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.

Advances in mechanical metamaterials for vibration isolation: A review

“…After solving the equations, an example was introduced to examine the results of the sample, which was proven to be a good absorber. To this end, the differential quadrature method was recently developed by Liang et al 75 to introduce a numerical technique for solving the elastic bandgaps of periodic structures, especially those distinguished with local resonant. In addition to the DQM, other methods like matrix-partitioning and variable substitution were involved in the developing process of the proposed numerical procedure.…”

A Wave-Based Vibration Analysis of a Finite Timoshenko Locally Resonant Beam Suspended with Periodic Uncoupled Force-Moment Type Resonators

“…Hajhosseini [12] studied an infinite Euler-Bernoulli LR beam consisting of concentrated rigid masses and tapered beam elements with a linearly variable width using a generalized differential quadrature method. Liang [13] proposed an improved differential quadrature method to obtain the band-gap properties of a Euler-Bernoulli LR beam with spring-mass resonators. Lepidi [14] studied a kind of nonlinear LR beam under the micro-scale using the method of multiple scales.…”

Analysis of complete vibration bandgaps in a new periodic lattice model using the differential quadrature method

“…In recent years, a lot of studies have been carried out on the vibration bandgap analysis of periodic structures using the differential quadrature method (DQM) (Cheng et al, 2018; Liang et al, 2019; Xiang and Shi, 2009). In this method, the differential equations are written as a set of linear algebraic equations.…”