The heave stability of an air cushion vehicle with a flexible peripheral skirt

h JAN. 1970

. M S E HOGESCHOOL DELFi

VLiSGTUIGSOUWKUNDE

BIBLIOIHEEK

P ^ C O A ' R E P O R T AERO No.

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE HEAVE STABILITY OF AN AIR CUSHION VEHICLE

WITH A F L E X I B L E P E R I P H E R A L SKIRT

by

CoA R E P O R T AERO No. 210 August, 1969

THE C O L L E G E O F AERONAUTICS

C R A N F I E L D

THE HEAVE STABILITY O F AN AIR CUSHION VEHICLE

WITH A F L E X I B L E P E R I P H E R A L SKIRT

by

CONTENTS 1. 2. 3. 4. Symbols Introduction

General analysis of the motion A new type of skirt

Heave stability of a vehicle fitted with a porous skirt system Conclusions F i g u r e s Page No. 1 1 2 3 5

SYMBOLS

A The internal c r o s s - s e c t i o n of plenum chamber taken in a plane p a r a l l e l to the supporting surface.

a The r a t e of change of a i r supply r a t e with change in plenum chamber p r e s s u r e .

b A constant proportional to the fixed leak a r e a . C A constant proportional to the r a t e of change of leak

a r e a with change in height. g The acceleration due to gravity. H The depth of the fixed side walls.

h The height of the roof of the plenum c h a m b e r above the supporting surface.

h, The value of h when the s k i r t just touches the surface.

lu The height at which a i r s t a r t s to leak from under the edge of an undulating skirt hem.

P The plenum chamber p r e s s u r e above atmospheric p r e s s u r e . P The atmospheric p r e s s u r e in absolute units.

p The peturbation in p r e s s u r e from the equilibrium value, Q The r a t e of inflow into the plenum chamber.

Q The r a t e of inflow into the plenum c h a m b e r when its p r e s s u r e is a t m o s p h e r i c ,

Q. An equivalent leak r a t e , see equation (14),

t The t i m e , m e a s u r e d from initial lift off.

W The weight of vehicle, l e s s that of self-supporting s k i r t e l e m e n t s . X The height of the plenum c h a m b e r roof above that corresponding to

the vehicle resting on the ground.

X The perturbation in X from the equilibrium condition,

y The gap between the undersurface of the s k i r t elements and the supporting surface.

1

-1, Introduction

It has been found that many a i r cushion vehicles with flexible s k i r t s suffer from a violent instability in heave. This note d i s c u s s e s the reasons for this and suggests a means of obtaining a well damped movement.

2. General analysis of the motion.

Figure 1 shows a simple, s y m m e t r i c a l l y loaded, a i r cushion vehicle with a flexible impervious m e m b r a n e forming the lower part of the p e r i p h e r a l wall. At r e s t the vehicle stands on the hard peripheral side walls with the roof of its plenum chamber a distance H above the supporting surface.

As a i r is pumped into the plenum chamber the p r e s s u r e there r i s e s until it is sufficient to give the nriain body a slight upwards acceleration. As soon as the vehicle begins to lift it will quickly stabilise to a vertical velocity equal to the a i r supply r a t e , Q, divided by the c r o s s - s e c t i o n a l a r e a of the plenum chamber. A, m e a s u r e d parallel to the supporting surface. Thus for a craft with negligible leakage

dh _ Q . . (1) dt ' A

where h is the height of the roof of the plenum chamber above the supporting surface and t is the time. This condition corresponds to figure 1(a) and the p r e s s u r e in the chamber P, m e a s u r e d relative to the atmospheric p r e s s u r e surrounding the vehicle, is simply the weight of the vehicle, W, divided by A. Strictly, the weight is increasing with height a s m o r e of the skirt is lifted clear of the supporting surface. Eventually the height becomes such that the edge of the skirt is just touching the surface and a further increase in height would leave a gap between the skirt hem and the surface. In this condition, as for all con-ditions when the s k i r t i s in contact with the surface and h is g r e a t e r than H,

dh _ Q J „ W

TT ' A and P = -A

dt A A As Boon as the vehicle r i s e s above this c r i t i c a l height, as shown in figure 1(b), the plenum chamber is vented by an area equal to the gap, y, times the peripheral length, 3. Air from the plenum chamber escapes through this vent at a velocity proportional to the square root of the chamber p r e s s u r e . In most c a s e s this vent flow r a t e , V, considerably exceeds the input flow rate, Q, the p r e s s u r e in the chamber drops rapidly and the vehicle experiences a marked vertical deceleration. It stops r i s i n g and then falls back t o w a r d s the supporting surface. At the. moment when the skirt just touches the supporting surface -r- is negative and the plenum chamber pressui-e is less than iL .

