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Unformatted text preview: 23 APPENDIX 2: ADDED MASS VIA LAGRANGIAN DYNAMICS The development of rigid body inertial dynamics presented in a previous section depends on the rates of change of vectors expressed in a moving frame, specifically that of the vehicle. An alternative approach is to use the lagrangian , wherein the dynamic behavior follows directly from consideration of the kinetic co-energy of the vehicle; the end result is exactly the same. Since the body dynamics were already developed, we here develop the lagrangian technique, using the analogous example of added mass terms. Among other effects, the equations elicit the origins of the Munk moment. 23.1 Kinetic Energy of the Fluid The added mass matrix for a body in six degrees of freedom is expressed as the matrix M a , whose negative is equal to: Z Y X u X v X w X p X q X r u Y v Y w Y p Y q Y r u Z v Z w Z p Z q Z r M a = N M K u K v K w K p K q K r u M v M w M p M q M r u N v N w N p N q N r , (260) (Continued on next page) 23.1 Kinetic Energy of the Fluid 125 where ( X, Y, Z ) denotes the force, ( K, M, N ) the moment, ( u, v, w ) denotes the velocity and ( p, q, r ) the angular velocity. The sense of M a is that the uid forces due to added mass are given by X am Y am Z am u v w d = M a . (261) K am M am N am dt p q r The added mass matrix M a is completely analagous to the actual mass matrix of the vehicle, m z G y G m z G x G m y G x G I M = , (262) y G I xx I xz z G xy x G I xy I yy I z G yz I xz I yz I zz x G y G where [ x G , y G , z G ] are the (vessel frame) coordinates of the center of gravity. The mass matrix is symmetric, nonsingular, and positive definite. These properties are also true for the added mass matrix M a , although symmetry fails when there is a constant forward speed. The kinetic co-energy of the uid E k is found as: 1 E k = q T M a q (263) 2 where q T = ( u, v, w, p, q, r ). We expand to find in the non-symmetric case: 2 E k = X u u 2 + X v uv + X w uw + X p up + X q uq + X r ur + Y u uv + Y v v 2 + Y w vw + Y p vp + Y q vq + Y r vr + Z u uw + Z v vw + Z w w 2 + Z p...
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- Fall '04