A unified formulation for vibration analysis of open shells with arbitrary boundary conditions

• A unified Chebyshev–Ritz formulation is developed for different types of open shells.
• The proposed method is appropriate for arbitrary boundary conditions.
• New results for open shells with elastic restraints.
• New results for open shells with circumferentially varying geometry.
• Open cylindrical, conical and shells with arbitrary boundary conditions are discussed in detail.

This paper presents a unified formulation for the free vibration analysis of open shells subjected to arbitrary boundary conditions and various geometric parameters such as subtended angle, conicity. Under the current framework, a general classical shell theory in conjunction with Chebyshev polynomials and Rayleigh–Ritz procedure is developed. Each displacement components of the open shell, regardless of shell types and boundary conditions, is expanded as Chebyshev polynomials of first kind in both directions. All the Chebyshev expanded coefficients are treated equally and independently as the generalized coordinates and solved directly by using the Rayleigh–Ritz procedure. The convergence, accuracy and reliability of the current formulation are validated by comparisons with existing experimental and numerical results published in the literature, with excellent agreements achieved. A considerable number of new vibration results for open cylindrical, conical and spherical shells with various geometric parameters and boundary conditions are presented, which may be used for benchmarking of researchers in the field. The effects of boundary stiffness, subtended angle and conicity on the frequency parameters of different open shells are also discussed in detail. The presented formulation is general compared to the existing literature. Different boundary conditions and geometric dimensions (shallow or deep), different types of shells (cylindrical, conical or spherical), can be easily accommodated in this formulation. It also offers a unified operation for the entire restraining conditions and the change of boundary conditions from one case to another is as easy as changing structural parameters without the need of making any change to the solution procedure. In addition, it can be readily applied to open shells with more complex boundaries such as point supports, non-uniform elastic restraints, partial supports and their combinations.