The Longitudinal Static Stability of an Aerodynamically Alleviated Marine Vehicle, a Mathematical Model

Abstract

An assessment of the relative speeds and payload capacities of airborne and waterborne vehicles highlights a gap that can be usefully filled by a new vehicle concept, utilizing both hydrodynamic and aerodynamic forces. A high-speed marine vehicle equipped with aerodynamic surfaces is one such concept. In 1904, Bryan & Williams (Bryan & Williams 1904 Proc. R. Soc. Lond. 73, 100–116 (doi:10.1098/rspl.1904.0017)) published an— article on the longitudinal dynamics of aerial gliders, and this approach remains the foundation of all the mathematical models studying the dynamics of airborne vehicles. In 1932, Perring & Glauert (Perring & Glauert 1932 Reports and Memoranda no. 1493) presented a mathematical approach to study the dynamics of seaplanes experiencing the planing effect. From this work, planing theory has developed. The authors propose a unified mathematical model to study the longitudinal stability of a high-speed planing marine vehicle with aerodynamic surfaces. A kinematics framework is developed. Then, taking into account the aerodynamic, hydrostatic and hydrodynamic forces, the full equations of motion, using a small perturbation assumption, are derived and solved specifically for this concept. This technique reveals a new static stability criterion that can be used to characterize the longitudinal stability of high-speed planing vehicles with aerodynamic surfaces.

Full text

Page 1

The Longitudinal Static Stability of an
Aerodynamically Alleviated Marine
Vehicle, a Mathematical Model
By Maurizio Collu, Minoo H. Patel and Florent Trarieux
Department of Offshore Engineering & Renewable Energy, School of Engineering,
Cranfield University, Cranfield MK43 0AL, UK
An assessment of the relative speeds and payload capacities of airborne and wa-
terborne vehicles highlights a gap which can be usefully filled by a new vehicle
concept, utilizing both hydrodynamic and aerodynamic forces. A high speed ma-
rine vehicle equipped with aerodynamic surfaces is one such concept. In 1904, Bryan
& Williams published an article on the longitudinal dynamics of aerial gliders, and
this approach remains the foundation of all the mathematical models studying the
dynamics of airborne vehicles. In 1932, Perring & Glauert presented a mathematical
approach to study the dynamics of seaplanes experiencing the planing effect. From
this work, planing theory has developed. The authors propose a unified mathemat-
ical model to study the longitudinal stability of a high speed planing marine vehicle
with aerodynamic surfaces. A kinematics framework is developed. Then, taking into
account the aerodynamic, hydrostatic and hydrodynamic forces, the full equations
of motion, using a small perturbation assumption, are derived and solved specifi-
cally for this concept. This technique reveals a new static stability criterion that
can be used to characterize the longitudinal stability of high speed planing vehicles
with aerodynamic surfaces.
Keywords: marine vehicle; dynamics; stability; aerodynamic alleviation; wing
in ground; planing
1. Introduction
Context During the last five decades, interest in High Speed Marine Vehicles
(HSMV) has been increasing for both commercial and military use, leading to new
configurations and further development of already existing configurations (Clark
et al. 2004). To create vehicles capable of carrying more payload both farther and
faster, many concepts have been proposed, and they can be classified analysing
the force that can be employed to sustain the weight of a HSMV: hydrostatic lift
(buoyancy), powered aerostatic lift, hydrodynamic lift.
Buoyancy is the lift force most commonly used by ships. Marine vehicles that
exploit only buoyancy to sustain their weight are usually called displacement vessels.
For high speed marine vehicles it is not feasible to use only buoyancy, as in this
case the buoyancy force is proportional to the displaced water volume, and at high
speed it is better to minimize this parameter, since as more vehicle volume (and
the wetted surface) is immersed in the water the higher the hydrodynamic drag will
be.
Article submitted to Royal SocietyTEX Paper
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,
Volume 466, Number 2116, 8 April 2010, Pages 1055-1075

