NON-LINEAR DYNAMICS OF AN ARTICULATED TOWER IN THE OCEAN

Transcription

1 Journal of Sound and Vibration (1996) 190(1), NON-LINEAR DYNAMICS OF AN ARTICULATED TOWER IN THE OCEAN P. BAR-AVI AND H. BENAROYA* Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, New Jersey 08855, U.S.A. (Received 28 June 1994 and in final form 3 May 1995 This paper presents studies on the response of an articulated tower in the ocean subjected to deterministic and random wave loading. The tower is modeled as an upright rigid pendulum with a concentrated mass at the top, having one angular-degree-of-freedom (planar motion) about a hinge with Coulomb friction, and viscous structural damping. In the derivation of the differential equation of motion, non-linear terms due to geometric (large angle) and fluid forces (drag and inertia) are included. The wave loading is derived using a modified Morison s equation to include current velocity, in which the velocity and acceleration of the fluid are determined along the instantaneous position of the tower, causing the equation of motion to be highly non-linear. Furthermore, since the differential equation s coefficients are time-dependent (periodic), parametric instability can occur depending on the system parameters such as wave height and frequency, buoyancy, and drag coefficient. The non-linear differential equation is then solved numerically using ACSL software. The response of the tower to deterministic wave loading is investigated and a stability analysis is performed (harmonic, subharmonic and superharmonic resonance). To solve the equation for random loading, the Pierson-Moskowitz power spectrum, describing the wave height, is first transformed into an approximate time history using Borgman s method with slight modification. The equation of motion is then solved, and the influence on the tower response of different parameter values such as buoyancy, initial conditions, wave height and frequency, and current velocity and direction, is investigated Academic Press Limited 1. REVIEW AND PROBLEM DEFINITION Compliant platforms such as articulated towers are economically attractive for deep water conditions because of their reduced structural weight compared to conventional platforms. The foundation of the tower does not resist lateral forces due to wind, waves and currents; instead, restoring moments are provided by a large buoyancy force, a set of guylines or a combination of both. These structures have a fundamental frequency well below the wave lower-bound frequency. As a result of the relatively large displacements, geometric non-linearity is an important consideration in the analysis of such a structure. The analysis and investigation of these kinds of problems can be divided into two major groups: deterministic and random wave and/or current loading. Work in this area is briefly reviewed in the next two subsections DETERMINISTIC LOADING Chakrabarti and Cotter [1] analyzed the motion of an articulated tower fixed by a universal joint having a single degree of freedom. They assumed linear waves, small *Corresponding author X/96/ $12.00/ Academic Press Limited

2 78 P. BAR-AVI AND H. BENAROYA perturbations about an equilibrium position, a linear drag force and that the wind, current and wave are collinear. The resulting equation of motion is I +B( )+D +C =M 0 e i( t), (1) where I is the total moment of inertia including added mass, B( ) is the non-linear drag term, D is the structural damping, C is the restoring moment due to buoyancy and M 0 is the wave moment. An analytical solution is then compared to experimental results, showing good agreement as long as the system is inertia dominant and not drag dominant. In a later paper, Chakrabarti and Cotter [2] investigated transverse motion, the motion perpendicular to the horizontal velocity. The tower pivot is assumed to have two angular degrees of freedom and is taken to be frictionless. It was also assumed that the motion is not coupled, so the inline solution is obtained (the same as in the previous paper), from which the relative velocity between the tower and the wave is obtained. The lift force (in the transverse direction) can then be obtained and the linear equation of motion solved analytically and compared to experimental results. The comparison shows good agreement, especially when the drag terms are small. Jain and Kirk [3] investigated the dynamic response of a double articulated offshore structure to waves and current loading. They assumed four-degrees-of-freedom, two angular for each link. The equations of motion were derived using Lagrange s equations. In deriving the equations of motion the following assumptions were made: drag and inertia forces tangent to the tower are negligible, and the wave and current velocities are evaluated at the upright position (small angles assumption). The linearized equations were solved to find the natural frequencies of the system and then numerically solved to find the response due to collinear and non-collinear current and wave velocities. They found that when the wave and the current velocities are collinear, the response of the top is sinusoidal, while for non-collinear velocities the response is a complex three dimensional whirling oscillation. Thompson et al. [4] investigated the motions of an articulated mooring tower. They modeled the structure as a bilinear oscillator which consists of two linear oscillators having different stiffnesses for each half cycle, mx +cx +(k 1, k 2 )x=f 0 sin t, (2) where k 1, k 2 are the stiffnesses for x 0 and x 0, respectively. The equation is solved numerically for different spring ratios and, as expected, harmonics and subharmonic resonances appeared in the response. A comparison between the response and experimental results of a reduced-scale model showed good agreement in the main phenomenon. Choi and Lou [5] have investigated the behavior of an articulated offshore platform. They modeled it as an upright pendulum having one-degree-of-freedom, with linear springs at the top having different stiffnesses for positive and negative displacements (bilinear oscillator). The equation of motion is simplified by expanding non-linear terms into a power series and retaining only the first two terms. They assumed that the combined drag and inertia moment is a harmonic function. The discontinuity in the stiffness is assumed to be small, and thus replaced by an equivalent continuous function using a least-squares method to get the Duffing equation I +c +k 1 +k 2 2 +k 3 3 =M 0 cos t, (3) where k 1, k 2, k 3 are spring constants depending on buoyancy, gravity and the mooring lines. The equation of motion is solved analytically and numerically, and stability analysis is performed. The analytical solution agrees very well with the numerical solution. The

