Hybrid frequency–time domain models for dynamic response analysis of marine structures

doi:10.1016/j.oceaneng.2007.11.002

Ocean Engineering

Volume 35, Issue 7, May 2008, Pages 685-705

https://doi.org/10.1016/j.oceaneng.2007.11.002 Get rights and content

Abstract

Time-domain models of marine structures based on frequency domain data are usually built upon the Cummins equation. This type of model is a vector integro-differential equation which involves convolution terms. These convolution terms are not convenient for analysis and design of motion control systems. In addition, these models are not efficient with respect to simulation time, and ease of implementation in standard simulation packages. For these reasons, different methods have been proposed in the literature as approximate alternative representations of the convolutions. Because the convolution is a linear operation, different approaches can be followed to obtain an approximately equivalent linear system in the form of either transfer function or state-space models. This process involves the use of system identification, and several options are available depending on how the identification problem is posed. This raises the question whether one method is better than the others. This paper therefore has three objectives. The first objective is to revisit some of the methods for replacing the convolutions, which have been reported in different areas of analysis of marine systems: hydrodynamics, wave energy conversion, and motion control systems. The second objective is to compare the different methods in terms of complexity and performance. For this purpose, a model for the response in the vertical plane of a modern containership is considered. The third objective is to describe the implementation of the resulting model in the standard simulation environment Matlab/Simulink.

Introduction

Dynamic response of marine structures to wave excitation is commonly analyzed in the frequency domain by a first-order potential theory approach and by assuming the wave process as Gaussian, see for instance Newman (1977) and Faltinsen, 1990, Faltinsen, 2005. The response statistics is then obtained using the well-established theory for Gaussian processes. This approach is based on linear theory, which implies that the wave steepness is small and also that the response due to wave excitation is proportional to the wave amplitude (Faltinsen, 2005). The frequency response functions of the structure can be obtained by using the standard hydrodynamic codes (e.g. WAMIT, 2006). This method is particularly efficient since for each frequency considered, the motion response is obtained by solving a set of simultaneous linear equations with constant coefficients.

When nonlinear effects such as viscous forces, water entry and exit are considered, however, the linearity assumption is no longer valid. One approach to overcome the difficulties of full nonlinear time-domain analysis is to apply a higher order frequency-domain approach, e.g. by using simplified bilinear and trilinear frequency response functions based on Volterra functional representations (Bendat, 1998). However, frequency-domain approaches are limited to steady-state processes. Besides this fact, higher order frequency-domain methods can be cumbersome to implement and computationally inefficient.

A different approach consists of using a linear time-domain model based on the Cummins equation which will be referred to as a hybrid frequency–time domain model throughout this paper. The resulting linear model is a vector integro-differential equation which involves convolution terms. Nonlinear effects can be introduced to this model at a later stage. In this regards, Wu and Moan (1996) introduced a hybrid frequency time domain approach by first solving the linear problem in the frequency domain and then transforming the input–output into time domain and accounting for nonlinear effects as “additional” loads.

The convolution terms in a time-domain model are not convenient for analysis and design of motion control systems (Fossen, 2002, Perez, 2002). Moreover, time-domain simulations of linear transient or nonlinear problems with convolution terms are computationally demanding and their implementation in standard simulation packages is inconvenient (Kashiwagi, 2004). For these reasons, different methods have been proposed as approximate alternative representations of the convolutions. Because the convolution is a linear operation, different approaches can be followed to obtain an approximately equivalent linear system in the form of either transfer function and state-space models. This process involves the use of system identification, and several options are available depending on the way in which the identification problem is posed. This raises the question whether one method is better than the others.

Hence, the main purpose of this paper is to investigate efficient alternative formulations of the dynamic equations of motion that avoid convolution terms. 2 The Cummins equation and the frequency response model, 3 Relation between time and frequency domain models, 4 Alternative representations of the radiation force of this paper deal with description of the Cummins equation and the convolution integrals. Different representations of the linear systems, with focus on the state-space models, is the subject of Section 5. Convolution replacement alternatives are investigated in Section 6. Sections 7 and 8 deal with different identification approaches that are commonly used to obtain the state-space models. Application and discussion regarding the use of these methods for a simple memory function is carried out in Section 9. Dynamic response simulations for a container vessel is performed in Section 10 by using state space models. Finally, comments and conclusions are given in Section 11.

Section snippets

The Cummins equation and the frequency response model

Cummins (1962) considered the behavior of the fluid and the structure in time domain. He assumed linear behavior and considered impulses in the components of motion. This resulted in a boundary value problem in which the potential was separated into two parts: one valid during the duration of the impulses and the other valid afterwards. By expressing the pressure as a function of these potentials and integrating it over the wetted surface of the vessel, he obtained a vector integro-differential

