Applying the above defined methods a linear control and response analysis was performed using the OPTIPOD designs. The results of the analysis were then compared to free-running model tests results to assess the usefulness of the model.

　

5.1 Second Order Equations

　

 The general form of the equations of motion in the horizontal plane form a coupled pair of simultaneous equation, as shown in Eqs.(29) and (30). These equations can be rearranged into a pair of decoupled second order equations [19]; one in terms of sway velocity the other in terms of yaw rate. In practice, yaw rate is far easier to measure than sway velocity and is thus used for this analysis as shown in Eq.(31); where the time constant and gain terms are function of the previously defined derivatives [11].

　

　

　

5.2 Simple Stability Analysis

　

 For the system to posses dynamic stability the roots of Eq.(31) must have negative real parts. The equation describes the ship system without control inputs from either helmsman or auto-pilot, thus the roots of this 'open-loop' system will be real rather than complex. The roots in terms of time constants are given by Eqs.(32) and (33). The stability boundary is reached when either σ1 or σ2 become zero which is true when either the first or second time constant becomes infinite. It can be shown [19] that the time constants become infinite when the simple stability criteria is satisfied, Eq.(34). Thus, C must be greater than zero for the system to poses dynamic stability.

　

　

　

 This methodology is applied to the OPTIPOD designs to assess their stability characteristics. The results are compared to the results of the free-running spiral tests and are presented in Table 2. Both the Cargo and the Supply ship are predicted as unstable and both the Ropax and the Cruise liner as stable. In direct agreement, both the Cargo and Supply ship showed negative spiral loop gradient, insinuating instability, and both Ropax and Cruise liner showed positive spiral loop gradient, insinuating stability.

　

 This type of analysis can easily be utilised in an optimisation routine to find feasible design solutions [20]. However, the magnitude of the result is not a measure of stability. To investigate both the sign and magnitude the roots should be obtained using Eq.(35). The results are shown in Fig. 9 demonstrating the same conclusion as with the simple analysis. However, it is now apparent that the Ropax should be the most stable while the Supply ship should be the least stable.