An Unsteady Vortex Lattice Method to Assess Aspects
of
Safety of Operation for Hydrofoil Craft
E)eft IJniV2TSity of TchnClOY
Sai? HyromcZflCS
Laboratori
Frans van Wairee' and Tan Seng Gie'
LiLTY
Mekelweg 2 - 2628 CD IDeift
The Netherlands
PPone. 31 76373 - Fax: 31 15 781836
The dynamic stability of hydrofoil craft is investigated by means of an unsteady vortex lattice method. The vortex lattice method is applied in a non-linear time domain simulation method insix
degrees of freedom. The computational method includes a control system actuating trailing edge flaps and impact forces acting on the hull bottom when the hull hits the water suiface during a
touchdown manoeuvre. Two validation cases are discussed. The simulation method is appliedto
investigate the dynamic stability and accelerations during an anti-collision manoeuvre, a crash-stop manoeuvre and a flap failure during a turning manoeuvre in wind and waves. The computed accelerations are compared with criteria specified in the International Maritime Organization
High-Speed Craft Code.
1. INTRODUCTION
The use of high speed craft for transport of
passengers and time sensitive goods has considerably
increased during the last decade. Transport of passengers needs ferries with a high degree of comfort, also under less favourable weather conditions. Not only from the viewpoint of passengers, which can do without seasickness, but also from an economic viewpoint of the ferry operator. Furthermore, the use of high speed craft in confined waterways and/or with a high traffic density sets high requirements to the manoeuvrability of these craft types. Manoeuvrability concerns amongst others
the ability to
change course and to perform an
emergency stop within a specified distance. Especially for fully submerged hydrofoil craft, relying on the use
of a ride control system for seakeeping and
marloeuvring, stringent requirements must be set to the
safety of operation during normal and emergency
conditions.
The High-Speed Marine Vehicles Committee of the
ITTC (1996), following the 1MO Code of Safety for High-Speed Craft (1994), recommends the development,
validation and application of computational methods to investigate the comfort and safety of operation of high speed craft. For hydrofoil craft computational methods
'Maritime Research Institute Netherlands (MARIN)
161 ABSTRA CT
are especially needed since performing model lests is complicated, if not impossible, and thus costly. This is due to the high speed which needs large model basins with high speed carriages and oblique wave generators, the costs of manufacturing foil systems at model scale, the use of ride control systems which require expensive and complicated actuation mechanisms for flap control and the need for model scale propulsion units for self propelled tests.
The 1MO HSC Code does not specify specific
requirements to the dynamic stability of hydrofoils. The
complexity of this item is recognized and further
research into this field is recommended. The HSC Code merely states that the safety of operation should initially
be demonstrated by means of model testing and/or
mathematical analysis and finally by means of full scale
trials. For hydrofoil craft with fully submerged foils operating in the transient (take-off) and foilborne modes,
the HSC Code states: "The stability should be examined by the use of verified computer simulations to evaluate the craft's motions, behaviour and responses under the normal conditions and limits of operation and under the influence of any malfunction". For instance directional
control systems should be included in a failure mode and effect analysis whereby adverse effects on the
controllability
in the event of a involuntary
totaldeflection of any control system device must be
investigated. Annex 8 gives a list of possible failures in
propulsion, control and electrical systems to be
examined. Annex 3 describes criteria in terms of
maximum horizontal accelerations not to be exceeded
for 'minor' and 'major' effects resulting from system malfunctions.
For hydrofoil craft the following handling aspects
during normal operation and subsequent to control
system failures are of interest: the stability in the take-off and foilborne modes in heave, roll, pitch and yaw,
turning, and normal and emergency stopping. For
hydrofoil craft a critical condition may be for instance
a control system failure while performing a turning manoeuvre whereby the craft assumes a certain roll
angle to counteract the centrifugal forces. If one or more
flaps become disabled, the craft must not capsize but
must be able to resume a safe position.
A number of such aspects are investigated in the
present paper on basis of results from a computational
method. This method is briefly described in the next
section. For a more detailed description one is referred
to Van Walree (1999). Next, two validation cases on
basis of full scale trials and model tests are discussed.
For the present problem a time domain simulation method in six degrees of freedom is required to include
non-linearitjes and transient effects. From a
hydrodynamic point of view, free surface and foil interaction effects need to be included to obtain an accurate description of forces acting on foil systems.
