Journal of Sound and Vibration (1982) 83(1), 37-51
ANALYTICAL A N D EXPERIMENTAL EVALUATIONS OF
SPACE SHUT T L E TPS TILE V I B R A T I O N RESPONSE
A. G. PIERSOL AND L. D. POPE
Boh Beranek and Newman Inc., Canoga Park, California 91303, U.S.A.
(Received 8 June 1981, and in revised form 10 October 1981)
Analytical studies and laboratory experiments have been performed to evaluate the vibration response of the Space Shuttle Thermal Protection System (TPS) tiles due to the intense rocket generated acoustic noise during lift-off. The TPS tiles are mounted over the exterior of the Space Shuttle orbiter structure through Strain Isolation Pads (SIP) which protect the tiles from thermal induced shear loads at their interface. The analytical predictions indicate that the response of a typical tile is governed by the structural vibration inputs through the SIP under the tile at frequencies below 250 Hz, and by the direct acoustic excitation over the exterior surface of the tile at frequencies above 250 Hz. An evaluation of the laboratory test data for this same tile, in which conditioned (partial) coherent output spectral analysis procedures were used, leads to exactly the same.conclusion. The results demonstrate the power of conditioned spectral analysis procedures in identifying vibration response mechanisms when two or more of the inputs are highly correlated.
1. INTRODUCTION
The Space Shuttle orbiter vehicle is protected from the heat of re-entry by a Thermal Protection System (TPS) consisting of a collection of small tiles attached through an interface material to the exterior of the aluminum skin, The TPS tiles, which are fabricated from a silica material, have spectacular thermodynamic properties, but are physically fragile and must be protected from the shear loads that would arise if they were directly attached to the aluminum skin because of the widely different coefficients of thermal expansion of the tile and skin materials, This alleviation of shear loads is accomplished by a Strain Isolation Pad (SIP) as illustrated in Figure I, The shear and extensional moduli of the SIP material are very low relative to the tile material, Figure 2 shows the non-linear behavior of the SIP in tension and compression, Various experiments have been performed on sample TPS tiles mounted through the SIP to simulated skin structure with both acoustic excitation in a traveling wave acoustic facility and aerodynamic boundary layer excitation in a wind tunnel, This paper is concerned with an evaluation of the results from acoustic tests in which the dynamic loads on the tiles due to rocket noise excitation during lift-off were simulated, Of specific interest is the mechanism causing the tile response: i,e,, is the tile motion caused by a resonant response to the panel vibration under the tile or by a direct response to the acoustic pressures over the exterior surface of the tile? To answer this question, the response of a selected tile is first predicted by using an analytical model with experimentally measured input para- meters, The experimental data are then evaluated by using (conditioned) coherent output spectral analysis procedures, for comparison with the analytical results,
37 0022--460X/82/130037.+ 15502.00/0 O 1982 Academic Press Inc. (London) Limited
38 A . G . PIERSOL A N D L. D . P O P E
~ ~ A d j o c e n t tiles
Propagating ~ / ~ ~ ~.v.~ AdJacent tiles
~ ~ Adjacent tiles
V l V x
Microphones
Tile ~ , , / - - Tile occelerometer �9 -- -T - ~ / ~ Tile density = y 0"0,57 m , /m 3 144 kg
4 mm 1 ~ , - ~ |
Panel "~' ~ Panel ~ Strain Isolation accelerometer Pod (SIP)
z Figure 1. Experimental set-up for tile vibro-acoustic tests.
105 I i
5 X 104 L
A g- o
v
b
-5 x 10 4
-105-0~4 -1~-2 O l:
i i / 0~2 0"14 0"5
Figure 2. Typical stress--strain relationship for strain isolation pad (SIP).
2. EXPERIMENTAL PROCEDURES AND RESPONSE DATA
The experiment (referred to as the Peelable SIP Test) was designed by personnel of the Rockwell International Space Systems Division in Downey, California, and carried out by personnel in the Rockwell International B-1 Division in Los Angeles, California. The basic data reduction to auto- and cross-spectral density estimates was also performed by the Rockwell International B-1 Division.
