Vibration response of spacecraft shrouds toin-flight fluctuating pressures

Journal o f Sound and Vibration (1974) 33(4), 399-425

VIBRATION RESPONSE OF SPACECRAFT SHROUDS TO IN-FLIGHT FLUCTUATING PRESSURES J. A. COCKnURNt AND J. E. ROBERTSON Wyle Laboratories, 7800 Governors Drive, West Huntsville, Alabama 35807, U.S.A.

(Received 12 July 1973, and ht revised form 1 November 1973) Results are presented for the vibration response of a spacecraft shroud to a range of inflight fluctuation pressures. An Atlas-Agena 15 degree cone-cylinder shroud was analyzed during the present study, and three critical flight Mach numbers were considered. At transonic Mach numbers considered during this investigation (Moo= 0-7 and Mo0 = 0.8), the aerodynamic flow over the shroud is complex, involving zones of regular attached flow, separated flow, shock wave oscillation and modified attached flows induced by local thickening of the boundary layer. The overall shroud vibration levels for a particular Mach number were determined by initially calculating the mean square acceleration levels induced by the fluctuating pressures distributed over an individual zone, and then summing mean square acceleration levels in one-third octave bands over all zones. Over most of the frequency range of interest, the vibration levels induced during transonic flight are considerably higher than the vibration levels induced during maximum dynamic pressure (qm~.) at Moo = 2.0. At frequencies well above the ring frequency of the shroud, however, this situation is reversed, and vibration levels during q,~x are higher than those during transonic flight. This is shown to be due to hydrodynamic coincidence effects where matching between the flexural and pressure wavelengths results in a number of near-coincident modes contributing significantly to the vibration levels. A discussion is given of the relative effects of the various fluctuating pressure environments distributed over discrete zones on the shroud surface. For the shroud analyzed, the separated flow at transonic Mach numbers contributes little to the overall vibration response in the lower third octave bands, most of the vibration being induced by the thickened boundary layer aft of the re-attachment point. At frequencies well above the ring frequency of the shroud, however, the overall vibration response is induced almost exclusively by the separated flow. Shock wave oscillation was found to contribute very little to the vibration levels of the shroud analyzed during this study, primarily because of the mis-match between the low-frequency spectral content of this environment and the lower bound frequencies of the shroud modes. For certain spacecraft shrouds or space vehicle structures where the structural modes have much lower resonant frequencies than the present shroud, the shock wave oscillation may possibly be more severe in terms of induced vibration response. 1. INTRODUCTION The primary function of a spacecraft shroud is to protect the spacecraft from aerodynamic, acoustic and thermal loads during launch and subsequent flight. Typical shrouds consist of a cone-cylinder body which extends aft of the spacecraft and is mated to the final stage of the launch vehicle. The in-flight random vibration of the spacecraft is caused primarily by the external pressure fluctuations which impinge upon the shroud. During launch and the initial phase of flight, the fluctuating pressures are associated with noise radiated from the propulsion systems, whereas during high speed flight, the pressure fluctuations are generated by unsteady oerodynamic flows over the vehicle. The characteristics of the acoustic environment at lift-off t Present Address: International Research and Development Co. Ltd., Fossway, Newcastle upon Tyne, NE6 2YD, England. 399

400

J . A . COCKBURN AND J. E. ROBERTSON

depend chiefly upon the size of the rocket engines, the exhaust deflector configuration, and the distance between the shroud and the base of the launch vehicle. During transonic flight, which for the purposes of the present study is assumed to extend from about M~ = 0.65 to _M,~= 1.0, the aerodynamic flow over the shroud begins to separate locally, and this is accompanied by the formation of local shock waves which tend to oscillate back and forth. The magnitudes of the external pressure fluctuations associated with the separated flows and the shock wave oscillations are highly dependent upon the shroud and vehicle geometry. When the vehicle accelerates through the maximum dynamic pressure phase offlight, normally occurring in the Math number range of M~ = 1-25 to M~ = 2.0, the aerodynamic flow over the shroud has, in most cases, progressed to a fully attached turbulent boundary layer. The resulting excitation of the shroud is caused almost exclusively by the pressure fluctuations beneath this attached flow. The external fluctuating pressure fields which surround the vehicle during flight cause significant vibration of the shroud, which in turn sets up an internal acoustic field around the spacecraft. The spacecraft receives further vibrational energy indirectly in the form of vibrations transmitted from the shroud via mechanical paths. In addition to the vibration environment resulting from direct acoustic and hydrodynamic excitation of the shroud, low frequency vibrations caused by excitation of the basic modes of the entire launch vehicle are transmitted upwards to the spacecraft via the vehicle structure and the spacecraft adapter. Because of the differences between the various in-flight fluctuating pressure environments, their often localized nature, and the fact that the shroud can be subjected to several of these environments simultaneously, it is essential that the vibration responses be accurately known during all phases of flight. The differences between these environments include not only the overall fluctuating pressure level and the power spectrum, but also the pressure correlation characteristics, since the spatial correlation of the fluctuating pressure field determines the degree of coupling between the excitation and the structure. The object ofthis paper is to present results which illustrate the effects of in-flight fluctuating pressure correlation characteristics upon the vibration response of the shroud. In addition, it is important from the standpoint of structural fatigue that the vibration response of the shroud is analyzed for each significant fight event, from launch to maximum dynamic pressure, since several of these flight events involve more than one type of fluctuating pressure environment impinging upon the shroud simultaneously; these results are also presented. The theoretical results presented here were derived for an Atlas-Agena launch vehicle fitted with a standard payload shroud comprising fiberglass skin and aluminum ring-frame stiffeners. The standard shroud is a 15 degree cone-cylinder body 1.676 m (66 in) diameter by 5.791 m (228 in) long; the length of the cylindrical section is 3.302 m (130 in), and the skin thickness tapers from 2"54 x 10-3 m (0.1 in) at the forward end to 3"556 x 10-3 m (0"14 in) at the aft end. These results are based on a larger study [1] which was carried out to investigate the problems involved in qualifying spacecraft systems. 2. IN-FLIGHT FLUCTUATING PRESSURE ENVIRONMENTS ~-.1. INTRODUCTION