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2M. Collu et al.
The Air Cushion Vehicles (ACV) class use a cushion of air at a pressure higher
than atmospheric to minimize contact with the water, thus minimizing hydrody-
namic drag. The air cushion is not closed, and an air flux keeps the pressure in the
cushion high. This system is called ‘powered aerostatic lift’.
At high speeds a marine vehicle experiences ‘hydrodynamic lift’, due to the fact
that the vehicle is planing over the water surface. This hydrodynamic lift supports
the weight otherwise sustained by buoyancy or, through increasing the speed, can
also replace in part or wholly the buoyancy force. Planing craft, high speed cata-
marans and other similar configurations use this principle to attain high speeds. If,
instead of a simple planing hull, a surface similar to an aerofoil is used underwater,
a hydrofoil is obtained. Basically, while in planing mode the hydrodynamic lift is
generated by only one surface, the wetted surface of the hull, hydrofoils experience
an effect similar to aerofoils, since the hydrodynamic lift is the difference between
the pressure acting on the lower surface and the pressure on the upper surface.
A high speed marine vehicle can use two or all these three kinds of forces to
sustain its weight. For example, a SES (Surface Effect Ship) consists of a catamaran
hull configuration plus a powered air cushion with a front and a rear skirt in the
space between the hulls. Therefore it experiences both hydrostatic and powered
aerostatic lift.
There is another lift force that can be exploited to ‘alleviate’ the weight of the
vehicle, leading to reduced buoyancy and therefore to decreased hydrodynamic drag:
this is aerodynamic lift. There is an extreme case where the aerodynamic forces are
sustaining 100% of the weight of the vehicle: WIG (Wing In Ground effect) vehicles.
A WIG vehicle is a vehicle designed to exploit the aerodynamic effect called
“wing in ground effect”. Extensive literature can be found on this effect and the
vehicles exploiting it (Rozhdestvensky 2006), and only a brief introduction is given
here. Given a conventional aerodynamic surface of fixed geometry, aerodynamic
forces (lift, drag, and moment) acting on it depend on two variables: the speed of
the surface relative to the air, and the angle of attack, defined as the angle between
the chord of the wing and the speed direction. When this aerodynamic surface
operates at a height above the surface, equal or lower than, roughly, one third of
its span length, the aerodynamic forces experienced start to be dependent not only
on the aforementioned parameters, but also on the height above the surface. The
quality (plus or minus) and the quantity of these changes depend on the geometry
of the wing, but in general it can be said that, reducing the height above the surface,
with other parameters fixed, increases lift and decreases drag, leading to enhanced
aerodynamic efficiency.
‘Aerodynamically Alleviated Marine Vehicles’ (AAMV)
An Aerodynamically Alleviated Marine Vehicle (AAMV) is a high speed marine
vehicle designed to exploit, in its cruise phase, aerodynamic lift force, using one or
more aerodynamic surfaces.
2. Problem Statement
This work illustrates a mathematical method to study the dynamics of an AAMV; a
vehicle designed to exploit hydrodynamic and aerodynamic forces of the same order
of magnitude, to sustain its weight. Methodologies for aircraft and marine craft exist
and are well documented, but air and marine vehicles have always been investigated
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Longitudinal Static Stability of an AAMV3
with a rather different approach. Marine vehicles have been studied analyzing very
accurately hydrostatic and hydrodynamic forces, approximating very roughly the
aerodynamic forces acting on the vehicle. On the contrary, the dynamics of Wing In
Ground effect (WIG) vehicles has been modelled focusing mainly on aerodynamic
forces, paying much less attention to hydrostatic and hydrodynamic forces.
An AAMV experiences aerodynamic and hydrodynamic forces of the same or-
der of magnitude, therefore neither the high speed marine vehicles nor the airborne
vehicles models of dynamics can cover and fully explain the AAMV dynamics. The
main objective of this work is to bridge this gap by developing a new model of dy-
namics, in the small disturbances framework, consisting of a system of equations of
motion that take into account the equal importance of aerodynamic and hydrody-
namic forces. This mathematical model is further developed to estimate the static
and dynamic stability of an AAMV.
3. Literature review
The hybrid nature of the model developed, hybrid between model of dynamics used
for WIG vehicles and for planing craft, is mirrored by the literature review below.
Furthermore, a section on literature review studying vehicles that can be classified
as AAMV is presented.
Wing In Ground (WIG) vehicles Research on WIG vehicles has mainly been
carried out in the former Soviet Union, where they were known as ‘Ekranoplans’.
The Central Hydrofoil Design Bureau, under the guidance of R. E. Alekseev, de-
veloped several test craft and the first production ekranoplans: Orlyonok and Lun
types (Kolyzaev et al. 2000).
In the meantime, several research programs were undertaken in the west to
better understand the peculiar dynamics of vehicles flying in ground effect (IGE).
Irodov (1970) and Rozhdestvensky (1996) made important contributions to the
development of WIG vehicles dynamic models.
In the 60’s and the 70’s Kumar (1968a, 1968b) started research in this area
at Cranfield University. He carried out several experiments with a small test craft
and provided the equations of motion, the dimensionless stability derivatives and
studied the stability issues of a vehicle flying IGE.
Staufenbiel & Bao-Tzang (1977) in the 70’s carried out extensive work on the
influence of aerodynamic surface characteristics on the longitudinal stability in wing
in ground effect. Several considerations about the aerofoil shape, the wing planform
and other aerodynamic elements were presented, in comparison with the experi-
mental data obtained with the experimental WIG vehicle X-114 built by Rhein-
Flugzeugbau in Germany in the 70’s. The equations of motion for a vehicle flying
IGE were defined, including non linear effects.
Hall, in 1994, extended the work of Kumar, modifying the equations of motion
of the vehicle flying IGE, taking into account the influence of perturbations in pitch
on the height above the surface.
