Experimental Study of Nonlinear Moored-Buoy Responses

Objectives:• To Identify and Classify Highly

Nonlinear Experimental Structural Responses Under Combined Deterministic and Random Waves

• To Validate Analytical Model Predictions and Investigate Fluid-Structure Interactions

Principal Investigator:• Prof. Solomon C.S. Yim

Civil Engineering Department Oregon State University

Approach: • Modifying Existing ID Techniques and/or

Developing New Tools to Classify Degree of Nonlinearity

• Comparing Overall Behaviors of Experimental Data and Simulations to Validate Analytical Model

• Conducting Sensitivity Study to examine Hydrodynamic Properties

Experimental Configuration (SDOF)

Highly Nonlinear Experimental Structural Responses

Observations: • Primary and Secondary Resonances in

Frequency Response Diagram

• Harmonic, Subharmonic, Superharmonic, and Possibly Chaotic Responses

• Transition Behavior of Multiple Coexisting Response Attractors

Possible Chaos (Poincare Map)

Possible Chaos (Poincare Time History)Coexisting Harmonics and Subharmonics

Poincare Analysis of Multiple Coexisting Responses

Sections II & VI Sections IV & VIII

Sections III & VIISections I & V

Comparisons of Experimental Results and Analytical Predictions

Numerical Model:

f(t)(t).x(t)

.x

4

2πDd

ρC

R(x(t))(t).xsC(t)

..x)aM(M

where

u(t)u(t)4

2πDd

ρC(t).umC3D

6πρf(t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

1

2

3

4

5

6

7

Freq (Hz)

Am

pl R

atio

Frequency Response Diagram Distributions of Large Excursions

Time-Averaged PDF

On-Going/Future Research

On-Going Research:• Identifying Best-Fit Fluid-Structure

Models

• Investigating Hydrodynamic Properties

• Predicting Occurrence of (Noisy) Chaos

• Verifying Inter-Domain Transitions as Analytically Predicted

Future Research: • Formulating Models Based on Large

Body Theory

• Extending Analysis Procedure to MODF Experimental Results

• Comparing SDOF and MDOF Results

0 0.5 1 1.5 2 2.5

x 105

0

1

2

3

4

5

6

7

8

9

10

Re

Cd

Cd versus Reynolds Number

0 0.5 1 1.5 2 2.5

x 105

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Re

Cm

Cm versus Reynolds Number

Stochastic Analysis of Nonlinear System under Narrowband Excitation

Principal Investigator:• Prof. Solomon C.S. Yim

Civil Engineering Department

Oregon State University

Objectives:• Improve accuracy of prediction using

semi-analytical method

• Apply semi-analytical method to nonlinear-structure nonlinearly-damped (NSND) model

• Compare prediction with experimental data

Approach:• Identify typical nonlinear response

behavior under deterministic excitation

• Employ semi-analytical method to predict system response

• Validate prediction method through comparison with experimental data

Coexisting Attraction Domain

Fig 3. Small amplitude harmonic, 1/3 subharmonic, 1/2 subharmonic and large amplitude harmonic

Response Amplitude Curve

Fig 4. Response amplitude curve of system in subharmonic region

Response under Deterministic Excitation

Typical Nonlinear Response Behavior

Fig.2 Four different response behavior under same excitation

100 105 110 115 120

-10

-5

0

5

10

excitation

100 105 110 115 120

-10

-5

0

5

10

excitation

100 105 110 115 120

-5

0

5

large harmonic

100 105 110 115 120

-5

0

5

small harmonic

100 105 110 115 120

-5

0

5

1/2 subharmonic

100 105 110 115 120

-5

0

5

1/3 subharmonic

Progress:System Configuration

(a) Plan view (b) Profile view Fig.1 Experimental model of nonlinear system

Stochastic Analysis of Nonlinear System under Narrowband Excitation

Where, A(1),A(2) = excitation amplitude of current and next cycles, = phase angle difference, (1)-(2)

Intra-Domain Transition

Fig 7. Intra-domain transitions within four different attraction domains

Response under Narrowband Excitation

Stochastic behavior of the excitation parameter

)sin()cos(22

1exp

2

),,(

)2()1(2)2(2)1(2)2()1(

)2()1(

ijf AAAAAA

AAp

Jump Phenomena

Fig 5. Amplitude jump from large amplitude to small amplitude harmonic domain

Fig 6. Schematic diagram of inter-domain transitions

Inter-Domain Transition

100 150 200 250-2

-1.5

-1

-0.5

0

0.5

1

1.5

2excitation

time

100 150 200 250-10

-5

0

5

10response

time

Stochastic Analysis of Nonlinear System under Narrowband Excitation

Future study:• Apply semi-analytical method to

NSND model

• Predict response of NSND model with coefficient determined by system identification technique

• Verify prediction using experimental dataFig 10. Overall response amplitude probability

distributions (compared with simulation result)

Result Overall Response Amplitude Probability Distribution

Numerical simulation

Large amplitude harmonic response

1/2 subharmonic response1/3 subharmonic responsesmall amplitude harmonic response

Fig 8. Time series of narrowband excitation amplitude (top) and corresponding response amplitude (bottom)

Fig 9. Amplitude response map correspond to time series shown in Fig 8.

