A numerical method for flexural vibration band gaps in a phononic crystal beam with locally resonant oscillators

Xu Liang, Titao Wang, Xue Jiang, Zhen Liu, Yongdu Ruan, Yu Deng

Research output: Contribution to journalArticle

Abstract

The differential quadrature method has been developed to calculate the elastic band gaps from the Bragg reflection mechanism in periodic structures efficiently and accurately. However, there have been no reports that this method has been successfully used to calculate the band gaps of locally resonant structures. This is because, in the process of using this method to calculate the band gaps of locally resonant structures, the non-linear term of frequency exists in the matrix equation, which makes it impossible to solve the dispersion relationship by using the conventional matrix-partitioning method. Hence, an accurate and efficient numerical method is proposed to calculate the flexural band gap of a locally resonant beam, with the aim of improving the calculation accuracy and computational efficiency. The proposed method is based on the differential quadrature method, an unconventional matrix-partitioning method, and a variable substitution method. A convergence study and validation indicate that the method has a fast convergence rate and good accuracy. In addition, compared with the plane wave expansion method and the finite element method, the present method demonstrates high accuracy and computational efficiency. Moreover, the parametric analysis shows that the width of the 1st band gap can be widened by increasing the mass ratio or the stiffness ratio or decreasing the lattice constant. One can decrease the lower edge of the 1st band gap by increasing the mass ratio or decreasing the stiffness ratio. The band gap frequency range calculated by the Timoshenko beam theory is lower than that calculated by the Euler-Bernoulli beam theory. The research results in this paper may provide a reference for the vibration reduction of beams in mechanical or civil engineering fields.

LanguageEnglish
Article number293
Number of pages17
JournalCrystals
Volume9
Issue number6
DOIs
Publication statusPublished - 5 Jun 2019

Fingerprint

Numerical methods
Energy gap
oscillators
vibration
Crystals
crystals
Computational efficiency
quadratures
matrix methods
mass ratios
stiffness
Stiffness
Euler-Bernoulli beams
Timoshenko beams
mechanical engineering
Periodic structures
Mechanical engineering
Civil engineering
Lattice constants
finite element method

Keywords

  • band gap
  • differential quadrature method
  • locally resonant
  • phononic crystal

Cite this

Liang, Xu ; Wang, Titao ; Jiang, Xue ; Liu, Zhen ; Ruan, Yongdu ; Deng, Yu. / A numerical method for flexural vibration band gaps in a phononic crystal beam with locally resonant oscillators. In: Crystals. 2019 ; Vol. 9, No. 6.
@article{dab07aecb7d14182a0a43e60405b054d,
title = "A numerical method for flexural vibration band gaps in a phononic crystal beam with locally resonant oscillators",
abstract = "The differential quadrature method has been developed to calculate the elastic band gaps from the Bragg reflection mechanism in periodic structures efficiently and accurately. However, there have been no reports that this method has been successfully used to calculate the band gaps of locally resonant structures. This is because, in the process of using this method to calculate the band gaps of locally resonant structures, the non-linear term of frequency exists in the matrix equation, which makes it impossible to solve the dispersion relationship by using the conventional matrix-partitioning method. Hence, an accurate and efficient numerical method is proposed to calculate the flexural band gap of a locally resonant beam, with the aim of improving the calculation accuracy and computational efficiency. The proposed method is based on the differential quadrature method, an unconventional matrix-partitioning method, and a variable substitution method. A convergence study and validation indicate that the method has a fast convergence rate and good accuracy. In addition, compared with the plane wave expansion method and the finite element method, the present method demonstrates high accuracy and computational efficiency. Moreover, the parametric analysis shows that the width of the 1st band gap can be widened by increasing the mass ratio or the stiffness ratio or decreasing the lattice constant. One can decrease the lower edge of the 1st band gap by increasing the mass ratio or decreasing the stiffness ratio. The band gap frequency range calculated by the Timoshenko beam theory is lower than that calculated by the Euler-Bernoulli beam theory. The research results in this paper may provide a reference for the vibration reduction of beams in mechanical or civil engineering fields.",
keywords = "band gap, differential quadrature method, locally resonant, phononic crystal",
author = "Xu Liang and Titao Wang and Xue Jiang and Zhen Liu and Yongdu Ruan and Yu Deng",
year = "2019",
month = "6",
day = "5",
doi = "10.3390/cryst9060293",
language = "English",
volume = "9",
journal = "Crystals",
issn = "2073-4352",
number = "6",

}

A numerical method for flexural vibration band gaps in a phononic crystal beam with locally resonant oscillators. / Liang, Xu; Wang, Titao; Jiang, Xue; Liu, Zhen; Ruan, Yongdu; Deng, Yu.

