lec4 - 4 4.1 VESSEL DYNAMICS LINEAR CASE Surface Vessel...

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4 VESSEL DYNAMICS: LINEAR CASE 4.1 Surface Vessel Linear Model We first discuss some of the hydrodynamic parameters which govern a ship maneuvering in the horizontal plane. The body x -axis is forward and the y -axis is to port, so positive r has the vessel turning left. We will consider motions only in the horizontal plane, which means χ = ω = p = q = w = 0. Since the vessel is symmetric about the x z plane, y G = 0; z G is inconsequential. We then have at the outset �u X = m �t rv x G r 2 (55) �v �r Y = m + ru + x G �t �t �r �v N = I zz + mx G + ru . �t �t Letting u = U + u , where U >> u , and eliminating higher-order terms, this set is �u X = m (56) �t �v �r Y = m + rU + x G �t �t �r �v N = I zz + mx G + rU . �t �t ( Continued on next page )
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18 4 VESSEL DYNAMICS: LINEAR CASE X A number of coefficients can be discounted, as noted in the last chapter. First, in a homoge- neous sea, with no current, wave, or wind effects, { X x , X y , X δ , Y x , Y y , Y δ , N x , N y , N δ } are all zero. We assume that no hydrodynamic forces depend on the position of the vessel. 1 Second, consider X v : since this longitudinal force would have the same sign regardless of the sign of v (because of side-to-side hull symmetry), it must have zero slope with v at the origin. Thus v = 0. The same argument shows that { X r , X v ˙ , X r ˙ , Y u , Y u ˙ , N u , N ˙ = 0. Finally, since u } fluid particle acceleration relates linearly with pressure or force, we do not consider nonlin- ear acceleration terms, or higher time derivatives. It should be noted that some nonlinear terms related to those we have eliminated above are not zero. For instance, Y uu = 0 because of hull symmetry, but in general X vv = 0 only if the vessel is bow-stern symmetric. We have so far, considering only the linear hydrodynamic terms, ( m X u ˙ ) ˙ u = X u u + X (57) ( m Y v ˙ ) ˙ v + ( mx G Y r ˙ ) ˙ r = Y v v + ( Y r mU ) r + Y (58) ( mx G N v ˙ ) ˙ v + ( I zz N r ˙ ) ˙ r = N v v ( N r mx G U ) r + N . (59) The right side here carries also the imposed forces from a thruster(s) and rudder(s) { X , Y , N } . Note that the surge equation is decoupled from the sway and yaw, but that sway and yaw themselves are coupled, and therefore are of immediate interest. With the state vector γs = { v, r } and external force/moment vector γ F = { Y , N } , a state-space representation of the sway/yaw system is m Y v ˙ mx G Y r ˙ dγs Y v Y r mU = γ s + γ mx G N v ˙ I zz N r ˙ dt N v N r mx G U F , or (60) s ˙ = Pγs + γ F γs ˙ = M 1 Pγs + M 1 γ F γs ˙ = Aγs + B γ F. (61) The matrix M is a mass or inertia matrix, which is always invertible. The last form of the equation is a standard one wherein A represents the internal dynamics of the system, and B is a gain matrix for the control and disturbance inputs.
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