Trans RINA, Vol 152, Part A2, Intl J Maritime Eng, Apr-Jun 2010
depending on symmetry of the body [4, 16]. The model test data available in [11] includes only 21 linear HDCs. The number
of second-order HDCs and various
correction terms may vary from model to model. The mathematical models described in sub-sections 3.2(a) (Model 1) and 3.2(b) (Model 2) contain 55 and 24 HDCs, respectively.
The values of the hydrodynamic coefficients (HDCs) in the equations of motion have their own levels
of
uncertainty depending upon the model testing methods and facilities used to determine them. For the SUBOFF body, [11] the reported levels of uncertainty are listed in Table 3.
Static HDCs: Zw’, Mw’, Yv’, Nv’ Rotary HDCs: Zq’, Mq’, Yr’, Nr’ Control HDCs
Table 3
HDC Estimated Uncertainty 4 – 5 % 10%
Added mass HDCs: Zw.’, Mq.’, Yv.’, Nr.’
6 – 10% 7%
Uncertainty estimated for various HDCs for SUBOFF body obtained by model testing [11]
The error level in one coefficient may be masked by errors in others, such that
computed give misleading results, or such that the trajectories computed are
the margins of similar for
combinations of HDC values. 5.1 SENSITIVITY STUDIES - BACKGROUND
Sensitivity studies have been carried out for HDCs of underwater vehicles in order to determine their relative importance and hence, the error bounds permissible for their values. In general, sensitivity has been defined as the ratio of the relative change in some output variable, O, and the relative change in an input parameter, I, each compared to nominal values (Onom, Inom). For each case the sensitivity, S, of the response to the variation in parameter can be calculated as [10]:
S =
()/ ()/
OO O −
II I −
nom nom nom nom
(26)
The influence of various hydrodynamic coefficients on the predicted manoeuvrability of submerged bodies has been examined [9, 10] and it is reported that for a submarine-like body, trajectories are most sensitive to linear damping coefficients.
For a bare hull axisymmetric body, the linear inertial coefficients were found to be the most significant. In another study
[17], the sensitivity of geometrical
characteristics of an AUV vis-à-vis its added mass HDCs have been explored. Sensitivity analysis using Genetic
stability different
Algorithms [18] for various manoeuvres has revealed that difference in the model geometry caused HDCs to have different tendencies in sensitivity change. Also, the nature of sensitivity changes depending on the trials executed.
Considering the focus of this study on vertical plane manoeuvres, the most significant coefficients for axisymmetric body ,
M Z Z M MMZδδ, ,,, q s ws w , q
have q
been identified [9] as: . For submarine-like body,
these are: ,, ,sδδ , qs , w, w, q q , w M M Z Z ZZZ M M . The q
difference in relative sensitivity has been attributed to the difference in the mathematical model used for the two bodies, but may also be the result of the difference in their geometries.
In order to ensure that the vertical plane trajectories do not vary by more than 10%, it is reported [9] that the HDCs that need to be determined within about 10% accuracy are ,, ,
body. Based on a study using
Z MZ Z s ,δδ M M for axisymmetric Genetic Algorithm
ws q q , w
technique [18], Mw and Mq were found to be the two most significant HDCs.
5.2 SENSITIVITY STUDIES - AXISYMMETRIC BODY
For this study, the HDCs considered most significant for the axisymmetric body were varied, one at a time, by ±25 to 50% of their initial values and the trajectory simulation was repeated using the linear mathematical model (Model 3). Although these values are higher than the uncertainty levels estimated for model test data (Table 3), the aim was to explore the relative and cumulative effect of change in various HDCs on the trajectory simulation parameters for the zigzag manoeuvre in vertical plane. Further, as an extreme case, all
the HDC values were simultaneously changed by ±50% from initial values repeated.
and trajectory simulation
Results of these sensitivity studies are summarised in Table 4. The results are graphically shown in Figure 6, wherein the cumulative effect of changes in each HDC is presented for the five parameters of the vertical plane zigzag manoeuvre.
It is seen that the three most important HDCs for the case considered are MδS (change in pitching moment due to
plane deflection), ZδS (change in heave force due to plane deflection) and Mq (change in pitching moment due to rate of change in pitch angle). Change in value of these
HDCs by ±50% causes change in at least one of the vertical plane zigzag manoeuvre parameters by more than 50%. The marked influence of HDCs MδS, ZδS and Mq are clearly evident from Figure 6.
©2010: The Royal Institution of Naval Architects
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