More recently Chun & Chang (2002) evaluated the stability derivatives for a 20
passenger WIG vehicle, based on wind tunnel results together with a vortex lattice
method code. Using the work of Kumar and Staufenbiel, the static and dynamic
stability characteristics were investigated.
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4M. Collu et al.
Planing craft Research on high speed planing started in the early twenti-
eth century for the design of seaplanes (Perring & Glauert 1932). Later research
focused on applications to design planing boats and hydrofoil craft. During the pe-
riod between 60’s and 90’s, many experiments were carried out and new theoretical
formulations proposed.
Savitsky (1964) carried out an extensive experimental program on prismatic
planing hulls and obtained some empirical equations to calculate forces and mo-
ments acting on planing vessels. He also provided simple computational procedures
to calculate the running attitude of the planing craft (trim angle, draught), power
requirements and also the stability characteristics of the vehicle.
Martin (1978) derived a set of equations of motion for the surge, pitch and heave
degrees of freedom and demonstrated that surge can be decoupled from heave and
pitch motion.
Troesch and Falzarano (Troesch 1992, Troesch & Falzarano 1993) studied the
nonlinear integro-differential equations of motion and carried out several exper-
iments to develop a set of coupled ordinary differential equations with constant
coefficients, suitable for modern methods of dynamical systems analysis.
Hicks et al. (1995) later extended their previous work and expanded the nonlin-
ear hydrodynamic force equations of Zarnick (1978) using Taylor series up to the
third order, obtaining a form of equation of motion suitable for path following or
continuation methods.
Aerodynamically Alleviated Marine Vehicles (AAMV) In 1976, Shipps
analysed a new kind of tunnel hull race boat. The advantages of this new configu-
ration come from the aerodynamic lift. In 1978, Ward et al. published an article on
the design and performance of a ram wing planing craft: the KUDU II. This vehi-
cle, which consists in two planing sponsons separated by a wing section, was able
to run at 78 kts (almost 145 km/h), thanks to the aerodynamic lift alleviation. In
1978, Kallio performed comparative tests between the KUDU II and the KAAMA.
The KAAMA is a conventional mono hull planing craft. The data obtained during
comparative trials showed that the KUDU II pitch motion, in sea state 2, at about
40 to 60 knots, was about 30% to 60% lower than the conventional planing hull
KAAMA.
In 1997, Doctors proposed a new configuration called ‘Ekranocat’ for which he
mentioned the ‘aerodynamic alleviation concept’. The weight of the catamaran was
alleviated by aerodynamic lift, thanks to a more streamlined superstructure than in
traditional catamarans. The theoretical analysis and computed results showed that
a reduction in the total drag of around 50% can be obtained at very high speed,
above 50 knots (93 km/h).
In these references some experimental data, theoretical and computed results
on vehicles which can be classified as AAMV are presented, but none of them
present an analysis of the static stability of a vehicle having both hydrodynamic
and aerodynamic surfaces. This is the gap the present work is aimed to bridge.
4. The model
In order to develop a mathematical model for the dynamics of an ‘Aerodynami-
cally Alleviated Marine Vehicle (AAMV)’, a kinematics framework is proposed to
describe the motion of the AAMV and the forces acting on it. Once a reference
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Longitudinal Static Stability of an AAMV5
O
WATERLINE
Ox
z
ξ
ζ
η1
3
η
5
η
τ
(t)
η
0
1
3(t)
η(t)
τ
h(t)
h0
V0
Figure 1. AAMV axis system
framework is established, it is necessary to narrow down possible configurations of
the AAMV, since the qualitative and quantitative nature of the forces and moments
acting on the AAMV depend on the elements that comprise its configuration.
(a) Kinematics
The study of kinematics requires a definition of coordinate frames, a notation
to represent vehicle motion and a technique for a transformation between fixed and
moving frames. To describe the motion of an AAMV and the forces acting on it,
a number of different axis systems are used. Starting from the axis systems used
for planing craft (Savitsky 1964, Martin 1978) and for WIG vehicles (Irodov 1970,
Staufenbiel & Bao-Tzang 1977), an earth-axis system and two body-axis system are
presented below. They are all right-handed and orthogonal, as shown in figure 1.
Dashed lines represent the vehicle in a disturbed state (rotation and displacements
have been emphasized for clarity).
For body-axis systems, the origin O is taken to be coincident with the centre of
gravity (CG) position of the AAMV in equilibrium state. The x and z axis lie in the
longitudinal plane of symmetry, x positive forward and z positive downward. The
direction of the x-axis depends on the body-axis system. Two are considered: aero-
hydrodynamic axes (η1Oη3), the direction of the x-axis η1being parallel to the
steady forward velocity V0, and geometric axes (ξOς), the direction of the x-axis
ξ being parallel to a convenient geometric longitudinal datum (as the keel of the
hull). Aero-hydrodynamic axes are used here as the counterpart of aerodynamic
axes (called wind or wind-body axes in UK and stability axes in USA) used for
airplanes. Usually the stability derivatives are calculated in this axis system.
The direction of the earth-axis systems (xOz) are fixed in space. The z-axis is
directed vertically downward, the x-axis is directed forwards and parallel to the
undisturbed waterline and the origin at the undisturbed waterline level.
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6M. Collu et al.
HYDRO-PROP. SYSTEM
MAIN AER. SURFACE
AERO-PROP. SYSTEM
SECONDARY AER. SURFACE
HYDRODYN. SURFACE
Figure 2. Class of configurations for the AAMV - Final choice
Table 1. AAMV configuration elements
Element
Aerodynamic surface
Hydrodynamic surface
Number
2
1
Note
with/out control surfaces
with/out control surfaces,
prismatic planing hull
aero- and/or hydro- propulsion system Propulsion system 1/more
(b) Configuration
The general approach of this work is to start from studies on WIG vehicles and
high speed marine vehicles to derive an integrated system of equation of motion for
an AAMV. This approach can be applied also to the choice of the AAMV configura-
tion. A WIG vehicle’s fundamental elements are the aerodynamic surfaces and the
aero-propulsion system, while on a high speed marine vehicle the elements would
