In this paper, a modified transfer matrix method is proposed to solve the transverse vibration bandgaps (BGs) in phononic crystal (PC) Euler beams on a Winkler foundation. An equivalent critical frequency is introduced to facilitate the calculation of the end frequency of the first BG. The band structures of a lead–steel PC Euler beam either on a Winkler foundation or not show that the existence of a foundation has a significant influence on the distribution of BGs. A new BG that starts at 0 Hz appears. Additional results demonstrate that a very narrow passband could exist between the first two BGs. This study is aimed at searching for large BGs, especially at low frequencies.
Band structures for the lead–steel PC Euler beam on a Winkler foundation (continuous line) or not (dash line). A new BG that starts at 0 Hz appears in the case of having a foundation.