Local resonance phononic crystals can be constructed by adding resonance elements periodically, and low frequency band gap can be generated. In this paper, a phononic crystal composed of one-resonator and one-resonator with elastic support has been presented. The band gap characteristics and band gap regulation characteristics of the structure were calculated and analyzed by using phononic crystal theory and vibration theory. The results show that the initial frequency of the first band gap is 0 Hz. The attenuation of vibration in the band gap is determined by the attenuation factor and period number of the finite structure. There are one tunable band gap and three untunable band gaps in such structure. By changing the structural parameters of one-resonator and one-resonator with elastic support, the tunable band gap can be tuned. The untuable band gaps which include the first band gap can attenuate elastic wave stably. The band gap characteristics of the structure have potential applications in the field of vibration reduction of pipelines, bridges and automobiles.