Twenty-First Symposium on Naval Hydrodynamics (1997) / Chapter Skim
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Third-Order Volterra Modeling Ship Responses Based on Regular Wave Results
Third-Order Volterra Modeling Ship Responses Based on Regular Wave Results
Pages 189-204
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From page 189... ...
Abstract A third order Volterra modelling was used to calculate the nonlinear vertical hull girder loads in irregular head waves in two slender models with different degrees of bow flare. The first order and the required approximations of the second and third order frequency response functions were derived from systematical experiments in regular waves only.
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From page 190... ...
This paper shows the applicability of an approximate third order Volterra modelling to analyze the statistics of the vertical hull girder loads in irregular waves. The numerical results are compared with extensive model tests in irregular waves.
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From page 191... ...
The high-frequency higher order components cause the response spectra to become wider, which is another complication in extreme statistics. In the following section a theory is outlined to derive approximations for the linear, quadratic and cubic frequency response functions using which it is possible to derive the response statistics in irregular waves for those nonlinear responses in which the first order term is still dominant.
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From page 192... ...
= ~ hi It—tights (5) /J : Ant so 0 to Hog [Nml Se9 Figure 4: Probability density functions of the recorded bending moments in the original Wigley (upper)
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From page 193... ...
2 4 (Do |rad/s] Figure 5: Equivalent RAO's derived from the irregular wave tests compared with the regular wave results ear, quadratic and cubic frequency response functions respectively are given by the single, double and triple Fourier transform of the time-domain kernels and show the same symmetry relations with respect to the variables as the time-domain kernels.
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From page 194... ...
as derived from the second harmonic. It was observed in the analysis of the regular wave results that the modulus of the experienced second harmonic and mean bending moment were generally of the same magnitude, especially for the Wigley with bow flare which shows the severest nonlinear behaviour i21]
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From page 195... ...
4 The Resulis In this section, reconstructed time histories resulting from the first, second and third order Volterra modellings are analysed and compared in detail with experiments in irregular waves. The main purpose of these analyses is to investigate whether or not it is possible to make a proper prediction of the probability densities and of the statistics of the maxima, minima and ranges of the hull girder loads.
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From page 196... ...
The reconstructed motion responses of the Wigley with bow flare are shown in figure 6 and 7. Figure 8 and 9 show some examples of the measured and numerically reconstructed vertical bending moments in the same Wigley model.
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From page 197... ...
This is the explanation for the large spreading in the normalised first harmonic responses in regular waves, which showed an increase of the normalised first harmonic response amplitude in the frequency response peak of more than 20 To over the range of tested wave amplitudes, see figure 5. The low frequency and double frequency peak are very well predicted by applying the second order Volterra modelling.
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From page 198... ...
mO ml m2 ma m4 125 1.40e3 3.99e4 2.24e7 1.79e8 119 955 8.43e3 8.82e4 1.52e6 129 l.OOe3 8.26e3 7.59e4 1.13e6 133 1.03e3 8.71e3 8.39e4 1.29e6 113 883 7.64e3 7.84e4 1.29e6 mO ml m2 ma m4 14.8 250 9.64e3 5.84e5 4.11e7 12.3 115 1.33e3 1.85e4 3.36eS 6.1 54 500 4.98e3 6.62e4 9.1 80 864 1.07e4 1.72e5 12.4 116 1.36e3 1.92e4 3.52e5 mO ml m2 ma m4 48.6 458 5.91e3 1.73eS 1.17e7 35.3 337 5.73e3 1.69eS 8.94e6 48.2 437 4.43e3 5.10e4 7.69e5 34.2 328 3.50e3 4.21e4 6.32eS 46.9 417 4.07e3 4.39e4 6.27eS 31.2 286 2.84e3 3.09e4 4.38e5 49.5 439 4.40e3 4.95e4 7.34eS 35.4 323 3.38e3 3.96e4 6.13e5 49.9 453 4.68e3 5.45e4 8.28eS 41.5 385 4.06e3 4.90e4 7.75eS _ 7 mO m: r.~2 ma m4 28.9 330 9.53e3 6.02e5 4.70e7 46.2 662 2.57e4 1.70e6 1.27e8 27.8 234 2.23e3 2.43e4 3.78e5 41.3 344 3. l9e3 3.52e4 6.62eS 30.5 251 2.25e3 9.21e4 3.13e5 36.6 298 2.55e3 2.40e4 3.47e5 32~0 265 2.67e3 2.58e4 3.81e5 38.7 316 2.81e3 2.84e4 4.36eS 27.2 294 2.17e3 9.48e4 3.97e5 34.4 293 2.84e3 3.35e4 5.81e5 mO me m2 ma m4 Table 2: Comparison of the measured and simulated spectral moments of the vertical hull girder loads ire both Wigley variants, all simulations filtered at 4 Hz simulations gave the best results for the higher order spectral moments.
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From page 199... ...
07 7.476 88.42 0.016 0.047 -0.426 7.617 93.59 -0.217 0.402 -0.426 7.020 78.13 -0.135 0.179 ~ ~1 ~ K3 |
From page 200... ...
4.4 Peak-Peak Probability Densities The final test for the proposed Volterra modelling was the comparison of the probability density of the peak-peak values. The time histories as well as the power spectra showed a wide band hull girder load response, which was not the result after the linear simulation, and was only partly estimated by the second order simulation.
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From page 201... ...
This was confirmed by the experimental results. 5 Discussion and Conclusion A third order Volterra modelling was developed for the calculation of the hull girder load responses in irregular waves.
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From page 202... ...
An investigation of midship bending moments experienced in extreme regular waves by a Mariner-type ship and three variar~ts. Technical Report SSC-155, Ship Structure Committee, 1964.
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From page 203... ...
Experimental investigation of the influence of bow flare and forward speed on the nonlinear vertical motions, bending moments and shear forces in extreme regular waves. Technical Report 993 MEMT 32, Delft University of Technology, Ship Eydromechanics Laboratory, February 1994.
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From page 204... ...
To explain this phenomenon, it is necessary to explain the physical origin of the loads. In general, hull girder loads are defined as the forces and moments in a hull cross-section, i.e., the "internal loads," which make equilibrium with the net external forces and moments and the inertial reactions of the ship.
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