mathematical modelling and roll motion

Mathematical Modeling and Understanding Roll
Motion of A Ship
Lt Cdr Swarup Das
sdas [email protected]
TSO-II (Navy), MILIT, Pune 411025
Abstract
The stability of a ship’s motion is very crucial and a better understanding of ship’s stability
in high sea is required to prevent losses due to unstable motion. Mathematical model of a
ship involving factors such as hydrodynamic forces and moments, environmental uncertainties
/ disturbances like wind, wave and ocean currents, unsteady forces due to rudder movement
accounting for all rigid body motion of a ship is discussed in the paper. It is evident that nonlinearities of ship’s fluid system arising from above mentioned factors play in important role in
determining ship’s dynamics or stability. A bifurcation is observed before a ship capsizes when
confronted with wave of very high amplitude. Finally paper brings out clear understanding of
these non linearities and inclusion of these criteria in implementation of control algorithm.
1
Introduction
The righting lever curve of static stability calculated for calm water is still considered as criteria
regarding stability or capsizes of a ship by IMO. Righting lever curves are obtained by dividing
the righting moment by weight of a ship, which is shown at fig 1. This notion of stability is well
accepted but they don’t account for dynamic load, a ship is subjected to when moving at rough sea
and maneuvering with steep rudder angle. A high amplitude wave leads to bifurcation in the motion
Figure 1: Righting lever curve
and stability as predicted conventionally. Model tests conducted at few maritime institutes around
the world showed that current stability criteria do not always correspond to danger of capsizing.
Hydrodynamics forces and environmental factors like currents, wind and wave which can neither
1
be predicted, calculated and incorporated in any model accurately by static criteria nor by linear
theory very much contribute to the non linear dynamic effects and in turn responsible for bifurcation.
The remainder of this article is organized as follows. Section 2 reviews the mathematical model
of a ship describing movement of ship in six degree of freedom. Section 3 and section 4 describes
the thrust components due various hydrodynamic and forces and moments and also contribution of
uncertainty like environmental factor which will eventually modify the model. In section 5 importance of movement of rudder in dynamic loading of ship is discussed. Finally Section 6 summarizes
and explains the importance of these factors to be considered in application of control strategies.
2
Mathematical Model: Newtonian Approach
In this section, mathematical model describing motion of ship in six degree of freedom (DOF)is
derived using Newtonian approach. Translation motion or position of a ship is considered in three
direction namely surge, sway and heave. Similarly rotational motion or orientation of a ship is
considered about three axis namely roll, pitch and yaw.
Figure 2: Standard notation and sign convention for ship motion description
Two frames of references are considered to determine the equation of motion of a moving ship
in 6 DOF. The inertial or fixed to earth frame 0 that may be taken to coincide with the ship-fixed
coordinates in some initial condition and the body-fixed frame 0. While analysing the motion of
a moving ship in 6 DOF, it is convenient to define two co ordinate frames and different variables
are defines according to SNAME notation. For surface ships, the most commonly adopted position
for the body-fixed frame is such it gives hull symmetry about the x0 z0 - plane and approximate
symmetry about the y0 z0 - plane, while the origin of the z0 axis is defined by the calm water surface
[2].
The magnitudes describing the position and orientation of the ship are usually expressed in the
T
T
inertial frame and the coordinates are noted: [x y z] and [φ θ ψ] respectively, whilst the
T
T
T
forces [X Y Z] , moments [K M N ] , linear velocities [u v w] , and angular velocities
T
[p q r] are usually expressed in the body-fixed frame.
Let us define the position-orientation vector η with respect to the inertial frame ν and the
2
linear-angular velocity vector ν with respect to the body-fixed frame as
∆
η = [x
y
∆
ν = [u
v
z
w
φ
p
θ
q
ψ]
T
(1)
r]
T
(2)
Then, the position-orientation rate vector η˙ is related to ν via:
η˙ = J(η)ν
(3)
where J(η) is a transformation matrix that depends on the Euler angles (φ
form as follows[3]:
J1 (φ, θ, ψ)
03X3
J(η) =
03X3
J2 (φ, θ, ψ)
θ
ψ) and is of the
(4)
where


