The variation principle widely used in structural dynamics analysis allows us to transform the differential equation of the boundary value problem into a functional with extreme value. In recent years, it has been applied to the band-gap analysis of periodic structures. However, for periodic beam-plate composite structures with periodic and ordinary boundaries, it is relatively difficult for the traditional energy methods, such as the Rayleigh–Ritz method, to construct the displacement functions that satisfy boundary conditions. Hence, a hybrid solution is proposed in this paper to account for various boundary conditions of the periodic beam-plate composite structure. Specifically, the displacement functions constructed by the plane wave series can automatically satisfy the periodic boundary conditions. With the ordinary boundaries modeled by artificial springs, the spectral functions conforming to arbitrary boundary conditions are used to represent the displacement functions. The proposed solution is used to solve the band-gap problems of CRTS-III and CRTS-II slab ballastless tracks in China, and the accuracy of the solution is verified by comparing the calculated results with numerical simulations. In addition, through band-gap formation mechanism analysis, the frequency range of propagating flexural waves in each track component is accurately evaluated, which provides a theoretical basis for refined structural vibration reduction in the future. The solution proposed in this paper is flexible and convenient, which can be extended to a more complex band-gap analysis of periodic structures.
The effective mass problem for the Landau-Pekar equations