A pre-requisite to the prediction of vibration response of a structure exposed to random ~ressure fields is the definition of the fluctuating pressure loads acting on the external surface. Fhe key statistical properties necessary for a complete definition of the fluctuating pressure oads are (i) the overall level, (ii) the power spectrum and (iii) the narrow band space correlaion coefficients along the principal axes of the structure. Once these characteristics have been lefined, the vibration responses can be computed by modal superposition or by statistical

VIBRATIONOF SPACECRAFTSHROUDS

401

energy methods. The remainder of this section is devoted to a discussion of the statistical properties of the various in-flight environments encountered by spacecraft shrouds of conecylinder geometry. 2.2. GENERALFEATURESOF THE AERODYNAMICFLUCTUATINGPRESSUREENVIRONMENTS Previous wind tunnel and flight data show that fluctuating pressures are proportional to free-stream dynamic pressure (q=) for a given unsteady flow phenomenon. However, peak fluctuating pressures do not necessarily occur at maximum q~ofor certain regions of a launch vehicle due to the non-homogeneous nature of the flow field. For example, regions of the vehicle exposed to separated flow and impingement of oscillating shock waves will experience fluctuating pressures at least an order of magnitude greater than regions exposed to attached flow. Thus, if separated flow and oscillating shock waves are present, say at Mach numbers other than the range ofq . . . . then peak fluctuating pressures will also be encountered at conditions other than at q. . . . Consequently, launch vehicle geometry is very important in the specification of fluctuating pressure levels since the source phenomena are highly geometry dependent, in addition to varying with Mach number and angle of attack. Practically all experimental data for unsteady aerodynamic flow have been acquired for bodies of revolution which are typical of missile or launch vehicle configurations. As a result of these studies, it is known that certain basic unsteady flow conditions will occur regardless of the detailed geometry of the vehicle. Furthermore, several fluctuating pressure environments having different statistical properties may exist over a portion of the vehicle surface at any given instant in the flight trajectory. The basic unsteady flow conditions are illustrated in Figure l, which shows the subsonic, transonic and supersonic flow fields for three common shroud geometries; a cone-cylinder body, a cone-cylinder-flare combination, and a cone-cylinder-boat-tail body. At high subsonic speeds (M.~ - 0.6), all three shroud geometries experience regions of attached and separated flow. The flow is attached over the conical section of each shroud shown in Figure 1, and separated flow is induced immediately aft of the cone-cylinder junction for cones having half-angles greater than approximately 15 degrees. Re-attachment occurs within approximately one diameter aft of the shoulder (depending on the cone angle) for the cone-cylinder and boat-tail configurations, but for the flare body, separation may continue over the flare. Subsonic

(b)

,o,

Transonic

Supersonic

Shoulder separation

Shock wave oscillation with separated flow

Attached flo~

Shoulder and flareInduced separation

Shock wove oscillation with flare- induced separated flow

Attached flow with floreInduced separation and shock wow oscillofion

Shoulderond boot-toilinduced separation

$hockwove oscilloiion with boot-tail -induced separation

Attached flow with b~t-toil-induced separation and shock wove oscillation

Figure 1. Subsonic, transonic and supersonic flow fields for three common shroud geometries. (a) Conecylinder; (b) cone-cylinder-flarecombinations; (c) cone-cylinder-boat-tail(bulbous).

402

J. A. COCKBURN AND J. E. ROBERTSON

Both the flare and boat-tail induce separation for typical shrouds. At high transonic speeds, the flow negotiates the shoulder of a cone-cylinder body without separating, reaches supersonic speed immediately aft of the shoulder and produces a near-normal, terminal, shock wave a short distance aft of the shoulder. The boundary layer immediately aft of the shock may or may not separate depending upon the strength ofthe shock wave. At transonic speeds, the boat-tail and flare region produce setJarated flow which may be accompanied by weak shock waves in the vicinity of the separation and re-attachment points. At supersonic speeds, the cone-cylinder bodies experience fully attached flow, as shown in Figure 1. For the flare geometry, the separated flow is now bounded by shock waves at the separation and re-attachment points, whereas for the boat-tail geometry separation occurs at the shoulder of the boat-tail (expansion region) and is bounded at the re-attachment point by a shock wave. It is evident that even simple geometries, such as cone-cylinders, produce complex and highly non-homogeneous flow fields at certain Mach numbers--particularly at subsonic and transonic speeds. The unsteady flow phenomena are of particular importance at transonic speeds, since in this range fluctuating pressures generally reach maximum values due to their proportionality to dynamic pressure. The remainder of this discussion is restricted to the definition of the fluctuating pressures existing at the surface of 15 degree cone-cylinder shrouds. M~= 0.7

T D/2

Zone I: separatedfbv

"-~ D/2 t

Zone2: modified afloched flow Zone 5 : modified ]ttached flow

Mm :0.8

Moo= 2.0

zgn~l J

Zone 2

.'.~

:! Zone 3: 7, modified ~. attached flow 3 .i\

dtached flow

], Final stage of launch vehicle

Zone I:attoched "flow Zone2:seporatedflow plus shockwave oscillation

Figure 2. Characteristicsof the aerodynamicfluctuatingpressures for a 15 degreecone-cylinderpayload shroud at three flightMach numbers.

