O. Gottlieb
Faculty of Mechanical Engineering,
Technion--lsrael Institute of Technology,
Haifa 32000, Israel
Bifurcations of a Nonlinear
Small-Body Ocean-Mooring
System Excited by FiniteAmplitude Waves
We investigate the response of a nonlinear small-body ocean-mooring system excited
by finite-amplitude waves. The system is characterized by a coupled geometrically
nonlinear restoring force defined by a single elastic tether. The nonlinear hydrodynamic exciting force includes both dissipative and. convective terms that are not
negligible in a finite wave amplitude environment. Stability of periodic motion is
determined numerically and the bifurcation structure includes ultrasubharmonic and
quasi-periodic response. The dissipation mechanism is found to control stability
thresholds, whereas the convective nonlinearity governs the evolution to chaotic
system response.
Introduction
Nonlinear aperiodic responses have been documented in various deterministic mooring system models for over a decade.
These results are based on approximate theoretical analyses of
relatively simplified models complemented by extensive numerical analyses (eg., Thompson, 1983; Papoulias and Bernitsas,
1988; Sharma et al., 1988; Bishop and Virgin, 1988). Furthermore, a limited number of reported experimental studies have
been reported describing subharmonic (Thompson et al., 1984)
and chaotic response (Isaacson and Phadke, 1994).
Ocean mooring systems are characterized by a nonlinear restoring force, a structural damping force, and a coupled velocitydisplacement exciting force. The restoring force includes material discontinuities and geometric nonlinearities associated with
large-amplitude motion. The exciting force includes nonlinear
effects and periodic components governed by steady (current)
and unsteady (wave-induced) viscous drag, radiation damping,
and convective inertial effects (e.g., Sarpkaya and Isaacson,
1981; Chakrabarti, 1990). The response of these nonlinear systems to external and parametric excitation consists of periodic
and aperiodic (amplitude-modulated quasi-periodic and chaotic) motion. Coupling of degrees of freedom and the shedding
of vortices further complicate system behavior.
Mechanical systems typically depict global aperiodic behavior for relatively small damping values (Moon, 1992). However, the magnitude of the dissipation mechanisms governing
ocean mooring system response is not always small. Consequently, in order to investigate the bifurcation structure of a
mooring system where hydrodynamic damping effects are not
small, a finite-amplitude wave theory is needed. This requirement is enhanced by development of deepwater mooring systems designed for operation in more severe environmental conditions.
Past research has demonstrated the need to incorporate convective terms (incorporating spatial gradients) in the inertial
part of the exciting force for body motion in unsteady nonuniform flows (Newman, 1977). The inclusion of convective terms
was also carried out for inertial force calculations of fixed struc-
Contributed by the OMAE Division and presented at the 15th International
Symposium and Exhibit on Offshore Mechanics and Arctic Engineering, Florence,
Italy, June 16-20, 1996, of THE AMERICANSOCIETYOF MECHANICALENGINEERS.
Manuscript received by the OMAE Division, August 1996; revised manuscript
received June 29, 1997. Associate Technical Editor: S. K. Chakrabarti.
tures in a nonlinear wave regime (Isaacson, 1979). Recently,
modified potential flow wave-loading calculations on fixed
(Rainey, 1988) and moving (Manners, 1992; Foulhoux and
Bernitsas, 1993 ) small-body structures (where the characteristic
body dimension (b) is small with respect to the flow wavelength
(L) or ( b / L ) < 0.1 ) reveal conditions where the convective
terms cannot be neglected as in a standard Morison formulation.
Furthermore, the total (steady and unsteady) pressure forces on
deformable bodies in nonuniform inviscid flows have recently
been derived by Miloh (1994), including a limiting case of a
moving rigid sphere in axisymmetric nonuniform flow.
In this paper, we consider a single-point mooring configuration where a submerged small body is excited by finite-amplitude waves and is restrained by a massless elastic tether. The
system restoring force consists of a geometric nonlinearity of
an inverted extensible (spring) pendulum; cf., Nunez-Yepez et
al. (1990) for the complex Hamiltonian structure and Bayly
and Virgin (1993) for stability of periodic motion of a standard
harmonically forced spring pendulum. The system dissipation
function includes linear structural damping and coupled quadratic wave-induced damping governed by the relative motion.
The inertial exciting force is formulated to account for convective terms. Stability thresholds from periodic response are obtained numerically and the system bifurcation structure is investigated to reveal ultra-subharmonic and aperiodic response.
The Dynamical System
The dynamical system is formulated utilizing a Lagrangian
approach in cartesian coordinates (X, Y) where the origin (Fig.
