2nd edn. Wiley
Dong B, Parker RG (2022) Vibration of multi-stage systems with arbitrary symmetry of stages: A group theory approach. Journal of Sound and Vibration 524:116,738
Zucco G, Weaver PM (2020) The role of symmetry in the post-buckling behaviour of structures. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476(2233):20190,609
Samaranayake S, Samaranayake G, Bajaj AK (2000) Resonant vibrations in harmonically excited weakly coupled mechanical systems with cyclic symmetry. Chaos, Solitons & Fractals 11(10):1519–1534
Zingoni A (2009) Group-theoretic exploitations of symmetry in computational solid and structural mechanics. International Journal for Numerical Methods in Engineering 79(3):253–289
Vakakis AF (1992) Dynamics of a nonlinear periodic structure with cyclic symmetrys. Acta Mechanica 95(3):197–226
Srinivasan AV (1984) Vibrations of Bladed-Disk Assemblies—A Selected Survey (Survey Paper). Journal of Vibration, Acoustics, Stress, and Reliability in Design 106(2):165–168
Thomas D (1974) Standing waves in rotationally periodic structures. Journal of Sound and Vibration 37(2):288–290
Bobylyov VN, Grebnev PA, Erofeev VI, Kuzmin DS, Monich DV (2020) Sound insulation of frameless sandwich panels with groove-type connection of the middle layer. Privolzhsky Scientific Journal 8(3):9–18
Hu G, Tang L, Das R, Gao S, Liu H (2017) Acoustic metamaterials with coupled local resonators for broadband vibration suppression. AIP Advances 7(2):025,211
Qureshi A, Li B, Tan KT (2016) Numerical investigation of band gaps in 3D printed cantilever-in-mass metamaterials. Scientific Reports 6:28,314
Zhou X, Wang J, Wang R, Lin J (2017) Band gaps in grid structure with periodic local resonator subsystems. Modern Physics Letters B 31(25):1750,225
Vasiliev AA, Pavlov IS (2021) Discrete and generalized continuum dynamical models of tetrachiral cosserat lattices with finite-sized particles. Mechanics Research Communications 115:103,732
Bobrovnitskii YI (2022) Impedance theory of wave propagation on infinite periodic structures. Journal of Sound and Vibration 525:116,801
Banakh LY, Kempner ML (2010) Vibrations of Mechanical Systems with Regular Structure. Foundations of Engineering Mechanics, Springer, Berlin Heidelberg
Weyl H (2007) The Theory of Groups and Quantum Mechanics. Kessinger Publishing
Hammermesh M (1962) Group Theory and its Application to Physical Problems. Addison-Wesley
Zlokovic GM (1989) Group Theory and G-vector Spaces in Structural Analysis. Ellis Horwood, Group C, Chichester
Zienkiewicz OC (1971) The Finite Element Method in Structural and Continuum Mechanics. McGraw-Hill, London
Oden JT (1971) Finite elements: An introductions. In: Handbook of Numerical Analysis, Finite element methods (Part I), Handbook of Numerical Analysis, Finite element methods (Part I), Amsterdam, pp 3–12
Ewins DJ, Imregun M (1984) Vibration Modes of Packeted Bladed Disks. Journal of Vibration, Acoustics, Stress, and Reliability in Design 106(2):175–180
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Banakh, L.Y., Pavlov, I.S. (2023). Dynamic Properties of Periodic Structures with Symmetric Inclusions. In: Altenbach, H., Irschik, H., Porubov, A.V. (eds) Progress in Continuum Mechanics. Advanced Structured Materials, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-031-43736-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-031-43736-6_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-43735-9
Online ISBN: 978-3-031-43736-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)