Journal of Sound and Vibration (1996) 190(1), 77–103

NON-LINEAR DYNAMICS OF AN ARTICULATEDTOWER IN THE OCEAN

P. B-A H. B*

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 subjectedto deterministic and random wave loading. The tower is modeled as an upright rigidpendulum 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. Inthe 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 derivedusing a modified Morison’s equation to include current velocity, in which the velocity andacceleration 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 differentialequation’s coefficients are time-dependent (periodic), parametric instability can occurdepending on the system parameters such as wave height and frequency, buoyancy, anddrag coefficient. The non-linear differential equation is then solved numerically using‘ACSL’ software. The response of the tower to deterministic wave loading is investigatedand a stability analysis is performed (harmonic, subharmonic and superharmonicresonance). To solve the equation for random loading, the Pierson-Moskowitz powerspectrum, describing the wave height, is first transformed into an approximate time historyusing 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, isinvestigated.

7 1996 Academic Press Limited

1. REVIEW AND PROBLEM DEFINITION

Compliant platforms such as articulated towers are economically attractive for deep waterconditions 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 ora combination of both. These structures have a fundamental frequency well below the wavelower-bound frequency. As a result of the relatively large displacements, geometricnon-linearity is an important consideration in the analysis of such a structure. The analysisand 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 brieflyreviewed in the next two subsections.

1.1.

Chakrabarti and Cotter [1] analyzed the motion of an articulated tower fixed by auniversal joint having a single degree of freedom. They assumed linear waves, small

*Corresponding author.

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0022–460X/96/060077+27 $12.00/0 7 1996 Academic Press Limited

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perturbations about an equilibrium position, a linear drag force and that the wind, currentand wave are collinear. The resulting equation of motion is

Ic� +B(c� )+Dc� +Cc=M0 ei(a−bt), (1)

where I is the total moment of inertia including added mass, B(c� ) is the non-linear dragterm, Dc� is the structural damping, Cc is the restoring moment due to buoyancy and M0

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 motionperpendicular to the horizontal velocity. The tower pivot is assumed to have two angulardegrees of freedom and is taken to be frictionless. It was also assumed that the motionis not coupled, so the inline solution is obtained (the same as in the previous paper), fromwhich the relative velocity between the tower and the wave is obtained. The lift force (inthe transverse direction) can then be obtained and the linear equation of motion solvedanalytically and compared to experimental results. The comparison shows goodagreement, especially when the drag terms are small.

Jain and Kirk [3] investigated the dynamic response of a double articulated offshorestructure to waves and current loading. They assumed four-degrees-of-freedom, twoangular 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 inertiaforces tangent to the tower are negligible, and the wave and current velocities are evaluatedat the upright position (small angles assumption). The linearized equations were solved tofind the natural frequencies of the system and then numerically solved to find the responsedue to collinear and non-collinear current and wave velocities. They found that when thewave and the current velocities are collinear, the response of the top is sinusoidal, whilefor non-collinear velocities the response is a complex three dimensional whirlingoscillation.

Thompson et al. [4] investigated the motions of an articulated mooring tower. Theymodeled the structure as a bilinear oscillator which consists of two linear oscillators havingdifferent stiffnesses for each half cycle,

mx+cx+(k1 , k2 )x=F0 sin vt, (2)

where k1, k2 are the stiffnesses for xq0 and xQ0, respectively. The equation is solvednumerically for different spring ratios and, as expected, harmonics and subharmonicresonances appeared in the response. A comparison between the response andexperimental results of a reduced-scale model showed good agreement in the mainphenomenon.

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 springsat the top having different stiffnesses for positive and negative displacements (bilinearoscillator). The equation of motion is simplified by expanding non-linear terms into apower series and retaining only the first two terms. They assumed that the combined dragand inertia moment is a harmonic function. The discontinuity in the stiffness is assumedto be small, and thus replaced by an equivalent continuous function using a least-squaresmethod to get the Duffing equation

Iu� +cu� +k1u+k2u2+k3u

3=M0 cos vt, (3)

where k1 , k2 , k3 are spring constants depending on buoyancy, gravity and the mooringlines. The equation of motion is solved analytically and numerically, and stability analysisis performed. The analytical solution agrees very well with the numerical solution. The

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main results of their analyses are that as damping decreases, jump phenomena and highersubharmonics occur, and chaotic motion occurs only for large waves and near the firstsubharmonic (excitation frequency equals twice the fundamental frequency); the system isvery sensitive to initial conditions.

Seller and Niedzwecki [6] investigated the response of a multi-articulated tower in planarmotion (upright multi-pendulum) to account for the tower flexibility. The restoringmoments (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. Eachlink is assumed to have a different cross section and density. The equations of motion arederived using Lagrange’s equations, in which the generalized co-ordinates are the angulardeflections of each link. The equations in matrix form are

[M]{u� }+[K]{u}={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 dragforces are not included in the model. The homogeneous equations for a tri-articulatedtower are numerically solved to study the effects of different parameters, such as linklength, 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 freedomarticulated tower. In the derivation of the equation, the expressions for the buoyancymoment arm, added mass term, and drag and inertia moments were evaluated along thestationary upright tower position and not at the instantaneous position of the tower. Thegoverning equation is of the form

u� +gu� +R(u)=M(u� , t), (5)

where R(u)=a sin u and a is linear function of buoyancy and gravity, M(u� , t) is the dragmoment. Approximated harmonic and subharmonic solutions are derived using a finiteFourier series expansion, and stability analysis is performed by a Lyapunov functionapproach. The solution shows a jump phenomenon in the primary and secondaryresonances.

