ANALYTICAL SOLUTION OF HEAVE VIBRATION OF TENSION LEG PLATFORM

J. Hydrol. Hydromech., 54, 2006, 3, 280–289 ANALYTICAL SOLUTION OF HEAVE VIBRATION OF TENSION LEG PLATFORM M. R. TABESHPOUR, A. A. GOLAFSHANI, B. ATA...
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J. Hydrol. Hydromech., 54, 2006, 3, 280–289

ANALYTICAL SOLUTION OF HEAVE VIBRATION OF TENSION LEG PLATFORM M. R. TABESHPOUR, A. A. GOLAFSHANI, B. ATAIE ASHTIANI and M. S. SEIF Department of Civil Engineering, Sharif University of Technology, Teheran, Iran; mailto: [email protected]

An analytical solution for the two-dimensional tension leg platform (TLP) interacting with ocean wave is presented. The legs of TLP are considered as elastic springs. The flow is assumed to be irrotational and single-valued velocity potentials are defined. The effects of radiation and scattering are considered in the boundary value problem. Because of linear behavior of legs during wave excitation, ignoring coupling effects with other degrees of freedom, the analytical solution of heave response has good agreement with the real behavior of the structure. KEY WORDS: Tension Leg Platform, Heave Motion, Wave Radiation, Wave Scattering. M. R. Tabeshpour, A. A. Golafshani, B. Ataie Ashtiani, M. S. Seif: HYDRODYNAMICKÁ ANALÝZA VERTIKÁLNÍHO POHYBU PEVNĚ UKOTVENÉ PLOŠINY S PŘIHLÉDNUTÍM NA RADIAČNÍ A ROZPTYLOVÉ VLIVY. Vodohosp. Čas., 54, 2006, 3; 8 lit., 3 obr. Práce uvádí analytické řešení působení vln na dvoudimenzionální pevně ukotvenou plošinu (TLP). Úchyty plošiny jsou uvažovány jako elastické pružiny. Předpokládá se nevírový tok a je definováno jednoznačné rychlostní pole. Okrajové podmínky zahrnují vlivy radiace a rozptylu. Předpoklad lineárního chování úchytů při excitaci mořskými vlnami i zanedbání některých dalších efektů v analytickém řešení odezvy TLP na vznášení ukázaly, že existuje soulad mezi naším řešením a skutečným chováním konstrukce. KLÍČOVÁ SLOVA: vrtná plošina, vztlak, radiace vln, rozptyl vln.

1. Introduction TLP is a suitable structure for oil exploitation in deep water. A schematic view of the system is shown in Fig 1. Many numerical studies have been carried out to understand the structural behavior of TLP and determine the effect of several parameters on dynamic response and average life-time of the structure. The legs are the most important part of the TLP in design point of view. Also among the various degrees of freedom, vertical motion (heave) is very important because of the direct effect on the stress fluctuation that leads to fatigue and fracture of the legs. Therefore the conceptual studies to understand the dynamic vertical response of TLP, can be useful for designers. An analytical solution for surge motion of TLP was proposed and demonstrated (Lee and Lee, 1993, 1994), in which the surge motion of a platform with pre-tensioned tethers was calculated. In that study, however, the elasticity of tethers was 280

only implied and the motion of tethers was also simplified as on-line rigid-body motion proportional to the top platform. Thus, both the material property and the mechanical behavior for the tether incorporated in the tension leg platform system were ignored. When this simplification was applied, no matter what the material used was or what the dimension of tethers was, the dynamic response of the platform would remain the same in terms of the vibration mode, periods and the vibration amplitude. An important point in that study was linearization of the surge motion. But it is obvious that the structural behavior in the surge motion is highly nonlinear because of large deformation of TLP in the surge motion degree of freedom (geometric nonlinearity) and nonlinear drag forces of Morison equation. Therefore the obtained solution is not true for the actual engineering application. For heave degree of freedom the structural behavior is linear, because there is not geometric nonlinearity in the

Analytical solutin of heave vibration of tension leg platform

heave motion degree of freedom and drag forces on legs have no vertical component.

several complicated factors such as buoyancy, scattering, radiation and simulated ocean wave load are considered. 2. General wave theory For the inviscid and incompressible fluid and irrotational flow, a single-valued velocity potential φ can be defined as: T