A

On touching the supporting surface the skirt seals the periphery and the plenum chamber p r e s s u r e r i s e s rapidly because of the inflow, Q, and the effective inflow, ' ^ . associated with the reduction in plenum chamber volume as the depth of the skirt d e c r e a s e s . As the p r e s s u r e i n c r e a s e s , the vertical acceleration of the vehicle i n c r e a s e s , the descent stops, and the vehicle s t a r t s to r i s e again. As a r e s u l t it r e a c h e s the fully extended skirt condition with a positive velocity.

2

-Thus the vehicle oscillates about the fully extended skirt height because it can

only achieve an equilibrium position when the gap, y, is very s m a l l , but in this

position it always has a high positive or negative vertical velocity. Although there is some damping in both phases of the motion, in general this is insufficient to

a b s o r b the work input into the motion associated with the volume i n c r e a s e in phase (a) and the volume d e c r e a s e in phase (b). Only when the velocity through the equilibrium height position and the venting velocity a r e small will the damping be sufficient to allow an equilibrium position to be reached and maintained. These conditions r e q u i r e a low inflow r a t e , Q. and a low plenum chamber p r e s s u r e , W.

A A The basic cause of this heave instability is the discontinuity in the leak r a t e

at a vehicle height which must be exceeded for the equilibrium condition to be reached, It follows that e i t h e r the discontinuity must be removed or else the vehicle must be made to operate in a height range sufficiently below that at which the discontinuity o c c u r s to eliminate the possibility of instability.

A v e r y rapid i n c r e a s e in vent r a t e with i n c r e a s e in height must always be a

feature of a finite length skirt. Varying the length of the s k i r t around the periphery

will cause the leak r a t e to vary smoothly over a height range corresponding to the range in s k i r t lengths. The g r e a t e r the variation in length the s m a l l e r the mean value of d(Q-V) and hence the g r e a t e r the probability of stability. Figure 2

dh

indicates the r a t e of change of net flow r a t e with height for various types of skirt.

It can be seen that an i n c r e a s e in stability is only likely to be achieved if the motion is such that the skirt ie never fully c l e a r of the supporting surface. It would seem that what is required is a small i n c r e a s e in the leak rate with i n c r e a s e in height and an equilibrium position at a height much l e s s than h^. as depicted by the dotted line on figure 2. With the types of flexible skirts shown this would correspond to operating at or n e a r the equilibrium height with most of the s k i r t rubbing the surface. Wear would be very s e v e r e under such a condition.

3, A new type of skirt.

Figure 3 shows a form of flexible s k i r t made by a s e r i e s of rigid segments which fit, like curtain runners to the fixed sidewalls of the vehicle. Their curved undersurface e n s u r e s that, above a minimum p r e s s u r e in the plenum chamber, they lift c l e a r of the supporting surface and find an equilibruim height just a fraction of a m i l l i m e t r e above the surface. At this height the low p r e s s u r e s induced by the venturi type flow beneath the segments offset the direct cushion p r e s s u r e effect and the v e r t i c a l force they produce just balances the weight of the segments. Should the gap, y, vary, the change in the p r e s s u r e distribution beneath the segments is such as to provide a restoring force. The viscous nature of the flow in the channel provides damping.

When the vehicle i s at its minimum height the segments completely block the vents in the fixed walls, but as the height of the vehicle i n c r e a s e s , X i n c r e a s e s and the vents a r e gradually exposed. As a consequence the vent shape can be such a s to give the vehicle an equilibrium position before the s k i r t is fully extended, but without any part of the skirt rubbing on the supporting surface. The vent a r e a can be varied with change in height to give satisfactory c h a r a c t e r i s t i c s to the system.

4. Heave stability of a vehicle fitted with a porous s k i r t system .

In this analysis it is assumed that the motion resulting from a disturbance never changes the vehicle height sufficiently to fully extend the skirt and change the gap, y.

In this c a s e the balance of forces in a v e r t i c a l s e n s e yields

A P - W - W d ^ = 0 - - - (2) g dt2

where W is the weight of the vehicle without the s k i r t segments, if these a r e self supporting, and A c o r r e s p o n d s to the inside wall dimensions. Consideration of the volume flow through the s y s t e m yields

Q = V + A dX + A(H+X) _dP . . . (3) dt 7 ( P + P a ) dt

where the last t e r m r e p r e s e n t s the effects of p r e s s u r e change on the density of the

a i r within the plenum chamber, assuming adiabatic flow conditions. P is the

atmospheric p r e s s u r e . ^ Since the delivery from s e v e r a l types of pumps falls with i n c r e a s e in the

supply p r e s s u r e required, a simple attempt to allow for this is to assume that Q = Qa - a P , (4)