1.2.

Muhuri and Gupta [8] investigated the stochastic stability of a buoyant platform. Theyused 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 stochastictime-dependent function due to buoyancy. It is assumed that G(t) is a narrow-bandrandom process with zero-mean. A criterion for the mean square stability is obtained fromwhich the following results are found: for cq1 the system is always stable, and for 0QcQ1there are regions of stability and instability.

Datta and Jain [9] investigated the response of an articulated tower to random waveand wind forces. In the derivation of the single-degree-of-freedom equation of motion thetower is discretized into n elements having appropriate masses, volumes and areas lumpedat the nodes, with viscous damping. The equation of motion is

I(1+b(t))u� +cu� +R(1+n(t))u=F(t), (7)

where Ib(t) is the time varying added mass term, Rn(t) is the time varying buoyancymoment and F(t) is the random force due to wave and wind. The Pierson-Moskowitzspectrum is assumed for the wave height and Davenport’s spectrum is assumed for the

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wind velocity. The equation is solved in the frequency domain using an iterative method,which requires that the deflection angle u(t) and the forcing function F(t) be decomposedinto Fourier series. The coefficients of the sin and cos are then found iteratively. From theirparametric study, they concluded the following:

1. Non-linearities such as large displacements and drag force do not influence the responsewhen 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 ofsolution to investigate the response due to random wave and current loading. The waveloadings (drag, inertia and buoyancy) are evaluated using numerical integration. Thefollowing results were obtained from the parametric study:

1. The dynamic response is very small since its fundamental frequency is well below thewave’s fundamental frequency.

2. Non-linear effects (drag force, variable buoyancy) have considerable influence on theresponse.

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 toweris represented by a lumped parameter model consisting of discrete masses. Each mass hasthree-degrees-of-freedom, two translations and one rotation about the vertical axis. Theexternal forces on the structure are approximated by concentrated forces and torques atthe nodal points. The equation of motion is

[M]{u}+[C]{u}+[K(u)]{u}={P(t, u, u)}, (8)

where [M] is the total mass matrix including added mass terms, [C] is the structuraldamping matrix assumed to be proportional to the mass matrix and [K(u)] is the totalnon-linear stiffness matrix that includes mooring lines effects, soil stiffness and geometricstiffness. {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 iscalculated 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 flexuremodes 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 selectingthe parameters for the best cable mooring array. The idea was to find a cable configurationso that the tower’s r.m.s. deflection is minimized. The tower was assumed rigid with a pivotat the sea floor. Only planar motion was assumed. The equation of motion was derivedand forces due to wind, wave, and six cables attached to the tower were considered. Theoptimization problem was formulated and solved for normal operating conditions andthen for storm conditions. They showed that a stiff cable array is needed for normalcondition while a softer system is preferred for storm conditions.

Kanegaonkar and Haldar [13] investigated the non-linear random vibration of a guyedtower. They included non-linearities due to guyline stiffnesses, geometry, load andmaterial. The simplified planner equation of motion is

Iu� +cu� +Ku+k1u3=M(t), (9)

where K is a spring constant depending on buoyancy, gravity and guyline horizontalstiffness, and k1 is a constant depending on the guyline vertical stiffness. M(t) is the random

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wave loading. The equation is then solved numerically where the wave height is definedby the Pierson-Moskowitz spectrum. It was seen that the response is non-Gaussian forsignificant wave heights greater than 5 m.

Gerber and Engelbrecht [14] investigated the response of an articulated mooring towerto irregular seas. It is an extension of earlier work done by Thompson et al. [4]. The toweris modeled as a bilinear oscillator, that is, a linear oscillator with different stiffnesses forpositive and negative deflections,

mx+cx+(k1 , k2 )x=F(t). (10)

The random forcing function F(t) is assumed to be the sum of a large number of harmoniccomponents, each at different frequencies, a procedure similar to that proposed byBorgman [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 oscillationcan occur only in one half of the cycle. For future study they suggest inclusion of non-linearstiffness and/or using a different spectrum to describe the wave height.

1.3.

In this paper, the planer response of an articulated tower submerged in the ocean isinvestigated. The non-linear differential equation of motion is derived, includingnon-linearities due to geometry, Coulomb damping, drag force, added mass, andbuoyancy. All forces/moments are evaluated at the instantaneous position of the towerand, therefore, they are time-dependent and highly non-linear. The equation is thennumerically solved using ‘ACSL’—Advanced Continuous Simulation Language [16], asoftware language, for deterministic and random wave loading using the Pierson-Moskow-itz wave height spectrum. Harmonic, superharmonic, and subharmonic solutions fordeterministic wave heights are obtained. The response to random wave heights for differentsignificant wave heights is then investigated, the influence of Coulomb damping andcurrent velocity and direction on the response is analyzed, and chaotic regimes of behaviorare 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 termsin the governing differential equation of motion are analytically derived; Coulomb frictionin the tower hinge is added; usage of ‘ACSL’ for the numerical solution provides an easyway 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 inthe ocean having a concentrated mass at the top and one degree of freedom u about thez-axis. The tower is subjected to wave and current loading. Two coordinate systems areused; one fixed (x, y, z) and the second attached to the tower (x', y', z'). Allforces/moments are derived in the fixed coordinate system, which means that the towerrectilinear velocity is resolved into x, y coordinates. The motion of the tower is assumedto 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 presenceof gravity waves, additional considerations are included:

(1) A buoyancy force T0 , keeps the pendulum in a stable upright position.

