K=
Ke 0 0
0 Ka 0
0 0 KT
2 4
3 5;C=
Ce 0 0
0 Ca 0
0 0 CT
2 4
3
5 ð11Þ
Figure8 shows 36 s of simulation with frames every 2 s, considering an ocean current acting on the cable with speed of 3.5 m/s. Drag forces was considered proportional to the square of the relative velocities betweenfluid and structure. The cable is initially at rest in its vertical position and with free terminal load. The ocean current starts from zero initial time and then imposes a significant dynamic dis- turbance on the cable. If the cable terminal load is a remotely operated vehicle (ROV), the ocean current can to induce dynamic disturbances to the cable that would be transmitted to the vehicle, thus hindering the performance of any control strategy.
Fig. 5 Free fall simulation, underwater, from zero to 22 s, frames every 2 s
Fig. 6 Free fall simulation, out of water, from zero to 22 s, frames every 2 s
Fig. 7 Free fall simulations, out of water and underwater, from zero to 40 s, frames every 1 s
136 S. C. P. Gomes et al.
The following simulation was performed considering the terminal loadfixed to the seabed. The cable was placed in an initial spatial configuration (first frame of Fig.10). The position of the terminal load at this initial configuration has been taken as a reference to a Proportional and Derivative (PD) control, applied to maintain the terminal loadfixed in this reference (constraint forces). These forces were applied in the CM of the terminal load and in the three dimensions, generating torques in the last joint, which enter as external torques in Eq. (1). Figure9shows the elevation angles and Fig.10shows the cable spatial configuration, with frames every 2 s. It is observed that the control was effective in maintaining the terminal load in its original initial position, which could be on the seabed. This simulation is in a single plan, so that all azimuth angles are zero. We verified that the cable stabilization trend in search of itsfinal spatial configuration (catenary static equi- librium), which is fully achieved after 180 s.
spatial cable configurations (m), (frames every 2s)
-1200 -1000 -800 -600 -400 -200 0
-50 50 0
-60 -40 -20 0 20 40 60
Fig. 8 Cable under the action of underwater current, from 0 to 36 s, frames every 2 s
time (s)
0 10 20 30 40 50 60 70 80
0 0.5 1
elevation angles
rd
21 22 23 link 24
Fig. 9 Elevation angles, terminal loadfixed to the seabed, considering 24 links
Cable Dynamic Modeling and Applications… 137
5 Conclusions
Nature follows growth patterns in their phenomena, but we cannot always identify them. The discrete approach proposed in the modeling formalism approximates the real continuousflexibility when increasing the number of links. Each new added link increases in three degrees of freedom the cable dynamics and its equations grow considerably in size and complexity. It was possible to identify patterns of this growth, allowing to the proposition of algorithms that automatically generate dynamic models for any number of links considered in discrete approximation.
The simulations were chosen in order to know a priori what should be the dynamic behavior of the cable and, in all cases the results were in agreement with the physically expected. Torsion motion was considered in cable modeling because in future applications, there is an interest in having a ROV as a cable terminal load.
In this case, forces applied to the ROV can generate torsion in the cable. It has been specially developed software that performs three-dimensional animations of the cable’s spatial configuration, for better visualization and analysis of simulation results. Three-dimensional animations allowed us to identify a great sense of physical reality. Increasing the number of links implies a better discrete approxi- mation of the continuous flexibility. It was observed that over forty links the discrete model closely matches the continuousflexibility.
In future works it is planned to build an experimental support sensed by digital cameras to validate simulations and physical parameters identification strategies.
Fig. 10 Frames showing spatial cable configuration considering the terminal load fixed to the seabed (from zero to 30 s, frames every 6 s, 24 links)
138 S. C. P. Gomes et al.
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