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LOUGHBOROUGH UNIVERSITY OF TECHNOLOGY
LIBRARY
AUTHOR/FILING TITLE
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1
AC
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A STUOY DJ:"' THE HYOROOVNAMICS OF AC\I HIILLS
WI TH PART I Cl ILA R EMPHA S I S ON PLDUGH- I N
by
Rudrasena Aditya Prasad B.Tech
Submitted for the M.Sc
of Loughborough University of Technology
July, 19.78
Supervisors: Mr.D.Waters, M.Sc.,C.Eng.,M.R.Ae.S.,M.C.A.S.I.
Mr.E.Jenkins, B.Tech •• M.Tech.,C.Eng.,M.I.E.E.
~ by Rudrasena Aditya Prasad~ 1978
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Loughborough University
of T~.:hnoiogV Library
Dlte ~~ Cla~s
I Ace. \ '1"\ ()S1 J 01 Ne.
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SUMMARY
Since the major calm-water capsizes occurred in 1965/66,
much experimental work has been done to establish better
operational margins of safety. The general approach has
been to establish well defined limits of manoeuvrability
based upon available model and full-scale data. Mathematical
modelling of the ACV motion was used as a secondary approach
because the expressions involved are high in non-linearities . involving aerodynamic and hydrodynamic force terms of
similar orders of magnitude.
In this study, a numerical technique for the solution
of ACV non-linear equations is proposed and a two degree-
of-freedom model is built up using the digital simulation
language, SLAM. The simulation involved the use uf the
technique of storing values of the various non-linear functions
over a defined regime and then using these to provide updated
inputs as the craft changed its state •
. :An ACV overturn sequence is studied by developing,
simulating and testing of equations describing the roll and
sideslip motion of the craft. In particular, the equations
take into account stiffness and damping forces associated
with both the hard structure and the craft cushion system;
inertial coupling effects due to craft deceleration are also
incorporated; induced trim effects due to the position of
the cushion wave system below the craft is modelled and
suitable phase lag is employed depending upon the deceleration
of the craft.
A basic configuration craft is chosen based upon critical
design parameters such as hull depth, skirt depth, VCG height
and hull angle of inclination.Extensive numerical testing of this
configuration is carried out involving a systematic method
of variation of the critical design parameters.
Results indicate that it is good philosophy to design ACV
hulls with planing capability applied to all faces of the hard
structure. The results also allow a set of ranges to be
established for the critical design parameters, which, if adhered to will minimise the possibility of capsize for a craft
configuration of the type chosen for the studYe
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ACKNOWLEDGEMENTS
The author would like to express his sincere thanks to
Mr.O.Waters for his generous advice and counsel throughout
the period of this ~tudy.
Grateful thanks are also giVen to Mr.f.Jenkins for his
guidance in the initial stages of the work and for offering
useful suggestions on the presentation of material.
Thanks also go to Professor D.J.Johns, the Director of
Research for his encouragement and to the Staff of the Loughborough University Computer Centre, especially to
Dr.B.Negus who helped overcome computational problems.
(iii)
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TABLE OF CONTENTS
Summary
Acknowledgements
Table of Contents
Index of Figures
1. GENERAL INTRODUCTION
1.1. General Remarks Concerning Ploughing-in Phenomena
ii
iii
iv-
viii
1
1
1.1.1. The Mechanism of Plough-in and Overturn 2
1.2. Justification for Further Investigation 3
1.3. Necessary Approaches to the study and Associated Problems 4
1.3.1.
1.3 0 2.
1.3.3.
1.3.4.
1.3.5.
General Remarks on Stability
Critical Uesign Parameters
t.;raft Sensitivity to Critical Uesign Parameters
Problems Associated with the 1'10 d e 11 in g of ACV Motions
The Formulation of Equations Describing the Lateral notion of ACV1s ~Capsize
4
5
6
7
Mode) 9
Problem Solution 11
2. PREVIOUS AND PRESENT INVESTIGATIONS
2.1. Previous Investigations
12
12
2.1~1. notion Studies of Skirted Craft with emphasis on Plough-in 12
2.1.2. urag Forces ~ssociated with ACV's 16
2.1.3. The Hydrodynamics of ACV-Type Hulls 19
2.1.4. Studies Associated with the Cushion System 21
2.2. The Present Investigation 23
26
26
3. THEORY
3.0. Introduction
(iv)
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3.1. uerivation of the Equation Describing the Roll l'lotion of the Craft 27
3.1.1. Moment due to Change in VCG Position 30
3.1.2. Moment due to Loss of Cushion Area 30
3.1.3. Moment due to Buoyancy Forces acting on the Immersed Hull structure 33
3.1.4. Uevelopment of uynamic Force Coefficients for Flat ~urfaces at High Angles of Incidence 41
3.1.5. uetermination of Centre of Pressure Location 43
3.1.6. Moments due to uynamic Forces acting on the Hull Face 44
3.1.7. Moments due to uynamic Forces acting on the Hull Top 47
3.1.8. uerivation of the Inertial ~oupling Moment 49
3.1.9. Derivation of the Holl uamping Expressions 50
3.1.10. uetermination of Lraft Inertia in Roll 60
3 e 1.11. Assembling the Holl Equation 67
3.2. uerivation of the Equation Describing Motion of the Lraft in ~ideslip 68
3.2.1. Aerodynamic urag 69
3.2.2. Lushion Air nomentumurag 69
3.2.3. llletting Drag 70
3.2.4. Induced Wave Drag and Wave Slope 70
3.2.5. Drag due to Hydrodynamic Forces acting on the Hull face 71
3.2.6. Drag due to" Hydrodynamic Forces acting on the Hull Top 72
3.2.7. Assembling the Equation Describing Motion in Sideslip 73
4. SIMULATION PROGRAM STRUCTURE 74
4.1. Subroutine HCVDATA and its Function Segments 74
4.2. Master Program ACVDYNAMICS 77
4.2.1. The INITIAL Hegion
(v)
77
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4.2.2. The DYNAMIC Region
4.2.3. The TERMINAL Region
5. DISCUSSION AND ANALYSIS OF RESULTS
5.0. Introduction
5.1. Development of the Theoretical Model
5.1.1. ~uasi-Static Terms
5.1.2. Craft Dynamic Terms
5.2. ~imulation Program Development
78
80
82
82
84
84
91
106
5.2.1. Function Generation 107
5.2.2. Lhoice of Integration Algorithm 109
5.2.3. Problem ~onstraints 110
5.3. Model Performance 112
5.3.1. ~ensitivity to Variation of Initial ~tate 114
5.3.2. ~ensitivity to Variation of ~hape Parameters 116
5.3.3. Sensitivity to Variation of Fan Characteristics 124
5.4. Final Evaluation of Craft Design Parameters 127
6. GENERAL CONCLUSIONS AND RECOMMENDATIONS FOR -FURTHER lJORK
6.1. Major Conclusions
130
131
6.1.1. Conclusions on Model Perfo~mance 132
6.2. Hecommendations for Further lJork
7. REFERENCES
8. NOTATION
9 •.. APPENDICES
I. Evaluation of the Main Geometric Terms associated with the Buoyancy Forces Generated
135
137
141
146
by the Immersed Portion of the Hull 146
11. Lift and Drag Coefficients Associated with the Movement of a Flat Plate through a Fluid Medium at High Angles of Incidence 152
(vi)
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Ill. Determination of Geometric Expressions Associated with the Damping of the Hull