be hydrostatic surfaces, hydrodynamic surfaces and a hydro-propulsion system.
During the course of the work, it was decided to limit the type of hydro-surfaces
to only a prismatic planing hull (figure 2). Therefore the possible configurations
was narrowed down, as shown in table 1.
The main difference is that, with the first class of configurations, hydrofoils could
be represented. With the latter, only planing surfaces can be taken into account.
Authors considered that the AAMV should also have the capability of free flight (or
wing in ground flight), therefore the configuration with hydrofoils as hydrodynamic
surfaces was not considered suitable.
Among all the other possible hydrostatic/hydrodynamic surfaces, a prismatic
planing hull has been chosen, and the Savitsky planing hull model is used for this
configuration (Savitsky 1964). The available literature on planing craft dynamics is
extensive (Blake & Wilson 2001) and the approaches used are somewhat similar to
the approach used for WIG vehicles: this aspect makes the coupling of the airborne
and waterborne dynamics simpler. Also if the majority of planing hulls used are non-
prismatic, it has been demonstrated that the Savitsky approach is suitable also for
non-prismatic hulls (Savitsky et al. 2007), and in particular in the preliminary phase
of design. To estimate the equilibrium state, the starting point of the static and
dynamic stability analysis presented in this work, the Savitsky method is chosen,
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Longitudinal Static Stability of an AAMV7
WATERLINE
ξ
ζ
CG
τ
HC
AC
L
DM
T
N
D
L
ε
W
a1
DF
Dws
a1 a1
a2
Ma2
a2
AC1
2
Dah
aerodynamic
hydrodynamic
Figure 3. Forces and moments acting on a AAMV
Table 2. Forces and moment acting on an AAMV
Force
Gravitational
Hydrostatic and hydrodynamic
Aerodynamic
Symbol
W
N, Dws, DF
Lai, Dai,
Mai, Dah
Acting on
Centre of gravity
Hull
aerodynamic surfaces
Aerodynamic and hydrodynamic
control systems
aerodynamic surfaces
and hull (control fixed analysis,
are supposed constant, see § 4d)
thrust pointAero- and/or hydro-propulsionT
and along with it the prismatic planing monohull as the hydrostatic/hydrodynamic
surface of the AAMV. The equilibrium state estimation method has been previously
developed by the authors (Collu et al. 2008, Collu 2008).
(c) Forces and Moments acting on the AAMV
The forces and moments acting on the vehicle, after an external disturbance,
are listed in table 2 and illustrated in figure 3.
Decoupling of Equations of Motion The AAMV, represented as a rigid body
in space, and with control surfaces fixed, has 6 degrees of freedom. To describe its
motion a set of six simultaneous differential equations of motion is needed. However,
a decoupled system of equations of motion can be derived. For airplanes, in the
frame of small perturbations approach, the lateral-longitudinal coupling is usually
negligible. This is still valid for WIG vehicle (Chun & Chang 2002). For planing
craft, as demonstrated by Martin (1978), not only the lateral-longitudinal coupling
is usually negligible, but also the surge motion can be decoupled from the heave and
pitch motion. Therefore it is assumed that the AAMV has a negligible longitudinal-
lateral coupling. In this work, the longitudinal motion of the AAMV is analysed,
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8M. Collu et al.
thus only the forces and moments acting in the longitudinal plane xOz are taken
into account: surge, heave forces and pitch moments. Following the nomenclature
used for ships and airplanes, the force in the x direction is X, in the z direction is
Z and the moment about the y axis is M.
(d) Forces and moments expressions
The total force vector acting on the AAMV can be expressed as:
F = Fg+ Fa+ Fh+ Fc+ Fp+ Fd
(4.1)
where the components of each force are
Fi
=[ Xi
Zi
Mi]T
The total force is the sum of gravitational (g) force, aerodynamic (a) and hydrody-
namic (h) forces, control (c) systems forces, propulsion (p) force and environment
disturbances (d) forces.
When considering the motion of an airplane or a marine vehicle, after a small
perturbation from a datum motion condition, it is usual to express aerodynamic
and hydrodynamic forces and moments in Taylor expansions about their values at
the datum motion state. The expansion can be nonlinear and expanded up to the
n-th order, but in this work a linear expansion will be used. The linear approach
has some limitations, but it is the result of a trade off. To study the dynamics
of modern airborne vehicles, a non linear approach is used, but not in the early
design phases (see, for example, Roskam 1989). In fact, in these phases a basic
but solid understanding of the physics underpinning the static stability of a new
concept vehicles is needed, and the linear approach is perfect for this task. The non
linear approach would unnecessarily complicate the approach, hiding important
aspects, and giving information too detailed at this stage. As regard hydrodynamic
derivatives, linear methods are generally considered good enough to estimate the
static stability boundaries (Troesch & Falzarano 1993, Hicks et al. 1995).
As for airplanes and planing craft, forces and moments are assumed to depend
on the values of the state variables and their derivatives with respect to time. Then,
each force and moment is the sum of its value during the equilibrium state plus its
expansion to take into account the variation after a small disturbance, which is:
F
F0
F?
=
=
=
F0
[ X0
[ X?
+
Z0
Z?
F?
M0]T
M?]T
(4.2)
where the subscript (0) denotes starting equilibrium state and superscript (?) de-
notes perturbation from the datum. Initially, the AAMV is assumed to maintain
a Rectilinear Uniform Level Motion (RULM) with zero roll, pitch and yaw angles.
In this particular motion, the steady forward velocity of the AAMV is V0and its
components in the aero-hydrodynamic axis system are [˙ η1,0, ˙ η3,0], with ˙ η1,0= V0
and ˙ η3,0= 0, since this is a level motion (constant height above the surface).
Control, power and disturbances forces
that the controls are fixed (similar to the“fixed stick analysis”for airplanes). Then
In this analysis, it is assumed
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Longitudinal Static Stability of an AAMV9
controls’ forces and moments variations are equal to zero. The thrust is assumed
not to vary during the small perturbation motion and it is equal to the total drag
of the vehicle. The effects of environmental disturbances, like waves, are beyond the
scope of this work; so a stable undisturbed environment is assumed.