1) is located at the equilibrium position of a body submerged
at depth d - 1 ( d is the water depth and 1 is the equilibrium
mooring length extending from Y = 0 to the bottom at Y =
- 1). The system Lagrangian L = K - P is assembled from the
following kinetic and potential energies where the hydrostatic
body force is included in the potential:
K = m ( ~ 2 4 1~2)
2
k
P = ~ [I(X, Y) - lo1 = + (m
-
pV)gY
(1)
where m and V are the body mass and volume; k and 1o are the
tether stiffness and dry massless length; p and g are the water
234 / Vol. 119, NOVEMBER 1997
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0.20
z
D
y
0.10
0.00
d
Q.
>-
-0.10
S
f
~\\\\\\\'~
-0.20
Fig. 1
Definition sketch
-0.30
i i i i i l , l , l , , , , , , l l l l
0.20
0.18
mass density and gravitational acceleration; l is the tether extension and l is the equilibrium length.
[= lo + ( p V -
l = ~/X2 + ( [ + y)2,
m)-~g
Qx = Dx + E x - CsX, Qr= Dr+ E y - Cs}" (3)
where Cs is a linear damping coefficient. The quadratic hydrody-
,,,,1
=,,,,,,III
0.24
×p
=111,,,,1
0.26
[ , , , l l l l * ' l
0.28
0.50
(a) Poincar6 map
0.o
(2)
The generalized forces Qx.vinclude both dissipative and inertial components of the hydrodynamic exciting force and the
effects of linear structural damping. The hydrodynamic damping depends on the relative velocities between the exciting field
and the body, whereas the inertial components include spatial
gradients induced by the finite amplitude wave excitation.
LLI,I
0.22
-20.0
-40.0
m
-60.0
(.f)
-80.0
-100.0
-12o.o
0.10
0.0
. . . . . . .
. . . . . 1'.O'
'2.O . . . . . . .
(b)
5.0' . . . . . . .
s.o
'¢.O . . . . . . . . . . . . . . . . . . .
6.0
Spectra
0.05
Fig. 3
0-
0.00
Quasi-periodic response (1' = 0,01, 9 = 0.5)
namic damping force (Dxx) is formulated to include the respective projections of the relative velocities.
>-
>~-o.o5
( D x ) = ~p CoS ( UV- ~ )
~/(U-f()2+(V-~)2
-0.10
Co and S are the drag coefficient and projected area and
U(X, Y) and V(X, Y) are the horizontal and vertical field
where
-0.15
-0.20
-0.30
-0.10
0.00
x,
0.10
0.20
Xp
(a) Phase plane, Poincar6 map
0.0
velocities, respectively. The inertial force is formulated under
a weak nonuniform flow assumption including second-order
terms, but neglecting the effects of higher-order spatial gradients
(Miloh, 1994). This formulation is consistent as it includes
quadratic terms in both dissipative and convective terms.
ox
-100.0
er
+ u OU + v OV
OY
-0
OY/
-200.0
U3
-500.0
-400.0
0.0
,,, .....
=. . . . . . . . .
i .........
i .........
i, ........
i .........
i
2.0
4.0
6.0
8.0
10.0
12.0
where CM is the added mass coefficient (0.5 for a sphere).
The velocities ( U, V) can be obtained from a nonlinear finite
amplitude (N > 1 ) Stokes potential (cf., Sarpkaya and Isaacson,
1981).
CO
(b)
Fig. 2
\ V ( X , Y)
Spectra
Subharmonic response
(rain
= 2/1)
(1' = 0 . 0 1 , 9
= 1.0)
Journal of Offshore Mechanics and Arctic Engineering
cosh O cos x.)
/nl,
,=1 \
sinhn~(Y)sinnO(X,t)
(6)
where ~ and a are the wave frequency (T = 27c/~) and ampli-
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0.50
-0.00
E3
=~
(]o)
OU
+
/
-0.50
the hydrodynamic dissipation D,,3
O_
>- -1.0o
(D~)D3 = 6 ~ ( u - x ' ) 2 +
(v-y')2(
11)
u-x')v y'
--1.50
and the velocity field is
-2.00
(:)=~y~
-2.50
IIlll
-3.00
,lllllllll
-2.00
J,,lllll,,,,,
-1,00
,I ,,L,'''l
0.00
x,
(a) Phase
'Ill'
1.00
~[coshnK(I+y)cosn(KX--~T)]'
,' Jll'"l
2.00
n=l n l . ~ sinh nK(1 + y) sin n(Kx - ~2v) /
12)
Xp
plane, Poincard
The system parameters are
map
cS/wl
+ pVCM)
20.0
(m
0.0
a
6=
-40.0
(./3
-60.0
~ = - -wl,
pCDS[
2(m + pVCM) '
.