⎧ ∂φ ∂φ ⎫ u = { ux , uz } T = − ⎨ , ⎬ , ⎩ ∂x ∂z ⎭

(1)

where u is the flow velocity vector. The velocity potential satisfies the Laplace equation:

∂ 2φ ∂x 2

+

∂ 2φ ∂z 2

=0

(2)

and the Bernoulli equation −

Fig. 1. Schematic view of Tension Leg Platform (TLP). Obr. 1. Schéma pevně ukotvené plošiny (TLP); (trup plošiny – kotevní úvazy – základy).

A continuous model for vertical motion of TLP considering the effect of continuous foundation has been reported (Tabeshpour et al., 2004a). The effect of added mass fluctuation on the heave response of tension leg platform has been investigated by using perturbation method both for discrete and continuous models (Tabeshpour et al., 2004b, 2004c). An analytical heave vibration of TLP with radiation and scattering effects for undamped systems has been presented (Tabeshpour et al., 2005). The effect of structural and radiation damping on the response of the structure has not been considered and therefore the amplitude of the heave motion was over estimated. In this study the equation of the motion, and the corresponding solution for heave motion of the tension leg platform system subjected to sea wave, is derived and solved analytically considering structural and radiation damping. Based on Lee and Lee (1993) results, first the scattering problem is solved and the results were used to calculate the forcing function for the radiation problem and then both solutions were used for the solution of the tether motion. The structural model is very simple but

∂φ p + + gz = 0 ∂t ρ w

(3)

in the flow field, where p is the pressure and ρ w is the water density. A two-dimensional tension leg platform interacting with a long crested linear wave propagating in the x-direction is considered here as is shown in Fig. 2. The wave form and the associated velocity potential are given accordingly as,

ηi = −iAi exp[−( K1x + iωt )]

(4)

and

φi =

Ai g cos[ K1 ( z + h)] exp[−( K1 x + iωt )] , cos( K1h) ω

(5)

where Ai is the wave amplitude, g – the gravitational constant, h – the water depth, ω = 2π T – the angular frequency with T as the period, and K1 = −ik , where k = 2π L is the wave number with L as the wave length. K1 satisfies the dispersion relation

ω 2 = gK1 tan( K1h) .

(6)

3. Boundary value problem

In the platform system, the motion of the structure induced by the small amplitude incident wave is assumed to be small. The wave induced structural motion can be solved from the imposed boundary 281

M. R. Tabeshpour, A. A. Golafshani, B. Ataie Ashtiani, M. S. Seif

problem. Because of the linearity of the problem, the problem can be incorporated into a scattering and a radiation problem (Black et al., 1971). The wave force calculated from the scattering problem provides the force function in the radiation problem, and the forced oscillation then generates outgoing waves. A tension leg platform system is illustrated in Fig. 2, where the flow field is divided into three regions with two artificial boundaries at x = −b and x = b . In region I, the total velocity potential φI consists of incident waves φi , scattered waves φIs , and radiated waves φIw .

φI = φi + φIs + φIw .

(7)

In region II and III, the total velocity potential φII and φIII consists of scattered waves φIIs and φIIIs ,

and radiated waves φIIw and φIIIw . The subscript s denotes the scattering problem and w denotes the radiation (wave making) problem

φII = φIIs + φIIw ,

(8)

φIII = φIIIs + φIIIw .

(9)

All of the velocity potentials satisfy the Laplace equation. Furthermore, Sommerfeld’s radiation condition is satisfied at the infinity of region I and III to secure unique solutions ⎡ ∂φ 1 ∂φ ⎤ lim ⎢ ± ⎥ =0, x→±∞ ⎣ ∂x Cw ∂t ⎦

(10)

where Cw is the wave celerity.

Fig. 2. Illustration for the scattering problem (Lee et al., 1999). Obr. 2. Ilustrace k problému rozptylu vln (Lee et al., 1999).