Gravitational force The gravitational contribution to the total force can be
obtained by resolving the AAMV weight into the body axis system. Since the origin
of the axis system is coincident with the CG of the AAMV, there is no weight
moment about the y axis. Remembering that the equilibrium state pitch angle
is equal to zero and the pitch angular perturbation θ?is small, the gravitational
contribution is
Fg
=
Fg
0
Fg
0
=[ 0
Fg?
=[ − mgθ?
Aerodynamic forces Usually, to evaluate aerodynamic forces and moments,
the state variables taken into account in their Taylor linear expansion are the veloc-
ity along the x and z axes (˙ η1and ˙ η3) and the angular velocity about the y axis (˙ η5).
Among the accelerations, only the vertical acceleration (¨ η3) is taken into account
in the linear expansion. Since the dynamics of a vehicle flying IGE depends also on
the height above the surface, Kumar (1968, 1968b), Irodov (1970) and Staufenbiel
& Bao-Tzang (1977) introduced for WIG vehicles the derivatives with respect to
height (h).
These derivatives can be evaluated knowing the geometrical and aerodynamics
characteristics of the aerodynamic surfaces of the AAMV (Hall 1994). As shown by
Chun and Chang (2002), the Taylor expansion to 1st order (linear model) is a good
approach to have a first evaluation of the static and dynamic stability characteristics
of the WIG vehicle.
The expansion of the generic aerodynamic force (moment) in the aero- hydro-
dynamic axis system (η1Oη3) for a AAMV with a longitudinal plane of symmetry
is
Fa
=
Fa
0
Fa
0
=[ Xa
0

