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Dynamic analyses of floating fish cage collars in waves
Aquacultural Engineering 47 (2012) 7–15
Contents lists available at SciVerse ScienceDirect
Aquacultural Engineering journal homepage: www.elsevier.com/locate/aqua-online
Dynamic analyses of floating fish cage collars in waves Shixiao Fu a,b,∗ , Torgeir Moan b a b
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, 20030 Shanghai, China Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 2 February 2011 Accepted 5 December 2011 Keywords: Interconnected floating collars Hinge connector Linear hydroelasticity Frequency domain
a b s t r a c t An extended 3D hydroelasticity theory is used in the frequency domain to predict the dynamic response of 5 by 2 floating fish farm collars in regular waves. Hinge modes in two directions and flexible torsional modes are all considered in the analyses. The dry and wet resonant natural frequencies for each rigid relative motion and the flexible modes are also calculated. The results show that the modes, flexible bending and torsional modes, as well as their interactions, greatly contribute to the dynamic response of the floating collar system. From a design point of view, the effects of the connector rotational stiffness on the dynamic response of the structures are also analysed. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Because of the increasing demand of ocean fish resources and the environmental and nearshore space limitation requirements, traditional nearshore fish farming is moving into the open ocean. Efforts have been made to develop traditional fish cages for use in open ocean environments. However, due to the more severe environmental loads from the high energy sea states, the design for future offshore fish farms requires more accurate analyses and calculations (Lader et al., 2003). Currently, fish farm structures can be split into two subsystems: floating collar structures and net cages. The main loads on the system result from ocean currents and waves. Analyses of fish farm structures exposed to currents and waves have often been performed using Morison’s equation and by considering that the rigid floating collars attached by the nets are fixed. When the focus is mainly stressed on the net cage deformations in the current, this assumption may be acceptable. However, regarding the influence of waves on the net cage system, this assumption is unacceptable. The floater motion in the waves creates slack in the net and therefore causes large forces at the joint between the bottom weight and the net, which can result in net fault. Research has shown that the floater motion is the main contributor to the forces and tensions in the net, while the forces on the net from the current are much smaller (Lader et al., 2001). Based on the linear potential wave theory and the linearised drag forces on the net, a 2D analytical solution for a tension
∗ Corresponding author at: State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, 20030 Shanghai, China. Tel.: +86 1350 1947 087. E-mail address: [email protected] (S. Fu). 0144-8609/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.aquaeng.2011.12.001
leg-type fish cage in waves was performed (Lee and Wang, 2005), where the interaction between the top platform structure, incident wave, and motion of both the tether and net were considered. However, these authors mainly focused their research on the surge motion of the system. Fredriksson (Fredriksson, 2001; Fredriksson et al., 2003, 2005) has conducted measurements of waves spectra, current profiles, forces in moorings and the motion responses of a realistic fish cage during an extreme storm. Numerical simulations were also performed using these measured data sets as inputs to a dynamic finite element model (FEM). Comparisons between these measured and simulated results showed good agreement. Moe et al. (2010) used ABAQUS to analysis the deformation of net cage under different water current velocities, the numerical result compared with the tests in the tank was good. Kristiansen (2010) studied the wave induced effects on floaters of aquaculture plants using CFD. From experimental point of view, Lader (Lader and Fredheim, 2006; Lader et al., 2007; Lader and Olsen, 2007; Lader and Dempster, 2008) studied the waves and current induced net deformations. In the last few decades, various theories (Bishop and Price, 1979; Wu, 1984; Newman, 2005) have been developed to predict the hydroelastic responses of continuous or multimodule-connected flexible structures in waves. Recently, an extension was added to the 3D hydroelasticity theory to account for the rigid hinge modes of these interconnected flexible structures (Fu et al., 2007). In this paper, to predict the dynamic response of 5 by 2 floating fish farm collars in regular waves, the extended 3D hydroelasticity theory is adopted. Hinge modes in two directions and flexible torsional modes are considered. The effects of the connector rotational stiffness, the rigid relative motion modes, the flexible bending and torsional modes, as well as their interactions to the dynamic response of the floating collar system, are studied.
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S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
2. Basic theory The fluid around the interconnected flexible floating collars at zero forward speed is assumed to be ideal. Hence, the fluid behaviour can be completely described by the velocity potential governed by the Laplace equation. The unsteady velocity potential around the structures may be decomposed into three parts and expressed as:
= Re
ϕI + ϕD +
m
ϕr pr
e
iωt
,
(1)
r=1
where ω is the wave circular frequency; ϕI (x, y, z, ω) and ϕD (x, y, z, ω) are components of the incident wave velocity potential and the diffraction wave potential, respectively; ϕr (x, y, z, ω) (r = 1, . . . , m) are the components of the radiation wave potential arising from the rth dry mode of the flexible structure with unit amplitude; m is the number of modes; and pr is the complex amplitude of the generalised coordinate. ϕD (x, y, z, ω) and ϕr (x, y, z, ω) can be solved by Green’s function method under the following governing equation and corresponding boundary conditions:
⎧ [˝] : ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [SF ] : ⎪ ⎪ ⎪ ⎨ [S] :
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [SB ] : ⎪ ⎪ ⎩
[S∞ ] :
∇ 2 ϕv = 0 ∂ϕv − ω2 ϕv = 0 ∂z ∂ϕD ∂ϕr ∂ϕI ៝ 0r n ៝, =− , = −iωu ∂n ∂n ∂n ∂ϕv =0 ∂z ϕv → 0 g
(2)
where [˝], [SF ], [S], [SB ], and [S∞ ] are the fluid field, free surface, hull surface, sea bed, and far field cylinder, respectively. ϕv can be ៝ denotes the normal vectors of the body’s wetted surface; ϕD or ϕr ; n ៝ 0r denotes the rth modal shape g is the gravity acceleration; and u vector of the point in the wetted elements. We assume that the structures are composed of flexible interconnected elements with axial, shear and rotational stiffness in their connection. However, for structures with a very soft connector, it will be impossible to directly solve the eigenvalue problem through traditional methods. Therefore, to obtain the eigenvalue and corresponding eigenvectors of the interconnected structures and to ensure the orthogonality conditions between the modes, we can transform the eigenvalue problem into the following one by introducing an artificial stiffness proportional to the mass into the system. The eigenvalue and corresponding eigenvectors of the floating collar structures can then be found using the following transformed eigenvalue problem: (−(ω02 + )[M] + ([K] + [M])){X} = {0},
(3)
where [K] and [M] are the stiffness and mass matrix of the structure, respectively and ω0 is the eigenvalues of the original interconnected structure, which should satisfy the following equation: (−ω02 [M] + [K]){D} = {0}.