Previous wind tunnel studies [2] have shown that for 15 degree cone-cylinder bodies, the most significant fluctuating pressures occur at Mach 0.7, Mach 0.8, and the flight Mach number corresponding to maximum dynamic pressure. Typically, maximum dynamic pressure occurs in the range of 1.25 ~
403

V I B R A T I O N OF S P A C E C R A F T S H R O U D S

previous wind tunnel studies by Robertson [2] involving a wide range of cone-cylinder geometries at transonic speeds. A typical result from these studies is shown in Figure 3, which describes the axial distribution of the fluctuating pressures over a 15 degree cone-cylinder model for a range of free-stream Mach numbers. Attachment ,~paration

01

i

i

I

0-1

I

~8

0-0(

"~. J~

00'( 0.02

I

I

I

ee

eee

!

f

I

~>8 OOE

0-0(

' " x .....,~. I

1

0-02

o-o-e-e-e. I

OOE 0.04

I

eo,,~ I

I

0-4 0"8 I-2 I-6 2 0

0 4 0 8 1 - 2 16 2-0

X/D (a)

'\ X/D

--~.~c-~;~

(b)

Oscillofing shock

0.1 ,-

""

I

I

I

I

006

~# ~ 1 7 6 0.02 9

2.o

X/D

(cl Figure 3. Axial distribution of fluctuating pressure for blunt-body separated flow; 15 degree conecylinder[2]. (a) M',o= 0-65; (b) M= = 0.7; (c) ?,t= = 0-8. At Mach 0.7, the flow separates at the cone-cylinder junction and re-attaches within onehalf of the shroud diameter. The flow continues to be attached over the remainder of the shroud since the transition from the shroud to the final stage (Agena) of the launch vehicle is continuous. However, this attached flow is characterized by a thickened boundary layer because of the flow separation upstream. The resulting turbulent boundary layer is approximately three times thicker than that resulting from regular undisturbed attached flow [1]. Moreover, along the flow axis the thickened boundary layer is divided into two distinct regions having different overall fluctuating pressure levels. These regions, or zones, extend from 0.5 D to 1.0 D and from 1.0 D to the aft edge of the shroud, respectively, as shown in Figure 2. Thus although these two zones have basically the same type of flow phenomenon (that is, a modified attached flow), the overall levels differ and, from a vibration analysis standpoint, it is convenient to consider them separately. At Mach 0.8, the flow negotiates the shoulder without separating, reaches supersonic speed immediately aft of the shoulder and produces a terminal shock wave as shown in Figure 2. The fl0w separates immediately aft of the shock wave at approximately 0.1 D, and re-attaches downstream at 0-4 D, where the flow continues to be attached over the remainder of the shroud. This downstream attached flow region again consists of local thickening of the boundary layer, similar to that for the previous Mach number. Interaction between the separated flow and the foot of the terminal shock wave causes the shock to oscillate back and forth; the portion of the shroud which is affected by this oscillatory movement of the shock may be considered to extend from 0.1 D to 0.4 D. As a result, this section of the shroud, designated zone 2 in Figure 2, is subjected to the combined

404

J . A . COCKBURN AND J. E. ROBERTSON

influence ofthe separated flow and the shock wave oscillation. For free-stream Mach numbers greater than Moo -- 1.0, the flow is attached over the whole of the cylindrical section of the shroud, and local thickening of the boundary layer does not occur. Generally, shock wave oscillation produces the most intense fluctuating pressure levels encountered by a launch vehicle. Typical shock waves encountered by launch vehicles include terminal shocks, displaced oblique shocks, re-attachment shocks and impingement shocks. However, all shock waves may be expected to produce similar fluctuating pressure environments since the movement of the shock wave results from interaction with the turbulent boundary layer [4] and the fluctuating pressure is the result of the motion of the shock wave. 2.3.

OVERALL LEVELS

The overall (or broadband) fluctuating pressure level is generally expressed in terms of the root-mean-square fluctuating pressure level, V ~ , normalized by the free-stream dynamic pressure, q=. The effects of Mach number on the normalized r.m.s, intensities of the fluctuating pressures in attached flows have been reported by several investigators [4, 5]. For the Mach number range for which wind tunne, and flight data are available, the fluctuating pressure varies from 7V~/q~ ~ 0.006 at subsonic Mach numbers to 0.002 at supersonic Mach numbers in the vicinity of M~ = 3.0 [4]. Lowson [5] derived the following semi-empirical equation which agrees reasonably well with the general trend in the data:

q'-~/q~ = 0.006](1 4- 0"14M2).