3 ARTICULATED TOWER DYNAMICS 79 main results of their analyses are that as damping decreases, jump phenomena and higher subharmonics occur, and chaotic motion occurs only for large waves and near the first subharmonic (excitation frequency equals twice the fundamental frequency); the system is very sensitive to initial conditions. Seller and Niedzwecki [6] investigated the response of a multi-articulated tower in planar motion (upright multi-pendulum) to account for the tower flexibility. The restoring moments (buoyancy and gravity) were taken as linear rotational springs between each link, although the authors state that non-linear springs are more adequate for this model. Each link is assumed to have a different cross section and density. The equations of motion are derived using Lagrange s equations, in which the generalized co-ordinates are the angular deflections of each link. The equations in matrix form are [M]{ }+[K]{ }={Q}, (4) where [M] is a mass matrix that includes the actual mass of the link and added mass terms, while the stiffness matrix [K] includes buoyancy and gravity effects. Damping and drag forces are not included in the model. The homogeneous equations for a tri-articulated tower are numerically solved to study the effects of different parameters, such as link length, material density and spring stiffness, on the natural frequency of the system. Gottlieb et al. [7] analyzed the non-linear response of a single degree of freedom articulated tower. In the derivation of the equation, the expressions for the buoyancy moment arm, added mass term, and drag and inertia moments were evaluated along the stationary upright tower position and not at the instantaneous position of the tower. The governing equation is of the form + +R( )=M(, t), (5) where R( )= sin and is linear function of buoyancy and gravity, M(, t) is the drag moment. Approximated harmonic and subharmonic solutions are derived using a finite Fourier series expansion, and stability analysis is performed by a Lyapunov function approach. The solution shows a jump phenomenon in the primary and secondary resonances RANDOM LOADING Muhuri and Gupta [8] investigated the stochastic stability of a buoyant platform. They used a linear single-degree-of-freedom model, x +2cx +(1+G(t))x=0, (6) where x is the displacement, c is the damping coefficient and G(t) is a stochastic time-dependent function due to buoyancy. It is assumed that G(t) is a narrow-band random process with zero-mean. A criterion for the mean square stability is obtained from which the following results are found: for c 1 the system is always stable, and for 0 c 1 there are regions of stability and instability. Datta and Jain [9] investigated the response of an articulated tower to random wave and wind forces. In the derivation of the single-degree-of-freedom equation of motion the tower is discretized into n elements having appropriate masses, volumes and areas lumped at the nodes, with viscous damping. The equation of motion is I(1+ (t)) +c +R(1+ (t)) =F(t), (7) where I (t) is the time varying added mass term, R (t) is the time varying buoyancy moment and F(t) is the random force due to wave and wind. The Pierson-Moskowitz spectrum is assumed for the wave height and Davenport s spectrum is assumed for the

4 80 P. BAR-AVI AND H. BENAROYA wind velocity. The equation is solved in the frequency domain using an iterative method, which requires that the deflection angle (t) and the forcing function F(t) be decomposed into Fourier series. The coefficients of the sin and cos are then found iteratively. From their parametric study, they concluded the following: 1. Non-linearities such as large displacements and drag force do not influence the response when only wind force is considered. 2. The random wind forces result in higher responses than do wave forces. 3. The r.m.s. response due only to wind forces varies in a linear fashion with the mean wind velocity. In a later paper, Jain and Datta [10] used the same equation and the same method of solution to investigate the response due to random wave and current loading. The wave loadings (drag, inertia and buoyancy) are evaluated using numerical integration. The following results were obtained from the parametric study: 1. The dynamic response is very small since its fundamental frequency is well below the wave s fundamental frequency. 2. Non-linear effects (drag force, variable buoyancy) have considerable influence on the response. 3. Current velocity has a large influence on the response. Hanna et al. [11] analyzed the non-linear dynamics of a guyed tower platform. The tower is represented by a lumped parameter model consisting of discrete masses. Each mass has three-degrees-of-freedom, two translations and one rotation about the vertical axis. The external forces on the structure are approximated by concentrated forces and torques at the nodal points. The equation of motion is [M]{ü}+[C]{u }+[K(u)]{u}={P(t, u, u )}, (8) where [M] is the total mass matrix including added mass terms, [C] is the structural damping matrix assumed to be proportional to the mass matrix and [K(u)] is the total non-linear stiffness matrix that includes mooring lines effects, soil stiffness and geometric stiffness. {P(t, u, u )} is the non-linear dynamic load vector due to wave, current and wind. The equation is then solved using normal mode superposition and the response is calculated at each time step. This method is good only if the non-linearities are not large. Deterministic and random loading are considered. The solution shows insignificant flexure modes while the torsional one has a noticeable effect on the deck rotational response. Wilson and Orgill [12] presented a study which deals with the methodology for selecting the parameters for the best cable mooring array. The idea was to find a cable configuration so that the tower s r.m.s. deflection is minimized. The tower was assumed rigid with a pivot at the sea floor. Only planar motion was assumed. The equation of motion was derived and forces due to wind, wave, and six cables attached to the tower were considered. The optimization problem was formulated and solved for normal operating conditions and then for storm conditions. They showed that a stiff cable array is needed for normal condition while a softer system is preferred for storm conditions. Kanegaonkar and Haldar [13] investigated the non-linear random vibration of a guyed tower. They included non-linearities due to guyline stiffnesses, geometry, load and material. The simplified planner equation of motion is I +c +K +k 1 3 =M(t), (9) where K is a spring constant depending on buoyancy, gravity and guyline horizontal stiffness, and k 1 is a constant depending on the guyline vertical stiffness. M(t) is the random