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Figure 1. Model and coordinate frames.

(2) Drag forces proportional to the square of the relative velocity between the fluid andthe tower are considered.

(3) Fluid inertia forces due to fluid acceleration and lift forces due to vortex shedding arepart 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 asingle-degree-of-freedom: a rotation u about the z-axis (planar motion). The equation isderived for large displacements under certain assumptions that are listed below.

3.1.

The tower stiffness is infinite (EI=a): Coulomb friction in the pivot and viscousstructural damping are included; the tower has a uniform mass per unit length, m and isof length l and diameter D; the tower diameter is much smaller than its length, D�l; thetower is a slender smooth structure with uniform cross section; the end mass M isconsidered to be concentrated at the end of the tower (It has no volume); the tower lengthis greater than the fluid depth, but the dynamics is not limited to the case of M alwaysbeing above the mean water level; the structure is statically stable due to the buoyancyforce; the waves are linear having random height; Morison’s fluid force coefficients CD andCM are constant; the center of mass (c.g.) of the tower is at its geometric center.

3.2. ’

The general form of Lagrange’s equations is

ddt 01KE

1qi 1−1KE

1qi+

1PE

1qi+

1DE

1qi=Qqi , (11)

where KE is the kinetic energy, PE is the potential energy, DE is the dissipative energy andQqi is the generalized force related to the qi generalized coordinate.

The model consists of a single-degree-of-freedom, thus there is one generalizedcoordinate, u. The generalized force in the relevant direction is derived using the principle

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Figure 2. Generalized force for u.

of virtual work by first deriving its general form assuming an external force having twocomponents,

s Fe=s Fx x+s Fy y. (12)

From Figure 2 the virtual work done by Fe due to a virtual displacement du,

Fu du=s Fxx'[cos (u+du)−cos u]+s Fyx'[sin (u+du)−sin u], (13)

and using appropriate trigonometric identities,

Fu du=s Fxx'[cos u cos du−sin u sin du−cos u]

+s Fyx'[sin u cos du+cos u sin du−sin u].

Since virtual work is being considered, the virtual displacement du�1, and x'=x/cos u.Thus, the generalized force per unit length for the u coordinate is

Fu=−s Fxx tan u+s Fyx. (14)

Finally, the generalized moment is evaluated by integrating Fu along the tower,

Qu=gL

0

Fu dx=gL

0 0−s Fxx tan u+s Fyx1 dx, (15)

. - . 84

where L is the projection, in the x direction, of the submerged part of the tower. It dependson the angle u as follows:

L=6l cos u,d+h( y, t),

if d+h( y, t)ql cos u,if d+h( y, t)Ql cos u,

(16)

where h( y, t) is the wave height elevation to be defined later.

3.3. ,

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 thissubsection, the tower absolute velocities, linear and angular, and accelerations aredetermined in the fixed coordinate system (x, y, z) attached to earth. Then in section 3.4the fluid moments are evaluated.

3.3.1. Tower kinematicsThe tower is assumed to be oriented along a unit vector 1 with the directional cosines

(see Figure (1))

1=cos ux+sin uy, (17)

so that the tower’s radius vector R is

R=x'1=x' cos ux+x' sin uy. (18)

Its velocity V, relative to the wave’s velocity, is found by taking the time-derivative ofthe radius vector,

dRdt

=V=−x'u� sin ux+x'u� cos uy, (19)

and the acceleration V� by taking the time-derivative of the its velocity,

dVdt

=V� =−x'(u� sin u+u� 2 cos u)x+x'[u� cos u−u� 2 sin u]y. (20)

Since the equation is derived in the fixed coordinate system x, y, x'=x/cos u giving,

R=xx+x tan uy, V=−xu� tan ux+xu� y,

V� =−x(u� tan u+u� 2)x+x(u� −u� 2 tan u)y. (21)

Finally, the tower total angular velocity is

V=u� z. (22)

3.3.2. Wave and current kinematicsIn this study linear wave theory is assumed; therefore the horizontal and vertical wave

velocities are (Wilson [17]):

uw=12

Hvcosh kxsinh kd

cos (ky−vt), ww=12

Hvsinh kxsinh kd

sin (ky−vt), (23)

and the respective accelerations:

uw=12

Hv2 cosh kxsinh kd

sin (ky−vt), ww=−12

Hv2 sinh kxsinh kd

cos (ky−vt), (24)

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where H is the wave height, v the wave frequency, k the wave number, and d the meanwater level, which are related by

v2=gk tanh (kd). (25)

Without losing generality it is assumed that the wave propagates in the y-direction sothat the horizontal velocity u is in that direction, and w is in the x-direction, although itshould be noted that random waves are not uni-directional but that is beyond the scopeof this study.

