4.6 Alternative procedures

 The procedure consists of the sequential evaluation of:

• .1 roll damping (Bertin’s coefficient N ) from roll decay test in calm water;

• .2 effective wave slope coefficient r from roll tests in beam waves; and

• .3 the “regular waves roll-back angle” Φ _1r .

  4.6.1.1 Execution of roll decay tests

  4.6.1.1.1 To obtain the roll damping characteristics of the ship, a series of roll decay tests for the scaled model in calm water should be carried out. The model is initially inclined up to a certain heel angle. This initial angle should be larger than about 25°. If the mean roll angle between the initial angle and the next peak angle is smaller than 20°, the initial angle should be increased to obtain a mean angle of 20° or over. When the initial roll angle is given to the model, additional sinkage and trim should be minimum. The model should be released from an initial angle with zero roll angular velocity. During this test, no disturbance including waves propagating in the longitudinal direction of the basin and reflected by its end should be given to the model. At least four tests with different initial angles are required. If the roll damping is very large, the number of tests should be increased to obtain sufficient number of peaks of the roll angle. Recording of the roll time history should start before the release of the model to confirm that no angular velocity is given when releasing. Recording should continue until the model has reached rolling angles smaller than 0.5°. This eventually requires that the length of the basin should be sufficiently large.

  4.6.1.1.2 Full details of the experiments, including time histories, should be included in the report.

  4.6.1.2 Determination of Φ _1r

  4.6.1.2.1 First step

The aim of this step is the determination of the Bertin’s extinction coefficient curve and the roll period as a function of roll amplitude. Assuming that the absolute values of measured consecutive extremes (one maximum and following minimum or vice-versa) of roll angle during roll decay are Φ ₁, Φ ₂, ... (deg), the mean roll angle and the decrement δΦ _i = Φ _i — Φ _i-1 are calculated. Bertin’s extinction coefficient, N, as a function of Φ _m is obtained by . It should be noted that N depends on roll amplitude. The obtained raw data for should be fitted by a smooth curve. In addition, periods from peaks to peaks should be calculated as a function of mean roll angle, which is necessary for step 2.

An equivalent linear damping coefficient defined as:

where Φ is in degrees, can be used as an alternative to the Bertin's coefficient. When the equivalent linear damping coefficient is used, all the formulae involving N(Φ) should be modified accordingly.

In case frictional correction on roll damping is required in paragraph 4.3.2, the above value of N should be reduced by the value from the following formula, which represents the model-ship correlation on frictional damping:

where:

All variables should be in model scale and the symbols in the above formulae are defined as follows:

Alternatively a numerical calculation with unsteady boundary layer can be used to the satisfaction of the Administration.

Alternatively, a forced roll test may be used to determine the N(Φ) coefficient by using an internal or external roll motion generator.

The former requires measurement of roll angles and the latter does that of roll moment. The experimental procedure and the subsequent analysis of data should be subject to the satisfaction of the Administration. In order to decide on the suitability of experimental and analysis procedure, as a guide, a reasonable agreement between results from forced roll tests and N(Φ) from roll decay tests, can be considered a good indication.

  4.6.1.2.2 Second step

The aim of this step is the determination of the effective wave slope coefficient r. The following two methods are provided:

• .1 The resonant roll amplitude in regular waves is determined according to the procedure described in paragraph 4.5.2 but using a wave steepness which should be smaller than 1/20. Regardless of the requirement in paragraph 4.5.2, a used wave period should be the same as the given natural roll period. Once the steady roll amplitude is obtained, the natural roll period for this amplitude should be estimated with the results of roll decay test. If this period is significantly different from the wave period, roll angle measurement should be repeated but by using the newly estimated period as the input to the wave maker. Then the effective wave slope coefficient, r, is determined as follows:

  

  where T_wave,r and H_r are the wave period in seconds and the wave height in meters respectively used in the test, and g is the gravitational acceleration in m/s² In equation (4.6.1.2.2-1) the wave steepness is assumed to be related to wave height and wave period by . The effective wave slope is assumed to be independent on Φ_r .

