## Friday, May 10, 2013

### DYNAMIC RESPONSE OF ARTICULATED TOWER PLATFORMS

INTRODUCTION
The demand for oil and gas has brought the offshore drilling and production of hydrocarbon deposits to greater water depths. As the depth of water increases, the size of conventional fixed leg platforms is approaching the economic limit. As a result, several new structural systems have been developed to improve the water depth capability of offshore structures. Some of the promising concepts are tension leg platforms, guyed and articulated tower platforms which take advantage of the effect of compliance, i.e., yield to the environmental forces. Articulated towers belong to the class of compliant towers which have been found quite suitable for deep water applications. Similar to a reed which “bends but does not break” the articulated structures withstand with suppleness the combined effects of the waves, wind and currents.
A typical mobile loading and storage system is the articulated buoyant loading tower (Fig. 1.2a) which may have either a single universal joint or a greater number of joints in the intermediate level which can be used at very deep water depths. The articulated tower with universal joints in the intermediate level is called a multi-hinged articulated tower (Fig. 1.2b). The extension of the concept of the single-leg articulated tower led to the development of a new type of platform with several columns which are parallel to one another. The columns are connected by universal joints, both to the deck and to the foundation. The use of universal joint ensures that the columns always remain parallel to one another and the deck remains in horizontal position. There is no rotation about the vertical axis of the columns. This type of platform is called a multi-leg articulated tower which has three or more columns (Fig. 1.2c). The advantage of this system is that the pay loads and deck areas are comparable with the conventional production platforms in moderate water depths. As the connection to the seabed is through the articulation, the structure is free to oscillate in any direction and does not transfer bending moment to the base. Since the articulated tower is a compliant structure and it freely oscillates along with the waves, the wave force on the structure is much less than that of a fixed structure. The dynamic amplification factor is low compared to the other fixed structures since its natural frequency is much less than the frequency of the wave.
Fig.1.1 Articulated tower
Articulated towers are quite susceptible to wind induced oscillations than is the conventional fixed-bottom platforms. This is especially true in the low frequency range of wind spectra to which this type of platform is exposed. Since wind is the major source of sea wave generation, consideration of wave alone in an open sea environment is not a realistic proposition. Therefore, in the present study, the response of articulated tower to random forces generated by wind and wave is investigated. The exposed portion of the tower is subjected to the action of turbulent wind while the submerged portion is subjected to random wave forces. The response analysis is performed by a time domain iterative procedure which includes the structural as well as forcing nonlinearities. Using the proposed method of analysis, a comparative study of Single Hinged Articulated Tower (SHAT) and Double Hinged Articulated Tower (DHAT) is conducted under various ocean environments.
(a) single leg articulated tower (b) Multi-hinged hinged articulated tower (c)Multi-legged articulated
Tower
Fig.1.2 Types of articulated towers
2. ARTICULATED TOWER MODEL
Fig.2.3 shows the idealized configuration of double hinged articulated tower. The complete nonlinear model for the dynamic analysis involves the formulation of a non linear stiffness matrix consisting of fluctuating buoyancy, formulation of a mass matrix consisting of structural mass and added mass due to the tower motion, and the formulation of a damping matrix. The tower structure is idealized by replacing its mass distribution with discrete masses located at the centroid of a series of small cylindrical elements of equivalent diameter Di representing inertia, added mass and buoyancy. All forces are assumed to act at these centroids and include weight, inertia, buoyancy, and fluid forces on the submerged parts and wind forces on the above water parts of the structure. The exposed area has been transformed into an equivalent projected area assuming that wind loads are associated with drag loads only. The model also involves the selection of wave theory that reasonably represents the water particle kinematics. Water particle velocities and accelerations are calculated at the geometric centroids of each element and are assumed not to be influenced by the presence of the structure. The error introduced with this assumption is small as long as the ratio of element diameter to wave length is small. In the computerized analysis, the equations describing motions and loads of the articulated tower are based on Morison’s equation applied to a moving system. Moments due to these forces about the joints are determined by multiplying the differential force equation by the appropriate moment arms and then integrating over the length of each element to obtain the total moment. All the ALP members are divided into the finite number of elements for the determination of the wave force and moments. The total force is obtained by the summation of all these elemental values. In order to incorporate high degree of nonlinearities associated with the system, a time domain numerical integration scheme (Newmark- beta) is used to solve the equations of motion.