Fc
Fp
Fd
=
=
=
Fc
Fp
0
0
0
(4.3)
+
mg
0
Fg?
0 ]T
0 ]T
(4.4)
+
Za
Fa?
Ma
00]T
(4.5)
Fa?
=


Xa
Za
Ma
Xa
˙ η5
Za
˙ η5
Ma
˙ η5
h
h
h






h?+
+


Xa
Za
˙ η1
Ma
˙ η1
Xa
Za
˙ η3
Ma
0
0
0
˙ η3
˙ η1
˙ η3




˙ η1
˙ η3
˙ η5
¨ η1
¨ η3
¨ η5




?
+
+


Xa
Za
¨ η3
Ma
¨ η3
0
0
0
¨ η3
?
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10M. Collu et al.
The superscriptadenotes “aerodynamic forces”. Fj denotes the derivative of the
force (or moment) F with respect to the state variable j, it corresponds to the
partial differential ∂F/∂j.
Hydrodynamic forces In Hicks et al. (1995), the nonlinear integro-differential
expressions to calculate hydrodynamic forces and moments are expanded in a Tay-
lor series, to third order. Therefore, equations of motion can be written as a set of
ordinary differential equations with constant coefficients. The planing craft dynam-
ics are highly non-linear, but the first step is to linearize the non-linear system of
equations of motion and to calculate eigenvalues and eigenvectors, where variations
are monitored with quasi-static changes of physical parameters, such as the position
of the CG. This approach seems reasonable as a first step for the analysis of the
AAMV dynamics too, for which a linear system of equations is developed.
The derivatives are usually divided into restoring coefficients (derivatives with
respect to heave displacement and pitch rotation), damping coefficients (derivatives
with respect to linear and angular velocities) and added mass coefficients (deriva-
tives with respect to linear and angular accelerations). Martin (1978) and Troesch &
Falzarano (1993) showed that the added mass and damping coefficients are nonlin-
ear functions of the motion but also that their nonlinearities are small compared to
the restoring forces nonlinearities : therefore added mass and damping coefficients
are assumed to be constant at a given equilibrium motion. Their value can be ex-
trapolated from experimental results obtained by Troesch (1992). For the restoring
coefficients, the linear approximation presented in Troesch and Falzarano (1993)
will be followed:
Fh, restoring− Fh, restoring
The coefficients of [C] can be determined using Savitsky’s method for prismatic
planing hull (Savitsky 1964) or the approach presented by Faltinsen (2005).
An approach to estimate added mass, damping and restoring coefficients is pre-
sented by Martin (1978). Furthermore, an alternative approach is to compute the
added mass and damping coefficients as presented in Faltinsen (2005).
Then the expansion of the generic hydrodynamic force (moment) with respect
to the aero-hydrodynamic axis system η1Oη3is:
0
∼= −[C] η
(4.6)
Fh
Fh
=
=
Fh
[ Xh
0
+
Zh
Fh?
Mh
0000]T
(4.7)
Fh?
=






0
0
0
Xh
Zh
˙ η1
Mh
Xh
¨ η1
Zh
¨ η1
Mh
Xh
Zh
η3
Mh
η3
Xh
Zh
η5
Mh
η5
η3
Xh
Zh
Mh
Xh
Zh
¨ η3
Mh
η5
Xh
Zh
Mh
Xh
Zh
¨ η5
Mh


˙ η5






η1
η3
η5
˙ η1
˙ η3
˙ η5
¨ η1
¨ η3
¨ η5






?
+
+
˙ η1
˙ η3
˙ η3
˙ η5
˙ η1
˙ η3
˙ η5




?
+
+
¨ η3
¨ η5
¨ η1
¨ η3
¨ η5
?
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Longitudinal Static Stability of an AAMV11
The superscripthdenotes “hydrodynamic forces”. Xη1, Zη1and Mη1are equal to
zero since surge, heave and pitch moment are not dependent on the surge position
of the AAMV.
Equilibrium state The equilibrium state has been already analysed (Collu
et al. 2008, Collu 2008): here it is only briefly presented to make the necessary
simplifications.
When an equilibrium state has been reached, by definition, all the accelerations
are zero as well as all the perturbations velocities and the perturbation forces and
moments. Then, using eq.s 4.3, 4.4, 4.5 and 4.7 in eq. 4.2:



0
0
0
=
=
=
Xa
mg + Za
Ma
0+ Xh
0+ Xc
0+ Zh
0+ Mc
0+ Xp
0+ Zc
0+ Mp
0
0+ Zp
0
0+ Mh
0
(4.8)
(e) System of Equations of Motion
The generalized system of equations of motion (in 6 degrees of freedom) of a
rigid body with a left/right (port/starboard) symmetry is linearized in the frame of
small-disturbance stability theory. The starting equilibrium state is a RULM, with
a steady forward velocity equal to V0. The total velocity components of the AAMV
in the disturbed motion are (evaluated in the Earth-axis system):



˙ η1
˙ η2
˙ η3
˙ η4
˙ η5
˙ η6



=



V0+ ˙ η?
˙ η?
˙ η?
˙ η?
˙ η?
˙ η?
1
2
3
4
5
6



(4.9)
By definition for small disturbances, all the linear and the angular disturbance
velocities (denoted with?) are small quantities: therefore, substituting eq. 4.9 in the
generalized 6 degrees of freedom equations of motion, and eliminating the negligible
terms, the linearized equations of motion can be expressed as















m¨ η?
6V0)
5V0)
1
=
=
=
=
=
=
X
Y
Z
L
M
N
m(¨ η?
m(¨ η?
I44¨ η?
2+ ˙ η?
3− ˙ η?
4− I46¨ η?
I55¨ η?
I66¨ η?
6
5
6− I64¨ η?
4
(4.10)
If the system of equations is decoupled, the longitudinal (symmetric) linearized
equations of motion are



m¨ η?
5V0)
I55¨ η?
1
=
=
=
X
Z
M
m(¨ η?
3− ˙ η?
5
(4.11)
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Page 12