Current velocity magnitude is calculated assuming that it consists of two components(Issacson [18]): the tidal component, Ut

c , and the wind-induced current Uwc . If both

components are known at the water surface, the vertical distribution of the current velocityUc (x) may be taken as

Uc (x)=Utc (x/d)1/7+Uw

c (x/d). (26)

The tidal current Utc at the surface can be obtained directly from the tide table, and the

wind-driven current Uwc 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 velocityon the wave field is treated by applying wave theory in a reference frame which is fixedrelative to the current velocity. For a current of magnitude Uc propagating in a directiona relative to the direction of the wave propagation, the wave velocity defined as c0=v0 /kwithout current is modified and becomes

c=c0+Uc cos a, v=ck. (27)

The velocities used to determine wave loads are the vectorial sum of the wave andcurrent velocities:

w=ww , u=uw+Uc cos a, (28)

where w and u are the total velocities in x, y directions, respectively.

Figure 3. Wave and current fields.

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To consider geometrical non-linearities, the velocities and accelerations are evaluated atthe instantaneous position of the tower. Replacing y=x tan u in the velocity andacceleration expressions (equations (23) and (24)) yields velocities

w=12

Hvsinh kxsinh kd

sin (kx tan u−vt),

u=12

Hvcosh kxsinh kd

cos (kx tan u−vt)+Uc cos a, (29)

and accelerations

w=12

Hv 0−v+u�kx

cos2 u1 sinh kxsinh kd

cos (kx tan u−vt),

u=−12

Hv 0−v+u�kx

cos2 u1 cosh kxsinh kd

sin (kx tan u−vt). (30)

The influence of current on the wave height depends on the manner in which the wavespropagate onto the current field. An approximation to the wave height in the presence ofcurrent is given by Isaacson [18],

H=H0z2/(g+g2) , (31)

where H0 , H are the wave heights in the absence and presence of current respectively, andg is

g=z1+(4Uc /c0 )cos a , for (4Uc /c0 ) cos aq−1. (32)

3.4.

Figure 4 depicts the external forces acting on the tower: T0 is a vertical buoyancy force;Ffl are fluid forces due to drag, inertia, added mass and vortex shedding; Mg, mlg are theforces due to gravity.

These forces and moments are described and developed next.

Figure 4. External forces acting on the tower.

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3.4.1. Buoyancy momentThe restoring moment is achieved via the buoyancy force

Mb=T0 lb . (33)

T0 is the buoyancy force and lb is its moment arm; both are time-dependent, where

T0=rgV0=rgp(D2/4)Ls . (34)

V0 is the volume of the submerged part of the tower, r is the fluid density and Ls , whichis the length of the submerged part of the tower, is

Ls=[d+h( y, t)]/cos u, (35)

where h( y, t) is the wave height elevation evaluated at the instantaneous position of thetower and at x=d with y=d tan u,

h(u, t)=12H cos (kd tan u−vt+e). (36)

The buoyant force acts at the center of mass of the submerged part of the tower. If thetower is assumed to be of cylindrical cross-section then the center of mass in the x', y'coordinates is

ly'b =D2

16Lstan2 u, lx'b =

12

Ls+D2

32Lstan2 u. (37)

Transforming to x, y coordinates the moment arm lb ,

lb=D2

16Lstan2 u cos u+012 Ls+

D2

32Lstan2 u1 sin u, (38)

and finally the buoyancy generalized moment is then

Mub=rgp

D2

4 $D2

32tan2 u(2 cos u+sin u)+

12 0(d+h( y, t))

cos u 12

sin u% . (39)

3.4.2. Morison’s equation for wave forcesIn 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 thefluid and the tower, and the inertia force is proportional to the fluid acceleration. The dragand inertia forces per unit length are approximated by Morison’s equation,

Ffl=CDrD2

=(Vrel )=(Vrel )+CMrpD2

4(U� w ), (40)

where Ffl is the fluid force per unit length normal to the tower. (Vrel )=(Uw−V) is therelative 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. CD and CM are the drag and inertiacoefficients, respectively. The relative velocity and fluid acceleration normal to the towercan be decomposed to their components as

[Vxrel , Vy

rel ]T=1×(Uw−V)×1, [U� xw , U� y

w ]T=1×U� w×1. (41)

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Using Morison’s equation (40), the tower velocity equation (21), and fluid velocity andacceleration equations (29) and (30), the fluid force components are: the drag force,

[FxD , Fy

D ]T=CDrD2

=1×(Uw−V)×1=(1×(Uw−V)×1)

=CDrD2

(+z(Vxrel )2+(Vy

rel )2)[Vxrel , Vy

rel ]T, (42)

and the inertia force,

[FxI , Fy

I ]T=CMrpD2

4(1×U� w×1)=CMrp

D2

4[w, u]T. (43)

3.4.3. Vortex shedding momentThe lift force FL 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, onlyshedding forces in the direction of the wave propagation due to current velocityperpendicular to the waves (a=90°) are considered. Different models of lift force exist inthe literature; see especially Billah [19]. Initially a simple model will be used,

FL=[FxL , Fy

L ]T=CLrD2

cos vs t=1×UT =(1×UT ), (44)

where UT , the vector of the maximum fluid velocity, is

UT=[wm , um ]T, (45)

CL is the lift coefficient which depends on the Reynolds number Re, and vs is the vortexshedding frequency that depends on the Keulegan-Carpenter number K as follows(Issacson [18]):

CL=0·2, for Ree1·5×105; vs=2v, for K=5–16. (46)

3.4.4. Total fluid momentThe moment due to the fluid forces (drag, inertia, and lift) is evaluated by substituting

the sum of all fluid forces, defined by equation (47):

s Fxfl=FxD+Fx

I +FxL , s Fyfl=Fy

D+FyI+Fy

L , (47)

into equation (15). Therefore, the fluid moment Mufl is

Mufl=g

L

0 0$s Fxfl , s Fy

fl% · [−tan u, 1]1x dx, (48)

which is evaluated using ‘MAPLE’ and is not given here because of its length andcomplexity.

