Heave stability of air cushion vehicles hovering over deep water

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August, 1982

HEAVE STABILITY OF AIR CUSHION VEHICLES

HOVERING OVER DEEP WATER

by

M.

J. Hinchey

TECH

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E HOGESCHOO

L DELFT

LUCHTVAJ\fl

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1983

UTIAS Technical Note No. 236

.

CN ISSN 0082-5263

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HEAVE STABILITY OF AIR CUSHION VEHICLES

HOVERING OVER DEEP WATER

by

M.

J.

Hinchey

Submitted March, 1982

August, 1982

UTIAS Technica1 Note No. 236

CN ISSN 0082-5263

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Acknowledgement

This work was funded by the National Research Council of Canada and by the Canadian Transportation Dp.velopment Centre. The support of these agencies is sincerely appreciated.

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Abstract

The heave stability of air cushion vehicles hovering over infinitely deep water is examined theoretically. When applied to the Canadian National Research Council craft, HEX-S, the theory predicts that the volume modulation produced by the spatially averaged deflection of the plenum water free surface would increase the stabIe operating reg ion relative to overland operation. The air flow modulation produced by deflection of the water free surface at the lip of the plenum on the other hand would cause unstable behaviour at low cushion pressures.

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Contents

Acknow1edgement Abstract Symbo1s 1. INTRODUCTION

2. PRESSURE-WATER SURFACE INTERACTION

2.1 Spatia11y Averaged Def1ection of Plenum Water-Free Surface

2.2 Def1ection of Water-Free Surface at Lip 3. HEAVE STABILITY ANALYSIS

3.1 Lumped Capacitance Model 3.2 Lip Flow Model

3.3 Stabi1ity Criterion 4. RESULTS FOR HEX-S

4.1 Effect of Spatia11y Averaged Def1ection

4.2 Effect of Flow Modu1ation Produced by Def1ection at Lip

4.3 Spatia11y Averaged Def1ection and Lip Flow Modu1ation

S. CONCLUSIONS REFERENCES TABLE FIGURES

APPENDIX A: RESPONSE FUNCTIONS IN THE LIMIT OF ZERO FREQUENCY

ii iii v 1 1 2 2 3 3 4 4 5 5 5 6 6 6

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a b C e C. ~ C _m C_pQ fez) , g G(iw), h i R, m M N p Q R a R e si t x y z CL a y n v g(z) H(iw) Symbols

sound speed in air

vehiele semi-width

lumped pneumatie eapaeitanee

eosine integral funetion

diseharge eoeffieient

slope of fan pressure-f1ow eharaeteristie

auxi1iary exponential integra1 funetions

gravitationa1 aeeeleration

transfer funetion in Nyquist bloek diagram

hover gap

eushion perimeter

suspension mass

transfer funetion relating the mot ion of the water surfaee at the lip of the eushion to the eushion pressure

transfer funetion relating the averaged mot ion of the water surfaee inside the plenum to the eushion pressure

pressure

volumetrie flow

outlet orifiee resistanee

in1et orifiee resistanee

sine integra1 funetion

time

volume

horizontal eoordinate

vertieal eoordinate

x + iy

hover gap flow parameter

po1ytropie eoeffieient 1.4 (isentropie)

perturbation

def1eetion of wave free surfae~ wave length

wave number

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11 p X w Subscripts a c e t w y 3.14159 density heave displacement angular frequency air cushion equilibrium

partial derivative w.r.t. time water

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1. INTRODUCTION

The dynamic stability of air cushion vehicles

hovering over land has been studied quite exten-sively [1,2,3]. It has been found th at the vehicles are subject to self-excited instabilities due primarily to a force-motion lag associated with cushion air compressibility, although other factors such as the inertance of the air in supply ducts, the local slope of fan pressure-flow characteristics, the viscoelastic properties of the flexible skirt material and the flows within the cushion volume also have significant effects. However, very little work has been done on the dynamics of air cushions hovering over water. But Fowler [4] has recently shown experimentally that significant destabilizing effects can occur in practice. His experiments on a small Canadian research craft known as HEX-5 showed that over land the vehicle would not self-oscillate at any point in its operating range whereas over water it showed heave oscillation clearly and systematically. So understanding the over-water stability problem is important, especially for Canadian operations where th ere are many poten-tial uses for slow moving air cushion platforms that can operate over water as weIl as over land.

One important theoretical work is that of Ogilvie [5]. He used classical linearized deep water wave theory [6,7] to derive a response function relating the deflection of the water free surface directly beneath a vehicle to a sinusoidal cushion pressure variation. He used this to cal-culate added mass and damping coefficients for a two-dimensional air cushion having zero inflow and outflow. The present author [8] added to Ogilvie's model a simple orifice flow model for the flow under the lip of the cushion to atmosphere which ignored flow modulation due to water motion. It was found that the analysis predicted trends which were opposite to field observations made by Fowler

[4]. It was suggested that the lip flow model was probably responsible for the discrepancy. The present Note extends the work of [8] by presenting an improved lip flow model which accounts for modulation due to water mot ion.