IV. Determination of Geometric Expressions Associated with Added Inertia in Roll
V. Flowcharts and Listings associated with Program ACVDYNAMICS
VI. Flowcharts" and Listings associated with Program ACVDATA
VII. Definition of Main Computer Variables used in the Study.
(vii)
156
158
164
170
186
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Fig. i~o •
1
2
3
4
5
6
7
8
9
10
11 a
11 b
12
13
14a
14b
14c
15
16
17
. 18
19
INDEX OF FIGURES
Description
Demonstration of Full-scale Plough-in Phenomena for SRN-6 Craft
Line Film Recording of the Capsize Sequence of a Model of ~RN-6 Craft
~asic Craft Configuration used in the ~tudy
The Main Forces acting on an RCV in a Heam-on Roll Condition
Uverturning of SRN-5 Craft - a Diagrammatic Sequence of Events
The Components of Drag acting on a'Hovercraft
wave Resistance for Different Craft Planforms Computed by Baratt
wave Resistance in Accelerated Motion computed by uoctors and jharma
Basic configuration Craft
Moment about the Trailing ~dgB due to craft weight acting through the C of G
Moment arisin~ from the Lraft ~ushion :Jystem (~ 11A)
Main Geometric uefinitions
1'10 men t due to 13 u 0 y a n c y For c e s (~A < ~ < J1 B ) l'loment due to Buoyancy Forces (I1 B < 11 < I -ex.) condition at which ~ = -!-~
l'loment due to Buoyancy Forces (~> ~ -oc)
Moment arising from uynamic forces acting on the Hull Face
Moment arising from uynamic Forces acting on the Hull Top
Moment arising from the Inertial Coupling t:.ffect
t:.stimation of ~ushion Volume (}1 .0 A) (viii)
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Fig. IIJO. uescription
20 fan ~tatic ~haract8ristics - Linearised ~urve
21 uetermination of Uamping Contribution due to Hull ~ace
22 uetermination of uamping ~ontribution due to Hull fop
23 uetermination of the ~omponents of the ~raft Holl lnertia in Air
24 uetermination of the urag ~omponent acting on the ~raft due to uynamic ~orces Associated with the Hull ~ace
25 uetermination of the urag ~omponent acting on the Lraft due to uynamic ~·orces Associated with the Hull fop
26 Graph of Change in 'Primary Hump ~peed i- plotted against Craft ueceleration (g's)
27 Graph of Total Non-uimensional ~oment plotted against Craft Roll Angle
28 Graph repre~enting Terms affecting the Moment Contribution due to the ~ushion (Static Lase)
29 Graph of Lhange in Buoyancy and Total Moment Terms plotted against Craft Roll Angle for Pre-fransition and Aft-Iransition Conditions ~Static Case)
30 Graph of Change in Force Coefficient due to the Movement of a flat ~urface through a tluid Medium plotted against Incidence Angle
31 -llraph of ~hange in Moment due to t low Impingement on the Hull plotted against Sideslip Velocity
32 ~raph of Lhange in Damping ~oefficient due to"Flow Impingement on the HUll plotted against ~raft Holl Angle
33 Performance Curves for a Multi-wing Fan
34 Graph of Non-uimensional ~ushion Pressure Variation plotted against Craft Roll Angle
35 Graph Representing Terms affecting the ~oment Contribution due to the Lushion (Dynamic ~ase)
36 Graph of Variation of Wave Slope plotted against Lraft Speed
37 Graph of Variation of Craft Total lnertia in Roll plotted against Craft Holl Angle
(ix)
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Fig. NO.
38
39
40
41
42
43
44
uescription
Graph of Total Non-uimensional urag Force plotted against Sideslip velocity
uutput Graphs used for a Comparison of Integration Houtines
l'lotion l..urves (see Legend)
l'lotion Curves ~see Legend)
l'lotion Curves (see Legend)
l'lotion l..urves (see Legend)
uetermination of Neutral Stability Line w.r.t Lhanges in VCG Height and Hull Angle of Inclination
45 Effect of Varying ~kirt Depth on the Position of the Neutral Stability Line
46 Effect of Varying Hull Uepth on the Position of the Neutral ~tability Line (Thin ~ectibn)
47 Effect of Varying Hull Uepth on the Position of the Neutral ~tability Line ~Thick §ection)
48 Motion Curve: Comparison Plot ~see Legend)
49 Plot of Craft Motion in Roll ~Demonstrating Motion Phases)
50 Plot of Craft Motion in Roll \Demonstrating the Effect of the Transition to Fully Attached Flow)
51
52a
52b
53
54a
54b
55a
55b
56
fllotion Curve: Comparison Plot
l'lotion· Curves (see Legend)
l'lotiofl Curves ~see Legend)
IYJotion Curves: Comparison Plot
rlotion Curves (see Legend)
Motion Curves \see Legend)
i'lotion Curves ~see Legend)
l'lotion Curves \see Legend)
Performance Curves: l'lul ti-wing the Fan Design Operating Point
tsee Legend)
(see Legend)
Fan - Shifting
57 ~lot of ~raft Motion in Roll \Basic Configuration Craft)
(x)
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fig. No~ uescription
58 Plot of Craft Motion in Holl \Effect of Shifting the fan Uesign Uperating Point)
59 Graph of Variation of Cushion Pressure plotted against Craft Roll Angle - iffect of Shifting the fan uesign uperating Point
Basic Configuration Craft uata:
A1 Storage of Values of Induced Trim Angle
A2 storage of Values of Non-Uimensional Moment due to ~ushion
A3 Storage of Values of Craft Total Inertia in Roll for Detached and Attached flow Assumptions
A4 storage of Values of Total Non-uimensional Moment acting on the Craft
A5 ~torage of Values of Total Non-uimensional urag acting on the ~raft
A6 ~torage of Values of Moment Derivative -Damping Coefficients
A7 Input Data to Program ACVuYNAMICS.
(xi)
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1 • GENERAL INTRODUCTION
The evaluation of air cushion vehicle (ACV) dynamics
is going through a process of change mainly due to the fact
that air cushion technology is very much in its youth and
not enough research has been done to· cover all the areas
of: major concern.
Mathematical modelling therefore serves as a useful
tool in these early stages of development of ACV's as it
provides the designer with valuable insight into the principal
factors which influence the general behaviour of the craft
when subjected to va~ious initial coridiiions and constraints.
1.1. General Remarks concerning Ploughing-in Phenomena
In April 1965, an SRN-5 craft overturned in roll off
the coast of Norway and in May of the same year another such
incident occurred in San Francisco Bay. A third incident
was later recorded in 1966 off the British coast. On each
of these three occasions, the craft had developed yaw of
45 degrees or more at about 40 knots waterspeed; the
skirt plough-in and overturn finallt occurred at around la
knots waterspeed when travelling approximately beam-on
(reported in CAA paper on Hovercraft Stability and Control,
1975, re~ 1).
In all three occasions, the accidents were of a
plough-in tYRe associated with tight turns initiated at
high speed in glassy calm sea conditions.
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1.1.1. The Mechanism of Plough-in and Overturn
The hydrodynamic mechanisms of plough-in and overturn
are intimately bound up with the presence of the skirt~
Usually, movement of the craft contrOl surfaces result in
some change of trim;.at high speeds, or in choppy conditions,
parts of the soft structure or 'skirt' tend to brush the
water surface. In some cases if the change in craft trim is
large enough, part of the lower skirt may collapse ~th a
. tendency to fold inwards or 'tuck' under the hard structure.
The effect of 'skirt tuck' is to cause cushion area to reduce
and the centre of pressure of the cushion to shift rearwards
resulting in a bow down motion.
If the motion is strictly in a longitudinal sense, the
craft rapidly decelerates and this deceleration is coupled
with a large nose down movement known as ploughing-in.
A multiexposure photographic recording of an actual
ploughing-in sequence is shown in Fig. 1.
In many cases, say in initiating a turn, there is a
tendency for the stern to break away so that large angles
of yaw rapidly develop well before the full turn is completed.
This is typic~lly characteristic of ACV motion and if skirt
tuck and plough-in ~ occurs, the craft is in many cases
unable to restore itself resulting in an almost beam-on
capsize. This phenomena has also been recorded on film
for an SRN-6 model and a multiexposure photograph is shown
in Fig. 2.
*.Hereafter, references will be indicated as superscripts unless otherwise indicated.
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1.2. Justification for Further Investigation
In the ARB special committee report on Hovercraft
Stability and Contro11 , an attempt was made to summarise
all the information available on Hovercraft capsize. The
report suggested that a theoretical research program of some
magnitude was necessary to establish satisfactory design
guidelines on which future hovercraft designers will base
their models.
Since the 1965/66 capsizes, a set of well defined limits
of manoeuvrability were established in order to ensure better
operational margins of safety; present day hovercraft pilots
are required to conform to these rules. However, this is no
justification for assuming that further accidents are
impossible and there are two main reasons to support this:
(a) if the skirts are damaged sufficiently for the ba9
internal pressure to approach that of the cushion,
then skirt stability is reduced and tuck under
could easily follow.
(b) secondly, much of the vehicle control is today still
left largely in the hands of the pilot. It is quite
possible that errors of judgement could be made,
especially in going through tight turns where proper
pre~ision control is required. In this mode, as
demonstrated before, a beam-on plough-in condition
could easily be initiated •
. It is therefore not only necessary to safeguard against
these conditions occurring, but primarily, if they do occur,
to ensure by proper design that the craft is capable of
r~storing itself.
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1.3. Necessary Approaches to the Study and Associated Problems
1.3.1. General Remarks on Stability
In the design of any engineering system, there are
generally two approaches to the problem of dynamic stability_
The first and "oremost is usually carried out in the early
development stages of the system and this involves the use
of fundamental stability criteria to keep the general
structural configuration within certain specified limits.
Typically for ACV's, a VCG location range and a skirt depth
range will dictate agreat deal of the initial design approach.
Of course a secondary approach that has also achieved
wide application, is the use of feedback control to improve
the dynamic characteristics of the system if for instance it is
thought that the basic design has unstable tendencies built
into it. This will suggest for instance:
(a) . the use of motion sensors to detect any undesirable
changes in the operation of the system
(b) use of a controller to interpret the signals
coming from the sensors and using a
feedback law to generate a countera~tive signal
with the aim of combating the changes detected.
(c) the counteractive signal from the controller would
then be used to drive· force or moment generators
to return the-system to an equilibrium state.
In the case of ACV's various force and moment generators
like puff ports, multidiroctional rotors, rudders and skirt
lifting systems are generally used to contrel the motion.
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The problem of ploughing-in however is sesn as a result
of basic design features and it was thought that preventative
measures could be sought by structural configuration changes
rather than by seeing the former as a control problem.