5 ARTICULATED TOWER DYNAMICS 81 wave loading. The equation is then solved numerically where the wave height is defined by the Pierson-Moskowitz spectrum. It was seen that the response is non-gaussian for significant wave heights greater than 5 m. Gerber and Engelbrecht [14] investigated the response of an articulated mooring tower to irregular seas. It is an extension of earlier work done by Thompson et al. [4]. The tower is modeled as a bilinear oscillator, that is, a linear oscillator with different stiffnesses for positive and negative deflections, mx +cx +(k 1, k 2 )x=f(t). (10) The random forcing function F(t) is assumed to be the sum of a large number of harmonic components, each at different frequencies, a procedure similar to that proposed by Borgman [15]. The equation is then solved analytically, since it is linear for each half cycle. The solution is obtained for different cases: linear oscillator (both stiffnesses are the same), bilinear oscillator, and for the case of impact oscillator (a rigid cable) in which oscillation can occur only in one half of the cycle. For future study they suggest inclusion of non-linear stiffness and/or using a different spectrum to describe the wave height PROBLEM DEFINITION In this paper, the planer response of an articulated tower submerged in the ocean is investigated. The non-linear differential equation of motion is derived, including non-linearities due to geometry, Coulomb damping, drag force, added mass, and buoyancy. All forces/moments are evaluated at the instantaneous position of the tower and, therefore, they are time-dependent and highly non-linear. The equation is then numerically solved using ACSL Advanced Continuous Simulation Language [16], a software language, for deterministic and random wave loading using the Pierson-Moskowitz wave height spectrum. Harmonic, superharmonic, and subharmonic solutions for deterministic wave heights are obtained. The response to random wave heights for different significant wave heights is then investigated, the influence of Coulomb damping and current velocity and direction on the response is analyzed, and chaotic regimes of behavior are identified. The distinctions between this study and the literature of which we are aware are that: a sound and exact derivation of the non-linear equation of motion is provided; all terms in the governing differential equation of motion are analytically derived; Coulomb friction in the tower hinge is added; usage of ACSL for the numerical solution provides an easy way to modify parameters and perform sensitivity studies. 2. PROBLEM DESCRIPTION A schematic of the structure is shown in Figure 1. It consists of a tower submerged in the ocean having a concentrated mass at the top and one degree of freedom about the z-axis. The tower is subjected to wave and current loading. Two coordinate systems are used; one fixed (x, y, z) and the second attached to the tower (x', y', z'). All forces/moments are derived in the fixed coordinate system, which means that the tower rectilinear velocity is resolved into x, y coordinates. The motion of the tower is assumed to be only in plane (x, y) but the wave and current can be three dimensional. This problem has similarities to that of an inverted pendulum, but due to the presence of gravity waves, additional considerations are included: (1) A buoyancy force T 0, keeps the pendulum in a stable upright position.

6 82 P. BAR-AVI AND H. BENAROYA Figure 1. Model and coordinate frames. (2) Drag forces proportional to the square of the relative velocity between the fluid and the tower are considered. (3) Fluid inertia forces due to fluid acceleration and lift forces due to vortex shedding are part of the loading environment. (4) Fluid added mass is directly included in the inertia forces. (5) Current influence on the wave kinematics is considered. 3. EQUATIONS OF MOTION The equation of motion is derived using Lagrange s equation. The model consists of a single-degree-of-freedom: a rotation about the z-axis (planar motion). The equation is derived for large displacements under certain assumptions that are listed below ASSUMPTIONS The tower stiffness is infinite (EI= ): Coulomb friction in the pivot and viscous structural damping are included; the tower has a uniform mass per unit length, m and is of length l and diameter D; the tower diameter is much smaller than its length, D l; the tower is a slender smooth structure with uniform cross section; the end mass M is considered to be concentrated at the end of the tower (It has no volume); the tower length is greater than the fluid depth, but the dynamics is not limited to the case of M always being above the mean water level; the structure is statically stable due to the buoyancy force; the waves are linear having random height; Morison s fluid force coefficients C D and C M are constant; the center of mass (c.g.) of the tower is at its geometric center LAGRANGE S EQUATIONS The general form of Lagrange s equations is d dt K E q i K E + P E + D E =Q q i q i q qi, (11) i where K E is the kinetic energy, P E is the potential energy, D E is the dissipative energy and Q qi is the generalized force related to the q i generalized coordinate. The model consists of a single-degree-of-freedom, thus there is one generalized coordinate,. The generalized force in the relevant direction is derived using the principle

7 ARTICULATED TOWER DYNAMICS 83 Figure 2. Generalized force for. of virtual work by first deriving its general form assuming an external force having two components, F e = F x xˆ + F y yˆ. (12) From Figure 2 the virtual work done by F e due to a virtual displacement, F = F x x'[cos ( + ) cos ]+ F y x'[sin ( + ) sin ], (13) and using appropriate trigonometric identities, F = F x x'[cos cos sin sin cos ] + F y x'[sin cos +cos sin sin ]. Since virtual work is being considered, the virtual displacement 1, and x'=x/cos. Thus, the generalized force per unit length for the coordinate is F = F x x tan + F y x. (14) Finally, the generalized moment is evaluated by integrating F along the tower, Q = L 0 F dx= L 0 F xx tan + F y x dx, (15)

8 84 P. BAR-AVI AND H. BENAROYA where L is the projection, in the x direction, of the submerged part of the tower. It depends on the angle as follows: L= l cos, d+ ( y, t), if d+ ( y, t) l cos, if d+ ( y, t) l cos, (16) where ( y, t) is the wave height elevation to be defined later TOWER, WAVE AND CURRENT KINEMATICS To derive the equation of motion using Lagrange s equations requires that the kinetic, dissipative, and potential energies be evaluated, as well as the generalized forces. In this subsection, the tower absolute velocities, linear and angular, and accelerations are determined in the fixed coordinate system (x, y, z) attached to earth. Then in section 3.4 the fluid moments are evaluated Tower kinematics The tower is assumed to be oriented along a unit vector 1 with the directional cosines (see Figure (1)) so that the tower s radius vector R is 1=cos xˆ +sin yˆ, (17) R=x'1=x' cos xˆ +x' sin yˆ. (18) Its velocity V, relative to the wave s velocity, is found by taking the time-derivative of the radius vector, and the acceleration V dr =V= x' sin xˆ +x' cos yˆ, (19) dt by taking the time-derivative of the its velocity, dv dt =V = x'( sin + 2 cos )xˆ +x'[ cos 2 sin ]yˆ. (20) Since the equation is derived in the fixed coordinate system x, y, x'=x/cos giving, R=xxˆ +x tan yˆ, V= x tan xˆ +x yˆ, Finally, the tower total angular velocity is V = x( tan + 2 )xˆ +x( 2 tan )yˆ. (21) = zˆ. (22) Wave and current kinematics In this study linear wave theory is assumed; therefore the horizontal and vertical wave velocities are (Wilson [17]): u w = 1 cosh kx H cos (ky t), 2 sinh kd w and the respective accelerations: u w= 1 cosh kx 2 H 2 sin (ky t), sinh kd w w= 1 2 sinh kx H sin (ky t), (23) sinh kd w= 1 sinh kx 2 H 2 cos (ky t), (24) sinh kd

