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# lec2 - 2 VESSEL INERTIAL DYNAMICS We consider the rigid...

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2 VESSEL INERTIAL DYNAMICS We consider the rigid body dynamics with a coordinate system aﬃxed on the body. A common frame for ships, submarines, and other marine vehicles has the body-referenced x - axis forward, y -axis to port (left), and z -axis up. This will be the sense of our body-referenced coordinate system here. 2.1 Momentum of a Particle Since the body moves with respect to an inertial frame, dynamics expressed in the body- referenced frame need extra attention. First, linear momentum for a particle obeys the equality d γ F = ( mγv ) (19) dt A rigid body consists of a large number of these small particles, which can be indexed. The summations we use below can be generalized to integrals quite easily. We have F i + γ d γ R i = ( m i γv i ) , (20) dt where γ R i is the net force exerted by all the F i is the external force acting on the particle and γ other surrounding particles (internal forces). Since the collection of particles is not driven apart by the internal forces, we must have equal and opposite internal forces such that (Continued on next page)

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6 2 VESSEL INERTIAL DYNAMICS N γ R i = 0 . (21) i =1 Then summing up all the particle momentum equations gives N N d γ F i ( m i γv i ) . (22) = dt i =1 i =1 Note that the particle velocities are not independent, because the particles are rigidly at- tached. Now consider a body reference frame, with origin 0 , in which the particle i resides at body- referenced radius vector γ r ; the body translates and rotates, and we now consider how the momentum equation depends on this motion. z, w, y, v, x, u, θ φ ψ Figure 2: Convention for the body-referenced coordinate system on a vessel: x is forward, y is sway to the left, and z is heave upwards. Looking forward from the vessel bridge, roll about the x axis is positive counterclockwise, pitch about the y -axis is positive bow-down, and yaw about the z -axis is positive turning left. 2.2 Linear Momentum in a Moving Frame The expression for total velocity may be inserted into the summed linear momentum equation to give N N d γ F i = ( m i ( γv o + γ� × γ r i )) (23) dt i =1 i =1 N �γv o d γ m i γ r i = m + × , �t dt i =1 i =1 m i , and γv i = γv o + γ where m = N × γ r i . Further defining the center of gravity vector γ r G such that
2.3 Example: Mass on a String 7 N r G = m i γ r i , (24) i =1 we have F N i = m ( γ γ �γv o + m d × γ r G ) . (25) �t dt i =1 Using the expansion for total derivative again, the complete vector equation in body coor- dinates is γ �γv o F = N = m + γ × ( γ × γ r G ) . (26) × γv o + r G + γ �t dt × γ i =1 Now we list some conventions that will be used from here on: γv o = { u, v, w } (body-referenced velocity) γ r G = { x G , y G , z g } (body-referenced location of center of mass) γ� = { p, q, r } (rotation vector, in body coordinates) γ F = { X, Y, Z } (external force, body coordinates) .

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lec2 - 2 VESSEL INERTIAL DYNAMICS We consider the rigid...

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