A new method for band gap analysis of periodic structures using virtual spring model and energy functional variational principle

Owing to the advantage in converting the boundary value problem of a differential equation into the extreme value problem of a functional, the energy method is widely applied in structural dynamic analysis. Recently, it has also been introduced to calculate the band gap of periodic structures. However, because of the relative complication in the boundary conditions of periodic structures, it is difficult to construct a displacement function using the traditional energy method such as Rayleigh-Ritz method for analysis. Besides, as the constructed displacement function contains wavenumber, when it is used to calculate the band gap by scanning the wavenumber, both the mass and stiffness matrices must be repeatedly calculated, leading to a large amount of calculation. In view of this, a new band-gap calculation method based on the basic framework of the energy method is proposed in this study. In this method, a virtual spring was introduced to simulate the boundary conditions of a periodic structure so that there is no need for a displacement function satisfying the boundary conditions. Thus, the boundary constraints were converted into the elastic potential energy of the spring. For each energy distribution, only the stiffness matrix corresponding to the periodic boundary elastic potential energy contains the wavenumber term and should be repeatedly calculated every time the wavenumber is scanned; the other stiffness and mass matrices require only one time of calculation. The amount of calculation is thus reduced. The results show that the method proposed in this study is precise, reliable, and has a higher calculation efficiency compared with the traditional energy method. The advantage of high calculation efficiency of this method is even more pronounced when the dimensionality of the mass and stiffness matrix or the number of scanning wavenumber increases. Moreover, the virtual spring is flexible, convenient, and widespread in application, thus it can be extended to analyze the band gap of periodic composite structures.