a d

F €

cr: U

E G UNDER RAXDO AD1NG

https://ntrs.nasa.gov/search.jsp?R=19650001025 2020-05-14T14:25:17+00:00Z

C P

MODELLING OF SPACECRAFT UNDER RANDOM LOADING

By Joshua E, Greenspon

Distribution of this report is provided in the interest of information exchange. Responsibility for the contents r e s ides in the author or organization that prepared it.

Prepared under Contract No, NAS 5-3132 by J G ENGINEERING RESEARCH ASSOCIATES

Baltimore, Maryland

f o r

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sa le by the Off ice of Technical Services, Department of Commerce, Washington, D.C. 20230 -- Pr ice $1.00

ABSTRACT

This report gives modelling laws for space vehicles under random loading where the model is constructed of a different material than the full scale vehicle.

ACKNOWLEDGMENTS

This work was done under the sponsorship of the Goddard Space Flight Center, NASA. This author would especially like to acknowledge the helpful suggestions and advice of Mr. Thomas G. Butler of Goddard.

i

.. ,

CONTENTS

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11. Approximate Formulation of the Modelling Laws .

A. Forces and Motions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Joint or Interface Damping

111. More General Formulation . . . . . . . . . . . . . . . IV. Discus s ion . . . . . . . . . . . . . . . . . . . . . . . . . . V. Appendix . . . , . , . . . . . . . . . . . . . . . . . . . . .

Page

iii

1

2

2

8

11

19

20

ii

LIST OF SYMBOLS

4' t '

k'

4 t

k A 7

E E'

P

P'

U

uE

4 f

f'

U

P

- V

a

C P.S. S.S.

character is t ic length in prototype

character is t ic t ime in prototype

character is t ic force in prototype

charac te r i s t ic length in model

character is t ic t ime in model

character is t ic force in model

length scale factor

t ime scale factor

force scale factor

Lame 's elastic constants

modulus of elasticity of model mater ia l

modulus of elasticity of prototype mater ia l

m a s s density of model mater ia l

mass density of prototype mater ia l

Poisson ' s ratio

scale factor for modulus of elasticity

scale factor for solid density ps'/ps

frequency of model

frequency of prototype

p r e s su re

direct s t r e s s in x direction

vel0 city

ac c e le r a tion

correlat ion

power spec t ra l density

shear s t r e s s

normal s t r e s s

coefficient of friction for model

coefficient of friction for prototype

scale factor fo r coefficient of friction

power spec t ra l density of displacement

iii

w

kf P

power spectral density of velocity

power spectral density of acceleration

Frequency response function for r t h mode

mode shape of r t h mode

correlation integral

generalized m a s s for nth mode

mater ia l damping coefficient for nt

natural frequency of nt mode

logarithmic decrement in n t h mode

constant mater ia l damping factor

mode

= r/p forcing frequency

body force per unit volume of solid

body force per unit mass of fluid

displacements of solid o r fluid

temperature (absolute)

scale ra t io for temperature

thermal conductivity of the solid

specific heat of solid (per unit volume)

reference temperature above which temperature

changes a r e measured

density of the fluid

velocity components

viscosity

specific heat at constant pressure

for fluid (per unit mass )

thermal conductivity of fluid

force

iv

c 1

GENERAL

All pr imed quantities refer to the prototype, a l l unprimed quantities

refer to the model. All scale factors a r e denoted by u ; e.g.

is the rat io of the specific heat at constant p re s s of the fluid pro- *CP f

totype mater ia l to the fluid model material. All s subscripts refer to

solid; all f subscripts refer to fluid.

