Two-dimensional mooring dynamics for wave energy converters

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Two-dimensional mooring dynamics for wave energy converters

Raman-Nair, W.; Boileau, R.; National Research Council of Canada. Institute

for Ocean Technology

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DOCUMENTATION PAGE

REPORT NUMBER NRC REPORT NUMBER DATE

REPORT SECURITY CLASSIFICATION DISTRIBUTION

UNCLASSIFIED UNLIMITED

TITLE

Two-Dimensional Mooring Dynamics for Wave Energy Converters

AUTHOR(S)

W. Raman-Nair and R. Boileau

CORPORATE AUTHOR(S)/PERFORMING AGENCY(S)

PUBLICATION

SPONSORING AGENCY(S)

IOT PROJECT NUMBER NRC FILE NUMBER

PJ42_2520_16

KEYWORDS PAGES FIGS. TABLES

mooring, wave energy converter, lumped mass, Kane’s method, waves

iii, 27 9 0

SUMMARY

Mooring analysis algorithms developed at the NRC have been modified to evaluate the per-formance of mooring designs for wave energy devices through numerical simulations. The equations of the coupled two-dimensional motion of a generic wave energy device and mul-tiple mooring lines are formulated using Kane’s formalism. The lines are modelled using lumped masses and tension-only springs including structural damping. Surface waves are described by Stokes’ second order wave theory. The hydrodynamic loads are applied via a Morison’s equation approach using the instantaneous relative velocities and accelerations between the fluid field and the bodies (floating body and lines). The mathematical model and associated algorithm are validated by comparison with special cases of an elastic catenary mooring line.

ADDRESS National Research Council

Institute for Ocean Technology Arctic Avenue, P.O.Box 12093 St. John’s, NL A1B 3T5

National Research Council, Institute for Ocean Technology, St. John's, NL March 2012

Natural Resources Canada (NRCan), ecoENERGY Initiative (ecoEII) TR-2012-04

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National Research Council Canada Institute for Ocean Technology Conseil national de recherches Canada Institut des technologies océaniques UNCLASSIFIED

Two-Dimensional Mooring Dynamics for Wave Energy

Converters

TR-2012-04

W. Raman-Nair and R. Boileau

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TR-2012-04

Abstract

Mooring analysis algorithms developed at the NRC have been modified to evaluate the performance of mooring designs for wave energy devices through numerical simula-tions. The equations of the coupled two-dimensional motion of a generic wave energy device and multiple mooring lines are formulated using Kane’s formalism. The lines are modelled using lumped masses and tension-only springs including structural damping. Surface waves are described by Stokes’ second order wave theory. The hydrodynamic loads are applied via a Morison’s equation approach using the instantaneous relative velocities and accelerations between the fluid field and the bodies (floating body and lines). The mathematical model and associated algorithm are validated by comparison with special cases of an elastic catenary mooring line.

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TR-2012-04 Contents

1 Introduction

Wave energy converters are one class of marine renewable energy systems, an ap-proach to alternative energy sources that is strongly dependent on design constraints for an intimidating environment. WEC designers are challenged to effectively address mooring issues to ensure survival of their equipment and efficiency of power generation and to prevent costly failures.

The moorings for wave energy converters (WEC)are known to be a major compo-nent of the cost for such systems. A traditional mooring arrangement is shown in Fig-ure 1. The primary function of a mooring system is to keep the WEC on station in the most severe storm conditions. The mooring line dynamics is also an integral part of the performance of the WEC. Johanning, Smith and Wolfram raised issues in the design of moorings for wave energy converters in their work on generic station-keeping ar-rangements. Of special interest are potential additional considerations when designing WEC moorings, such as the interplay between mooring response characteristics and WEC efficiency in extracting power from waves. NRC1 Institute for Ocean Technology is taking the initial steps in developing methods to design moorings for WECs.

Figure 1: SyncWave conceptual mooring design (image c DSA Dynamic Systems Analysis)

Project deliverables: In this work, mooring analysis algorithms developed at the NRC have been modified to evaluate the performance of generic mooring designs for WEC through numerical simulations. The new code is capable of time domain pre-dictions of mooring line tensions and WEC motions in typical and extreme wave en-vironments. NRC purchased ANSYS AQWA c, a suite of commercial software tools, to determine WEC hydrodynamic and mooring parameters and received training in its use.

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TR-2012-04 2 System Configuration

It is the purpose of this paper to describe a method of analysis for the two-dimensional coupled dynamics of a generic WEC and multiple mooring lines based on Kane’s for-malism ([4]).