2 . 1 . D E S C R I P T I O N O F T H E E X P E R I M E N T
A total of 20 TPS tiles were mounted on a ribbed aluminum panel simulating a typical orbiter skin section. The tiles and the panel were instrumented at various locations with accelerometers. The entire assembly was placed in a traveling wave acoustic test facility and microphones were located over selected tiles. The data of interest here are from a single tile representing a typical installation, as shown in Figure 1. The tile had a density of 144.2 kg/m 3 (9 lb/ft 3) and was attached to the panel through a layer of SIP material
SPACE SHUTTLE TPS TILE VIBRATION 39
with a thickness of 4.06 mm (0.160 in). The tile side of the panel assembly was exposed to a propagating acoustic wave with grazing incidence to the tile surface, simulating the rocket noise during lift-off.
2.2. DATA ANALYSIS PROCEDURES
The basic data analysis was accomplished by using a GenRad two channel FFT signal processor. Auto- and cross-spectral density functions were computed by conventional FFT procedures [1]. Specifically, given two time history signals, a (t) and p (t), representing acceleration and acoustic data, the one-sided auto- and cross-spectra (defined for f>~ 0) were estimated with
2 -d 2 G~ =-----j,~l la'(f)l~' OP(f) = n---~,=l ~ IP'(f)l~'
2 nd G~p(jO = ~aT ,~1 a* (JOP,(~, (1)
where the asterisk (*) denotes the complex conjugate, T is the length of individual records, and na is the number of disjoint (statistically independent) records used in the averaging operation. The Fourier transforms of the individual records in equation (1) are given by
T I , T
a,(]') = Io ai(t) e -i2~'f' dt, Pi(j0 = Jo pi(t) e -i2"*f' dt. (2)
The coherence function between two signals a (t) and p(t) was estimated with
-- ( f )6p(f ) - -I ao.(f)l=/ o (f)Gp(f), (3) where the spectral density estimates are as defined in equation (1). The individual record lengths for the analysis were T = 0.5 s giving a basic frequency resolution of f = 2 Hz in the spectral estimates. However, the spectral results were frequency averaged over 10 contiguous components to provide a final resolution of Be = 20 Hz in the results. The equivalent number of disjoint averages was na = 1800. This means that the random error (coefficient of variation) is e =0 .024 in the auto-spectra estimates and e = 0.024/1~,(f)1 in the cross-spectra magnitude estimates [1]. The random error in the coherence estimates is given by e = 0.03311 -
i05 I I I I I I I I I __-_
N " l -
v
' 0 4
10 3
o .
i0 2 0 lO0 200 300 400 500 600 "tO0 800 900 I000
Frequency (Hz)
F i g u r e 3. A u t o - s p e c t r u m o f a c o u s t i c p r e s s u r e i m m e d i a t e l y a b o v e t i le . Be = 2 0 H z , n a -- 1 8 0 0 a v e r a g e s .
40 A. G. P I E R S O L A N D L. D. POPE
2.3. EXCITATION AND RESPONSE SPECTRA The auto-spect rum of the acoustic excitation applied to the tile is shown in Figure 3.
The auto-spectra of the acceleration response levels of the panel and tile are shown in Figure 4. Note that the response spectral levels of the panel and tile are similar at
A N I 0
-lr-
g - - I - 0 o
(3L
0 -1
I I I I I I I I
/"k
E I I I I I I I I I J0O. 20o 30o 400 .500 600 too 8oo 9oo fO00
F r e q u e n c y ( H z )
Figure 4. Vibration response of panel and tile due to acoustic excitation. Be = 20 Hz, na = 1800 averages. , Tile; - - - , panel.
frequencies below 300 Hz, suggesting that the tile rides on the panel as a fixed rigid body at these lower frequencies. Above 300 Hz, however, the panel and tile response levels are qu!te different. It is known that the first resonance frequency of the tile body is well above 100 Hz. Hence, the difference in the panel and tile response between 300 and 1000 Hz must involve strain in the SIP material. The issue of interest is as follows: how much of the tile response below 1000 Hz is due to the mechanical excitation f rom the panel through the SIP material and how much is due to the direct acoustic excitation on the exterior surface of the tile?
3. ANALYTICAL EVALUATIONS
For the basic tile response model that is developed here it is assumed that the tile is driven by a fluctuating pressure over its exterior surface and a fluctuating panel mot ion f rom below, the panel itself responding to the excitation fluctuating pressure. The excitation pressure is due to an acoustic field passing diagonally across the tile as shown in Figure 1. The static pressure is allowed to increase across the tile to simulate a pressure gradient. At the leading edge, the static pressure is taken to be pl and at the trailing edge p2. The pressure beneath the tile can range between pl and P2. This pressure appears in the void of the SIP which has 90% porosity. Thus the model allows for a net "l if t" on the tile. In the case of no pressure gradient, pl = p z and there is no net lift.