In any case, uhile present day motion controlling systems
for ACV's may be capable of enforcing constraints on the motion
in order to avoid a plough-in being initiated for instance,
their usefulness will almost inevitably be shortlived in
actually preventing a plough-in once initiated.
1.~.2. Critical Design Parameters
If there is to be any change in the basic design
philosophy applied to the ACV, it must happen while this
mode of travel is still passing through a relatively early
phase of development. An attempt uas made in 1975, by the
ARB special committee to supply a range of suitable values
fo~ the most critical design parameters affecting craft
stability1.
It appears that the most sensitive factors influencing
craft overturn are VCG height, skirt depth and hull depth.
Large VCG heights and skirt depths would obviouslY result
in a tendency for large angles of rotation to be developed~.
The hull depth is important from the point of view of providing
adequate buoyancy restoring forces should the craft develop
large angles of roll.
ACV plough-in is quite common both in the longitudinal
and lateral mode. No capsizes however, have been known to
occur in a strictly longitudinal sense. This is perhaps
partly because craft inertia is usually high in
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the longitudinal mode and . partly because the hull
face of the craft bow is usually hydrodynamically designed to
give positive restoring momenE~ about the C of G.
As a result of these observations, it was believed that .craft
ability to resist capsize would be greatly enhanced by proper
hydrodynamic design ~ppli~d to ~ll faces of the hull structure.
This therefore introduces a fourth design parameter which
could possibly influence craft behaviour. in the capsize
mode and that is the hull angle of inclination, sometimes
referred to as the planing angle.
These four design parameters allows a basic craft
configuration to be defined and this is shown in Fig. 3.
1.3.3. ~raft Sensitivity to Critical Design Parameters
There are several approaches to the way in which the
craft sensitivity to change in the critical design parameters
may be assessed.
As the study would involve basic structural changes,
work with full scale craft would be almost impossible due to
the eitremely high costs of modification that would frequently
be incurred.
A second approach could of course be the use of scale
models to perform the required, tests. However, in this case
one is faced with the added problems of scaling the dynamics
of the cushion system especially the inertia and friction
associated with the skirt.
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A third approach which has also been favoured in this
study is the development and use of a mathematical model to
achieve the desired objectives. Naturally in mathematical modelling
it is necessary to be very careful about the way in which the
problem is formulated; it will be important to
constantly bear in mind the limits and assumptions made
regarding the- various .. expressions involved~ However, -0'
once the problem is formulated obtaining a solution is generally
a routine matter and a large number of cases could be evaluated
in a relatively short period of time compared with other
approaches.
1.3.4. Problems associated with the ~odellin9 of ACV Motions
Mathematical modelling of ACV motion presents an
interesting challenge to the engineer because the various
terms involved are extremely rich in non-linearities. The
problem is highly complex involving both aerodynamic and
hydrodynamic force terms of similar orders of magnitude and
motion of the craft can take place in six degrees of freedom,
in roll, pitch, yaw, sideslip, heave and surge.
Thus it is _not unusual to find that motion studies are
often simplified. A typi~al assumption for instance made
with vehicles having symmetry in the longitudinal plane is
that there is no coupling between longitudinal and lateral
motion as is done in the case of aircraft and ship motions.
Using a similar approach, the ACV plough-in and capsize
in roll when treated as a sequence of events, may be looked
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upon as a series of unconnected motion ~hases. This is
demonstrated more clearly in some statements on the-observed
capsize sequence recorded in-the AR8 special committee report 1•
These are:
(a) initially, the craft is assumed to be on cushion
and operating sat~sfactorily
(b) the craft is put into some situation due to engine
failure, lift power reduction, skirt failure or
manoeuvring which results in leading skirt tuck-
under.
(c) following leading skirt tuck-under, craft hard
structure immersion occurs.
(d) following hard structure immersion, the capsize
occurs. It is also important to note that development ~f high angles
of yaw are also usually associated with (b) above. The above indicates two major regimes of study
and these are:
(1) to establish the onset of skirt tuck-under
(2) to establish the dynamic stability of the craft
with hard structure immersion and assuming that
skirt tuck under had already occurred.
In the case of capsize due to a turn being initiated,
skirt tuck under usually ensues during the transition from
tha initiation point to the time when a very high yaw angle
approaching 90 degrees has developed. This is observed from
the motion of the model in Fig. 2.
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The modsl then enters a second regime, as indicated
in (2) above, uhere the hard structure begins to influence
the craft behaviour and where--the following motion is almost
beam-on.
Modelling the craft behaviour in the second phase of
the capsize sequence therefore need not include all the
degrees-of-freedom of the steady level motion. In fact
the ~otion during this phase is an almost pure sideslipping
and rolling of the craft coupled with some heave as the
craft leading edge sinks at high angles of roll.
It is seen from Fig. 2 that even at very much aggravated
angles of roll, the trailing edge of the model appears to
remain f.ixed in space at least vertically; this allows a further
reduction of the problem to that of a 2 degree~of-freedom
model describing roll and sideslip motion.of the craft at
least up to about ~O degrees of roll.
1.3.5. The Formulation of £ uations describin the lateral ~ion of ACV's \capsize mode.
It may have become clear that during the second phase
of an ACV capsize sequence, ~described above), the hydro-
dynamics of the hull mU9t necessarily play an important
part in determining the general behaviour of the craft;
this of course is in addition to the contribution from
aerodynamic and cushion forcing terms which are always
associated with the motion anyway.
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Thus if the main forces and moments associatod with
the craft ltaking into account the Interaction of the hull),
could be estimated during this phase of the motion , the model
can be assumed to match fairly closely the condition of a beam-
on capsize.
Fig. 4 shows diagrammatically a general breakdown of the
main forces involved in a beam-on roll condition. It is seen
that it will be necessary to estimate the buoyancy of the
. immersed portion of the hull and this will act as a restoring
force. The action of water impingement on the hull face will
also contribute to the rolling moments and this will obviously
become aggravated as the craft Igunwale l moves below the ·water
surface. The hull will also have some damping associated
with it since energy is imparted to the water as the leading
edge portion of the structure sinks.
other terms in the equations must include the effects
'of stiffness and damping due to the cushion and the
destabilising effect of the VCG shift as the craft rolls.
Aerodynamic force terms will also come into play mainly as
orag terms in influencing the rate of deceleration of the
craft while in this phase of the motion.
The main coupling term between the roll and sideslip
equations ~ill be an induced downward roll due to the inertia
of the craft as it decelerates.
Non-linearities will come from geometric effects at
large angles of roll together with sudden changes in the
pressure distribution on the craft hull especially as the
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-craft 'gunwale' s1nks below the water surface. Other sources
of non-linearity will be the ·effect of the changing cushion
volume of the craft on fan characteristics and from transitions
in the flow pattern around the hull as the craft speed changes.
1.3.6~ Problem Solution
Once the equations describing the roll and sideslip
motion of the craft in a beam-on condition have been
. formulated, it will then be necessary to apply some problem
solving technique in order that the craft motion may be
observed.
There are two basic approaches to the solution and these
are respectively by analogue simulation and by digital
simulation. However, analogue machines though. speedy, are
generally unable to handle problems where the non-linearities
are extensive. On the other hand, the digital machine has
the qualities of versatility and far less hardware to deal
with and will be favoured h~re.
This will of course raise the question of deciding upon
a suitable numerical integration technique and also the
development of a method of dealing with the non-linear
terms in the equations. Usually the task is to find an
accurate and reliable approach, which exhibits good stability
-properties without requiring excessive computer time.
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2. PREVIOUS AND PRESENT INVESTIGATIONS
2.1. PREVIOUS INVESTIGATIONS
Studies so far into air cushion vehicle dynamics have
generally been accompanied by very little published material
as much of the work is usually done through commercial
or military interests. In particular, the availability of
material on the actual mechanism of ACV plough-in is severely
limited and therefore alternative means had to be sought in
retrieving the information required. Thus~it will be seen
from the following sections that a series of independent
studies were made on various aspects of the ACV in order that
.a more coherent picture may be built up illustrating the
state of the art.
2.1.1. Motion Studies of Skirted Craft with emphasis on Plough-in.
-Addition of 'flexible extensions' or 'skirts' to
ACV's not only enhanced the obstacle clearance capability
of the craft but also the wave riding capability. However,
as would be expected, along with these major advantages came
a series of problems, three of which have formed the basis
for further research and development.
2 by Crago as:
These were outlined
(a) ~ound skirt design consistent with the need for
it to adequately maintain its shape while at the
same time be pliable enough to reduce rough water
drag
(b) studies of the phenomena of plough-in and overturn
(c) studies involved with the reduction of skirt oscill-
ation and uear.
-
-13-
So it is seen that the plough-in phenomena right from the
start has been intimately bound up with the skirt system.
The plough-in phenomenon was first ·observed and studied
with SRN-1 craft as early as 1961 3• It was not given a great
deal of attention as this only occurred in the longitudinal
mode where resistance to c~psize is greatest. In the
case of SHN-1, a hydrodynamic bow had already been fitted
to cope with intermittent wave contact and this fortunately
gave the craft good restoring capability against capsize.
