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.

No abstract

In this paper, a 3D semi-analytical method is proposed by introducing the Durbin's Laplace transform, as well as its numerical inversion method, state space approach and differential quadrature method to analyse the transient behaviour of functionally graded material cylindrical panels. Moreover, to investigate the effectiveness of the proposed semianalytical solution, four boundary conditions are used to undertake the analyses. Comparing the proposed approach with other theoretical methods from the literatures, we see better agreements in the natural frequencies. Besides, the semi-analytical solution acquires nearly the same transient response as those obtained by ANSYS. Convergence studies indicate that the proposed method has a quick convergence rate with growing sample point numbers along the length direction, so do layer numbers increase along the radial direction. The effects of thickness/outer radius ratio, length/outer radius ratio and functionally graded indexes are also studied. When carbon nanotube is added to functionally graded material cylindrical panel, the composite structures have been reinforced greatly. The proposed 3D semi-analytical method has high accuracy for the analysis of composite structures. This study can serve as a foundation for solving more complicated environments such as fluid-structure interaction of flexible pipe or thermal effect analysis of functionally graded material in aerospace field.