Pipeline systems are widely used in industrial fields. To effectively control the vibration of the pipes, a periodic pipeline structure with a single degree-of-freedom resonator was designed in this paper, and the propagation characteristics of flexural waves were investigated. The dispersion relationship between the transmission constant and frequency was derived using the transfer matrix method combined with Bloch’s theorem, and the bandgap results were verified through finite element simulation. The results indicate that there are three bandgaps and three passbands within the frequency range of 0~250 Hz, with the first passband having a very narrow frequency range, the second and third passbands having a wider frequency range. In order to interpret the influence of pipeline parameters on transmission characteristics, the control variable method was used to analyze parameters such as unit cell length, pipe thickness, and inner diameter size. The results indicate that an increase in the unit cell length leads to a reduction in bandgap width, while increases in pipe thickness and inner diameter result in an expansion of bandgap width. Finally, to achieve control over the passbands, a single degree-of-freedom resonator was introduced to effectively control specific passbands, and the optimal installation position of the resonator was determined through finite element analysis. The results show that the resonator significantly reduces the width of the passbands. This study provides theoretical support for vibration control of pipeline structures, and can be applied to the design of low-frequency vibration isolation.