Frequency Splitting Characteristics and Evolution in Bolted Flange-Joined Conical-Cylindrical Shells

Bolted flange-joined conical-cylindrical shells (BFJCCS) are widely used in aerospace and marine engineering, where circumferentially discrete bolt constraints and service-induced bolt loosening may cause frequency splitting and modal evolution. However, the formation mechanism and evolution of frequency splitting under different bolt states remain insufficiently clarified. To address this issue, a unified semi-analytical model is developed by treating the flanges as independent substructures and representing the bolted joints with circumferentially distributed artificial springs. The model is validated through convergence studies, geometric degeneration verification, and finite element comparisons. It is then used to investigate BFJCCS under intact, single-bolt loosening, and multiple-bolt loosening conditions. The results show that intact bolts induce frequency splitting associated with the matching between bolt-constraint periodicity and modal wavenumber. The bolt influence region and cylindrical-shell length further affect the splitting amplitude. Single-bolt loosening breaks the original circumferential periodicity of bolt constraints, leading to splitting in nearly all studied nearly degenerate modal pairs and modal localization accompanied by wavenumber redistribution and mixing. For multiple-bolt loosening, the circumferential periodicity of the bolted-joint stiffness distribution governs the dominant split modal families, while the stiffness distribution within each periodic unit regulates the splitting amplitude. Once the periodic repetition is broken, frequency splitting extends to more modal pairs. From a unified physical perspective, frequency splitting under different bolt states originates from the interfacial connection-energy difference between two branches of a nearly degenerate modal pair.