(4)
Here, {D} is the eigenvector of the equation. From Eq. (3), we 2 + , can obtain the corresponding positive eigenvalues, r = ω0r and the corresponding eigenvectors, {X}, through traditional methods. The orthogonality conditions with respect to [K] + [M] and [M] are automatically satisfied. Thus, these values can also satisfy the orthogonality conditions with respect to [K] and [M] for the original interconnected structure. Therefore, the eigenvalues and corresponding eigenvectors of the original structure should be 2 = − and {D } = {X }. ω0r r r r Based on the FEM and mode superposition method, we usually select the first several oscillatory modes to present the structural
dynamic responses. Therefore, we assume that the nodal displacement of the structure can written as a superposition of the first m modes of the structure as {U} =
m
{Dr }pr (t) = [D]{p},
(5)
r=1
where pr (t) refers to the rth generalised coordinate. For r = 1 − 6, {Dr } represents the vector of the first six rigid modes as defined in the sea-keeping theory, and pr (t) is the magnitude of the rigid displacement about the centre of mass, (xG , yG , zG ). When r > 6, {Dr } represents the hinge and flexible modes of the structure calculated from Eq. (4). Then, from the motion equations of the structure based on FEM, we have the equations of motion for solving the generalised coordinates, {p}, in the frequency domain as [−ω2 ([a] + [A]) + iω([b] + [B]) + ([c] + [C])]{p} = {F},
(6)
where [a] = [D]T [M][D], [b] = [D]T [C][D] and [c] = [D]T [K][D] are the generalised mass matrix, damping matrix and stiffness matrix of the structure, respectively; and [A], [B] and [C] are the generalised added mass matrix, added damping matrix and restoring matrix, respectively. {F} is the generalised wave exciting forces and may be further expressed as
⎧ 1 0 ⎪ ៝ ៝ n A = Re i · u ωϕ dS ⎪ rk k r ⎪ ⎪ ω2 ⎪ S¯ ⎪ ⎪
⎪ ⎪ ⎪ ⎨ B = i Im i 0 ៝ ៝ n · u ωϕ dS rk k r ω . S¯ ⎪ ⎪ ⎪ ⎪ ៝ ·u ៝ 0r gw0 dS Crk = n ⎪ k ⎪ S¯ ⎪ ⎪ ⎪ ⎪ ⎩ 0 ∂ Fr =
៝ ៝
n · ur S¯
∂t
(7)
[I + D ]dS
When the responses of the principal coordinates have been obtained from Eq. (6), the displacements can be obtained using the following equation:
u(t) =
m
{Dr }pr (t).
(8)
r=1
To obtain the bending and torsional moments or stress responses of the structures, vector {Dr } can be replaced by the corresponding moments or stress modes, which can be easily obtained from the corresponding displacement modes, {Dr }. The wet resonant frequency for the rth mode, ωwr , can be obtained by solving the following equation with the iterative procedure: ωwr =
c + C 1/2 rr rr arr + Arr (ω)
,
(9)
where crr , Crr , arr and Arr (ω) are the generalised structural restoring force, generalised hydro-restoring force, generalised structural mass and generalised added mass of the rth structural mode, respectively. As for the specified rth mode, crr , Crr and arr have determinate values, and the added mass, Arr (ω), will have different values with different vibration frequencies (ω). To find the wet resonant frequency, ωwr , which will be determined with added mass Arr (ω), the iterative method should be used, wherein the relationship |ωwr − ω| ≤ ε should be satisfied for every wet resonant mode of the structures.
S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
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Fig. 1. A sketch of the floating collars and connectors (black dots mark the hinge connectors).