Again, the vent r a t e can be e x p r e s s e d as

V = (b + cX) P^ (5) where the exit velocity is proportional to the s q u a r e root of the plenum chamber

p r e s s u r e , b c o r r e s p o n d s to a fixed leak a r e a and c depends upon the r a t e of variation of leak a r e a with height, i. e. upon the total width of the slots. The l i n e a r form of equation (5) gives the simplest form of vent a r e a relationship which satisfies the r e q u i r e m e n t that the vent a r e a v a r i e s with height and c o r r e s p o n d s to slots of constant width. Substituting for Q and V equation (3)

becomes

i

Q, - a P -(b + cX)P^ - A dX - A(H + X) d P = 0 - - - (6)

dt 7 ( P + P ) dt

An equilibrium condition exists when the conditions denoted by suffix o, are given by ° A W - - - (7) Q = a P - (b + cX ) P ^ a o o o If p - P + p and o X = X + x o

Then from equations (2), (7) and (8)

}

}

p = W d_x_ . . . (9) g"^ dt^

and from equations (6), (7) and (8) ap+(^b + cX^ + c x ) ( l + P ^ P ^ ' - ( ^ b + c X ^ ^ P ^ ' O + A dx ^ A(H + X^ + x) dp ^ Q _ _ _ ^ Q J dt 7 (P + P + p) dt O 3. F o r s m a l l perturbations in p r e s s u r e , relative to P ^ o ' ' l . f p - - - ( 1 1 ) o

and, a s s u m i n g s m a l l perturbations, equation (10) becomes

fa . ^ I f k 1 P - A-1 . cP/x . ^üllV ^ - 0

L 2 p i J dt • ' ( P + P J "it

o U o.

and substituting for p from equation (9)

(H + X ) W d^x r b + cX T W d^x dx i — - - -Y (P + P ) g dt O &

w

1

This i s the equation of motion d e s c r i b i n g the perturbation heave motion about an equilibrium position. The first t e r m allows for the effect of changes of density a s s o c i a t e d with changes in p r e s s u r e . If the perturbations a r e s m a l l and P i s s m a l l

o

in comparison to P the effect of changes in density will be negligible. In this c a s e

Si

equation (12) can be written a s

2

r2aP^ +(b + c X ^ ) P ^ ^ j ^ + 2gA ^ + 2gcP^^x = 0 - - - (13)

It is of i n t e r e s t to note that

a) the stiffness of the motion is directly proportional to the width of the ventilating s l o t s , and

5

-b) the damping t e r m , relative to the o t h e r s , d e c r e a s e s a s the plenum chamber W

equilibrium p r e s s u r e P . or -^ , i n c r e a s e s .

o A

The motion becomes a damped subsidence when

(2cP ' ) (2aP + (b + cX ) P ^) > 1 * o o o o

i

Now (aP + (b + cX ) P ^) is the steady state pump delivery r a t e into atmospheric p r e s s u r e , Q , Thus (2aP +(b + cX ) P ^) can be thought of as an effective leak

. ,r, 7^. o o o " r a t e . Writing % 1 2aP + (b +cX ) P ^ - - - (14) o o o

the condition for a subsidence after a disturbance becomes

(^)

^a)

%

dXo

is a m e a s u r e of the mean flow velocity in the plenum chamber. It will A

d e c r e a s e with an i n c r e a s e in the size of the vehicle, a d e c r e a s e in the porosity of the s k i r t and a d e c r e a s e in the size of the gap, y. It will i n c r e a s e with i n c r e a s e in plenum chamber p r e s s u r e , P , A pump with a delivery which does not v a r y significantly with p r e s s u r e r e s u l t s in a lower value of Q^ .

5. Conclusions

A plenum chamber type a i r cushion vehicle with an impervious flexible skirt m u s t always be subject to instability in heave if the supply r a t e and cushion

p r e s s u r e a r e not v e r y small.

Heave stability can be obtained by using a porous s k i r t . The porosity r e q u i r e d depends upon the sensitivity of the valve controlling the a i r supply and upon the magnitude of the likely d i s t u r b a n c e s . The porosity must be big enough to e n s u r e that the g e n e r a l pumping effect associated with the vehicle moving over an uneven supporting surface does not cause changes in vehicle height g r e a t e r than that corresponding to the depth of the porous p a r t of the s k i r t . Again porosity must be such that, when the a i r supply valve i s opened, any overshoot in height associated with the initial r i s e to the equilibrium condition does not exceed the design variation in height associated with the depth of the porous part of the skirt, The s m a l l e r the pumping effect of t r a v e r s i n g an i r r e g u l a r surface and the g r e a t e r the sensitiveness of the control valve the s m a l l e r the porosity required of the s k i r t and the lower the power required for hovering. In many c a s e s a skirt such as illustrated in figure 3 may need no ventilating slots in the fixed sidewalls; the s m a l l gaps between consecutive s e g m e n t s may give sufficient leakage variation with height to e n s u r e adequate stability in heave,

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