Exact expression for dispersion relation of an infinitely long beam with periodical attachments and supports

Periodic beam-type structures have wide applications in engineering practice. The dispersion spectra of periodic structures show the band structure, consisting of stopping bands and passing bands. Various analytical and numerical techniques have been applied to obtain dispersion spectra of periodic beam-type structures. The aim and novel contribution of this work is to provide an analytical and exact expression which describes the dispersion relation of a periodic beam, either with periodical supports or repetitive attachments or both, regardless whether Euler–Bernoulli or Timoshenko beam theories are used. It is found that the dispersion relation can be expressed by a “zero stiffness/stiffness-matrix condition” in which the host beam is represented by an equivalent dynamic stiffness. This dispersion relation obtained is exact without any truncation or approximation and thus it provides a fast and accurate prediction of wave dispersion in periodic beams comparing to commonly used finite element method and plane wave expansion. Using the “zero stiffness/stiffness-matrix condition”, dispersion spectra of periodic beams can be readily obtained regardless of the type of supports and attachments once their dynamic stiffness are known. The proposed exact dispersion relation is validated by different cases. Two engineering examples, namely a continuously supported beam with periodical resonator and railway track with frequency-dependent support are illustrated to reveal wave dispersion characteristics of these two cases. It is found that the dispersion relation is much dependent on relative value between the natural frequency of the oscillator and the cut-off frequency of the host beam for the continuously supported beam with periodic oscillators. As for periodic railway track with frequency-dependent support stiffness, it is found that the frequency dependences enlarge the bandwidth of bandgaps related to local resonance and pinned–pinned resonance. In addition, vibrations attenuate more efficiently when frequency-dependent fastener stiffness is considered.