(1)

An important feature ofthis formula is that it has some theoretical basis and is not strictly an empirical approximation of measured results [5]. The variation of overall fluctuating pressure level with free-stream M a t h number for various separated flow environments downstream of expansion corners is illustrated in Figure 4. The regions aft of cone-cylinder junctions, and rearward-facing steps and in the near wake of boat-tail configurations are represented by the data presented in Figure 4. The variation of x/-~/q~ for attached flows is also shown in this figure for the purposes of comparison. In general, the largest fluctuating pressure levels occur at low Mach numbers, and decrease as free-stream Mach number increases. These data represent the region of plateau static pressure and the tolerance brackets on the data represent the variations due to nonhomogeneous flow within the region of constant static pressure rather than scatter in the measurements [6]. 0.06

i

I

l

i

I

i

I

i

I

i

I

0-05 0.04 003 0-02 0'01 9

0 04

0 8

I-2

1"6

2-0

2-4

2"8

3-~

M~ Figure 4. Variation of normalized fluctuatingpressure level with free-stream Math number for expansion induced separated flows [6]. A, 34 degree boat-tail; o, 25 degree cone-cylinder; ~, 30 degree cone-cylinder; i, 17-5 degree boat-tail;l, rearward-facingstep;------, attached turbulent boundary layer, equation (1). The axial distribution of r.m.s, fluctuating pressure resulting from terminal shock wave oscillation [2] is shown in Figure 3. A special case of terminal shock wave oscillation results when the terminal shock wave moves forward to the expansion shoulder of a cone-cylinder.

~1~ o , o

i --~

LO

I~...I

0

0

II

I o~ o l

kO~

I

~D

I ~

~~1 666

~0

I ~6 6 ~

iI ii~iI 0

0

' o

66

~

-

~

= o

,-I

o

0

=

0

0

o 0

0

~

o~~.

0

N

~o

~

o

"~

0

"~ ~oO

o

406

J . A . COCKBURN AND J. E. ROBERTSON

For this case, the flow intermittently fluctuates between the blunt-body separated flow condition and the attached flow condition at high subsonic (low transonic) Mach numbers. This condition represents an alternating unbalance between the large pressure rise through the shock wave that exceeds the values required to separate the flow and the small pressure rise that is too small to maintain fully separated conditions. As Mach number is increased above the range of alternating flow, the localized oscillation of the shock wave produces intense fluctuating pressures for the region in close proximity to the shock wave as shown in Figure 3. The shock wave moves aft with diminishing strength as the Mach number is increased, and the r.m.s, fluctuating pressure levels also decrease. In defining the overall fluctuating pressure levels for the 15 degree cone-cylinder shroud Robertson's model data [2] were used extensively, since the average levels in each zone were the primary requirement for subsequent vibration response studies. The average overall fluctuating pressure levels existing over the various zones for the three Mach numbers considered are summarized in Table I. These fluctuating pressure levels are consistent with the axial distribution of the fluctuating pressures shown in Figure 3. For the Mach 0.8 case, the average total fluctuating pressure level over zone 2 due to the shock wave oscillation superimposed on separated flow was taken to be Vr~/q~o = 0.06. For separated flow in the absence of the shock, the fluctuating pressure level is, from Figure 4, approximately 0.026; thus the fluctuating pressure level due to the shock alone is 0.054. For the Mach 0.7 free-stream conditions, the average total fluctuating pressure level over zone I was taken to be V~[qo o = 0.03__(see Figure 3); this includes the contribution from homogeneous separated flow (%/P/qoo ,~,0-026) plus a small contribution from a weak shock due to re-attachment of the flow ( V~[qr ,~, 0"015). 2.4.

POWER SPECTRA

Power spectra, which represent the frequency distribution of the mean square fluctuating pressure, have been found to scale on a Strouhal number basis: that is, the frequency is normalized by multiplying by a typical length and dividing by a typical velocity. A semiempirical equation for the power spectrum of the fluctuating pressures beneath a homogeneous attached turbulent boundary layer has been proposed by Lowson [5], based primarily upon the experimental results of Speaker and Ailman [7]. In comparing this equation with other data, and in particular, with recent measurements at supersonic speeds, Robertson [6] found that the Lowson prediction underestimated the spectral levels at low Strouhal numbers and also gave too large a roll-offat high Strouhal numbers. A refined equation was therefore developed [6], which appears to be more representative of the experimental findings throughout the Mach number range. The power spectral density, non-dimensionalized by local velocity and local boundary layer thickness at the point of consideration, and the free-stream dynamic pressure, is given by

9 (f) v,

2

2

I1

(P /q~o)a

(2)

where the superscript H denotes homogeneous flow conditions, and the subscript A denotes attached flow. In the above relation, (p2[q~)~ represents the normalized overall mean square fluctuating pressure beneath the homogeneous attached flow (see equation (1)), and the characteristic frequency, fo, is equal to 0.346 Ud6~. The most comprehensive available data for the power spectrum of the fluctuating pressures within separated flows have been obtained for the homogeneous region of compression corners at supersonic M ach numbers [4, 8, 9 and 10]. All of these measurements, which were obtained

407

VIBRATION OF SPACECRAFt S H R O U D S

for forward facing steps, wedges, and conical frustums, showed a distinct similarity in spectral characteristics when compared on the basis of spectrum level and frequency, nondimensionalized by local velocity (U0, free-stream dynamic pressure, and local boundary layer thickness (~) [6]. Specifically, it was found that the power spectrum of the fluctuating pressures within the homogeneous region of separated flows was given by the following relation:

9 (f) U,

(p2/q~)~

(3)

{l + ~/fo) ~ where the superscript H again denotes homogeneous flow conditions, and the subscript S denotes separated flow. In the above relation (P/q=)s 2 z , represents the normalized overall mean square fluctuating pressure beneath the homogeneous separated flow, and the characteristic frequency, fo, is equal to 0.17 Ut/(5~. Comprehensive data concerning the spectral characteristics of shock-wave oscillation have only recently become available. Experimental work by Coe and Rechtien [9], involving supersonic flow separation ahead of a 45 ~ wedge, has given a clearly defined spectrum for shock wave oscillation at M~ = 2.0. However, power spectra for other Mach numbers have not been published. Nevertheless, experimental results obtained for three-dimensional protuberance flows [I 1] generally show good agreement with the Coe and Rechtien data. The normalized power spectra for shock-wave oscillation for both two- and three-dimensional protuberances [9, 11] are presented in Figure 5. The power spectra show a relatively steep roll-off starting at a Strouhal number (ftSo/Uo) of 1 x 10-2, where the subscript 0 denotes I*0

