On natural frequencies of non-uniform beams modulated by finite periodic cells

• The transfer matrix method to study the natural frequencies of the finite periodic non-uniform beams is derived.
• The transfer matrix method to study the band gaps of the infinite periodic non-uniform beams is derived.
• The effects of the periodic cells on the natural frequencies are explored.
• The relationships of the natural frequencies and band gap boundaries are investigated.

It is well known that an infinite periodic beam can support flexural wave band gaps. However, in real applications, the number of the periodic cells is always limited. If a uniform beam is replaced by a non-uniform beam with finite periodicity, the vibration changes are vital by mysterious. This paper employs the transfer matrix method (TMM) to study the natural frequencies of the non-uniform beams with modulation by finite periodic cells. The effects of the amounts, cross section ratios, and arrangement forms of the periodic cells on the natural frequencies are explored. The relationship between the natural frequencies of the non-uniform beams with finite periodicity and the band gap boundaries of the corresponding infinite periodic beam is also investigated. Numerical results and conclusions obtained here are favorable for designing beams with good vibration control ability.