A new dynamic model of towing cables

doi:10.1016/j.oceaneng.2020.107653

Ocean Engineering

Volume 220, 15 January 2021, 107653

https://doi.org/10.1016/j.oceaneng.2020.107653 Get rights and content

Highlights

•
Dynamic cable modeling.
•
Proposition of a new discreet dynamic modeling formalism.
•
Application to problems related to towing vessels.
•
Proposition of a method to calculate the dynamic tension in the cable.

Abstract

This article proposes a new formalism for the dynamic modeling of a cable towing system, in which both the tugboat and the towed vessel are subject to forces from waves on the sea surface. The continuous flexibility of the cable is approximated by a discrete equivalent, formed by rigid links connected by fictitious elastic joints that allow elevation movements, since the dynamics are restricted to the vertical plane. The Euler-Lagrange formalism was used to determine the dynamic models considering two, three and four links. The growth patterns of the differential equations of the models were found, thus allowing the development of generic algorithms, capable of generating dynamic models considering any number of links. Vertical forces obtained from proportional and derivative control were applied to the tugboat and the towed vessel, thus simulating the wave motion of the sea surface. Simultaneously, a motor thrust is applied to the tugboat. An algorithmic procedure is also proposed to determine the dynamic tension in the cable. Simulation results showed dynamic responses as physically expected. Computer animations made from simulation data show a great sense of physical reality.

Introduction

The significant growth in applications involving flexible cable-type structures has been observed mainly in the offshore oil exploration industry (Fig. 1) and this is one of the main reasons for the increase in scientific papers on cable modeling. Finite elements are widely used in cable modeling, including dynamic and static analysis. Fang et al. (2012) proposed a new finite element method to modeling cable deformation in towed system, using cubic spline interpolation functions. Bamdad (2013) proposed an analytical solution for the dynamics of manipulators suspended by cables. Bi et al. (2013) applies the Finite Element Method to study the tensions present in submarine umbilical cables. Eidsvik and Schjølberg (2018) presented a novel three-dimensional cable model for Remotely Operated Vehicles using Euler-Bernoulli beam theory, with the resulting equations discretized spatially by finite elements.

Some authors use other methods to study the dynamics of cables. Masciola et al. (2012) study a solution method for the cable static configuration, from a discrete approximation. Liu et al. (2013) study dynamic characteristics of cables in transmission lines, based on the Lagrangian formulation. Guo et al. (2016) studied the influence of cable support on the dynamics of the cable itself, considering that the support has mass, stiffness and friction known. Zhu and Yoo (2017) propose a new element reference frame for the lumped-mass model by which the dynamics of marine cables can be analyzed efficiently. Vu et al. (2017) propose to use the catenary equation method to identify the efforts applied by the cable in an underwater vehicle. González et al. (2017) proposed two new simulation methods based on simplified cable models for real-time simulation of fast cable pay-out and reel-in manoeuvres with towed fishing gears.

The main subject of this article is the towing of vessels and thus, some articles involving cable modeling in the context of this application were consulted. Buckham et al. (2003) describe the development of a numerical model that accurately captures the dynamics of the mine hunting systems and use lumped mass approximation for the tow cable. Sun et al. (2011) has introduced a new nodal position finite element method to cable towed body modeling. Central implicit finite-difference method is used by Srivastava (2014) to solve a three-dimensional model of underwater towed system. Yuan et al. (2014) use finite difference method for solving the nonlinear dynamic equations of the towed system, including gains in terms of computational efficiency. Park and Kim (2015) introduced an absolute nodal coordinate formulation to modeling a towing cable and showed simulation results. Finite difference method is used by Wang and Sun (2015) to integrate the differential equations of a cable towing system dynamics, as well as showed a study on the influence of some parameters on the dynamic behavior of the model. Lumped mass is also used by Gao and Wang (2018) to simulate a towing cable system. Du et al. (2019) used the lumped mass approach to modeling a sonar tow cable and showed simulations that included the effects of hydrodynamic drag. Sun et al. (2019) proposed a tug–cable–barge coupling motion model for bridle towed system using finite differences to perform numerical simulations.