The computational method has to be suited for handling transient and oscillatory motions simultaneously: manoeuvres are usually performed while waves are present.
Furthermore, the method must be suited to take into account finite aspect ratio hydrofoil configurations with
a varying planform due to taper, sweep, and dihedral
angles, and the presence of supporting struts and partial span trailing edge flaps. A ride control system, actuating
trailing edge flaps, needs to be included. Also, the
forward speed is not necessarily constant.
The hydrofoil craft is assumed to operate in a
foilborne mode, although impacts between the hull
bottom and wave crests are accounted for, for instance for analysing touchdown procedures.
Mathematical formulation
The problem is described in a space-fixed Cartesian coordinate system, see Figure 1. A thin, finite aspect
ratio lifting surface of arbitrary planform is considered
that performs arbitrary motions in
six degrees of
freedom, below the free surface. Parts of the lifting
surface may be surface piercing. Only the submerged
parts are considered here, although the submergence
may vary in time.
z0
-SF-A vortex sheet exists, consisting of a bound vortex sheet representing the lifting surface and a free vortex
sheet that represents the wake. The position and velocity of the bound vortex sheet are assumed to be known
from the equations of motions of the hydrofoil craft.
The formulation of the problem is based on a
mathematical formulation for large-amplitude ship
motions by Lin and Yue (1990). Their formulation is
based on the use of impulsive strength sources to
represent the unsteady flow about ship hulls. Here a formulation based on impulsive strength vortex elements
is derived.
The motions in the fluid are described by a velocity
potential D:
4 (x ,t) = I(t) +
T - 1(x,t)where is the space-fixed position vector, t is time, c1 is the disturbance potential associated with the vortex
sheets and c is the incident wave potential. The
incident wave potential is known a priori and can be
shown to satisfy the conditions in eq. 's (3) and (4). The definition of a potential function for sinusoidal waves is as follows:
X0
(I) Figure 1 Coordinate axes and vortex sheets 2. UNSTEADY VORTEX LATTICE METHOD
R /(x0
-
)2 +(y0 -11)2 + (z0 - Ç)ZR0=(x0 - )2
+ (y 11) +(z0 +Ç)- (9)
r/(x)2
+(y0-i)2
A boundary integral formulation for the problem is derived by applying Green's second identity to the
potential and Green's function in the fluid domain. By using the tangential flow condition, eq. (5) on the lifting surface, the following formulation may be derived, see
Van Walree (1999):
V -
'
I =_{f f
(q,t) dS + a 4itanan
p S,.,(:) p g (10) dS}fdcf
f
(qt)
aran ¿In o S,,lt> p qEquation (10) must be solved for the unknown doublet
strength .i(q,t). First,
a wake model needs to be
established in which the vortex strength in and thelocation and shape of the wake sheet are specified. Wake model
The Kutta condition for steady flow is that the velocity along the trailing edge of lifting surfaces
remains finite. The thin wing theory equivalent is that
the vortex strength at the trailing edge must be zero. This is satisfied in a vortex lattice method by
representing the continuous vortex distribution by a set of vortex elements with a bound vortex segment at the quarter chord position and by requiring tangential flow at the three-quarter chord position of the eJement.
In unsteady flow the same approach can be used. At the same time vorticity must be shed from the lifting surface into the wake sheet in order to satisfy the Kelvin
condition: in a potential flow the circulation F around
a contour enclosing the lifting surface and its wake must
be conserved. By using vortex elements on the lifting surface and on the wake sheet as a discretization of a continuous vortex sheet and by transferring at each time step the circulation at the trailing edge elements into the wake elements, this requirement is satisfied. Once shed, the circulation strength of wake sheet elements remains
constant.
