15 SEP. 972
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BibIioheek van de epsbouwkunde Ondrfdir\g L. nische Hogeschoo DCUMEN!A1IE I:DAT UM: j 9 OY.1. 9?3
MOTIONS OF MOORED SHIPS IN SIX DEGREES OF FREEDOM
J-Min Yang
Tetra Tech, Inc. Pasacicna, California
U. S. A. 91107
For presentation at the
9th SYMPOSIUM ON NAVAL HYDRODYNAMICS
PARIS, FRANCE
August 1972.
Lab. y. Scheepsbouwkunde Technische Hogeschooj
ABSTRACT
The equations of motions of a moored ship having six degrees of freedom were formulated. The mooring force is nonlinear and asymmetrical. The effects of shape of the ship, water depth, side walls and the nonlinear fender system have also been included in the analysis. These equations were solved through use of an equivalent linearization technique.
INTRODUCTION
Recently the development of large ships has attracted many investigators to study motions of moored ships The introduction of container ships and the increase of oil exploration in deeper water depths make it necessary to have a thorough understanding of motions of moored structures. For container ships, the operation of lOading
and unloading containers are controlled by land based huge cranes, extensive ship motion may greatly reduce container loading and
unloading rate. Oil explor ation in deep sea needs to drill through the ocean floor from a moored ship (or other moored
structures), large
motions of the ship may hinder drilling operation. Another related problem is that of a moored buoy system. The effective design and development of such system also require an ability to predict its oscillatory motions.
The study of motions of a moored ship is usually limited to the surge motion [lJ that is, the moored ship is considered as having only one degree of freedom, and it is known that the force-elongation rela-tionship of a mooring line is highly nonlinear [Z]. Kaplan and Putz [3],
and Muga [4] have investigated moored structures in six degrees of freedom. In their study, the force-elongation relationship of mooring lines is assumed to be linear, thus the problem is linear and the solution can be readily obtained, In this paper, we consider a more general six degrees of freedom problem. The mooring force is a nonlinear function of elongation; since a ship can not be symmetrically moored, and fender s are only at one side of the ship, motions of the ship are asymmetric. An approach has been developed to generate an approximate steady- state
solution to this nonlinear asymmetric problem. FORMULATION
The motion of a moored ship in waves is an oscillating system Numbers in brackets designate References at the end of the paper.
-2-with six degrees of freedom corresponding to surge, heave, sway, roll, pitch, and yaw. The ship is considered as a rigid body and its deformation is neglected. Usually the length of a ship is much longer thai its beam, and the slender body theory can be applied to find the hydrodynamic properties of a moored ship. According to this theory,
for an elongated body where lateral dimensions are small compared to its length, the flow field at any cross-section is independent of that at any other sections. Hence, the flow field of an elongated body, like a ship, is reduced to a two dimensional probalm of
its
cross-sections. The total hydrodynamical properties is found by integrating over the length of the body
The six dynamic variables surge, sway, heave, roll, pitch, and yaw of a ship are expressed in terms of two right-hand cartesian coordinate systems. A moving system which is fixed in the ship, and a fixed system which is fixed in space. The moving system (y1, y2 3) has its origin at the center of gravity of the ship and its three axes
(y1, y2 and y3) coincident with the three principal axes of the ship. The y1-axis is positive toward the bow, the y2-axis is positive to port and the y3-axis is positive upward. This system will move with the
ship and the angular displacements about the three axes are respectively, the roll, the pitch and the yaw of the ship. They are positive for
rotations about the positive directions of y1, y2 and y3 in a counter -clockwise direction. The fixed system (x1, x2, x3) is chosen such that the two coordinate systems are coincident when
the ship is at rest.
Then the three components of translational motion of the origin of the moving system relative to the iixed system are defined as the surge, the sway, and the heave of the ship. The positive directions of forcesand moments are defined in the same way as their corresponding
dis placements.
The forces and moments on a moored ship can be divided into four categories: inertia, damping, restoring and exciting forces and moments. The details of determining these forces and moments are
-3-discussed in [5], and only a brief discussion will be given below.
The inertia forces and moments arise from change of velocities of the ship and water particles around it. Although the change of
velocity of water particles depends on their position in relation to the
ship, it can be assumed that a certain amount of water behaves as if integral with the ship and moves with it. The amount of entrained water is different for different components of motion and the mass or the
moment of inertia of such entrained water is called the added mass or added moment of inertia. By applying the slender body theory, the inertia force or moment on a section of the ship is then equal to the product of the virtual mass (the sum of natural mass and added mass) or the virtual moment of inertia (the sum of natural moment of inertia and added moment of inertia) and acceleration. The added mass and added moment of inertia depend on wave frequency, water depth, shape of ship sections and the clearance between ship sections and side walls.
Damping forces and moments arise from wave generation and arc proportional to the relative velocity between the ship and water
particles. The proportional constants are called the damping coefficients and depend on wave frequency, water depth, shape of ship sections anà
the clearance between ship sections and side walls.
Restoring forces and moments come from three different origins and will be discussed separately in the following.
Hydrostatic restoring forces and moments are due to the buoyancy effect resulting from ship displacement. The total hydrostatic restoring force has only one component in the vertical direction and the hydrostatic restoring moments has components in the roll and pitch direction.
The second restoring forces and moments come from mooring lines. The behavior of a mooring line under tension has been
investi-gated by Wilson [Z]. The relationship between force and elongation is
-4-.
T
c(
L
)'ifLL>O
O
ifALO
where L is the moored length of a mooring line, the change in length due to a tensile force T is ¿L, and c and n are two constants depending on the type of mooring lines. The total restoring force and moment due to all mooring lines have components in all three directions of
trans-lation and rotation0
Another restoring forces and moments come from fenders which will be in action only when they are in contact with the ship. The total restoring force has only one component in the sway directiDn and the total restoring moment has components in the roll and yaw directions.