3.1. M O D E L F O R M U L A T I O N
If the static pressure is not uniform over the tile, since the SIP has a non-l inear stress-strain curve, each elemental area of the SIP has a different stiffness due to the rotat ion imposed on the tile. To account for this effect, the tile is considered to rest on a number of different springs, each located a distance yi away from the x axis, which is defined to be perpendicular to the direction of propagat ion of the acoustic wave. For simplicity in the present case, only two degrees of f reedom are considered: namely, (1) vertical translation, and (2) rocking about the transverse (x) axis. The tile itself is considered rigid and broken down into N elements. Each ith element has area At. Acting
S P A C E S H U T I ' L E T P S T I L E V I B R A T I O N 4 1
on top of each tile element is a force F~ given by
F~ (pi -i i = + p )Ai = p Ai, (4)
where pi is the static pressure on top of the ith element (pl<~Pi<~p2), and pi is the fluctuating pressure on top of the ith element of the tile. Similarly the bottom of element i has a force due to the pressure on the bottom equal to
F~ = (P~, +p~,)A, ' =p't,A,, (5)
and a force F / d u e to the SIP stress o-t acting over the elemental area At such that
F / = t r~i . (6)
Summing forces in the z direction over the N elements gives
N N
- ~, F/+ E (F~-Fib) =my.. (7) i = l i = 1
Summing moments about the center of mass of the tile yields
N N - F F/y/+ Y. (F~-Flb)yi=IO. (8)
t=l i = 1
NowF/=F/ (&) , where & = z~ - z p; zt is the tile deflection relative to its equilibrium position, and z~' is the panel deflection relative to its equilibrium position. From Figure 2, the z~ deflection due to the dead weight of the tile is negligible.
If F~ is negative, the SIP is in tension. For simplicity, a curve can be fitted to the stress-strain data by assuming tangent elasticity given by
tri = S tan ('a'ei/2) = S tan {('rr/2L)(zi -z~)} , (9)
where L is the thickness of the SIP. A least squares fit of the curve of Figure 2 gives S = 9 3 . 1 • (13.5psi) when tension only is considered, and S = 8 2 . 0 x 1 0 3 p a (11.9 psi) for both tension and compression. With this model
F~ = SAt tan {Or/2L )(zi - z~')}. (10)
At this point the analysis can be simplified by letting z~' = zp. It is thus assumed that the bending wavelength in the panel Ap >> l, where l is the maximum dimension across the tile. Thus,
Ft -- SAt tan {( zr/ 2L )(zt - zp)}. (11)
Since all elements A~ are the same, say A, and since the tile is assumed rigid, the equations of motion become
N 77" N m . .
- S ~, tan-~-s ~, (pt--p~b)=-7-Z, (12) iffil i=l Y-t
7r - -Zp ) y i + 1 1 - S ~ t a n ~ ( z + y t 0 ~ (pi-pib)yi=--~O. (13) i = 1
Suppose now that the equilibrium position for an assumed static pressure distribution is desired. Then k" = 0" = 0, and with Zp = 0, pi = pJ, and p/b = P~, it follows that
N 7r N ~ [ ~. ] N S E tan~--~(z+y,0)= F. (Pi-Pib), S tan~-~(z+y,O) yt = E (Pi--Pib)Yi.
i = 1 iffil iffil t=l
(14, 15) The solutions of equations (14) and (15) are z =Zo and 0 = 0o.
4 2 A . G . P I E R S O L A N D L. D. P O P E
One can now reconsider equations (12) and (13). Let z = Zo+5 and 0 = 0o+ 0. Also pt=pJ+pi andp~ i -i = P b +pb. Then equation (12) becomes (upon noting that 3o = 0)
DI .. - S ~ tan ~-~ (Zo+2+y, Oo+y,O-zp)+~. (P i -P /b) +X (P i -P~ ) = X S .
" i i
But since by equation (14)
(16)
Z ( P i - P ~ ) = rr S ~ tan m (Zo + y~Oo), t 2L
it follows that equation (16) reduces to
, [ 2L 2L (z~176 - S ~ tan--(zo+y~Oo+5+y~O-zp)-tan~Tr - 7r (P-i-Pb)=-~z m ._.