3. Numerical example 3.1. The model The methods mentioned above have been validated with floating flexible hinge-connected plates in regular waves (Fu et al., 2007). In this paper, numerical examples for specified open ocean fish farming floating collars with 5 by 2 fish cages in waves were performed. These structures are based on the structural designs of traditional fish farming structures. The specifications of the 5 by 2 unit module of the floating collar system are listed in Table 1. The floating collars are composed of 10 modules, with 5 and 2 modules arranged in rows and columns, respectively, as shown in Fig. 1. These modules are connected by hinge connectors, which have zero rotational stiffness along the horizontal axis and will be denoted as krot = 0 in the following text. The modules exhibit large stiffnesses in their translational degrees of freedom. The module positions are marked by the black dots in Fig. 1. The 1 m × 1 m (top and bottom) and 1 m × 0.5 m (side and ends) shell elements are used to model the dry structure while simultaneously considering the membrane and bending forces. The ABAQUS (Hibbit et al., 2004) is used as the eigensolver. Similar panel elements are used as the hydrodynamic mesh (Fig. 2). The displacement on the six points is determined and is marked as P1–P6 in Fig. 1. The system damping is very complicated, and determining this damping in the numerical model, including the damping from fluid motions and structural deformations, is difficult. The radiation damping from the water induced by the motions and deformations of the system is precisely estimated by Eq. (7). The damping contributed from other aspects, for example, the damping from net motions in the water and from the mooring lines, is not precisely
calculated in this model but is roughly estimated together with the structural damping by 5% of the structural critical damping. The damping from the structural deformations is usually determined to be 1.0–2.0% in the numerical simulations. However, to include the damping from the nets and the mooring line motions, the damping ratio in this paper is assumed to be 5%, which means that the damping from the nets and the mooring lines are crudely estimated as 3.0–4.0%. Of course, the exact damping from the nets and mooring lines should be measured directly from experiments, which is not in the scope of this paper. The research of this paper will mainly focus on the dynamic responses of the flexible system in wave frequencies; thus, the low frequency and mean motions (both are mainly affected by the mooring lines) will not be considered in this study. Therefore, the mooring line will not be included in the simulations.
3.2. Floating collar system modes The first to sixth mode shapes correspond to surge, sway, heave, roll, pitch and yaw motions, respectively, of the entire floating collar system. Fig. 3 shows the first 8 non-global mode shapes of the floating collar system. The corresponding generalised mass and dry Table 1 General specifications of the unit module of the floating collar system. Property
Symbol
Quantity
Length (m) Beam (m) Height (m) Draft (m) Width (m)
L B T D W
32 16 1 0.5 2
Fig. 2. An illustration of the hydrodynamic element.
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S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
Fig. 3. The first 14 modes of the structure.
and wet resonant natural frequencies are listed in Table 2. Notably, for a given mode, the structure will have the same mode shape and generalised mass regardless of the stiffness of the structure. Consequently, in Table 2, only one generalised mass is listed for a different structural stiffness. Thus, except for the first 6 zero frequency motion modes of the system, 5 zero frequency modes that represent the hinge motions of the floating collars and some nonzero frequency flexible modes due to the flexibility of the structures exist. Because of the effects of the radiation and hydrostatic forces, the wet resonant natural frequency is different from that of the dry frequency, whereas for the non-zero frequency modes, the former will be lower than that of the latter. 3.3. Dynamic analyses The generalised coordinate responses and wave exiting forces of the floating collars subjected to regular waves with various periods and incident wave angles are shown in Figs. 4 and 5, respectively. To enlarge and study the contributions from the rigid relative motion modes in the beam direction and the torsion flexible modes, the 75-degree incident angle is chosen as an example in an oblique sea. Fig. 4 demonstrates that the torsional modes will only be excited in oblique seas. Fig. 4 shows that the exciting wave frequencies that correspond to the response peak values for every mode are not matched with their corresponding wet or dry resonant natural frequencies, as listed in Table 2. Two reasons for this result are
as follows: firstly, the radiation wave damping will limit or eliminate resonance; secondly, the generalised wave exciting forces will reach their maximum values at their corresponding frequencies and peak value wave angles. The second explanation can be further explained with Figs. 4 and 5 in which each peak of the generalised exciting force in Fig. 5 will correspond to a peak of the generalised coordinate response in Fig. 4. Furthermore, some sub-peaks of the generalised coordinate response in Fig. 4 do not correspond to any of the generalised wave exciting force sub-peaks in Fig. 5 but occur at their corresponding wet resonant frequency (e.g., mode numbers 5, 7, 9 and 13). This fact indicates that resonance occurs for these modes. Considering Figs. 4 and 6, the rigid relative motions between the modules (modes 7–11) will greatly contribute to the final response of the system, which indicates the significance of the interactions between the modules. In head seas, because of the high longitudinal stiffness of each module, compared with that of the rigid body motion modes (first 6 rigid modes and 5 more rigid relative motion modes), the flexible modes (modes 12–14), which mainly belong to the torsional modes and will not be excited in head seas, have negligible contributions to the motion of the structure. However, in oblique seas, the torsional flexible modes and transverse rigid hinge mode (e.g., mode numbers 11–14), will be excited and produce more contributions to the motion, as shown by comparing (a) and (f) with frequencies of approximately 0.4–1.2 rad/s in Fig. 6. From Fig. 6, at a wave exciting frequency of approximately 0.9 rad/s in head seas, the entire system will almost reach its maximum
Table 2 Generalised mass and dry and wet natural frequencies of the structure for different structural stiffnesses. Mode number
GM.