2. PRESSURE-WATER SURFACE INTERACTION

When an air cushion vehicle is hovering stati-cally over water, it depresses the water free surface , according to Archimede's' principle, as shown in Fig. 1. When it is heaving, it acts as a kind of pneumatic wave maker. For the present work, the air cushion vehicle is approximated by the two-dimensional inverted box configuration shown schematically in Fig. 2. Using this model, Ogilvie [5] developed a frequency response function relating cushion pressure and water mot ion which allowed for the lip being immersed.in the water. He concluded however that for large low pressure air cushion vehicles lip immers ion itself would not have a large effect.

The frequency response function developed by Ogilvie [5] for the case where the lip is not immersed in the water is used here to examine the heave stability of an air cushion vehicle hovering over water. As mentioned, this function is based on classical linearized water wave theory which is in turn based on potential flow theory for an incompressible ideal fluid where, briefly, one is

1

interested in finding a velocity potential ~ which satisfies Laplace's equation

for y < 0 (2.1)

subject to the dynamical free surface or pressure condition,

(2.2)

and to the kinematic free surface or displacement condition, where t>p

o

~ y lxi < b lxi > b (2.3) (2.4)

where t>n is the deflection of the water surface from its reference (y

=

0) position, b is the vehicle semi-width, t>po is the amplitude of the cushion pressure oscillation, g is the acceleration due to gravity, and Pw is the density of water. The subscripts "t" and "y" indicate partial deriva-tives with respect to time and the vertical coor-dinate respectively. Equation (2.2) into Eq. (2.3) gives

(2.5)

By letting

~(x,y,t)

=

~l (x,y)sinwt + ~2(x,y)coswt (2.6)

and substituting into Eq. (2.5), one obtains

o

-00 < X < 00, y = 0 (2.7.1) = 0 where \! 2 w g lxi < b, y lxi > b, Y

is the wave number of generated waves. is related to the wavelength À by

À 27T \!

o

(2.7.2)

o

(2.8) The latter (2.9)

To obtain a unique solution, Ogilvie [5], following Stoker's work [6], assumed the water to be initially at rest and enforced the condition that there be only outgoing waves at lxi = 00. This is in contrast

to Lamb [7] who used fictitious damping forces to suppress free oscillations and thus achieve a unique solution. The lxi = 00 conditions together

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with the requirement th at the f1uid velocity vanish in the limit as y tends to minus 00 gave [5]

where

and

-26p

4>1 (x,y) = _ _ 0 sinvbe vy cosvx Pww fez) 26p Re [f (z)] - _ _ 0 sinvbe vy sinvx Pww z

f

eivtLn

~:~

dt ioo z

=

x + iy (2.10.1) (2.10.2) (2.11) (2.12) Substitution into the dynamica 1 condition gave for

lxi < b the water surface def1ection [5]

[ -6p

6n(x, t) = _ _ 0 (1+2sinvbsinvx) + Pwg

i

Re[f(X-iO)]] sinwt +

[~::o

sinvbcosvX] coswt (2.13)

2.1 Spatia11y Averaged Def1ection of Plenum Water-Free Surface

Ogi1vie [5] showed how for the air cushion problem one cou1d integrate over x to obtain an average piston-1ike qeflection

b

6n(t) = ib

J

6n(x,t)dx -b

2 lip [ 2 1

=

~ sin vbcoswt + 2TI [Ce Pwgv

+ Ln(2vb) + g(2vb)]sinwt - sinvbcosvbsinwt] (2.14)

where Ce Eu1er's constant

=

0.577216 and

co

J

-xu ue g(x)

=

-2-- du u +1 o (2.15)

Equation (2.14) can be rewritten in the complex form (see Fig. 3)

6n(iw) = N(· )

lip (iw) ~w (2.16) and

N(iw) A + Bi (2.17) where

A =

o--k

[21TI [Ce + Ln(2vb) + g(2vb)] - sinvbcosvb

l

Pwg\!

'J

(2.18) 2 . 2 b

B = - - b s~n v

Pwgv

Note that in the limit as w tends to zero, B tends to zero and A tends to -(l/Pwg). In other words, the water-free surface responds quasi-statica11y to cushion pressure variations, as expected (see Appendix A).