Upon using the identity
tan x - t a n y = tan ( x - y ) [ 1 + tan x tan y],
this becomes
(17)
(18)
(19)
~- - [ ~r - ~- ] - S ~ tan ~zz. (2 + y~O - z~) 1 + tan ~ (Zo + Ydo + 5 + y~O - z~) tan ~ (Zo + y~Oo)
+Z (P~-P~,) m .. = ~ - 2 .
] (20)
Now one can let 5, O, and zp represent small oscillations such that tan{(~r/2L) (2 + y f i - zp)} = (~ /2L)(5 + y i 0 - zp). Then equation (20) reduces to
m._. }] " Xz+~-s ~--s Oo) (~+yd-z~)= Z (P~-P~).
i ~ l I = l (21)
Similarly, the moment equation becomes
- ~ O + - ~ Lfxy, sec 2 ~-~(zo+y, Oo) ( 2 + y f i - z p ) = Y. (/~i-P~)Yi. (22) / = l
It is convenient to define ki = (Sz'/2L) sec2{(~r/2L)(zo+ yi0o)}. Then, in matrix notation, the dynamic equations reduce to
(23)
The fluctuating pressure on the underside of the tile ff~, could be due to the presence of a contained volume of air in the SIP (acting as an airspring) in which case, one might model it as p~ = kcA V = k,.A(z - zp), where ft, = NA, kc = pc2~ V, p is the density of air in the contained volume, c is the speed of sound in the air, and V is the enclosed air volume in the SIP in some deformed state. Here, however, it is assumed that ,O~ is much less then pi, and thus can be neglected. Then, with the definitions
m / A = Mz, I /A = Mo, ~ ki = k11, ~ kiy~ = k12 = k2b Z kiy~ = k22,
SPACE SHU'IWLE TPS TILE VIBRATION 43
equation (23) becomes
+ [ k 2 1
The resonance frequencies are given by
Mzk22+Mok,1+l M~k22+Mokll 2 1/'~11 22-- '2~1 (27rf) '= ~ ~[( ~ !) - 4 " " k k 2 " " / 2 ~ M---~ )J '
w h e r e / ' = f a and f2, with f~ <]'2. The associated mode shapes are
_12} 1 s :11 1 1 , {,0,,z,,} { J , ,co, ,(0,,,,,, O,/Zl ={k11-(2~f,)2Mz}/(-kx2), 02/Z2 = { k l l 2 - (2#/'2) M.}l(-kl2).
(24)
(25)
3.2 . RESPONSE PREDICTIONS (SMALL OSCILLATIONS)
Let ZTT(f) and 0 r ( / ) be finite Fourier transforms of the fluctuating (random) displace- ments observed over time T) as defined in equation (2). Then
2 2 ~r(JO = E ~:,T(D0~, t~r(/0 = E ~:,r(D~O;, (26, 27)
s=l s=l
where ~s~-(]') is the finite Fourier transform of the modal co-ordinate for mode s, i.e.,
#sT(lO = M,(2~rL)2[l_(flL)2 + jm] , (28)
=Mz(~b~) + Mo(Oo) �9 A modal loss and M, is the modal mass for mode s given by 3//, �9 2 �9 2 factor of ,7, = 0.7 (35% of critical damping) is assumed for the tile/SIP combination, based upon independent data supplied by Rockwell International.
Now one can suppose that the fluctuating pressure at all exterior points is in phase, which is an acceptable assumption for frequencies below 1000 Hz. It follows that/7 i = p i _. t7, and therefore, by symmetry, Y.itSJyi = 0 and ~ip ~= Np. Thus equation (26) becomes
rN(r . _ f ( , P ' ~ ) 2 k ~ + ~ , P ' o k ~ 2 (~)2k,,+,p2,p2k,2] Z r ( D = [ Y--~-D - Y2(f) JPr(t)+[ Yx(D § ~ .j zv~(f),
(29)
where Y, = M,(2~-f,)2[1 - (f/[~)2 +jr/s]. Similarly, equation (27) becomes
rNe, le,~o +NO2O21 _ __ [ololoklx +(0~)2k,2.~ 0202k, , + (tp2)2k~2] ~(D=L Y,(D Y'~-DJ pr(D+L Y,(f) Y2-~ j zvT(f).
