###### Abstract  
Phononic crystals (PCs) are a novel class of artificial periodic structure, and their band gap (BG) attributes provide a new technical approach for vibration reduction in piping systems. In this paper, the vibration suppression performance and natural properties of fluid-conveying pipes with periodically varying cross-section are investigated. The flexural wave equation of substructure pipes is established based on the classical beam model and traveling wave property. The spectral element method (SEM) is developed for semi-analytical solutions, the accuracy of which is confirmed by comparison with the available literature and the widely used transfer matrix method (TMM). The BG distribution and frequency response of the periodic pipe are attained, and the natural frequencies and mode shapes are also obtained. The effects of some critical parameters are discussed. It is revealed that the BG of the present pipe system is fundamentally induced by the geometrical difference of the substructure cross-section, and it is also related to the substructure length and fluid-structure interaction (FSI). The number of cells does not contribute to the BG region, while it has significant effects on the amplitude attenuation, higher order natural frequencies and mode shapes. The impact of FSI is more evident for the pipes with smaller numbers of cells. Moreover, compared with the conventional TMM, the present SEM is demonstrated more effective for comprehensive analysis of BG characteristics and free vibration of PC dynamical structures. periodic structure; fluid-conveying pipe; band gap; free vibration; spectral element method +
Footnote †: journal: Applied  
doi: 10.3390/10485. https://doi. BG modulation technique for one- and two-dimensional spring-mass systems based on parametric excitations. [PERSON] et al. [7] realized the metamaterials with tunable BGs by introducing periodically arranged negative-stiffness inclusions. [PERSON] et al. [8] considered manipulating transverse waves using longitudinal vibrations, which in essence provide a pseudo-stiffness. [PERSON] et al. [9] designed a new periodic composite with quasi-phononic crystals, which can open the ultra-low frequency BG. [PERSON] et al. [10] proposed two novel phononic unit-cells and demonstrated that they can provide over 200% BG coverage factor over the low-frequency range of 0-12 kHz. Experiment was also conducted to validate the BGs of PC pipes [11]. Furthermore, acoustic black hole (ABH) models were concerned to adjust the BGs of PCs and AMs [12; 13; 14; 15], and periodic tubular structures were used to design PC sensors [16; 17; 18]. In addition to the applications in beam and plate structures, PCs and AMs are also considered in the design of fluid-transporting systems in order to control vibration and eliminate noise. However, owing to the practical difficulty of fabricating periodic thin-walled structures, an extensive utilization of such structures has yet been conducted, and associated theoretical and experimental research is also limited. [PERSON] and [PERSON] [19] early determined the BG distribution of fluid-conveying pipes with periodic supports, and their theoretical findings were further verified by experiments. [PERSON] et al. [20] used the FEM to study the dynamics of a two-section cantilevered pipe conveying fluid composed of different materials. [PERSON] et al. [21] achieved the flexural vibration BGs of periodic composite pipes conveying fluid by applying both the Bragg scattering and locally resonant (LR) mechanisms, and further attained the wave propagation and attenuation characteristics of various PC pipes conveying fluid in the subsequent years [22; 23; 24]. [PERSON] et al. [25] and [PERSON] et al. [26] validated the wave attenuation ability of periodic fluid-filled pipes by performing vibration experiments. [PERSON] et al. [27] and [PERSON] et al.