˛ 0.2
3 5 7 8 9 10 11 12 13 14
– – 1.97 2.26 2.86 2.57 4.73 1.52 2.37 1.41
1.0
10.0
DF. (rad/s)
WF. (rad/s)
DF. (rad/s)
WF. (rad/s)
DF. (rad/s)
WF. (rad/s)
0 0 0 0 0 0 0 1.59 2.47 3.20
2.07 1.61 2.67 2.13 1.88 2.70 1.63 1.03 1.46 2.11
0 0 0 0 0 0 0 3.61 5.62 7.26
2.07 1.61 2.67 2.13 1.88 2.70 1.63 3.20 3.47 4.11
0 0 0 0 0 0 0 11.23 17.48 22.59
2.07 1.61 2.67 2.13 1.88 2.70 1.63 10.51 10.41 13.42
Mn. denotes the mode number, ˛ = EI/0.159 × 1011 ; EI denotes the bending stiffness of the modulus in N m2 ; DF. denotes the dry natural frequency of the structure in rad/s; WF. denotes the wet natural frequency of the structure in rad/s; and GM. denotes the generalised mass × 106 .
S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
Fig. 4. Generalised coordinate response amplitude of the first 14 modes with incident wave (—0◦ , - - -75◦ ).
11
12
S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
Fig. 5. Generalised wave exciting amplitude of the first 14 modes with incident waves of (—0◦ , - - -75◦ ).
S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
13
Fig. 6. Vertical response amplitudes of points 1–6 in various regular waves with incident angles of (—0◦ , - - -75◦ ).
vertical response amplitude. This result can be explained by coupling the four rigid relative motion modes to the heave motion mode. Notably, some peak values at other wave frequencies, which may have significant contributions to the velocities and relative motion responses of the floating collars, exist. Therefore, when the tension forces between the net and the bottom weight of the system are considered, more attention to these peaks, which will be further analysed in our future works, should be extended. The effects of the incident wave frequency, wave incident angle, structural module stiffness, and connector rotational stiffness on the response amplitudes of each mode’s generalised coordinate and vertical motion are shown in Figs. 7 and 8, respectively. For the hinge-connected system, as shown in Figs. 7 and 8, the torsional modes will not be excited in head seas; therefore, their response amplitudes in oblique seas are shown in Fig. 7(f)–(h). As shown, increasing module stiffness decreases the response amplitude of each flexible mode; the more flexible the module is, the greater the contribution its flexibility will make. This contribution will not always increase the final response amplitude of the
structure, which depends strongly on the relative phase angle of the response. Fig. 7(a)–(e) shows that the stiffness of the modules will not affect the response amplitude of the rigid body modes. That is, the rigid modes will contribute equally to the dynamic response of the flexible floating collar structure whether it is assumed to be rigid or flexible. As shown in Fig. 7, increasing the connector stiffness will reduce the generalised coordinate response amplitude, indicating that the restoring forces from the connectors will reduce the response. This correlation is also demonstrated in Fig. 8. As shown in Fig. 8, because the flexible modes will not be excited by head seas and the rigid hinge modes exhibit the same contribution to the structure responses, the bending stiffness will not affect the vertical motion in head seas. However, regarding oblique seas, the flexible modes will be excited, and their corresponding contributions can be found from the vertical motion amplitudes. In the domain of relatively low-incident wave frequencies, increasing the module stiffness will generally lead to lower response amplitudes. (Here, the flexibility of the modules will increase the response
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S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
Fig. 7. The effects of bending and connector stiffness on the generalised coordinate response amplitudes.
S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
15
Fig. 8. The effect of bending stiffness on vertical motions.
amplitudes of the structures.) This result will differ from the conclusions of Fu et al. (2007), where the flexible modes will be excited not only in head seas, but also in oblique seas. 4. Summary and conclusions An extended 3D hydroelasticity theory is used to predict the dynamic responses of 5 by 2 floating fish farm collars in frequency domain regular waves. The hinge modes in two directions and the flexible torsional modes are considered. Dynamic response analyses of specified floating collar systems are performed. In head seas, the rigid relative motion modes will dominate the dynamic responses of the floating collar systems when compared with torsional flexible modes. However, in oblique seas, the flexibility of the structure will provide a greater contribution to the response, indicating that the flexibility should be considered. Meanwhile, from the floating collar point of view, the nets and mooring lines attached to the fish farm structures will mainly affect the damping properties in this system. However, determining the degree of influence of these structures will be challenging because the damping from the nets strongly depends on its motion amplitude and frequency and can only be determined with special forced vibration experiments.
References Bishop, R.E.D., Price, W.G., 1979. Hydroelasticity of Ships. Cambridge University Press, UK. Fredriksson, D.W., 2001. Open ocean fish cage and mooring system dynamics. Ph.D. Thesis. University of New Hampshire, USA.
