Theoretical analysis of a loop type flexible skirt air cushion

THEOR.E:rICAL ANALYSIS OF A LOOP TYPE FLEXIBLE SKIRT AIR CUSHION

TECHNISCHE HOGESCHOOL DELft LUCHTVAART - EN RUIMTEVAARTIECHNIEK

BIBLIOTHEEK

Kluyve

r

weg 1 - DELFT

by

ANALYSIS OF A LOOP TYPE FLEXIBLE SKIRT AIR CUSHION

by

P. A . Sullivan r;md G. J. Parravano

Submitted June,

1977

Acknowle~eme~ts

TlU~ work was supported by the National Research Council of Canada through Opel'ating Grapt No. A.,

3378.

Also one of us (G. J. Parravano ) was award,.ed

a.n

NRC scholarship d,.uring ;part of the period in wlrlch this work was done.

- - - -

~--Summary

A theory is proposed for the loop or bag type flexible skirted air cushion or air bearing. It is shown that the nozzle formed by the skirt in the region where the air escapes from the cushion plays an essential role in the skirt action. Consequently earlier attempts to model the skirt by ignoring this region, on the ground that it is usually small, are incomplete.

Some numeri cal results are obtained for the case where the skirt behaves as

a membrane and the flow in the nozzle can be considered inviscid and one-dimensional. The stiffness curves are compared wi th the corresponc;ling curves for a simple rigid walled plenum chamber fed from the same source. These show that the skirt can be relatively stiff in heave, but also that if very low mass flow rates are practical it will be much less stiff than the simple

plenum chamber. It is suggested thatviscous floweffects are very impoftant, and that an alternative simplification, namely th at the flow ca,n be treated as inertialess, may not be valid.

Contents ~ Acknow1edgements i i S~y i i i Nomenclature v 1- INTRODl)"CTION 1

2. BASIC ELEMENrS OF A TWO-DnENSIONAL MODEL 2

3.

INVISCID MODEL FOR FLOW IN NOZZLE REGION

₅

4.

SYSTEM EQ,UATIONS AND TREIR SOLlJrION

₇

5.

RESULTS ₉

6.

DISCUSSION AND CONCLUSIONS 10

REFERENCES 12

Nomenclature

Al' ~, ~, A4 are coefficients describing effect of nozzle region

on the main part of the skirt. They are defined in Eqs.

3.6, 3.7,

3.9,

3.11, respectively

Bi Coerficients describing air source flow characteristics C Bearing hard structure geometry as defined in Fig. 2 CD Mass disCharge coefficient

D Half width of cushion hard structure

h Gap betweenlowest point of cushion and ground K Membrané material stiffness

LA Bearing geometry as defined in Fig. 2

Lo Unstretched length of cross section of skirt material ..e

m Stretched length of cross section of skirt material

..es Length of straight porti on of membrane that occurs when ~ :::: Pc ..e

v Length of membrane in nozzle region Be

me Cushion air mass flow rate per uni t length of cushion p Hydrostatic pressure

r Radius of curvature of membrane

T Hoop tension per unit length of membrane V Cushion volumes as defined in text

W Weight per unit length supported by cushion x Half width of footprint ofskirt

y Height of hard structure above ground

ex Angle that membrane makes with l1orizontal at outer attachment point. Defined in Fig. 2

Angle subtended by circular part of membrane expesed to atmosphere. Defined in Fig. 2

Pc Cushion air densi ty

(J" !).ngle that membrane makes wi th horizontal at inner attachment point. Def'ined in Fig. 2

S,1')

Coordinate system with origin on ground at narrowest part of' nozzle. Used to define nozzle geometry as shown in Fig. 2

CPs

Fluid seJ)aration angle as def'ined in Fig. 2

Subscripts

a

b

c v

Denotes atmospheric conditions Denotes conditions in bag or loop

Denotes conditions in cushion area where f'luid is at rest Denotes nozzle region

1. INTRODOCTION

An essential ingredient in the application of the air cushion principle

to a variety of transportation and materials handling problems has been the development of various types of flexible extensions frequently called "skirts".