V

t 1

MODELLING O F SPACECRAFT UNDER RANDOM LOADING

Joshua E. Greenspon, Dr. Eng. J G ENGINEERING RESEARCH ASSOCIATES

3831 Menlo Drive Baltimore, Maryland 2 12 1 5

I. INTRODUCTION

As spacecraft become la rger -and more complicated, it becomes

more difficult and expensive to tes t the full scale structures. In addi-

tion, i f specifications for a spacecraft a r e to be obtained before the

vehicle and payload a r e constructed, the only way that predicted r e -

sponse can be obtained is for scale model tes t s to be conducted and

calculations performed on the proposed design. I f small models could

be constructed of mater ia ls which were easier to handle in manufactur - ing and l e s s expensive than the full scale, a great deal of information

could be obtained efficiently on a proposed space vehicle.

of this report is to present a derivation of the modelling laws for

random dynamic loading on an elastic structure such a s a spacecraft-

booster combination which is constructed of a different mater ia l than

the prototype.

The purpose

1

11. Approximate Formulation of the Modelling Laws

A.. Forces and Motions

The theory presented here is an application and extension of the

basic theory of modelling presented by Weber (Ref. 1).

associated with the full scale be pr imed and those associated with the

model unprimed.

scale, t ' the full scale t ime, and k' the force in the full scale ,

Let the quantities

Thus i f 8' is the characterist ic length in the full

where A i s the linear dimension scale factor , 7 is the t ime scale factor,

and x is the force scale factor.

The phenomenon that is to be scaled is random loading and response

of an elastic structure.

the model is geometrically similar to the prototype and is subjected to

a geometrically similar load distribution.

The only assumption that will be made is that

The basic equations governing the elastic motions of the s t ructure

in the absence of body fo rces a r e the three dimensional equations of

elasticity (Ref. 2).

2

w h e r e x and F are the Lame constants, A is the cubical dilatation, and p,

is the mass density of the solid material . The displacements a r e u, v, w.

The cubical dilatation and the Laplacian a r e

aU av a w ax ay aZ

A = - + - + -

The Lame constants h and jY can be writ ten in t e r m s of the modulus

of e las t ic i tyE and Poisson 's ratio v as follows:

E V - E - A = 9 p=- ( l t v ) ( 1 - 2 V ) 2 ( l + V )

So the x component of the equations of motion becomes

and there a r e two other equations for the y and z components.

scaling laws can be derived by mere ly working with one of these equa-

tions. In accordance with our notation, equation (5) is for the model

since it contains unprimed quantities.

The

The equivalent equation for the

. I

3

prototype, assuming a different mater ia l with approximately the same

Poisson's ratio but different E and p," is

Le

aZui a 2 U ) t - E' (- 2 ( l t v ) a x f 2

the scale factors for E and pebe

By the choice of model mater ia l and linear scale this will f i x u E , D

and A.

basic quantities.

follows :

ps ' The r e s t of the scale factors will then be derived f rom these

Thus the time scale factor 7 will be determined a s

*Usually the Poisson ratio is of secondary importance. Moreover, the Poisson ratio for the model probably will not vary greatly from the full scale material.

4

Thus,

and

(frequency is therefore modelled as 1/r = ( l / A ) , / x ). E ps

Equation (6) f o r the prototype can then be written

DEE 1 a% +-+-+- a% aZu 1 (%+- a 2 V +-j aZw

2 ( 1 + v ) [z a y 2 a z 2 i - 2 v a x 2 ax ay ax aZ

Then all scaling t e r m s a r e collected into a single coefficient on the

left. In order for the model to obey the same dynamic equations a s the

prototype, the scaling relation must be a s follows:

uE/A = D X/r2 ps

which reduces to

E'/E t 1 2 / t2 --=1 P,/P, .et2/.e2

o r

5

o r finally f't1/f- = f&/* where f is the frequency.

The physical number f 4 / G s is a type of Strauhal number. If the

same mater ia l were used in the model and prototype, then Junger 's

relations (Ref. 3) would hold; i.e.,

F o r Junger 's case, frequency is scaled inversely to length. Equations (8)

and (10) a r e the basic relations for modelling with different mater ia ls .

Variations of this scaling law a r e contained inReferences (4-7). The other

quantities involved in the problem canbe derived f rom these relations.

Thus

P res su re : p'/p = X / A 2 P f = DEP

(for the same mater ia l , the velocities scale directly).

