TimeWhipping/springing response in the time series analysis series of ship structural stresses in marine science and applicat Wengang Mao*, Igor Rychlik ions for industry Chalmers University of Technology, Göteborg, Sweden wengang@chalmers.se Logonna -Daoulas (France) «Time-series analysis in marine science and applications for industry» 17-22 Conference in Logonna-Daoulas, France, 17-22 sept. 212
Outline Background Full-scale measurements from containerships Time series of structural stresses (sample frequency 25 Hz) Wave height, ship speed,heading... (each 3 minutes) Wave (WF) and high frequency (HF) signals from the stresses Fatigue and extreme response due to WF and HF signals Modeling of whipping/springing by LMA Spectrum and kernel for whipping signals Modeling of HF + WF response
Wave frequency (WF) ship response Vertical 2-node vibrations of ship as a beam P a] S tre s s [M P a ] W F 5-5 1 2 3 4 5 6 7 8 9 T im e [ s ] WF signals: the response frequency is closehtof the encountered wave frequency 5 1
HF signals whipping/springing W F S tre s s [M P a ] 5-5 Whipping signals: due to transient loads 1 2 3 4 5 6 7 8 9 1 such as slamming, green water T im e [ s ] HF S tre s s [M P a ] 5-5 1 2 3 4 5 6 T im e [ s ] W F + HF 7 8 5 Springing signals: resonance response 9 1
Fatigue problems and extreme loadings Fatigue cracks observed in ship structures with only about 2-5 years service (Gaute Storhaug). Slamming loads applied on ship s bow section and effect of extreme loadings (photos from internet)
Containerships in the future One of the biggest container vessel 35 m long (more than 12 TEU containers)
The full-scale measurements Measurements from 2 different container ships Based on the time series of data, we will study If WF signals in a stationary sea condition are Gaussian How much fatigue damage caused by WF signals HF effect to ship s fatigue and extreme response How to model HF signals by LMA
Measurement instruments Strain sensors Wave radar Wave buoys Satellites/hindcast GPS Wind sensor Accelerometer Rudder angle RPM... Gaute Storhaug
Ship sailing routes 28TEU containership 7 voyages from EU to NA 7 voyages from NA to EU Time in total 6 months 44TEU containership 2 voyages from EU to NA 2 voyages from NA to EU Time in total 2 months
Measured time series of stresses 28 TEU 44 TEU Measured times series of ship structural stresses in 3 minutes (a stationary sea state)
Spectra of measured ship responses The real signals of all sea states contain 3 peaks: i. Wave frequency signals (about 97% energy) ii. High frequency signals (3%) iii. Measurement noise High frequency signals Transient oscillation-whipping, and resonance vibration -- springing. Hard to compute! Response spectrums at different sea sates during one winter voyage
Definition and time series of HF response Spectral density 16 7 S(w) [m 2 s / rad] 6 5 W ave induced response High frequency response 14 Wave induced fp1 = 1.1 [rad/s] Whipping contri fp2 = 4.6 [rad/s] 12 Time series 1 8 4 6 3 Whipping/springing: High frequency (HF) signals 2 4 2 1-2 2 1 2 3 4 5 6 2.1 7 Frequency [rad/s] Separated signal with wave frequency & high frequency Total response = WF + HF (whipping/springing) 1. WF signals: ω [, 2] [rad/s] 2. HF signals: ω [2, 7] [rad/s] Page 12 3. Measurement noise: ω > 7 rad/s 2.2 2.3 2.4 2.5 2.6 x 1 4
Fatigue damages due to HF response Step 1: Dt Total fatigue damage Step 2: Dwf Extract WF signals and get the damage Step 3: Dhf = Dt-Dwf
WF response caused fatigue Fatigue Damage = f V, θ, T, S ( H s, T p ), C Operation profiles V - Ship speed θ - Heading angle T - Sailing time Wave environment Hs Wave height Tp Wave period Structural response characteristics, got from engineering analysis
Measured Hs in one storm
Calibration of wave measurements 1. 2. 3. Onboard wave measurements include some uncertainties; Radar measurements should be calibrated before practical applications; Some statistical model may be used to interpolate Hs for missing data.