4. Scattering and radiation problems

ξ = S exp(−iωt ) ,

In the scattering (diffraction) problem, the incident wave is considered to be diffracted by a fixed structure. The corresponding boundary value problem was also shown in Fig. 2. In the radiation problem the structure is considered to be forced into motion by the wave force induced by incident waves and scattered waves. The corresponding boundary value problem is illustrated in Fig. 3. The displacement of the dragged heave motion is given by

where S is the unknown amplitude of the heave motion. Applying the method of the separation of variables, matching the horizontal boundary conditions in each region, and applying Sommerfeld’s condition to regions I and III, the corresponding surface elevation and velocity potential of scattering (s) and radiation (w) problems can be found as follows (Lee and Lee, 1993): In region I:

282

(11)

Analytical solutin of heave vibration of tension leg platform

AIs / wj g cos[ K j ( z + h)]



φIs / w = ∑

ω

j =1

cos( K j h)

exp[−( K j ( x + b) − iωt )]

(12)

and ∞

η Is / w = −i ∑ AIs / wj exp[−( K j ( x + b) − iωt )] .

(13)

j =1

In region III: AIIIs / wj g cos[ K j ( z + h)]



φIIIs / w = ∑

ω

j =1

cos( K j h)

exp[−( K j ( x − b) + iωt )]

(14)

and ∞

η IIIs / w = −i ∑ AIIIs / wj exp[−( K j ( x − b) + iωt )] ,

(15)

j =1

where the eigenvalues K j can be solved from the dispersion equation

ω 2 = gK j tan( K j h)

(16)

with K j = −ik , for j = 1;

(2 j − 3)

π 2

< K j h < ( j − 1)π , for j ≥ 2

In region II:

φIIs =

ig ⎡ ⎛ x ⎞ ⎜ AIIsP1 + AIIsN 1 ⎟ cos K II 1 ( z + h) + ⎢ b ω ⎣⎝ ⎠

⎤ j −1 AIIsPj exp(− K IIj ( x + b)) + AIIsNj exp( K IIj ( x − b)) cos K IIj ( z + h) ⎥ exp(−iωt ) ∑ (−1) j =2 ⎥⎦ ∞

(

)

(17)

where the eigenvalues K IIj can be solved from the dispersion equation K IIj =

( j − 1)π , h−d

j ≥1

⎧⎪ ( z + h)2 − x 2 ig ⎡⎛ x ⎞ + ⎢⎜ AIIwP1 + AIIwN 1 ⎟ cos K II 1 ( z + h) + ω ⎣⎝ b ⎠ ⎪⎩ 2( h − d )

φIIw = ⎨ ∞

∑ (−1)

j =2

j −1

(

⎤ ⎫⎪ AIIwPj exp(− K IIj ( x + b)) + AIIwNj exp( K IIj ( x − b)) cos K IIj ( z + h) ⎥ ⎬ exp(−iωt ) ⎦⎥ ⎭⎪

(18)

)

The series of four unknowns AIs / wj , AIIs / wPj , AIIIs / wj and AIIs / wNj can further be solved from the following four equations derived from the four boundary conditions on the two vertical boundaries of region II. They are, for α ≥ 1

283

M. R. Tabeshpour, A. A. Golafshani, B. Ataie Ashtiani, M. S. Seif

∞ d ⎡1 d AIs / wα − i ⎢ Z II 1Zα AIIs / wP1 + ∑ (−1) j −1K IIj Z IIj Zα ⎣b j =2

Kα Zα Zα cos Kα h

(

⎧ K ZZ δα 1 exp( K1b) 1 1 1 Ai s ⎪ ⎤ ⎪ cos K1h − AIIs / wPj + exp(−2 K IIj b) AIIs / wNj ⎥ = ⎨ ⎥⎦ ⎪ iω 2 o w ⎪ g Zα S ⎩

(19)

)

d



Z IIα Z j

j =2

cos( K j h)

−i ∑

AIs / wj − Z IIα Z IIα

d

⎧ ⎪⎡ α −1 − δα 1 ⎤ AIIs / wPα + ⎨ ⎣(1 − δα 1 )(−1) ⎦ ⎪⎩

d ⎧ Z IIα Z1 ⎫ ⎡ (1 − δα 1 )(−1)α −1 − δα 1 ⎤ exp(−2 K IIα b) AIIs / wNα ⎪⎬ = ⎪⎨i exp( K1b) cos( K h) Ai s 1 ⎣ ⎦ ⎪⎭ ⎪ w 0 ⎩

i

Z II 1 Z α b

d

AIs / wP1 + i

∑ [(−1) ∞

j −1

K IIj Z IIj Z α

d

(exp(−2K

IIj b) AIIs / wPj

)]