The earliest detailed study made of ACV plough-in and
overturn was carried out in the USA by walker and Rood 4
immediately following the SRN-S capsizes in 1965. . In this
report"it was suggested that with a small change of trim,
it was possible for water to attach to the fl~xible skirt
causing a hydrodynamic drag, a suction on the lowest part
of the skirt and a higher pressure rather higher up (See
fig. 5).
R. Stanton Jones records some work done on the effect
of reducing cushion pressure on craft heeling stability5.
It was found that the moment required to overturn an SRN-6
model reduced by up to 50% if cushion pressure was reduced
by only 20%.
In the Walker study4, an attempt was made to analyse
the effect of the forces on the skirt. It was suggested that the
net result would be to cause a downward pitching moment and
the subsequent collapse of the skirt. This would be coupled
-
-14-
with a violent nose doyn motion until the hard structure
contacts the water. As soon as the speed reduces suffici~ntly,
the skirt will ~edeploy restoring the craft to a full
hovering position once again; this phenomenon is popularly
known today as the tuck under and plough-in sequence.
Of course the skirt may not redeploy at all as in cases 5 where the bow contact is asymmetrical. R. Stanton Jones
records that the plough-in can proceed to a lateral roll
over, which, in actual full-scale incidents, has not reached
a completion until the craft has been moving sideways at
about lateral 'hump! speed. Thus two things can be
inferred:
(a~ In all the capsizes observed, the craft is travelling
approximately beam-on and this has been confirmed
in the ARB Special Committee report1
and (b) the critical region of capsize occurs when the
craft is travelling at around hump speed.
In 1968, Silverleaf in his review of hovercraft research
in Britain6 indicated that the lateral antisymmetric motions
of hovercraft in calm water had been analysed by using the
equations of motion for longitudinal force, sideforce, rolling
moment and yawing moment assuming zero pitch trim, small roll
-angle -and axes coincident with the principal axes of the
motion.
-
-15-
Obviously, this model l:Jill not --have been sui table for
the study of craft capsize which involves development of
relatively large angles of roll. Also, as the work was
done purely by analogue techniques, it is possible that
the many non-linearities of the motion could not have been
adequately represented •
. M.J. Saratt7 formulated lateral equations of motion
based on derivative notation widely used in aircraft dynamic
studies. However, this depends very much upon how the
solution to the expressions for these derivatives is approached
as they will·conta~n many non-linear terms and will therefore
be di ff icul t to sol v·e.
The general approach to the solution of ACV motion
equations has more usually been a numerical one. The
iequired data is accumulated for the particular regim~ under
study, and this information is then used for providing updated
inputs as the craft changes its stage in a simulation run.
Crag02 reported on dynamic studies carried out at SHC
where this principle was employed. As a result, time
histories of·craft attitude, speed and longitudinal deceleration
were obtained. This enabled a number of plough-in boundaries
to be calculated for zero yaw.
other reports written on this principle of motion study
only cater for specific cases. Fein et a1 8 studied air
cushion performance using BH-7 data as part of a computer
simulation. A four degree-of-freedom mathematical model
-
-16-
was used incorporating hydrodynamic experiments, aerodynamics,
propulsive forces and a control system. The equations for
yaw, sway, surge and roll. were integrated numerically using \
a computer program to provide time histories of the craft
motion. The program however was capable of simulating only
on-cushion condi ti o'ns andcalm-wa ter manoeuvres.
~ .'
2.1.2. Drag forces associated with ACV's
Another area which has stimulated great interest for
further research and development concerns the drag forces
associated with ACV's. The approach .has usually been
to separate the drag into major components and R. stanton
Jones lists theseS as:
(1) Air momentum drag
(2) Air profile drag
(3) water wave-making resistance
(4) water wetting resistance
(S) Increase of resistance due to encountered waves
A typical plot of the relative importance of the various
components is given in fig. 6.
Water wave-Making Resistance
It is noticed that the induced wave drag is a
dominant term influencing the total drag force acting on the
ACV and by far the majority of research work has been directed
towards a better understanding of the factors which have a
bearing on this drag.
Havelock9 was one of the first to treat the theoretical
problem of the wave resistance associated with a pressure
.1
-
-17-
distribution moving over water at constant speed.
Lunde 10 extended the theoretical treatment to cover
the case of an arbitrary distribution moving over.a finite
depth.
. 11 Baratt computed the wave resistance of rectangular
and elliptical pressure distributions. In deep water, the
main 'hump' on the drag curve occurs at a Froude number ~iven
by fR' = 1//Tf. In water of finite depth, this hump is shifted to a lower froude number, and for sufficiently
shallow water occurs at a depth/Froude number ratio equal
to unity.
The results obtained so far were based on the assumption
of a constant pressure distribution over the planform
and indicated that the resistance coefficient oscillates
rapidly at lower speeds (Fig. 7), a trend which is not
observed in experimental results.
fortunately, operating speeds of most craft are high
and this deficiency in the theory is generally of minor
importance. Houever, Doctors and Sharma 12 to some extent
were able to overcome this deficiency by introducing a more
'realistic' pressure distribution where a hyperbolic fall
off of the pressure at the edges was assumed. This allowed
further studies to be made for the cases of accelerating
craft and a typical wave drag plot obtained is shown in
fig. 8.
-
-18-
As to the computation of surface waves generated by
13 ~cVts, Hu and wang presented numerical results for craft
with rectangular planforms travelling at three different 14 speeds. Yeung computed surface elevations as well as
induc~d wave forces.
On several occasions, extensions to general trajectories
and to arbitrary time-dependent pressure distributions were
by now being inhibited by the computational complexities.
However, Haussling and Van Eseltine 15showed that a fourier
series approach lead to an efficient computatio~al scheme
for the analysis of wave resistance, side force, yawing
moment and wave elevations associated with ACV's.
A number of experimental programs were formulated to
check the above theoretical results obtained by Barat0 1
and Everest and Hogben16,17,18,19,20are the chief workers in
this field and the main question pointed out in these papers
is the resolution of the total drag acting on the ACV into
its components. Thus a short summary of the way the other
components are generally defined will be given here •
.8.!r r~omentum Drag
The momentum drag is that resulting from a change in
direction of the air flow, as 'the latter moves through the
c~shion system. The drag is in fact mainly made up of two
components usually referred to as the inlet and outlet momentum
drag respectively. The outlet momentum drag may sometimes 12 act as a thrust , depending upon the trim of the craft.
-
-19-
Air Profile drag
The air profile or aerodynamic drag is assumed to be
that resistance acting on a model if it were tested in a wind
. tunnel with engines not running.
Water wetting drag
This is usually referred to as water contact drag as
it is that due to any contact of the louer edge of the skirt
with the water surface. Spray drag is usually included in
this estimation as both have a very non linear nature and
therefore very difficult to predict. Everest 16 estimated
the water wetting resistance by eliminating it - using a
thin polythene sheet floating on the water surface.. The
technique however raises the .qu8stion about the tensile
forces on the sheet.
2.1.3. Hydrodynamics of ACV-type hulls
Very little theoretical work has been done on the
hydrodynamics of ACV hulls possibly because the hull shapes
in use today are too varied and possibly because it may be
considered that work done in other fields will be adequate in
providing information of the behaviour of a particular hull
shape.
However, experimental work has in fact shown that proper
hull design is vital to providing the craft with adequate
restoring capability in the event of a beam-on plough-in
(see Hefs. 4 ~ 21).
-
-20-
22 Shuford made a study of various planing surfaces and
found that an optimum planing angle of about 4-5 degrees
is typical. However it is expected that with ACVt s , this will be unachievable considering that some buoyancy must be
built into the hard structure and this usually implies steep
angles of hull incllnation.
It may also be possible for the surfaces of the hard
structure to be considered as flat plates. A relatively
large volume of work is available on the study of flat
bottomed planing craft which might therefore be of use~
• 23 24 Notably Shoemaker and Murray have made some worthwhile
contributions to the experimental study of the hydrodynamics
of planing surfaces.
Most texts on aerodynamics or hydrodynamics, notably
references 25 & 26, treat the theoretical study of flow "
i~pingement on flat surfaces using a two-dimensional approach.
However these methods will "generally be limited by assu~ptions
of small relative angles of incidence of the surface to the
stream flow.
On the other hand use of the 2-0 approach to the
theoretical treatment of ACV hulls is still justified as
the relative aspect" ratios of surfaces involved are generally
large (> 10) and therefore some extension to the theory will
have to be made to cover large angles of incidonce (>10 0 ).
Another approach which would also conveniently take into
account the bends at the top and lower edges of the hard
structure would be to use oonformal mapping techniques and
again olassical methods are found in most texts, notably the
-
-21-
uork of Prantle and Tietjens 27 •
28 The latter technique was used by Hogben reg~ding
the flow past a two-dimensional curved plate at a free
surface. One of the aims of the work was to make an
analysis of the flo~ past the rounded portion of the skirt.
Of course Hogben's work could be utilised in the aggravated
case where the skirt has tucked back close to the hull;
however, it is likely that the assumptions of a rigid boundary
and detached flow at the lower edge will impose too many
limitations on arriving at a suitbble solution.