(30) The auto-spectral density of the tile acceleration response is defined by
2 . G , ( f ) = l i rn -~ E [ a r ( f ) a r ( f ) ] , (31)
w h e r e ar(f)=(27T/.)2~.r(f) and 7.r(f)----gx(f)pr(f)+g2(f)zpr(f). The terms PT(D and
4 4 A . G . P I E R S O L AND L. D. P O P E
Zpr (13 are Fourier transforms of the fluctuating (random) pressure and panel displacement observed over time T, and E denotes the expected value of the quantity in the brackets, determined in practice by an ensemble average over independent records of ar(]') as detailed in equation (1). Also
N(0I )2 + N(0~)2 [ (0'.)2kH + 010~k,2 (0~)2k,, + 0~020k,2]
g,(D Yl(f) Y2(13' g2(D = t ~'~-~ q -~2(~ .!" (32)
Then ar(D=(27r13~gl(13pr(D+g2(13C~r(13, where ar(13=(2~rD2zpT(D is the Fourier transform of the acceleration of the panel below the tile. This gives
G.(D = (2~r1341gl]2Gp(13+lg2]2G~(D+(2~r132[g,g*G~p(D+g~g2G*p(D], (33)
where G~(f) is the spectral density of the pressure on top of the tile, G,,(D is the spectral density of the acceleration of the panel below the tile, and G,~p(D is the cross-spectral density of the exciting pressure on top of the tile and the panel acceleration below the tile.
With 3~p(D the coherence function for the exciting pressure and the panel acceleration, and q~,,p the associated phase, then
Go~ (1314G (I3 O~ (13 = ,/-~r~. (D e-J~~ Therefore
Ga(D = (27rD41gd2G,(D+lg21~O.(13+(2~rD2O~.(13[g,g * +g'g2 ei2~ (34)
Now, to obtain preliminary estimates of the tile response, the cross terms in equation (34) can be ignored, and this simplified prediction model gives
G~ (D = (27rf)4lg,12G, (13 + Ig 12 o,, (13. (35)
By using equation (35), the tile acceleration spectral density can be estimated from a measured pressure spectral density above the tile and a measured 15anel acceleration spectral density below the tile. Except in rare circumstances, one or another of the two terms on the right-hand side of equation (35) will dominate and yield the estimate for the left-hand side. Of course G~(13, the panel acceleration spectral density, is ultimately dependent on Gp(13 and this dependence must be considered in a final prediction of the tile response.
3.3. D E T E R M I N A T I O N OF D O M I N A N T S O U R C E
The tile acceleration spectral density level in dB re 1 g can be defined as
ALr = 10 log [G~(13/g2]. (36)
Similarly the panel acceleration spectral density level in dB re 1 g can be defined by
ALp = 10 log [O.(13/g2], (37)
and the excitation pressure spectral density level in dB re 20/zPa by
SPL = 10 log [Gp(DIp~]. (38)
According to equation (35), if the panel acceleration input dominates the tile response, the tile response is related to the panel response by the result
ALr - A L . = 20 log Ig=l. (39)
On the other hand, if the tile response is due to the pressure fluctuations on its top surface (assumed to be in phase), the difference between the tile acceleration spectrum
SPACE SHUTTLE TPS TILE V I B R A T I O N 45
level and the pressure spectrum level should be given by
A L T - SPL = 20 log [(27rD21gllpo/g], (40)
where Po = 20/xPa (2.9 x 10 -9 psi) and g = 9.81 m/s 2 (386,4 in/s2). Hence, if
ALp - S P L > 20 log [(2~D21gdpo/g]-20 log Ig21, (41)
then the panel input below the tile dominates the fluctuating pressure input on top of the tile. Otherwise the opposite is true.
Analysis of the test data may now be considered. To determine if the tile is being shaken from above or below, the quantity on the right side of equation (41) must be calculated. For the tile under consideration, the computed results are as shown in Figure 5.
-i io I I I I I I I I I I I I i i I
-120 -- . ~
Measurement
m
~ j - 1 4 0 - -
I
-- -- Panel drives ~ Pressure tile / drives ti le
r n / -160 -- .,J
-170 - / Criterion curve
-lec i / , , ~ ~ t , d , i , , ~i IO I 0 0 IOOO
Frequency (Hz)
Figure 5. Cr i ter ia curve for t i le v ibrat ion response due to acoustic test.