IO-'

16 z

i o -4

10"3 Sfrouhol

I0 -z

No. f ~o/Llo

I0-I

I'0

I0

hosed on cond~fJons upsfreorn of shock w(~ve

Figure 5. Comparison between power-spectra for shock-wave oscillation induced by two- and threedimensional protuberances. , Curve through Coe's data [9] for 45 degreewedge,A/~ = 2.0; [2, cylindrical protuberance [11], M~ = 1.4; e, cylindricalprotuberance lll], M= = 1"6.

J. A. COCKBURN AND J. E. ROBERTSON

408

local velocity and boundary layer thickness immediately upstream of the shock wave. The roll-offis 8 dB per octave for the range 1 • 10-2 <<.f6ofUo~<2 • 10-a and above this range the roll-offchanges suddenly to 4 dB per octave. These unique spectral characteristics of shockwave induced fluctuating pressures are explained by the physical behavior of the shock-wave oscillation and the resulting pressure time history. The shock wave is basicaily a pressure discontinuity which becomes slightly distorted by the boundary layer such that a finite gradient through the shock wave is observed at the surface. Oscillation of the shock wave produces a wave form which approaches a random-rectangular wave as the displacement of the oscillation increases. Superimposed upon this signal is the low amplitude, high frequency disturbance associated with the attached boundary layer (for that portion of the signal when the shock wave is aft of the measurement point) and the moderate amplitude and frequency disturbances associated with separated flow (for that portion of the signal when the shock wave is forward of the measurement point). The roll-off rate of the power spectrum for a random-rectangular wave form is 6 dB per octave which is 2 dB lower than the experimentally observed value. Abovef6o[Uo= 2 x 10-a, the power spectral density for the random modulation of the shock wave diminishes below the power spectral density for the turbulence portion of the signal. Thus, the roll-offrate changes to a value roughly equal to that for separated flow since this environment is the larger of the two turbulence generating mechanisms (the other being attached flow). Coe's results for supersonic flow separation ahead of a wedge [4] are shown in Figure 6. The longitudinal distribution of pressure fluctuations, shown in the upper half of Figure 6, displays a sharp peak at the foot of the shock wave, with non-homogeneousattached and separated flow conditions immediately upstream and downstream of the shock, respectively, This non-homogeneity may result from intermittency of the shock wave oscillation or from a more basic modification to the turbulence structure of attached and separated flow due to the motion of the shock wave. The variation in the power spectrum with position relative to the shock wave is clearly evident in Figure 6. Power spectra for homogeneous attached and separated flows are also shown in Figure 6 to illustrate the presence ofadditional low frequency energy due to the shock wave. Robertson [6] carried out an extensive investigation ofthe power spectra in the vicinity of the shock foot and in the regions ofn0n-homogeneous attached and separated flows. As a result of this study, empirical equations were derived for the power spectra beneath the shock, and immediately in front of and behind the shock. It was found that for peak overall levels of shock wave oscillation (corresponding to a point located at the mean position of the shock wave on a 15~cone-cylinder) the contribution of the non-homogeneous attached flow is negligible in comparison to that for non-homogeneous separated flow. For the 15~ cone-cylinder shroud considered during this study, the average fluctuating pressure environment over zone 2 for M= = 0.8 consists primarily of non-homogeneous separated flow: that is, the fluctuating pressures due to homogeneous separated flow plus the contribution from the shock. The power spectrum for the fluctuating pressures over zone 2 can be determined from the following relation, derived by Robertson [6];

('ff~lq~)~

It'(f) Uo] Nn =

q~6o

Js

(~O~

+-

(4)

+(.f]fo)os3}T M

where the superscript NH denotes non-homogeneous, and the subscripts S and SW denote separated flow and shock wave, respectively. In the above relation, (fi~/q=)~nrepresents the

409

VIBRATION OF SPACECRAFT SHROUDS

total mean square fluctuating pressure beneath the non-homogeneous separated flow: i.e., the total of the contributions from the homogeneous separated flow and the shock wave. The characteristic frequency, fo, (for the portion of the power spectrum defined by the shock wave), is equal to 0.001 [lo/6o. The power spectrum derived from equation (4) is plotted in Figure 6 and shows good agreement with Coe's results for locations immediately behind the shock. The empirical equations for homogeneous attached (equation (1)) and separated (equation (3)) floware also plotted in Figure 6 for the purposes of comparison, though experimental results for the homogeneous separated flow immediately ahead of the wedge are unavailable.

0,08

II

i

I~

~s

i

i

i (a)l

2

I

u..z.o

0"04

~

0

5

4

3

o

Xs/h I

i i I lliiiI

I I|1111

i

a a llir~

--.':., Eq. (4)

(b)

=-

"Z

i

n6z

~,B ~. ~o v

163

- **~o o~176 ~o-'~.,,

=.__ ~

Eq() \

~ -%.

o ~ __..