Fig.2.3 Idealized configuration of DHAT
3. FORMULATION OF EQUATIONS OF MOTION
The equations of motion are obtained using Lagrangian approach. This approach provides several advantages over the Newtonian method, like eliminating free body diagrams with interaction forces between the members. The tower model consists of two-degree-of-freedom system. Thus, there are two generalized coordinates; rotations y1 and y2 about the vertical axis. The equations are derived for large displacements under certain assumptions that are listed below. In the derivation of equationsofmotion,‘1’ stands for the lower shaft, while‘2’ stands for the upper shaft.
3.1 Assumptions And Structural Idealizations
The flexural deformations of the platform are assumed to be small as compared to its displacement as a rigid body.
The platform has uniform properties over the segments of uniform diameters.
Diffraction effects are assumed to be insignificant as the member dimensions are small compared to wavelengths.
Morison’s fluid force coefficients CD and CM are constant.
The motion of the tower is only in the plane of fluid loading.
3.2. Lagrange’s Equation
The general form of Lagrange’s equation is
d/dt (∂T/∂ϴi)-∂T/∂ϴi+∂V/∂ϴi=Qϴi (3.1)
where Qϴi represents the kinetic energy, the potential energy and the generalized force, respectively. Thus, the objective of the analysis which follows is to obtain analytical expressions for the energy and generalized forces. In the following subsection, the tower absolute velocities are determined in the fixed coordinate system attached to the earth which means that the tower rectilinear velocity is resolved into x, y coordinates. Then, in Section 4, fluid forcing is evaluated.
3.3. Tower Kinematics
A schematic of the double hinged articulated tower as shown in Fig. 3 has been considered for the present study. The kinematic relationships for the tower model can be expressed as
〖x=L_1 sin〖θ_1+r_2j sinθ〗〗_2 (3.2)
y=L_1 cos〖θ_1 〗〖+r_2j cos〖θ_2 〗 〗 (3.3)
where L1 is the length of the bottom shaft; r2j the position vector of an element in the top tower measured from the mid hinge; ϴ1 and ϴ2 are tilt angles of the bottom and top tower. The inter dependence of upper and lower tower coordinates results in lengthy, nonlinear derivatives to describe the system motions. The velocity vectors in x and y is found by taking the time-derivative of the displacement vector:
〖 ẋ=L_1 cos〖θ_1 〗〖(θ_1 ) ̇+r2jsinθ_2 〗 θ ̇〗_2 (3.4)
y ̇=-L_1 sin〖θ_1 〗〖(θ_1 ) ̇+r2jsinθ_2 (θ_2 ) ̇ 〗 (3.5)
The resultant velocity is given by
V=〖L_1〗^2 〖θ_1〗^2+〖r_2j〗^2 〖θ ̇_2〗^2+2L_1 r_2j cos〖(θ_1-θ_2)(θ_1 ) ̇(θ_2 ) ̇ 〗 (3.6)
3.3.1.Kinetic energy
The kinetic energy, T, for the tower can be expressed in terms of the generalized coordinates and their first derivatives as
T=T_1+T_2 (3.7)
where T1 and T2 are the kinetic energies of the lower and upper towers:
T_1=1/2 ∑_(i=1)^Np▒〖(m_1t 〗)〖(r_1i (θ_1 ) ̇)〗^2=1/2 [∑_(i=1)^Np▒〖(m_1t ) 〖r_1i〗^2 〗] (θ_1 ) ̇^2=1/2 I_1t (θ_1 ) ̇^2 (3.8)
where m1t is the mass of an element in tower ‘1’and r1i is its position vector.
Kinetic energy of top tower, T2, consists of three components, viz: kinetic energy of the elements submerged in water (T2water), kinetic energy of the elements between water level and under side of the deck (Tair) and kinetic energy of the deck (Tdeck), respectively. Therefore, the total kinetic energy of the top tower will become
T_2=T_2water+T_2air+T_deck (3.9)
The resulting kinetic energy of the top tower may now be expressed as
T_2=1/2 [∑_(j=1)^Nw▒(m_2t ){〖〖〖(L〗_1 (θ_1)) ̇〗^2+(r_2j (θ_2 ) ̇ )〗^2+2L_1 r_2j cos(θ_2-θ_1 ) (θ_1 ) ̇(θ_2 ) ̇ } ]
+[∑_(j=N_W+1)^Nw▒(m_2t ){〖〖〖(L〗_1 (θ_1)) ̇〗^2+(r_2j (θ_2 ) ̇ )〗^2+2L_1 r_2j cos(θ_2-θ_1 ) (θ_1 ) ̇(θ_2 ) ̇ } ]
+m_d {〖(L_1 (θ_1 ) ̇)〗^2+〖(L_P (θ_2 ) ̇)〗^2+2L_1 L_p cos((θ_1 ) ̇-(θ_2 ) ̇)(θ_1 ) ̇(θ_2 ) ̇ } (3.10)
where m2j is the mass of an element in the top tower; Np the number of parts of the tower. The total mass m_it=(m_i+am_i) is used to calculate the kinetic energy of the elements account for the structural mass, mi and the inertial added mass, ami due to the motion of the structure; a is the added mass; md is the mass of the deck; Id is the moment of inertia of the deck; L_P=L_2+P_cm is height of the c.g of the deck above mid hinge; L2 is the length of the top shaft, and Pcm is the height of the c.g above the deck.