These ,allow the air cushion devices to have relatively large effective or

"hard structure" clearances, while at the same time, keeping the air gap sufficiently small to enable the lifting power expended to be kept within

reasonable limits. The geometry of a particular skirt design depends critically on the intended application so that, although there are many common features, each design will require its own formulation in order to predict its suspension characteristics.

This artiele discusses some aspects of the theory for the static stiffness of a type of skirt which is of particular interest in the movement

of loads over relatively smooth surfaces. Figure 1, which is a cross-sectional

view of the skirt, shows the basic geometry. Although several designations of a commercial nature have been given this, it will be described here as a loop skirt. In its two-dimensional form, i t is a length of flexible material attached

to the vehicle base or "hard-structure" at points A and H. Air from a suitable

source is fed into the volume VI at E and then passes through holes F into the volume VII. The air in VII escapes to atmosphere through the converging passage

formed by the skirt and the ground. Usually the skirt material is thin enough to be modelled as a membrane, in which case its shape may be conveniently

considered to camprise three parts. The part AB is a circular arc of radius

~, since there is a constant pressure across it, namely the difference between

the pressure in VI, denoted by Pb' and atmospheric pressure Pa. In the section BC, the membrane curvature gradually changes from i ts value for the arc AB to a smaller value as the pressure difference across it changes from (Pb - Pa) to (Pb - Pc), where Pc is the cushion pressure. In the passage bounded by BC and the ground, the fluid accelerates from rest in VII and escapes from the cushion; this passage will therefore be called the nozzle region. Owing to the presence

of viscous effects , the fluid separates from the membrane at some angle CPs

> 0

as shown in Fig. 2, and forms a semiconfined free jet. The section CH also has constant curvature. Of ten the area of the orifices F is sufficiently large so

that Pb is very close to Pc and CH is very nearly a straight line. Also, for

many applications the flexible material is effectively inextensible.

Patents relating to various forms of the concept were registered

as early as

1963

(Ref.

1).

Tt has been proposed for use on tracked air cushion

vehicles (Ref. 2) and is commonly applied to small hovercraft. Perhaps its widest application is to materials handling problems on very smooth concrete

floors where effective air gaps as low as 0.05 mm have been obtained (Refs. 1,

3).

For TACV's, levitation powers of the order of

3

kw/tonne have been quoted

(Ref.

4).

In spi te of the wide range of potential applications, especially in the materials handling field, it has received relatively little attention in the available literature • Two exceptions appear to be the work by Jindxa

(Ref. 2) and Casey (Ref.

5).

Both considered the static behaviour of the

bearing and investigated the he ave instabili ty or bounce problem, but they approached the mechanies of the flow in the nozzle region from entirely

different viewpoints. Casey (Ref.

5)

assumed that it is inviscid, whereas

Jindxa (Ref. 2) assumed that the Reynolds mmlber of the flow is much less

than unity so that it is inertialess. Both also assumed that CPs = 0, and

Casey assumed that the length of the nozzle region was so small that i t could be omitted from the analysis of the statie heave stiffness. Thus his model takes the form shown in Fig.

3.

It can however be shown that this

assumption leads to sn inconsistency, because, although the length of the nozzle region may be small, it cannot be omitted from the analysis. Casey bypassed the problem by assuming that the cushion pressure Pc was gi ven as a design variable. Thus his model csn be viewed as incomplete. Jindra also has a similar incomplete-ness in his analysis. Although he wrote down a set of coupled differential equa-tions for the pressure distribution and membrane shape, an estimate of the membrane tension was obtained by assuming a simplified membrane shape and that Pc was given. No solutions to the coupled equations are given. In the analysis given here t-he membrane tension and the cushion pressure are determined as part of an overall

solution. For simplicity and for insight purposes it is assumed here that the flow in the nozzle region is inviscid, since this leads to a particularly simple solution. Also for simplicity it is assumed that

CPs

= 0, although there is evidence that this also may be questioned (Ref.

6).