Finally there are the exciting forces and moments due to water waves. The waves are assumed to be sinusoidal standing or
progres-sive waves and have a unique frequency, then forces and moments
on ship sections can be obtained from certain wave potential.
The six equations of motion for a moored ship are obtained by balancing the various forces and moments discussed above and may
be represented in the following matrix form:
M+C+K+7()= ¡(t)
(1)w h e r e
M = virtual mass and moment of inertia matrix
C = damping matrix
K stiffness matrix due to linear restoring forces and o
moment s
T()
= force vector due to nonlinear restoring forces
-5-¡(t)
= force vector related to water waves = q.cos (ut + ), i 1, . . . , 6= (xi, x2, x3, , , 3)
The first three elements x1, x2, x3 in the displacement vector x represent the surge, the sway and the heave of the moored ship, and the next three elements @, , and e3 represent respectively the roll,
the pitch and the yaw of the ship. The force vector
T()
is nonlinear. If its argumentis replaced by -, T(-) will in general differ from
i(i) in magnitude as well as in sign. HenceT()
is asymmetric. Thequantities q. and represent wave frequency, force or moment amplitude and phase angle for the ith element of the vector
¡(t).
METHOD OF SOLUTIONSince the excitation vector is harmonic, we assumed
tha(an
approximate steady-state solution for the response of the system (1) may take the following form:
x. = z. + y. (Z)
i i i
= z + y. cos (wt + cp) i.
1,... , 6
(3)z. is a constant introduced to account for the asymmetry of
T()
in (1).If?() is symmetric, then
will vanish. y is a harmonic function whoseamplitude and phase are y. and .
Consider a linear system defined by
(4)
where K is an unknown matrix. If K is known, this linear system can readily be solved to give:
in which
V
K+K -Mw2
i o
V2 = Cu.j (9)
and the superscript -1 for a matrix denotes its inverse.
If the exact solution for the linear system is used as an ap-proximate solution for the nonlinear system, direct substitution give s
(z., k..)i
= T() -
K + K13 0
where k.. are the (i, j) element of K and denotes the error vectorb The unknowns k.. are chosen in such a way that the average mean-square error over one cycle defined by the integral
/
W12/
¡z
z Y1 =VW+ w+6
- i i + 6 tan w. iwhere w11s are given by
-6-i = 1,..., 6
(5) (6) (8) (10)Vi
-V2 V2 -V1 -1 q1 Cos q6 cos q1 sin q6 sin)
(7)a a ak.. 13 k.. = O, 13 ZITK z + o
is a minimum. This leads to
= o
i,j
=l,...,6
(12)However, it can be shown that not all k.. s are independent. In order to
13
avoid this difficulty, we choose in this case,
-7-Ut (11)
ifi
j (13)Then, k.. can be uniquely deyermined as follows:
Zu k.. __L
L
cos(G + cp.) de11 Tri'.
1
Six more equations are furnished to determ me z by averaging the nonlinear equations over one cycle which leads to
fi()de
= ojo
(14)
i =
1,... 6
(15)Thus the solution ai the nonlinear system is reduced to the solution of. Equations (5) - (7), (14) and (15). Thcy are nonlinear algebraic equatIons but can be solved numerically by the following iteration approach.
First, set z.
= O and assume a set of values k... Then Equation (7)can be solved by simple matrix inversion and y and zp are determined from Equations (5) and (6), Now a new set of values of k11 and z are
calculated from (14) and (15). This procedure can be repeated until required accuracy is reached.
SUMMARY AND DISCUSSION
An approach to the determination of an approximate solution for the steady-state response of moored ships in six degrees of
freedom has been formulated. This approach can be applied to moored structures in open sea as well as moored ships in harbors. In this paper, the degrees of freedom of the system is specified as six. However, this approach is still valid for degrees of freedom other than six.
The accuracy of an approximate analysis is difficult to predict in general. A different version of this approach
where f() is symmetric
has been employed to problems which possess known exact solution[6], it shows that the accuracy of this approach is well within the limits of practical engineering usefulness.
R EFER ENCES
Wilson, B. W. , "The Energy Problem in the Mooring of
Ships Exposed to Waves, " Proceedings of Princeton Conference on Berthing and Cargo Handling in Exposed Locations, October, 1958. Wilson, B. W. » "Elastic Characteristics of Moorings, n
Journal of the Waterways and Harbors Division, ASCE, WW4,
November, 1967.
Kaplan, P. » and Putz, R. R., HThe Motions of a Moored
Construction-Type Barge in Irregular Waves and Their Influence on
Construction Operation, " Contract NBy-32206, an investigation con-ducted by Marine Advisers, Inc., La Jolla, California for U.S. Naval Civil Engineering Laboratory. Port Hueneme, California, 1962.
Muga, B. J. » "Hydrodynamic Analysis of a Spread-Moored
Platform in the Open Sea, " U. S. Naval Civil Engineering Laboratory, Port Hueneme, California, August, 1966.
-8-
-9-Hwang, L. S., Yang, I. , Divolcy, D. » and Yuen, A.
'A Study of Wave and Ship Behavior at Long Beach Harbor with
Applica-tion to a Modern Container Ship, " An investigaApplica-tion conducted by Tetra Tech, Inc. , Pas aclena, California for the Port of Long Beach, Long Beach, California, 1972.
Yang, I. , 'Stationary Random Response of Multidegree-of-freedom Systems,' California Institute of Technology, Dynamic Laboratory, Report No. DYNL-100, June, 1970.