c(ψ)c(θ) −s(ψ)c(φ) + c(ψ)s(θ)s(φ) s(ψ)s(φ) + c(ψ)c(φ)s(θ)
J1 (φ, θ, ψ) = s(ψ)c(θ) c(ψ)c(φ) + s(ψ)s(θ)s(φ) −c(ψ)s(φ) + s(ψ)c(φ)s(θ)
−s(θ)
c(θ)s(φ)
c(θ)c(φ)
(5)
and

1
J2 (φ, θ, ψ) = 0
0

s(φ)t(θ)
c(φ)t(θ)
c(φ)
−s(ψ) 
s(ψ)/c(θ) c(ψ)/c(θ)
(6)
where s(·) = sin(·), c(·) = cos(·) and t(·) = tan(·)
Using the Newtonian approach, the equations of motion of vehicle in the body fixed frame are
given in a vector form by:
MRB ν˙ = τ (ν,
˙ ν, η) − CRB (ν)ν
ν˙ = J(η)ν
(7)
(8)
where MRB is the matrix mass and inertia due to rigid body dynamics, the term CRB (ν)ν arise
from the coriolis and centripetal forces and moments also due to rigid body dynamics, and J(η) is
given in (3). The forces and moments vector τ is defined as
τ = [X
Y
Z
K
M
T
N] ,
(9)
Motions in pitch and heave can generally be neglected in comparison with the other motions for
conventional surface ships; thus, ship motion modeling can be considered only 4-DOF: surge, sway,
yaw and roll. Therefore, from (6) the following approximations can be made:
φ˙ = p
ψ˙ = rcos(φ)
(10)
Considering very small heave and pitch and neglecting them we get equation of motion of ship in 4
DOF as follows:

    