2.2 Def1ection of Water-Free Surface at Lip Equation (2.13) is singular at lxi

=

b. How-ever, as noted by Ogi1vie [5], the singu1arity is very weak and any error shou1d be 10cal in scope. If it can be assumed th at Eq. (2.13) is approxi-mately correct at each of the lips, then the tota1 def1ection of the water-free surface at lxi

=

b is

Now where 1₁ lIn(b,t) + 6n(-b,t) 2l1p _ _ _ 0 sinwt pwg + ~ Re[f(b-io)]sinwt g + ~ Re[f(-b-io)]sinwt g 4 lip + ___ 0_ sinvbcosvbcoswt pwg f(b-io) f(-b-io) W6po - - - I TIPwg 1 w6po - - - I _7ïP wg 2 b-io -ivb

i

dt eivt t-b e Log t+b ~oo -b-io t-b I ₂

=

e ivb

~

dt eivt _{Log t+b}

ioo

Upon evaluating the integrals one obtains [5]

1 1

~

[

I -

si(vb) - iCi(vb) + i[C e + Log(2vb)] e-iVb[2f(Vb) + 4TIisinvb] (2.19) (2.20) (2.21)

ie-2iVb [Ci(2Vb) - Ci(vb) + isi(2vb) - iSi(Vb)]] (2.22) 1 2

= ~ [

I -

si(vb) + iCi(vb) - e iVb 2f(Vb) - i[C e + Log(2vb)]

+ ie2iVb[Ci(2Vb) - Ci(vb) - isi(2vb) + iSi(Vb)]] where

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---

----

---

--

---

- - - --- - -si (z) Ci (z) and -f(z)cosz - g(z)sinz f(z)sinz - g(z)cosz fez) g(z)

=

00

J

-zt _e_ _dt t2+1 o 00

J

te-zt dt t2+1 o

Substitution back in Eq. (2.19) gives

2tlp l'.n 1lp " (t)

= - __

pwg 0 sinwt l'.p { _ _ _ 0 7T-2si(vb) - 4f(vb)cos(vb) 7TP_wg - 47T sin(vb)sin(vb)

- 2sin(2vb) [Ci(2vb) - Ci(vb)]

+ 2cos(2vb) [si(2vb) - Si(Vb)]} sinwt

4l'.p + _ _ 0 sin(vb)cos(vb)coswt pwg (2.23) (2.24) (2.25)

Equation (2.25) can be rewritten in the complex

form (see Fig. 4)

where where and l'.n_lip(iw) l'.p(iw) M(iw) M(iw)

x

+ Yi - 4f(vb)cos(vb) - 47T sin(vb)sin(vb)

- 2sin(2vb) [Ci(2vb) - Ci(vb)]

+ 2cos(2vb) [si(2vb) - si(vb)]}

Y

=

-±-

sin(vb)cos(vb)

Pwg

(2.26)

(2.27)

(2.28)

Note th at in the limit as w tends to zero Y tends

to zero _{and X tends to -(2/pwg). In other words,}

the water surface at the cushion lip responds quasi-statica11y to cushion pressure variations, again as expected (see Appendix A).

3

3. HEAVE STABILITY ANALYSIS 3.1 Lumped Capacitance Model

The 1umped capacitance model for a unit 1ength of an infinite 1ength vehic1e is used. For this the thermodynamic processes within the air cavity are assumed to behave isentropica11y and the f10ws into and out of the cavity are assumed to be gov-erned by inviscid incompressib1e orifice flow 1aws. In 1inearized form, the governing equations are [1] :

Conservation of Mass

P

pRT

Newton's Second Law

where a where 2bl'.pc + Damping Forces V-ce

=

- - 2 P a a R c

-~I

CPQ - dQ

Lumped Capacitance

Air Sound Speed

In1et Flow

Lip Flow

e

Equilibrium Air Flow

(3.1)

(3.2)

(3.3)

(3.4)

A1though in practice the damping forces in (3.2) may be quite significant, for the present work they are assumed to be neg1igib1e. Two possible sources for these are skirt-water contact hysteresis and spray formation.

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3.2 Lip Flow Model

Observations show that the flow under the lip of an air cushion vehicle hovering over water usually consists of an air-water spray. It may even be in-herently unsteady. Probably, the simplest concep

-tual model for steady flow is the one shown in Fig.

5j this model assumes that the water-free surface forms one side of a channel through which the cushion air flows and the hydrostatic pressure impresses itself along the channel. The latter implies th at the air in the channel must accelerate according to Bernoulli's law, and this in turn implies that the channel must be converging as shown. One would expect that for steady lip flow one could use the Bernoulli orifice flow law

Q

=

C ih

J2IÇ

a m

/-f

(3.5)

provided Cm could be measured experimentally. Here, it is assumed th at the Bernoulli law is not only adequate for steady flow but for unsteady flow alsoj the governing equation has the linearized form

where and where lIQa lIpc _lI_h

R

_a + -u _a R 2pce a

_T

h e u Qe a

lIh

=

lIX - lIn .

hp

(3.6)

(3.7)

(3.8)

In [8] the flow modulation produced by deflection of the water-free surface at the lip of the plenum was ignored and lIh was set equal to lIX.

