17_add_Added Masses of Ship Structures.pdf - 1.3 Transformation of Added Masses under a Change of Coordinate System 5 arbitrary body motion can be

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1.3 Transformation of Added Masses under a Change of Coordinate System 5 arbitrary body motion can be written as follows: 2 T = λ 11 u 2 1 + λ 22 u 2 2 + λ 33 u 2 3 + 2 λ 12 u 1 u 2 + 2 λ 13 u 1 u 3 + 2 λ 23 u 2 u 3 + 2 u 1 14 u 4 + λ 15 u 5 + λ 16 u 6 ) + 2 u 2 24 u 4 + λ 25 u 5 + λ 26 u 6 ) + 2 u 3 34 u 4 + λ 35 u 5 + λ 36 u 6 ) + λ 44 u 2 4 + λ 55 u 2 5 + λ 66 u 2 6 + 2 λ 45 u 4 u 5 + 2 λ 46 u 4 u 6 + 2 λ 56 u 5 u 6 . (1.13) The values for the AM λ ik depend on the shape of the body, chosen coordinate system and fluid density ρ . The variables λ ik ( i,k = 1 , 2 , 3) have dimension of mass, values for λ ik ( i = 1 , 2 , 3; k = 4 , 5 , 6) have dimension of static moment, values for λ ik ( i,k = 4 , 5 , 6) dimension of moment of inertia. Similarly to the formula ( 1.13 ), the expression for kinetic energy of the body of mass m , with coordinates of the center of inertia x c , y c , z c , with diagonal mo- ments of inertia J x , J y , J z and products of inertia J xy , J xz , J yz , is represented in the following form: 2 T 0 = mu 2 1 + mu 2 2 + mu 2 3 + 2 u 1 (mz c u 5 my c u 6 ) + 2 u 2 (mx c u 6 mz c u 4 ) + 2 u

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