+ AIIIs / wNj =

j=2

⎧0 ⎪ ⎨ iω 2 o ⎪− g Z α S ⎩

Z IIα Z j

j =1

cos( K j h)

AIIIs / wj − Z IIα Z IIα

d

⎡(1 − δα 1 )(−1)α −1 − δα 1 ⎤ × ⎣ ⎦ ⎧0 ⎩0

( exp(−2 K IIα b) AIIs / wPα + AIIs / wNα ) = ⎨

Fig. 3. Illustration for the radiation problem. Obr. 3. Ilustrace k radiaci vln.

284

s

(21)

w

d



i∑

(20)

(22) s w

Analytical solutin of heave vibration of tension leg platform

where δ is the Kronecker delta, and the notations of Z* Z*

d

o

, Z*Z* , and Zα

are defined in the

Appendix (A). It is clear that Eqs (19) and (21) are obtained from the kinematic boundary conditions, and (20) and (22) from the dynamic boundary conditions. Eqs (19) – (22) can then be solved for the four series of the unknowns AIsj , AIIsPj , AIIsNj and AIIIsj in the scattering problem and substituted into the corresponding equations to calculate the following properties. However, for the radiation problem Eqs (19) and (21) involve the unknown S, and therefore an additional equation is required to resolve all unknowns AIwj , AIIwPj , AIIwNj and AIIIwj . More

Fam = −ma

d 2ξ dt 2

mass,

dξ – the radiation damping force from dt fluid-structure interaction; ma and Cr will be determined in this paper. The equivalent stiffness of the platform system is presented, when the material property and the tether dimension are taken into account, as Frd = −Cr

K e = Kt + Kb , AE AE Kt = t = t l h−d

5. Simple model for motion of the platform

and

2

m

d ξ dt

2

+ Ce

dξ + K eξ = Fwz + Fam + Frd , dt

(23)

where m – the mass of the platform structure, Ce – the equivalent viscous structural damping, K e – the equivalent stiffness of the platform, Fwz – the vertical wave force acting on the horizontal side of the bottom of the structure,

Kb = ρ wgbc ,

0 Fwz = Fwz exp(−iωt ) ,

)

5.1 Added mass and radiation damping

Cr = Im(ω F0 )

Added mass and radiation damping are obtained as follows

in which

ma = Re( F0 ) ,

(29)

ˆ F0 = ∫S φIIwvds 0

exp(2 K IIj b) − 1 ⎫⎪ j −1 AIIwPj + AIIwNj cos K IIj (h − d ) ∑ (−1) ⎬ K IIj j =2 ⎪⎭

(

(26)

(27)

(28)

(30)

(31)

or

⎧⎪ b3 F0 = ρ ⎨[ (h − d ) + 2 AIIwN 1 ] b − + 3(h − d ) ⎪⎩ ∞

(25)

where At is the total area of the tethers cross section, E – the Young modulus of the tether material and l – the length of the tether. The wave force Fwz can be obtained through the integration of the total hydrodynamic pressure over the horizontal surfaces of the bottom of the structure (from the linearized Bernoulli equation), as

∞ exp(2 K IIj b) − 1 ⎪⎫ ⎧⎪ 0 = ρ iω ⎨2 AIIsN 1 b + ∑ (−1) j −1 AIIsPj + AIIsNj cos K IIj (h − d ) Fwz ⎬. K IIj j =2 ⎩⎪ ⎭⎪

(

(24)

where

calculations are presented in Appendix (B).

