Hydrodynamic derivatives

1. Static drift

Sway velocity derivatives are determined by curve fitting the static drift test data measured over a range of drift angle β.

2. Dynamic tests

Hydrodynamic derivatives can be evaluated by using either the 'Multiple-Run' or 'Single-Run' method.

Hydrodynamic derivatives are evaluated by curve fitting the harmonics data obtained from a multiple number of dynamic PMM tests over a range of PMM motion parameters.

1) Sway derivatives

2) Yaw derivatives

3) Cross-coupled derivatives

'Single-Run' (SR) method:

Hydrodynamic derivatives are evaluated by solving the mathematic model equations using the harmonics data from a single number of dynamic PMM tests for a fixed PMM motion parameter value.

Ex) pure sway test:

1) Sway derivatives

2) Yaw derivatives

3) Cross-coupled derivatives

3. Reconstruction Error

The validities of hydrodynamic derivatives determined using the MR and SR methods are evaluated by examining the errors in reconstruction of forces and moment data time-histories through the mathematic model.
Reconstruction error is defined as

where

D_i is the measured data

R_i is the reconstructed data

1) Sway derivatives reconstructing the pure sway time histories:

2) Yaw derivatives reconstructing the pure yaw time histories:

3) Cross-coupled derivatives reconstructing the yaw and drift time histories:

For MR method, the reconstruction error in general E_R (%) < 20 over the entire range of the tested PMM parameters.
For SR method, the error value is typically huge, E_R (%) < 600, when PMM motion is small, and relatively large, E_R (%) < 50, as the PMM motion becomes large.
Consequently, the MR method is more rigorous than the SR method, and the SR method is only suggested when PMM motion is large enough.