TABLE 1
Calculated values of tile and panel acceleration levels minus sound pressure level
Freq. (Hz)
Tile Panel Pressure accel, accel.
PSD level SPL PSI) level ALr PSD level ALp (dB re 20 v.Pa) (dB re 1 g) ALT-SPL (dB re 1 g) AL~-SPL
100 125 160 200 250 315 400 500 630 800
1000
139.4 8"3 -131-1 7.1 139.7 3.8 -135.9 3.7 138.5 0.8 -137.7 1.5 137.3 -0.5 -137.8 -0.5 134.4 -3 .2 -137.6 -4.8 133.3 -3 .9 -137.2 -6.6 132.6 0.8 -131.8 -4.2 132-2 3.8 -128.4 -4.0 131.4 0.0 -131.4 -7.0 125.4 -2 .3 -127.7 3.8 123.7 -4.3 -128.0 -3.3
-132.3 -136.0 -137.0 -137.8 -139.2 -139.9 -136.8 -136.2 -138.4 -121.6 -127.0
4 6 A . G . P I E R S O L A N D L. D. P O P E
(3
~ q 4 ~
. o . = o ~ o o g ~
ao~
~
o
t ~
~ ~ 1~-~
o c ' , l
0
o.~
_oo d ~ -~- 0
~ 0 0 0 0 ~ 0 0
I I I I I I I + + +
9 9 ~ T ~ 9 ~ 9 9 ~
T T l i i i i
I I I I I I I I I I I
A ~ 6 6 6 6 6 6 6 6
I I I I I T I ?
9 9 9 ~ 9 9 ~ 9 ~
ill liFT
�9 ~
o.i o.)
v ~+ ~ m
r . ~
< -#-- - H - ~.o~.
SPACE SHUTTLE TPS TILE VIBRATION 47
Next, by using the sound pressure levels from Figure 3 and the'panel response data from Figure 4, the left side of equation (41) is computed with the results shown in Table 1. In the last column of Table 1, the difference between the measured panel acceleration spectrum level re 1 g and the measured sound pressure level re 20tx Pa is given. This is also plotted in Figure 5. According to the criterion set by equation (41), below about 250 Hz, the panel input dominates the tile response; above 250 Hz the pressure drives the tile.
3.4. PREDICTED TILE RESPONSE
The net response of the tile, calculated from equation (34), is shown in Table 2 and plotted in comparison to the measured tile response in Figure 6. Note that the predicted and measured results are in reasonable agreement, except in the regions around 200 and 250 Hz where the predicted results fall significantly below the measured response. These frequencies are just below the resonance frequencies of the tile on the SIP material, which are computed to be about 300 Hz in rotation and 340 Hz in translation. Hence, the discrepancies could be due to an incorrect loss factor estimate, the non-linear stiffness
"1-
~ " 1 0 j ~ t I I I I I I I ~ ~ >, I 1
I "~ Predicted response "1
0 I:O - / ,,..
"~ o-i I I I I I I I I " - / " I 0 I00 200 300 400 500 600 700 800 900 I000
Frequency (Hz)
Figure 6. Measured and predicted vibration response of tile. - - , Measured response; ---O---, predicted response.
of the SIP, a significant rotational response not predicted by the analytical calculations, or some combination thereof.
4. COHERENT OUTPUT SPECTRAL ANALYSIS
The second analysis approach employed to identify the mechanisms causing the tile response involves the use of coherent output spectral calculations as detailed in references [1] and [2]. Specifically, the problem is considered to be a two input/one output system as shown in Figure 7. The first input p( t ) is the acoustic pressure over the top surface
n ( t )
p{t) ~ ~ P o(t) a ( f ]
Figure 7. Two input/one output system with uncorrelated inputs.
48 A. G. P I E R S O L A N D L. D. P O P E
of the tile as measured by the microphone shown in Figure 1, the second input re(t) is the vibration of the panel below the tile as measured by the panel accelerometer shown in Figure 1, and the output a(t) is the tile response as measured by the tile accelerometer shown in Figure 1.
4 . 1 . ANALYSIS A P P R O A C H
If the acoustic excitation and the panel vibration in the experiment were uncorrelated, the tile response due to the acoustic excitation and the panel vibration would be given directly by the coherent output power formulas [1]
Oa:p(jO = aa:a (.f)= y,~. ( f ) G . (.f), (42 )
where the colon (:) reads "due to". The spectral density and coherence quantities in equation (42) are as defined in section 2.2.