***

_~

-., \

_%_ ~_

~

-

-% V~,

tt6 ~L

:

_-"

"-X E

l~

~

\

"

-

\

E Iff~ t 10-3

i wllud 10-z

Strouhal NQ f S o / U o ,

a I 01vwvd I0"n

J 0 al0tllI ~10n 1.0

I0

based on c o n d i t i o n s upstream of shockwave

Figure 6. Longitudinal distribution of pressure fluctuations and typical power spectra in vicinity of supersonic flow separation ahead of a 45 degree wedge [4]. e, Attached turbulent boundary layer; <>,non-homogeneous attached turbulent boundary layer; vzx, non-homogeneous separated flow; i , shock wave oscillation.

410

J . A . COCKBURN AND J. E. ROBERTSON

2 . 5 . NARROW BAND SPACE CORRELATION COEFFICIENTS

The final requirement in determining the characteristics of the fluctuating pressure fields is to define the narrow band, space correlation function, or co-power spectral density, along the longitudinal and circumferential directions of the shroud. This parameter is the key function needed to describe an impinging pressure field in order to calculate the induced mean-square vibration response of the structure. Measurements by several investigators [5] have shown that the co-power spectral density of turbulent boundary layer pressure fluctuations in the direction of the flow can be approximated by an exponentially damped cosine function, and the lateral co-spectral density can be approximated by an exponential function. Under the assumption that the pressure field is homogeneous, in the sense that the correlation coefficient is a function only of the separation distances and independent of the actual position on the structure, and also that the correlation function is separable into longitudinal and lateral (circumferential) components, the following empirical equations define the narrow band correlation coefficients [6]: C((; co) = exp

-

[

C(~; co) = exp -

Uc + 0"27 ~-71j j cos -u-c '

0"72--~-- + 2.0

,

longitudinal,

circumferential,

(5)

(6)

where ~ and tl are the separation distances along, and at right angles to, the flow direction, respectively, U~ is the convection velocity and 6~ is the local boundary layer thickness. To account for the variation'of convection velocity with frequency, Bies' empirical equation [12] may be used. It should be noted that the assumed separable form for the correlation function leads to the prediction that the magnitude is constant along straight lines on the surface, forming a diamond pattern surrounding the origin. In fact the lines of constant broadband correlation usually swell out away from either the lateral or longitudinal directions [5]. However, experimental results for the amplitude of the narrow band cross-spectrum at an angle to the flow agree reasonably well with the separable form, at small spacings [5]. Since the use of the separable form provides considerable analytical simplification for the vibration response predictions, it was adopted for the present study. Cross-power spectra for the homogeneous region of two-dimensional separated flows have been investigated by Chyu and Hanly [8], and Coe and Rechtien [9]. Their results have shown that the longitudinal space-correlation in attached flows is significantly higher than that in separated flows. Conversely, the lateral space-correlation in attached flows is lower than that in separated flows. The spatial coherence exhibited by separated flow was found to be similar to that of attached flow; however, the exponential decays which represent the envelope of the coherence functions, at various spatial distances, vary with free-stream Mach number and are applicable only at high frequencies because of a loss ofcoherence at low frequencies. This loss of coherence precluded a general collapse of data with a single decay constant; however, a non-dimensional attenuation coefficient, related to the normalized modulus of the crosspower spectral density, was derived [6]. Empirical approximations of the attenuation coefficient, based upon the results reported in reference [9], have been derived by Robertson [6], and are summarized in Table 2. The resulting normalized correlation coefficients for homogeneous separated flow may be expressed as follows: C [(;-U-7) = exp [-~:(]c~ u T '

longitudinal,

(7)

411

VIBRATION OF SPACECRAFT SHROUDS

(

fS,~

C tr/;-~-~/] = exp [---a,r/],

circumferential,

(8)

where ctc and ct, are the attenuation coefficients in the longitudinal and circumferential directions, respectively, as shown in Table 2. Empirical approximations of the convection velocities, which have been found to vary considerably with frequency [9], are also given in Table 2. TABLE2

Longitudinal and lateral attenuation coe~cients and convection velocity as a fimction of Strouhal number for homogeneous separated flow [6]. Note: (f~l/Ul)o = 0.006 Attenuation coefficients ~; and ct, (m-z)

Convection velocity Uc (m/s) U~

<0"006

o.ooo to

29.5

29-5

29.5

006 >0.06

29.5

<0.1

/t-o-://

o.,

[, tV,/oj

, o

59.0

to

0-3 Ul

.

U, 0"3.-t- l~

>1.0

t--~-')

}

0.8 Ut

Verylittle data has been published in the form of cross-power spectra of fluctuating pressures beneath oscillating shock waves. Because oscillating shock waves at a given flight condition are confined to relatively small areas of the vehicle surface, it is extremely difficult to define the spatial characteristics of the attendant fluctuating pressures. The only significant results on the spatial coherence of fluctuating pressures in the vicinity of shock waves are those obtained by Coe and Rechtien [9]. Their results indicate that the fluctuating pressures generated by the shock wave are related only at frequencies belowf6o/Uo = 0.08, for the region immediately downstream of the mean location of the shock wave. For the region immediately upstream of the shock wave, a small degree of coherence is also evident in this frequency range as well as atf6o/Uo >i 0.2. Robertson [6] fitted an empirical equation to the coherence data of reference [9] and derived the following narrow band longitudinal correlation coefficient: C(~; o9) = exp [[ - 3.18 ~Uo J] cos ~-o' ~

longitudinal.

(9)

Published data are not available on the transverse spatial characteristics of shock-induced fluctuating pressures. However, it is anticipated that these disturbances will be reasonably correlated over much larger distances in the transverse direction than in the longitudinal direction because of the continuity of the shock wave in the plane normal to the flow. It appears physically reasonable to assume that because of vehicle symmetry the correlation coefficient will be unity over approximately one-quarter of the shroud circumference. Thus in the circumferential direction, the shroud would be subjected to four un-correlated loads of equal amplitude.