12M. Collu et al.
(f ) Longitudinal Linearized System of Equations of Motion
Taking into account eq. 4.8, the longitudinal linearized equations of motion (eq.
4.11) written in the aero-hydrodynamic axis system can be rearranged as:
[A] ¨ η + [B] ˙ η + [C] η + [D]h = 0
(4.12)
where
η
=


η1
η3
η5


and h is the (perturbated) height above the waterline.
The matrix [A] is the sum of the mass matrix, the hydrodynamic added mass
derivatives and the aerodynamic“added mass”terms (usually in aerodynamics they
are referred to as simply “acceleration derivatives”), whereas in the hydrodynamic
case these terms are much more significant.

−Mh
[B] is the damping matrix and is defined as:

−Ma
[C] is the restoring matrix and is defined as:

0
−Mh
The matrix [D] represents the wing in ground effect, to take into account the
influence of the height above the surface on the aerodynamic forces.

[A]=

m − Xh
−Zh
¨ η1
−Xa
m − Za
−Ma
¨ η3− Xh
¨ η3− Zh
¨ η3− Mh
¨ η3
−Xh
−Zh
I55− Mh
¨ η5
¨ η1
¨ η3
¨ η5
¨ η1
¨ η3
¨ η5


(4.13)
[B]=

−Xa
−Za
˙ η1− Xh
˙ η1− Zh
˙ η1− Mh
˙ η1
−Xa
−Za
−Ma
˙ η3− Xh
˙ η3− Zh
˙ η3− Mh
˙ η3
−Xa
−Za
−Ma
˙ η5− Xh
˙ η5− Zh
˙ η5− Mh
˙ η5
˙ η1
˙ η3
˙ η5
˙ η1
˙ η3
˙ η5


(4.14)
[C]=

0
0
−Xh
−Zh
η3
−mg − Xh
−Zh
−Mh
η5
η3
η5
η3
η5


(4.15)
[D]=

−Xa
−Za
−Ma
h
h
h


(4.16)
(g) Cauchy standard form of the Equations of Motion
By defining a state space vector ν as
ν
=
?
˙ η1
˙ η3
˙ η5
η3
η5
η0
?T
(4.17)
the system of equations eq. 4.12 can be transformed to the Cauchy standard form
(or state-space form). The state space vector has six variables while the system of
equations eq. 4.12 has only 3 equations. The remaining 4 equations are:


Article submitted to Royal Society



∂(η3)
∂ t
∂(η5)
∂ t
∂(h)
∂t=∂(η0)
=
=
=
˙ η3
˙ η5
− ˙ η3+ V0η5
∂ t
(4.18)

Page 13

Longitudinal Static Stability of an AAMV13
Therefore the system is:
[ASS]˙ ν = [BSS] ν
(4.19)
where
[ASS]=



[A][0]3×3
1
0
0
[0]3×3
0
1
0
0
0
1



(4.20)
and
[BSS]=



−[B]
0
−mg
−C35
−C55
0
0
V0
−C33
−C53
−[D]
0
0
0
1
0
0
1
0
0
0
0
0
0
0
−1



(4.21)
The system of equations of motion in state-space form is:
˙ ν = [H] ν
(4.22)
where
[H]=[ASS]−1
[BSS]
(4.23)
Now it is possible to analyse the static and dynamic stability of a AAMV con-
figuration and the influence of the configuration characteristics on the AAMV dy-
namics.
5. Static stability
Analyzing the forces and moments under the small disturbances hypothesis, the
static stability of an AAMV is derived using the Routh-Hurwitz criterion.
In particular, Staufenbiel & Bao-Tzang (1977) showed how the last coefficient
A0of the characteristic polynomial of a WIG vehicle can be used to estimate its
static stability. In general, given the characteristic polynomial of a system:
Ansn+ An−1sn−1+ ... + A1s1+ A0= 0(5.1)
if the condition
A0
An
> 0(5.2)
with An> 0, is fulfilled, the system is statically stable.
(a) AAMV characteristic polynomial and static stability condition
In § 4 the mathematical model is developed to study the longitudinal dynamics
of an AAMV. A system of ordinary differential equations of motion is derived for
the longitudinal plane in the frame of small-disturbance stability theory. Starting
from this model, the Routh-Hurwitz condition is used to derive a mathematical
expression to estimate the AAMV static stability.
Article submitted to Royal Society

