Table 3.1 summarizes the measurements of the responses and shows that these 
thirteen representative runs cover a good variety of sea conditions, wave heights, and 
directions. This is reflected in the variation of normalized correlations and spectra, for 
example in the rolling, pitching, stress, and relative wave heights in Fig. 3.14 and 
ign sels: 
If we assume a variety of environmental conditions, it is possible that various sea 
conditions might appear ideally by the same chance, and the average of all normalized 
wave spectra might give us a nearly white spectrum, or the spectrum of white noise. Then 
the averaged response might show the characteristics of the response to the white noise, 
i.e., the characteristics of the response functions themselves, as was discussed by 
Yamanouchi, et al.*° The average normalized correlation diagram or correlogram and 
averaged spectra are shown in Fig. 3.16. 
These averaged rolled and pitch spectra are not the ideal ones, but quite reasonably 
show the smooth peak at its natural frequency. Moreover, the correlation functions of 
roll, pitch and stress show the beautiful forms of damped oscillation. The averaged spec- 
trum of stress has a smooth peak at twice the frequency of the roll natural frequency. The 
natural frequency of the stress response should be in a higher frequency range, but must 
have been cut off by the filters and should not appear in the averaged spectra. The peak 
that does appear at the double frequency of the roll natural frequency might indicate that 
the transverse stress induced at the web frame is quadratic nonlinear to the rolling motion. 
To check this possibility, an artificial process of rolling square was made as shown 
in Fig. 3.17. This naturally has a biased mean as shown in its variational form in the time 
series and also in the spectrum in Fig. 3.18, where the power value near w = 0 is large. In 
the next step, a digital high pass filter, shown by Eq. 2.132, was applied to this roll 
1 
squared process as X,(f) = ry {xa =x) 4}. The filtered roll squared process appears at the 
bottom of Fig. 3.17 and its spectrum in Fig. 3.18. In Fig. 3.18, we find the shape of the 
spectrum of the filtered roll squared process is similar to that of the averaged spectrum 
of the stress, which validates the assumption that the stress is a quadratic and nonlinear 
response to rolling motion. 
The lower two graphs in Fig. 3.19 show the results of single input-output spectrum 
analysis, with the stress as output and relative wave height or rolling as a single input. 
Coherencies 7” values are so low that the stress cannot be the output of only the relative 
wave height nor only of rolling. The top of Fig. 3.19 shows on the contrary that relative 
wave height is fairly well explained just by rolling; the coherency, especially in the 
frequency range in which rolling has reasonable power, has a value pretty near 1. 
Since we found that the stress was quite possibly quadratic and nonlinear to the 
rolling, the next step was to find the response of stress to the single roll squared process, 
(roll)?. To eliminate the effect of large bias on (roll), the response of stress to the single 
(roll)? filtered process was also obtained. The results are shown in Fig. 3.20, and the 
coherency is again rather low. 
To check the coexistence of many other inputs, multiple input analysis was intro- 
duced. Stress was considered as the output of rolling, pitching, relative wave height, and 
79 