~-
(m
÷
k
RVCM)
# - pV(1 + CM)
(m + pVCM)
K = h[,
13)
and an example set of amplitude coefficients F, for a secondorder (N = 2) Stokes wave (where the dispersion relation remains that of linear waves) is
-80.0
-100.0
, , , , , r E , , l , , , J , , ,
0.0
, , i , , , , , , , , , l , , , , , , ,
4.0
2.0
6.0
,,
I
8.0
0.40
co
(b) Spectra
Fig. 4
w
T = 7,
-20.0
0.20
Ultrasubharm0nic response (rnln = 3•2) (3' = 0.07, ~ = 0.75)
0.00
O..
tude ( H = 2a); ® = (XX - wt), • = X(Y + 0 and X is the wave
number obtained from the appropriate dispersion relationship (
= k(w, a, d; N ) ) ; F. is a nondimensional coefficient ( F =
F ( h a , Xd)). Note that N = 1 corresponds to linear wave theory
(w 2 = g k tanh (kd), F1 = sinh l(~kd)).
The equations of motion are obtained by substitution of L =
K - P from (1) and Qx,Y from (3) into Lagrange's equations.
>- -0.20
-0.40
-0.60
-0.80
` ~ ' ~ t ~ ' ~ ` ~ = l ~ t ~
-1.20-1.00-0.80-0.60-0.40-0.20
H
dt \ Oqj ] - ~qj = QJ
(7)
(a) Phase
~
;
~
H
~
H
0.00
x,
~
r
~
0.20
H
~
0.40
0.60
Xp
plane, Poincard
map
0.0
where j = X, Y. We scale the displacement by the equilibrium
length (x, y) = (X, Y ) / i and scale time by the natural frequency
of linearized vertical equation ~- = wit where w,2 = k/(m +
p VCM). Consequently, the equations of motion (7) are obtained
in nondimensional form.
-100.0
.£3
"O
-200.0
x" = Di + Ei - fix' - R ,
co
y"
= D3 + E3 -
fly'
-
R3 +
ot
(8)
-300.0
where the restoring forces Ri,3 are given by
(R)=[,
R3
1o
~/x2 + (-i'+ y ) 2
](x)
1 +y
-400.0
0,0
(9)
,,,,,,,,i,,,
2.0
......
i .... ,,,,,i .........
i .......
4,0
6.0
8.0
,,1,,,,,,,,,i
10.0
12.0
60
(b) Spectra
the exciting force E~,3
2 3 6 / Vol. 1 1 9 ,
NOVEMBER
Fig. 5
1997
Subharmonic response (rnln = 4/1 ) (y = 0.07, fl = 1.25)
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of the ASME
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0,10
o
0.06
//-,
p0.04
"'\
//
'\S/
'
°'°°o,5d "~.'~,~ ~',3~" ~.~,3'" ~~'""L'6~'"'~'.~'"'~'.':~'"'h'.'~o
Fig. 6 Stability boundary of periodic response ( m = n = 1 ) : dashed and
solid lines correspond to with and without convective terms, respectively
1
F~
sinh
(Kd/D
'
3
K'y
F2 - 8 sinh 4 (Kd/[)
(14)
The unforced ( 7 = 0), undamped (fl = 6 = 0), Hamiltonian
subsystem is governed by a single parameter a that is bounded
from above and below: 0 -< a < 1, by the spring stiffness and
the system buoyancy ( a = 0 for m = pVor k ~ oo; a ~ 1 with
k -~ 0). Similarly, the mass parameter # is also bounded (CM
= 0.5) 1 --< /.z < 3 (m = pV; m ~ 0). The exciting force
consists of combined external and parametric excitation (cf.,
Yagasaki et al., 1990), and resonantly excited coupled mechanical systems (cf., Bajaj and Tousi, 1990) reveal a complex bifurcation structure including ultrasubharmonic and aperiodic response.
structural or hydrodynamic damping was found to raise the
minimal threshold. Consequently, the width of the unstable
wedge becomes narrower for a given wave amplitude. The nonlinear, damping mechanism produces a bias which governs loss
of symmetry (cf., Gottlieb and Yim, 1993), whereas the primary contribution of the convective terms is the combined parametric and external system excitation (cf., Gottlieb, 1992) enhancing the bifurcation sequence culminating with chaotic response (Fig. 7). A numerical investigation of system response
without the convective terms in (10) revealed a higher threshold
curve (Fig. 6, solid line). This difference may become more
pronounced for different parameter regimes (i.e,, smaller hydrodynamic damping; nonresonant fluid-structure interaction).
