For comparison, the spectra calculated by the nonparametric method (Blackman— 
Tukey method) are also shown in this figure. The maximum lag numbers are 50 and 100, 
which give a fairly high confident estimate since these numbers are 1/20 and 1/10 of the 
total number of observations N=1,000. It is interesting to note that the result for lag=50 
looks much like the one for AR(20). The impressive point here is, of course, that the esti- 
mate from the model fitting technique, which determines the order from the minimum 
AIC criteria, gives a result that checks the theoretical process very well. 
Figure 7.6 compares the estimates from AR model fitting and the B—T method for 
the same simulated process (Fig. 7.4) for a wider range of order (AR 1—16) and maximum 
lag numbers of 20-500, for reference. 
Figure 7.7°* shows a few examples of the analysis of actual at—sea ship performance 
data. Rolling, wave height measured by buoy, and the horizontal acceleration of the en- 
gine bed (as of different frequency characteristics, because the engine bed is mostly 
excited by the prime movers of the ship) were analyzed by three different methods: the 
AR model fitting method, the F.F.T. method, and the Blackman—Tukey method. The sam- 
pling time interval was Ar=1 sec for roll and wave height, Ar=0.02 sec, much smaller for 
acceleration. The spectrum of the horizontal acceleration of the engine bed estimated by 
the AR model fitting method gives very reasonable results, compared with the ones ob- 
tained by other methods, the former clearly showing the existence of multiple natural 
frequencies. 
Figure 7.8>° shows other examples of the spectra of the horizontal acceleration of 
the engine bed estimated by the AR model fitting method at orders n of 5, 10, 15, 20, 30, 
and 55. The result at n=10, where the AIC shows minimum values, looks most reasonable 
and clearly shows the existence of multiple natural frequencies in horizontal vibration. 
7.3. EXAMPLES OF PARAMETRIC ANALYSIS OF RESPONSE 
CHARACTERS OF MARINE VEHICLES AND STRUCTURES 
Figure 7.9°? shows an example of simple response, the case of yaw angle versus 
rudder angle of a seagoing ship. The results for the spectra of input and output and the 
frequency responses from two methods, the AR model fitting and nonparametric meth- 
ods, are compared. When the spectrum and response characteristics do not show any 
abrupt changes with frequency, as in the example, most of the results from the two meth- 
ods look similar. However, when we look at the Nyquist diagram or the mode diagram of 
the frequency response characteristics, the results from the AR model fitting are much 
better and smoother and more reasonable, physically, than those from the nonparametric 
method. 
Figures 7.10 and 7.11*° show the results of seakeeping data for a model ship in 
waves produced in the towing tank, also analyzed by two different methods. 
From these figures we see that the AR—model fitting techniques give smoother 
curves for the spectra and for the frequency responses. Also, in the important range of 
frequencies, the coherency values are closer to 1, showing that the response characteris- 
tics are more reliable when estimated by this method. 
In both figures, the cross spectra are shown by their real and imaginary parts, i.e., 
by co— and quadrature—spectra. The shape of these spectra, which show very sharp peaks, 
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