A General Framework for Material Properties Calculation and the Free Vibration Analysis of New Three-Phase Composite Cylindrical Shell Structures

New three-phase composite structures reinforced synergistically by nano-fillers and macroscopic fibers have great application potential. This paper presents a general framework for material properties calculation and the free vibration analysis of three-phase composite shell structures. Based on this methodological system, the free vibration characteristics of three types of nano-enhanced functionally graded three-phase composite cylindrical shells are investigated. First, the equivalent mechanical properties of these three three-phase composites were evaluated using the Halpin–Tsai and Mori–Tanaka models. The governing equations for the cylindrical shells were derived based on the first-order shear deformation theory (FSDT) and Hamilton’s principle. The equations were discretized using Galerkin’s method and solved to obtain the natural frequencies and mode shapes. The finite element simulation results and existing literature verified the accuracy and reliability of the method in this paper. The synergistic effects of nano-reinforced fillers and macroscopic fibers on the free vibrations of these structures were also analyzed. Among them, the natural frequency of the three-phase composite cylindrical shells was the highest when graphene platelets (GPLs) were used as the nano-reinforced fillers, which was 150.32% higher than that of fiber-reinforced epoxy composite cylindrical shells. These studies provide theoretical guidance for the design and manufacture of such symmetric or antisymmetric structures in the future.