A final approach to the prediction of the pressure
distribution around the ACV hull could be taken by adopting
the theory of SquirJ9 who considered the motion of a
simple wedge along the water surface. However the use of
this work will once again be limited by the assu~ption of
low trim angles and separation of the flow at the bottom edge
for all speeds above zero. It is thought that the flow must
detach at some speed since a point will be reached where it
will be impossible to achieve the proper formation of a wave
system.
2.1.4. Studies associated with the Cushion System
studies on the cushion system form a~other area which
has occupied a great deal of interest in the total research
and development of ACV's. Available material on the cushion
system may include studies on the fan, internal ducting and
-
-22-
the skirt all of which play an important part especially in
determining the heaving dynamics of th~ craft.
An important attempt to correlate the heaving and
pitching motion of ACV's with the cushion system was made
by Wheatley30. Ho~ever, his theory was based upon
restricting the craft to small motions. Important conclusions
made were that the stiffness of the cushion will be inversely
proportional to the daylight clearance at the bottom of the
skirt while the damping will be inversely proportional
to the flow rate.
Richardson 31 advanced the principle that the cushion
can be regarded as a suspension system similar in many
respects to that of a motor vehicle and essentially ~.
consisting of a spring and dashpot. This was later verified
in a report made by Wheeler 32 on research and development
of ACV's in Britain.
. Also the damping contribution of the cushion appears
to be intimately bound up with the shape of the fan
characteristic curve. It is therefore found that in most
theoretical work, linear or quadratic assumptions are made
in describing the slope of the fan characteristic curve
(see Ref. 33).
-
-23-
2.2. PRESENT INVESTIGATION
The present investigation involved:
(a) the theoretical development of a two degree-of-
freedom ACV modal describing the roll and
sideslJp motion in the beam-on leading skirt
collapsed phase.
(b) the simulation of the model using digital
techniques
(c) the subsequent testing of the model and the
final evaluation of the major craft design·
parameters.
The. equation describing the roll motion of the craft
was developed to take into account the following factors:
(1) static moment contributions due to VCG shift
and buoyancy of the immersed portion of the hard
structure
(2) dynamic moment contribution due to flow
impingement on the hard structure
(3) dynamic moment contribution due to inertial
coupling coming from the deceleration of the
craft
(4) damping forces due to the flow over the hard
structure at aggravated angles of roll.
(5) stiffness and damping contributions of the cushion
system.
-
-24-
(6) an estimate of craft inertia and added inertia
terms due to fluid displacement as the craft rolls
and (7) modelling of the induced trim due" to" the position
of the cushion wave system below the craft and
applying suitable phase lag to this system
depending upon the deceleration of the craft.
The equation describing the motion of the craft in side-
slip took into account the following factors:
(1) hydrodynamic drag forces associated with the flow
over the hard structure
(2) aerodynamic drag forces dUB to the air flow over
the craft
(3) cushion air momentum drag
(4) water wetting and spray drag
and (5) induced wave drag due to the cushion.
The simulation language SLAM, compatible with ICl , machines was used to model the motion of the craft. The
simulation of the model involved the use of the technique
of storing values of the various non-linear functions over
a defined regime and then using these to provide updated
inputs as the craft changed its state. This allowed fairly
accurate "solutions to the non-linear problem to be obtained.
A basic configuration craft was chosen as shown in Fig.9
and the following initial values were assigned. to the
major design parameters:
-
-25-
hB/B = 0.1
h /B = 0.1 5
hG/ B = 0.2
8 = 21 0
Extensive testing of the model which involved a
systematic method of variation of the above parameters
finally enabled a set of 'safe' ranges to be decided upon
for the major design parameters.
Also various aspects of the testing have brought out
some interesting features on motion characteristics for
instance the relationship of cushion damping with flow rate
through the fan which agree very well with previous.
b t · 30 o serva ~ons •
-
-26-
3. THEORY
3.0 Introduction
In this ~ection, the Hall and jideslip Equations des-
cribing the model dynamics are developed. It is considered
that inclusion of a Heave Equation in this initial study would
greatly increase the complexity of the problem as the dynamics
of the cushion system at large clearance heights (h> .058), is
extremely difficult to predict mainly due to non-linear cross-
flow effects. lt is thought that the effect of including a
Heave £quation will in general be to reduce the relative roll
angles achieved at least over the regime of rol~ angles considered
in this study and therefore the results using a 2 degree-of-
freedom model is likely to be on the pessimistic side.
five main assumptions are made based upon notes and obser-
vations made in ~ections 1 & 2 and these are:
(1) the craft is assumed to be in a condition where it is
travelling beam-on and possessing some initial pre-
determined sideslip velocity
(2) the leading-edge skirt in the beam-on condition is assumed
to be fully collapsed exposing a planing hull face
(3) the water flow is assumed to detach about the lower
edge of the hard structure at higher speeds, ~point p,
fig.4). A transition si~eslip speed is then defined
at which the flow becomes fully attached
(4) no coupling exists between the roll motion and heave
motion, or between sideslip motion and heave motion up to about 30 0 of roll. Thus for all purposes the craft
trailing-edge ~point U, Fig.4) remains fixed vertically . in time
-
-27-
(5) the trailing skirt is considered tb be rigid so
that for co~venience, body axes are set up about
the craft trailing edge and these axes coincide with
the space axes at the start of the motion.
3.1 DERIVATION OF THE ROLL EQUATION
The general equation describing rotation in any
engineering system usually has terms dependent on angular
acceleration, angular velocity and angle of rotation. If
the forcing function is given by some moment.Jt, the linearised
equation is of the form:
+ + !i. rJ -- . It • • • ••• (1)
where 1 , C and f
-
-28-
If cross coupling and added mass effects are included,
the expression becomes:
• • • ~ •• (3)
This expression is more easily formulated if the terms
on the right hand side are further broken down as functions
of one or two variables only.
Thus Equation (3) can be written in the form:
• • • • • • ( 11 )
Again ~he dashed term denotes derivative w.r.t ft and the coupling term (M 1) and the cushion forcing term (M 0 d) have been cp cp separated from the expression for total moment in Equation 3.
The dynamic equation written in this form makes it easier to
describe the roll motion of ACV's. However, although it
is possible to write this equation'o describing o
the rolling motion of the craft directly into a computer
simulation program, grouping the expressions derived
will be too cumbersome and lengthy.
Instead, the various terms are derived and put in a
suitable non-dimensional form, and a computer program is then
used to determine numerical values forO these terms over
a defined regime of study. This approach is discussed in
greater detail °in Section 4.
Section 3.1.1 - 3.1.3 inclusive involves the derivation
of the static moment terms and these are dependent on roll
angle (,£1) only.
-
-29-
Section 3.1.4 involves a brief study of lift and drag
coefficients associated with the movement of a flat plate
through a "fluid medium"Bt high angles of incidence as is
typically the case with a planing ACV hull.
Section 3.1.5 involves a short study of centre of pres-
sure location asso~iated with the movement of a flat plate
through a fluid medium at high angles of incidence.
Section 3.1.6 - 3.1.7 inclusive concerns the derivation
of the dynamic moment terms and these are functions of
both roll angle (~) and sideslip velocity (y).
Section 3.1.8 concerns the derivation of an inertial
coupling moment expression arising from the deceleration
of the craft after release from some initial speed. This
term is" dependent upon both roll angle (~) and sideslip
deceleration (V).
Section 3.1.9 involves the derivation of suitable
damping expressions for the rolling model. Two major areas
of contribution were identified and these are:
(a) that arising from the change in flow rate as the
cushion undergoes compression and expansion and this is
dependent upon both roll angle (~) and rate of roll (~)
and (b) the second term is that due to an induced incidence
" effect of combined sideslip and roll rate as water
impinges on the hard strutture, and this is~dependent
on "both roll angle (~) and sideslip velocity (~) of the craft.
Section 3.1.10 concerns the derivation of a suitable
ex~ression for the craft inertia and also an extra inertia
-
-30-
term due to added mass effects experienced as the craft:-rolls.
As a result, the inertia of the craft is dependent upon the
roll angle (~) of the craft.
Finally, in Section 3.1.11 the full equation describing
the roll motion of the craft is assembled based on the
various terms already derived in Sections 3.1.1 - 3.1.10,
inclusive.
3.1.1 Moments due to change in VCG position
ALL i v-From Figure to , the moment about 0 is given by:
= ••• ••• (5)
Non-dimensionalising by dividing throughout by LlB,
gives:
MOG [~ hG tanp J cosp LJrB = + B = cosW + hG sin,0 ••• ••• (6) 2 B
3.1.2 Moments due to Loss of Cushion Area
It will help at this stage tb"consider the contribution
of the cushion from a stri~tly static point of view,
although the expressiqns derived are not actually
incorporated in the model. A more rigid study is
made in Section 3.1.9 where the effects
of compression and expansion of the cushion are taken
-
-31-
into consideration.
It is assumed that the cushion pressure remains
constant and that cushion area is red~ced due to
leading edge skirt collapse. The effective plan area thus
has a reduced beam, shown as OP' in Figure 11a.
