Wavenumber analysis on vibrations of inner and outer shells in functionally graded porous doubly combined shells

Objective This study aims to investigate the vibrational wavenumber spectrum characteristics of functionally graded porous (FGP) doubly combined shells, consisting of an inner cylindrical shell and an outer conical shell connected by annular plates. The focus is to reveal the mechanism by which annular plate constraints affect the dispersion characteristics and modal coupling between the inner and outer shells from the perspective of elastic wave propagation.

Method A theoretical model is established based on the first-order shear deformation theory and the generalized variational principle. The FGP material follows a power-law distribution along the thickness direction with porosity characterized by a distribution coefficient. An interface potential energy term is introduced via the sub-domain generalized variational principle combined with the least squares weighted residual method to ensure continuity compatibility at substructural interfaces, which relaxes variational constraints and allows flexible selection of admissible functions. Displacement and rotational components are expanded using Chebyshev polynomials along the meridional direction and Fourier series along the circumferential direction, enabling analytical extraction of the wavenumber spectrum. The equations of motion are derived by variational operations and reduced via Krylov subspace-based model order reduction for computational efficiency. The model reliability is validated through convergence analyses and comparisons with reference results and finite element simulations.

Results Modal analysis reveals that the vibration modes can be categorized into global and local modes. Annular plate connections induce elastic wave reflection at interfaces, where incident and reflected waves superimpose to form standing waves with wavelength λ = 2Δx (Δx being the spacing between adjacent plates), corresponding to high-order local modes. Wavenumber spectrum analysis shows that coupling between the inner and outer shells is predominantly concentrated in the low-wavenumber region (circumferential order n ≤ 2). As frequency increases, short-wavelength local vibrations (n ≥ 4) become increasingly prominent, leading to significant discrepancies between the two shells' responses. Annular plate constraints cause distinct changes in phase and group velocities, manifesting as additional dispersion branches and intensified wavenumber coupling. Porosity design primarily induces frequency shifts in the wavenumber domain without altering fundamental dispersion characteristics.

Conclusion The proposed theoretical approach is validated as accurate and efficient for analyzing FGP doubly combined shells. Practical design guidelines include controlling annular plate spacing to avoid matching between elastic and acoustic wavelengths, thereby suppressing high-radiation-efficiency modes, and utilizing porosity design to tune characteristic frequencies away from periodic excitations such as propeller blade passage frequencies. The research can serve as a reference for the vibration and acoustic design of doubly combined shell.