In this article is presented a new method for the dynamic modeling of vessel towing systems, consisting of a cable and two masses at its ends (tugboat and towed vessel). The cable is modeled using a discrete formalism similar to that used by Gomes et al. (2016). However, in the present case the cable has its ends connected to platforms that can move vertically and horizontally. Disturbances arising from the wave motion on the sea surface are also considered. Another important contribution is the proposal of an algorithm for determining the dynamic tension in the cable. As will be exposed, the results are impressive due to their realism, especially when viewed in computer animations.

Section snippets

Cable dynamic modeling

In the present work, a discrete formalism is used for dynamic cable modeling. Fig. 2 illustrates the main idea of the method. It is assumed that the cable is approximated by $n_{e}$ rigid links $(l_{i}, i = 1, \dots, n_{e})$ connected by fictitious elastic joints and each of these joints has an associated elastic constant and friction coefficient $(k_{i} a n d c_{i}, i = 1, \dots, n_{e})$ . Mass of each link $(m_{i})$ are considered to be concentrated in their respective centers of mass. The cable is connected at its ends to two vessels, as

Generic algorithms

The analytical development of the dynamic cable model equations, for the cases of two, three and four links, allowed the identification of the growth patterns of the vectors and matrices of the models. Once identified, these growth patterns allowed the generation of generic algorithms, capable of automatically generating dynamic models, for any number of links considered in the discreet approximation of the continuous flexibility of the cable.

Inertia matrix $I (\vec{r})$ is symmetric $n$ order $(n = n_{e} + 2),$

External forces

The purpose of this section is to show how the external forces acting on the tugboat will act in the dynamic system, generating forces and torques in the other masses of the system. It is important to remember that the external forces acting on the towed vessel entered the Euler-Lagrange equations and therefore their dynamic effects are naturally incorporated into the dynamical system. As shown in Fig. 2, the external forces acting on the tugboat can be written in the form: ${\vec{F}}_{c} = (F_{t} + F_{h t}) \vec{i} + F_{y t} \vec{j} = F$

Simulations

In the two simulations presented below, the system is initially at rest, in the spatial configuration shown in Fig. 2, with $D = 30 m$ and $d = 2 m$ . A parabolic curve was considered for this initial spatial configuration of the cable in static equilibrium (Benedettini et al., 1995): $y (x) = \frac{4 d}{D} (x - \frac{1}{D} x^{2})$

A motor thrust is applied to the tugboat, so that its time derivative obeys a trapezoidal profile, whose objective is to bring this motor thrust from zero to 320,000 N in 80 s. After $t = 80 s$ the motor thrust

Dynamic cable tension

The cable will be dynamically subject to two possible situations: stretched and not stretched. The way to calculate the dynamic tension in the cable changes radically in these two situations. It is considered that the cable is initially at rest, in the spatial configuration shown in Fig. 2. The span is $D = 30 m$ (distance between tugboat and towed vessel) and the sag is $d = 2 m$ .

The cable was considered to be in the not stretched situation if the distance between the tugboat and the towed vessel is

Conclusions

A new method for the dynamic modeling of cable towing systems has been proposed in this article. In addition to the horizontal movement of the entire system, the tugboat and the towed vessel were also considered to move vertically due to the wave motion of the sea surface. Qualitative analyzes of the simulation results indicate dynamic behavior that is quite consistent with what is physically expected. It is important to explain that many simulations were carried out assuming different types of

CRediT authorship contribution statement

Samuel da Silva Gomes: Programming generic algorithms, performing simulations. Sebastião Cícero Pinheiro Gomes: Conceptualization, Supervision, Methodology, Writing - original draft.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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View full text

A new dynamic model of towing cables

Highlights

Abstract

Introduction

Section snippets

Cable dynamic modeling

Generic algorithms

External forces

Simulations

Dynamic cable tension

Conclusions

CRediT authorship contribution statement

Declaration of competing interest

J. Sound Vib.

Ocean Engineering

Ocean Engineering

Ocean Engineering

Ocean Engineering

J. Sound Vib.

Ocean Engineering

Appl. Ocean Res.

Analytical dynamic solution of a flexible cable-suspended manipulator

Front. Mech. Eng.

Dynamic simulation and tension compensation research on subsea umbilical cable laying system

J. Mar. Sci. Appl.