The second requirement for the wake model is that the wake sheet should be force free as it is not a solid surface; no pressure difference must be present between
(Dl =
.._e'sin(k_x0cor)
(2)where Ç, is the wave amplitude, g is the gravitational constant, w is the wave frequency and k is the wave number(k-w2/g).The disturbance potential (D satisfies
the Laplace equation:
V2« =0
(3)On the undisturbed free surface SF (t), the following
linearized condition is imposed (for t>0):
rO (4)
On the instantaneous lifting surface S8(t)the tangential flow condition is imposed (for t>0):
y
= + (5)¿In ¿In
where V, is the instantaneous velocity of the lifting
surface in a direction normal to its camber surface. The
tangential flow condition is applied at the reference
plane of the lifting surface. The a/an operator denotes the derivative in normal direction, a/an=n.V. The unit normal vector is positive into the fluid domain. The conditions at infinity (S... ) are (for t>0):
(D -40 and 2. o
(6)Apart from incoming waves, the fluid is at rest at the start of the process, the initial conditions on the free
surface 5F (r) are then (for t=0):
(D = - =o (7)
¿It
The transient free surface Green's function is introduced for a submerged vortex with an impulsive strength:
G(p,r;q,t) = =
±
-
±
+R R0
(8)
2S{ 1 -cos
(/(t -t))]
e°'J0(kr)dk
for pq,
tt
where p(x0,y0,z0)
and q(,i,Ç)
are the field and singularity point coordinates respectively,t
is a pasttime variable, G° is the vortex plus biplane image part
and G is the free surface memory part of the Green's
the upper and lower sides of the sheet. This can be
accomplished by displacing the vortex element corner
points with the local fluid velocity. This requires
however an impractically large amount of computer time. Therefore, wake sheet elements are kept in a
stationary position, once shed. This simplification turns out to have only little effect on the forces acting on the lifting surface.
Discretization
Instead of using doublet elements for discretization
of the vortex sheets, vortex ring elements carrying a
circulation strength F are used. These vortex elements consist of four discrete, straight vortex lines of constant strength which enclose the quadrilateral element area, see Figure 2. The induced velocity due to a vortex ring element is identical to that of a constant strength doublet
element if F=1.i.
to
LOCATION OF TRAILING EDGE AT SUCCESSIVE TIME STEPS
WAKE VORTEX aEMENTS
/ /4'
/
/7 /A'
z0
'o
CONTROL POINT
BOUND VORTEX ELEMENTS
Figure 2 Bound and wake vortex lattices The vortex elements are uniformly distributed over
both the chord and the span. The vortex elements are
located on the reference plane of the lifting surface. The
leading segment of the vortex ring is placed on the
elements quarter chord line. The control point where the
tangential flow condition is applied is at the three-quarter point of the vortex element, at the spanwise centre.
Time stepping process
At t=0 the lifting surface is impulsively set into motion. At this instant, the circulation on the lifting surface is
determined for the condition without wake vortex
elements. The last spanwise vortex line just behind the
trailing edge represents the
startvortex. At each
subsequent time step, the lifting surface is advanced to a new position with an instantaneous velocity. Both the
position and velocity are known from the equations of
motion. The gap between the instantaneous trailing edge vortex element on the lifting surface and the wake
vortex element shed in the previous time step is filled
with a new wake vortex element. In this way, the wake
vortex element at the
trailing edge has the same
orientation as the flow leaving the trailing edge, to a
first order approximation.
The circulation strength of the new wake vortex
elements is set equal to that of the trailing vortex
element of the previous time step. With a known wake vortex position and circulation, lifting surface position and velocity, the tangential flow condition, eq. (10) can
be solved for the unknown circulation on the lifting
surface. Once the circulation is known, the forces acting on the lifting surface are determined by using the
Kutta-Joukowsky law. Control system
Foils are generally equipped with trailing edge flaps
actuated by a control system. The control system used in the computational method is
based on Linear
Optimum Control theory, see Anderson and Moore
(1989), and can be applied to arbitrary hydrofoil craft. For the purposes of determining the optimum control model, the linearized hydrodynamic characteristics of the craft are used. The control system actuates the flaps
by using Proportional (P), Integral (1) and Differential (D) coefficients as follows:
ro'
+JJ[x-x]dt
+(Il)
I)
where x is the required position vector, , is the actual position vector, i is the required velocity vector and
i
is the actual velocity vector. Hull forcesWhen operating in a foilborne mode impact forces on the hull bottom due to contact with the water surface are determined from a method described by Matusiak and Rantanen (1986). Hereby the hull is represented by
a number of sections
with known hydrodynamiccharacteristics. The impact force F, is obtained from the rate of change of the momentum of the flow, expressed
in terms of the added mass and the velocity of a
section relative to the water surface Vr:
dV dm
F.=m_-L+V
° dt
rdt
Besides impact forces, only buoyancy and quasi-steady resistance forces are taken into account
to end
touchdown simulations. Wave induced loads and
motions for craft operating in the huilborne mode are not of interest for the present purpose.