3.3 Stability Criterion

By introducing the Laplace variabie

"5"

and noting th at at a stability boundary 5

=

iw, one

obtains where APc (iw) C iwllp (iw) = -c c Rc-CpQ

Ap

(iw) c R .a

lIn(iw) (A+Bi)lIp (iw)

c

(3.9)

(3.10)

(3.11)

Substitution of Eqs. (3.10) and (3.11) into (3.9) gives

o

where _ 1 1 2b 1 lI_{R - - - - +}

R -

--2

a

+ B2bw Rc-CpQ a mw a X u a (3.12) (3.13) -A2bw _ (2b) 2 + C w _ y mw c u a where (3.14 )

is the system characteristic equation with 5

= i

w.

For a nontrivial solution to exist, both the real and imaginary parts of the characteristic equation must separately vanish [9]. Thus

(3.15)

When the equilibrium gap he is specified, the re-maining unknowns in Eqs. (3.15) are the equilibrium cushion pressure Pce and the oscillation frequency

w. A numerical search of the (Pce-w) plane could be used to obtain a solution. Unfortunately, the technique of setting 5 = iw to find a stability boundary has the limitation th at a solution of Eqs. (3.15) is not necessarily associated with critical stability. This is because Eqs. (3.15) are satisfied when any root of the system charac-teristic equation lies on the imaginary axis in the s-plane. Thus the system could already be asymptotically unstable because there could al ready be one or more roots in the right half of the 5

plane. The problem could be circumvented by multi-plying the coefficients in the water compliance transfer functions by a factor

8.

With

8 =

0, the water compliance effect is effectively removed, and one should be left with the case of an air cushion vehicle hovering over a rigid surface. The value

of Pce for critical stability for this case as

obtained by the rigid surface lumped capacitance

Routh-Hurwitz criteria should agree with the pre-diction of Eqs. (3.15). As

8

is increased from zero to one, a continuous change in the values of Pce and w at which Eqs. (3.15) are satisfied should occur which would indicate th at the va lues obtained are associated with critical stability. This was found to be the case here when the piston effect was studied.

The stability question can also be dealt with the aid of the Nyquist plot concepts and the block diagram representation given in Fig. 6. The open loop frequency response function of the system is G(iw)H(iw) where G(iw) C iw c -

~

- 2biw u a 1 1 X+Yi + + -Rc-CpQ Ra ua H(iw) _{- 2}2b mw 2biw(A+Bi) (3.16 )

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real-imaginary plane. Theory shows that s

=

iw when IG(iw)H(iw)I = 1 and the phase angle of G(iw)H(iw)

is -180°. In other words, it occurs when the

G(iw)H(iw). plot passes through the -1 + io point in

the complex plane. When the phase of G(iw)H(iw) at

IG(iw)H(iw) I = 1 is less negative than -180°, the force acting on the suspension mass leads the

negative of 6X whereas when the phase is more

negative than -180° it lags. Simple arguments

based on the work done on the suspension mass during an oscillation cycle show the lead case is stabIe whereas the lag case is unstable.

Note that it has been implicitly assumed that the water response functions N(iw) and M(iw) are unaffected by transients in the basic air cushion. Thus one has a characteristic equation with com-plex coefficients where C m Co

=

2 ~

- 2bN(iw)

2~

1 Cl. a

o

(3.17) (3.18)

If the coefficient of the imaginary part of any

root of this equation is equal to the frequency

used to evaluate N(iw) and M(iw) , then that

par-ticular root is a valid root if its real part is close to zero; otherwise the roots mean

nothing. This fact is used in the following

section to determine the nature of stability immediately next to a particular stability boundary.

4. RESULTS FOR HEX-5

The theory was applied to the HEX-5 research

craft (see Fig. 7 [4]). lts dimensions are given

in Table 1.

4.1 Effect of Spatially Averaged Deflection

For a typical operating gap of he = 4.57 mm*

(0.015 ft) rigid surface lumped capacitance cal-culations indicate th at the system would be un-stabIe for cushion pressures Pce greater than 6.84 KPa; note in Table 1 that a typical operating

pressure is 1 KPa. So the HEX-5 should be very

stabIe over land, which in fact it is. The theory

of Section 3 indicates that the spatially averaged

deflection acting by itself would cause an approxi

-mately S-fold increase in critical stability pressure. The behaviour at other gaps is similar.

Figure 9 gives Nyquist plots for selected values of equilibrium cushion pressure showing the piston effect.

Rigid surface Nyquist plots are given for

*For a cushion pressure of 1 KPa, this gap gives a

typical operating flow. ₅

reference in Fig. 8. Note in Fig. 9 the good

agree-ment between Lamb's model (Appendix B) and the

Ogilvie-Stoker model.

