scholarly journals A Numerical Method for Flexural Vibration Band Gaps in A Phononic Crystal Beam with Locally Resonant Oscillators

Crystals ◽  
2019 ◽  
Vol 9 (6) ◽  
pp. 293 ◽  
Author(s):  
Xu Liang ◽  
Titao Wang ◽  
Xue Jiang ◽  
Zhen Liu ◽  
Yongdu Ruan ◽  
...  
Keyword(s):  
Band Gap ◽  
Computational Efficiency ◽  
Differential Quadrature Method ◽  
Beam Theory ◽  
Band Gaps ◽  
Quadrature Method ◽  
Stiffness Ratio ◽  
Mass Ratio ◽  
Resonant Structures ◽  
Matrix Partitioning

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.

Vibration and buckling analysis of laminated sandwich conical shells using higher order shear deformation theory and differential quadrature method

2017 ◽  
Vol 21 (4) ◽  
pp. 1445-1480 ◽  
Author(s):  
M Nasihatgozar ◽  
SMR Khalili
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Differential Quadrature Method ◽  
Shear Deformation Theory ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Stiffness Ratio ◽  
Conical Shells ◽  
The Core ◽  
The Face

Vibration and buckling analysis of laminated sandwich truncated conical shells with compressible or incompressible core are presented in this work considering curvature effects. The formulation uses the quadratic and cubic functions for transverse and in-plane displacements of the core and the first-order shear deformation theory for the face sheets. The motion equations of each individual layer are derived according to the principle of minimum total potential energy considering the continuity of the displacements and the internal stress fields at the interfaces. Differential quadrature method is applied in order to obtain the frequency and buckling load of the sandwich structure. The effects of different parameters such as core to face sheet stiffness ratio, number of layers of the face sheets, boundary condition, geometrical parameters of the core and the face sheets, semi vertex angle of the cone, trapezoidal shape, and in-plane stresses of the core are examined on the vibration and buckling response of sandwich truncated conical shells. Comparison of the present results with those reported in the literature confirms the accuracy of the proposed theory. Numerical results indicate that the effects of in-plane stresses of the core significantly affect the frequency with increasing the core to face sheet stiffness ratio.

Numerical solution for dynamic analysis of semicircular curved beams acted upon by moving loads

2014 ◽  
Vol 228 (13) ◽  
pp. 2314-2322 ◽  
Author(s):  
Ali Nikkhoo ◽  
Hassan Kananipour
Keyword(s):  
Numerical Solution ◽  
Differential Quadrature Method ◽  
Moving Load ◽  
Beam Theory ◽  
Differential Quadrature ◽  
Steel Bridge ◽  
Quadrature Method ◽  
Curved Beams ◽  
Various Boundary Conditions ◽  
Structural Problems

The present study proposes a dynamic numerical solution for deflections of curved beam structures. In order to extract characteristic equations of an arch under an in-plane constant moving load, an analysis procedure based on the Euler–Bernoulli beam theory considering polar system is conducted. A prismatic semicircular arch with uniform cross section, in various boundary conditions, is assumed. Radial and tangential displacements, as well as bending moments are obtained using differential quadrature method as a well-known numerical method. In addition to parametric studies, a curved steel bridge as an actual application is analyzed by the mentioned method. By using this differential quadrature technique, the function values and some partial derivatives are approximated by weighting coefficients. Convergence study is carried out to demonstrate the stability of the present method. In order to confirm the high level of accuracy of this approach, some comparisons are made between the results obtained by selected methods such as differential quadrature method, Galerkin method, and finite element method. The results show that in the structural problems with specific geometry, using differential quadrature method, which is independent of domain discretization, is proven to be efficient.

Analysis of Periodicity Termination in Phononic Crystals

2011 ◽  
Author(s):  
Bruce L. Davis ◽  
Andrew S. Tomchek ◽  
Edgar A. Flores ◽  
Liao Liu ◽  
Mahmoud I. Hussein
Keyword(s):  
Band Gap ◽  
Timoshenko Beam ◽  
Periodic Structures ◽  
Beam Theory ◽  
Phononic Crystals ◽  
Timoshenko Beam Theory ◽  
Band Gaps ◽  
Pass Band ◽  
Unit Cells

While resonant propagation modes are non-existent within band gaps in infinite periodic structures, it is possible for anomalous band-gap resonances to appear in finite periodic structures. We establish two criteria for the characterization of band-gap resonances and propose approaches for their elimination. By considering flexural periodic beams, we show that as the number of unit-cells is increased the vibration response corresponding to band-gap resonances (1) does not shift in frequency, and (2) drops in amplitude. Both these outcomes are not exhibited by regular pass-band resonances, nor by resonances in finite homogenous beams when the length is changed. Our conclusions stem from predictions based on Timoshenko beam theory coupled with matching experimental observations.

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