2. BASIC ELEMENTS OF A TWO-DlMENSIONAL MODEL

A model which would properly account for the statie stiffness in heave of the skirt geo~try shown schematically in Fig. 1 requires the following seven elements:

(i) A specific souree charaGteristic which relates the mass flow, ml, into the cushion system at E as a function of the bag pressure Pb in VI, (ii) A law relating the mass flow, Irl;2, into VII as a funGtion of the pressure

difference Pb - Pc across the orifices,

(iii) A rela-tionship between the shape of the skirt and the distribution of pressure difference ~ across it,

(iv) Constituency laws relating stresses and strains in the skirt material, (v) A force balanee for -the hard structure ,

(vi) An appropriate description of the fluid mechanics in the nozzle region so that the pressure distribution along i t can be related to the skirt shape in the region, and the mass flow ma out of the skirt can be calculated,

(vii) An integral relatingthe pressure distribution on the ground to the weight W carried by the cushion.

Continuity requires mI

=

~

= IDa

= IDe.

Items (i) and (ii) present no problem. The source charaGteristic will be some gi ven funGtion

me

= mc(Pb), and a

simple orifice law such as that available for incompressible flow

where A is the orifice area, can be used to describe the flow from loop to cushion.

Item (iii) can be complex. However, gi ven that the skirt material is thin compared to the local radius of curvature, then it may be regarded as a memb;rane so that bending moment effects may be ignored. It is shown in standard texts on shells (q.v., e.g., Ref.

7,

p.

109)

that for a cylindrical shell, if z is the distance along a longi tudinal generator and f3 is the angle of a tangent plane to the surface , both measured relati ve to some arbi trary datum, the differenifial equations for the loop and longi tudinal forces per unit length Tf3 and T

_z

respectively, and the shear NZf3 = Nf3z are given by

(2.2a)

(2.2b)

CYr

_z

dz = -

Pz - (2 .2c)

The quantities Pr' Pf3 and Pz are the unit surface loadings in the nOrmal, hoop, and longitudinal directions respectively. For a two-dimensional skirt, derivatives in the z-direction are zero, and the only loadings would be the hydrostatic pressure difference Pr

=

!:sp and possibly shear Pf3

=

T W due to fluid viscous effects in the nozzle region, so that Eqs. (2.2a-c) simplify to 1

CYrf3

T + - --.:;:'D

=

0 w r CP N

=

0 zf3 (2.3a) (2.3c)

If the flow in the nozzle can be assumed to be inviscid, then

TW = 0, ~d CYr~ = 0 and

T = !:sp r = constant = T

f3 (2.4)

It fOllOws immediately from (2.4) that in the regions where l5p i s constant then r ;is also constant. This implies that the arcs AB and He are circular.

For a membrane, the appropriate constituency relationship assumes a particularly simple form. Usually it will suffice to assume that the material is linear so that in the case of inviscid flow:

(2.5)

The hard structure force balarlCe also assumes a simple form when the skirt is a membrane. From Fig. 2 i t can be seen that

W/2

+

TH sin cr + TA siza

=

Pb(D - L ) _a

+

P L - P D _{c a a} (2.6) where TH and TA are the values of T~ at Hand A respectively. For the inviscid case TH = TA

=

T.

Item (vi) can be a complex problem since the shape of the nozzle region is not initially known, but is determj.ned by the fluid mechanics

coupled to the ~mbrane equations. Also, apart fro~the fa ct that .viscous effects determine the magnitude of C/Js,the gap h may well be small enough that the flow in the nozzle region is entirely .viscous. A simple theory for the

case when the flow is inviscid and one dimensional is presented in the next section.

Casey (Ref.

5)

attempted to circumvent item (vi) for the case when h is very small by arguing that, since the region

Be

can 'he expected to scale with h as h/D ~O, then its length

irv

could be assumed to be zero. However, as noted above, this leads to a fundamental inconsistency. This is most easily demonstrated in the case that the fluid is inviscid, C/Js = 0, and the area A of the bag orifices per unit 1ength of cushion is sufficiently large compared to the gap h that

Pb

~ Pc' In this case the are CH is a straight line and, if

tv

=

0, the skirt assumes the shape shown in Fig.

3.

From the geometry of this shape,

D - x = r_b sin~

Equation (2.6) reduces to

W/2

+

T(sincr

+

siza)

=

(p - P )D

c a (2.8)

and the cushion lift integral becomes simply

..

W/2

=

(p - p _{c a}

)x

From Fig. 2, a

=

7r - 9, and since T = (pc Pa)rb' Eq. (2.8) reduces to

(2.10)

which is incompatible wi th Eq. (2.7). Casey circumvented the prob1em by ignoring the hard structure force equation and by arbitrarily specifying the va1ue of Pc' since his formulation consequent1y contained one more unknown than equations.

