307_add_Added Masses of Ship Structures.pdf - 6.9 Added Masses at Complex Structure Motion 297 are determined by formulas(6.60 and J1 and J2 are the

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6.9 Added Masses at Complex Structure Motion 297 are determined by formulas ( 6.60 )), and J 1 and J 2 are the inertia moments of the cross-sections of the respective cross joist and the joist of the principal direction. Substituting the expressions for the kinetic T Σ and potential P G energies in the Lagrange equation d dt ∂T Σ ˙ w i + ∂P G ∂w i = 0 (the potential energy of the fluid is not taken into account since it remains constant) and introducing the notation A = i,j X(x i )Y(y j ) 2 , A 3 = M pl 13 + M 1 M 3 ; A 1 = M pl 12 + M st 12 + M 1 M 2 , A 4 = M pl 23 + M 2 M 3 ; A 2 = M pl 12 + M st 2 + M 2 ; A 5 = M pl 3 + M 3 ; B 1 = M ad + ( M pl 1 + M st 1 + M 1 ) A, (6.75) we obtain the system of differential equations for the oscillatory motion of the gril- lage: B 1 ¨ w 1 + AA 1 ¨ w 2 + AA 3 ¨ w 3 + C ad w 1 = 0 ; A 1 ¨ w 1 + A 2 ¨ w 2 + A 4 ¨ w 3 + C 1 w 2 = 0 ; A 3 ¨ w 1 + A 4 ¨ w 2 + A 5 ¨ w 3 + Cw 3 = 0 . Looking for the solution of this system in the form w i = f i sin (λt) and equating the determinant to zero, we obtain the frequency equation λ 6 ( A 2 A 3 B 1 A 2 4 B 1 AA 2 1 A 5 + 2 AA 1 A 3 A 4 AA 2 A 2 3 ) λ 4 ( A 5 B 1 C 1 + A 2 B 1 C + A 2 A 5 C js A 2 4 C js AA 2 1 C AA 2 3 C 1 ) + λ

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