The statist ical quantities will then be scaled a s follows:

< p12> Mean square p re s su re : - - CP2 > -0'E

6

Correlation full scale Correlation model Correlation function:

(b) Displacement: <w;w;>/<wlw2> = A2

(d) Acceleration: <a;a;>/<a,a2> = -

Power spectral density: i.e. ,

(P.S.)' - Power spectral density full scale (P. s. ) - -

Power spectral density model

(b) Displacement: A 2 r = A2A F = OE OE

(c) Velocity: - aE r = - uE AE U 0- P, P,

(d) Acceleration: I (;I r = ($1 AE =tF h2 OPS

7

1. Joint or Interface Damping

One of the big problems in using scale models is damping. Here

we will analyze how Coulomb damping and mater ia l damping enter into

the problem. Coulomb damping will a r i s e as external damping through

rubbing between attached pieces.

ing states that the frictional shearing s t r e s s is proportional to the normal

s t r e s s ; i.e.,

The basic relation for Coulomb damp-

\ -

( S . S . ) = (N.S.)

- where i s the coefficient of sliding friction between the two surfaces.

F o r the prototype this relation will be

( S . S . ) ' = E' (N.S. ) ' (15)

The previous scaling laws derived in section A stated that the s t r e s s

must scale by a factor uE; thus

Thus the frictional shearing s t r e s s in the model must be adjusted to

1/uE of the frictional shearing s t r e s s in the prototype. This can be

done by adjusting the coefficient of friction and the normal bearing s t r e s s

in the model so that ( 1 6) is satisfied.

bolt fittings a t the joints so that the normal p re s su re could be adjusted.

The model could be constructed with

8

2. Material and Air Damping

For mater ia l damping the problem is not as direct , and we must

r e s o r t to some assumptions regarding the response of a s t ructure to

random loading. Take the case in which the mater ia l and a i r damping

can be completely described by a viscous damping coefficient.

power spectral density of the displacement of a s t ructure under random

loading can, for practical purposes, be written a s (Ref. 8)

The

where{ is associated with the mode shape of the s t ructure , C,, i s the

correlation integral which for all practical purposes can be considered

independent of mater ia l damping, and Y i s the response function which

i s cri t ically dependent on damping. (To make this paper self -contained,

the origin of equation (17) i s explained in the appendix.)

function Y can be written

The response

where M, is the generalized m a s s for the rth mode and P, is the damping

coefficient for this mode. The logarithmic decrement 6, can be written

9

Fung et a1 (Ref. 9 ) found that p r was approximately a constant for var i -

ous modes of aluminum cylindrical shells.

w = w ,

Thus at resonance, where

O r

In a structure subjected to random loading, the most severe r e -

sponse in the spectrum will occur a t the frequencies corresponding to

resonances of the modes. Therefore, a t these frequencies

1 (P.S.), - - = 62 B2

So, f o r a model of different mater ia l than the full scale,

Thus, in order to scale mater ia l damping properly a t the cr i t ical

f r equencie s ,

10

e 1

The ratio c'/c must be determined experimentally for the different

mater ia l s . However, a relatively straightforward experiment might be

used to do this; possibly an experiment such a s the one performed by

Fung et al. (Ref. 9) could be performed on cylindrical shell models made

of the prototype and model mater ia ls .

A more basic assessment of mater ia l damping and its dependency

on level of excitation is offered by Crandall (Ref. 10). However more

recent experiments have been conducted by Granick (Ref. 11) ( in an

extension of Crandall 's study) at Goddard Space Flight Center which

should throw a great deal of additional light on the mater ia l and air

damping problem.

111. More General Formulation

In the previous section the modelling laws for the s t ructure were

formulated in an approximate fashion considering mater ia l and a i r damping

a s lumped parameters in an effort to derive some pract ical modelling

relationships.

that ma te r i a l damping a r i se s out of a thermal relaxation phenomenon first

Consider now a more exact formulation by assuming

11

discussed by Zener (Ref.