3.1.2a Case for yi
-
-32-
= -t
3.1.2b Case for [5> -liA
In this case, the effective cushion is reduced even
further as the crait rolls through angles greater than -A.
Thus the leading edge of the cushion is moved further inboard
as the craft rolls, shown by the movement of point M in
Figure11b.
The effective beam OM is given by:
B. h S.1 OM = s~n Assuming once again that the pressure force acts at
half the effective beam, the moment about 0 is given by:
= [ ~2 SI"nfJ] ••• . ... (10)
Using Equation (8), Equation ·(10) can be
non-dimensionalised by dividing throughout by WTB.
Thus:
= ·2
.[~J • • • • • • • • • (11 )
-
-33-
3.1.3 Moments due to buoyancy forces acting on the
immersed hull structure
The theory for both the detached and the attached flo~
cases will be derived in this section. Some of the geometric
terms associated with the buoyancy forces are derived
in Appendix I and these will be referred to from. time to time.
Definitions:
(See Fig.12) Let:
5 ~
~A 4
~S ~
(See Fig.13) Let:
T ~
V ,;~
. 3.1.3a Case for _ < ~A
h S/ S 1· - tanS
tan -1 hS/ S S
tan -1 [:S + :S~
(Stan~ hS) cos~ S S Ttan~ cos~
· ... • •• (12)
• • • • • • '(13)
• •• • • • (14 ).
• 0 • (15)
• •• (16)
There is no contribution of buoyancy from the hard
structure.
Thus:
= o • • • • • • (17a)
and the non-dimensional quantity:
= o ••• " . . (17b)
-
-34-
Figure.13 shows craft immersion at an angleji, CI1 A
-
-35-
moment, a relative density term «(reI) is defined given by:
('reI MT
= L.S.h S
= ~;r/9 (19) L.S.h S • • • • ••
D i vi din g E q u a t ion (1 8~ t h r 0 u g h 0 u t by LJ T San d sub s tit uti n g
Definition (19) gives:
=
= _1.'&. 1 6 .(rel hs/ S • [
T2 J [ T l tan(S-0) • 3V + tan(8-0~
• • • ••• (20)
3.1.3b(ii) Attached flow case (Fig.13)
. . If the flow is assumed to remain attached as could be
the case at low speeds, the area under consideration ought
to be that given by b. PMN and not b. PXN as assumed for the
detached case.
But: . b. PMN = b. PXM of- A PXN
Thus only the extra contribution of APXM need be
considered here as APXN has already been taken into account
in the provious section. The area of interest therefore is
as shown below.
-
~36-
Thus the moment contribution due to APXM is given by:
• • • ••• (21c) •
Non-dimensionalising Equation (21q by dividing throughout
bYWTB and using Definition (19) as before gives:
= .1. L5! . -L.. 6 I'rel hB/B
T 1 ... (22) tan~J
The total moment contribution due to buoyancy assuming
fully attached flow is found by summing Equations (20) and
(22) giving:
• 0 .. ••• •• (23)
-
-37-
3.1.3c Case for n> ,08 figure 14 shows craft immersion at an angle ~ degrees
to the water line (11 > ,08)' the latter crossing the hull structure in two points M and N.
3.1.3c(i) Detached "case
In evaluating the buoyancy terms as the top of the craft
becomes immersed below the water line, the assumption is
made that flow separation occurs along the line PXin the
detached case (fig.14a). The angle ~ is as defined in
figure 14a. It should be noted that as the craft sinks'
deeper, (i.e ~ increases), a point is rea~hed where PN and
PX coincide and 0(= (~ -¥5),(see fig.14b).
A new assumption is then made that separation occurs
along the line PN as _ increases further (see fig~4c ).
Definitions:
Let: W ~ h8Ls+ hS/ S .. . . • • • (24) tanj1
0( ~ h
sLs • • • • • • (25) 5 - lJ
In this case, the immersed portion of the hull under
consideration is the area PXNQ (fig.14a). This can be divided
along PN and represented by triangles PXN and PNQ.
~PXN .
f-r 0 m A pp end i x I • 2 • 1 ., the are a 0 f t his t r i a n g 1 e i s
-
-38-
given by:
=
The moment arm (OU) is also found in Appendix I.2.3
to be given by,
=
.6. PNQ
iBLlt C"OSff +
2/3 {8T~ tan (
-
=
where
1t Case for g1 > 2" -0(
A' =
-39-
~ tw . ~'[A'+ S' J frel B/B
. [ 3lJ + cos~
• • • •
2T 1 tan{d. +~ )J
••• (27)
As 'mentioned before, for y1 > (1-0
-
-40-
at the lower speeds, the area under consideration ought
to be that enclosed by PMNQ (Fig.14c).
But: Area PMNQ = APNM + A PNQ
The contribution of Area PNQ has already bean considered
in Section 3.1.3c(i) for ~> H-ex). Thus only the extra
contribution due to Area PNM need be considered here.
From Appendix I:
Area pNM =
The moment arm due to a PNM is given by:
ApNM hs
+ ~(BS -t:~Ji)'( = sin~
Let: G ~. S - ~ ... - •••• (30) tan The moment contribution due to area PNM is then given
by:
= +
Non~dimensionalising Equation (31) by dividing
throughout by IJTB and using O'efini tion (19) gi ves:
~l°B 1 .&. . --L. C' = - 6" • • • • •• WTB Irel hB/B h .
. [3hs/~ 2G. Sino
-
-41-
The net moment contribution of Area PMNQ is found by
summing Equations (29) and (32) and non-dimensionalising
by dividing throughout by WS;-
Thus:
1 - '6 ~ .~[ C'+5']
(reI hs/S • • •
where C'is as defined in Equation (32)
and 5' is as defined in Equation (29).
• • •
3.1.4 Development of Dynami c Force Coe ff icients for 'Flow
over Flat Surfaces at High Angles of Incidence
As much of the dynamics of the craft depend upon the
(33)
lifting 'properties of the surfaces presented to the stream,
it is important that realistic expressions be used for the
various lift and drag coefficients associated with these
surfaces.
The surfaces presented to the flow possess properties
of high aspect ratio (>10), and high angles of relative
incidence (>15 0 ). This has therefore dictated a 2-Dimensional
approach to the problem and this is outlined in greater detail
in Appendix 11.
In the extreme case where the surface is set broadside
to the stream, the force coefficient assumes a maximum value
of about 2.025 •
-
-42-
Ctc -
Diag.3
for the case shown in uiagram 3,
Total force coefficient
lift coefficient
drag coefficient
CN ~ 2·0
C ~ 0·0 L .. C ~ C
D N
CL and Co are defined perpendicular and parallel to the
stream flow respectively.
ixpressions were developed in Appendix 11 for surfaces
at both positive and negative angles of incidence and three
constraints were used,
(a ). CN:j. +2·0 (for all positive angles of incidence)
(b) C {-O·1 (for all negative angles of incidence) deN
(c) .-1! "'" 1t ( for small angles
-
-43-
::: Sir )] ... (36)
3.1.4(ii) Negative ~ngles of Incidence
---"-~. Oiag.5 Expressions for these coefficients were found in Appendix 11
to be given by:
Cl = o • 1 4 [ e 1 0 T{{l - e ltfolcosf • • • • •• \37)
as in t:.:q~34) I,; -0- cL:Tanf • • • • • • \38)
dCl [ 1011f ;:lint) -e Y: (ltCO¥ - ~if}t3 B err = 0·14 e .• (10ncosr- • • • • • • (39)
3.1.5 Centre of ~ressure Location
It is difficult to predict the movement of the centre of
pressure on thb craft hull as the incidence angle changes,
though, it has been shown that the value stays around the
quarter-chord position at least for small angles of incidence29
•
To simplify expressions describing the hull dynamics which
are derived in the next two Sections, the quarter-chord assumption
was made for all cases studied. fhe further implications of
this assumption is discussed. in greater detail in Section 5.1.2a.
-
I .
-44-
3.1.6 Water Impingement of the Hull ~ace
The moments due to water impingement on the hull face
will be derived in this ~ection. tigure 15 shows the force
components to be considered and PC is treated as a section
of a flat plate of length L making an incidence angle of (8-~)
with the stream flow. The assumptions made for derivation of
force coefficients and centre of pressure position in Sections
3.1.4 and 3.1.5 apply here.
3.1.6a Case for p < PH
There is no contribution from the hull and therefore,
3.1.6b Case for @A < ~ < ~t.l
from Fig.15,
A
(B-~) PCX =
.: .. PC PX
= ~int8-~)
= {BT} ~in~8-~)
= .. 8(TF)
'where(TF)~ T ~in~8-.0) • • • ••••
The area ~SH) presented to the flow is given by:
,sH = L.(PC)
= L8(TF)
(40)
-
-45-
Also from ~ection 3.1.5, the centre of pressure location
is assumed to be at the quarter-chord position,(point T in
fig.15).