Following Johanning et al. (2006) [3], the WEC is modeled as a floating cylinder. For the mooring lines, the lumped mass approach is attractive because of its intuitive simplicity and ability to tackle problems with complex geometry and varying material properties and constitutive behaviour. Problems such as line touchdown can be mod-eled in a straightforward manner and the large motion dynamic response is captured. No difficulties are encountered when the mooring line becomes slack, in which case the tension is set at zero. The bending stiffness of the mooring lines is assumed to be negligible. We assume that the hydrodynamic loads are due primarily to the Froude-Krylov force, added-mass effects and viscous drag. In this regard, we allow for loading due to an arbitrary fluid velocity and acceleration field which is assumed to be undis-turbed by the system. This allows for the inclusion of wave and current effects via the use of the Morison et al. approach (Sumer and Fredsoe,1997).

This work presents a method for modeling multiple elastic and inelastic lines. The typical multi-line system configuration on which this is based is described in §2, fol-lowed by the descriptions of the forces acting on the WEC and the lines (§3) from which the equations of motion are derived (§4). The test problem presented in §5 vali-dates this method. The conclusions (§6) summarize the progress on these objectives. Proposed future work to support development of WEC moorings in conjunction with WEC design is recommended in §7.

Derivation of static equations for an inextensible catenary with touchdown are pro-vided in Appendix A. The formulae representing the relations between certain dimen-sionless characteristics for mooring systems lead toward a method to simplify design choices. AppendixB presents dynamic results, including motions and line tensions for a simple buoy in a typical wave environment.

2 System Configuration

A diagram of the system to be analysed is given in Figure 2. The origin of inertial coordinates is an arbitrary pointO on the seabed and the inertial frame is denoted by N with unit vectors −→N1,−→N2. Cylinder B has a body-fixed frame at its centre of mass G with unit vectors−→b1,−→b2 parallel to its central principal axes. Lineα (α = 1 . . . ν) is attached to B at point Pα

0.

The line is modelled bynαlumped massesmαk(k = 1, . . . , nα) at points P1α, P2α, . . . , Pnαα.

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mo-TR-2012-04 2 System Configuration G 1 br 2 br

Mean Sea Level

α 0 P Seabed α 2 P α 1 P α α+1 n P α 1 m α 2 m α α n m ) ,..., 1 ( Line α α = ν B Body 1 Nr 2 Nr O Frame Inertial Line Typical and Buoy 1 Fig. α α n P θ

Figure 2: Buoy and Typical Line tion of end-pointsPα nα+1as −−→ OPα_nα+1 = 2 X i=1 cα_i(t)−→Ni, (α = 1, . . . ν) (1) wherecα_i(t), ( α = 1, . . . , ν; i = 1, 2) are prescribed functions of time t . Fixing these end-points would represent the multi-point mooring system. The system to be analysed consists of the following sub-systems

• Rigid Body B with 3 degrees of freedom • Mooring lines Lα _with_2n

α degrees of freedom (α = 1 . . . ν) . The total number of degrees of freedom is

m = 3 + 2 ν X

α=1

nα (2)

Let the inertial coordinates ofG at time t be q₁B, q₂B and let qB

3 = θ where θ is the angle between the−→b1 and−→N1 axes. Let

−−−→ GP₀α= 2 X i=1 pα_i−→bi (α = 1, . . . , ν) (3) wherepα_i(i = 1, 2) are constants that specify the location of the attachment point P₀α relative to the centre of mass G of the body. The inertial coordinates of the lumped

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TR-2012-04 2 System Configuration

mass locationsP_kαon mooring lineα are denoted byqL,α_2k−1, qL,α_2k , k = 1, . . . , nα. The m generalized coordinates qr, (r = 1, . . . , m) are defined as

qr= qBr (r = 1, 2, 3) qmα+r = q L,α r (r = 1, . . . , 2nα ; α = 1, . . . , ν) (4) wheremα is defined as mα= 3 (α = 1) 3 +Pα−1 r=1 2nr (α = 2, . . . , ν) (5) The generalized speeds (Kane and Levinson [4]) are defined for the sub-systems as uB_r =q·B_r (r = 1, 2, 3) and uL,αr =

·

qL,α_r (r = 1, . . . 2nα ; α = 1, . . . , ν). For the whole sys-tem the generalised speeds are

ur = uBr (r = 1, 2, 3) umα+r= u L,α r (r = 1, . . . , 2nα ; α = 1, . . . , ν) (6) i.e.ur= · q_r(r = 1, . . . , m). 2.1 Kinematics

Following Kane and Levinson [4], the partial velocities are the coefficients of the gener-alised speeds in the expressions for the velocities of the system components and are written by inspection. The results are

• Rigid Body B : partial velocities −→ωB_r and −→vG_r − →_ωB r = ( −→ Nr (r = 1, 2) − →₀ _otherwise (7) − →_vG r = ( −→ N 3 (r = 3) − →₀ _otherwise (8)

• Connection points P0α (α = 1, . . . , ν): partial velocities −→v Pα 0 r − →_vPα 0 r =          − →_N 1 (r = 1) − →_N 2 (r = 2) ζ₁α−→N1+ ζ2α−→N2 (r = 3) − →₀ _otherwise (9) where ζ₁α _{= − p}α₁ sin qB₃ + pα₂cos q₃B ζ₂α = pα₁ cos qB₃ _{− p}α₂sin q₃B (10)

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TR-2012-04 3 Forces acting on the system • Lumped Masses at Pα k(α = 1, . . . , ν ; k = 1, . . . , nα) : partial velocities −→vP α k r − →_vPα k r =      − → N1 (r = 2k − 1) − → N2 (r = 2k) − →₀ _otherwise (11)

3 Forces acting on the system

The forces acting on the system include gravity, buoyancy, seabed reaction and hydro-dynamic loads.

3.1 Generalized inertial forces

The non-hydrodynamic generalized inertia forceF∗B

r on bodyB is F_r∗B = −→ωB_r _· −I3 · uB₃−→N3 + −→vG_r _{· (−M}0−→aG) (r = 1, 2, 3) (12) whereM0 is the mass of B , I3 is the moment of inertia of B about axis through G normal to the plane of rotation, u·B₃ is the angular acceleration of B and −→aG _{is the} acceleration of the center of massG. The hydrodynamic inertia forces contribute to the added mass effects of the buoy motion in water and will be discussed later. Equation (12) may be written in the form

F∗B_{= − M}B · uB (13) where MB_{is a 3 × 3 diagonal matrix with diagonal entries M}

0, M0, I3. The vector

· uB

is a _{3 × 1 column vector with entries} u·B_r(r = 1, 2, 3) . For line Lα with lumped massesmα_kthe non-hydrodynamic generalized inertia force is

F_r∗α = nα X k=1 − →_v_rPα k · (−mα k−→aP α k) (14) (r = 1, . . . , 2nα;α = 1, . . . , ν) This may be written in matrix form as

{F∗α} = −ML,α · uL,α (15)

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TR-2012-04 3 Forces acting on the system where · uL,α

is the 2nα × 1 column vector of · ur

L,α

values, ML,α_{is a 2n}

α × 2nα diagonal matrix with diagonal entries

M_{2k−1,2k−1}L,α = M_2k,2kL,α = mα_k (k = 1, . . . , nα) (16)

3.1.1 Gravity and buoyant force on cylindrical body

The gravity force on_{B is −M}0g−→N2 atG, where g is the acceleration due to gravity. To estimate the buoyant force we define the functionVs(d) to be the submerged volume ofB as a function of draft d when it is vertical (i.e. when θ = 0). The buoyant force is then approximatelyρfVs(dK) g−→N2at the metacentreM , where dKis the submerged depth of the baseK of B and ρf is the water density. For small θ, the quantity dK is given by

dK= dW + η − qB2 + hK (17) where η is the wave elevation above mean sea level at the location of G and hK = GK. The heave damping force and roll damping couple are represented respectively as _−chuB2−→N2 and -cruB3−→N3where ch is a heave damping coefficient and cr is a roll damping coefficient. The effect onB due to gravity and buoyancy with heave and roll damping is a force−→FGatG with couple−M→b where

− →_F G = ρfVs(dK) g − M0g − chuB2 −→ N2 (18) −→ Mb = −hMρfVs(dK) g sin qB3−→N3 (19) and hM = GM is the metacentric height. The generalised active force, denoted by FrGB/B, is found as

F_rGB/B =−→FG· −→vrG+−M→b· −→ωBr (20) wherehM is the metacentric height GM .

3.1.2 Gravity, buoyant force and seabed reaction on mooring lines

If we denote the volume of the portion of lineLα associated with lumped massmα_k by Vα

k, the net force on lumped massmαk due to gravity and buoyancy is(mαk− ρfVkα)g − → N3 . To allow for the possibility of contact between any portion of the mooring lines and the seabed (”touchdown”) we assume that the seabed normal reaction force is directly proportional to the depth of lumped mass penetration into the bed surface. Hence the vertical touchdown reaction force onmα_k has magnitude

1 2kE(|h α k| − hαk)−→N3= 0 if hα_k _{≥ 0} kE|hα_k| if hα_k < 0

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TR-2012-04 3 Forces acting on the system wherehα k = −−→ OPα k · − →

N2 is the height of pointP_kα above the seabed andkE is a seabed stiffness coefficient. The net force onmα_k due to gravity, buoyancy and seabed normal reaction at touchdown is written as −→χα

k = χαk − →_N 2 where χα_k _{= −(m}α_k _{− ρ}fVkα)g + 1 2kE(|h α k| − hαk) (21) The generalised active force due to −→χα_k is

F_rGBT /Lα = nα X k=1 χα_k−→N2· −→vrP α k (22)

3.2 Line tension and structural damping

The segment of lineLαbetween pointsP_k−1α andP_kαis denoted byS_kαand its stiffness kSαk is defined in the usual way as

kSkα =

Aα_0,kE_kα lα

k

(k = 1, . . . , nα+ 1)

whereAα_0,k, E_kα, lα_k are, respectively, the area of cross-section, modulus of elasticity and unstretched length of the segment. To define the unit vectors along the line segments the position vectors of the pointsP_kαcan be written in the form

−−→ OPα_k = 2 X i=1 Y_i,kα−→Ni (k = 0, . . . , nα+ 1) (23)

where the quantitiesY_i,kα are functions of the generalised coordinates. The unit vector along segmentSα k fromPk−1α toPkαis then − →_tα k = Pi=2 i=1Zi,kα − →_N i Zα 3,k (24) whereZα

i,k = Yi,kα − Yi,k−1α ,(k = 1, . . . , nα+ 1; i = 1, 2) and Z3,kα = r Zα 1,k 2 +Zα 2,k 2 is the instantaneous length of the segment. We allow for line tension but not for com-pression. To this end, we define the elongation of segmentSα

k as Z_4,kα = 1 2 Z_3,kα _{− l}_kα + Z_3,kα − lα_k (25) (k = 1, . . . , nα+ 1)

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TR-2012-04 3 Forces acting on the system

which is identically zero if the instantaneous segment length becomes less than the unstretched length. The magnitude of the tension in segmentS_kαis thus

Z_5,kα = kSkαZα

4,k (k = 1, . . . , nα+ 1)

The line tensions act on B at points P₀α (α = 1, . . . , ν) in the directions of the unit vectors −→t1α. The generalized active force due to line tension on bodyB is therefore

F_rT /B,α= Z_5,1α −→tα_k _{· −}→vP

α 0

r (r = 1, . . . , 2nα) (26) The force on the lumped mass atPα

k in lineLαdue to tension is −

→_FT /Pα

k = −Zα

5,k−→tαk + Z5,k+1α −→t αk+1 (27) and the generalised active force is

FT /P α k r =−→FT /P α k · −→vP α k r (r = 1, . . . , 2nα) (28) In any line segment, the damping force on the end masses is of the form_±cs

−

→_v_R_·−→_t−→ t , where −→vRis the velocity of one mass relative to the other, −→t is the unit tangent vector along the segment andcs is a structural damping coefficient. The velocity of P_kα can be written in the form

− →_vPα k = 2 X i=1 ξ_i,kα −→Ni (k = 0, . . . , nα+1; α = 1, . . . , ν) (29) where the quantitiesξ_i,kα are functions of the generalised coordinates and generalised speeds. The structural damping force due to lineLα onB at P₀α is

− →_DB,α_{= c}α s −→vP α 1 − −→vP α 0 · −→tα 1sign Z4,1α −→ tα₁ (30) where cα

s is the structual damping coefficient, and the associated generalised active force is

F_rSD/B,α =−→DB,α_{· −}→vP

α 0

r (31)

The line structural damping force on mass atP_kαis − → DPkα = −Zα 6,k−→tαk + Z6,k+1α −→tαk+1 (k = 1, . . . , nα) (32) where Z_6,kα = cα_s →−vPkα− −→vP α k −1 ·−→tα_ksign Z_4,kα (k = 1, . . . , nα+ 1) (33) and the generalised active force is

FSD/P α k r =−→DP α k · −→vP α k r (34)

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3.3 Fluid drag

Fluid drag is considered on the moored body and on the lines independently.

3.3.1 Drag on cylindrical body

If the dimensions of body B are small compared to the length of the surface waves we can assume that the fluid velocity field is not affected by the presence of the body. where

For the drag resisting translational motion we need the velocity of the fluid relative toB

− →

VG_R =−→UG_F _{− −}→vG (35) where−→UG_F is the fluid velocity at the location of the body’s centre of massG. The drag on the body is − →_FB D = 2 X i=1 − →_F(i) D (36)

where the drag in direction−→Niis given by Morison’s formula (Sumer and Fredsoe,1997) − →_F(i) D = 1 2ρfA (i) BC (i) D − →_VG R·−→Ni −→ VG_R_·−→Ni −→ Ni (37)

HereA(i)_B is the projected surface area of body normal to−→NiandC_D(i)is the associated drag coefficient. The wetted areas are estimated using the submerged depth with q₃B= 0. The generalised active force due to viscous drag on body B is

F_rD/B_{· −}→vG_r (38)

3.3.2 Drag on mooring lines

Consider segment S_kα, diameter dα_k, (k = 1, . . . , nα + 1). Assume that the segment Sα

k has a velocity − →_VSα

k equal to the velocity of its mid-point and let the fluid velocity

at the segment mid-point be −→US

α k F . Let C Sα k DT, C Sα k

DN be the tangential and normal drag coefficients for segmentS_kα. The associated areas are

AS α k T = πdαkZ3,kα andA Sα k N = dαkZ3,kα (39)

The velocity of the fluid relative toSα k is − →_VSα k R = − →_USα k F − − →_VSα k (40)

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The unit vector normal toSα k in the

− →_N

1−−→N2 plane is −→nα_k =−→N3×−→tα_k. The components of−→VS

α k

R parallel and normal to segment Sαk are−→V Sα k t = −→ VS α k R ·−→tαk −→ tα_k and −→VS α k n = −→ VS α k R · −→nαk − →_nα

k respectively, and the drag force onSkαis − →_FSα k D = 1 2ρfA Sα k T C Sα k DT − →_VSα k t − →_VSα k t + 1 2ρfA Sα k N C Sα k DN − →_VSα k n − →_VSα k n (41)

The drag force onPα k is − →_FPα k D = 12 −→ FS α k D + − →_FSα k+1 D

and the generalised active force is FD/P α k r =−→FP α k D · −→v Pα k r (42)

The generalised active force due to fluid drag on lineLα is F_rD/Lα = nα X k=1 FD/P α k r (r = 1, . . . , 2nα) (43)

3.4 Hydrodynamic pressure forces

Hydrodynamic pressure is considered on the moored body and on the lines indepen-dently.

3.4.1 Hydrodynamic pressure on cylindrical body

As before, we assume that the fluid velocity field is unaffected by the presence of body B. The fluid acceleration at G is −→aG_F = _DtD −→UF

evaluated atG where D Dt = ∂ ∂t+ − → UF · −→▽ (44)

Let _{[A] be the 2 × 2 added mass matrix of body B in the inertial frame. Define the} inertia matrix[E] in the inertial frame by

[E] = ρfVs[I] + [A] (45)

where Vs = submerged volume of body B and [I] is the 2 × 2 identity matrix. The submerged volume Vs is approximated by its value when q3B = 0.The hydrodynamic pressure force−→HBon bodyB may be written (Landau and Lifshitz [6])

−

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where −→HI/B_{is due to fluid inertia and} −→_HA/B _{is due to added mass. Let the vectors} −

→_HI/B_and−→_HA/B _{have inertial components}_HI/B_{and H}A/B_{. Then} n HI/Bo = [E]aG F (47) n HA/Bo _{= −[A]}aG (48) whereaG

F is the fluid acceleration at G in the absence of the body and aG is the acceleration ofG, both in the inertial frame.The generalised active forces on B due to fluid inertia and added mass are

FrI/B =−→HI/B· −→vGr Fr∗A/B =−→HA/B · −→vGr

(r = 1, 2, 3) (49)

The vectorF∗A/B_{can further be written in the form} n F∗A/Bo_{= −}MB a · uB (50) whereMB

a is the 3 × 3 added mass matrix of B augmented to include added rota-tional inertia. These terms (row 3 and column 3) have been assumed negligible in the simulations.

3.4.2 Hydrodynamic pressure on mooring lines

The added mass of segmentS_kαfor lateral motion is (Blevins,1990)mS

α k

a = ρfπ 1₂dαk 2

Z_3,kα . The added mass matrix of segmentS_kα can be expressed in the inertial frame as

ASα k = mS α k a   nα 1,k 2 nα 1,knα2,k nα_1,knα_2,k nα_2,k2   (51) wherenα

i,k (i = 1, 2) are the comonents of the unit normal vector −→nαk. Define the fluid inertia matrix ofSα k by [E]Skα = ρ fVS α k[I] + [AS α k] (52)

whereVSkα is the volume ofSα

k. The hydrodynamic force onPkα due to fluid inertia and added mass is − →_HPα k =−→HI/P α k +−→HA/P α k (53)

The vectors−→HI/Pkα and−→HA/P α k have components n HI/Pkα o = 1 2 n [E]Sαk + [E]S α k+1 o n {aP α k F } o (54) n HA/Pkα o = −[MP α k a ]{aP α k} (55)

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TR-2012-04 4 Equations of motion

where we have included the contributions from segmentsS_kαandS_k+1α . Here the matrix [MP α k a ] = 1₂ n [A]Sα k + [A]S α k+1 o , the vector_{aP α k

F } is the fluid acceleration at Pkα and the vector{aPα

k} is the acceleration of Pα

k. The associated generalised forces are FI/P α k r = −→HI/P α k · −→vP α k r (56) FA/P α k r = −→HA/P α k · −→vP α k r (57)

The generalised active forces onLα due to fluid inertia and added mass are then FI/L α r =Pn_k=1α FI/P α k r F∗A/L α r =Pn_k=1α FI/P α k r (r = 1, ..., 2nα) (58)

The vectornFr∗A/Lα o can be written n F_r∗A/Lαo_{= −[M}A/Lα] · uL,α (59) where [MA/Lα] is a 2nα × 2nα block diagonal matrix with block diagonal elements consisting of the matrices[MP

α k

a ].

4 Equations of motion

We recall that the system has m degrees of freedom as defined in equation (2). To write the equations of motion we assemble the components of the generalized inertia and active forces for the system as_{m × 1 vectors such that the first 3 components refer} to the degrees of freedom of bodyB, the next 2n1components refer to lineL1, the next 2n2 components refer to lineL2,etc. The generalised inertia forces for the system can be written in the form

n

F∗Systemo_{= −}hMSystemi nu·o (60) where the matrixMSystem_{is a m × m block diagonal matrix representing the sum} of mass and added mass matrices. The generalised active forces due to gravity, buoy-ancy, fluid drag, line tension and structural damping, fluid inertia and mooring line touchdown are added to form the m × 1 vector FSystem_{. We are now able to write} the system of2m coupled nonlinear equations of motion of buoy B and its ν mooring lines (Kane and Levinson,[4]) as

{F∗System} + {FSystem} = {0} (61) from which we find

n_·

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TR-2012-04 5 Test problem

Define the_{2m × 1 column vector {x} by}

{x} = {q} {u}

(63) The system of equations to be solved is then

{x} =· ( {q}· {u}· ) (64)

When appropriate initial conditions are specified, equation (64) is solved by the imple-mentation of the Runge-Kutta algorithm “ode45” provided in MATLAB [7].

5 Test problem

To test the algorithm we consider the case of a solid cylindrical buoy moored by a single line at its centre of massG to the seabed in water depth 50 m. Seawater density is ρf = 1025 kg/m3. The buoy has diameter D = 12 m, height h = 6 m, mass M0 = 3.5 × 105 kg, sway drag coefficient CD = 1.5 and is subject to a surface current ofVc = 1 m/ sec in the −→N1 direction. The mooring line has unstretched length60 m, diameter0.1 m, mass per unit length 150 kg/m and modulus of elasticity 10 GP a. The mooring line is modeled using10 lumped masses. A 300 sec simulation takes 9 min on a Z400 workstation. The steady state equilibrium configuration obtained from the simulation is illustrated in Figure 3. The equilibrium position of the buoy is found as (28.8873, 50.3312) . The draft of the buoy at equilibrium is thus c = 2.6688 m and the submerged volume is V0 = 301.8342 m3. We then determine the horizontal current force on the buoy to be

FH = 1

2ρfcDCDV 2

c = 2.4620 × 104N (65)

The vertical supporting force is

FV = (ρfV0− M0) g = 9.2018 × 104N (66) For comparison purposes, we make use of the closed form solution for an elastic cate-nary presented by Irvine [8]. This solution was re-written for the geometry illustrated in Figure 4. For an anchored line of unstretched length L0, area of cross-sectionA0, modulus of elasticityE, submerged weight per unit length w, the stretched line profile

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TR-2012-04 5 Test problem 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 x axis (m) z axis (m)

Steady State Profile

Mooring Line

Elastic Catenary

Figure 3: Steady state profile as a function of unstretched arc lengths is given by

x(s) = Hs EA0 +H w sinh−1 V − wL0+ ws H − sinh−1 V − wL_H 0 (67) z(s) = s EA0 V − wL0+ 1 2ws +H w    " 1 + V − wL0+ ws H 2# 1 2 − " 1 + V − wL0 H 2# 1 2    (68) where the top end is supported by a force whose horizontal and vertical components areH and V as shown in Figure 4. The distances a and b (Figure 4) are obtained from equations (67)and (68) by puttings = L0, i.e.

a = HL0 EA0 +H w sinh−1 V H − sinh−1 V − wL0 H (69) b = L0 EA0 V −1₂wL0 +H w    1 + V 2 H2 1₂ − " 1 + V − wL0 H 2# 1 2    (70)

Witha = 28.8873 and b = 50.3312, equations (69) and (70) are solved numerically to yield_{H = 2.2908 × 10}4 _{N and V = 9.1435 × 10}4 N which compare well with the theo-retical valuesFH andFV respectively. The mooring line profile is then calculated from

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TR-2012-04 6 Conclusions x a z b H V B A s

Figure 4: The elastic catenary

equations (67) and (68) and compared with the profile obtained from the simulation as shown in Figure 3.

Typical dynamic results from a realistic example of the use of this tool is provided in Appendix B.

6 Conclusions

The intent of this work was to develop a tool for evaluating the performance of the coupled dynamics of a generic wave energy converter mooring system. We have de-scribed the formulation of a numerical method for simulating the two-dimensional dy-namics of a floating generic WEC and multiple mooring lines. The results have been tested using known formulae for a special analytic case of the equilibrium of a cylinder moored by a single elastic catenary.