(Ref. 14) and others and considered m o r e recently by Lazan and Goodman

(Ref. 15), Crandall (Ref. lo ) , and Granick (Ref. 11). Fur thermore , con-

sider that the vibrating s t ructure is moving in a viscous heat conducting

fluid so that fluid damping may a r i s e out of radiation and viscous losses .

12) la te r generalized by Mason(Ref. 13) Biot

The equations governing the behavior of the solid a r e

-Az T h e equation of motion of the solid (Ref. 16) (x component with

body forces)

aZu a2u a2u 1 a aU t-t-) a v a w

ax2 a y 2 a z 2 1-2v a,Q ay a Z - t - + - t - -

X = b o d y f o r c e p e r u n i t volume

B. The heat conduction equation of the solid (Ref. 14) (whichassumes

material damping by the mechanism of thermal relaxation)

(25) K s (2 t3 t$)=Cs Ft aT To -- UE a aU t-t-) av aw 1 - 2 Y a t ( x ay aZ

C. The equations governing behavior of the fluid a r e

1. The equation of continuity (Ref. 17)

12

2. The Navier Stokes equations of motion (Ref. 17) (x component

with body forces)

a; 2 a; a; a+ = p f F x - - + - a [ 2 p - - - p ax 3 ( ax ax

a; . a6 . a;

Fx = b o d y f o r c e p e r u n i t mass

3. The equation of heat t ransfer in the fluid (Ref. 17, 18)

(28)

The main boundary conditions between fluid and solid to consider

for modelling a r e

a t f l u i d - s o l i d i n t e r f a c e ( T ) f l u i d = (T)solid

a t f l u i d - s o l i d i n t e r f a c e

(Disp lacement ) f = (Displacement)solid a t f l u i d - s o l i d i n t e r f a c e

(Normal v e l o c i t y ) f l U i d = (Normal ve loc i ty ) so l id a t f l u i d - s o l i d i n t e r f a c e

- (Stress)fluid - (StreSS)solid a t f l u i d - s o l i d i n t e r f a c e

13

If modelling solids and fluids a r e chosen which a r e different f rom the

prototype then the following scaling parameters must hold:

For the solid

E'/E = crE, p : /ps =5P , K:/Ks ~5~ , a' /a =u2, C:/Cs = X ' / X = ~ ~ ( 2 9 ) S S 5cs

F o r the fluid

p ; /p , =CTpf, p ' / p =5$ cp I/Cp = D c 1 q/k, = C J k f 9 P;/Pf =apf' F./Fx =5 f f P f F x

( 3 0 )

These relations a r e the fundamental parameters once the model mater ia l

and model fluid a r e chosen.

Equation (24) and ( 2 5 ) can then be written for the solid model a s

( see Eq. (9 ) and again assume same v in model and prototype)

Thus the following relations must hold among the scaling factors in

order that the basic differential equations will be the same fo r model

and prototype

14

The relation cE/h = up, h/r2 leads to equation (10). The additional

relation oE/h = oE 52 B/h leads to

1 e = - ull

This says that the temperature of the model will be scaled by l /ua.

F r o m (34 ) we have the relations

( 3 5 )

It is plainly seen that i f the same mater ia ls a r e used in model and

prototype and testing is done in the same fluids, the first of the equations

in (36) is inconsistent with the elastic scaling law ( 3 3 ) . In fact the only

way that thermoelastic damping can be scaled is by choosing the length

scale and the mater ia l for the model in such a manner that the coef-

ficient of expansion, heat t ransfer coefficient and specific heat of the

model and prototype obey equation (36) . This does not s eem to offer

much hope.

r ialdamping scaling seems to be by u s e of a relation s imi la r to ( 2 3 ) .

In cases of built up s t ruc tures containing many connections, the joint

o r interface damping will undoubtedly over shadow the mater ia l damping

so that this inconsistency will play a minor role.

Thus the only hope for obtaining sensible resu l t s for mate-

. *

15

.