3.4.5. Added mass momentThe fluid added mass force per unit length Fad is

Fad=CArpD2

4V� , (49)

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where CA=CM−1 is the added mass coefficient. Substituting expression (21) for thetower acceleration into equation (49) leads to expressions for the forces in the x, ydirections,

Fxad=−CArp

D2

4x(u� tan u+u� 2), Fy

ad=CArpD2

4x(u� −u� 2 tan u). (50)

Substituting these added mass forces into the generalized moment equation (15), andintegrating to yield the generalized moments due to fluid added mass, results in

Muad=

13

CArpD2

4L3(1+tan2 u)u� . (51)

3.4.6. Friction momentThe 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 thenormal force N and the coefficient of friction m. It is assumed to be independent of thevelocity, once the motion is initiated. Since the sign of the damping force is always oppositeto that of the velocity, the differential equation of motion for each sign is valid only fora half cycle interval. The friction force is

F ufr=Nm[sgn (u� )]. (52)

The normal force is

N=s Fx cos u+s Fy sin u, (53)

where a Fx , a Fy are the total forces due to gravity, buoyancy and tower acceleration inthe x, y directions, respectively. The fluid forces (drag, inertia and vortex shedding) do notinfluence the friction force since they are perpendicular to the tower. Thus,

s Fx=T0−Fg+Fxac , s Fy=Fy

ac , (54)

where T0 is the buoyancy force given in equation (39), Fg is the gravitational force,

Fg=(ml+M)g, (55)

and the forces due to the tower acceleration Fxac , Fy

ac are

Fxac=$18 CArpD2L2+

12 012 ml+M1 l�% 1

cos u(u� tan u+u� 2),

Fyac=$18 CArpD2L2+

12 012 ml+M1 l�% 1

cos u(−u� +u� 2 tan u), (56)

. - . 90

where l� is the projection of the tower’s length l in the x-direction, i.e., l�=l cos u. Assuminga hinge radius Rh , and substituting for N, the generalized damping moment is

Mufr=0018 CArpD2 L2

cos2 u+

14

ml2+12

Ml1u� 2+(T0−Fg ) cos u1Rhm[sgn (u� )]. (57)

The only term remaining in the acceleration forces is the centrifugal one which is tangentialto the tower, namely lu� 2.

3.5.

The dynamic moments Mudy , those which are evaluated in the left hand side of Lagrange’s

equation (11), are found using the kinetic, dissipative, and potential energies,

KE=12 IzV

2, DE=Cu� 2, PE=(12ml+M)gl cos u, (58)

where C is the structural damping constant and Iz is the moment of inertia of the towerabout the z axis, given by

Iz=13ml3+Ml2. (59)

Substituting equations (59) and (22) into (58) leads to the expression for the kinetic energy,

KE=12 (

13ml3+Ml2)u� 2. (60)

The dissipative energy due to Coulomb friction is not velocity dependent, but the viscousdamping is. Substituting the energies into the left hand side of equation (11) leads to Mu

dy ,

Mudy=(1

3ml3+Ml2)u� +Cu� −(12ml+M)gl sin u. (61)

3.6.

The governing non-linear differential equation of motion is found by equating thedynamic moment, Mu

dy , to the applied moment, Muap , which is the sum of all external

moments:

Mudy=Mu

ap , (62)

where the applied moment is found by adding equations (39), (51), (57), and (48):

Muap=Mu

fl−Muad−Mu

b+Mufr . (63)

Substituting equations (61) and (63) into (62) and rearranging leads to the governingnon-linear time dependent differential equation of motion for the tower:

Jeffu� +Cu� =gL

0 0$s Fxfl , s Fy

fl% · [−tan u, 1]1 x dx−Mugb−Mu

fr , (64)

where Jeff is the effective moment of inertia,

Jeff=013 ml3+Ml21+13

CArpD2

4L3(1+tan2 u), (65)

91

T 1

Tower’s properties

Property Value

Tower’s length 400 mTower’s diameter 15 mTower’s mass 20×103 kg/mEnd mass 2·52105 kgFriction coefficient 0·1 to 0·4Pivot radius 1·5 m

and Mgb is the moment due to gravity and buoyancy,

Mugb=rgp

D2

4 $D2

32tan2 u(2 cos u+sin u)+

12 0d+h( y, t)

cos u 12

sin u%−012 ml+M1 gl sin u.

(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 isperformed for deterministic as well as for random wave heights. The physical parametersused in the simulation are shown in Tables 1 and 2.

4.1.

The non-linear differential equation for the single-degree-of-freedom system is solved forseveral cases: equilibrium position of the tower in the presence of current; fundamentalfrequency, and damping (drag, viscous and friction) effect; response to wave excitation;superharmonics, harmonic and subharmonics resonances; chaotic regions and influence ofcurrent velocity and direction.

