Three-dimensional free vibration analysis of doubly-curved shells

The vibration analysis of the natural frequencies and mode shapes of a class of doubly-curved shells with different boundary conditions is presented. The doubly-curved shells are geometrically taken from various parts of a hollow torus with annular cross-section. The small strain, three-dimensional, linear elasticity theory is adopted to establish the governing equations of the problem in terms of the toroidal coordinate system ( r, θ, ϕ). The Chebyshev-Ritz method is used to set up the eigenvalue equation: displacement in each direction is taken as a triplicate product of the Chebyshev polynomials in r, θ and ϕ, multiplied by a boundary function along with a set of generalized coefficients to yield upper bound values of the natural frequencies. The natural frequencies converge monotonically to the exact values as more terms of Chebyshev polynomials are included in the Ritz approximation. The effects of thickness ratio, radius ratio, toroidal angle in ϕ direction, initial angle and subtended angle in θ direction on natural frequencies and mode shapes are discussed in detail.