The major reason for the large stabilizing tendency is the oscillation frequency at the critical stability pressure is typically weIl beyond the break frequency associated with the

cushion pressure-water surface interaction. In

fact, the water piston motion is typically close to being in phase with the cushion pressure. This is very much like a skirt flexibility which causes the cushion volume to decrease when the cushion pressure increases. As has been shown elsewhere [1,8], such a flexibility is stabilizing because it counteracts the destabilizing lag effect

of cushion air compressibility.

The Routh-Hurwitz stability criterion for the rigid surface case is

(4.1)

The criterion with quasi-static piston-like water mot ion is (4.2) [ hs 1

J

1 > ~2+ Pwg Cl. a a

In this case the water compliance piston term would dominate the air compressibility term and cause a large decrease in critical stability

pressure. So at very low frequencies the piston

-like water motion would reduce stability. It would add to the destabilizing lag effect of cushion air compressibility.

4.2 Effect of Flow Modulation Produced by Deflection at Lip

The theory of Section 3 indicates th at the flow modulation produced by deflection of the water surface at the lip acting alone causes unstable behaviour at low cushion pressures but greatly improved stability at high cushion pressures. Nyquist plots illustrating this are given in the following section.

The Routh-Hurwitz criterion with quasi-static lip flow modulation is

(4.3)

> [ hs 2

J

1

~2- Pwg Cl.

a a

In this case the lip flow modulation term would again dominate the air compressibility term but unlike the water compliance piston term it would cause a very large increase in critical stability pressure. This suggests that the improved stability at high cushion pressures is due to the water sur-face responding approximately quasi-statically.

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4.3 Spatially Averaged Deflection and Lip Flow Modulation

When both flow modulation and spatially aver-aged deflection are considered together it is found that in general flow modulation dominates. Figure 10 shows Nyquist plots for this and the previous case. These show improved stability at high cushion sures and unstable behaviour at low cushion pres-sures. Again note how Lamb's model agrees closely with the Ogilvie-Stoker model.

The Routh-Hurwitz criterion with quasi-static water mot ion gives for this case

[ ; : 2-hs CLwg

1) 1

CL_a

a

(4.4 )

Again, the quasi-static water term would dominate the air compressibility term and stability would be greatly improved.

Equation (3.17) shows that for pressure slightly greater than the critical pressure at approximately 1.5 KPa the system is unstable whereas for pressures slightly less the system is stabie. The work done argument also suggests this. However the work done argument also predicts that the system is very stabie at high cushion pres-sures. This means that there must be a stability boundary at some pressure greater than 1.5 KPa where the system passes into the stabie region at high cushion pressures. Using Eq. (3.17), it was found that at this boundary s ~ iw. In other words, the boundary is not one at which the damping associated with a particular root changes sign. It was found th at this boundary occurred at a cushion pressure where the frequency asso-ciated with the potentially unstable root failed to match the frequency used to evaluate N(iw) and M(iw), no matter what value of w was used, while at slightly lower pressures the frequencies did match. In other words as the cushion pressure was increased beyond this value instability would suddenly disappear. This behaviour is a conse-quence of having a characteristic equation with complex coefficients.

5. CONCLUSIONS

An important fact is th at the HEX-5 has never been unstable overland. However, it has been unstable when hovering with zero forward motion over water. From the results presented in Section 4, it appears that this stability problem is mostly a lip flow modulation effect. However, it must be pointed out th at the theory of Section 3 can at best be of only a qualitative value. This is because

(i) TIle theory was developed for an infinitely long two dimensional cross section air cushion. The HEX-5 for obvious reasons has a finite length.

(ii) The lip flow modulation model is questionable because the water wave theory is suspect in the neighbourhood of the lip.

(iii) At the higher pressures considered, the lip immers ion assumption is probably not

justi-fied. Lip immers ion generally reduces the amplitude of generated waves but for certain wavelengths there can be wave amplitude augmentation or resonance [5].

(iv) The notion of practical stability should be considered because it is possible for an equilibrium to be unstable in the linear sense discussed above and yet be practically stabie. This is because the system could, because of nonlinearities, enter a limited amplitude oscillation or limit cycle which is of sufficiently small amplitude that the basic operation of the system is not ad-versely affected. Similarly, it is also conceivable th at an equilibrium which is stabie in a linear sense could be practically unstable.

REFERENCES

1. Ribich, W. A., Richardson, H. H., "Dynamic Analysis of Heave Motion for a Transport Vehicle Fluid Suspension", Dept. of Mech-anical Engineering, MIT, 1967, Report DSR-76110-3.

2. Sweet, L. M., Richardson, H. H., Wormley, D. N., "Plenum Air Cushions/Compressor-Duct Dynamic Interactions", ASME Paper No. 75-WA/AUT-23, 1975.

3. Hinchey, M. J., Sull ivan , P. A., "Duct Effects on the Heave Stability of Plenum Air Cushions", Journalof Sound

&

Vibration, Vol. 60, 1, 1978.

