Segment-dependent hierarchical frequency tailoring of rotating double-layer three-phase FGM joined conical–cylindrical shells via matrix gradation and GPL distribution

Abstract

To achieve resonance-safe frequency tailoring of rotating joined shell structures, this study investigates the traveling wave vibration characteristics of joined conical–cylindrical shells (JCCSs) reinforced with graphene platelets (GPLs) and made of double-layer three-phase functionally graded materials, from the perspective of hierarchical material–structural design. A semi-analytical model is developed based on the first-order shear deformation theory, in which the boundary and junction constraints are represented by artificial springs, and the traveling wave frequencies are determined using the improved Fourier series method combined with the Rayleigh–Ritz procedure. The results show that the frequency tailoring behavior of the rotating JCCS exhibits a clear hierarchical structure. Within the investigated parameter range, the matrix-type combination establishes a baseline three-level frequency platform hierarchy. The segmental role is found to be geometry-dependent: For the present baseline configuration, the cylindrical segment mainly governs the platform ordering, whereas the conical segment mainly provides within-platform adjustment. The double-layer gradient exponents further regulate the accessible frequency range and the transitions between adjacent platform regions. GPL reinforcement mainly provides global frequency amplification and secondary fine-tuning without altering the matrix-controlled hierarchy; specifically, within the considered low-speed range, a GPL mass fraction of 1% increases the traveling wave frequencies by approximately 15%–18%. Under rotating conditions, the forward and backward traveling wave splitting, the critical speed intersections, and the resonance safety margin are strongly affected by rotational speed and boundary restraint. These results reveal a segment-dependent hierarchical tuning mechanism and provide a design-oriented basis for resonance-safe frequency tailoring of rotating joined shell structures.