The nonlinear responses of ship rolling motion characterized by a roll damping moment are of great interest to naval architects and ocean engineers. Modeling and identification of the nonlinear damping moment are essential to incorporate the inherent nonlinearity in design, analysis, and control of a ship. A stochastic nonparametric approach for identification of nonlinear damping in the general mechanical system has been presented in the literature (Han and Kinoshits 2012). The method has been also applied to identification of the nonlinear damping moment of a ship at zeroforward speed (Han and Kinoshits 2013). In the presence of forward speed, however, the characteristic of roll damping moment of a ship is significantly changed due to the lift effect. In this paper, the stochastic inverse method is applied to identification of the nonlinear damping moment of a ship moving at nonzeroforward speed. The workability and validity of the method are verified with laboratory tests under controlled conditions. In experimental trials, two different types of ship rolling motion are considered: timedependent transient motion and frequencydependent periodic motion. It is shown that this method enables the inherent nonlinearity in damping moment to be estimated, including its reliability analysis.
Ship motions are defined by the six degrees of freedom that a ship can experience at sea. Among them, rolling motion, which is defined by a rotational motion around the longitudinal axis of a ship, has been attracting considerable research attention over the years because large roll motions might be a serious threat to the safety of a ship such as ship capsizing or structure failure. Therefore, an appropriate model to describe the rolling motion is crucial for accurate predictions of ship’s roll response in a given sea state.
There are considerable studies on the modeling of the rolling motion [
A large number of studies on this subject have been concentrated on the parametric identification mentioned above. In contrast, there are few studies involving nonparametric identification where the prior knowledge for the nonlinearities of damping moments is not necessary. For example, Haddara and Hinchey [
In recent years, inverse methods have been presented [
In the literature [
In this paper, attention is focused on the real practical application of the stochastic inverse method [
The outline of this paper is as follows. The stochastic inverse model for nonlinear damping of a ship is derived in Section
It is assumed that the rolling motion
The schematic of the experiment.
The roll damping moment is expressed as a positive nonlinear function of the roll angle and angular velocity. The roll damping moment
From the mathematical manipulation based on the concept of variation of parameters, the following relationship is obtained [
The purpose of the study is to inversely identify the nonlinear damping moment given a measured response data
The identification of the nonlinear damping can be achieved by inverting the matrix system (
It is worth noting that the system (
To ensure a stable solution procedure, the unknown damping moment
Using adequate probabilistic models, the probabilistic expression (
To extract information on the damping moment from the constructed stochastic inverse model, it is necessary to employ the simulation technique such as Markov chain Monte Carlo, whose aim is to draw an identical independent distributed set of samples from a target density. The hierarchical model (
Initialize
For
Sample
Sample
Sample
Sample
Sample
if
else
Sample
Sample
if
else
The sampled set of realizations
In summary, the damping moment of a ship moving with nonzeroforward speed can be identified through the following steps. (1) Derive the inverse model (
The workability and accuracy of the present method were verified by damping moment identification of the test model. The related experiments were conducted at the Ocean Engineering (OE) Basin of the University of Tokyo. The basin, which is often called as the towing tank, is a physical water tank to perform hydrodynamic tests with ship models. Figure
Particulars of the test model.
Length 

Breadth 

Draft 

Displacement volume 

The distance between metacentric height and center of gravity GM 

Vertical position of the center of buoyancy KB 

Vertical position of the transverse metacenter above the keel line KM 

Natural frequency 

Overview of (a) the test model and (b) experimental setups.
Body plan of the test model.
Figure
Layout of ocean engineering basin at the University of Tokyo.
The model was first tested without any appendages such as bilgekeels to consider the hull damping characteristics. After that, for the purpose of assessing the workability, two bilgekeels (BK) with about 1 m long are attached to both sides of the test model at the turn of the bilge as in Figure
Attachment of bilgekeels to the both sides of the test model.
For the experimental application, two different types of motions, timedependent transient motion and frequencydependent periodic motion, are considered. Firstly, the transient motion caused by an initial roll angle is considered. An external moment is first applied to the test model by static means to give an initial roll angle. Then the moment is eliminated and the decaying roll motion is measured. Secondly, the forced motion caused by a periodic excitation is considered. Monofrequency sinusoidal roll motion is first imposed by the vertical force generated by the force oscillating device in Figure
Forced oscillating devices: (a) side view. (b) front view.
It should be noted that a linear approximation for the restoring moment is valid only for sufficiently small roll angle. As pointed out earlier, in this study, we restrict our attention to the small amplitude motion so that the restoring moment can be expressed by the linear form.
As a first application, a transient motion induced by an initial roll angle is considered. This motion is referred to as the freedecay rolling motion. For the trial, an initial roll angle is first given as
Recorded roll responses are presented in Figure
Roll decay curve of the test model for different forward speeds.
Without BK
With BK
It is worthwhile to note that the physical coefficients are not necessary for the case of freedecay rolling. The only thing that is required for applying the present method is the natural angular frequency. The natural frequency for the test model is shown in Table
To illustrate the method, through application to experimental data in Figure
The converted quantity
Without BK
With BK
Before illustrating results from MCMC simulation, the leastsquares estimation for the inverse solution of (
Inverse solutions through leastsquares estimation.
Without BK
With BK
We consider now the stochastic inverse model
Solutions from stochastic inverse modeling with MCMC simulation.
Without BK
With BK
It is worth to note that MCMC simulation takes a while to properly sample the target distribution. In this study, we used the evolution of components to check if the chain works properly. Figure
Typical examples of the trace plots for two components.
Once the probability density function
Identified nonlinear roll damping moments.
Without BK
With BK
It is also important to check the accuracy of the identified model. For this purpose, the roll motion is resimulated by using the identified damping moment. Figure
Resimulated roll response with the identified damping moment.
Without BK
With BK
Finally, the preceding procedures are applied to all other experimental data for the case of freedecay rolling motion. The results are summarized in Table
Identified nonlinear damping for the case of freedecay rolling.
Fr  Trials with BK  Trials without BK  