It is clear that

tv

may well be small relative to

tso

But changes

in

tv

are not small compared to changes in

t

_s

or

tAB,

and it is changes in

the lengths of these three components of the skirt length which are associated with the skirt action.

3. INVIscm MODEL FOR FLOW IN NOZZLE REnION

In the event that the flow in the nozzle region BC can be treated

as incompressible, inviscid and one-dimensional, it is possible to formulate a very simple model for its shape in terms of the simplified membrane equations

and the assumed flow equations. In this treatment it is also assumed that

Pb ::: Pc, and

CPs

= O.

A (s,~) coordinate system is located with or1g1n on the ground at the point of minimum gap as shown in Fig. 2. The equations describing the flow are

energy: (3.la)

continuity: ~v( s) == m _c /2p _c (3.lb)

This is item (vi) above. The membrane equation, item (iii) becomes

(3.2) Note that the radius of curvature of the membrane r can be approximated by

[d2~/ds2]-1 since we must have dyJds

«

1 to be consistent with the one-dimensional flow assumption.

From Eqs. (3.la, b, and 3.2) we obtain the following equation for the menibrane shape:

where the mass flow per unit length of cushion

me

= _{2CDh 1I/2pc Pc , and T can}

be conveniently expressed in terms of rb and Pc'

Equation

(3.3)

can be solved by using the substitution À ::: d'lld~

to reduce the problem to two successive standard integra'tions subject to the

initial condi tions ~ ::: 0 and d1'll ds ::: 0 at ~ = h. We have then

and f'inally the shape of' the section BC of' the membrane is given by

c .

_D

_'VhiÇ

f2 ,

"J~

_h

-

21

_h

+.en

{.!iJ

_'V"h

+

_{-yE -}

~}

.L

(3.5)

We have now to use this expression to determine the coordinates

(sv' ~) of' the point C which def'ines the end of' the nozzle region. In the

constant pressure or cushion region the membrane slope ds/d~ is constant,

but Eq.

(3.5)

onlyapproaches a constant slope asymptotically as s or 1'\~-oo:.

This of' course is expected since the equations

(3.1)

inherently r-equire lJ ~ '00

as p ~ Pc . Hence we define the nozzle region as being tha t in whi ch the

pressure p increases to within a given f'raction 5 of' Pc, that is p =. (1 - 5)Pç

at C in Fig. 2 where 5 is, say, 0.01. It f'ollows immediately f'rom Eqs. (3.la)

and

(3

.lb) that at the point C,

(3.6)

We then have

The length of' the membrane tv and the contribution to the cushion lift in the nozzle region can now be computed. We have

J

I

r {

J-l

t.

= . l + _ b _

l-~

v 2C2 h l)

h

D

This can also be reduced to a standard f'orm, but the result is a complicated

f'unction of' À. = _2CD2h/rb. Since the quantity h/~

«

1 usually, the integral

can be simplif'ied by approximating the integrand by two terms of' a Taylors

series in À.. We then have

(3.8)

where

The contribution to the lift Wv in the nozzle region is given by integrating the ground pressure distribution

This can also be reduced to a standard form to give

(3.10)

where

(3.11)

In summary, the effect of the nozzle on the remainder of the. skirt is described by the coefficients Al(5) to A4(5), which are functions onlyof the degree of accuracy desired in selecting the end point C, together with the quantities À = 2Cn2h/rb and Cn2 . Typically for 1% accuracy we find

A:t.

=

10,

~

=

11.3053,

This greatly simplifies the subsequent formulation .

4.