4.1.1. Response in the absence of wavesIn 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

T 2

Fluid properties

Property Value

Mean water level 350 mDrag coefficient 0·6 to 1·0Inertia coefficient 1·2 to 1·6Lift coefficient 0·8 to 1·2Water density 1025 kg/m3

Wave frequency 0·03 to 1 rad/sSignificant wave height 0 to 15 mCurrent velocity 2 m/sStructural damping 0·02

. - . 92

Figure 5. Free vibration of the tower.

fundamental frequency, the tower response to non-zero initial conditions with zerodamping is found. Figure 5 describes the response of the tower for u� (t=0)=0·001 rad/sin the time and frequency domain. From the figure it can be seen that the fundamentalfrequency is vn=0·17701 rad/s=0·028 Hz. Calculating the frequency analytically usingequation (67) yields vn=0·17704 rad/s=0·028 Hz,

vn=12pXMu

gb

Jeff. (67)

Adding damping to the system (drag, friction or viscous) causes a decay in the responseas can be seen from Figure 6. A typical decay for each damping mechanism is clearly seen:hyperbolic decay, proportional to ua1 /t, for drag damping, linear decay, proportional to(u−a2 t), for Coulomb damping, and exponential decay, u exp (−a3 t), for viscousdamping. a1 , a2 , a3 are the decay constants for each damping mechanism. The response forthe first two damping mechanisms consists of the fundamental frequency and its oddmultipliers, as can be seen from the frequency domain figures. The reason for the oddmultipliers 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) CD=1; (c, d) m=0·1; (e, f) z=0·2.

93

Figure 7. Damped, free vibration in the presence of current; ——, Uc=1 m/s; – · – · , Uc=2 m/s.

anti-symmetric. On the other hand the response for viscous damping is linear and thereforeonly the fundamental frequency is seen.

Figure 7 shows the damped, free vibration of the tower in the presence of current. Thefigure 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 (2CDUc cos a)u�which is similar to linear viscous damping.

The influence of the current velocity on the equilibrium position is found from the steadystate solution. Figure 8 describes the tower position in the presence of current velocitiesof Uc=1 and 2 m/s, with drag (CD=1) and with friction damping (m=0·1). In both casesa=0. Setting u� =0 and u� =0 in the non-linear governing equation and solving for u withUc=1, 2 m/s leads to the following equilibrium positions: u(Uc=1)=0·0035 rad andu(Uc=2)=0·0141 rad, with simulations leading to the same results. As can be seen, thedeflection angle for Uc=2 m/s is four times the deflection angle for Uc=1 m/s, and thereason is that the equilibrium position is proportional to Uc cos a =Uc cos a =.

4.1.2. Response in the presence of wavesIn 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, a, on the

response is investigated next. Figure 9 shows the response of the tower for a=0°, 45°(Figure (a)) and 135°, 180° (Figure (b)). Here, the wave height is H=1 m, v=0·1 rad/sand Uc=1·5 m/s. From the figure it is seen that the steady state response for a and 180−a

Figure 8. Tower equilibrium position in the presence of currents with (a) drag only, CD=1; (b) friction only,m=0·1; ——, Uc=1 m/s; ·– · – ·, Uc=2 m/s.

. - . 94

Figure 9. Influence of current direction: (a) ----, a=0°; - - -, a=180°; (b) ----, a=45°; - - -, a=135°.

Figure 10. Response in (a) time and (b) frequency domain for current direction a=90°.

has the same magnitude but opposite sign, and for a=90° the equilibrium position is zero.This is because the steady state response is proportional to Uc cos a =Uc cos a =. For a$0,the response has two frequencies, v and 2v, due to the vortex shedding model used inthe analysis; see equation (46).

The highest response is achieved when a=90°, as can be seen when comparing Figures9 and 10. The reason is that an additional lift (transverse) force is added when the angleis not zero, and this force is maximum for a=90°. From the frequency domain response,the two excitation frequencies are clearly seen, v=0·016 Hz and 2v=0·032 Hz.

The response due to resonance wave excitation; superharmonics, harmonic andsubharmonics is analyzed. Figure 11 shows the tower’s response to wave excitation at

Figure 11. Undamped response to subharmonic excitation v=2vn : (a) time domain and (b) frequencyresponse.

95

Figure 12. Undamped response to subharmonic excitation v=2vn with current velocity Uc=2 m/s. (a) Timedomain and (b) frequency response.

about twice its fundamental frequency, the subharmonics. Here the wave height isH=10 m, the current velocity and all damping forces are set to zero. Thefigure demonstrates beating with high amplitudes, u10·4 rad, implying an unstableregion. The frequency response includes the excitation frequency v=0·056 Hz, thefundamental frequency v=0·028 Hz and its multipliers. This is due to the systemnon-linearity, where subharmonics cause a response at the exciting frequency as well asin the fundamental frequency.

The subharmonic response in the presence of current is shown in Figure 12. ComparingFigures 11 and 12 it is seen that the highest response is for zero current velocity. ForUc=2 m/s, the response is much smaller. The reason is that the excitation frequencydepends on the current velocity as shown by equation (68). For Uc=0 m/s, the excitationfrequency is exactly twice the fundamental frequency thus causing the highest response.The exciting frequency and the current velocity are related by

v=v0 (1+v0Uc /g). (68)

Adding drag, CD=0·6, to the system reduces the amplitude of the tower’s response andeliminates the beating phenomenon, as can be seen from Figure 13 (time domain). In thefrequency domain the exciting frequency, the fundamental frequency and its multipliersare as seen in Figure 11. The difference is that the amplitude of the fundamental frequencyv=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 theabsence of damping is shown in Figure 14. The response is much higher than forsubharmonic excitation. The beating phenomenon is clearly seen and the system jumpsbetween two amplitudes. The frequency domain curve shows the fundamental frequencyv=0·028 Hz as well as its multipliers.

