Vibration problems wildly exist in beam-foundation structures. In this paper, finite periodic composites inspired by the concept of ideal phononic crystals (PCs), as well as Timoshenko beam theory (TBT), are proposed to the beam anchored on Winkler foundation. The bending vibration band structure of the PCs Timoshenko beam-foundation structure is derived from the modified transfer matrix method (MTMM) and Bloch's theorem. Then, the frequency response of the finite periodic composite Timoshenko beam-foundation structure by the finite element method (FEM) is performed to verify the above theoretical deduction. Study shows that the Timoshenko beam-foundation structure with periodic composites has wider attenuation zones compared with homogeneous ones. It is concluded that TBT is more available than Euler beam theory (EBT) in the study of the bending vibration characteristic of PCs beam-foundation structures with different length-to-height ratios.

We introduce the periodic composite materials, so-called phononic crystals, to the flexible mechanical arms systems. Due to the transfer matrix method and the Bloch theorem, the theoretical solution of band structure of the two-link model is deduced and then verified by the frequency response by the finite element method. The influence of the included angle of arms to vibration characteristics is analyzed. The frequency response of the two-link flexible arms with/without phononic crystals is investigated. The results illustrate that, by using the periodic composite materials, some frequency ranges with strong attenuation can be obtained. This study provides a new way to eliminate vibrations in flexible mechanical arms.