SYSTEM EQUATIONS AND TREIR SOLurION

We may now write down the equations describing the simplified model present here. All unknown variables are lower case letters while given para-meters are upper case. First consider the length of the membrane; we have

Now from Fig. 2, i t can be easily seen that, wi th CH straight,

~~

= ( x

(4.1)

Since ~ is given by Eq. (18), and~.A:B = ~8t, we have

~Hb

ji}

h

₃

_,

~ + - --+AC - . +r A. =1, S C

n

2

3

n

rb b-b m

(4.2)

Since the tension can be conveniently written in the form T

=

Pcrb' the mem-brane constituency relation equation (2.5) can be written

p r

=

K(~ - L )

c b m 0

(4.3)

The lift generated by the cushion is equal to the integral of the footprint pressure distribution:

p

(x

The integral has been expressed in ter.ms of Eq. (3.10), so we find

(4.4)

The membrane force equation

(2.6)

becomes,

{

y - ~h - C }

D - _{r b}

sin~

+ ] = 2 W

s Pc

(4.5)

Some geometry conditions are

(4.6)

(4.7)

FinaJ.ly, we have the air supply and cushio~ flow relations. For computationaJ. purposes it has been assumed that most supply characteristics can be approximated with acceptable accuracy by a cubic fit:

m _c

The cushion exit flow rela-tion is

(4.8)

Equatiom (4.1) to

(4.9)

constitute a closed system for the nine unknowns • For a gi ven skirt geometry, that is with D, C, La and Lo fixed, and for a given skirt materiaJ., that is K fixed, the quantities of principaJ. interest are the hard structure height y, the mass flow mc' and cushion

pressure, Pc, as a function of the weight supported, W. Sblutions of the above system of equations have to ~e obtained numerically.

A reduction in the number of parameters is obtained, and there is some benefitto numerical computation resulting from recasting the above equations in a nondimensional form. Lengths are referred to D, pressures to W/2D and mass flows to "V"WDPc. Note that W/2D is the pressure that would be required to lift

a simple plenum chamber of width 2D. The nUIOOrical results presented below are for an inelastic membrane (K ::;. 00) and for a discharge coefficient CD

=

1. In

this case the nondimensional form of the equations is

(4.11) x - (4.12) 1 (4.13) = -(4.14) (4.15) (4.16) with (4.17)

as an auxiliary relation. Also B

B

=

Bl JW

-B

2

Jl-.

- - B31*-..

Bo

=

0 B₂

-'4 - - ,

B

_{3 -}

_ö

T

(4.18)

~

1 2 D P

3

_c D5 Pc _Pc

Selution of the above system of equations was found to oe straight-forward when a Newton technique with a convergence criterion was used (Ref. 8). The Jacobian matrix was derived explicitly.

5.

RESULTS

Some numerical results for cushion stiffness were obtained for two

values of membrane length, LID

=

1.50 and 1.88 and for a source flow curve of

the form

m - = k { -B - B - -2 }

c o,r 2,r Pc

where for convenience Bo and B2 have been expressed in terms of a reference

weight Wref used -to nondimensionalize the stiffness curves discussed below.

In the present calculations Bo r = 0.290 and B2 r = 0.186, while k was varied

between 0.1 ~ k ~ 1 to establi~h the effect of ~ss flow on the cushion behaviour.

In order to provide a.n insight of the effect of the cushion, the results for

rigid wall plenum chamber having the same hard structure width 2D and operating from the same mass flow source, but with a mass discharge coeffic~ent CD

=

0.70. For the case of inviscid flow it seems reasonable that this is a meaningful

comparison. If the flow is treated in the manner described by Jindra (Ref. 2) then a typical rigid air bearing might be a more appropriate reference.

Figure 4 shows a typical computed skirt geometry for LWD = 1.88,

W/Wref

=

1.0 and k

=

m/mref

=

1.0. The most interesting feature is the relatively large extent of the nozzle region, which clearly indicates that it cannot be

omitted from the model as originally proposed by Casey (Ref.

5).

The stiffness curves are given in Fig.

5.

The quantity of interest is the change in load W as a function of the hard structure height y. However, in order to make the comparison with the equivalent plenum chamber mentioned above, the change in y was referred to the quantity yo' which was that value of y corresponding to

me

= O. At this limit, both h and

tv

are zero so that the skirt geometry takes on the shape assumed by Casey as shown in Fig.

3.