Fredriksson, D.W., Swift, M.R., Irish, J.D., Tsukrov, I., Celkkol, B., 2003. Fish cage and numerical models with field measurements. Aquacultural Engineering 27 (2), 117–146. Fredriksson, D.W., Swift, M.R., Eroshkin, O., Tsukrov, I., Irish, J.D., Celikkol, B., 2005. Moored fish cage dynamics in waves and currents. IEEE Journal of Oceanic Engineering 30 (January (1)), 28–36. Fu, S.X., Moan, T., Chen, X.J., Cui, W.C., 2007. Hydroelastic analysis of flexible floating interconnected structures. Ocean Engineering 34, 1516–1531. Hibbit, Karlsson, Sorenson, 2004. “ABAQUS Users’ Manual 6.5,” ABAQUS Inc. Kristiansen, D., 2010. Wave induced effects on floaters of aquaculture plants. Ph.D. Thesis. Department of Marine Hydrodynamics, Nowegian Institute of Technology. Lader, P., Dempster, T., 2008. Current induced net deformations in full-scale seacages for Atlantic salmon. Aquacultural Engineering 38, 52–65. Lader, P.F., Fredheim, A., 2006. Dynamic properties of a flexible net sheet in waves and current—a numerical approach. Aquacultural Engineering 35, 228–238. Lader, P.F., Olsen, A., 2007. Experimental investigation of the interaction between waves and net structures – damping mechanism. Aquacultural Engineering 37, 100–114. Lader, P.F., Fredheim, A., Lien, E., 2001. Modelling of net structures exposed to waves and current. In: Proceedings of OOA IV; Open Ocean Aquaculture IV Symposium, St. Andrews, New Brunswick, Canada, June 17–20, 2001. Lader, P.F., Enerhaug, B., Fredheim, A., Krokstad, J., 2003. Modeling of 3D Net Structures Exposed to Waves and Current, 3rd International Conference on Hydroelasticity in Marine Technology, 15–17 September 2003, Oxford, UK. Lader, P.F., et al., 2007. Experimental investigation of wave forces on net structures. Applied Ocean Research 29, 112–127. Lee, H.H., Wang, W.S., 2005. The dragged surge motion of tension-leg type fish cage system subjected to multi-interactions among wave and structures. IEEE Journal of Oceanic Engineering 30 (1), 59–78. Moe, H., Fredheim, A., et al., 2010. Structural analysis of aquaculture net cages in current. Journal of Fluids and Structures 26, 503–516. Newman, J.N., 2005. Efficient hydrodynamic analysis of very large floating structures. Marine Structures 16, 169–180. Wu, Y.S., 1984. Hydroelasticity of floating bodies. Ph.D. Thesis. Brunel University, UK.
Contents lists available at SciVerse ScienceDirect
Aquacultural Engineering journal homepage: www.elsevier.com/locate/aqua-online
Dynamic analyses of floating fish cage collars in waves Shixiao Fu a,b,∗ , Torgeir Moan b a b
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, 20030 Shanghai, China Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 2 February 2011 Accepted 5 December 2011 Keywords: Interconnected floating collars Hinge connector Linear hydroelasticity Frequency domain
a b s t r a c t An extended 3D hydroelasticity theory is used in the frequency domain to predict the dynamic response of 5 by 2 floating fish farm collars in regular waves. Hinge modes in two directions and flexible torsional modes are all considered in the analyses. The dry and wet resonant natural frequencies for each rigid relative motion and the flexible modes are also calculated. The results show that the modes, flexible bending and torsional modes, as well as their interactions, greatly contribute to the dynamic response of the floating collar system. From a design point of view, the effects of the connector rotational stiffness on the dynamic response of the structures are also analysed. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Because of the increasing demand of ocean fish resources and the environmental and nearshore space limitation requirements, traditional nearshore fish farming is moving into the open ocean. Efforts have been made to develop traditional fish cages for use in open ocean environments. However, due to the more severe environmental loads from the high energy sea states, the design for future offshore fish farms requires more accurate analyses and calculations (Lader et al., 2003). Currently, fish farm structures can be split into two subsystems: floating collar structures and net cages. The main loads on the system result from ocean currents and waves. Analyses of fish farm structures exposed to currents and waves have often been performed using Morison’s equation and by considering that the rigid floating collars attached by the nets are fixed. When the focus is mainly stressed on the net cage deformations in the current, this assumption may be acceptable. However, regarding the influence of waves on the net cage system, this assumption is unacceptable. The floater motion in the waves creates slack in the net and therefore causes large forces at the joint between the bottom weight and the net, which can result in net fault. Research has shown that the floater motion is the main contributor to the forces and tensions in the net, while the forces on the net from the current are much smaller (Lader et al., 2001). Based on the linear potential wave theory and the linearised drag forces on the net, a 2D analytical solution for a tension
∗ Corresponding author at: State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, 20030 Shanghai, China. Tel.: +86 1350 1947 087. E-mail address: [email protected] (S. Fu). 0144-8609/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.aquaeng.2011.12.001
leg-type fish cage in waves was performed (Lee and Wang, 2005), where the interaction between the top platform structure, incident wave, and motion of both the tether and net were considered. However, these authors mainly focused their research on the surge motion of the system. Fredriksson (Fredriksson, 2001; Fredriksson et al., 2003, 2005) has conducted measurements of waves spectra, current profiles, forces in moorings and the motion responses of a realistic fish cage during an extreme storm. Numerical simulations were also performed using these measured data sets as inputs to a dynamic finite element model (FEM). Comparisons between these measured and simulated results showed good agreement. Moe et al. (2010) used ABAQUS to analysis the deformation of net cage under different water current velocities, the numerical result compared with the tests in the tank was good. Kristiansen (2010) studied the wave induced effects on floaters of aquaculture plants using CFD. From experimental point of view, Lader (Lader and Fredheim, 2006; Lader et al., 2007; Lader and Olsen, 2007; Lader and Dempster, 2008) studied the waves and current induced net deformations. In the last few decades, various theories (Bishop and Price, 1979; Wu, 1984; Newman, 2005) have been developed to predict the hydroelastic responses of continuous or multimodule-connected flexible structures in waves. Recently, an extension was added to the 3D hydroelasticity theory to account for the rigid hinge modes of these interconnected flexible structures (Fu et al., 2007). In this paper, to predict the dynamic response of 5 by 2 floating fish farm collars in regular waves, the extended 3D hydroelasticity theory is adopted. Hinge modes in two directions and flexible torsional modes are considered. The effects of the connector rotational stiffness, the rigid relative motion modes, the flexible bending and torsional modes, as well as their interactions to the dynamic response of the floating collar system, are studied.