Now consider the fluid. If the equations of the model fluid a r e

written in a similar manner a s those for the solid, we obtain

( 3 7 ) -- % P f Pf t u - ~a [= (Pf ; ) + , ( P f G ) a +z a ( P f i ) ] = O 7 a t p f A~

cr P f aP

f fP f (3) ( L e f t Side) = crp UFx Pf Fx -h ax

The basic scaling relations that mus t hold a r e therefore

r 7 h2 r2

The relation cr A / r 2 = f f P / A 7 leads to the constant Reynold's number

s c aling Pf

A " r

(noting that 5- = - )

16

This relation inherently contains the viscous damping offered by the

fluid to the solid. Noting the fluid-solid boundary conditions and intro-

ducing (33)

If the same fluids and solids a r e used for the model and prototype,

this will lead again to an inconsistency in (43) i f A f 1. The only way

that we can scale fluid viscous damping is to choose a length scale and

model mater ia l s which obey (43). The second scaling relation f rom (40)

i s

where p denotes p re s su re and P denotes force.

This is Newton's Universal Similitude Law (Ref. 1) which musthold be-

tween the model and full scale fluids. It inherently contains the scaling

relations for m a s s loading and radiation damping offered by the fluid

to the solid.

Now going back to the boundary condition between solid and fluid,

it is seen that s t r e s ses and p res su res must be scaled by wE. Thus

17

(45) r 2 h

and noting that h2/72 = crE/up f r o m elastic scaling, we obtain finally S

w = f f P f ps

This says that the scale factors for the density of the fluids and

solids for model and prototype must be the same in order for m a s s

loading and radiation damping to scale properly. If the same density

fluids a r e used for the model testing then the same density solids must

be used.

The fluid heat t ransfer equation gives scaling parameters which

must satisfy

The same inconsistency between these relations and the elastic

scaling relation is obtained when the same mater ia ls a r e used for model

and full scale. In cases where heating of the fluid is not a pr imary prob-

l em, the heat transfer character is t ics of the fluid can be neglected com-

pletely and it is only necessary to consider the fluid-solid problem

without Eq. (28).

18

IV. DISCUSSION

The use of very smal l scale models of different mater ia l than the

prototype offers a very efficient and inexpensive way to obtain o rde r

of magnitude answers which would ordinarily be very difficult o r im-

possible to obtain.

use much lower p re s su re excitation levels than the full scale.

This type of scaling also offers an opportunity to

There

might a lso be a possibility of using light weight, easy to construct,

plastic models to perform initial tes ts , in the same manner a s those

performed by Sankey and Wright (Ref. 19) but with a more complete

a s ses smen t of damping.

Invariably, damping offers a big problem i n using scale model r e -

sults to extrapolate full scale response. If we limit our discussion to

the resonance response, then damping could be introduced approximately

in the scaling a s shown in the previous section i f 5 were constant for a

given mater ia l .

ing, since the major response a t each frequency of interest is usually

composed of primatily resonant contributions in modes which a r e close

to this frequency.

This resonant limitation is not ser ious in random load-

As shown in the third section of the repor t mater ia l damping of the

thermoelastic type cannot easily be scaled nor can viscous a i r damping

ar is ing f rom a i r viscosity unless tes ts are conducted in "thinner" a i r

o r in different fluids.

tu res under random loading is that joint o r interfacial slip will be the

main source of damping both in the model and in the full scale whether

they be of the same o r different materials.

The main hope with model testing built-up s t ruc-

If this were t rue and if

19

frictional s t r e s ses were scaled properly according to (16), then re la -

tions (13) and (13a) will hold for the random response. If one uses

model materials with high mater ia l damping such a s plastics, he must

be sure that the joint damping is modelled properly before resorting

to approximate equations of the form of (23). Equations (23) mus t be

used with extreme caution since they will hold a t resonant frequencies

and they will hold only i f 6 can be shown to be a function of mater ia l only.