Closing R e m a r k s
A nonlinear, single-point, small-body mooring dynamical
system excited by finite-amplitude waves has been formulated
and investigated. The nonlinear hydrodynamic exciting force
incorporated both dissipative and convective terms that are not
negligible in a finite wave amplitude environment. Stability of
periodic motion was determined numerically and revealed a
bifurcation structure including ultrasubharmonic and quasi-periodic response. The hydrodynamic dissipation mechanism was
found to control stability thresholds, whereas incorporation of
convective terms enhanced the onset of secondary resonances
culminating with chaotic motion. The instabilities found were
more pronounced for a mooring configuration near an internal
resonance resulting in a strong nonlinear coupling between the
vertical and horizontal system modes. Consequently, excitation
by finite amplitude waves may generate a complex transfer of
1.50
1.00
The Bifurcation Structure
We numerically integrate the dynamical system (8) and investigate stability of the periodic solutions excited by a secondorder Stokes wave (14) via Floquet analysis (cf., Nayfeh and
Balachandran, 1995). We consider a " s o f t " tether configuration (c~ = 0.25) near an internal resonance where the natural
frequency of the vertical motion (eel) is twice the frequency of
the sideways motion (cv~ = c~cv~ oc g/~). The structural and
hydrodynamic damping coefficients are fixed (/3 = 0.05, 6 =
0.1 ) and we investigate stability thresholds near the primary
system resonance.
The fundamental periodic solution ( z ( t ) = z ( t + T); z =
(x, y) r ) governs system response for relatively small amplitude
( 7 ~ 0.01) between the system resonances (0.5 < f~ < 1:co2
< cv < 0vj for a = 0.25 or COl = 2cv2). However, the periodic
solution loses its stability to a period-doubled solution ( z ( t ) =
z ( t + mT), rn = 2) at f2 = 1 a n d loses its periodicity to a
quasi-periodic torus at f2 = 0.5. These solutions are depicted
in Figs. 2 and 3 by their phase plane (y(x)), Poincar4 map
(Yp(Xp)) stroboscopically sampled each forcing period (27r./
~ ) and power spectra (S(cv)). The quasi-periodic solution is
identified (Fig. 3 ( a ) ) by the closed curve of Poincar6 of points
organized in an invariant geometric shape. An increase in wave
amplitude reveals additional bifurcations of the periodic motion
to higher order ultrasubharmonic solutions (cf., Wiggins,
1990). Figures 4 and 5 depict unsymmetric ultrasubharmonics
of order m/n = 3/2, 4/1. The number of Poincar6 points (m)
define the solution period (mT), whereas the order n can be
seen in the frequency content of the spectra.
Stability of the fundamental periodic solution near the primary resonance (~2 = 1 ) is summarized in Fig. 6. The threshold
of stability (7(f2)) is controlled by the resonant frequency for
a given damping (Fig. 6, dashed line). An increase in either
Journal of Offshore Mechanics and Arctic Engineering
"..~
0.50
"..':':'
•
.
o_
>#
0,00
-0.50
i,,,
-1.00
--I .50
ii1
illl
iiiii1,,111,,i
--1.00
iii,
--0.50
ii,
0.00
,,1
iii
f Ii,
iiii
0.50
×p
iiii
,,i
i,ii
1.00
,11,111111
1.50
Ill
2.00
la) Poincar6 map
0.0
-20.0
__Q
-40.0
03
-60.0
-50.0
-100.0
.........
0.0
, .........
2.0
, .........
4.0
, .........
6.0
8.0.
60
(b)
Fig. 7
Spectra
Chaotic response (3' = 0 . ! , fZ = 0.9)
NOVEMBER 1997, Vol. 119 / 237
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energy between the modes of motion for wave frequencies that
are integer multiples of the system natural frequencies. While
small and large-body mooting systems differ in their dissipation
mechanisms, they equivalently incorporate the nonlinearities of
the mooring restoring force and the ine~ial form of the exciting
force. Consequently, a similar bifurcation structure may evolve
for different environmental conditions where higher-order terms
are not negligible.
Acknowledgment
The author is greatful to Professors M. Bernitsas and T. Miloh
for pointing out the controversy that exists in the selection of
the inertial exciting force and motivating the investigation of
its influence on mooring system dynamics.
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