TC = tB(Tf)
Hence the moment-arm due to the drag forces (JT) is
given by:
JT = tl:l(TF). Sin(8-~)
. = tlBT)
I"loment arm due to the lift force lDJ) is given by:
DJ = 01'1 + I'IX + XJ hS
+ ~ + f(BT) = sin~ fan lan~B-~) ,. .," .. Let (T1 ) ~ ~ T ~41) .. Sin + T~ • • • • • •
DJ = B.[CTI) + f( TF,~
The lift force is given by:
LH = !rW"U:,~,CLH . (42) · -- • •• CLH follows the definition given in Equation 34, where:
-~(8-~) 0-2(1 - e ).Cos(8-~)
The drag force is given by:
Substituting for CO H using Equation 35, gives:
= • • • · -.
-
-46-
The forces due to skin friction is small compared with
LH and OH and will be neglected here. Thus, the total moment
due to the hydrodynamic forces acting on the hull face is
given by:
.
= ~}f' U:LB2.( TF). CLH{~ TI) + 0' 75 (Tf ~ - [O.25(T)Tan(8-j1~ ••• (44)
Non-dimensionalising ~quation 44 by dividing throughout
byWrB and substituting lquation(B) forWT on the right hand
side gives:
2 2 -tPw Uw LB ( ) r 2. TF .CLH
Pc L.B . {
- [oe25(T) Tan~8-J6)]
+ 0'75(Tf l] - [0'25(T) Tan(S->1B} • e • • • •
3.1.6c Case for ,0>}ifB
For ~ > ~lj' water impinges upon the whole of the hull
face so that,"
PC = PLl
t4S)
Thus the lTF) term in lquation ~44) is substituted by
hS/B and by induction, the moment expression is found to be: SinS
• CLH" {~TJ) + 0. 75:~~~] - [0. 25( T) • Tan\ 8-,11~}
o • .. • • • • • l46)
-
-47-
~imilarly, the non-dimensional moment becomes:
- [0' 25 (T) Tan (a-pH} •••
It should be not~d that it is possible for incidence
angles to go negative at high roll angles and hence the-
following constraints on CLH are observed:
••• ~47)
From £q.~34) (for 8 ~ ~)
3.1.7 water Impingement on the Top structure
In cases where roll angles are large (~> ~ti)' water
impinges on the top structure, and this increases the over-
turning moment acting on the craft.
from fig.16:
OH' =
= (SW) ,definition (24-)
OQ' = 8
R'Q' = RQ = 8(1-w)
"The area (3 T) presented to the flow is given by:
Assuming the centre of pressure to be at the quarter-chord
position, (see Section 3.185), the force will act at point K,
-
-48-
where,
RK = iRQ
The arm due to the lift force (DC) is given by:
OC = OR + HC
he + hS -+ RK.Cos.0 = Sinj!)
tJ + 0·75(1-W).Cos~ = Cos~
The arm due to the drag force (CK) is given by:
CK = HK.Sin~
= 0.75 RQ.5in~
CK = 0-75 B( 1-W). Sin~
The lifting force on the top structure is given by:
L.T = ••• • • • • (48)
CLT follows from the definition given in t::quation (34j,
where,
CLT = 2· D (1··--it~
e 2 ).Cos~
The drag force is given by:
=
Substituting for .COT using Equation (35) gives:
• • • •••
Again it is assumed that the forces arising from skin
f~iction will be small compared with the normal pressure
forces acting on the hull top and is neglected here.
(49)
-
-49-
Thus the total moment arising fr6rn the hydrodynamic forces
acting on the hull top is given by:
=
(LT". DC) + (D T • CK)
o/wU~~B2(1-1J).CLT{[~~S~ + O.75(1-1J).CO.~ I +' [O.75C.1-IJ). Sin~. ran~
which reduces to,
• • • t50)
Non-dimensionalising Equation 50: by dividing by IJrB and
making the subs ti tution for IJ T on the right hand side given
by £quation ~8) gives:
• • •
In this case, no constraints are put on ~LT regarding
incidence angles achieved as ~ will always be positive for
the water to impinge on the top structure.
3.1.8:Derivation of the Inertial coupling Moment
As the craft decelerates from some initial speed, there
will be a force generated at the C of G due to the inertia
of the former; AS the resulting moment is dependent upon
the dynamics of the motion in sideslip, it is here referred
(51)
to as a coupling moment. It is required to derive an expression
for the moment about 0, the trailing edge of the skirt.
from figure 17, the inertial force at the ~ of G parallel
to the water line is given by:
-
-50-
FI -M T ...
~-.- --~--
= • y
where MT is the total mass of the .. and y is the craft deceleration
2' also OG Jh G
2 = + tj 2
=
=
The moment arm (GK) is given by:
GK- OG.Sin(¥-.£1)
=
The moment about 0 is given by:
frlCPL MT
.. GK = .y •
lJT ... GK = - .y. g
Non-dimensionalising tquation ~52) by dividing throughout
=
3.1.9 uerivation of the Koll uamping Expressions
It is expected that the major contribution to craft
damping in roll will come from the Ipumping' effect of the
• • •
craft
~53)
-
-51-
cushion and from the dynamics of the hull.
3.1.9a Cushion uamping
As the craft rolls, the cushion volume will change. ~ince
the cushion system i~ not a perfect seal," this will result in
some amount of leakage through the skirt. accompanied by
changes in flow through the fan. It is assumed that leakage
through the skirt will be negligible and that"the fan will
respond instantaneously to any flow demand; this will result
in a change in pressure rise across the fan and consequently
the craft cushion pressure will change.
3.1.9a~i) Estimate of Change in Cushion Volume
as Craft Rolls lCase for p1
It is assumed that althoug~ the leading edge skirt has
collapsed, it still operates as an effective seal as shown
diagrammatically in Figure 18.
The cushion volume per unit length is taken as the
area enclosed by OQPl.
But area UQPL = A POQ +" A POl
Consider 6.POl
let APOL = A(1)
It is seen that this area does not depend upon the roll
angle ~ and is therefore a constant.
Consider 6.POQ
PQ = hS - (!:is) Tan~
Hrea POQ = A(2) = -KBS).[hS - (8S) Tan~J
-
The Lushion Volume (V)
V = L [A (1 )
= L[ k1
-52-
is given by:
+ A(2)]
+ k2 - ~82~2 2 .:l
where:
Tan,O]
k1 = 11(1)
k2 = ~B~hs
• • • • • • (54)
The change in cushion volume w.r.t time can be written in
the form:
dV dV • .2if dt = d}f dt.
Equation (54) is then differentiated to give:
dV (ft=
2 2 2,,( ri{ ~L.! ~.5 ~ec p • ? • • • •
3.1.9a(ii) Estimate of Change in Cushion Volume
as Craft Rolls tCase for .0 > PAl
· ...
from figure 19, the cushion volume per unit length is
given by area OLM.
Now OM' =
LM =
=
Area DLI'I =
=
The Cushion Volume is given by:
V =
= + h 2]
t Ta~;0 • • • • • •
(55)
(57)
-
-53-
where =
uifferentiating Equation l56) w.r.t time gives:
hS 2
dV 2 cosec2~ % ....... = -L.t.B B • • dt
3.1.9~liii) ~xpression for the Moment arising
from Cushion Forces \Dynamic Case)
• • • • • • l57)
It is necessary to define some of the fan characteristics
for a given KPM, in order to obtain an estimate of changes
in cushion pressure.
Let the fan design operating pressure = PT o Let the fan design operating flow rate = Qo
let the duct diameter = dc
Also, assume that the characteristic curve can be linearised
and that. any pressure rise will be accompanied by a propor-
tional decrease in flow rate. This allows a simple relation-
ship to be developed relating total pressure rise across
the fan to its flow rate and is given by a curve with a 450
slope as shown in figure 20. The equation of the line will
therefore be given by:
PT = k1~ + C • • • • • • • l58) -PT
where: k1 0 = ~
C = 2PTo
The assumption may also be made that the total pressure
loss in the cushion system is approximately equal to the . .
dynamic pressure 34a , (~fouJ). This assumption is typical for
small craft where air flow is made to pass direetly into the
-
-54-
plenum area. thus encountering a sudden -enlargement.
The mean duct velocity, Ud.!_ will be proportional to the
volume flow Q, where,
=
This then allows the total pressure rise across the fan to
be related to the cushion pressure by the equation:
PT = Pc + -!(i ~d
= - Pc + k Q2 2
2
• • •
where: =
• • • (59)
14-A 2
d.
jubstituting Equation (58) in Equation (59) gives:
• • • • • • (60/
Also by ignoring any additional flow under the trailing
skirt, the following equation can be written down:
Q =Q + dV o dt •••
,; .. (61) where: (~~ is the change in flow rate
Que to craft roll and is related by the expressions given in Equations (55) & (57).
~ubstituting Equation (61) in Equation (60) yields the final
expression for the cushion pressure (pc)' where,
or
p = c
p . = c + lSDL)2
P~cr£ • • • • • • 0 • (62)
-
-55-
In the dynamic case, this expression for Pc is substi-
tuted in the equation for the moment contribution due to
the cushion derived in ~ection 3.1.2.
Case for !1 < @'I\
Substituting Equation (62) in ~quation (7) gives:
Case for 0> 01\
Substituting ~quation ~62) in tquation t10) gives:
• • •
It should be noted that the correct expression for
••• ~63)
(64)
&~ must be substituted using either of Equations (55) or ~57} depending upo~ the roll position of the craft.