For the problem in hand, the acoustic pressure over the tile and the panel vibration under the tile are strongly correlated, as shown by the coherence data in Figure 8. This
1.0
"9
8 ~ . 0.5
o
I I 1 I I I I I
/x ,.-..._----"'- ..... . M
I # - tl
I I v I I I I I I
I
\
! I
0 IO0 z00 300 400 500 600 700 cO0 900 I000
F r e q u e n c y ( H z )
Figure 8. Ordinary coherence functions among measurements . B, = 20 Hz, na = 1800 averages. , Acous- tic noise versus panel vibration; - - - - , panel vibration versus tile response; - - - , acoustic noise versus tile response.
would be expected since the acoustic pressure over the tile and the panel vibration under the tile originate from the same source, namely the acoustic noise excitation used for the test. For the situation where two inputs are correlated, it is convenient to view the problem in terms of a conditioned two input one output system as shown in Figure 9. The first input p(t) is the acoustic pressure over the exterior surface of the tile as before. The second input a~.p(t) is the panel vibration below the tile with all linear contributions of the acoustic pressure over the tile removed from the data by least squares procedures: i.e., the dot (.) reads "with p removed". It follows that the two inputs in Figure 9 are
n ( t )
p(t) ~ ~ ) olt) a . . p ( t )
Figure 9. Two inpu t /one output sys tem with originally correlated inputs.
S P A C E S H U T r L E TPS T I L E V I B R A T I O N 49
uncorrelated and, hence, the problem reduces to a pair of single input/single output systems. It can be shown [1, 2] that the contribution of each input to the output a(t) is now given by
yL.p(/?G~ Ga:p(D ~--" y~CI')G.(D, a.:~.pCf) = (43)
where the conditioned terms in the second half of equation (43) are defined as follows:
vL.(/3 1G ""(/312 - G~.p(/)G..p(D'
G a.p(f) = [1- "~2a(jO]G'a(f),
G,~,,.p(f)= G,,,O0 G,w(DGp,,(D, G,,.p(D= [1-r~(D]G,,(D. (44)
The term y]~.p(f) is the conditioned (or partial) coherence function between or(t) and a (t) with the linear effects of p(t) removed, and G...(/O is the conditioned auto-spectrum of the output a(t) with the linear effects of p(t) removed.
4.2. ORDERING OF INPUTS
The coherent output power terms in equation (43) define all linear contributions of the two inputs p(t) and a(t) to the output a(t), that is,
G a ( j O = ~ ' a : p ( D " [ - ~ ' a : ~ . p ( j O 4 " ~ n ( f ) , (45)
where G,(f ) is the noise: i.e., all effects not accounted for by linear operations on p(t) and a (t), including other inputs, extraneous measurement noise and/or non-linear effects. This relationship can also be viewed in terms of the multiple coherence function between the two inputs and the output given by [1]:
~,a:ap(D= l__On(]~/~,a(D= l__[l__ 2 ",'~o (/')][1 --'/L..(f)]- (46)
Inherent in equation (45) is the assumption that the acoustic noise is responsible for that portion of the tile response due to the redundant (correlated) contribution of the two inputs. The panel vibration could also be made responsible for the redundant contribution to the response by simply reversing the order of the inputs to obtain
(47)
Of course, equation (47)~vill yield a substantially different indicated contribution from each input.
In many applications of conditioned coherence analysis, a physically meaningful order for the measured inputs is ambiguous or at least not obvious. A good example of this situation is the aeroacoustic experiment involving a flight vehicle instrument panel structure presented by Barrett [3]. In such cases, Bendat [2] has suggested that the correlated inputs should be ordered in terms of the ordinary coherence value between each input and the output, with the input producing the largest coherence being first in order. For the problem in hand, however, there is a physically meaningful basis for ordering the two inputs. Specifically, it is obvious that both the acoustic pressure over the tile and the vibration under the tile originate from the same source, namely the acoustic noise in the test facility. It follows that the acoustic measurement over the tile more accurately represents the source of the redundant component in the panel vibration, and hence should be first in order as given in equations (43) through (45).