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S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
2. Basic theory The fluid around the interconnected flexible floating collars at zero forward speed is assumed to be ideal. Hence, the fluid behaviour can be completely described by the velocity potential governed by the Laplace equation. The unsteady velocity potential around the structures may be decomposed into three parts and expressed as:
= Re
ϕI + ϕD +
m
ϕr pr
e
iωt
,
(1)
r=1
where ω is the wave circular frequency; ϕI (x, y, z, ω) and ϕD (x, y, z, ω) are components of the incident wave velocity potential and the diffraction wave potential, respectively; ϕr (x, y, z, ω) (r = 1, . . . , m) are the components of the radiation wave potential arising from the rth dry mode of the flexible structure with unit amplitude; m is the number of modes; and pr is the complex amplitude of the generalised coordinate. ϕD (x, y, z, ω) and ϕr (x, y, z, ω) can be solved by Green’s function method under the following governing equation and corresponding boundary conditions:
⎧ [˝] : ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [SF ] : ⎪ ⎪ ⎪ ⎨ [S] :
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [SB ] : ⎪ ⎪ ⎩
[S∞ ] :
∇ 2 ϕv = 0 ∂ϕv − ω2 ϕv = 0 ∂z ∂ϕD ∂ϕr ∂ϕI ៝ 0r n ៝, =− , = −iωu ∂n ∂n ∂n ∂ϕv =0 ∂z ϕv → 0 g
(2)
where [˝], [SF ], [S], [SB ], and [S∞ ] are the fluid field, free surface, hull surface, sea bed, and far field cylinder, respectively. ϕv can be ៝ denotes the normal vectors of the body’s wetted surface; ϕD or ϕr ; n ៝ 0r denotes the rth modal shape g is the gravity acceleration; and u vector of the point in the wetted elements. We assume that the structures are composed of flexible interconnected elements with axial, shear and rotational stiffness in their connection. However, for structures with a very soft connector, it will be impossible to directly solve the eigenvalue problem through traditional methods. Therefore, to obtain the eigenvalue and corresponding eigenvectors of the interconnected structures and to ensure the orthogonality conditions between the modes, we can transform the eigenvalue problem into the following one by introducing an artificial stiffness proportional to the mass into the system. The eigenvalue and corresponding eigenvectors of the floating collar structures can then be found using the following transformed eigenvalue problem: (−(ω02 + )[M] + ([K] + [M])){X} = {0},
(3)
where [K] and [M] are the stiffness and mass matrix of the structure, respectively and ω0 is the eigenvalues of the original interconnected structure, which should satisfy the following equation: (−ω02 [M] + [K]){D} = {0}.
(4)
Here, {D} is the eigenvector of the equation. From Eq. (3), we 2 + , can obtain the corresponding positive eigenvalues, r = ω0r and the corresponding eigenvectors, {X}, through traditional methods. The orthogonality conditions with respect to [K] + [M] and [M] are automatically satisfied. Thus, these values can also satisfy the orthogonality conditions with respect to [K] and [M] for the original interconnected structure. Therefore, the eigenvalues and corresponding eigenvectors of the original structure should be 2 = − and {D } = {X }. ω0r r r r Based on the FEM and mode superposition method, we usually select the first several oscillatory modes to present the structural
dynamic responses. Therefore, we assume that the nodal displacement of the structure can written as a superposition of the first m modes of the structure as {U} =
m
{Dr }pr (t) = [D]{p},
(5)
r=1
where pr (t) refers to the rth generalised coordinate. For r = 1 − 6, {Dr } represents the vector of the first six rigid modes as defined in the sea-keeping theory, and pr (t) is the magnitude of the rigid displacement about the centre of mass, (xG , yG , zG ). When r > 6, {Dr } represents the hinge and flexible modes of the structure calculated from Eq. (4). Then, from the motion equations of the structure based on FEM, we have the equations of motion for solving the generalised coordinates, {p}, in the frequency domain as [−ω2 ([a] + [A]) + iω([b] + [B]) + ([c] + [C])]{p} = {F},
(6)
where [a] = [D]T [M][D], [b] = [D]T [C][D] and [c] = [D]T [K][D] are the generalised mass matrix, damping matrix and stiffness matrix of the structure, respectively; and [A], [B] and [C] are the generalised added mass matrix, added damping matrix and restoring matrix, respectively. {F} is the generalised wave exciting forces and may be further expressed as
⎧ 1 0 ⎪ ៝ ៝ n A = Re i · u ωϕ dS ⎪ rk k r ⎪ ⎪ ω2 ⎪ S¯ ⎪ ⎪
⎪ ⎪ ⎪ ⎨ B = i Im i 0 ៝ ៝ n · u ωϕ dS rk k r ω . S¯ ⎪ ⎪ ⎪ ⎪ ៝ ·u ៝ 0r gw0 dS Crk = n ⎪ k ⎪ S¯ ⎪ ⎪ ⎪ ⎪ ⎩ 0 ∂ Fr =
៝ ៝
n · ur S¯
∂t
(7)
[I + D ]dS
When the responses of the principal coordinates have been obtained from Eq. (6), the displacements can be obtained using the following equation:
u(t) =
m
{Dr }pr (t).