APPENDIX A

The general variational equation of motion for any elastic s t ructure

in the absence of body forces can be writ ten a s (Ref. 8)

[p(USu t VSv t WSw) t GwIdv 151 V

where

mass density of body

displacements at any point . variations of displacements

surface forces

elemental surface a rea

elemental volume

variation of s t ra in energy function.

20

b

f n accordance with Love’s analysis let the displacements in the

normal modes be described by

where y, = A, cos

Now let the forced motion of the system be described by

p, t , p, being the natural frequency of the r t h mode.

where u , , v , , w , , are the mode shapes and yr is a function of time. In

accordance with Love let

Substituting into the variational equation of motion, we obtain

However, since the modal functions satisfy the equation for f r ee

vibration

2 1

and the orthogonality of the normal modes implies that

So the final equation of motion becomes

where

+ v: + w:)dV (the generalized m a s s for the r t h mode)

V

and

F r ( t ) =L [[[xv(t)ur t yv ( t )vr t zv(t)wrldS (A9) s

Mr

(the generalized force for the r t h mode)

If s t ructural damping is taken into account, it can be written a s

another generalized force which opposes the motion

22

S

where K is the damping force per unit volume per unit velocity. Finally,

the equation of motion becomes

It will be convenient to employ the vector notation; thus, let the dis-

placement functions in the rth mode be written as

+ + + where i , j , k are the unit vectors in the x, y, z directions, respectively.

Let

Thus,

where D means surface.

2 3

Now the Fourier t ransform of 3 (a, t ) is

F’ (c, t) e-iot d t 1 s- (a, w) = - fi F

and the Fourier t ransform of cp, can be written

yr ( t ) e-iot d t 1 s (w) = -

‘Pr

S. (w) = i w S (w) ‘Pr ‘Pr

sa. (w) = - a2 s ( w ) ‘Pr ‘Pr

and

+a

cp, ( t ) e- iwtdt

Thus the t ransform of the equation of motion becomes

2 4

Thus

Now, wri te the auto correlation function of the displacement in vector

notation as follows:

A--(V, 7) = lim - 1 [:' ; ( t ,V) - ; ( t t r , V ) d t T - W 2T q q

where < denotes the complete displacement vector

Now

o r

25

so

r t a

Now the power spectral density of the displacement:: P;; is defined in

te rms of the autocorrelation function as

P - - ( u , V ) e i W T du 1 A - - ( 7 , V ) =- q q

Thus,

or

But

*The notation (P. S. )d i s used for power spectral density in the main body of the report.

26

a

so

Therefore

where

Let Crk(w) represent the double surface integral in (A31). C r k ( w > ,

when writ ten out, gives

27

c I

The te rms in the brackets become

r s , s , * ~ ~ ~ ~ A I. s ~ s ~ * ~ ~ ~ ~ s ~ s , * ~ ~ ~ ~

t S y S X * V r U k t S Y S y * V r V k t S Y S Z * V r W k

S Z S X * W r U k t s z s y * w r v k t

For the special case of only vertical forces in 2 direction, a l l

t e rms disappear except the las t ( S, S,* w r w k ). This las t equation gives

the coupling between the forces when there a r e exciting forces in more

than one direction. It is seen that the general solution for forces in

any direction involves c r o s s correlations between a l l pa i r s of compo-

nents of the forces in a l l directions.

Equation (31) can now be written a s

which corresponds to equation (17) in the body of the report .

28

L

1.

2.

3.

4.

5 .

6.

7 .

8.

9.

10.

11.

12.

13.

r)

REFERENCES

M. Weber, "The Universal Principle of Similitude in Physics and its Relation to the Dimensional Theory and the Science of Models,'' David Taylor Model Basin Translation 200, Sept., 1949.

A. E. H. Love, "The Mathematical Theory of Elasticity," Dover Publications, p. 293, 167, 179.

M. C. Junger, "The Scaling Laws Governing the Dynamic Response of a Structure to a Turbulent Boundary Layer , ' ' U. 5. Navy Journal of Underwater Acoustics, April, 1963, p. 439.