Ogilvie relations

The relationship between the parameters of Eq. (1) and those of Eq. (2) were established by Ogilvie (1964): $A (ω) = A - \frac{1}{ω} \int_{0}^{\infty} K (t) \sin (ω τ) d τ, B (ω) = \int_{0}^{\infty} K (t) \cos (ω τ) d τ,$ from which it follows that $K (t) = \frac{2}{π} \int_{0}^{\infty} B (ω) \cos (ω t) d ω$ and $A = \lim_{ω \to \infty} A (ω) = A (\infty) .$ Using these relations, Eq. (1) can be written as $[M + A (\infty)] \ddot{x} (t) + \int_{0}^{t} K (t - τ) \dot{x} (τ) d τ + Cx (t) = f^{exc} (t) .$ A related form of the above by Wehausen (1967) is $[M + A (\infty)] \ddot{x} (t) + \int_{0}^{t} L (t - τ) \ddot{x} (τ) d τ + Cx (t) = f^{exc} (t),$ where $K (t) = \frac{d}{d t} L (t) .$ The velocity formulation of Eq. (10) is more convenient for numerical

Alternative representations of the radiation force

Within the scope of fluid memory effects, the identification of state-space models can be posed for not only the retardation functions, but also for the other representatives of the radiation force. Therefore, we devote this section to clarify these alternatives before explaining different convolution replacement approaches in Section 6.

The total hydrodynamic radiation force vector in the frequency domain can be expressed as $F^{R} (j ω) = - [B (ω) \dot{X} (j ω) + A (ω) \ddot{X} (j ω)] .$ Replacing the acceleration vector by

State-space models for linear systems

Before proceeding with different representations of the convolution terms and of the Cummins equation (Eq. (10)), linear system representations are briefly revisited. This will then be used throughout the rest of the paper.

Convolution replacement

Time-domain simulations of the dynamic response of marine structures by direct evaluation of the convolution integrals may be computationally demanding depending on the time step, simulation length and degrees of freedom of the model (Holappa and Falzarano, 1999, Kashiwagi, 2004). The state-space representation provides an attractive alternative for simulation due to the simple form of its solution i.e. Eq. (32). For zero input, $u (t) = 0$ , this indicates that dynamic variables, in this

Identification methods for convolution replacement

Fig. 1 shows the components defining an identification method in the context of this paper. Every identification method consists of selecting a series of data, choosing a model structure (model type and order) and a fitting criterion. The unknown parameters in the prospective model are obtained by a parameter estimation method and using the specified fitting criteria (Ljung, 1999). The objective of system identification is to obtain the lowest order model possible that is able to reproduce the

Parameter estimation methods

In this section, we will describe three methods by which the unknown model parameters resulting from the identification methods are obtained:

(i)
Impulse response curve fitting.
(ii)
Realization theory.
(iii)
Regression in the frequency domain.

The performance and use of each method will be discussed in Section 9.

Application example

Herein, we consider a simple retardation function as an example to compare the performance of the different identification methods. The example is based on pure analytical relations and satisfies all the properties of the convolution terms given in Section 3. It is standard practice in system identification to compare different estimators with scenarios for which the values of the true quantities to be estimated are known and this is the motivation for this example. We should mention, however,

Specification of the case

Numerical studies are performed in this section for a container vessel at zero forward speed. The specifications of the ship are given in Table 1. Fig. 6 shows the body plan of the hull.

Two different scenarios are analyzed:

(i)
The objective of the first one is to examine the steady-state part of the time-domain response of the container vessel by using two different state-space models. The focus is on the models obtained by identifying the retardation functions based on the realization theory and

Concluding remarks

In this paper, different alternatives to replace the convolution terms in the Cummins equation have been reviewed. By this way, the integro-differential equations of motion have been written in the form of differential formulations. The emphasis has been on the methods that replace the convolutions by the state-space models. It was discussed how to obtain the replacing models via system identification techniques based on the hydrodynamic data from the standard software. Identification methods

Acknowledgements

We are grateful to Professor Johannes Falnes for his invaluable comments. We also thank Mr. Ingo Drummen at Centre for Ships and Ocean Structures for providing the data of the container ship. The authors appreciate the financial support of the Norwegian Research Council which has been granted through Centre for Ships and Ocean Structures.

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Hybrid frequency–time domain models for dynamic response analysis of marine structures

Abstract

Introduction

Section snippets

The Cummins equation and the frequency response model

Ogilvie relations

Alternative representations of the radiation force

State-space models for linear systems

Convolution replacement

Identification methods for convolution replacement

Parameter estimation methods

Application example

Specification of the case

Concluding remarks

Acknowledgements

Ocean Engineering

Applied Ocean Research

Ocean Engineering

Ocean Engineering

Marine Structures

Applied Ocean Research

An error bound for a discrete reduced order model of a linear multivariable system

IEEE Transactions on Automatic Control

Nonlinear Systems Techniques and Applications

Linear System Theory and Design

The impulse response function and ship motions

Schiffstechnik

Absorption of outgoing waves in a numerical wave tank using a self-adaptive boundary condition

International Journal of Offshore and Polar Engineering

Modeling and Simulation for Automatic Control

Sea Loads on Ships and Offshore Structures

Hydrodynamic of High-speed Marine Vehicles

Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles

A nonlinear unified state-space model for ship maneuvering and control in a seaway

International Journal of Bifurcation and Chaos

Effective reconstruction of linear state-variable models from input/output functions

Regelungstechnik

Application of extended state space to nonlinear ship roling

Ocean Engineering

Time domain models from frequency domain descriptions: application to marine structures

International Journal of Offshore and Polar Engineering

Linear Systems

Transient response of a VLFS during landing and take-off of an airplane

Journal of Marine Science and Technology

Complex curve fitting

IRE Transactions on Automatic Control