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Based on the concept of generalized phononic crystals (GPCs), a type of 1D cylindrical shell of generalized phononic crystals (CS-GPCs) where two kinds of homogeneous materials are arranged periodically along radial direction was proposed in this paper. On the basis of radial, torsional shear and axial shear vibrational equations of cylindrical shell, the total transfer matrix of mechanical state vector were set up respectively, and the bandgap phenomena of these three type waves were disclosed by using the method of transfer matrix eigenvalue of mechanical state vector instead of the previous localized factor analyses and Bloch theorem. The characteristics and forming mechanism of these bandgaps of CS-GPCs, together with the influences of several important structure and material parameters on them were investigated and discussed in detail. Our results showed that, similar to the plane wave bandgaps, 1D CS-GPCs can also possess radial, torsional shear and axial shear wave bandgaps within high frequency region that conforms to the Bragg scattering effect; moreover, the radial vibration of CS-GPCs can generate low frequency bandgap (the start frequency near 0 Hz), as a result of the double effects of wavefront expansion and Bragg scattering effect, wherein the wavefront 1550176-1 Int. J. Mod. Phys. B Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SANTA BARBARA on 09/28/15. For personal use only. H.-S. Shu et al.effect can be the main factor and directly determine the existence of the low frequency bandgaps, while the Bragg scattering effect has obvious enhancement effect to the attenuation. Additionally, the geometrical and material parameters of units have significant influences on the wave bandgaps of CS-GPCs.

A two-dimensional cylindrical shell (2DCS) structure with radial and circumferential periodicities is proposed in this paper. The band gap characteristics of radial wave propagating radically from the inner circle of the structure are investigated numerically. According to the calculation and discussion of wave transmission behavior, radial wave in the structure is found to show distinct attenuation in certain frequency region, i.e. wave band gap. The internal mechanism of the band gap is then investigated and the Bragg scattering effect is illustrated. Furthermore, the detailed behaviors of radial wave within the band gap are studied, especially for the transmission peaks. Special localization phenomenon is observed where the mechanical energy localizes strongly in a certain circle of scatterers. Consistency is then found by comparing these localization modes of the whole 2DCS with those of corresponding single scatterers, which indicates that the localization phenomenon is essentially resulted from the local resonance behavior of certain scatterers.

Two alternating homogeneous materials are periodically introduced along the radial direction, forming a circular plate of radial phononic crystal (CPRPC). To illustrate the characteristics of the out-of-plane transverse wave and the in-plane longitudinal wave propagating along the radial direction, the transfer matrices are derived based on the basic wave equations of a thin circular plate in cylindrical coordinates. Localization factors are introduced to evaluate the average attenuation of the transverse and longitudinal waves in the structure, and corresponding bandgaps are obtained. Moreover, finite element method simulations, numerical analyses and the insertion loss method are combined to investigate the effects of the main parameters on these wave bandgaps. The results show that significant transverse and longitudinal wave bandgaps caused by the radial periodicity of the CPRPC exist, and the structural and material parameters have essential influences on them.

Piezoelectric rings are periodically introduced and inserted in a base plate along the radial direction, forming a one-dimensional circular plate of piezoelectric radial phononic crystals (CPPRPC). The transfer matrix of the torsional wave between adjacent units is derived in cylindrical coordinates. Then, by introducing Lyapunov exponents, the phenomenon of the torsional wave band gap is analyzed with consideration of outer control. Furthermore, the effects of some factors, namely, radial span ratio, inner radius of the plate, open-circuit, and short-circuit statuses of the piezoelectric ceramics, on the torsional wave band gap are also discussed in detail. The results show that the distinct band gaps are obtained for torsional waves propagating in CPPRPC. Together with the outer control gain, structural and piezoelectric parameters have significant effects on the band gaps. In particular, middle-low frequency band gaps can be effectively obtained by rationally varying the outer active control gain.