50 A.G. PIERSOL AND L. D. POPE
4.3. COHERENT OUTPUT SPECTRAL RESUL;I'S The multiple and partial coherence functions computed between the two inputs and
the tile response are shown in Figure 10. Note that the multiple coherence function is relatively strong (y2:ap>0.90) at most frequencies below about 900Hz. The major exception is a drop in the multiple coherence at frequencies around 300 Hz. This is near the resonance frequencies of the tile on the SIP computed from the analytical model in section 3, and also near the frequencies of significant discrepancies in the analytical prediction in Figure 6. The drop in coherence might reflect the non-linear stiffness characteristics of the SIP and/or coupling of the translational and rotational vibration modes of the tile.
1.0
g
c
o.5 c
u
m
D
_ L
1 0 0
I I I I I I I /
/ "
~ V - t ~--~ I A;'% / ~,// ~, ;~ I
i I I I I I 200 300 400 500 600 7o0 800 90o iO00
Frequency (Hz)
Figure 10. Multiple and partial coherence functions among measurements. B, = 20 Hz, na = 1800 averages. -, Multiple coherence between tile response and acoustic noise and panel vibration; - - - - , partial coherence
between panel vibration and tile response; - - - , partial coherence between acoustic noise and tile response.
The results of the coherent output spectral analysis computed by using equations (43) through (45) are shown in Figure 11. At frequencies below 250 Hz, the results suggest that the tile response is due almost totally to the panel vibration. This is exactly the prediction made with the analytical model in section 3. At frequencies between 300 and
10
N I
v >,
U, 143 g "o
o
0-1 .~_ o
u
0 . 0 1 I ~" i i I I I i I 1 0 0 2 0 0 3 0 0 4 0 0 500 6 0 0 7 0 0 B00 9 0 0 1000
Frequency (Hz)
Figure 11. Decomposition of tile response. B e = 2 0 H z , n a = 1 8 0 0 ave rages . - - , Total response; , coherent acoustic induced response; - - - - , partial coherent panel vibration induced response.
SPACE SHUTTLE TPS TILE VIBRATION 51
800 HZ, the coherent output spectral data indicate the tile response is due primarily to the acoustic pressure over the tile and not the panel vibration. This is the frequency range where the auto-spectra in Figure 4 show the tile response substantially higher than the panel vibration. Again, this is consistent with the analytical predictions in section 3. Finally, above 800 Hz, the coherent output spectral data reveal a return of significant panel vibration contributions to the tile response. Again referring to the auto-spectra data in Figure 4, one sees that this is the frequency range where the panel response shows a dramatic increase, probably due to higher order panel resonances. It is reasonable to expect that the panel vibration would tend toward dominance at such resonances. This conclusion is supported by the results in Figure 5 where the difference ALp - S P L almost climbs over the criterion curve at these frequencies.
5. CONCLUSIONS
Both analytical predictions and coherent output spectral analysis of experimental data indicate the response of a typical Space Shuttle Thermal Protection System (TPS) tile to a simulated lift-off acoustic noise environment is dominated by vibration inputs from the panel below the tile at frequencies below 250 Hz, and by the direct acoustic pressures above the tile at frequencies above 250 Hz. It is believed that all the Space Shuttle tiles would display a similar behavior although the cross-over frequency between the domin- ance of panel vibration from below versus acoustic pressures from above may shift depending on the mass density of the tile and the thickness of the tile and SIP, as well as the relative levels of the acoustic pressure above the tile and the structural acceleration below it.
ACKNOWLEDGMENTS
The data used for the evaluations in this paper came from acoustic noise experiments performed by the Rockwell International Space Systems Division in Downey and B-1 Division in Los Angeles, California. The authors are grateful to Messrs F. W. Schmedes, D. D. Bert, G. P. Hetzel, J. P. Chevez, R. E. Hill and I. R. Lesyk of Rockwell International for their effort in the design and execution of the experiments and the basic data analysis. The analytical predictions were performed under a contract from the Jet Propulsion Laboratory, Pasadena, California. The authors are also grateful to Messrs Charles Lifer and Ben Wada of JPL for their support of this work.
REFERENCES
1. J. S. BENDAT and A. G. PIERSOL 1980 Engineering Applications of Correlation and Spectral Analysis. New York: John Wiley and Sons.
2. J. S. BENDAT 1976 Journal of Sound and Vibration 44, 311-325. Solutions for the multiple input/output problem.
3. S. BARRE'VF 1979 Shock and Vibration Bullethz 49(1), 43-58. On the use of coherence functions to evaluate sources of dynamic excitation.
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