Figure 13. Damped response to subharmonic excitation v=2vn with CD=0·6; (a) Time domain and (b)frequency response.

. - . 96

Figure 14. Undamped response to harmonic excitation v=vn . (a) Time domain and (b) frequency response.

Figure 15. Undamped response to harmonic excitation v=vn for current velocity Uc=2 m/s. (a) Time domainand (b) frequency response.

Again, in the presence of current, the excitation frequency is modified and resulting indifferent response amplitudes, as can be seen in Figure 15. Here, the response is higherin the presence of a current velocity, Uc=2 m/s, but the difference is not large whencompared to Figure 12, since the region of instability around the fundamental frequencyis 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 CD=0·6 has a stabilizing effect on the system. The amplitude is lowerand the beating disappears as can be seen in Figure 16.

Similar results are found when the excitation frequency is about one half of thefundamental frequency (superharmonic excitation), as can be seen from Figures 17, 18and 19.

Figure 16. Damped response to harmonic excitation v=vn with CD=0·6. (a) Time domain and (b) frequencyresponse.

Figure 17. Undamped response to superharmonic excitation v=vn /2. (a) Time domain and (b) frequencyresponse.

97

Figure 18. Undamped response to superharmonic excitation v=vn /2 with current velocity Uc=2 m/s. (a) Timedomain and (b) frequency response.

Figure 19. Damped response to superharmonic excitation v=vn /2 with CD=0·6. (a) Time domain and (b)frequency response.

4.1.3. Quasiperiodic and chaotic behaviorThe 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 certainfrequencies 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) Poincare map.

. - . 98

Figure 21. Chaotic response: (a) Time domain, (b) phase plane and (c) Poincare map.

phase plane and Poincare map, for wave excitation v=0·06 rad/s, H=10 m. The dampingmoments, initial conditions and current velocity are set to zero. From the phase plane theresponse looks chaotic but the Poincare map shows that the response is in factquasi-periodic.

A chaotic region is shown in Figure 21 where the response to a wave frequency ofv=0·03 rad/s is shown. From the Poincare map it is clearly seen that the response ischaotic 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 additionaldamping of CD=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 pointsare highly organized, as described by Moon [20].

The influence of initial conditions is shown in Figure 23. The left figure shows a Poincaremap of a chaotic response with zero initial conditions. The right figure is also a Poincaremap but with the following initial conditions: u(t=0)=0·5 rad, u� (t=0)=0·05 rad/s. It isclearly seen in the latter case that the chaotic response has become quasi-periodic withlarger deflection.

4.2.

In this section the tower response to random wave height excitation is investigated. Thewave height distribution is generally expressed in the form of a power spectral density. Forsimulation 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.

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 h( y, t) can be expressed as

h( y, t)=12H cos (ky−vt+e). (69)

The Pierson-Moskowitz spectrum for the wave height is

Sh (v)=(A0 /v5) e−B/v4, (70)

where A0 and B are constants defined by

A0=8·1×10−3g2, B=3·11/H2s , (71)

where Hs is the significant wave height. Using Borgman’s method, the wave elevationh( y, t) can be approximated by

h( y, t)=a sN

n=1

cos (kny−vn t+en ), (72)

where the amplitude a, is constant and given by

a2=A0 /4BN. (73)

Figure 24. Influence of different wave heights on the tower response: (a) Hs=4, (b) Hs=9, (c) Hs=15.

. - . 100

The partition frequencies vn are

vn=(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 waveelevation. Thus, in the expressions for wave velocity and acceleration, the followingsubstitutions are made:

H czA0 /4BN , v c vn , k c kn , n=1, 2, . . . , N. (75)

For example, the horizontal velocity and acceleration will become

u=sN

n=1

A0

4BNvn

cosh knxsinh knd

cos (knx tan u−vn t),

u=sN

n=1

−A0

4BNvn 0−vn+u�

kn xcos2 u1 cosh knx

sinh kndsin (knx tan u−vn 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.

4.2.1. Effect of significant wave heightFigure 24 compares the tower response for three different significant wave heights:

Hs=4, 9, 15 m. It can be seen from the figure that lower significant wave height resultsin lower response. The reasons are: the larger the significant wave height the higher theinput; as Hs increases, the mean wave frequency approaches the fundamental frequencyof the tower.

To emphasize this second reason, the tower fundamental frequency is changed to aboutvn=0·8 rad/s and the responses due to Hs=4, 9 m are calculated and shown in Figure 25,from which it can be seen that the response for Hs=4 m beats, almost as in harmonicexcitation, and is higher than the response for Hs=9 m although the spectral peak is seventimes smaller.

4.2.2. Effect of currentThe influence of current velocity on the random response is shown in Figure 26 for the

case of collinear wave and current i.e., a=0. Here the significant wave height is Hs=9 mand the drag coefficient is CD=0·2. It is seen that the higher the current velocity the smallerthe response. The reason is that current in the direction of the wave’s propagation tendsto lower the wave height (see Isaacson [18]).

Figure 25. Influence of significant wave height with vn=0·8 rad/s. (a) Hs=4, (b) Hs=9.

101

Figure 26. Influence of current on the random response for a=0. (a) Uc=0 (m/s), (b) Uc=1 (m/s), (c) Uc=3(m/s).