9 ARTICULATED TOWER DYNAMICS 85 where H is the wave height, the wave frequency, k the wave number, and d the mean water level, which are related by 2 =gk tanh (kd). (25) Without losing generality it is assumed that the wave propagates in the y-direction so that the horizontal velocity u is in that direction, and w is in the x-direction, although it should be noted that random waves are not uni-directional but that is beyond the scope of this study. Current velocity magnitude is calculated assuming that it consists of two components (Issacson [18]): the tidal component, U t c, and the wind-induced current U w c. If both components are known at the water surface, the vertical distribution of the current velocity U c (x) may be taken as U c (x)=u t c (x/d) 1/7 +U w c (x/d). (26) The tidal current U t c at the surface can be obtained directly from the tide table, and the wind-driven current U w c at the surface is generally taken as 1 to 5% of the mean wind speed at 10 m above the surface. When current and waves coexist, the combined flow field should be used to determine the wave loads. Figure 3 shows a top view of the system. The influence of current velocity on the wave field is treated by applying wave theory in a reference frame which is fixed relative to the current velocity. For a current of magnitude U c propagating in a direction relative to the direction of the wave propagation, the wave velocity defined as c 0 = 0 /k without current is modified and becomes c=c 0 +U c cos, =ck. (27) The velocities used to determine wave loads are the vectorial sum of the wave and current velocities: w=w w, u=u w +U c cos, (28) where w and u are the total velocities in x, y directions, respectively. Figure 3. Wave and current fields.

10 86 P. BAR-AVI AND H. BENAROYA To consider geometrical non-linearities, the velocities and accelerations are evaluated at the instantaneous position of the tower. Replacing y=x tan in the velocity and acceleration expressions (equations (23) and (24)) yields velocities and accelerations u= 1 2 w= 1 2 sinh kx H sin (kx tan t), sinh kd H cosh kx sinh kd cos (kx tan t)+u c cos, (29) w = 1 2 H + u = 1 2 H + kx sinh kx cos (kx tan t), cos 2 sinh kd kx cosh kx sin (kx tan t). (30) cos 2 sinh kd The influence of current on the wave height depends on the manner in which the waves propagate onto the current field. An approximation to the wave height in the presence of current is given by Isaacson [18], H=H 0 2/( + 2 ), (31) where H 0, H are the wave heights in the absence and presence of current respectively, and is = 1+(4U c /c 0 )cos, for (4U c /c 0 ) cos 1. (32) 3.4. EXTERNAL FLUID FORCES AND MOMENTS ACTING ON THE TOWER Figure 4 depicts the external forces acting on the tower: T 0 is a vertical buoyancy force; F fl are fluid forces due to drag, inertia, added mass and vortex shedding; Mg, mlg are the forces due to gravity. These forces and moments are described and developed next. Figure 4. External forces acting on the tower.

11 ARTICULATED TOWER DYNAMICS Buoyancy moment The restoring moment is achieved via the buoyancy force M b =T 0 l b. (33) T 0 is the buoyancy force and l b is its moment arm; both are time-dependent, where T 0 = gv 0 = g (D 2 /4)L s. (34) V 0 is the volume of the submerged part of the tower, is the fluid density and L s, which is the length of the submerged part of the tower, is L s =[d+ ( y, t)]/cos, (35) where ( y, t) is the wave height elevation evaluated at the instantaneous position of the tower and at x=d with y=d tan, (, t)= 1 2 H cos (kd tan t+ ). (36) The buoyant force acts at the center of mass of the submerged part of the tower. If the tower is assumed to be of cylindrical cross-section then the center of mass in the x', y' coordinates is lb y ' = D2 tan 16L 2, s lb x ' = 1 2 L s+ D2 tan 32L 2. (37) s Transforming to x, y coordinates the moment arm l b, l b = D2 tan 16L 2 cos + s 1 2 L s+ D2 tan 32L 2 sin, (38) s and finally the buoyancy generalized moment is then Mb= g D2 4 D2 32 tan2 (2 cos +sin )+ 1 2 (d+ ( y, t)) sin (39) 2 cos Morison s equation for wave forces In general, the fluid forces acting on a slender smooth tower are of two types, drag and inertia. The drag force is proportional to the square of the relative velocity between the fluid and the tower, and the inertia force is proportional to the fluid acceleration. The drag and inertia forces per unit length are approximated by Morison s equation, F fl =C D D 2 (V rel ) (V rel )+C M D2 4 (U w ), (40) where F fl is the fluid force per unit length normal to the tower. (V rel )=(U w V) is the relative velocity between the fluid and the tower in a direction normal to the tower, and (U w ) is the fluid acceleration normal to the tower. C D and C M are the drag and inertia coefficients, respectively. The relative velocity and fluid acceleration normal to the tower can be decomposed to their components as [V x rel, V y rel ] T =1 (U w V) 1, [U x w, U y w ] T =1 U w 1. (41)

