1. Chalmers University of Technology Department of Mathematical Sciences Göteborg, Sweden Effect of whipping on ship fatigue- Gaussian VS non-Gaussian modelling Wengang Mao, Igor Rychlik

2. Measurement data Page 2 Measured voyaged in the North Atlantic during 0.5 year Investigated vessel and the measurement location Measurement location 1.Full-scale measurements of mid-section during half year 2.14 voyages: 7 EU-US and 7 US – EU 3.Rainflow (RFC) fatigue estimation as a reference 4.Total fatigue = wave induced fatigue (WF) + whipping fatigue (HF)

3. Scope of the project 1.From full-scale measurements: Our definition of whipping; How much fatigue is contributed from whipping; Investigation of Narrow band approximation (NBA) and Non-Gaussian contributions. 2.General methodology (NBA) for the ship fatigue design 3.Non-Gaussian effect on extreme responses 4.Discussions Page 3

4. RFC analysis of measure stresses Page 4 Winter voyages Summer voyages Fatigue estimated for different voyages (RFC) Spikes Measured stresses during one voyage (2 weeks) Standard deviation of responses in each individual sea state Overview of fatigue estimation 1. Fatigue in Winter from EU to US (Important); 2. Measurement errors should be cleaned 3. Large sea states for further investigation 4. Small sea states only contribute 3.7% fatigue damage Fatigue difference due to removing small response

5. Our definition of whipping Page 5 Whipping: High frequency response Separated signal with wave frequency & high frequency Whipping Definition: 1. Total response = Wave induced + Whipping 2. Wave induced response: f z  [0, 2] [rad/s] 3. Whipping response: f z  [2, 8] [rad/s] 4. Measurement noise: f z > 8 rad/s Spectrum

6. Investigation of whipping Page 6 Wave induced response Whipping response Normalized spectrums of response in a voyage Whipping induced energy: 1. Three peaks of measured spectrum; 2. Last peak is treated as measurement noise; 3. Whipping ratio = 4. Average whipping ratio is less than 3%.

7. Whipping effect on Fatigue Page 7 Fatigue components based on measurements 1.Wave induced fatigue (WF): 72%, 2.Whipping contributed fatigue (HF): 24%, 3.Some other contributions: 4%. Fatigue components Whipping contribution to total fatigue damage

8. Gaussian assumption Page 8 Non-Gaussian effect on fatigue: 1. Simulated process (the same spectrum); 2. Largest fatigue difference 5%; 3. Identical average fatigue 4.Gaussian model is available for applications. Non-Gaussian responses (whipping) contribute 24% of total fatigue! 1.Balanced between whipping contribution (30%) and NBA conservative part (33%); 2.For long-term fatigue estimation, NBA maybe a good choice! Simulated Gaussian process

9. Gaussian model and NBA Page 9 NBA for Gaussian response 1.Energy of ship response 3%: (less influenced by whipping!) 2.h s stress range little influenced by whipping! 3.No measurement available  Numerical analysis (linear). H σ (  ) – RAOs (transfer function)  h s Significant stress range (energy)  f z frequency of the mean level Strongly effected by noise (cut-frequency)

10. h s based on hydrodynamic numerical analysis A new simple fatigue model Polar diagram of the constant C (linear relation between h s and H s ) in terms of the ship speed U (radial direction) and the heading angle  (hoop direction). Page 10 H s – significant wave height T z – crossing period of waves U 0 – ship forward speed  – heading angle

11. Model for f z Page 11 1.f z is strongly influenced by whipping (measurement) 2.f z is computed by linear numerical analysis: f z (waveship) 3.Approximated by the encountered wave frequency: f z (mod) f z model (encountered wave frequency) Vibration period of  2 seconds Vibration of ship beam model (Hogging) f z comparison by different approaches Linear numerical underestimate f z ; Expected value of f z computed from model is close to f z.

12. Estimation of Safety index Total fatigue damage during one voyage Δt – time interval of one stationary sea state v s – ship forward speed  i – ship heading angle H s – significant wave height The model works well for the measurement: Errors for this ship are below 30%.; Errors are smeared out when compute the total damage; The proposed model depends on H s. Page 12

13. Page 13 Gaussian assumption for Extreme prediction Rice’s formula for Gaussian crossings: 1.Up-crossing rate of one stationary period (30min) 2.Up-crossing rate of half a year period Crossing of half a year interval: E[N hy + (X 100 )]=3.6*10 -4

14. Discussions Whipping contributed fatigue 30% Whipping induced average energy 3% h s can be computed by linear analysis Simple NB fatigue model works well wrt RFC Gaussian assumption will lead to large underestimation of extreme response Page 14

15. Thank you! http://www.chalmers.se/math/EN/research/research-groups3561/spatio-temporal Email: Wengang@chalmers.seWengang@chalmers.se