Hydsim computer program
The computer program Hydsim contains a
non-linear time domain simulation method in six degrees of
freedom. The unsteady vortex lattice method and the
control system model are used in Hydsim to determine
the forces acting on the foil system. Furthermore,
modules are available to compute the characteristics of propeller systems and the wind forces acting on hulls.
These force components are determined in a quasi-steady manner. Hydsim contains the equations of motion
for the propeller rate of rotation. The user must specify the torque acting on the propeller axis. By means of the open-water characteristics of propeller series the thrust force is determined which, together with the resistance of the craft, determines the speed of advance.
3. VALIDATION OF HYDSIM
The first case concerns the manoeuvring behaviour of the Jetfoil hydrofoil craft. Figure 3 shows the general lay-out of the Jetfoil with its control system
components.
(12)
Full-scale trials data are obtained from Saitoet al. (1990). Figure 4 shows time traces for a coordinated turning manoeuvre whereby the helmsman suddenly sets
a certain helm command H proportional to the required
yaw rate dqíId:. At this command, the entire forward foil
is rotated (strut angle 6) and the craft assumes a certain roll angle and yaw rate. It is seen that there exists a fairly good agreement between the trials and simulation data. The roll angle during the turn is reasonably well predicted, but in the simulation the roll response of the
craft is too fast. The same holds for the yaw rate and
the lateral acceleration at the bow, y. This discrepancy may perhaps be caused by wave disturbances during the trials. These are present at least before the start of the manoeuvre where the strut angle is equal to zero.
Figure 3 Jetfoil configuration Figure 5 Foil system geometry
Figure 4 Comparison turning manoeuvre
The second case deals with model test results for a
fully submerged, tandem foil system. Figure 5 shows the
foil system arrangement. For the development of a ride
control system an extensive series of oscillation and
Cire 00 - k 0.00 0.02 0.04 0.06 0.08 0.02 - Hydsim - - - Expenment o Forward foil O Aftfoil A Forward+aft foil A A
.
s
.
0.02 0.04 0.06 0.08 CK/rÇ 0.35 0.30 0.25 0.20 0.15 0.10 0.05 o.0&0o - Hydsim - - Experiment o Forward foil o Aft foil "A Forward+aft foilA A A A
s
e,.
.
0.02 0.04 0.06 0.08 kFigure 9 Comparison roll moment amplitude
Figures 6 and 7 show the amplitudes of the
normalized heave force and roll moment acting on the foil system during oscillatory pitch and roll motions,
respectively. The horizontal axis shows the reduced frequency k=wc/2U,
where w is
the oscillationfrequency, c is the mean chord of the foil system and U is
the speed of advance. The agreement between
experimental and calculation results is satisfactory. Figures 8 and 9 show the normalized wave induced
heave force and roll moment for the
foil systemoperating in regular beam waves. Here k is based on the wave frequency. The agreement is again satisfactory.
4. HYDSIM SIMULATIONS
A series of Hydsim simulations
have beenperformed for a hydrofoil craft equipped with the foil system shown in Figure 5. The main particulars of this craft are shown in Table 1.
Table 1 Main particulars of hydrofoil craft
Hull length 25.00 m
Hull beam 6.00 m
Weight 55 ton
Cruise speed 36 knots
Foil submergence 1.20 m Hull clearance 1.00 m 1.50 - Hydsim- - - Experiment o Forward foil 1.25 O Aft foil
.
A Forward+aft foil 1.00.
0.75i
.---
i
0.50 0.25 000.,
i k 0.02 0.00 0.04 0.06 0.08 0.10 0.12Figure 6 Comparison heave force amplitude
C KIr, 4.00 - Hydsim Experiment Forward foil A A---A--3.50 OAftfoil A Forward+aft foil 3.00 2.50 2.00 1.50
Figure 8 Comparison heave force amplitude
Figure 7 Comparison roll moment amplitude
C,ir. 0.14-0.12 0.10 0.06 0.04
.
s
The tip flaps (1,4,5,8) are used to control the roll
motion, the centre flaps (2,3,6,7) are used to control
heave and pitch while the rudders on the aft foil struts (9,10) are used to control the yaw motion. The craft is
equipped with two
propellers. Unless mentionedotherwise, all simulations were performed for the craft operating in wind and waves. A wind velocity of 10 mIs was used whereby low frequency variations in velocity and direction were applied. The significant wave height was LO m and the peak period was 6 sec. A Jonswap
type wave spectrum was used. The wind and wave
directions were 180 deg., i.e. on the bow at the start of the simulations.
