Effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam

Vladislav S. Sorokin; Jon Juel Thomsen

doi:10.1098/rspa.2015.0751

Effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam

Vladislav S. Sorokin, Jon Juel Thomsen

Research output: Contribution to journal › Journal article › Research › peer-review

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Abstract

The paper deals with analytically predicting the effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli– Euler beam performing bending oscillations. Two cases are considered: (i) large transverse deflections, where nonlinear (true) curvature, nonlinear material and nonlinear inertia owing to longitudinal motions of the beam are taken into account, and (ii) midplane stretching nonlinearity. A novel approach is employed, the method of varying amplitudes. As a result, the isolated as well as combined effects of the considered sources of nonlinearities are revealed. It is shown that nonlinear inertia has the most substantial impact on the dispersion relation of a nonuniform beam by removing all frequency band-gaps. Explanations of the revealed effects are suggested, and validated by experiments and numerical simulation.

Original language	English
Article number	20150751
Journal	Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume	472
Issue number	2186
Number of pages	22
ISSN	1364-5021
DOIs	https://doi.org/10.1098/rspa.2015.0751
Publication status	Published - 2016

Keywords

Elastic wave propagation
Dispersion relation
Frequency band-gaps
Weak nonlinearity
Periodic Bernoulli–Euler beam
Method of varying amplitudes

Cite this

Sorokin, V. S., & Thomsen, J. J. (2016). Effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2186), [20150751]. https://doi.org/10.1098/rspa.2015.0751

Sorokin, Vladislav S. ; Thomsen, Jon Juel. / Effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam. In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2016 ; Vol. 472, No. 2186.

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title = "Effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam",

abstract = "The paper deals with analytically predicting the effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli– Euler beam performing bending oscillations. Two cases are considered: (i) large transverse deflections, where nonlinear (true) curvature, nonlinear material and nonlinear inertia owing to longitudinal motions of the beam are taken into account, and (ii) midplane stretching nonlinearity. A novel approach is employed, the method of varying amplitudes. As a result, the isolated as well as combined effects of the considered sources of nonlinearities are revealed. It is shown that nonlinear inertia has the most substantial impact on the dispersion relation of a nonuniform beam by removing all frequency band-gaps. Explanations of the revealed effects are suggested, and validated by experiments and numerical simulation.",

keywords = "Elastic wave propagation, Dispersion relation, Frequency band-gaps, Weak nonlinearity, Periodic Bernoulli–Euler beam, Method of varying amplitudes",

author = "Sorokin, {Vladislav S.} and Thomsen, {Jon Juel}",

year = "2016",

doi = "10.1098/rspa.2015.0751",

language = "English",

volume = "472",

journal = "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",

issn = "1364-5021",

publisher = "The/Royal Society",

number = "2186",

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Sorokin, VS & Thomsen, JJ 2016, 'Effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam', Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 472, no. 2186, 20150751. https://doi.org/10.1098/rspa.2015.0751

Effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam. / Sorokin, Vladislav S.; Thomsen, Jon Juel.

In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 472, No. 2186, 20150751, 2016.

Research output: Contribution to journal › Journal article › Research › peer-review

TY - JOUR

T1 - Effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam

AU - Sorokin, Vladislav S.

AU - Thomsen, Jon Juel

PY - 2016

Y1 - 2016

N2 - The paper deals with analytically predicting the effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli– Euler beam performing bending oscillations. Two cases are considered: (i) large transverse deflections, where nonlinear (true) curvature, nonlinear material and nonlinear inertia owing to longitudinal motions of the beam are taken into account, and (ii) midplane stretching nonlinearity. A novel approach is employed, the method of varying amplitudes. As a result, the isolated as well as combined effects of the considered sources of nonlinearities are revealed. It is shown that nonlinear inertia has the most substantial impact on the dispersion relation of a nonuniform beam by removing all frequency band-gaps. Explanations of the revealed effects are suggested, and validated by experiments and numerical simulation.

AB - The paper deals with analytically predicting the effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli– Euler beam performing bending oscillations. Two cases are considered: (i) large transverse deflections, where nonlinear (true) curvature, nonlinear material and nonlinear inertia owing to longitudinal motions of the beam are taken into account, and (ii) midplane stretching nonlinearity. A novel approach is employed, the method of varying amplitudes. As a result, the isolated as well as combined effects of the considered sources of nonlinearities are revealed. It is shown that nonlinear inertia has the most substantial impact on the dispersion relation of a nonuniform beam by removing all frequency band-gaps. Explanations of the revealed effects are suggested, and validated by experiments and numerical simulation.

KW - Elastic wave propagation

KW - Dispersion relation

KW - Frequency band-gaps

KW - Weak nonlinearity

KW - Periodic Bernoulli–Euler beam

KW - Method of varying amplitudes

U2 - 10.1098/rspa.2015.0751

DO - 10.1098/rspa.2015.0751

M3 - Journal article

VL - 472

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences