Free Vibrations and Buckling of a Thin Cylindrical Shell of Variable ...

3 Asymptotic AnalysisAccording to a procedure proposed by Tovstik (2001), the asymptotic solution of the boundary value problem forequations (2), can be expressed as( ∫ i ϕ)∑ ∞w(s, ϕ, ε) = exp q(ϕ) dϕ ε k w k (s, ϕ) λ =ε ϕ 0 k=0∞∑ε k λ k (3)k=0whereIm q(ϕ 0 ) = 0{ } dqImdϕ (ϕ 0) > 0 (4)Function Φ has similar asymptotic expansion. It follows from conditions (4) that the functions w and Φ arelocalized near the line ϕ = ϕ 0 . By substituting expressions (3) into (2) and boundary conditions we get theequations for q(ϕ), w k (s, ϕ), Φ k (s, ϕ) and λ k . In the zeroth-order approximation we obtaind 2 Φ 0ds 2 − q4 g 3 w 0 + λ 0 Nw 0 = 0d 2 w 0ds 2+ q4g Φ 0 = 0 (5)where N = q 2 for the buckling problem, N = g for the vibration one. The elimination of the function Φ 0 fromsystem (5) gives the following equationd 4 w 0ds 4 − α4 w 0 = 0 α 4 q 4 N= λ 0 − g 2 q 8 (6)gWe suppose that the shell edges are clamped. Then the boundary conditions for equation (6) arew 0 = dw 0ds= 0 s = 0 s = l(ϕ) (7)The asymptotic solution in case of freely supported shell edges is analyzed in Eliseeva and Filippov (2003). Thechoice of boundary conditions for equation (6) is discussed in Tovstik (2001).The solutions of boundary value problem (6), (7) have the formw 0n = U(α n s)T (α n l) − T (α n s)U(α n l) n = 1, 2, . . .whereand α n is a root of the equationT (z) = sinh z − sin zU(z) = cosh z − cos zcosh(αl) cos(αl) = 1 (8)The values of α n depend on the boundary conditions. If the shell edges are freely supported then α n = πn/l.Taking into account second formula (6) we get that λ 0 is the function of the parameters q and ϕ:λ 0 = f(q, ϕ) = α4 gq 4 N + q4 g 3N . (9)As the zeroth-order approximation for the eigenvalue λ we selectThenλ 0 = minq,ϕ f(q, ϕ) = f(q 0, ϕ 0 ) (10)λ q = ∂f∂q = 0λ ϕ = ∂f∂ϕ = 0 for q = q 0 ϕ = ϕ 0 (11)3