(8)
r=1
To obtain the bending and torsional moments or stress responses of the structures, vector {Dr } can be replaced by the corresponding moments or stress modes, which can be easily obtained from the corresponding displacement modes, {Dr }. The wet resonant frequency for the rth mode, ωwr , can be obtained by solving the following equation with the iterative procedure: ωwr =
c + C 1/2 rr rr arr + Arr (ω)
,
(9)
where crr , Crr , arr and Arr (ω) are the generalised structural restoring force, generalised hydro-restoring force, generalised structural mass and generalised added mass of the rth structural mode, respectively. As for the specified rth mode, crr , Crr and arr have determinate values, and the added mass, Arr (ω), will have different values with different vibration frequencies (ω). To find the wet resonant frequency, ωwr , which will be determined with added mass Arr (ω), the iterative method should be used, wherein the relationship |ωwr − ω| ≤ ε should be satisfied for every wet resonant mode of the structures.
S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
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Fig. 1. A sketch of the floating collars and connectors (black dots mark the hinge connectors).
3. Numerical example 3.1. The model The methods mentioned above have been validated with floating flexible hinge-connected plates in regular waves (Fu et al., 2007). In this paper, numerical examples for specified open ocean fish farming floating collars with 5 by 2 fish cages in waves were performed. These structures are based on the structural designs of traditional fish farming structures. The specifications of the 5 by 2 unit module of the floating collar system are listed in Table 1. The floating collars are composed of 10 modules, with 5 and 2 modules arranged in rows and columns, respectively, as shown in Fig. 1. These modules are connected by hinge connectors, which have zero rotational stiffness along the horizontal axis and will be denoted as krot = 0 in the following text. The modules exhibit large stiffnesses in their translational degrees of freedom. The module positions are marked by the black dots in Fig. 1. The 1 m × 1 m (top and bottom) and 1 m × 0.5 m (side and ends) shell elements are used to model the dry structure while simultaneously considering the membrane and bending forces. The ABAQUS (Hibbit et al., 2004) is used as the eigensolver. Similar panel elements are used as the hydrodynamic mesh (Fig. 2). The displacement on the six points is determined and is marked as P1–P6 in Fig. 1. The system damping is very complicated, and determining this damping in the numerical model, including the damping from fluid motions and structural deformations, is difficult. The radiation damping from the water induced by the motions and deformations of the system is precisely estimated by Eq. (7). The damping contributed from other aspects, for example, the damping from net motions in the water and from the mooring lines, is not precisely
calculated in this model but is roughly estimated together with the structural damping by 5% of the structural critical damping. The damping from the structural deformations is usually determined to be 1.0–2.0% in the numerical simulations. However, to include the damping from the nets and the mooring line motions, the damping ratio in this paper is assumed to be 5%, which means that the damping from the nets and the mooring lines are crudely estimated as 3.0–4.0%. Of course, the exact damping from the nets and mooring lines should be measured directly from experiments, which is not in the scope of this paper. The research of this paper will mainly focus on the dynamic responses of the flexible system in wave frequencies; thus, the low frequency and mean motions (both are mainly affected by the mooring lines) will not be considered in this study. Therefore, the mooring line will not be included in the simulations.
3.2. Floating collar system modes The first to sixth mode shapes correspond to surge, sway, heave, roll, pitch and yaw motions, respectively, of the entire floating collar system. Fig. 3 shows the first 8 non-global mode shapes of the floating collar system. The corresponding generalised mass and dry Table 1 General specifications of the unit module of the floating collar system. Property
Symbol
Quantity
Length (m) Beam (m) Height (m) Draft (m) Width (m)
L B T D W
32 16 1 0.5 2
Fig. 2. An illustration of the hydrodynamic element.
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S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
Fig. 3. The first 14 modes of the structure.
and wet resonant natural frequencies are listed in Table 2. Notably, for a given mode, the structure will have the same mode shape and generalised mass regardless of the stiffness of the structure. Consequently, in Table 2, only one generalised mass is listed for a different structural stiffness. Thus, except for the first 6 zero frequency motion modes of the system, 5 zero frequency modes that represent the hinge motions of the floating collars and some nonzero frequency flexible modes due to the flexibility of the structures exist. Because of the effects of the radiation and hydrostatic forces, the wet resonant natural frequency is different from that of the dry frequency, whereas for the non-zero frequency modes, the former will be lower than that of the latter. 3.3. Dynamic analyses The generalised coordinate responses and wave exiting forces of the floating collars subjected to regular waves with various periods and incident wave angles are shown in Figs. 4 and 5, respectively. To enlarge and study the contributions from the rigid relative motion modes in the beam direction and the torsion flexible modes, the 75-degree incident angle is chosen as an example in an oblique sea. Fig. 4 demonstrates that the torsional modes will only be excited in oblique seas. Fig. 4 shows that the exciting wave frequencies that correspond to the response peak values for every mode are not matched with their corresponding wet or dry resonant natural frequencies, as listed in Table 2. Two reasons for this result are
as follows: firstly, the radiation wave damping will limit or eliminate resonance; secondly, the generalised wave exciting forces will reach their maximum values at their corresponding frequencies and peak value wave angles. The second explanation can be further explained with Figs. 4 and 5 in which each peak of the generalised exciting force in Fig. 5 will correspond to a peak of the generalised coordinate response in Fig. 4. Furthermore, some sub-peaks of the generalised coordinate response in Fig. 4 do not correspond to any of the generalised wave exciting force sub-peaks in Fig. 5 but occur at their corresponding wet resonant frequency (e.g., mode numbers 5, 7, 9 and 13). This fact indicates that resonance occurs for these modes. Considering Figs. 4 and 6, the rigid relative motions between the modules (modes 7–11) will greatly contribute to the final response of the system, which indicates the significance of the interactions between the modules. In head seas, because of the high longitudinal stiffness of each module, compared with that of the rigid body motion modes (first 6 rigid modes and 5 more rigid relative motion modes), the flexible modes (modes 12–14), which mainly belong to the torsional modes and will not be excited in head seas, have negligible contributions to the motion of the structure. However, in oblique seas, the torsional flexible modes and transverse rigid hinge mode (e.g., mode numbers 11–14), will be excited and produce more contributions to the motion, as shown by comparing (a) and (f) with frequencies of approximately 0.4–1.2 rad/s in Fig. 6. From Fig. 6, at a wave exciting frequency of approximately 0.9 rad/s in head seas, the entire system will almost reach its maximum
Table 2 Generalised mass and dry and wet natural frequencies of the structure for different structural stiffnesses. Mode number
GM.