In: Crystals, Vol. 9, No. 6, 293, 05.06.2019.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A numerical method for flexural vibration band gaps in a phononic crystal beam with locally resonant oscillators

AU - Liang, Xu

AU - Wang, Titao

AU - Jiang, Xue

AU - Liu, Zhen

AU - Ruan, Yongdu

AU - Deng, Yu

PY - 2019/6/5

Y1 - 2019/6/5

N2 - The differential quadrature method has been developed to calculate the elastic band gaps from the Bragg reflection mechanism in periodic structures efficiently and accurately. However, there have been no reports that this method has been successfully used to calculate the band gaps of locally resonant structures. This is because, in the process of using this method to calculate the band gaps of locally resonant structures, the non-linear term of frequency exists in the matrix equation, which makes it impossible to solve the dispersion relationship by using the conventional matrix-partitioning method. Hence, an accurate and efficient numerical method is proposed to calculate the flexural band gap of a locally resonant beam, with the aim of improving the calculation accuracy and computational efficiency. The proposed method is based on the differential quadrature method, an unconventional matrix-partitioning method, and a variable substitution method. A convergence study and validation indicate that the method has a fast convergence rate and good accuracy. In addition, compared with the plane wave expansion method and the finite element method, the present method demonstrates high accuracy and computational efficiency. Moreover, the parametric analysis shows that the width of the 1st band gap can be widened by increasing the mass ratio or the stiffness ratio or decreasing the lattice constant. One can decrease the lower edge of the 1st band gap by increasing the mass ratio or decreasing the stiffness ratio. The band gap frequency range calculated by the Timoshenko beam theory is lower than that calculated by the Euler-Bernoulli beam theory. The research results in this paper may provide a reference for the vibration reduction of beams in mechanical or civil engineering fields.

AB - The differential quadrature method has been developed to calculate the elastic band gaps from the Bragg reflection mechanism in periodic structures efficiently and accurately. However, there have been no reports that this method has been successfully used to calculate the band gaps of locally resonant structures. This is because, in the process of using this method to calculate the band gaps of locally resonant structures, the non-linear term of frequency exists in the matrix equation, which makes it impossible to solve the dispersion relationship by using the conventional matrix-partitioning method. Hence, an accurate and efficient numerical method is proposed to calculate the flexural band gap of a locally resonant beam, with the aim of improving the calculation accuracy and computational efficiency. The proposed method is based on the differential quadrature method, an unconventional matrix-partitioning method, and a variable substitution method. A convergence study and validation indicate that the method has a fast convergence rate and good accuracy. In addition, compared with the plane wave expansion method and the finite element method, the present method demonstrates high accuracy and computational efficiency. Moreover, the parametric analysis shows that the width of the 1st band gap can be widened by increasing the mass ratio or the stiffness ratio or decreasing the lattice constant. One can decrease the lower edge of the 1st band gap by increasing the mass ratio or decreasing the stiffness ratio. The band gap frequency range calculated by the Timoshenko beam theory is lower than that calculated by the Euler-Bernoulli beam theory. The research results in this paper may provide a reference for the vibration reduction of beams in mechanical or civil engineering fields.

KW - band gap

KW - differential quadrature method

KW - locally resonant

KW - phononic crystal

UR - http://www.scopus.com/inward/record.url?scp=85068485836&partnerID=8YFLogxK

U2 - 10.3390/cryst9060293

DO - 10.3390/cryst9060293

M3 - Article

VL - 9

JO - Crystals

T2 - Crystals

JF - Crystals

SN - 2073-4352

IS - 6

M1 - 293

ER -