289_add_Added Masses of Ship Structures.pdf - 6.7 Added Masses of Shells 279 Fig 6.13 The oscillation modes for the cylindrical shell of a finite length

289_add_Added Masses of Ship Structures.pdf - 6.7 Added...

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6.7 Added Masses of Shells 279 Fig. 6.13 The oscillation modes for the cylindrical shell of a finite length: left —antisymmetric; right —axial-symmetric; f and f 1 are the shell oscillation amplitudes The added mass per unit area of the surface is therefore m kn = μr sh ρ. (6.46) We use the following notation in expressions ( 6.44 )–( 6.46 ): μ is the dimension- less coefficient of the shell added mass, γ k = kπr sh /l , r sh and l are the respective radius and the length (the size along the generatrix) of the shell, and k is the number of half-waves of the oscillation mode along the shell generatrix (Fig. 6.13 a). The number of waves n in the circular dimension that correspond to lacking sta- bility and to the least eigenfrequency of the shell can be determined from Fig. 6.14 . For thin shells, formula ( 6.46 ) becomes m kn r sh ρ n 2 + γ 2 k , (6.47) because in this case the quantity γ 2 k (n 2 + γ 2 k ) 3 / 2 in expression ( 6.45 ) is small as compared to unity.
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