D. V. Wright, D. F. Miller, J. G. Akey, and A. G. Hogg, "Vibration Transmission and Impedance of Basic Foundation Structures ," Westinghouse Research Labs., BuShips Contract Nobs 72326, Oct. 1962.

J. 5. Mixson, J. J. Catherine, and A. Arman, "Investigation of the Lateral Vibration Character is t ics of a 1 / 5 scale Model of Saturn SA-1," NASA TN D 1593, Jan., 1963.

W. E. Baker (editor), "Use of Models and Scaling in Shock and Vibration," ASME .Colloquium, November, 1963, (published under auspices of the American Society of Mechanical Engineers).

J. M. Calligeros and J. Dugundji, "Similarity Laws Required for Experimental Aerothermoelastic Studies," Aeroelastic and Struc - tu res Research Laboratory, MIT, Tech. Rep. 75-1, May, 1959.

A. Powell, "On the Response of Structures to Random P r e s s u r e s and to J e t Noise in Par t icular ," Random Vibration, edited by 5. Crandall, John Wiley and Sons, 1958, p. 187. is derived in the Appendix using a variation of the Powell procedures.

F o r completeness this relation

Y. C. Fung, et. al. Jour. Aero. Sci., Sept., 1957, P- 650.

S. H. Crandall, "On Scaling Laws for Material Damping," NASA TN D-1467, December, 1962.

Granick's experiments a r e a s yet unpublished.

C. Zener, "Elasticity and Anelasticity of Metals, "University of Chicago P r e s s , 1948, Chap. VII.

W. D. Mason, "Piezoelectric Crystals and Their Applications to Ultrasonics,l' D. Van Nostrand Go., Inc., 1949, Chap. 3 , and Appendix.

14. M. A. Biot, "Thermoelasticity and I r revers ib le Thermodynamics ,I1

Jour. of Applied Physics , Vol. 27, No. 3, March, 1956, p. 240.

15. B. J. Lazan and L. E. Goodman, "Material and Interface Damping,' ' Chap. 36 of Shock and Vibration Handbook, (Vol. 2) Edited by C. M. Harr i s and C. E. Crede.

16. Handbook of Engineering Mechanics, edited by W. Flugge, Chap. 43, "Thermal S t resses" by H. Parkus.

17. Ibid, Chap. 68, "Fluid Mechanics - Basic Concepts," by C. E. Brown.

18. J. ivi. Kay, "An In t r~duc t ion to Fliiid Mechanics and Heat Transfer ," Cambridge University Press , 1963, p. 198.

19. G. 0. Sankey and D. V. Wright, "Vibration Character is t ics of Propulsion Machinery Subbases a s Determined by Plast ic Model Tests", Westinghouse Research Labs. , Buships Contract Nobs 86809, Nov. 1963.

SUPPLEMENTARY REFERENCES

1. H. M. Schauer and D. S. Cohen, "On the Feasibil i ty of Model Tes ts for Noise -Radiation Studies , I ' Journal of Underwater Acoustics , Vol . 10, No. 2, April , 1960 (CONFIDENTIAL - Title Unclassified)

2. W. G. Molyneux, "Scale Models for Thermoelastic Research," Royal Aircraft Establishment, Technical Note No. : Structures 294, March 1961, (DDC No. AD 258163).

3. M. Schlessinger, B. Omilian, J. Deutschman, and J. S. Isenberg, "Practical Aspects of Aerothermoelastic Modelling,'' ASD-TDR- 62-705, Dec. 1962 (DDC No. AD 297725)

4. L. A. Scipio, L. C. Teng, "Analytical Study of Similari ty Parameters for Aerodynamic Model Testing at High Tempera tures , ' ' WADC TR 57-496, Nov., 1957 (DDC No. AD-142264)

5. "Proceedings of the Symposium on Aeroelastic and Dynamic Model- ling Technology," 23-25 September, 1963, (DDC NO. AD 434798)

30 NASA-Langley, 1964