4. Fowler, H., "An Experiment on the Heave Oscillation of a Light Air Cushion Vehicle Over Water", Canadian Aeronautics and Space Journal, Vol. 2, No. 2, Second Quarter, 1980.

5. Ogilvie, T. F., "Oscillating Pressure Fields on a Free Surface", Dept. of Naval Architec-ture and Marine Engineering, College of Engineering, University of Michigan, Sept. 1969.

6. Stoker, J. J., "Water Waves - The Mathemati-cal Theory with Applications", Interscience Publishers Pure and Applied Mathematics Series, Vol. IV, 1957.

7. Lamb, H., "On Deep Water Waves", Proceedings of the London Mathematical Society, Ser. 2, Vol. 2, 1904.

8. Hinchey, M. J., "Heave Instabilities of Amphibious Air Cushion Suspension Systems", Ph.D. Thesis, UTIAS, University of Toronto, 1979.

9. Hahn, W., "Stability of Motion" , Springer-Verlag, 1967.

10. Fung, Y. C., "An Introduction to the Theory of Aeroelasticity", Dover Publications, 1964. 11. Abramowitz, M., Stegun, I., "Handbook of

Mathe-matical Functions with Formulas, Graphs, and Mathematical Tables", Dover Publications, 1972.

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Length Width Depth

Fan slope at Pce 1 KPa

Tab1e 1 HEX-5 Data 3.75m 2.5m 0.2m dP

dQcl = -1.875 - 3 - - per meter 1ength KPa

m /sec e

The fan is modelled as an orifice separating the cushion and a constant pressure source. The orifice size is based on the above fan slope.

Assumed orifice flow discharge coefficients Fan 0.61

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APPENDIX A: RESPONSE FUNCTIONS IN THE LIMIT OF ZERO FREQUENCY

In the limit of zero frequency, the frequency response functions developed in Section 2 should show the water-surface to respond quasi-statically to cushion pressure variations. This fact is used below to check the water surface-pressure inter-action model.

A.l Spatially Averaged Deflection The response function is

l'In(iw) = N(' ) = A + Bl.· l'Ip(iw) l.W where A

=

Pw~

V

b

[2lTI [Ce + Ln(2vb) + g(2vb)] - sinvb cosvbl and 2 . 2 b B = - - b Sl.n v PwgV

In the limit as w tends to 0, v tends to O. reduces to 2 2 2 B = P gvb (vb) = - vb = 0 w Pwg (Al) (A2.l) (A2.2) So B (A3) Also, in the limit as w tends to 0, g(2vb) reduces to [11] - [Ce + Ln(2vb) -

~

sin2vb] So Areduces to +2

r

b sin2vb "] A

=

pgvb [S invb cosv + 4 -w Thus _{l'In(io) _} l'Ip(io)

-which is the expected result.

A.2 Deflection at Lip

The response function is l'In

lip (iw) _M(iw) _{X + Yi} l'Ip (iw) where X _ _ 1 {3TI - 2si(vb) TIPwg - 4f(vb)cosvb (A4) (A6) (A7) Continued ...

- 4TI sinvb sinvb

- 2sin(2vb) [Ci(2vb) - Ci(vb)] + 2cos(2vb) [si(2vb) - Si(Vb)]} and

In the limit as w tends to 0, Y reduces to

Also in the limit as w tends to 0 [11] TI si(vb) tends to -

2

f(vb) tends to [C + e Ln (vb)] vb TI or

2

[Ci(2vb) - Ci(vb)] tends to Ln (2vb) or Ln (2)

[si(2vb) - si(vb)] tends to 0

So X reduces to TI +

2

- Ln(vb) X 1 [3TI + TI - 2TI - 0 - 0 + 0] TIPwg 2 pwg Thus l'In lip (io) l'Ip(io)

which is the expected result.

Sine and Cosine Integral Functions [11] fez)

=

Ci(z)sinz - si(z)cosz g(z)

=

-Ci(z)cosz - si(z)sinz TI si(z) = .5i(z) -

2

00 Si(z) \"'

_L

(_1)n z2n+l (2n+I)(2n+l)! Ci(z) n=O 00 Ce + Lnz +

L

n=l (_1)n z2n 2n(2n)! (A8.l) (A8.2) (Ag) (AIO) (All) (A12)

(16)

APPENDIX B: PRESSURE-WATER SURFACE INTERACTION ACCORDING TO LAMB

B.I Pressure-Water Surfaee Interaetion For the pressure oseillation

lxi

< b Lamb gives where or P i (wt-Kb) gpw~n

= -

b

e eOSKX KPe iwt

I

'"

e -mb eoshrnx +~ 2 2 dm o m +K 2 w K = -g _2e- iKb ---COSKX gpw 00 2K

J

+ lTgpw -mb e coshrnx dm 2

2

m +K o

Note that this is symmetrie w.r.t. x.