In the preceding expressions, number of submerged elements in the water Nw is defined by
N_W=L ̀□(/2)/ds_2 (3.11)
where L ̀□(/2) is the instantaneous submerged length of the top tower and ds2 the height of an element in the top tower.
3.3.2. Potential energy due to gravity and buoyancy
The total potential energy in the system, V, is due to the conservative forces of buoyancy and gravity which can be expressed as
V=V_Ieff+V_2 (3.12)
Where
V1eff = {∑_(i=1)^(N_p)▒〖f_1i r_1i-∑_(i=1)^(N_p)▒〖m_1i r_1i 〗〗}gcosθ_1 (3.13)
where f1i is the buoyancy of an element in the bottom tower and Np the number of parts. V2 consists of the effective potential energy due to the top tower and deck, respectively.
V_2=V_2eff+V_deck (3.14)
The resulting potential energy may now be expressed as
V_2={(∑_(J=1)^(N_W)▒〖f_2j-∑_(j=1)^(N_P)▒m_2j 〗)g(L_1 COSθ_1+r_2j cosθ_2)} (3.15)
where f2j is the buoyancy of an element in the top tower. Now, treating the kinetic and potential energies of the system as per Lagrange’s equation, we have:
〖d/dt (∂T/(∂(θ_1 ) ̇ ))=(I_1t+m_2t 〖L_1〗^2+m_d 〖L_1〗^2 ) (θ_1 ) ̈+m_2t L_1 c_2 sin(θ〗_2-θ_1)θ ̇_1 (θ_2 ) ̇
〖-m〗_2t L_1 c_2 sin(θ_2-θ_1)(〖θ_1〗^2+m_2t L_1 c_2 cos(θ_1-θ_2)(θ_2 ) ̈ ) ̇ (3.16)
Where
m_2t=∑_(j=1)^(N_w)▒(m_2j+am_2j ) (3.17)
and
m_2t c_2=∑_(j=1)^(N_w)▒〖m_2j r_2j+∑_(j=1)^(N_W)▒〖(am_2j 〗〗+r_2j) (3.18)
Similarly
∂T/(∂θ_1 )=m_2t L_1 c_2 sin(θ_1-θ_2)(θ_1 ) ̇(θ_2 ) ̇ (3.19)
∂V/(∂θ_1 )=[-∑_(i=1)^(N_P)▒〖m_1i r_1i+∑_(i=1)^(N_P)▒〖f_1i r_1i+(-∑_(j=1)^(N_P)▒〖m_2j+∑_(j=1)^(N_P)▒〖f_2j-m_d g〗〗) L_1 〗〗]gsinθ_1 (3.20)
Or
[(F_1 b_1-W_1 c_1 )+(F_2-W_2-W_d ) L_1 ]sinθ_1 (3.21)
where F1 and F2 are buoyancy forces in bottom and top tower; W1 and W2 are the weights of the lower and upper tower; Wd is the weight of the deck; b1 and c1 are the position of center of buoyancy and center of gravity in the lower tower from the bottom hinge; b2 and c2 are the center of buoyancy and center of mass in upper tower from the mid hinge.