Anti-collision manoeuvre
The first simulation cases concern a manoeuvre
whereby the craft suddenly changes course to avoida
collision. A required yaw
rateof 6 deglsec
in combination with a roll angle of 12 deg. to port are commanded to the controller at the startof the
manoeuvre. When a transverse displacementy0=5Om is reached the craft starts to return to its original course. Two cases are considered: a 'tight' (Case 1) and a more oose' (Case 2) control of the roll and yaw angles and the yaw rate. For Case I the P, I and D coefficientsare
determined on basis of a performance index in which the craft response to the required roll and yaw modes of motion has a relatively high weight in comparison to the
other modes of motion. For Case 2, the modes of
motion are more equally weighted.
Figure 10 shows the trajectories of the craft. The
overshoot for Case 2 is appreciably higher than for Case 1 where the craft is able to resume its original course when a transverse displacement of 100 m is reached.
Figure 11 shows the time traces of the roll and yaw
angles, 4) and
and the yaw rate diIdr. The tight
control action for Case I results in almost impulsive
craft responses to the required yaw rate and roll angle.
As a result the transverse acceleration at the bowy shows sharp peak values when the yaw rate is changed, see Figure 12.
In Annex 3 of the IMO-HSC Code horizontal
accelerations of 0.7g. as occurring for Case 1, are categorized as hazardous effects (Level 3) and
passengers must wear seat belts to prevent them falling out of their seats. For Case 2 the maximum acceleration
is 0.35g which falls in the category of major effects (Level 2). y0 (m) 140 120 lOO 80 60 40 20
- Case I
Case 2 E x0 (m) 0 100 200 300 400 500 600Figure lO Trajectories y0-x0
4(deg) 60 í(deg) 50 dW/dt (deg/sec) 30 "II Case 1 Case2 14! d14!/dt 20 10 -.10 -20 10 20 Time (Sec) 30
Figure 11 Roll, yaw and yaw rate vs. time
Crash stop
The next three cases show the craft response to a
crash stop procedure whereby the craft must transfer
from a foilborne to a huliborne position as quickly as possible. For all cases the torque acting on the propeller axis was reduced to zero wíthin 4 seconds after the start
of the procedure, at t=15 sec. Case I concerns a
simulation whereby the required hull clearance (1 m)
was not altered during the crash stop procedure. As speed reduces, the control system will then gradually
increase the centre flap angles until the maximum
deflection is reached. For Case 2, a faster touchdown is achieved by specifying a negative hull clearance (-1.0
m) at the start of the procedure. The third case is
identical to Case 1, but without the presence of wind
and waves.
The time traces of the speed of advance U are shown in Figure 13. Due to the absence of wind and waves, the speed increases initially
for Case
3.Commanding a negative hull clearance for Case 2 clearly reduces the speed more quickly than for the other cases. Note that at some instant the propeller thrust may be reversed to come to a complete stop.
Figure 14 shows the time traces of the forward central flap angle, The maximum flap angle used is 20 deg. Figure 15 shows the hull clearance h at the stern of the craft, relative to the water surface. When the clearance becomes negative the hull is in contact with the water
surface. For Case 2 this happens within one second
from the start of the procedure (rl6 sec), for the other cases it happens only after some 15 seconds(r=30 sec). The effects
of these
different strategies and conditions on the vertical acceleration at the bow z areshown in Figure 16. For Case 2 first a negative peak of 08g is seen due to the sudden negative flap deflections.
This peak is immediately followed by a positive acceleration peak of O.8g due to the impact loads as the hull touches the water surface. For Case 1 the negative
peak is absent while the positive peak is reduced to 0.2g. For Case 3, without wind and waves, the peak value reduces further. The 1MO HSC Code does not
specify safety levels based on vertical accelerations, but the 0.8g peak values for Case 2 are quite uncomfortable. The maximum horizontal acceleration is 0.3g for Case
2 and 01g for Cases i and 3 (not shown in the
Figures). The 1MO HSC Code specifies a maximum horizontal acceleration of 0.2g for normal stopping
procedures and O.35g for emergency stop procedures so that the crash-stop procedure used is acceptable for all cases, at least when judging the horizontal acceleration.