The method for the dynamic model is based on Kane’s formalism, which is well known to provide an efficient way of formulating the equations of motion of multibody systems.

The cost of energy drives the development of renewable energy such as wave en-ergy devices. Besides feed-in tariffs and political support of ecological choices for alternative energy sources, reducing the real costs by designing more efficient

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sys-TR-2012-04 7 Recommendations for future work

tems is critical to the acceptance and broader use of marine renewable energy in the market. Canadian developers need assistance to develop efficient, robust systems to protect Canadian investments in innovative technologies like this. This tool is a first step in aiding design.

(Appendix A provides a set of equations that can be used as an aid to selecting design parameters for a single line case.)

7 Recommendations for future work

The primary purpose of a mooring is station-keeping; however, moorings can also affect the performance of some WECs by restricting motion; the interaction of the de-vice with the mooring could be beneficial or detrimental to it’s efficiency. Harris (2004) [2] stressed the need to consider efficiency when designing moorings for WECs. He suggested that mooring design should be application-specific, considering both moor-ing cost and potential revenue from the interaction of the moormoor-ing with the particular power-generating device.

Numerical models can reduce the time and cost of mooring design. In addition, mooring design must be founded on real-world situations; physical models and field work are necessary to ensure results meet expectations. The following steps are rec-ommended for future work to achieve these goals:

1. Validate numerical method with physical testing: Perform tank testing and field trials of a generic moored wave energy device by measuring the motions and mooring tensions in several wave environments. Use the results to calibrate and validate the numerical model developed here.

2. Apply the current numeric model to risk scenarios: The mooring system should allow the removal of single devices without affecting the mooring of adja-cent devices. The existing tool can be used to model the system in a case where one of more mooring lines are disabled. This can advise operational decisions in the process of deploying or maintaining WECs.

3. Expand tool to floating bodies in three dimensions: The extension to 3D requires a significant effort, which would apply to a wider range of system ge-ometries, eg. for mooring lines out of plane.

4. Develop a user-friendly design tool: Agencies involved in the development of standards and codes for mooring design also require tools for guiding design engineers. The tool developed here can be further developed as a user-friendly aid to design.

In collaboration with a commercialization partner (TBD), a design tool should be developed to determine the performance envelope (tensions and motions)

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TR-2012-04 References

for wave energy conversion projects. Application of this tool will reduce the risk associated with this type of development.

5. Apply the tool to Canadian designs:The moorings for wave energy converters are known to be a major component of the cost for new designs. The validated numerical model should be used to simulate a variety of scenarios as design inputs to Canadian WEC designers.

6. Model complex multi-body scenarios: The interactions between an array of WECs or between a vessel and a WEC during berthing for inspection and main-tenance purposes is more complex. The modeling of such scenarios should be attempted with the assistance of ANSYS AQWA or a similar tool.

References

[1] R. Blevins,Flow-Induced Vibration, Krieger,2001,pg.25.

[2] R.E. Harris, L. Johanning and J. Wolfram,Mooring systems for wave energy con-verters: A review of design issues and choices, 3rd Intl. Conf. on Marine Renewable Energy (MAREC), 2004.

[3] L. Johanning, G. H. Smith and J. Wolfram, Mooring design approach for wave en-ergy converters J. Eng for the MarItime Environment, 2006, 220 Part M, 159-174. [4] T.R. Kane and D.A. Levinson , Dynamics : Theory and Applications ( McGraw Hill

Inc.,1985).

[5] B. Sumer and J. Fredsoe, Hydrodynamics Around Cylindrical Structures (World Scientific, 1997).

[6] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press,1959), pp 35. [7] MathWorks Inc.,MATLAB R2011b User’s Guide, Natick,MA,USA.

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TR-2012-04

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TR-2012-04 A The inelastic catenary with touchdown

A

The inelastic catenary with touchdown

This section presents the derivation of static equations for an inextensible catenary with touchdown. The formulae derived here can be used to determine the initial profile of the line. The equations for nondimenionalized parameters representing the relations between certain characteristics for mooring systems lead toward a method to simplify design choices.

Figure 5 defines the geometry for a mooring line anchored at one end and attached to a point mass at the water’s surface, suspended in a catenary for some part of the length. The touchdown pointO is not known. Suspended length Lo, horizontal distance between touchdown point and fairleadxo and horizontalH and vertical V loads at the fairlead are not known.

z o s H V ∇ z L_T = total length suspended length anchor L o = L T − Lo P (x(s), z(s)) O x_o x d

Figure 5: Definition of catenary mooring line profile

The solution for an arbitrary pointP (x(s), z(s)) along a suspended inextensible line expressed as parametric equations for a catenary are:

x(s) = H w sinh−1 V − wLo+ ws H − sinh−1 V − wL_H o (71) z(s) = H w   1 + V − wL o+ ws H 2!1/2 − 1 + V − wLo H 2!1/2   (72)

wherew = submerged weight per unit length.

