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Stochastic Predictions of Parametric Roll Motions of Ships

Stochastic Predictions of Parametric Roll Motions of Ships. J. Juncher Jensen Department of Mechanical Engineering, Technical University of Denmark, Denmark. Parametric Roll. Container Ship in Head Sea. Parametric Roll in Regular Waves. Linear Stochastic Waves. Equivalent Wave for GZ.

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Stochastic Predictions of Parametric Roll Motions of Ships

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  1. Stochastic Predictions of Parametric Roll Motions of Ships J. Juncher Jensen Department of Mechanical Engineering, Technical University of Denmark, Denmark

  2. Parametric Roll

  3. Container Ship in Head Sea

  4. Parametric Roll in Regular Waves

  5. Linear Stochastic Waves

  6. Equivalent Wave for GZ H(X,t): Stochastic wave profile h(t): Equivalent wave height used in calculation of GZ xc(t): Wave crest position

  7. FORM Analysis The design point D (*) and the associated value of βHLdetermine accurately the probability that the response exceed the prescribed value. Theycan be calculated by standard reliability programs.

  8. Conditional Most Probable Roll Motion • Roll period in waves must be in the range of twice the wave encounter peak period. • Simulation time t0 longer than the hydrodynamic memory. • Results to the right: • Design point wave elevation • Roll response for this wave • for Hs=6m, 12m and prescribed roll angle of 17 deg. (0.3 rad.) • A minimum wave height (3.7m) is needed to trigger parametric roll.

  9. Conditional Responses Conditional mean response = Most probable response

  10. Reliability Index βHL Linear response: βHL = constant*Roll/Hs

  11. Variation with V,Tz and Heave

  12. Mean Outcrossing Rates The mean outcrossing rate is given by

  13. Probability of Exceedance Given ship, speed, heading and zero-crossing wave period

  14. Decision Support Systems

  15. Conclusions • The First Order Reliability Method (FORM) provides a fast and accurate tool to predict parametric roll in stationary seaways: • The design point and reliability index follows from FORM • The mean outcrossing rates and probability of exceedance of a given roll angle within a given period is given analytically in terms of the design point and the reliability index. • It requires a realistic time-domain formulation of roll. • The procedure is well suited for generation of a operational polar diagram in an on-board decision support system.

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