The equation of motion of the platform ignoring structural damping of tethers is as follows

– the force related to the added

(32)

)

285

M. R. Tabeshpour, A. A. Golafshani, B. Ataie Ashtiani, M. S. Seif

where vˆ is the unit vertical vector and S0 – the integral area around the structure. Now the equation of motion is fully determined as ( m + ma )

d 2ξ dt

2

+ (Ce + Cr )

dξ + K eξ = Fwz . dt

(33)

ζ =

Im( F0 ) Cr = , 2( m + ma )ωs 2( m + Re( F0 ))ωs

ωs =

Ke m + ma

(35)

(36)

and considering Eqs (29) and (30) one obtains

Substituting Eq. (12) into (34) one obtains 0 ⎡ −( m + ma )ω 2 − i (Ce + Cr )ω + K e ⎤ S = Fwz . ⎣ ⎦

Defining

(34)

0 ⎡ −( m + Re( F0 ))ω 2 − i [ 2( m + Re( F0 ))ωsζ + Im( F0 )ω ]ω + K e ⎤ S = Fwz ⎣ ⎦

(37)

or S=

0 Fwz

( Ke − (m + Re( F0 ))ω )

2 2

. 2

+ ( 2( m + Re( F0 ))ζωs + Im( F0 )ω ) ω

(38)

2

6. Conclusion

REFERENCES

In this work the analytical solution of heave vibration of TLP was presented for a simple model considering the effects of radiation and scattering in the boundary value problem. The importance of this work goes back to the linear behavior of the vertically moored structures in heave motion rather than surge. Surge motion is affected by nonlinear terms both in Morison's equation and large deformation of the structure. But in heave motion the structural stiffness if linear and therefore a simple linear model has a good compatibility with the real structure. A set of equations to describe the motion of the platform subjected to the wave-induced heave motion and the flow-induced drag motion were derived, and the corresponding close form analytical solution was presented as an infinite series form for the dynamic behavior of the platform utilized for the tension leg platform structural system. Such a solution can be extended for heave and pitch motion of vertically moored structures such as reservoir, wave breaker as well as wind energy turbine structures. Other importance of the presented work is on the use of the results in fatigue analysis of heave dominated moored structures.

BLACK J.L., MEI C.C., BRAY M.C.G., 1971: Radiation and scattering of water waves by rigid bodies. J. Fluid Mechanics, 46, 151–164. LEE C.P., LEE J.F., 1993: Wave Induced Surge Motion of a Tension Leg Structure, Ocean Engineering, vol. 20, No 2, 171–186. LEE C.P., 1994: Dragged Surge Motion of a Tension Leg Structure, International J. Ocean Engng, vol. 21, No 3, 311– –328. LEE H.H., WANG P.-W., LEE C.-P., 1999: Dragged Surge Motion of Tension Leg Platforms and Strained Elastic Tethers. Ocean Engng, vol. 26, No 2, 579–594. TABESHPOUR M.R., GOLAFSHANI A.A., and SEIF M.S., 2004a: Simple Models for Heave Response of TLP under Harmonic Vertical Load. Proc 8th International Conference on Mechanical Engineering, ISME, Iran. TABESHPOUR M.R., GOLAFSHANI A.A., and SEIF M.S., 2004b: The Effect of Added Mass Fluctuation on Heave Response of Compliant Offshore Structures. (In Persian.) Proc 8th International Conference on Mechanical Engineering, ISME, Iran. TABESHPOUR M.R., SEIF M.S. and GOLAFSHANI A.A., 2004c: Vertical response of TLP with the effect of added mass fluctuation. 16th Symposium on Theory and Practice of Shipbuilding, SORTA 2004, Croatia. TABESHPOUR M.R., SEIF M.S., ATAIE ASHTIANI B., GOLAFSHANI A.A., 2005: Analytical Heave Vibration of TLP with Radiation and Scattering Effects. 16th International Conference on Hydrodynamics in Ship Design, HYDMAN 05, Gdańsk, Poland. Received 22. November 2005 Scientific paper accepted 14. June 2006

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Analytical solutin of heave vibration of tension leg platform

APPENDIX (A) Z j Zα = ∫−0h cos[ K j ( z + h)]cos[ Kα ( z + h)]d z = if j ≠ α ⎧0 ⎪ ⎨ h ⎡ sin(2 Kα h) ⎤ if j = α ⎪ 2 ⎢1 + 2 K h ⎥ α ⎦ ⎩ ⎣ d

