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176 ALEKSANDR KORJAKIN ET AL. vatures are given by (indices of the physical components of tensors are given in parentheses) [41]: 1 ∂v 1 ∂A Ω = + v − b w , (1↔2) (1) 1 (11) 1 2 (2) (11) A1 ∂x A1A2 ∂x 1 ∂v 1 ∂A 1 ∂v 1 2Ω = − + − ∂A (1) 2 (2) 1 (12) v 2 1 (2) v 1 2 (1) A2 ∂x A1A2 ∂x A1 ∂x A1A2 ∂x 1 ∂γ 1 ∂A χ = + γ (1) 1 (11) 1 2 (2) A1 ∂x A1A2 ∂x ⎛ 1 ∂v 1 ∂A ⎞ −b γ − b + v −b w⎟, (1↔2) ⎠ (1) 1 (11) (11) ⎜ 1 2 (2) (11) ⎝ A1 ∂x A1A2 ∂x (11) 1 ∂γ 1 ∂A 1 ∂γ 1 ∂A 2χ = + γ + − γ (1) 2 (2) 1 (12) 2 1 (2) 1 2 (1) A2 ∂x A1A2 ∂x A1 ∂x A1A2 ∂x ⎛ 1 ∂v 1 ∂A ⎞ ⎛ 1 ∂v 1 ∂A ⎞ −b ⎜ − v ⎟ −b ⎜ − v ⎝ ⎠ ⎝ ⎟ ⎠ (1) 2 (2) 1 (11) 2 1 (2) (22) 1 2 (1) A2 ∂x A1A2 ∂x A1 ∂x A1A2 ∂x 1 ∂w 2 Ω = γ + + b v , (1↔2) (13) (1) 1 (11) (1) A1 ∂x Here, the notations v (1) = u, v (2) = v are used for the physical components of the displacements. For shells of revolution it is convenient to use the coordinates x 1 = α, x 2 = ϕ (see Figure 1). In this case the curvature tensor is given by: b αβ ⎡ 1 ⎤ ⎢− 0 R ⎥ 1 = ⎢ ⎥ ⎢ sin α ⎥ ⎢ 0 − ⎥ ⎣ r ⎦ (12) The Lamé’s coefficients of a shell of revolution are A 1 = R 1 and A 2 = r, where the radii can be functions of the meridian coordinate, r = r(α) and R 1 = R 1 (α). From these expressions the kinematic equations for various shaped shells of revolution can be obtained. Thus, the general strain-displacement relations for cylindrical shells (see Figure 2) [39] are
Free Damped Vibrations of Sandwich Shells of Revolution 177 Figure 1. Geometry of a shell of revolution. Figure 2. Geometry of the sandwich cylindrical shell.
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