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

Vladislav S. Sorokin, Jon Juel Thomsen

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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 languageEnglish
Article number20150751
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume472
Issue number2186
Number of pages22
ISSN1364-5021
DOIs
Publication statusPublished - 2016

Keywords

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

Cite this

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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",
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language = "English",
volume = "472",
journal = "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",
issn = "1364-5021",
publisher = "The/Royal Society",
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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 journalJournal articleResearchpeer-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

SN - 1364-5021

IS - 2186

M1 - 20150751

ER -