At the mooring point on the surface,s = Lo,x = xoandz = zo=⇒ xo = H w sinh−1 V H − sinh−1 V − wL_H o (73)

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TR-2012-04 A The inelastic catenary with touchdown zo = H w   1 + V H 2!1/2 − 1 + V − wLo H 2!1/2   (74)

The tension at pointP is

T (s) = H2_{+ (V − wL}o+ ws)2 1/2 (75) From (71) and (72) dx ds = H (H2_{+ (V − wL} o+ ws)2)1/2 (76) dz ds = V − wLo+ ws (H2_{+ (V − wL} o+ ws)2)1/2 (77) Use (76) and (77) and the chain rule:

dz dx = dz ds ds dx = V − wLo+ ws H (78)

At the touchdown pointO, s = 0 and dz_dx = 0 so that V = wLo. Substitute forV in (73) and (74):

xo= H wsinh −1 wLo H (79) zo = H w   1 + wLo H 2!1/2 − 1   (80)

Define non-dimensional parameters for position, horizontal force and suspended length:

a = xo/zo, c = H/wzo, β = Lo/zo (81) Divide (79) and (80) byzoand use (81):

a = c sinh−1 β c (82) 1 = c 1 +β 2 c2 1/2 − 1 ! =⇒ 1 = (c2+ β2)1/2_{− c} (83)

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TR-2012-04 A The inelastic catenary with touchdown Solve (83) forβ: (1 + c)2= (c2+ β2_{) =⇒ 1 + 2c + c}2= c2+ β2 =⇒ β = (1 + 2c)1/2 (84) Substitute into (82): a = c sinh−1 √1 + 2c c (85) SinceV = wLo, the profile is found from (71) and (72):

xo= H wsinh −1ws H (86) zo = H w 1 +ws H 21/2 − 1 ! (87) T (s) = H2+ (ws)21/2 (88) From Figure 5: d = LT − Lo+ xo (89)

Define non-dimensional parameters for scope and total length:

λ = d/zo, γ = LT/zo (90)

Divide (89) byzo and use (90) and ()81):

λ = γ − β + a (91)

Substitute (84) and (85) into (91):

λ = c sinh−1 √1 + 2c c

−√1 + 2c + γ (92)

From (92), λ may be plotted vs c for a given γ, or equivalently H vs d, as shown in Figure 6. A curve fit may be applied to Equation 92 to estimate the increase in horizontal load with scope.

Alternatively, let_{p = γ − λ:}

p = −c sinh−1 √1 + 2c c

−√1 + 2c (93)

Solve (93) for c as a function of p. Figure 7 shows p evaluated over a range of c fitted to a fifth-order polynomial.

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TR-2012-04 A The inelastic catenary with touchdown 100 105 110 115 120 125 130 135 140 145 150 0 25000 50000 75000 100000 125000 150000 175000 Scope d (m) Horizontal load H (N)

Figure 6: Load excursion curve (depth ✺✵♠, uniform line weight ✾✶✽✳✼✺❦❣✴♠, length ✶✺✵♠)

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Figure 7: Relation of non-dimensionalized parameters for scope to horizontal load and mooring length

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Because the relation between c and p is exceptionally linear due to a non-expandable function (arcsin), simple exponential, polynomial or trigonometric functions will not approximate the function over the full range. Either spline fitting or a high-order polynomial (with caveats on the appropriate range) can give a reasonable estimate for this relation.

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TR-2012-04 B Dynamic results for buoy in sea state

B

Dynamic results for buoy in sea state

We consider the same buoy as in the test problem (§5) moored atG to the seabed by two mooring lines with the same properties as before. The anchor points are(0, 0) and (0, 20). The buoy is subject to a regular wave of period 7 sec, wavelength 76 m and height3 m. The motion of G and the line tensions at the buoy are shown in Figure 8 and 9 respectively.

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TR-2012-04 B Dynamic results for buoy in sea state 0 50 100 150 200 250 300 0 10 20 30 40

(a) Buoy x position

Time (s) x−component of OG (m)) 0 50 100 150 200 250 300 40 45 50 55 60 (b) Buoy z position Time (s) z−component of OG (m)

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TR-2012-04 B Dynamic results for buoy in sea state 0 50 100 150 200 250 300 0 1 2 3 4x 10

5 _{(a) Top Tension : Line 1}

Time (s) Tension (N) 0 50 100 150 200 250 300 0 1 2 3 4x 10

5 _{(b) Top Tension : Line 2}

Time (s)

Tension (N)