B.2 Spatially Averaged Defleetion By definition, this is

+b

-_~_n

₌

_2bI

I

_~n_dx

-b or, using Eq. (B3),

where

and

or

ön 2(eOSKb-isinKb) sinKb

öp = - bgpw K -K + -blTg p w +b +b

I

-b

I

OO -mb ~ eoshrnxdmdx m +K o

I

eoshrnxdx -b 2sinhmb m" sinhmb mb -mb e -e 2

ön 2(eosKb-isinKb) sinKb

~p

= -

bgpw K -2K + bngp w 00 -mb sinhmb

I

e 2 - m

2

dm o m +K (BI) (B2) (B3) (B4) (BS) (B6) (B7) (BS) B-I

B.3 Water Defleetion at Lip By definition, this is

or, using Eq. (B3)

where

~nlip

=

~p

4(eosKb-isinKb) eosKb gpw 00 -mb 4K

I

+ lTgpw e 2 2 coshmb dm m +K o coshmb mb -mb e +e 2

B.4 Quasi-Statie Limiting Cases: Spatially Averaged Defleetion

(B9)

(BID)

(BIl)

In the limit as w tends to

°

and b tends to 0,

Eq. (B.S) reduces to 00 -mb sinhmb 2K 2CosKbsinKb gPwKb + - -blTgpw

I

e m 2 2 dm m +K o 00 2Kb 2K + -gpwKb lTgpw

I

dm ---z--2 m +K o 2 2K lT + -gpw lTgp w 2K (BI2)

Water Defleetion at Lip

In the limit as

w tends to

°

and b tends to 0,

Eq. (BID) reduees to

00

~nlip

=

4cOS2Kb 4K

J

-mb eoshmb e dm + -~p gpw lTgpw _{m +K}2 2 0 00 00 -2mb 4 4K

[I

dm

I

e dm] - - + - - _{2 2} + 2 2 gpw lTgpw _{2(m +}_K₎ _{2(m +}_K₎ 0 0 (BI3)

(17)

Air Cushion

Air

/ '

Water

FIG. 1 AIR CUSRION ROVERING OVER DEEP WATER.

6P

=

~Po

Sin

wt

Inverted Box

y

---~X

~I

(18)

0.0 -50.0

AHP

1.0

10.0 _100.0

.J----=:::::J=:::::::::::::~___I_---~ F RE

Q

PHASE

360.0

180.0

0.0 _FAEQ

.0

(19)

AHP

0.0 -50.0

PHRSE

360.0

180.0

0.0

1.0

10.0

100.0 FREQ

FREQ

.0

(20)

t

Water

FIG.

5

LIP FLOW MODEL.

Air Cushion

Si de

Wall

Q

Input

~

~Pc

(iw)

---

G

(iw)

-""' ~

-I ~

H

(iw)

-.

--~X(iw)

(21)

ANTI-ICING AIR INTAKE FOR THRUST ENGINE

FIG. 7

45 kw ENGINE THRUST

SYSTEM-a _{ELF - PROPELLE}DUCTED FAN - ;~~PERIMENTS

S RECORDING INS TRUMENTATION S IN DETACHABLE POD

VISUAL INSTRUMENT DISPLAY ON PANEL

ANEMOMETER

I

Y NAMOMETER TOW-FORCE D ERIMENTS FOF! TOwED EXP

ON DRAG FORCE

(22)

lIllIG 1.11 PC! • II.U 111'11 RIGID SURFACE

---+---+---+::=4---t-st

REAL .. ~ 1.0 -1.0'-_ _ - -.. . , --5.0 8 Ca)

FIG. 8 NYQUIST PLOTS FOR RIGID SURFACE CASE

lIllIG 5.0 -5.0 -5.0 FIG. 8Ch) PCt: • 1.25 IIPA RIGID SURFACE IIlAL S.O

(23)

lIllIG s.o

Pa • 211.0 IU'II OOILVIE-STOKER

. . -I---_--+--+_----=t 0 IIfAL

FIG. 9 NYQUIST PLOTS SHOIHNG VOLUME MODULATION EFFECT

lIllIG

5.0

PeIE ... 0 KI'A

OOILVIE-STOKER

(24)

lIllIG 5.0 PCI • ..811 lCl'A OOILVIE-STOKER _~--~--~~--+---~~~----r---~---+----r---~o .~ VOLUME MODULAXION -5.0 FIG. 9(c) IMAG 5.0 PCI . . . "PA LAMB ::::.r+--i_--+---t--_----t 0 IIlAL -5.0 VOLUME MODULAXION FIG. 9(d)