m
0
0
0
u˙
X
m(vr + xG r2 − zg pr
0
    

m
−mzG mxG 
−mur

 v˙  =  Y  + 

(11)
 0 −mzG







Ixx
0
p˙
K
mzG ur
0
mxG
0
Izz
r˙
N
mxG ur
where m is the mass of the ship, Ixx and Izz are the inertias about the x0 and z0 axes, and xG
and zG are the coordinates of the center of gravity CG with respect to the body-fixed frame, i.e.,
CG = [xG , 0, zG ].
3
3
Hydrodynamic Forces and Moments
The radiation induced forces and moments can be identified as sum of three new components
which are Added mass due to inertia of the surrounding fluid, Radiation induced potential damping
due to energy carried away by generated surface waves, Restoring forces due to Archimedes (weight
and buoyancy). The contribution from these three components can be expressed mathematically as:
τR = −MA ν˙ − CA (ν)ν
{z
}
|
added mass
−DP (ν)ν
| {z }
−
potential damping
g(η)
|{z}
(12)
restoring f orces
In addition to radiation induced potential, damping effects like skin friction, wave drift damping and
damping due to vortex shedding are also considered. Linear skin friction due to laminar boundary
layer theory is important while considering low frequency motion. In addition to linear skin friction,
there will be a high frequency contribution due to turbulent boundary layer theory. This is usually
referred to as quadratic or non linear skin friction. Wave drift damping can be interpreted as added
resistance when a ship is moving ahead in waves. This type of damping is derived from second order
wave theory. This contributes maximum when ship is surging ahead at high sea states. This is due
to the fact that wave drift damping is proportional to the square of the wave height. Flow separation
often occurs at sharp corners of the ship and vortex shedding is observed. Mathematically these
damping can be expressed as follows:
−DW (ν)ν
| {z }
τD = −DS (ν)ν
| {z }
skin f riction
wave drif t damping
−
−DM (ν)ν
| {z }
(13)
dmaping due vortex shedding
Therefore hydrodynamic forces and moments τH can be written as the sum of τR and τD , that is:
τH = −MA ν˙ − CA (ν)ν − −CA (ν)ν − −D(ν)ν − g(η)
(14)
whereas total hydrodynamic damping matrix D(ν)ν is defined as:
∆
D(ν)ν = DP (ν) + DS (ν) + DW (ν) + DM (ν)
(15)
Incorporating the hydrodynamic forces and moments in to the motion of equation, we get
[MRB + MA ] + [CRB (ν) + CA (ν)]ν + D(ν)ν + g(η) = τη + τ
(16)
whereas τη is used to describe environmental forces and moments acting on vessel and τ is the
propulsion forces and moments.
4
Environmental Disturbances
We look further in to the aspects of environmental disturbances in a moving ship while carrying
out the modeling. There are in particular three types of environmental disturbances which are wave,
wind and ocean currents. In general these disturbances will be both additive and multiplicative to
the dynamic equation of motion. But here we assume linearity i.e., principle of superposition applies.
Wave forces and Moments are small wavelet due to wind appearing on water surface increases drag
force which in turn allow short waves to grow. These short waves continue to grow until they
finally break and their energy is dissipated. These waves form a spectrum which have been studied
and popularly known as Newman Spectrum, Bretschneider spectrum, Pierson Moskowitz spectrum,
JONSWAP spectrum. Simulation of motion of ship in ocean in the presence of irregular waves were
carried out by super positioning 1st and 2nd order model of wave disturbances. The forces due to
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waves at various directions are calculated as:
Xwave (t) =
N
X
ρgBLT cosβsi (t)
(17)
−ρgBLT sinβsi (t)
(18)
i=1
ywave (t) =
N
X
i=1
Nwave (t) =
N
X
1
ρgBL(L2 − B 2 )T ki 2 sin2βsi 2 (t)
24
i=1
(19)
Where L, B, T represent length, breadth and draft of a ship respectively. ρ is density, g is
acceleration due to gravity, β is the angle between heading and direction of waves and s is the slope
of the wave.
Wind Forces and Moments has significant impact on the values of the dynamic angles of heel[4].
Total wind speed contains slow varying components and high frequency components and the resultant
forces and moments acting on surface vessels are calculated using relative wind speed and angle of
incidence. Wind force and moments are written in the following manner:
1
CX (γR )ρw VR 2 AT
2
1
Ywind = CY (γR )ρw VR 2 AL
2
1
Nwind = CN (γR )ρw VR 2 AL L
2
Xwind =
(20)
(21)
(22)
Where CX CY are force coefficients and CN is moment coefficients. ρw is density of air, AT and AL
are lateral and transverse projected area.
Currents in the upper layer of ocean surface are due to atmospheric winds whereas heat exchange
at the sea surface together with salinity changes develops additional sea current components, usually
referred to as Thermohaline components. Tidal components arising from planetary interaction like
gravity have also an impact. Two distinct methods are there to derive current induced forces and
moments and both these methods are based on the assumptions that equation of motion can be
expressed in terms of relative velocity, i.e., νr = ν − νc where νc is the angular velocity vector due
to current and hence the equation (3) gets modified accordingly.
5
Rudder Forces and Moments
In this section model to calculate the force on the rudder, and then according to its position
and orientation on the hull, geometrically relate this force to the generated forces and moments that
produce motion of the ship.The mathematical model of rudder was first represented in simplified form
by Van Anerongen[9]. The total resulting hydrodynamic force acting on the rudder on a real fluid
T
acts on a single point on the rudder CP (centre of pressure) with coordinates CP = [xcp ycp zcp ]
expressed in fixed body frame. The magnitude of the forcxe can be considered as

δattack
1
ρCF Ar Vav 2 sin( π
if |δattack |<|δstall

2 δstall )

