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

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Figure 3. Buckling form of the cylindrical shell (η = 1.3)Figure 4. Buckling form of the cylindrical shell (η = 1.45)5 Vibrations of the Cylindrical ShellConsider free low-frequency vibrations of a thin cylindrical shell of a variable thickness with a curvelinear edge.Then formula (9) and equations (11) take the following formsf(q, ϕ) = α4q 4 + q4 g 2q 4 = α2g2l ′l= g′g(16)We find the first-order correctionusing formula (12).λ 1 = 2(m + 1)q 3 g 2√ 2Λ 1Λ 1 = 2g′′g− 4l′′l− g′2g 2As an example consider the vibrations of the cylindrical shell with slanted edge assuming that the function h(ϕ) isgiven by (1). The generatrix ϕ = ϕ 0 is the weakest one if ϕ 0 satisfies equationA(ϕ) sin ϕ = 0 A(ϕ) = γ/g − 2t/l. (17)andΛ 1 = 2A(ϕ 0 ) cos ϕ 0 − γ 2 sin 2 ϕ 0 /g 2 (ϕ 0 ) > 0The roots of equation (17) in the interval (−π, π] are 0, π and roots of equation A(ϕ) = 0. If ϕ 0 is the root ofequation A(ϕ) = 0 thenΛ 1 = −γ 2 sin 2 ϕ 0 /g 2 (ϕ 0 ) ≤ 0and the generatrix ϕ = ϕ 0 is not the weakest line. The vibration mode is localized near the top generatrix ϕ = 0if A(0) > 0. The last inequality is fulfilled if γ > γ 1 = 2t/(l c − t). The bottom generatrix ϕ = π is the weakestone if A(π) < 0. The inequality A(π) < 0 is true for all γ > 0 if t ≤ l c ≤ 3t. If l c > 3t then inequality A(π) < 0is fulfilled only for 0 < γ < γ 2 = 2t/(l c − 3t).Therefore, in case of vibrations, in contrast to buckling, the location of the weakest lines on the shell surface doesnot depend on ratio η = 1 + 2γ. On η depends only the number of such lines. If η < η 1 = 1 + 2γ 1 then bottom6
generatrix ϕ = π is weakest one. In case η 1 < η < η 2 , where η 2 = 1 + 2γ 2 , both lines ϕ = 0 and ϕ = π areweak. The number of the weakest lines for η > η 2 is a function of geometric shell parameters. If t ≤ l c ≤ 3t, thentwo weakest lines are on the shell surface, otherwise for η > η 2 only generatrix ϕ = 0 is the weakest line.The values of the fundamental frequency for the cylindrical shell of variable thickness with slanted edge are shownin Table 2. The mass density of the shell material ρ=7860 kg/m 3 . Others shell parameters have the same values asin the buckling problem.ηFundamental frequency (s −1 )AsymptoticFEM result1.0 24.833 24.1442.0 34.715 33.1234.0 48.585 45.0825.0 49.666 47.625Table 2. The values of the fundamental frequency vs. ratio η.The values of the fundamental frequency obtained by the asymptotic formulas and with the help of FEM are placedin the second column and third column respectively. The relative discrepancy in asymptotic and numerical resultsis less than 8%. The computation time of a value of frequency is approximately the same as of the value of pressure.The vibration mode shapes computed by FEM and plotted in Figures 5 and 6 amplify asymptotic results. In caseη < η 1 = 3 (Figure 5) the vibration mode, corresponding to the fundamental frequency is localized near thelongest bottom generatrix ϕ = π of the cylindrical shell.Figure 5. Vibration form of the cylindrical shell (η = 2)If η > η 1 = 3 then on the shell surface are two weakest lines ϕ = π and ϕ = 0 corresponding in general todifferent frequencies. In Figure 6 is shown the vibration mode corresponding to the fundamental frequency. Thismode is localized near the shortest top generatrix ϕ = 0.Figure 6. Vibration form of the cylindrical shell (η = 5)6 ConclusionBy means of both asymptotic solution and numerical analysis it is shown that vibrations and buckling of thincylindrical shell of variable thickness with curvelinear edge may be accompanied by the appearance of concavitieswhich are stretched along the shell surface. Near a weakest generatrix the concavities have the maximum depth.The depth of the concavities decreases fast away from the weakest lines.7
Page 1 and 2: TECHNISCHE MECHANIK, Band 25, Heft
Page 3: 3 Asymptotic AnalysisAccording to a
Page 8: In contrast to previously studied p

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

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