Stochastic Analysis of Nonlinear System under Narrowband Excitation

Modeling and Validation of Nonlinear Stochastic Barge Motions

Modeling and Validation of Nonlinear Stochastic Barge Motions

Objectives: - To examine predictive capability of coupled Roll-Heave-Sway models to estimate stochastic properties of nonlinear barge response behavior - To develop probability-based analysis and design methodology

FIG. 1. Roll-Heave-Sway Model

Approach: - Develop Roll-Heave-Sway barge-motion models (and lower order ones) - Identify system coefficients - Examine and compare numerical predictions with measured data - Develop nonlinear extreme-value prediction techniques

Principal Investigator:- Prof. Solomon C.S. Yim Civil Engineering Department Oregon State University

Identification of System Coefficients for Roll-Heave Model - Regular Waves

Comparison of Model Predictions with Measured Data - Measured Random Waves - Simulated Random Waves

Modeling and Validation of Nonlinear Stochastic Barge Motions

FIG. 1. Roll vs Roll Velocity (Regular Waves, Case SB27)

measured predicted

FIG. 2. Roll vs Heave (Regular Waves, Case SB27)

measured predictedFuture Research:- Examine complex nonlinear behavior including resonance and possible chaos - Perform and compare simulations- Use model to verify proposed theories on capsize

-10 0 10-15

-10

-5

0

5

10

15

Roll (deg)

Rol

l Vel

(ft/

s)

-10 0 10-15

-10

-5

0

5

10

15

Roll (deg)

Rol

l Vel

(ft/

s)

-10 0 10-5

0

5

Roll (deg)

Hea

ve (

ft)

-10 0 10-5

0

5

Roll (deg)

Hea

ve (

ft)

-15 -10 -5 0 5 10 150

100

200

300

400

Roll (deg)

Quantity

SB25 Measured Roll: Var= 15.673 Max= 11.8325 Min= -11.2845

-15 -10 -5 0 5 10 150

100

200

300

400

Roll (deg)

Quantity

Predicted Roll: Var= 17.1095 Max= 13.304 Min= -14.7709

FIG.3. Roll Distribution (measured random waves)

-5 0 50

200

400

600

Heave (ft)

Quantity

SB35 Measured Heave: Var= 1.3612 Max= 3.6192 Min= -3.5618

-5 0 50

200

400

600

Heave (ft)

Quantity

Predicted Heave: Var= 2.2245 Max= 4.6174 Min= -4.4626

FIG.4. Heave Distribution (measured random waves)

Modeling and Validation of Nonlinear Stochastic Barge Motions

-20 0 20-20

-10

0

10

20Test SB25: Roll vs Wave

Roll (deg)

Wav

e (ft

)

-20 0 20-20

-10

0

10

20Roll vs Wave, Hs=4.737 ft Tp=8.2 s

Roll (deg)

Wav

e (ft

)

measured predicted

FIG.5. Roll vs Wave (measured random waves)

-20 0 20-20

-10

0

10

20Test SB25: Roll vs Heave

Roll (deg)

Hea

ve (

ft)

-20 0 20-20

-10

0

10

20Roll vs Heave, Hs=4.737 ft Tp=8.2 s

Roll (deg)

Hea

ve (

ft)

measured predicted

FIG.6. Roll vs Heave (measured random waves)

-15 -10 -5 0 5 10 150

100

200

300

400

Roll (deg)

Quantity

SB25 Measured Roll: Var= 15.673 Max= 11.8325 Min= -11.2845

-15 -10 -5 0 5 10 150

100

200

300

400

Roll (deg)

Quantity

Predicted Roll: Var= 14.0997 Max= 11.3505 Min= -12.5418

-5 0 50

100

200

300

400

Heave (ft)

Quantity

SB35 Measured Heave: Var= 1.3612 Max= 3.6192 Min= -3.5618

-5 0 50

100

200

300

400

Heave (ft)

Quantity

Predicted Heave: Var= 0.95476 Max= 3.1609 Min= -3.0591

FIG.7. Roll Distribution (simulated random waves)

FIG.8. Heave Distribution (simulated random waves)

-10 0 10

-10

-5

0

5

10

Measured SB25: Roll vs Wave

Roll (deg)

Wav

e (ft

)

-10 0 10

-10

-5

0

5

10

Predicted: Roll vs Wave

Roll (deg)

Wav

e (ft

)

-10 0 10

-10

-5

0

5

10

Measured SB25: Roll vs Heave

Roll (deg)

Hea

ve (

ft)

-10 0 10

-10

-5

0

5

10

Predicted: Roll vs Heave

Roll (deg)

Hea

ve (

ft)

Modeling and Validation of Nonlinear Stochastic Barge Motions

FIG.9. Roll vs Wave (simulated random waves)

measured predicted

FIG.10. Roll vs Heave (simulated random waves)

measured measured