The curves show several interesting features. Firstly the upper limit on the value of (y - yo)/2D drawn on these curves occurs when ~s

=

O. For larger values of y a reformulation of the problem is required since there is, in effect, no longer a region of constant pressure Pc' This has not been attempted here. The second feature to notice is that the stiffness curves for the loop cushion seem to be much less dependent on source mass flow rates than does the curves for the simple plenum chamber. Also, if the skirt is operating at relatively low mass flow rates, the cushion is clearly much less stiff than the equivalent plenum chamber. Typically, at LrrfD

=

1.88, W)Wref = 0.75 and m/IDref = 0.1, the plenum chamber is about

5.6

times as stiff asthe loop skirt. However, for the large st mass flow rate used in the present calculations , the stiffnesses of the two systems are comparable. Finally i t should be noted that, in contrast to the plenum chamber, the skirt stiffness curve is convex.

6 .

DISCUSSION AND CONCLUSIONS

The major point to be made by the present analysis is the fundamental role played by the nozzle section in the skirt action. The numerical results which have been presented show that the nozzle can itself be large in comparison with the other parts of the skirt. However, even when the length of the nozzle section is small it cannot be omitted from the analysis.

Perhaps the major limitation of the present work is the omission of viscous floweffects . One starting point is the analysis undertaken by Jindra

(Ref. 2) that is, to assume the flow is inertialess, but with the differential equations he obtained coupled to the other algebraic equa'tions as was done here rather than solved separately. This would eliminate the necessi ty to assume a value for PC' The parameter which would provide some insight into the importance of viscous effects is a Reynolds number defined to scale the ratio of inertia forces in the nozzle flow direc'tion to shear forces associated with transverse velocity gradients . Following Schlichting in his discussion of very slow motion we have (Ref. 9, p. 109),

Re ~ ~

e

u

dU/Ox

2 2 ex:

(6.1)

where Vo

=

V at

'1

=

'0. For a test model having the geometri c proportions gi ven

in Fig.

4

with D = 0.305m and W = 89.2 kg/m,Re $':;$ 6000. This suggests that the

flow should be inertia dominated but there will probably be thick wall boundary

layers. For a cushion with h

=

0.13 mm., Pc RS 7200

N/rril-

($':;$ 1 psig),

Tb

= 3.0 cm,

Re RS 80. The latter might be typical of a bearing used in air cushion handling,

so that if viscous effects are important, V_{o might be considerably less than}

that es'timated in Eq. (6.1) by use .of the Bernoulli law. Hence the estimate of

Re should be high. Even so, this sUSgests that inertia effects rema.in important

for small gaps, so that the inertialess model developed by Jindra (Ref. 2) might only have a limited scope.

Finally, i t is worth noting that viscous effects might have a significant

effect on the membrane equation. From Eq. (2.3b), if ds

=

r~ where s is the

distance along the membrane , we have

~v

TH

=

TA +

J

~=O

T ds w

If the peak velocity is of the order of its inviscid value,

V RS -y'2p

7

p o c c cV (-1 T w '" I..l

dY ...

0 h ) as h -t 0 But 1/2 ~ ... Oeh ) as h ~ 0 so that (6.2)

1. Tho~s, T. R. 2. Jindra, F.

3 .

Wright, B. H.

4.

Bertin, J. Berthelot, J.

5.

Casey, B. L.

6.

Croix-Marie, F. C. 7 . Flugge,

w.

8.

Broyden,

c.

G.

9·

Schlichting, H. REFERENCES

Hoverpallets and Air Castors: State-of'-the-Art. Proceedings of' the Symposium on Air Cushion Handling, BHRA Fluid Engineering, London, Oct.

1974,

pp.

7:..18.

study of' the Potential of Hovair for High Speed Ground Transportati0n. General Motors Research Laboratories, Warren, Michigan, March

1968,

PB

177 523.

Heavy Lift wi th Fluid Bearings. Hovering Craft and Hydrofoil, Vol.

16,

No.

4,

Jan.

1977,

pp.

20-24.

Aerotrain Tridim for Urban Transportation: A System Designed to Spare Ini tial Costs and Energy. Proceedings of First International Hovering Craft, Hydrofoil and Advanced Transit Systems Conference, Brighton , England , May

1974,

pp.

307-315.

Static and Dynamic Behaviour of a Flexible Base Fluid Suspension. Massachusetts Institute of' Technology, Dept. of Mechanical Engineering Report DSR

76110-5,

Nov.

1966.

Load Handling in France. Hovering Craft and Hydrof'oil, Vol.

16,

No.

4,

Jan.

1977,

pp.

25-31.