˛ 0.2
3 5 7 8 9 10 11 12 13 14
– – 1.97 2.26 2.86 2.57 4.73 1.52 2.37 1.41
1.0
10.0
DF. (rad/s)
WF. (rad/s)
DF. (rad/s)
WF. (rad/s)
DF. (rad/s)
WF. (rad/s)
0 0 0 0 0 0 0 1.59 2.47 3.20
2.07 1.61 2.67 2.13 1.88 2.70 1.63 1.03 1.46 2.11
0 0 0 0 0 0 0 3.61 5.62 7.26
2.07 1.61 2.67 2.13 1.88 2.70 1.63 3.20 3.47 4.11
0 0 0 0 0 0 0 11.23 17.48 22.59
2.07 1.61 2.67 2.13 1.88 2.70 1.63 10.51 10.41 13.42
Mn. denotes the mode number, ˛ = EI/0.159 × 1011 ; EI denotes the bending stiffness of the modulus in N m2 ; DF. denotes the dry natural frequency of the structure in rad/s; WF. denotes the wet natural frequency of the structure in rad/s; and GM. denotes the generalised mass × 106 .
S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
Fig. 4. Generalised coordinate response amplitude of the first 14 modes with incident wave (—0◦ , - - -75◦ ).
11
12
S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
Fig. 5. Generalised wave exciting amplitude of the first 14 modes with incident waves of (—0◦ , - - -75◦ ).
S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
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Fig. 6. Vertical response amplitudes of points 1–6 in various regular waves with incident angles of (—0◦ , - - -75◦ ).
vertical response amplitude. This result can be explained by coupling the four rigid relative motion modes to the heave motion mode. Notably, some peak values at other wave frequencies, which may have significant contributions to the velocities and relative motion responses of the floating collars, exist. Therefore, when the tension forces between the net and the bottom weight of the system are considered, more attention to these peaks, which will be further analysed in our future works, should be extended. The effects of the incident wave frequency, wave incident angle, structural module stiffness, and connector rotational stiffness on the response amplitudes of each mode’s generalised coordinate and vertical motion are shown in Figs. 7 and 8, respectively. For the hinge-connected system, as shown in Figs. 7 and 8, the torsional modes will not be excited in head seas; therefore, their response amplitudes in oblique seas are shown in Fig. 7(f)–(h). As shown, increasing module stiffness decreases the response amplitude of each flexible mode; the more flexible the module is, the greater the contribution its flexibility will make. This contribution will not always increase the final response amplitude of the
structure, which depends strongly on the relative phase angle of the response. Fig. 7(a)–(e) shows that the stiffness of the modules will not affect the response amplitude of the rigid body modes. That is, the rigid modes will contribute equally to the dynamic response of the flexible floating collar structure whether it is assumed to be rigid or flexible. As shown in Fig. 7, increasing the connector stiffness will reduce the generalised coordinate response amplitude, indicating that the restoring forces from the connectors will reduce the response. This correlation is also demonstrated in Fig. 8. As shown in Fig. 8, because the flexible modes will not be excited by head seas and the rigid hinge modes exhibit the same contribution to the structure responses, the bending stiffness will not affect the vertical motion in head seas. However, regarding oblique seas, the flexible modes will be excited, and their corresponding contributions can be found from the vertical motion amplitudes. In the domain of relatively low-incident wave frequencies, increasing the module stiffness will generally lead to lower response amplitudes. (Here, the flexibility of the modules will increase the response
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S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
Fig. 7. The effects of bending and connector stiffness on the generalised coordinate response amplitudes.
S. Fu, T. Moan / Aquacultural Engineering 47 (2012) 7–15
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Fig. 8. The effect of bending stiffness on vertical motions.
amplitudes of the structures.) This result will differ from the conclusions of Fu et al. (2007), where the flexible modes will be excited not only in head seas, but also in oblique seas. 4. Summary and conclusions An extended 3D hydroelasticity theory is used to predict the dynamic responses of 5 by 2 floating fish farm collars in frequency domain regular waves. The hinge modes in two directions and the flexible torsional modes are considered. Dynamic response analyses of specified floating collar systems are performed. In head seas, the rigid relative motion modes will dominate the dynamic responses of the floating collar systems when compared with torsional flexible modes. However, in oblique seas, the flexibility of the structure will provide a greater contribution to the response, indicating that the flexibility should be considered. Meanwhile, from the floating collar point of view, the nets and mooring lines attached to the fish farm structures will mainly affect the damping properties in this system. However, determining the degree of influence of these structures will be challenging because the damping from the nets strongly depends on its motion amplitude and frequency and can only be determined with special forced vibration experiments.
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