3.4. Governing Equations Of Motion
The first nonlinear equation of motion takes the form as
〖(I〗_1t+m_2t 〖L_1〗^2+m_d 〖L_1〗^2)(θ_1 ) ̈-[m_2t c_2 L_1 (θ_2 ) ̇ sin(θ_2-θ_1 ) ] (θ_2 ) ̇+m_2t L_1 c_2 cos(θ_2-θ_1 ) (θ_2 ) ̈
+[{F_1 b_1-W_1 c_1)+(F_2-W_2-W_d L_P)L_1 } (sinθ_1)/θ_1 ] θ_1=Q_θ1 (3.22)
where Q_θ1 is the forcing function due to non-conservative forces and is described in Section 3. Similarly, the second equation of motion is as follows:
〖(I〗_2t+I_d+m_d 〖L_P〗^2)(θ_2 ) ̈+[m_2t c_2 L_1 cos(θ_2-θ_1 ) ] (θ_1 ) ̈+{m_2t L_1 c_2 (θ_1 ) ̇(θ_2 ) ̇ sin(θ_2-θ_1 ) } θ ̇_1
+[(F_2 b_2-W_2 c_2-W_d L_P ) (sinθ_2)/θ_2 ] θ_2=Q_θ2 (3.23)
The two equations of motion can now be written in matrix form as
[■(I_11&I_12@I_21&I_22 )]{■((θ_1 ) ̈@(θ_2 ) ̈ )}+[■(0&c_12@c_21&0)]{■((θ_1 ) ̇@(θ_2 ) ̇ )}+[■(K_11&0@0&K_22 )]{■(θ_1@θ_2 )}={■(Q_θ1@Q_θ2 )} (3.24)
I_11=I_1t+m_2t 〖L_1〗^2+m_d 〖L_1〗^2; I_12=I_21=m_2t L_1 c_2 cos(θ_2-θ_1 )
I_22=I_2t+I_d+m_d 〖L_P〗^2
c_12=-m_2t L_1 c_2 (θ_2 ) ̇ sin〖(θ_2-θ_1 ); 〗 c_21=[m_2t c_2 L_1 (θ_2 ) ̇ sin(θ_2-θ_1 ) ] (θ_1 ) ̇
K_11={F_1 b_1-W_1 c_1)+(F_2-W_2-W_d L_P)L_1 } (sinθ_1)/θ_1
K_22=(F_2 b_2-W_2 c_2-W_d L_P ) (sinθ_2)/θ_2
4. FLUID FORCING
The environmental forces of wind, waves and currents are categorized as non conservative forces. The forcing function Qϴ is the moment of the dynamic forces acting on the platform at any instant of time given by
Q_θ=F_a {u(z),(u,) ́x ̇ }+F_d ((u,) ̇v_c,x ̇ )+F_i (x ̈) (4.1)
in which is F_a {u(z),(u,) ́x ̇ } aerodynamicforceF_a {u(z),(u,) ́x ̇ } fluid drag forceand F_i (x ̈ ) fluid inertiaforce.Theseloadsaredefinedinthe followingsubsections.
The exposed portion of the tower is subjected to mean and fluctuating wind loads while, submerged substructure is acted upon wind driven waves. The steady mean wind velocity defines the mean static position of the tower about which it oscillates due to the fluctuating wind component. In the submerged portion of the tower, hydrodynamic forces develop due to wave structure interaction effects. The hydrodynamic load model duly considers the change in magnitude and direction of loading with the tower orientation.
4.1. Wind loads
The wind force per unit of projected area is given by
f(y,z,t)=0.5ρ_a C_p (y,z)[u(z)+u ́(y,z,t)-x(t) ]^2 (4.2)
where Cp(y,z) is the wind pressure coefficient; _x the structural velocity in the horizontal direction; ρ_a he air density; u(z) the mean wind velocity, and( u) ́(y,z,t) thefluctuatingwindvelocity. The total wind induced drag force, Fa, is then given by
F_a (y,z,t)=∫_Aa^0▒f(y,z,t)dydz (4.3)
in which Aa is the total projected area of the platform normal to the wind flow.
4.1.1. Mean wind estimation
The mean description of the turbulent wind field is assumed to be governed by the logarithmic law as
u(z)=u(z_ref )=(ln z/z_0 )⁄(ln z_ref/z_0 ) (4.4)
where zref is thereferenceelevationusuallytakenas10mabove mean sea level, z the vertical coordinate and z0 the roughness length which is provided by specifying the value of sea drag coefficient, defined as
C_Dsea=[K/(ln(10⁄z_0 )) (4.5)
where K is the Von Karman constant(K = 0.4).
The following empirical relations were proposed on the basis of large number of measurements for the range 4
CDsea=5.1×10-4[u(10) ]0.46 or (4.6)
CDsea=10-4[7.5+0.67u(10) ] (4.7)
For u(10) >20m=s, CDsea may be obtained by taking the average of the values given in Eqs.
(4.6) and (4.7).
4.1.2. Fluctuating wind simulation
In order to predict the response of an articulated tower to fluctuating wind forces, it is necessary to define the spectrum of wind fluctuations. Simiu and Leigh (1984) developed the spectrum for compliant platforms, expressions for which are
(nS_u (z,n))/〖U_*〗^2 ={a_1 f+b_1 f^2+d_1 f^3 }0

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