(25)

o IIIAG -5.0 FIG. lO(a) PCI • 0.70 KI'A OOILVIE-STOKER REAL

VOLUME MODULATION AND FLOW MODULATION TOOErHER

FIG. 10 NYQUIST PLOTS SHOWING FLOW 1·IODULATION EFFECT

IIIAG

s.O

PCI • I.SO KPA

OOILVIE-STOKER

If+-J'++--tt---"tl---~o IlEAL

-5.0

FIG. lO(b)

VOLUME MODULATION AND FLOW MODULATION TOOETHER

(26)

1 -5.0 -S.I! FIG. IO(e) 'MAG 5.0 -5.0 -5.0 FIG. lO(d) Pel • 2." !CPA ooILVIE-STOKER

PCI[ ... !CPA

00 ILVIE-STOKER

5.0 IIlAL

(27)

lIllIG· 5.0 PCI • ' •• 0 !IN OOnVIE-STOKER ·~~::~:::!::::!:::=!====~---+----~--~--~~--~I.O .~ -5.0 FIG. IO(e) lIllIG -5.0 FIG. lOef)

VOLUME MODULATION AND FLOW MODULATION TOOErHER

PC! • 0.70 KI'A

oonVIE-STOKER

III!AL

(28)

lIllIG s.o -5.0 FIG. 10 Cg) lIllIG s.o .... 0 -5.0 FIG. lOCh)

Pel • l.SO KPA

OOILVIE-STOKER

FLOW MODULATION ALONE

pcr ... KPA

LAMB

1.0 IIrAL

VOllJME MODULATION AND FLOW mDULATION TOOETHER

(29)

UTIAS Technical Note No. 236

Institute for Aerospace Studies, University of Toronto (UTlAS) 4925 Dufferin Street, Downsview, Ontario, Canada, M3H 5T6

HEAVE STABILITY OF AIR CUSHION VEHICLES HOVERING OVER DEEP WATER

Hinchey, M. J.

1. Pneumatic circuit stability; 2. Pneumatic wave maker; 3. CUshion volume modulation

4; Leakage flow modulation; S. Nyquist block diagram.

1. Hinchey, M. J. 11. UTIAS Technical Note No. 236

~

111e heave stability of air cushion vehicles hovering over infinitely dcep water is examined

theoretically. When applied to the Canadian National Research Council craft, HEX-S, the theory predicts that the volume modulation produced by the spatially averaged deflection of the plenum

water free surface would increase the stabie operating regian relati ve to over land operation. The air flow modulation produced by deflection of the water free surface at the lip of the plenum on the other hand would cause unstable behavioor at low cushion pressures.

Available copies of this report are limited. Return this card to UTIAS, if you require a copy.

Institute for Aero.pace Studies, University of Toronto (UTLAS) 49·25 Dufferin Street, Downsview, Ontario, Canada, M3H 5T6

Hinchey, M. J.

1. Pneumatic circuit stabilitYi 2. Pneumatic wave maker; 3. Cushion volume modulation 4; Leakage flow modulation; 5. Nyquist block diagram.

~

The heave stability of air cushion vehicles hovering over infinitely de ep water is examined

theoretically. When applied to the Canadian National Research Council craft, HEX-5, the theory

prediets that the volume modulation produced by the spatially averaged deflection of the plenum water free surface would increase the stabie operating region relative to over land operation. The air flow modulation produced by deflection of the water frec surface at the lip of the plenum on the other hand would cause unstable behaviour at low cushion pressures.

Available co pies of this report are limited: Return th is card to UTIAS, if you require a copy.

Institute for Acrospllce Studies, University of Toronto (UTLAS) 4925 Dufferin Street, Downsview, Ontario, Canada, M3H 5T6

Hinchey, M. J.

1. Pneumatic circuit stability; 2. Pneumatic wave maker; 3. Cushion volume modulation

1. Hinchey, M. J. II. UTIAS Tcchnical Noto No. 236

~

_V

theoretically. When applied to the Canadian National Research Council craft, HEX-S, the thcory

predicts that the volume modulation produced by the spatially averaged doflection of the plenum

water free surface would increase the stabie operating reg ion relative to overland operation.

The air flow modulation produced by deflection of the water free surface at the lip of the

plenum on the other hand would cause Wlstable behaviour at low cushion pressuros.

Available copies of this report are limited. Return th is card to UTIAS, if you require a copy.

Institute for Aerospace Studies, University of Toronto (UTIAS) 4925 Dufferin Street, Downsview, Ontario, Canada, M3H ST6

Hinchey, M. J.

1. Pneumatic circuit stabilitYi 2. Pneumatic wave maker; 3. Cushion volume modulation

~

theoretically. When applied to the Canadian National Research Council craft, HEX-5, the theory

prediets that the volume modulation produced by the spatiaHy averaged deflection of the plenum

water free surface would inerease the stabie operating région relative to overland operation. 111e air flow modulation produeed by defleetion of the water free surface at the lip of the plenum on the other hand would cause unstable behaviour at low cushion pressures.