12 88 P. BAR-AVI AND H. BENAROYA Using Morison s equation (40), the tower velocity equation (21), and fluid velocity and acceleration equations (29) and (30), the fluid force components are: the drag force, [F x D, F y D ] T =C D D 2 1 (U w V) 1 (1 (U w V) 1) and the inertia force, =C D D 2 (+ (Vx rel ) 2 +(V y rel ) 2 )[V x rel, V y rel ] T, (42) [F x I, F y I ] T =C M D2 4 (1 U w 1)=C M D2 4 [w, u ]T. (43) Vortex shedding moment The lift force F L due to vortex shedding is acting in a direction normal to the wave velocity vector and normal to the tower. In this section, since the motion is in plane, only shedding forces in the direction of the wave propagation due to current velocity perpendicular to the waves ( =90 ) are considered. Different models of lift force exist in the literature; see especially Billah [19]. Initially a simple model will be used, F L =[F x L, F y L ] T =C L D 2 cos s t 1 U T (1 U T ), (44) where U T, the vector of the maximum fluid velocity, is U T =[w m, u m ] T, (45) C L is the lift coefficient which depends on the Reynolds number Re, and s is the vortex shedding frequency that depends on the Keulegan-Carpenter number K as follows (Issacson [18]): C L =0 2, for Re ; s =2, for K=5 16. (46) Total fluid moment The moment due to the fluid forces (drag, inertia, and lift) is evaluated by substituting the sum of all fluid forces, defined by equation (47): F xfl =F x D+F x I +F x L, F yfl =F y D+F y I+F y L, (47) into equation (15). Therefore, the fluid moment Mfl is Mfl= L 0 Fx fl, Ffl y [ tan, 1] x dx, (48) which is evaluated using MAPLE and is not given here because of its length and complexity Added mass moment The fluid added mass force per unit length F ad is F ad =C A D2 V, (49) 4

13 ARTICULATED TOWER DYNAMICS 89 where C A =C M 1 is the added mass coefficient. Substituting expression (21) for the tower acceleration into equation (49) leads to expressions for the forces in the x, y directions, F x ad= C A D2 4 x( tan + 2 ), F y ad=c A D2 4 x( 2 tan ). (50) Substituting these added mass forces into the generalized moment equation (15), and integrating to yield the generalized moments due to fluid added mass, results in M ad= 1 3 C A D2 4 L3 (1+tan 2 ). (51) Friction moment The tower hinge is assumed to be governed by Coulomb friction. In this section this friction/damping moment is evaluated. The damping force is equal to the product of the normal force N and the coefficient of friction. It is assumed to be independent of the velocity, once the motion is initiated. Since the sign of the damping force is always opposite to that of the velocity, the differential equation of motion for each sign is valid only for a half cycle interval. The friction force is The normal force is F fr=n [sgn ( )]. (52) N= F x cos + F y sin, (53) where F x, F y are the total forces due to gravity, buoyancy and tower acceleration in the x, y directions, respectively. The fluid forces (drag, inertia and vortex shedding) do not influence the friction force since they are perpendicular to the tower. Thus, F x =T 0 F g +F x ac, F y =F y ac, (54) where T 0 is the buoyancy force given in equation (39), F g is the gravitational force, and the forces due to the tower acceleration F x ac, F y ac are F g =(ml+m)g, (55) F x ac= 1 8 C A D 2 L ml+m l 1 cos ( tan + 2 ), F y ac= 1 8 C A D 2 L ml+m l 1 cos ( + 2 tan ), (56)

14 90 P. BAR-AVI AND H. BENAROYA where l is the projection of the tower s length l in the x-direction, i.e., l =l cos. Assuming a hinge radius R h, and substituting for N, the generalized damping moment is M fr= 1 8 C A D 2 L 2 cos ml Ml 2 +(T 0 F g ) cos R h [sgn ( )]. (57) The only term remaining in the acceleration forces is the centrifugal one which is tangential to the tower, namely l LAGRANGE ENERGIES AND THEIR MOMENTS The dynamic moments M dy, those which are evaluated in the left hand side of Lagrange s equation (11), are found using the kinetic, dissipative, and potential energies, K E = 1 2 I z 2, D E =C 2, P E =( 1 2 ml+m)gl cos, (58) where C is the structural damping constant and I z is the moment of inertia of the tower about the z axis, given by I z = 1 3 ml 3 +Ml 2. (59) Substituting equations (59) and (22) into (58) leads to the expression for the kinetic energy, K E = 1 2 ( 1 3 ml 3 +Ml 2 ) 2. (60) The dissipative energy due to Coulomb friction is not velocity dependent, but the viscous damping is. Substituting the energies into the left hand side of equation (11) leads to M dy, M dy=( 1 3ml 3 +Ml 2 ) +C ( 1 2 ml+m)gl sin. (61) 3.6. GOVERNING EQUATION OF MOTION The governing non-linear differential equation of motion is found by equating the dynamic moment, M dy, to the applied moment, M ap, which is the sum of all external moments: M dy=m ap, (62) where the applied moment is found by adding equations (39), (51), (57), and (48): M ap=m fl M ad M b+m fr. (63) Substituting equations (61) and (63) into (62) and rearranging leads to the governing non-linear time dependent differential equation of motion for the tower: J eff +C = L 0 F fl x, Ffl y [ tan, 1] x dx M gb Mfr, (64) where J eff is the effective moment of inertia, J eff = 1 3 ml3 +Ml C A D2 4 L3 (1+tan 2 ), (65)