Results: Fatigue due to HF signals (1) Fatigue damage ( α D(t)) 6 x 1 1 5 T o ta l fa tig u e d a m a g e ( W F + H F ) 4 W a v e in d u c e d f a t ig u e ( W F ) 3 2 1 816 8117 8129 829 8218 831 8312 8321 841 8411 8424 863 8613 Difference Ratio (%) 4 3 HF fatigue: Whipping contributed 24% fatigue!! 2 1 816 8117 8129 829 9218 831 8312 8321 841 8411 8424 863 8613 Date of each voyage (identical with the column 1 in Table 1)
Results: Fatigue due to HF signals (2) General container/cargo 23% 28 TEU 28% 4 TEU 39-46% 44 TEU 37% (model test) 46 TEU 35% ( Rathje et al. 212) 44 TEU 26% 67 TEU 5% 86 TEU >6% (model test) 14 TEU 57% (Rathje et al. 212) The increase of extreme response follows similar trend! Gaute Storhaug
HF effect on extreme response --from the full-scale measurements of 2 container ships
Predict extreme response by upcrossings is the time series of stresses (signals) in 1 year (x) is expected no. of upcrossings during one year; + R e s p o n s e le v e ls ( M P a ) 2 x 1.5 T le v e l N + ( xt ) = 1 T - year response xt is at the crossing level 1/T m a x im u m re s p o n s e : M M le v e l 1 x.5 x 1 2 le v e l le v e l 5 1 T im e t 15 2
Extreme response due to HF effect Step 2: Upcrossing from extracted WF signals (observed and Rice s formula) Step 3: Upcrossings from HF+WF signals Step 1: Extract WF signals from the measurement (WF+HF signals) Step 4: Extrapolate the upcrossings to some extreme levels to get XT.
WF signals is it Gaussian? 4 Hogging Sagging 1 1 2 O b s e r v e d c r o s s in g W F c r o s s in g s a s s u m in g G a u s s ia n -5 1 u p c r o s s in g n u m b e r s u p c r o s s in g n u m b e r s 1 5 u p c r o s s i n g le v e ls ( M P a ) 28TEU in 6 months 1 1 4 Sagging Hogging 2 o b s e r v e d c r o s s in g W F c r o s s in g a s s u m in g G a u s s ia n -6-4 -2 2 4 c r o s s i n g le v e ls ( M P a ) 6 44TEU in 2 months
HF signals extreme stress Measurements of 28 TEU 1 1 1 4 Hogging Sagging 2 O b s e r v e d c r o s s in g W F c r o s s in g s a s s u m in g G a u s s ia n -5 u p c r o s s i n g le v e ls ( M P a ) WF signals 5 Sagging u p c r o s s in g n u m b e rs u p c r o s s in g n u m b e r s 1 5 1 Hogging o b s e r v e d u p c r o s s in g H F + W F u p c r o s s in g a s s u m in g G a u s s ia n -1-5 5 u p c r o s s i n g le v e ls ( M P a ) HF + WF signals
HF response extreme stress Measurements of 44 TEU 1 1 4 Sagging Hogging u p c r o s s in g n u m b e r s u p c r o s s in g n u m b e r s 1 1 2 o b s e r v e d c r o s s in g W F c r o s s in g a s s u m in g G a u s s ia n -6-4 -2 2 4 c r o s s i n g le v e ls ( M P a ) WF signals Sagging 1 1 6 4 Hogging 2 O b s e r v e d c r o s s in g H F + W F C c r o s s in g a s s u m in g G a u s s ia n -1-5 5 u p c r o s s i n g le v e ls ( M P a ) HF + WF signals 1
Modelling of ship response WF signals by Gaussian processes Response spectrum can be computed Gaussian process is simulated from response spectrum HF signals by LMA processes Hybrid model to combine the two processes (correlated/independent) WF signals Gaussian process (low frequency) HF signals -- Symmetric LMA (high frequency)
Wave frequency (WF) ship response 6 x 1 15 Hydrodynamic loads: Loading conditions Ship speed Heading angles Panel method for hydrodynamic analysis Energy density function HDG = 5 HDG = 4 HDG = 9 4 3 2 1.2.4.6.8 1 1.2 1.4 1.6 Angular frequency (rad/s) Transfer function (RAOs) Structural analysis: Global response Simple beam theory Direct Finite Element analysis Local structural details 1.8
Skewness and kurtosis of HF signals Skewness of HF signals Kurtosis of HF signals
Symmetric Laplace Moving Average (LMA) X (t ) = g (t u )db (u ) g (t ti ) Z i dt i
Spectrum and kernels for WF response WF response spectrums Kernel for WF signals simulation
Spectrum and kernels for HF response HF response spectrums Kernel for HF signals simulation
Comparison of fatigue damages Relative fatigue damage D=sum( σ 3 ) Observed 5.21e1 Simulated 5.34e1
Extreme prediction Upcrossings of simulated ship response using observed Kurtosis Upcrossings of HF signals Upcrossings of HF+WF signals
Conclusions HF response induces average energy 3% For extreme prediction, it is very sensitive to decide how to put on the whipping transient on the wave frequency response. It will affect significantly of the prediction. HF response contributes fatigue > 3% The wave frequency response is close to Gaussian The HF response is symmetric process The LMA modeling works well to simulate ship response for fatigue assessment
Thanks for your attention.