Bandgap analysis of cylindrical shells of generalized phononic crystals by transfer matrix method

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 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.