412

J.A. COCKBURNAND J. E. ROBERTSON

2.6. PREDICTEDIN-FLIGHTFLUCTUATINGPRESSURES Fluctuating pressure spectra for the Atlas-Agena 15 ~ cone-cylinder shroud were computed for flight Mach numbers Moo = 0.7, M= = 0-8 and Moo = 2.0, based upon the prediction methods discussed earlier. Dynamic pressure, altitude and Mach number time histories for the Atlas launch vehicle [3] were utilized in these calculations and the reference point on the shroud for the calculation of the boundary layer thickness was in each case taken to be the mid-point of the appropriate fluctuating pressure zone (see Figure 2). The predicted spectrum levels, converted to one-third octave band levels, are shown in Figure 7. The highest fluctuating pressure levels during flight occur in zone 2 of the cylindrical section at Mach 0.8; this particular zone is subjected to the combined influence of the separated flow and the shockwave oscillation. Significantly high fluctuating pressure levels occur in all three zones of the cylindrical section at Mach 0.7, while at maximum dynamic pressure, the fluctuating pressure levels are substantially lower in the low frequency region. The results shown in Figure 7 illustrate the significance of the transonic flight regime, and in particular, the high fluctuating pressure levels at low frequencies. 150

l

I

I

i

i

i

I

I

i

i

i

i

I

i

i

i

i

i

i

i

i

I

i

i

i

i

i

i

e "x.. N E

140

o

"~

= a

Z

130

-

~ . . . . . . . " . . - --"-7" " ' - . . . . . . . ~--'-.L~-.y.~" - --'~-~"~'~'~" ............

~ . _ _ _ _ I : ~ : . - -

--

. . . ." " - ' - - - - . " ~ -~ -'-'~""

. ....................

~

"---

..~

x od

m

120

~=

llo

4--

,oo

........

~---"

""

o..$5.---

i

b ~;'-J

d.i

-t

90

O 80

8

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I--16 31 63 125 250 500 100 2000 4000 8~ 00 One third octove bond center frequency (Hz)

M~=0-7 a, Zone I b, Zone 2 c,Zone 3

Meo--0 . 8 d,Zone I e,Zone 2 f,Zone 3

Figure 7. Predicted one-third octave band fluctuating Atlas-Agena shroud for three flight Mach numbers.

pressure

Moo=2-0 g, Whole cylindricol section

levels over

the cylindrical

section

of the

In the lower third octave bands, two of the spectra shown in Figure 7 (curves a and e) exhibit increasing fluctuating pressure levels with decreasing frequency. In the case of curve e, this is due to the effects of the shock-wave oscillation superimposed on the separated flow; in fact this spectrum peaks at a very low frequency, on the order of 3 Hz. Similarly, in the case of curve a, this effect is caused by the re-attachment of the flow at the aft edge of zone 1. Low frequency axial oscillation of the re-attachment point induces fluctuating pressures similar to those produced by shock-wave oscillation. Again this spectrum peaks at a very low

VIBRATION

(.4 E .%. ~)

150

~

l

I

I

I

I

OF I

I

I

I I

SPACECRAFT I

I

I

t

413

SHROUDS

I I I I I I I I I I I I I 11~

140

0

x 130

0

-

0

~j

120 -

"~E

~

__-_g / I / / /

I10

.o ,oo 8 Z

0

90

i

80 8

I

t6

I

I I I I I I i

51

65

i i i

125

i i

250

One - t h i r d o c t a v e bond r

i

I i

I

500

I I I I

I I

1(30 200

I I

4000 8000

frequency (Hz)

Figure 8. Comparison between extrapolated in-flight measurements and predicted fluctuating pressure spectra. , Predictedrange of fluctuatingpressuresfor transonic flight;---, predictedfluctuatingpressures for q~,,(M,~= 2.0); O, measured fluctuating pressures during transonic flight; e, measured fluctuating pressures at q.... Data from reference [13]. frequency. These relatively high fluctuating pressure levels in the lowest third octave bands are for the most part insignificant in terms of inducing radial shroud vibrations. Published in-flight measurements of the external fluctuating pressures acting on the AtlasAgena shroud are unavailable. Most of the in-flight fluctuating pressure measurements which have been published were obtained from Titan and Saturn launch vehicles [I], both of which have trajectories which differ substantially from the Atlas-Agena trajectory. However, a qualitative comparison between the predicted spectra and in-flight measurements may be obtained with the aid of internal sound pressure level measurements in conjunction with measured noise reduction through the shroud. Internal sound pressure levels measured within this type of shroud were reported by Williams and Tereniak [13] for a Delta launch vehicle (having a similar trajectory) during transonic and qmaxflight conditions. In addition, the noise reduction spectrum for the shroud was obtained from external and internal measurements during launch. Based upon the measured internal sound pressure levels and the noise reduction spectrum, the external fluctuating pressure levels during transonic and q,,ax conditions have been extrapolated as shown in Figure 8. In this figure, the extrapolated measurements are compared directly with the predicted spectra for transonic and qm~. conditions. However, some caution must be exercised in interpreting the results shown in Figure 8. It must be emphasized that the extrapolated measurements are based upon the shroud noise reduction which was measured during launch, where the excitation is primarily a reverberant acoustic field. In fact, laboratory and launch pad measurements have shown that for a given shroud, the noise reduction spectra vary significantly with the pressure correlation characteristics of the acoustic environment applied to the shroud [1]. Nevertheless, the comparison illustrated in Figure 8 shows that qualitatively the predicted in-flight fluctuating pressure spectra are in fair agreement with the extrapolated measurements. 3. IN-FLIGHT VIBRATIONS 3.1. THEORETICAL APPROACtt The vibration responses of the 15~ cone-cylinder shroud were predicted by using modal superposition. Since the cone-cylinder junction coincided with a heavy ring-frame stiffener which would effectivelyde-couple the conical and cylindrical sections, the shroud was analyzed