4.2.3. Effect of dampingDifferent damping mechanisms have different effects on the tower’s response. Figure 27shows the separate influence of drag, viscous, and Coulomb damping on the response. Thesignificant wave height is Hs=9 m and the damping constants are CD=0·6, z=0·02,m=0·1, respectively. As can be seen from the figure, although all damping mechanismscause the transient response to vanish after about 25 seconds, the ‘steady state’ responsein the presence of Coulomb damping m is almost one order of magnitude lower.

5. DISCUSSION AND SUMMARY

The non-linear differential equation of motion for an articulated tower submerged inthe ocean is derived including Coulomb and viscous damping. Geometric as well as forcenon-linearities are included in the derivation. The fluid forces, drag, inertia and lift dueto waves and current, are determined at the instantaneous position of the tower, adding

Figure 27. Influence of different damping mechanisms on the response: (a) CD=0·6, (b) z=0·02, (c) m=0·1.

. - . 102

to the non-linearities of the equation. The equation is solved numerically using ‘ACSL’for deterministic and random wave loading.

The equilibrium position (u� =u� =0) depends on the current velocity and direction,Uc cos a =Uc cos a =, and in the absence of drag the equilibrium position is u=0. Thecurrent’s direction affects the response greatly. For the same current velocity, the highestresponse is when the direction is perpendicular to the wave propagation, since the lift forceis then maximum.

The response of the tower to harmonic wave excitation at its ‘natural frequency’, andhalf and twice its ‘natural frequency’ demonstrates beating, where the amplitude variesbetween 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 hasa stabilizing effect on the system.

The system response depends on the wave frequency and amplitude. For mostfrequencies the response is quasi-periodic, but there are certain frequencies at which thesystem exhibits chaotic behavior.

To solve the equation for random wave loading, the Pierson-Moskowitz spectrum thatdescribes the wave height distribution was first transformed into a time history. Theequation was solved for three significant wave heights. For significant wave heights of 9and 15 m, the response was larger than that for 4 m, since in the former the tower’s ‘naturalfrequency’ is closer to the frequencies where most of the energy is located. Damping ofany kind (drag, friction and viscous) stabilizes the system with the greatest effect due tofriction damping. Notice that in order to reduce stresses in the structure, the frictionmoment has to be low enough so that the tower can comply with the wave loading. Currentvelocity tends to lower the response as long as aQ90°, because it lowers the wave height.

A more realistic model having two angular-degrees-of-freedom is being analyzed at thepresent time. The response due to wave, current (collinear and otherwise), vortex sheddingloading 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. N00014-93-1-0763.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. C and D. C. C 1979 Journal of the Waterway, Port, Coastal and OceanDivision, ASCE 105, 281–292. Motion analysis of articulated tower.

2. S. K. C and D. C. C 1980 Journal of the Waterway, Port, Coastal and OceanDivision, ASCE 107, 65–77. Transverse motion of articulated tower.

3. R. K. J and C. L. K 1981 Journal of Energy Resources Technology 103, 41–47. Dynamicresponse of a double articulated offshore loading structure to non-collinear waves and current.

4. J. M. T. T, A. R. B and R. C 1984 Journal of Energy ResourcesTechnology 106, 191–198. Stochastic and chaotic motions of compliant offshore structures andarticulated mooring towers.

5. H. S. C and J. Y. K. L 1991 Applied Ocean Research 12, 63–74. Non-linear behaviour ofan articulated offshore loading platform.

6. L. L. S and J. M. N 1992 Ocean Engineering 19, 1–20. Response characteristicsof multi-articulated offshore towers.

7. O. G, C. S. Y and R. T. H 1992 International Journal of Offshore and PolarEngineering 2, 61–66. Analysis of non-linear response of an articulated tower.

103

8. P. K. M and A. S. G 1983 Ocean Engineering 10, 471–479. Stochastic stability oftethered buoyant platforms.

9. T. K. D and A. K. J 1990 Computers and Structures 34, 137–144. Response of articulatedtower platforms to random wind and wave forces.

10. A. K. J and T. K. D 1991 Journal of Engineering for Industry 113, 238–240. Non-linearbehavior of an articulated tower in a random sea.

11. S. Y. H, A. M and R. S 1983 Journal of Energy ResourcesTechnology 105, 205–211. Non-linear dynamic analysis of guyed tower platforms.

12. J. F. W and G. O 1984 Journal of Dynamic Systems, Measurements and Control 106,311–318. Optimal cable configuration for passive dynamic control and compliant tower.

13. H. B. K and A. H 1987 Symposium of Non-linear Stochastic DynamicEngineering Systems 351–360. Non-linear random vibrations of compliant offshore platforms.

14. M. G and L. E 1993 Ocean Engineering 20, 113–133. The bilinear oscillator:the response of an articulated mooring tower driven by irregular seas.

15. L. E. B 1969 Journal of the Waterways and Harbors Division, ASCE 95, 557–583. Oceanwave simulation for engineering design.

16. ACSL 1992 Advanced Continuous Simulation Language. Mitchell and Gauthier Associates(MGA).

17. J. F. W 1984 Dynamics of Offshore Structures. New York: John Wiley.18. M. I 1988 Canadian Journal of Civil Engineering 15, 937–947. Wave and current forces

on a fixed offshore structure.19. K. Y. R. B 1989 A Study of Vortex-Induced Vibration. PhD thesis, Princeton University.20. F. C. M 1992 Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and

Engineers. New York: John Wiley.