Page 14

14M. Collu et al.
Complete order system By defining a state space vector ν as
?
the system of equations of motion can be rearranged in the Cauchy standard form
(or state-space form), showed in eq. 4.19. The characteristic polynomial of the
complete order system can be derived:
ν
=
˙ η1
˙ η3
˙ η5
η3
η5
η0
?T
(5.3)
A6s6+ A5s5+ A4s4+ A3s3+ A2s2+ A1s1+ A0= 0(5.4)
With A6= 1, the static stability is assured when:
A0=num0
∆
> 0(5.5)
where num0is equal to
num0= V0[D10(B31C53− B51C33) − B11(C35D50− C53D30)]
and ∆ is equal to
∆ = (I55+ A55)?m2+ m(A11+ A33) + A11A33− A31A13
Aij, Bij, Cij, and Dij stability derivatives are illustrated, respectively, in eq.
4.13, 4.14, 4.15, and 4.16.
Reduced order system This mathematical method has to be validated against
experimental data. Unfortunately, no experimental data on static stability of a
AAMV configuration is available in the public domain.
To plan experiments to obtain these data, it is necessary to have a physical
insight of the condition stated in eq. 5.2. This condition, applied to the complete
order system in eq. 5.5, is relatively complex. Assuming that the surge degree of
freedom (η1) can be decoupled from heave (η3) and (η5) pitch degrees of freedom,
a simplified version of the condition in eq. 5.5 can be obtained, leading to a better
physical insight.
The mathematical model of the dynamics here developed starts from the sys-
tems of equations of motion for WIG vehicles and planing craft. As regards the
dynamics of a planing craft, Martin (1978) demonstrated that the surge motion
can be decoupled from the heave and pitch motion. For the dynamics of WIG ve-
hicles, Rozhdestvensky (1996) proposed a reduced order system where the surge
motion is decoupled from heave and pitch motion.
By defining the reduced order state space vector ν as
?
the Cauchy standard form (or state-space form) of the reduced order system is
obtained. The characteristic polynomial can be derived:
(5.6)
?+
−(m + A11)A53A35− (m + A33)A51A15+ A53A31A15+ A51A13A35
(5.7)
ν
=
˙ η3
˙ η5
η3
η5
η0
?T
(5.8)
A5s5+ A4s4+ A3s3+ A2s2+ A1s1+ A0= 0(5.9)
With A5= 1, the static stability is assured when
A0=
V0(C33D50− C53D30)
(A55+ I55)(A3+ m) − A53A35
> 0 (5.10)
Article submitted to Royal Society

Page 15

Longitudinal Static Stability of an AAMV15
(b) Reduced order static stability: physical insight
Each coefficient in eq. 5.10 is the derivative with respect to: accelerations (Aij),
heave position(Cij), and height above the surface (Dij) of the sum of aerodynamic
and hydrodynamic forces (and moments). Referring to § 4.11, remembering that
the superscript ‘a’ stands for aerodynamic and ‘h’ for hydrodynamic, and that Z is
the heave force (positive downward) and M the pitch moment (positive bow up),
the coefficients are equal to:
A33
A35
A53
A55
=
=
=
=
Aa
Aa
Aa
Aa
33+ Ah
35+ Ah
53+ Ah
55+ Ah
33
=
=
=
=
−Za
−Za
−Ma
−Ma
¨
η3− Zh
¨
η3− Mh
¨
¨
η3
35
η5− Zh
¨
η5− Mh
¨
η5
53
¨
η3
55
¨
η5
(5.11)
C33
C53
D30
D50
=
=
=
=
Ch
Ch
Da
Da
33
=
=
=
=
−Zh
−Mh
−Za
−Ma
η3
,
,
,
,
Ca
Ca
Dh
Dh
33= 0
53= 0
30= 0
50= 0
53η3
30η0
53η0
(5.12)
The aerodynamic derivatives can be estimated with the approach presented in Hall
(1994) and the hydrodynamic derivatives with expressions presented by Martin
(1978) and by Faltinsen (2005). Using these expressions for the configuration pre-
sented in § 4b we have
(A55+ I55)(m + A33) − A53A35> 0
therefore, since the denominator of eq. 5.10 is greater than zero, the static stability
condition of the reduced order becomes
(5.13)
D50
D30
−C53
C33
> 0(5.14)
Similarity with WIG vehicles To better understand the condition expressed
in eq. 5.14, a parallel with WIG vehicles static stability criteria is illustrated. The
static stability condition derived by Staufenbiel & Bao-Tzang (1977) and Irodov
(1970) is:
Mw
Zw
that, using the present nomenclature corresponds to the condition
−Mh
Zh
< 0 (5.15)
B53
B33
−D50
D30
< 0 (5.16)
Mwand Mhare the derivatives of pitch moment with respect to the heave veloc-
ity and the height above the surface, Zwand Zhare the heave force same derivatives.
Staufenbiel & Bao-Tzang and Irodov define Mw/Zwalso as the aerodynamic cen-
tre of pitch and Mh/Zh as the aerodynamic centre in height. Remembering that
positive abscissa means ahead of the CG, the condition in eq. 5.15 and 5.16 can be
expressed as (Rozhdestvensky 2006):
Article submitted to Royal Society