U (ki)
2(deg)
Figure 14 Deflection angle of flap ô, vs. time 40 35 30 25 20 15 io 5 0 Case i Case 2 Case 3 Time (sec) 10 20 30 40 30-20 10 -10 -20 -30 -Case i Case 2 Case 3 Time (sec) 0 20 30 40
Figure 13 Speed of advance vs. time
Figure 15 Hull clearance vs. time
h,(m) 2 Case I Case 2 Case 3 Time (sec) iO 20 30 40
IO 20 Case 1 Case 2 Case 3 Time (sec) 30 40
Figures 17, 18 and 19 show the roll angle c and yaw
rate d/dt while Figures 20, 21 and 22 show the
transverse and vertical accelerations y and z at the
bow. Figure 17 shows that if no action is taken the ship will capsize: the roll angle rapidly increases to values
over 30 deg. For Case 2 this happens sooner than for Case I
due to
the beam wind which contributessignificantly to the roll moment. For Case I the positive
8-value counteracts the wind moment. The
accelerations show peak values at the instant that the hull hits the water surface. The peak values should be used in a qualitative manner since the roll angle is quite high at these instants and the method to compute impact forces is not very accurate for asymmetrical hull shapes.
For the Cases 3 and 4 the turning manoeuvre is
stopped by setting the required yaw rate and roll angle to zero in the controller, at the same instant that the flap
failure occurs. Furthermore, a normal touchdown procedure is initiated, similar to Case 1 in the crash stop
manoeuvre. Figure 18 shows that this is effective for Case 3, but for Case 4 the craft still capsizes. The
transverse acceleration peak is due to the sudden change in roll angle and yaw rate.
For Case 5
it is sufficient to stop the turning manoeuvre for a continuation of the flight without capsizing. Note that the craft responds with a largerovershoot in roll due to the flap failure. For Case 6, a
safe position can be attained by performing an emergency touchdown procedure, similar to Case 2 in
the crash stop manoeuvre. The vertical acceleration
shows again high peak values prior to and at the instant that the hull hits the water surface.
4(deg) dssIdt (deg/sec) 30 -30 ¡0 20 Case i Case 2 Time (Sec) 30 40 Case (deg) Action Effect +20 none capsize 2 -20 none capsize 3 +20 stop turning, touchdown safe 4 -20 stop turning, touchdown capsize
5 +20 stop turning safe
6 -20 stop turning,
emergency
touchdown
safe
Figure 17 Roll angle and yaw rate vs. time Figure 16 Vertical acceleration vs. time
Flap failure during turning manoeuvre
During a turning manoeuvre a flap failure occurs in
the sense that the port tip flap (ö) of the aft foil
suddenly is set at its maximum deflection angle, either upwards or downwards. Such a failure is recommended for investigation in the 1MO HSC Code, Annex 8. The
yaw rate is 6 deg/sec and the roll angle is 12 deg. to
port, similar to the anti-collision manoeuvre. The flap failure occurs at t=15 sec, when the yaw angle is about
90 deg. and the wind and waves come in from
starboard. Six simulations have been performed with different actions taken to bring the craft into a safe
position. These are summarized in Table 2. Table 2 Simulation cases
z (g) 1.0 0.5 T' 0.0 -0.5 20 10 d4/dt -Io o
(deg) dV/dt (deg/sec) (deg) dsy/dt (deg/sec) 30 20 10 -10 -20 30 Case 5 Case 6 5. CONCLUSIONS
The simulation results illustrate the possibilities of non-linear time domain simulation methods to
investigate the dynamic stability of hydrofoil craft under
normal and emergency conditions.
Impact loads on the hull bottom during emergency touchdown procedures result in large accelerations for passengers and crew.
In
some cases considered the
hydrofoil craftcapsizes when a flap failure occurs during a turning
manoeuvre in waves and wind. High vertical
accelerations due to an emergency touchdown have to be accepted to avoid capsizing.
y(g)
z (g) 1.0 0.5 0.0 -0.5 -1.0 Case5 Case6 y a. Time (Sec) IO 20 30 40 20 10 -10 20 -30 d41/dt Case 3 Case 4 Time (sec) 10 20 30 40Figure 18 Roll angle and yaw rate vs. time Figure 21 Transverse and vertical accelerations
vs. time
Figure 19 Roll angle and yaw rate vs. time Figure 22 Transverse and vertical accelerations
vs. time
Figure 20 Transverse and vertical accelerations vs. time
40
Time (sec)
References
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the Time Domain.
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Marine Dynamics Symposium, Society
of Naval
Architects of Japan, Japan 1990.
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