15 Property ARTICULATED TOWER DYNAMICS 91 TABLE 1 Tower s properties Value Tower s length 400 m Tower s diameter 15 m Tower s mass kg/m End mass kg Friction coefficient 0 1 to 0 4 Pivot radius 1 5 m and M gb is the moment due to gravity and buoyancy, Mgb= g D2 4 D2 32 tan2 (2 cos +sin )+ 1 2 d+ ( y, t) sin 2 cos 1 2 ml+m gl sin. (66) 4. NUMERICAL SOLUTION The governing non-linear differential equation of motion (64) is numerically solved using ACSL and the results are analyzed using MATLAB. The tower response to various waves and current is investigated. The analysis is performed for deterministic as well as for random wave heights. The physical parameters used in the simulation are shown in Tables 1 and RESPONSE FOR DETERMINISTIC WAVE HEIGHTS The non-linear differential equation for the single-degree-of-freedom system is solved for several cases: equilibrium position of the tower in the presence of current; fundamental frequency, and damping (drag, viscous and friction) effect; response to wave excitation; superharmonics, harmonic and subharmonics resonances; chaotic regions and influence of current velocity and direction Response in the absence of waves In this section, the free vibration of the tower, and the influence of current on the response are investigated. To do so, the wave velocities are set to zero. To find the Property TABLE 2 Fluid properties Value Mean water level 350 m Drag coefficient 0 6 to 1 0 Inertia coefficient 1 2 to 1 6 Lift coefficient 0 8 to 1 2 Water density 1025 kg/m 3 Wave frequency 0 03 to 1 rad/s Significant wave height 0 to 15 m Current velocity 2 m/s Structural damping 0 02

16 92 P. BAR-AVI AND H. BENAROYA Figure 5. Free vibration of the tower. fundamental frequency, the tower response to non-zero initial conditions with zero damping is found. Figure 5 describes the response of the tower for (t=0)=0 001 rad/s in the time and frequency domain. From the figure it can be seen that the fundamental frequency is n = rad/s=0 028 Hz. Calculating the frequency analytically using equation (67) yields n = rad/s=0 028 Hz, n = 2 1 M gb. (67) J eff Adding damping to the system (drag, friction or viscous) causes a decay in the response as can be seen from Figure 6. A typical decay for each damping mechanism is clearly seen: hyperbolic decay, proportional to a1 /t, for drag damping, linear decay, proportional to ( a 2 t), for Coulomb damping, and exponential decay, exp ( a 3 t), for viscous damping. a 1, a 2, a 3 are the decay constants for each damping mechanism. The response for the first two damping mechanisms consists of the fundamental frequency and its odd multipliers, as can be seen from the frequency domain figures. The reason for the odd multipliers is the fact that the drag and the Coulomb friction forces are non-linear and Figure 6. Time and frequency domain curves for damped free vibration: (a, b) C D=1; (c, d) =0 1; (e, f) =0 2.

17 ARTICULATED TOWER DYNAMICS 93 Figure 7. Damped, free vibration in the presence of current;, U c=1 m/s;, U c=2 m/s. anti-symmetric. On the other hand the response for viscous damping is linear and therefore only the fundamental frequency is seen. Figure 7 shows the damped, free vibration of the tower in the presence of current. The figure demonstrates that the higher the current velocity, the faster the decay to equilibrium. This is because one of the terms in the drag fluid force is proportional to (2C D U c cos ) which is similar to linear viscous damping. The influence of the current velocity on the equilibrium position is found from the steady state solution. Figure 8 describes the tower position in the presence of current velocities of U c =1 and 2 m/s, with drag (C D =1) and with friction damping ( =0 1). In both cases =0. Setting =0 and =0 in the non-linear governing equation and solving for with U c =1, 2 m/s leads to the following equilibrium positions: (U c =1)= rad and (U c =2)= rad, with simulations leading to the same results. As can be seen, the deflection angle for U c =2 m/s is four times the deflection angle for U c =1 m/s, and the reason is that the equilibrium position is proportional to U c cos U c cos Response in the presence of waves In this section, the tower s response to deterministic waves and current is investigated. Current direction, and super/subharmonic wave excitation is analyzed. The influence of the angle between the current and the wave propagation,, on the response is investigated next. Figure 9 shows the response of the tower for =0, 45 (Figure (a)) and 135, 180 (Figure (b)). Here, the wave height is H=1 m, =0 1 rad/s and U c =1 5 m/s. From the figure it is seen that the steady state response for and 180 Figure 8. Tower equilibrium position in the presence of currents with (a) drag only, C D=1; (b) friction only, =0 1;, U c=1 m/s;, U c=2 m/s.

18 94 P. BAR-AVI AND H. BENAROYA Figure 9. Influence of current direction: (a), =0 ; ---, =180 ; (b), =45 ; - --, =135. Figure 10. Response in (a) time and (b) frequency domain for current direction =90. has the same magnitude but opposite sign, and for =90 the equilibrium position is zero. This is because the steady state response is proportional to U c cos U c cos. For 0, the response has two frequencies, and 2, due to the vortex shedding model used in the analysis; see equation (46). The highest response is achieved when =90, as can be seen when comparing Figures 9 and 10. The reason is that an additional lift (transverse) force is added when the angle is not zero, and this force is maximum for =90. From the frequency domain response, the two excitation frequencies are clearly seen, =0 016 Hz and 2 =0 032 Hz. The response due to resonance wave excitation; superharmonics, harmonic and subharmonics is analyzed. Figure 11 shows the tower s response to wave excitation at Figure 11. Undamped response to subharmonic excitation =2 n : (a) time domain and (b) frequency response.