Z IIj Zα

= ∫−−hd cos[ K IIj ( z + h)]cos[ Kα ( z + h)]d z =

⎧ 1 ⎧⎪ sin( K IIj + Kα )(h − d ) sin( K IIj − Kα )(h − d ) ⎫⎪ ⎪ ⎨ + if K IIj ≠ Kα ⎬ K IIj + Kα K IIj − Kα ⎪ 2 ⎩⎪ ⎭⎪ ⎨ ⎪ h ⎡ sin[2 Kα ( h − d )] ⎤ if K IIj = Kα ⎥ ⎪ ⎢1 + 2 Kα h ⎦ ⎩2 ⎣ Z IIj Z IIα ⎧0 ⎪ ⎪ ⎨h − d ⎪h − d ⎪ ⎩ 2

(A1)

d

(A2)

= ∫−−hd cos[ K IIj ( z + h)]cos[ K IIα ( z + h)]d z = if j ≠ α

(A3)

if j = α = 1 if j = α ≠ 1

1 Z oj = ∫−0d cos[ K j ( z + h)]d z = [sin K j h − sin K j ( h − d )] Kj

(A4)

APPENDIX (B) The complete solution of φIIw is p h φIIw = φIIw + φIIw ,

(B1)

p h where φIIw and φIIw are the homogeneous and particular parts of the φIIw respectively. φIIw satisfies the Laplace Eq.

∂ 2φIIw ∂x 2

+

∂ 2φIIw ∂z 2

=0

(B2)

with the following boundary conditions

∂φIIw ∂φ = f ( x) , IIw ∂z ∂z z =− d

=0

(B3)

z=−h

In order to solve the Eq. (B2) the following relation is assumed

φIIw ( x, z ) = ψ IIw ( x, z ) + f ( x)

( z + h) 2 + g( x)h( z ) 2(h − d )

(B4)

h h The homogeneous parts of φIIw and ψ IIw are the same

287

M. R. Tabeshpour, A. A. Golafshani, B. Ataie Ashtiani, M. S. Seif h h φIIw = ψ IIw

(B5)

and the particular part of the φIIw is as follows p p = ψ IIw + f ( x) φIIw

( z + h) 2 + g( x)h( z ) 2(h − d )

Now considering the boundary condition h( z ) = cte = a p p = ψ IIw + f ( x) φIIw

(B6) ∂φIIw = 0 , one obtains ∂z z =− h

( z + h) 2 + ag( x) 2(h − d )

(B7)

Substituting Eq. (B7) into the Eq.(B2), one obtains ∂ 2φIIw ∂x 2

+

∂ 2φIIw ∂z 2

=

∂ 2ψ IIw ∂x 2

+

∂ 2ψ IIw ∂z 2

+ f ′′( x)

( z + h) 2 f ( x) + + ag′′( x) = 0 2(h − d ) (h − d )

(B8)

or g′′( x) = − f ′′( x)

( z + h) 2 f ( x) . − 2a ( h − d ) a ( h − d )

(B9)

If one considers f(x) as a constant value f ( x) = f 0 then f0 f0 g′′( x) = − and g( x) = − x 2 or a(h − d ) 2a ( h − d ) p p = ψ IIw + f0 φIIw

( z + h) 2 − x 2 2(h − d )

(B10)

and ∂ 2ψ IIw

+

∂x 2

∂ 2ψ IIw ∂z 2

=0

(B11)

with the following boundary conditions ∂ψ IIw =0 ∂z z =− h

(B12)

∂ψ IIw =0 ∂z z =− d

(B13)

The homogeneous part of solution of the eq. (B11) is as follows h ψ IIw = ∞

ig ⎡⎛ x ⎞ ⎜ AIIwP1 + AIIwN 1 ⎟ cos K II 1 ( z + h) + ⎢ ω ⎣⎝ b ⎠

∑ (−1)

j =2

288

j −1

(

⎤ AIIwPj exp(− K IIj ( x + b)) + AIIwNj exp( K IIj ( x − b)) cos K IIj ( z + h) ⎥ exp(−iωt ) ⎥⎦

)

(B14)