Stresses in Shells. Springer Verlag, Berlin,

1962.

A Class of Methods for Solving Nonlinear Simultaneous Equations. Mathematics of Computation, Vol.

19,

No.

22,

Oct.

1965,

pp.

577-591.

Boundary-Layer Theory. Sixth Edition, McGraw-Hill, New York,

1968.

Y

Air

Supply

\

E

LL...o~~

Pb

VI

~F

Vrr.

B

_Pc

~

< < < ( _{< <} < < < ( _{< < < <}

Fig. I Cross Section of a Two Dimensional Flexible Base

Air Bearing or Open Loop Ski rt.

I-

0

A

C

H

T

_Lo-l

---.---

(T

Pb

L,

Pc

I-< I-< < < < < < < <

, ,

,

< • < <

X

e

Fig.

2

Notation

I .

₀

1

Pb: Pc

rb/

----IT

t

s

P

9b

Yo Y

Pc

~

iJ

I

I.

"

"

,

,

"

"

\

,

\ \ " <

" "

"\""" ""\:''''''''t

x

·1

~---

0

---~ \ \ \ \ \ \ \ y=0.556 _~~I...

~-

_t

s

=

0.410

Pc

=

1.039 I \ \ \

\~

r

.... ---

X

=

1.05 - - - . . !

Fig. 4 Typical Skirt Geometry for Large Nozzle Length

f

2 . 0 t . . . . . . . r . . .

-1.5

,

0.5

o

\.

"-\.

0.05

\.

_ _ _ _ Loop Skirt

LMI

D

=

1.88

_ . _ . _ Loop Skirt LM/D

=

1.50

- - - - _ Plenum Chamber

- x

x-

Locus of Points for

which

R

₅

=0

\.

"-

,

_,

,

0.5

m/mref.

0.1

Fig. 5

Typical Stiffness Curves for Loop Skirt

and Comparable Plenum Chamber Cushion

..

L'1IAS TECHIUCAL NarE !lO. 204

~

Institute for Aerospe.ce studies, University of ToroJlto 4925 Dufferin street, Down.view, Ontario, Canada, lo\3H 5T6

THEORETICAL AllALYSIS Ol! A LOOP TYPE FLEICIBLE SICllIT AIR CUSHION

Sullive.."l, P. A. and Parravano, J. G. 22 pages 5 figure. 1. Air cusl".1 on 2. Loop or bag skirt 3. Air bearing

I. SuJ.livan, P. A., Parravano, J. G. II. l1rIAS Technical Note No. 204

A theory is proposed for the loop or bag type nex1.ble sldrted air cueh1.on or air bea.r1ng. It 1s

sho\.m tho.t the nozzle forr:;ed by the skirt in the reg10n ",here the air escapes tram the cuah10n

plays QJl. essential role in the skirt action . Consequently earlier attempts to model the skirt by

ignoring th1s regian, on the ground that it is usually smeJ.l, are incomplete. Some numer1caJ.

results are obtained for the case where the skirt behaves as a membrane and the flow in the nozzle

ca..'1 be con3id.ered inviscid and one-dimensional.. Tbe stiffness curves are compared w::1. th tbe

corrcspo!1.üil".g curves for a simple rigid walled plenum ch8lDber fed from the same source. These

s!'!o ... that ':;bc skirt CC'D be relatively stiff in heuve, but also tbat if very lew maas flow rates are pro.cticnl it \"il1 be much less stiff than the simple plenwn choxnber. It is suggested that

viscous floweffects are very important, and that on alternative simplification, namely that the 1'10".: CM bc treuted as inertialess , rnay nat be valid.

l1rIAS TECIINICAL NarE NO. 204

~

Inst1tute for Aerospace studie., University of Toronto 4925 Durferin Street, Downsview , Ontario, Canada, lo\3H 5T6 TllEORmICAL ANALYSIS Ol! A LOOP TYPE FLEICIBLE SICllIT AIR CUSHION