19 ARTICULATED TOWER DYNAMICS 95 Figure 12. Undamped response to subharmonic excitation =2 n with current velocity U c=2 m/s. (a) Time domain and (b) frequency response. about twice its fundamental frequency, the subharmonics. Here the wave height is H=10 m, the current velocity and all damping forces are set to zero. The figure demonstrates beating with high amplitudes, 0 4 rad, implying an unstable region. The frequency response includes the excitation frequency =0 056 Hz, the fundamental frequency =0 028 Hz and its multipliers. This is due to the system non-linearity, where subharmonics cause a response at the exciting frequency as well as in the fundamental frequency. The subharmonic response in the presence of current is shown in Figure 12. Comparing Figures 11 and 12 it is seen that the highest response is for zero current velocity. For U c =2 m/s, the response is much smaller. The reason is that the excitation frequency depends on the current velocity as shown by equation (68). For U c =0 m/s, the excitation frequency is exactly twice the fundamental frequency thus causing the highest response. The exciting frequency and the current velocity are related by = 0 (1+ 0 U c /g). (68) Adding drag, C D =0 6, to the system reduces the amplitude of the tower s response and eliminates the beating phenomenon, as can be seen from Figure 13 (time domain). In the frequency domain the exciting frequency, the fundamental frequency and its multipliers are as seen in Figure 11. The difference is that the amplitude of the fundamental frequency =0 028 Hz is lower due to the damping effect of the drag force. The response due to wave excitation at about the fundamental frequency and in the absence of damping is shown in Figure 14. The response is much higher than for subharmonic excitation. The beating phenomenon is clearly seen and the system jumps between two amplitudes. The frequency domain curve shows the fundamental frequency =0 028 Hz as well as its multipliers. Figure 13. Damped response to subharmonic excitation =2 n with C D=0 6; (a) Time domain and (b) frequency response.

20 96 P. BAR-AVI AND H. BENAROYA Figure 14. Undamped response to harmonic excitation = n. (a) Time domain and (b) frequency response. Figure 15. Undamped response to harmonic excitation = n for current velocity U c=2 m/s. (a) Time domain and (b) frequency response. Again, in the presence of current, the excitation frequency is modified and resulting in different response amplitudes, as can be seen in Figure 15. Here, the response is higher in the presence of a current velocity, U c =2 m/s, but the difference is not large when compared to Figure 12, since the region of instability around the fundamental frequency is wider than the one about the second harmonic and it can be seen the beating frequency (envelope frequency) has changed due to the modified excitation frequency. Damping with C D =0 6 has a stabilizing effect on the system. The amplitude is lower and the beating disappears as can be seen in Figure 16. Similar results are found when the excitation frequency is about one half of the fundamental frequency (superharmonic excitation), as can be seen from Figures 17, 18 and 19. Figure 16. Damped response to harmonic excitation = n with C D=0 6. (a) Time domain and (b) frequency response. Figure 17. Undamped response to superharmonic excitation = n /2. (a) Time domain and (b) frequency response.

21 ARTICULATED TOWER DYNAMICS 97 Figure 18. Undamped response to superharmonic excitation = n /2 with current velocity U c=2 m/s. (a) Time domain and (b) frequency response. Figure 19. Damped response to superharmonic excitation = n /2 with C D=0 6. (a) Time domain and (b) frequency response Quasiperiodic and chaotic behavior The response of the tower to an arbitrary frequency is investigated next. From a frequency sweep it is found that the response is mostly quasi-periodic, except for certain frequencies in which the response is chaotic. Figure 20 shows the time domain response, Figure 20. Quasi-periodic response: (a) time domain, (b) phase plane and (c) Poincaré map.

22 98 P. BAR-AVI AND H. BENAROYA Figure 21. Chaotic response: (a) Time domain, (b) phase plane and (c) Poincaré map. phase plane and Poincare map, for wave excitation =0 06 rad/s, H=10 m. The damping moments, initial conditions and current velocity are set to zero. From the phase plane the response looks chaotic but the Poincare map shows that the response is in fact quasi-periodic. A chaotic region is shown in Figure 21 where the response to a wave frequency of =0 03 rad/s is shown. From the Poincare map it is clearly seen that the response is chaotic since the points are scattered in an erratic fashion, unlike the quasi-periodic case. Figure 22 shows the Poincare map of the same response as in Figure 21 with additional damping of C D =0 2. The adjacent figure is a magnification of the response about zero. It describes a characteristic response of a chaotic region with damping in which the points are highly organized, as described by Moon [20]. The influence of initial conditions is shown in Figure 23. The left figure shows a Poincare map of a chaotic response with zero initial conditions. The right figure is also a Poincare map but with the following initial conditions: (t=0)=0 5 rad, (t=0)=0 05 rad/s. It is clearly seen in the latter case that the chaotic response has become quasi-periodic with larger deflection RESPONSE FOR RANDOM WAVE HEIGHTS In this section the tower response to random wave height excitation is investigated. The wave height distribution is generally expressed in the form of a power spectral density. For simulation of the response in the time domain, the wave height power spectrum is Figure 22. Influence of damping on the chaotic response: (a) full phase plane, (b) magnification around zero.

23 ARTICULATED TOWER DYNAMICS 99 Figure 23. Influence of initial condition on the chaotic response. (a) Chaotic motion with zero initial conditions, (b) quasi-periodic motion with non-zero initial conditions. transformed into a time history. This is accomplished using a method by Borgman [15], and Wilson [17]. The wave elevation ( y, t) can be expressed as The Pierson-Moskowitz spectrum for the wave height is where A 0 and B are constants defined by ( y, t)= 1 2 H cos (ky t+ ). (69) S ( )=(A 0 / 5 ) e B/ 4, (70) A 0 = g 2, B=3 11/H 2 s, (71) where H s is the significant wave height. Using Borgman s method, the wave elevation ( y, t) can be approximated by where the amplitude a, is constant and given by N ( y, t)=a cos (k n y n t+ n ), (72) n=1 a 2 =A 0 /4BN. (73) Figure 24. Influence of different wave heights on the tower response: (a) H s=4, (b) H s=9, (c) H s=15.