• .2 Alternatively it is possible to directly measure the roll excitation moment M_exc by means of a dynamometer. The model should be connected to the carriage by means of a guide allowing drift, sway, heave and pitch motions but fixing surge, roll and yaw. The dynamometer should measure the moment with respect to centre of gravity between model and the carriage. The dynamometer should be designed to limit the interaction between the detected force components within 2% of the resultant ones. Coefficient r is then determined as follows:

  

  4.6.1.2.3 Third step

The aim of this step is the prediction of the peak of roll for the steepness specified in table 4.5.1. By using the curve for N(Φ) and the estimated value for r from previous steps, and by using the wave steepness s obtained from table 4.5.1, the predicted angle of roll Φ _{1 r} can be calculated by the following formula:

Since this formula includes Φ _{1 r} in both its right- and left- sides, the calculation should be carried out with the following iterative procedure:

• .1 Φ _{1 r} is initially assumed to be 20°

• .2 the right-hand-side of this formula is calculated;

• .3 the obtained Φ _{1 r} should be substituted into the right-hand-side; and

• .4 when the value of Φ _{1 r} converges to a certain value, this should be regarded as the final value.

 The parameter identification technique (PIT) approach is outlined below, taking into account linear and nonlinear features of the mathematical model describing the roll motion in beam waves, with other forcing sources or roll decays. The basic structure of the method consists in the regression of the solution (exact or approximate, analytical or numerical) of the system of differential equations describing the time evolution of the system under analysis, containing as unknowns the characteristic parameters (coefficients of the mathematical model adopted to describe damping, restoring, forcing terms). The regression is considered to the experimental values of stationary roll amplitude versus frequency for forced roll. The basic idea on which the PIT relies is thus as follows: the solution of equation (4.6.2.1.1), for any consistent set of parameters and different wave frequencies allows to obtain a prediction for the roll response. The parameters of the model are modified systematically by the minimization procedure in order to obtain the best agreement between the predictions given by the model and measured experimental data. The “optimum” set of parameters is then obtained and used in solving equation (4.6.2.1.1) for the steepness required by table 4.5.1 and different wave frequencies, to obtain, finally, the peak Φ _{1 r} of the roll response curve. The angle of roll to windward due to wave action Φ ₁ is calculated according to paragraph 4.1.

When PIT is used, at least two response curves obtained for two different wave steepness are strongly recommended to be used.

  4.6.2.1 Modelling of roll motion in beam sea and determination of model parameters

  4.6.2.1.1 Recommended model in beam sea

The following differential equation is recommended as a suitable model for describing roll behavior in regular beam sea:

In the recommended model (4.6.2.1.1) the following parameters should, in principle, be considered as to be determined by the PIT: ω ₀, μ ,β,δ,γ ₃ ,γ ₅,α ₀,α₁,α₂. However, in certain cases, some of these parameters can be considered as constant and/or equal to zero.

  4.6.2.1.2 Definition of χ ²

  4.6.2.1.2.1 From a series of experiments in beam waves according to paragraph 4.5.2 (apart from required wave steepness), a value of roll amplitude C _{exp, ij} is obtained for each tested wave frequency ω _i and steepness s _j . It is recommended to determine the roll response curve for at least two different value of the wave steepness and a set of frequencies, for each wave steepness, as in paragraph 4.5.2. Given a tentative set of parameters { ω ₀ ,μ ,β ,δ ,γ ₃ ,γ ₅ ,α₀ ,α₁,α₂} , the value of roll amplitude C _{mod , ij} can be obtained (by numerical integration or analytical solution) as predicted by the model in equation (4.6.2.1.1) for each tested wave frequency ω _i and steepness s_j .

  4.6.2.1.2.2 The following function is used as a measure of the goodness of fit for the model:

As can be seen from equation (4.6.2.1.2.2), χ ² depends on the tentative values of the model parameters.

  4.6.2.1.3 Fitting of the model

The scope of the PIT is to determine a set of “optimum” parameters {ω ₀ ,μ ,β ,δ ,γ ₃ ,γ ₅ ,α ₀ ,α ₁,α ₂}_opt such to minimize χ ², that is:

Any numerical or analytical minimization procedure can be used, to the satisfaction of the Administration.

  4.6.2.1.4 Calculation of roll response’s peak Φ _1r

  4.6.2.1.4.1 When the “optimum” set of parameters {ω ₀ ,μ ,β ,δ ,γ ₃ ,γ ₅ ,α ₀ ,α ₁,α ₂}_opt is determined by the minimization procedure, the response curve for the steepness required in table 4.5.1 can be obtained as follows.

  4.6.2.1.4.2 Equation (4.6.2.1.1) is solved by means of standard numerical integration algorithms or analytical solution for different frequencies in order to obtain the roll response curve. The peak of such curve is Φ _1r .

  4.6.2.2 Additional comments

The framework of the methodology provided in paragraph 4.6.2.1 could be used, in principle, to obtain damping parameters from free roll decays or forced roll motion by means of roll moment generators (RMGs). Partially different modelling and/or definition of χ ² could thus be needed and can be used to the satisfaction of the Administration.