3.1.9b Oamping of the Hull
A certain amount of damping will come from the hull as
a result of the induced incidence effect arising from combined
sideslip and roll rate. The hull face and hull top are treated
as fl~t plates, as befo~e,:at relatively high angles of i··
incidence to the stream flow o
-
-56-
General ~olution
o Oia9·6
Consider the movement of the surface shown in uiagram 6.
Let the veloci ty of the stream flow. be l.J...> and l.et the angular
velocity of the surface about point 0 be w \positive downwards).
rhe velocity (w) at the 'centre of pressure of the surface
perpendicular to the stream flow is given 'by:
• • • ,I • • • • • • ~65)
The induced angle, e, is given by,
E = w = U
-
-57-
tram uiagram 4,
Differentiating this expression w.r.t gives:
dCN· cor + Cw -~inf·r ::: dCl .
dCN '" dCl 1
V = CN = df3 + Cl" Tant· cor ••• • •• (67)
The moment about o is given by:
M = N.X 2
~ -tr u; .S. eN .(;s + X1
-
-58-
XH, = B J S2 + • • • (70)
where, h (TG) A B S Sin8 + S.Cp_s8 ~71) • • • • • •
XH2 = x H 1 • Cos ( 8 - ¥') - .... • • • ~72) where: ¥'is as defined in App 111.1.1.
The incidence angle, f, is given by,
f = (8-}1) .... • •• ~73) Substituting £quation (73) in ~quation (57) and then
dCl making the relevant substitutions for CL and df given in
Equations l34) and ~36) respectively yi~lds the expression
for CN
for the hull face, where,
This simplifies to,
_ -~(8-~) CNH"" = It e ••• • • • l74)
Equation ~68) applied to the hull face may then be written
as:
= • • •
where the various terms on the right hand side are as
defined in ~quations t59) - l74).
[;ase for ~ ~ .fltj
for ~~"~~, water impinges upon the whole of the hull
face. The wetted length is then given by,
~75)
-
(TF) = hs/s SinB
-59-
. _--_ ..... _. • • • ~ 76)
lxpression l76) is substituted ~n Equations \69), \70)
.q' I I and l72) to give new terms ~, xH1' XH2 for this case ~see
Appendix 1I1.1.2).
I::..qua tion l 75) is· then of the form,
=
Also, it is possible for incidence angles to go negative
at high angles of roll. This is the case when 15 > 8. In this ....
case, the relevant expression for CNH is derived from
Equa tions'. ~ 37) and l39) giving,
• • •
3.1.9b(ii) Damping due to water Impingement on the Hull Top
figure 22 demonstrates the induced incidence effect
applied to the hull top. This will of course only come into
play when 11 > ,f1B.
The area presented to the flow is shown in Section 3.1.7
to be given b~,
Sr = L8(1-W) • • e • • • (7S)
where: W is as given in Definition (24)
The arms xT1 and xT2 measured w.r.t the trailing edge
of the craft is shown in Appendix 111.2. to be given
respectively by:
-
-60-
= + 2 (4W.Tan,0) •••• (79)
t:I = 4' (3+W) • ••• • •• ~80)
The incidence angle,Y'~ is given by,
• • • • • • • ~81)
,.. By induction, the expression for CN T can be obtained
using Equation (24) where,
-!!.Jf eNT = ite 2 • ••• • •• (82)
Equation (68) applied to the hull top is then of the
form:
= • • • (83)
where the various terms on the right hand side are as
defined in ~quations (78) (82).
3.1.10 uerivation of Expressions for Craft Inertia in .Holl
The craft Toll i~ertia will con~ist mainly of two terms,
(a) that due to the rotation of the craft about
its trailing edge in still air~
(b) an added masq term due to displacement of
fluid as the hard structure comes into
contact with the water.
These terms will have an additive effect on the total
craft inertia in roll and they are treated separately in the
forgoeing sections.
-
-61-
3.1.10a Craft Inertia in Hall associated
wi th I'lotion in ::Itill Air
In general, for a given craft weight, the inertia term
will change as the craft dimensions change. It was found
that a good estimate of the light hovercraft inertia in roll
could be obtained by making the following assumptions:
1. t".ngine
(a) Mass of engine (ME) = 30% MT-(b) The engine is fitted on the centre line and
represented by a rectangular block (dimensions
as given in Fig.23).
(C) C of G position at point x.
2. ~rew
(a) Crew mass (MC) = .30% Mr. (b) Crew mass is treated as lumped masses and the
craft takes a crew of two, symmetrically displaced
at n·3B from the centre line.
3. Hard ~tructure
(a) Mass of hard structure (M H) = 40% MT. lb) The hard structure is likened to a rectangular
block with dimensions as shown in fig.23.
tC) C of G position at point Y.
It is required to find the inertia about 0 where 10 is
given by:
I o =
=
IG + MT(O~)2
+ 2 ~2
MT(h G + '4)
-
.. . .
-62-
10 = IG + MT~2~hG/B)2 + ~
where: . 2
IG =.2, MrK
a _ •
The components of ~MTK2 are as given below.
from ~ig.2:S:
= o • 3", T • 2 ( 0 • 3 B ) 2
•••
••• \84)
••• (86)
+ [ o· 25 + ,,( h 8/ B ) 2.J 1 12 J
-.. _ •• (87)
3.1.10b uerivation of the Hdded Inertia ~xpression
General ;)olution
o
Diag. 7
Consider a mass of fluid mv being displaced as point P
rotates about 0 with angular acceleration ~(Diag. 7). The
force required to accelerate the mass is given by,
-
-63-
• • • • • • (88)
The moment about 0 is given by,
• • • • • • (89)
mvr2 is in effect an added inertia term, where,
• • • • • • t90)
Case for PH < 11 < .0 8
1. uetached Flow assumption (see ~ection 3.1.3b(i»)
from tig.13, for the detached case, the area under interest
is APXN. From Hppendix 1.1.1, this is given by,
A PXN = 1.'(8T)2 (91a) Tant8-#) • •••
. ~ mvpXN _ ! (8T)2 - fw L • Tan t 8-J¥) • • • • (91b)
From Hppendix 1V.1.1, the distance of the centre of
area of A PXN from 0 is gi ven by:
OZ' + 2 '
T • • • • (92)
Thus using Equation (90), the added inertia term due to
area PXN can be written as,
• ••• (93)
. where mVpXN and Ol' are as defined in Equations (91b)
and (92) respectively.
2. Attached Flow assumption (see ~ection 3.1.3b(ii)
The area under consideration is that given by APMN (fig&13).
-
-64-
tjut APMN = APXM + APXN
From Hppendix 1.1.2 area PXM is.given by,
= 1.. .uw.2 2 Tanl • • • • ••• (94)
~umming Equations (91~ and (94) and multiplying by I'wL
gives the expression for the total mass of fluid displaced.
Thus,
= ••• ~95)
Define rab as the distance of the centre.of area of
4P~N from the trailing edge of the craft. from Appendix Iv.1.3
rab is given by,
\
2 + T ... ~ 96) :z B jh3 V + T ( 1 ,_ 1 '0 2 3'. t . Tan~8-J1) 'f8"i1'W Thus the total added inertia for an attached flow assump-
tion is given by,
= • • • • .& •• \ 97 )
'where, mVPl"ilN and rab are as defined in I:.quations \95)
and \96) respectively.
Case for 11 > 0t:j
Again the form of the expressions involved here will
depend upon the assumptions made for the detached and
attached flow cases and these have already been treated in
~ection 3.1.3c.
-
-65-
1. uetached Flow assumption (see Jection 3.1.3c(i))
Case for Eo -C( > Rf > P 2 t:i
From Figure 14a, the areas under consideration are given
by ll. PAN and ll. PNQ. From Appendix 1.2.1,
=
,t. The mass of fluid displaced is given by,
= (BT)2
-~ fwL - -Tan (ex +.0) • • • • .... (98) Also, from Hppendix 1.2.2,
=
The mass of fluid displaced due to APXN is given by,
= • • • (99)
From f\ppendix IV.2.1, the distance of the centre of
area of A PXN from 0 is given by,
\
DU' 8 j(~ + 2T )2 T2 (100) = 3" lan(o;+~) + • • •
Also, from Hppendix IV.2.2, the distance of the centre of
area of A PNQ ·from 0 is given by,
OF = j(OH}2 + (FI;i)2' • • • • • • (101)
where: OH = f:m+ Cos a( 1-!J)[COS~ 2L3Sino
-
-66-
Thus the expression for the added inertia term is given
by,
= • • • ~102)
where the terms on the right hand side are as defined
in tquations (98) - ~101) inclusive.
Case for ~> ~ ~
In this case, ~PXN is non-existent and therefore only
APNQ contributes to the added inertia term in the detached
flow case~~se8 Fig. 14c).
• 2 (103)· .. IpNQ = mVPNQ·(OF) . . . , • • • where mvpr~Q and (or) are as defined in Equations (99)
and (101) respectively.
2. Rttached Flow assumption (see ~ection 3.1.3c(ii))
All cas e s (gf >