Sullivan, P. A. and Parrav811o, J. G. 22 pages 5 f1gures

1. Air cusbi on 2. Loop or bag skirt 3. Air bearing

I. SuJ.llvan, P. A., Parrav811o, J. G. II. l1rIAS Technical Note No. 204

A theory is proposed for the loop or bag type fiex1ble skirted air cushion or air bearing. It is shown that the nozzle formed by the skirt in the region where the air escapes from the cusr,iou

plays en essential. role in the skirt action , Consequently earlier attempts to model the skirt by

ignoring th1s region, on the ground that it is usUally small, are incomplete. Sor.e murerical

results are obtained for the case were the skirt behaves as a membrane and the flow in the nozzle

CM. be considered inviscid end one-dimensional. The stiffness curves are cornpared w1 th the

corresponding curves for a simple rig1d 'Walled plenum chamber fed from tbe SOll'.e source. These

shO't'l that 'tbe skirt can be relatively stift 1n heave, but eJ.so that if very low tr.8.SS now rates are prncticlll. it w111 be much less stiff than tbe simple plenum chamber. It is sugeested thnt

viscous floweffects are very important, and that sn alternatlve slmpllfication, namely that the flow can be treated as inertialess, may not be valid.

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

iJrIAS TECiiiJICAL tlCfl'E NO, 204

Institute for Aerospace Studies, University of Toronto 4925 Dufferin Street, DO',msview, Ontario, Canada, lGH 5T6

TJlEOfu."'rICAL AllJ\LYSIS Ol! A LOOP TYPE FLEXISLE SICllIT AIR CUSHION

Sulliva:l, P. A. and Parravnno, J. G. 22 pages 5 figures

1. Air cushion 2. Loop or bag skirt 3. Air bearing

I. Sul.livan, P. A., Parravano, J. G. II. l1rIAS Technical Note No. 204

~

_~

A theory is proposed for the loop or bag type flexible skirted air cushion or air oee.r1ng. It is sbol.T. that the nozzle formed by thc skirt in the region where thc air escapes from tbe cushion p!ays ~ essential role in thc ski!'t action, Consequently earlier attempts to model the skirt oy

~enori)'l..e thi,f: region, on the groWld that it is usually small, are incomplete. Same numeri cal rcsults ere obtained for the case where the skirt behaves as a membrane and the flow in the nozzle

CM be consitlercd inviscid and one-dimensional. '!'he stlffness curves are compared Vi th tbe corre.::po:l1inB curves for a sirnple rigid walled plenum chamber fed trom tbe same source, These !lhOH that -!;hc sltirt CM bc relatively stiff in heave, but also tbat if very low maas flow rates

.1.!'c pro.cticaJ. it • .... ill be much lcss stif!' thn.n the simple plenwn chan".bcr. It is suggested that viscO\:.s flow cf'fecta are very important, and that an aJ.ternative simplification, na.nely tbat tbe

f'l01-1 c~'1 "oe ";.t'ca.ted as inertialess, ma.y nat be valid,

l1rIAS TECHNICAL NarE NO. 204

Institute for Aerospace Studies, University of Toronto

4925 Durferin Street, Downsview, Ontario, Canada, lo\3H 5T6

~

...

-~

~HEORETICAL ANALYSIS Ol! A LOOP TYPE Fu::a:sLE SICllIT AIR CUSHION

Sulllvan, P. A. and Parravano, J. G, 22 pages 5 figures 1. Air cushion 2. Loop or bag skirt 3. Air bearing

I. Sul.livan, P. A., Parravano, J. G. II. l1rIAS Technical Note No. 204

A theory is proposed for the loop or bag type nexible skirted air cushlon or air bearing. It ls

shown that the nozzle formed by the skirt in the region where the air esca.pes from the cush10n plays an essential role in thc skirt act1on. Consequently earlier attempts to model the ski rt by ignoring this region, on the ground that lt is usunlly small, are incanplete. So:r.e n1..Ul:lerical

results are obtained for the case where the skirt bchaves as a membrane and the flow in thc nozzle

een be considered inviscid and one-dimensional. The stiffness curves are compnrcd wi th the

corresponding curves for a slmple rigid walled plenwn cbamber fed from the srune source. These

show that the skirt cen be relati vely stiff in heave, but also that if very 10'" mass flow rates

are practica.l it will be much less stiff than the slmple p1emun chambcr. It is suggested thllt

viscous floweffects are very importent, and that an alternative s1mplificat10n, nanely thllt thc

flow CM be treated as lnertialess, may nat be valid.

/ /