 
0  0.3796  0.1436  1.2408  0.6682 
0.1  0.5070  0.1016  1.2475  0.4852 
0.2  0.4203  0.1219  1.3262  0.6725 
0.3  0.7208  0.2027  2.5105  0.4689 
0.4  1.1007  0.1829  3.1812  0.6552 
It can be naturally concluded, based on the identified results in Table
Determination of the coefficient
As a second application, a periodic forced motion induced by periodic excitation is considered. The monofrequency periodic motion is imposed while ship is moving with a forward speed
For the forced oscillation test, the frequencydependent coefficients of the roll moment of inertia are generally obtained by Fourier analysis:
Nondimensional moment of inertia of the test model:
Without BK
With BK
Now we are ready to apply the present method to the measured data. For illustrating purposes, a particular case of experimental data was chosen as identification examples for the forced rolling motion. Figure
Measured roll response and exciting moment.
Stepbystep result for the identification of roll damping moment.
MCMC result
Resimulated roll response
It is worth noting here that, for the case of freedecay rolling motion, the effects on roll amplitude in a rolldecay are mainly due to the damping [
Instead, we performed here a quantitative analysis of the results obtained by applying the preceding procedure to all other experimental data of the forced rolling motion. The rationale behind this is that the relating results can be considered to be periodic with the exciting frequency since the monofrequency periodic motion was imposed. It is convenient to illustrate and compare the results in terms of frequency for this case. Figures
Illustration of the peak value of the identified result for the trial without BK.
Identified solution versus forward speed
Identified solution versus exciting frequency
Illustration of the peak value of the identified result for the trial with BK.
Identified solution versus forward speed
Identified solution versus exciting frequency
In this paper, a stochastic inverse method has been investigated for the identification of roll characteristics of a ship moving at nonzeroforward speeds. The rolling motion has been treated as a singledegreeoffreedom nonlinear equation of motion, uncoupled from other motions. On this basis, the stochastic inverse model for the nonlinear damping moment was derived as a probabilistic expression in terms of the observable parameter which is a function of the measurements of the roll angle and excitation. The stochastic inverse model contains the information of the nonlinear damping contribution as the multivariate random variables. Given measured data, the nonlinear damping moments were identified through the designed Markov chain Monte Carlo algorithm.
To ensure applicability, the proposed method has been applied to the experimental data for the two different mechanical phenomena regarding ship roll motions, that is, the transient motion and forced periodic motion. In nonlinear system identification, it is difficult to define the quality of the identified results because it depends on its purposes. The aim of the present study is to find the nonlinear system model which can reproduce the measured system response. In this sense, it can be concluded that the proposed method can accurately identify the nonlinearity in damping of a ship moving at nonzeroforward speeds.
The preset stochastic inverse method has the following limitations. Firstly, the method is derived based on the assumption of small amplitude of the rolling motion so that the restoring moment can be approximated by the linear form. In reality, the restoring moment is also nonlinear. The nonlinear contribution cannot be neglected. The extension to a system with the nonlinear restoring is not difficult mathematically. However, the new formulation needs additional information on the restoring nonlinearity for the identification purpose. Secondly, the experimental setups do not strictly reflect the actual motion of the ship. Rolling motion is generally coupled with other motions, such as sway and heave. In reality, the coupling effects should also be taken into account. However, for simplicity of the experimental application, in this paper, the test model was restrained in all degrees of motion except the roll motion while the model was moving at a constant speed.
The authors would like to thank the editor and anonymous reviewers for their valuable comments, suggestions, and constructive criticism, which were very helpful in improving the paper substantially.