24 100 The partition frequencies n are P. BAR-AVI AND H. BENAROYA n =(B/[ln (N/n)+B/F 4 ]) 0 25, n=1, 2,..., N, (74) with F being the upper limit frequency in the Pierson-Moskowitz spectra. The wave loading on the tower is a function of wave velocity and acceleration which in these numerical studies have to be expressed as functions of the approximate wave elevation. Thus, in the expressions for wave velocity and acceleration, the following substitutions are made: H A 0 /4BN, n, k k n, n=1, 2,..., N. (75) For example, the horizontal velocity and acceleration will become N u= n=1 A 0 4BN n N u = A 0 4BN n=1 n n+ cosh k n x sinh k n d cos (k n x tan n t), k n x cos cosh k n x 2 sinh k n d sin (k n x tan n t). (76) Next, the influences on the response of different significant wave heights, damping (drag, viscous and Coulomb) mechanisms, and different current velocities are investigated Effect of significant wave height Figure 24 compares the tower response for three different significant wave heights: H s =4, 9, 15 m. It can be seen from the figure that lower significant wave height results in lower response. The reasons are: the larger the significant wave height the higher the input; as H s increases, the mean wave frequency approaches the fundamental frequency of the tower. To emphasize this second reason, the tower fundamental frequency is changed to about n =0 8 rad/s and the responses due to H s =4, 9 m are calculated and shown in Figure 25, from which it can be seen that the response for H s =4 m beats, almost as in harmonic excitation, and is higher than the response for H s =9 m although the spectral peak is seven times smaller Effect of current The influence of current velocity on the random response is shown in Figure 26 for the case of collinear wave and current i.e., =0. Here the significant wave height is H s =9 m and the drag coefficient is C D =0 2. It is seen that the higher the current velocity the smaller the response. The reason is that current in the direction of the wave s propagation tends to lower the wave height (see Isaacson [18]). Figure 25. Influence of significant wave height with n=0 8 rad/s. (a) H s=4, (b) H s=9.

25 ARTICULATED TOWER DYNAMICS 101 Figure 26. Influence of current on the random response for =0. (a) U c=0 (m/s), (b) U c=1 (m/s), (c) U c=3 (m/s) Effect of damping Different damping mechanisms have different effects on the tower s response. Figure 27 shows the separate influence of drag, viscous, and Coulomb damping on the response. The significant wave height is H s =9 m and the damping constants are C D =0 6, =0 02, =0 1, respectively. As can be seen from the figure, although all damping mechanisms cause the transient response to vanish after about 25 seconds, the steady state response in the presence of Coulomb damping is almost one order of magnitude lower. 5. DISCUSSION AND SUMMARY The non-linear differential equation of motion for an articulated tower submerged in the ocean is derived including Coulomb and viscous damping. Geometric as well as force non-linearities are included in the derivation. The fluid forces, drag, inertia and lift due to waves and current, are determined at the instantaneous position of the tower, adding Figure 27. Influence of different damping mechanisms on the response: (a) C D=0 6, (b) =0 02, (c) =0 1.

26 102 P. BAR-AVI AND H. BENAROYA to the non-linearities of the equation. The equation is solved numerically using ACSL for deterministic and random wave loading. The equilibrium position ( = =0) depends on the current velocity and direction, U c cos U c cos, and in the absence of drag the equilibrium position is =0. The current s direction affects the response greatly. For the same current velocity, the highest response is when the direction is perpendicular to the wave propagation, since the lift force is then maximum. The response of the tower to harmonic wave excitation at its natural frequency, and half and twice its natural frequency demonstrates beating, where the amplitude varies between two extremes. This beating is due to the non-linear behavior of the system. Coulomb damping reduces the beating phenomenon and the response amplitude, so it has a stabilizing effect on the system. The system response depends on the wave frequency and amplitude. For most frequencies the response is quasi-periodic, but there are certain frequencies at which the system exhibits chaotic behavior. To solve the equation for random wave loading, the Pierson-Moskowitz spectrum that describes the wave height distribution was first transformed into a time history. The equation was solved for three significant wave heights. For significant wave heights of 9 and 15 m, the response was larger than that for 4 m, since in the former the tower s natural frequency is closer to the frequencies where most of the energy is located. Damping of any kind (drag, friction and viscous) stabilizes the system with the greatest effect due to friction damping. Notice that in order to reduce stresses in the structure, the friction moment has to be low enough so that the tower can comply with the wave loading. Current velocity tends to lower the response as long as 90, because it lowers the wave height. A more realistic model having two angular-degrees-of-freedom is being analyzed at the present time. The response due to wave, current (collinear and otherwise), vortex shedding loading and earth rotation is investigated and results will be published in the near future. Work is also proceeding on an elastic articulated tower. ACKNOWLEDGMENTS This work was supported by the Office of Naval Research Grant no. N The authors are grateful for this support from ONR and thank Program Manager Dr T. Swean for his interest in our work. REFERENCES 1. S. K. CHAKRABARTI and D. C. COTTER 1979 Journal of the Waterway, Port, Coastal and Ocean Division, ASCE 105, Motion analysis of articulated tower. 2. S. K. CHAKRABARTI and D. C. COTTER 1980 Journal of the Waterway, Port, Coastal and Ocean Division, ASCE 107, Transverse motion of articulated tower. 3. R. K. JAIN and C. L. KIRK 1981 Journal of Energy Resources Technology 103, Dynamic response of a double articulated offshore loading structure to non-collinear waves and current. 4. J. M. T. THOMPSON, A. R. BOKAIAN and R. CHAFFAI 1984 Journal of Energy Resources Technology 106, Stochastic and chaotic motions of compliant offshore structures and articulated mooring towers. 5. H. S. CHOI and J. Y. K. LOU 1991 Applied Ocean Research 12, Non-linear behaviour of an articulated offshore loading platform. 6. L. L. SELLER and J. M. NIEDZWECKI 1992 Ocean Engineering 19, Response characteristics of multi-articulated offshore towers. 7. O. GOTTLIEB, C. S. YIM and R. T. HUDSPETH 1992 International Journal of Offshore and Polar Engineering 2, Analysis of non-linear response of an articulated tower.