directly affected by its fundamental form and LCG as well as its rudder(s) size and location. Recent IMO requirements mandate performance in turns, zigzag maneuvers, and stopping. Thus, it is incumbent upon the designer to check basic maneuvering characteristics of a hull during the parametric stage when the overall dimensions and form coefficients are being selected. This subsection will illustrate a parametric design capability to assess course stability and turnability. This performance presents the designer with a basic tradeoff since a highly course stable vessel is hard to turn and vice versa.
Clarke et al (53) and Lyster and Knights (54) developed useful parametric stage maneuvering models for displacement hulls. Clarke et al used the
Figure 11.22 - Sample Propeller Optimization Program (POP) Output (40)
Figure 11.23 - Norrbin’s Turning Index versus |K’| and |T’|
linearized equations of motion in sway and yaw to develop a number of useful measures of maneuverability. They estimated the hydrodynamics stability derivatives in terms of the fundamental parameters of the hull form using regression equations of data from 72 sets of planar motion mechanism and rotating arm experiments and theoretically derived independent variables. Lyster and Knights obtained regression equations of turning circle parameters from full-scale maneuvering trials. These models have been implemented in the Maneuvering Prediction Program (MPP), which is also available for teaching and design (40). In MPP, the Clarke hydrodynamic stability derivative equations have been extended by using corrections for trim from Inoue et al (55) and corrections for finite water depth derived from the experimental results obtained by Fugino (56).
Controls-fixed straight-line stability is typically assessed using the linearized equations of motion for sway and yaw (57). The sign of the Stability Criterion C, which involves the stability derivatives and the vessel LCG position, can determine stability. A vessel is straight-line course stable if,
C = Yv' (Nr' – m'xg') – (Yr' – m')Nv' > 0 [75]
where m’ is the non-dimensional mass, xg’ is the longitudinal center of gravity as a decimal fraction of ship length plus forward of amidships, and the remaining terms are the normal sway force and yaw moment stability derivatives with respect to sway velocity v and yaw rate r.
Clarke (53) proposed a useful turnability index obtained by solving Nomoto’s second-order in r lateral plane equation of motion for the change in heading angle resulting from a step rudder change after vessel has traveled one ship length,
Pc = | ψ/δ | t' = 1 [76]
This derivation follows earlier work by Norrbin that defined a similar P1 parameter. Clarke recommended a design value of at least 0.3 for the Pc index. This suggests the ability to turn about 10 degrees in the first ship length after the initiation of a full 35 degree rudder command.
Norrbin's index is obtained by solving the simpler first-order Nomoto’s equation of motion for the same result. It can be calculated as follows:
P1 = | ψ/δ | t' = 1 = |K'|(1-|T'| (1–e–1/|T'|)) [77]
where K' and T' are the rudder gain and time constant, respectively, in the first-order Nomoto's equation,
T'dr'/dt' + r' = K'δ [78]
where r' is the nondimensional yaw rate and δ is the rudder angle in radians. Values for a design can be compared with the recommended minimum of 0.3 (0.2 for large tankers) and the results of a MarAd study by Barr and the European COST study that established mean lines for a large number of acceptable designs.
This chart is presented in Figure 11.23.
Clarke also noted that many ships today, particularly those with full hulls and open flow to the propeller, are course unstable. However, these can still be maneuvered successfully by a helmsman if the phase lag of the hull and the steering gear is not so large that it cannot be overcome by the anticipatory abilities of a trained and alert helmsman. This can be assessed early in the parametric stage of design by estimating the phase margin for the hull and steering gear and comparing this to capabilities found for typical helmsmen in maneuvering simulators. Clarke derived this phase margin from the linearized equations of motion and stated that a helmsman can safely maneuver a course unstable ship if this phase margin is above about –20 degrees. This provides a valuable early design check for vessels that need to be course unstable.
Lyster and Knights (53) obtained regression equations for standard turning circle parameters from maneuvering trials of a large number of both single- and twin-screw vessels. Being based upon full-scale trials, these results represent the fully nonlinear maneuvering performance of these vessels. These equations predict the advance, transfer, tactical diameter, steady turning diameter, and steady speed in a turn from hull parameters.
The input and output report from a typical run of the Maneuvering Prediction Program (MPP) is shown in Figure 11.24. More details of this program are available in the manual (40). The program estimates the linear stability derivatives, transforms these into the time constants and gains for Nomoto’s first- and second-order maneuvering equations, and then estimates the characteristics described above.
These results can be compared to generalized data from similar ships (57) and Figure 11.23. The example ship analyzed is course unstable since C < 0, with good turnability as indicated by Pc = 0.46, but should be easily controlled by a helmsman since the phase margin is 2.4º > –20º. Norrbin's turning index can be seen to be favorable in Figure 11.23. The advance of 2.9 L and tactical diameter of 3.5 L are well below the IMO required 4.5 L and 5.0 L, respectively. If these results were not acceptable, the design could be improved by changing rudder area and/or modifying the basic proportions of the hull.
11.4.3 Seakeeping Performance Estimation