1 2
F =
ρCF Ar Vav 2 sign(δattack )if |δattack | ≥ |δstall
(23)

2


where CF is the lift co efficient, Ar is rudder area, Vav is average flow and δstall , δattack are rudder
stall and attack angle which is calculated using rudder angle δ, v, u at stern produced by the turn
5
rate of the ship (xcp − xG )r as
δattack = δ − δf low
(24)
= δ − arctant(
v + (xcp − xG )r
)
u
(25)
The forces due to rudder on the hull are given
Xrudder = −F (u, Vav , v, r, δ)sin(δ)
Yrudder = F (u, Vav , v, r, δ)cos(δ)
(26)
Zrudder = 0
and the moments are
[Krudder
Mrudder
T
Nrudder ] = (CP − CG)[Xrudder
Yrudder
Zrudder ]
T
(27)
Since the rudder is located behind the propeller, the flow passing the rudder Vav is very much
influenced by the propeller which is shown by Van Berlekom[10].
6
Summary and Conclusions
Experiments have been conducted on both linearised and nonlinear model at various maritime
institutes around the world and ship model basin. The non-linear models describe the dynamic
response of the ships very accurately, and therefore can be utilized at first stage for testing control
strategies [5]. Similarly, the linear models presented are preferred for control application design
to models obtained from system identification. Path following technique was show that non linear
effects play an important role in ship dynamics. It turns out that coupling between roll and yaw
motion is important for capsizing [6]. Codependence between the ship’s position and orientation
with the ship’s stability is also known even at static condition [7]. The rudder speed has a huge
influence on roll motion as well as roll reduction capability[8].
In this paper a effort has been made to understand the kinematics motion of a moving ship in
6 DOF using Newtonian approach. Simple models for wave, wind and current induced force and
moments are understood and incorporated in the main model. The emphasis is placed on expressing
the multivariable non linear equation of motion. The current stability criteria for ships are not
sufficient to assess reliably ship’s stability. A systematic analysis of each ship’s dynamics is necessary
to provide criteria to prevent it from capsizing. Platform stabilisation by uncertainty parametric
estimation will require understanding of mathematical model and the entire factor contributing to
the dynamics of motion of ship. This understanding of this paper can be considered as the ab initio
stage or primary requirement for further application of control strategies for platform stabilisation.
The paper poses several demands on the ship design to enable a substantial roll reduction.
References
[1] IMO: Code on Intact Stability for all Types of Ships Covered by IMO Instruments. Resolution
A.749(18). London: International Maritime Organization (1995)
[2] Price, W. C. and R. E. D. Bishop. ”Probabilistic theory of ship dynamics”. Chapman and Hall,
London. (1974)
[3] Fossen, Thor I. ”Guidance and Control of Ocean Marine Vehicles”. John Wiley and Sons Ltd.
New York. (1994)
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[4] Waldemar Mironiuk ”The Initial Research On Air Flow’s Dynamic Impact On A Ship Model Of
888 Project Type” Journal of KONES Powertrain and Transport, Vol. 18, No. 4 2011
[5] Tristan Perez and Mogens Blanke ”Mathematical Ship Modeling for Control Applications”.
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Petersburg, Russia
[7] Catalin Bara, Mihai Cornoiu and Dumitru Popescu ”An optimal Control Strategy of Ballast
System Used in Ship Stabilisation” 20th Mediterranean Conference on Control Automation
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[8] PGM Van der Klugt, HR Van Nauta Lemke ”Rudder Roll Stabilisation and Ship Design: A
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Automation.
[9] Van Amerongen J ”Effects of propeller loading on rudder efficency” Proceding 4th ship control
system symposium, Haag, pp. 5.38-5.39
[10] van Berlekom, W. B. ”Effects of propeller loading on rudder efficiency.” Proceedings 4th Ship
Control Systems Symposium, Haag. pp. 5.835.98. 1975
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