Analytical solutin of heave vibration of tension leg platform

RESUME Tension leg platform (TLP) is a suitable structure for oil exploitation in deep water. Many numerical studies have been carried out to understand the structural behavior of TLP and determine the effect of several parameters on dynamic response and average life-time of the structure. The legs are the most important part of the TLP in design point of view. Also among the various degrees of freedom, vertical motion (heave) is very important because of the direct effect on the stress fluctuation that leads to fatigue and fracture of the legs. Therefore the conceptual studies to understand the dynamic vertical response of TLP can be useful for designers. An analytical solution for the two-dimensional tension leg platform (TLP) interacting with ocean wave is presented. The legs of TLP are considered as elastic springs. The flow is assumed to be irrotational and single-valued velocity potentials are defined. The effects of radiation and scattering are considered in the boundary value problem. Because of linear behavior of the legs during wave excitation, ignoring coupling effects with other degrees of freedom, the analytical solution of heave response has good agreement with the real behavior of the structure. In this work the analytical solution of heave vibration of TLP was presented for a simple model considering the effects of radiation and scattering in the boundary value problem. The importance of this work goes back to the linear behavior of the vertically moored structures in heave motion rather than surge. Surge motion is affected by nonlinear terms both in Morison's equation and large deformation of the structure. But in heave motion the structural stiffness if linear and therefore a simple linear model has a good compatibility with the real structure. A set of equations to describe the motion of the platform subjected to the waveinduced heave motion and the flow-induced drag motion were derived, and the corresponding close form analytical solution was presented as an infinite series form for the dynamic behavior of the platform utilized for the tension leg platform structural system. Such a solution can be extended for heave and pitch motion of vertically moored structures such as reservoir, wave breaker as well as wind energy turbine structures. Other importance of the presented work is on the use.

HYDRODYNAMICKÁ ANALÝZA VERTIKÁLNÍHO POHYBU PEVNĚ UKOTVENÉ PLOŠINY S PŘIHLÉDNUTÍM NA RADIAČNÍ A ROZPTYLOVÉ VLIVY M. R. Tabeshpour, A. A. Golafshani, B. Ataie Ashtiani and M. S. Seif Pevně ukotvená plošina (TLP) je konstrukce vhodná pro těžbu ropy v hlubokých vodách. K porozumění strukturálního chování TLP bylo již vypracováno mnoho numerických studií s cílem určit vliv některých parametrů na dynamickou odezvu a na trvanlivost konstrukce. Úvazy (nohy) jsou z konstrukčního hlediska nejdůležitější části TLP. Mimo jiné stupně volnosti je vertikální pohyb velmi důležitý, neboť má přímý vliv na napětí následkem fluktuací. To vede k únavě materiálu a praskání úvazů. Proto je pro projekt zařízení důležitá teoretická studie odezvy TLP na dynamické vertikální namáhání. Článek presentuje analytické řešení působení mořských vln na dvourozměrné TLP. Nohy TLP jsou uvažovány jako elastické pružiny. Předpokládá se nevírový tok a definováno je jednoznačné rychlostní pole. Okrajové podmínky zahrnují vlivy radiace a rozptylu. Náš předpoklad lineárního chování úchytů při excitaci vlnami i zanedbání některých dalších efektů v analytickém řešení odezvy TLP na vznášení vede k řešení, které je v souladu se skutečným chováním konstrukce. Analytické řešení v této práci je provedeno pro jednoduchý model. Byly při tom uvažovány vlivy radiace a rozptylu při řešení okrajových podmínek problému. Význam naší studie pramení z toho, že přednostně bere v úvahu lineární chování vertikálně ukotvené konstrukce při zdvihovém pohybu před vlivem nárazů vln. Rázový pohyb je ovlivněn nelineárními členy v Morisonově rovnici a velkou deformací konstrukce. Ale vertikální zdvihový pohyb tuhé konstrukce je lineární a proto jednoduchý lineární model je kompatibilní se skutečnou konstrukcí. V práci jsou odvozeny rovnice opisující pohyb plošiny následkem vlnami způsobené vznášivé síly a unášení vlivem toku vody. Odpovídající analytické řešení je ve tvaru nekonečné řady. Takové řešení může být rozšířeno na vznášení podobně vertikálně ukotvených konstrukcí jako jsou např. rezervoáry, vlnolamy nebo turbíny. Dále je možné použít výsledky pro zhodnocení vlivů z únavy upevněné konstrukce.

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