Band Gap Properties Of Periodic Beam Structures

Large space structures are usually periodic structures to facilitate manufacturing and in-orbit assembling, such as space truss structures, masts, honeycomb sandwich plates and solar panels. An attractive characteristic of periodic structures is the prevention of the wave propagation in specified frequency ranges resulting in the stop band. The main purpose of this paper is to reveal the band-gap properties of periodic beam structures combining the travelling wave method with Bloch theory. The main contents of this paper are as follows.Firstly, the traveling wave dynamic responses of tapered beams are analyzed. Based on the force equilibrium equation of the infinitesimal unit, the waveguide equations of the tapered beam with respect to axial, torsional and flexural deformations are deduced and the wave modes transition equations are extracted. According to the force equilibrium conditions and displacement coordination conditions, the scattering equations and transmission equations of the tapered beam are established. Combining with these equations, the wave modes representing the responses are calculated. The frequency responses of a cantilever tapered beam and a frame consisting of tapered beams are analyzed by the traveling wave method. The effects of the material and geometric parameters on frequency responses are revealed. The simulation results exhibit that the traveling wave method is the more accurate and higher computation efficiency than the finite element method for analyzing the dynamic responses of large space frame structures.Secondly, the band gap properties of one-dimensional periodic beam structures are investigated. Taking the displacement and force as the state vector, we deduce the frequency relationship between the input and output state vectors of one-dimensional periodic beam structures based on the travelling wave model. Introducing Bloch theory, we establish the wavenumber relationship between the input and output state vectors. On the basis of the two relationships, we obtain a universal equation of the band gap properties relating to the frequencies and wavenumbers. The band gap properties of periodic tampered and uniform beam structure are analyzed and compared. The influence of material and geometric parameters on band gap properties is analyzed, which provides the basis for band-gap property analysis and design of two-dimensional periodic beam structures.Finally, the band gap properties of two-dimensional periodic beam structures are explored. The mechanical equations of an orthogonal hinged periodic beam structure are derived including the displacement compatibility equations and force balance equations. Four types of wave propagation are respectively defined by different inputting wave vectors in four directions, and the corresponding reflection and scattering coefficients are deduced. Bloch boundary conditions that describe the wave propagation are established based on Bloch theory from phononic crystals. The relationship between the wavenumber and frequency are obtained with the wave transmission equations. The numerical analysis of the band gap properties of a two-dimensional periodic structure validates the feasibility of the proposed method for vibration isolation and filtering design in practical engineering.