414

J . A . COCKBURN AND J. E. ROBERTSON

in two stages: the lower cylindrical section, and the upper conical section. Only the results for the cylindrical section are reported here. The lower section of the shroud was treated as a simply-supported cylindrical shell for the purpose of computing resonant frequencies and mode shapes, and thc stiffeners were accounted for by computing equivalent orthotropic flexural rigidities. In predicting vibration response by superposition of the responses of the normal modes, it is implicitly assumed that the mean-square response amplitude of each mode can be obtained independently, and that the summation ofthese mean-square responses is insensitive to coupling between modes. In fact, the total mean-square response of a structure at any point depends upon the summation of the mean square modal responses and upon the summation ofthe cross-correlations between pairs ofmodes. The latter cross terms are in some cases significant; however, each term in this summation becomes equal to zero if the space average of the mean square response is obtained. The cancellation of modal cross-correlations for space average response is due to orthogonality between the modes. The analysis of structural response to random pressure fields by modal superposition was initially formulated by Powell [14, 15]; detailed results were derived for the response of structures to plane acoustic waves and to a two-dimensional reverberant acoustic field. The theory was extended to predict the response of panels to turbulent boundary layer pressure fluctuations by Wilby [16], and to a three-dimensional reverberant acoustic field by Crocker and White [17]. More recently, this work has been extended to predict the responses of cylindrical shells to random pressure fields [18, 19, 20]. It can be shown that Powell's final result for the space-averaged radial acceleration of a cylindrical shroud is

=

~

tim. H2

9j~2.(co),

(10)

al.l ?1=0

where S~(to) is the acceleration mean-square (power) spectral density in gZ/Hz, S~(to) is the pressure mean-square (power) spectral density in (N/m2)2/Hz, tl is the mass per unit area of the shroud surface in kg/m 2, ./.2 is the joint acceptance of the mn shrou d mode, and g is gravitational acceleration in m[s 2. The remaining dimensionless terms, tim, (introduced by the space averaging), and (the magnification factor of the mn mode at frequency co) are defined as follows:

H(o~.../w)

fl,.. = 2,

m = 1,2 . . . . , 4n 4

= il + nZ)2'

n=O,

m = 1,2,...,

n = 1,2,3 . . . . .

where Q,,, is the magnification factor at resonance, co.... of the mn shroud mode, and the subscripts iii and n refer to the number of axial half waves and the number of full circumferential waves, respectively. The joint acceptance term, j~,(co), refers to the direct joint acceptance of the mn mode of the shroud and is generally assumed to be separable into m (axial) and n (circumferential) components, such that j~,(to)=j2(o~).j~(to), as follows: L=

I

j~(co)=~--~ j" x-O

Lx

fC(~,to).~,,(x).~,.(x')dxdx', x'-O

(11)

VIBRATION OF SPACECRAFT SHROUDS Ly

u

415

L:,

y'-O

In the above relations, ~,,(x) and r represent the axial components of the mode shapes at points x and x' on the shroud, while ~,(y) and ff,,(y') represent the circumferential components at points y and y' on the Shroud. The quantities Lx and Ly denote the axial and circumferential lengths of the shroud, respectively, and the terms .C((; to) and CO1; to) represent the axial and circumferential narrow band space-correlation coefficients of the particular fluctuating pressure environment. Axial separation distances (x - x') are denoted by (, and circumferential separation distances (y - y ' ) by r/. Since the acceleration power spectral density, defined by equation (11 ), has been normalized by the pressure power spectral density, the problem of predicting the vibration response essentially reduces to the determination of the joint acceptances, j2 (to), for each fluctuating pressure environment. Computation of the response on this basis therefore provides a convenient means of comparing the effects of different pressure correlation characteristics upon vibration response. For conversion to absolute vibration response levels, the normalized response given by equation (11) is simply multiplied by the power spectral density of the fluctuating pressure field. For a particular structure and fluctuating pressure field, the joint acceptances are evaluated by substituting the axial and circumferential mode shapes and narrow band correlation coefficients into equations (11) and (12). Closed-form expressions for the joint acceptances of cylindrical structures to a number of random pressure fields have been published by several investigators [18, 19, 20]. For cylindrical structures subjected to localized random pressure fields, Bozich and White [20] derived joint-acceptances for attached turbulent boundary layers, a reverberant acoustic field and a progressive wave acoustic field. It can be seen from equations (1 I) and (12) that for a given structure the joint acceptances are determined by the functional form of the narrow band space correlation coefficients. For the separated flow environment considered during the present investigation, the functional forms of the narrow band space correlation coefficients (equations (7) and (8)) are similar to those for attached flows (equations (5) and (6)); thus the closed form solutions for the joint acceptances to attached flows derived in reference [l 8] are applicable to separated flows, with appropriate modifications to the constants. Similarly, for shock-wave oscillation, the functional form for the longitudinal correlation coefficient (equation (9)) is similar to that for attached flow; circumferentially, the joint acceptance is identical to that for a fourducted progressive wave acoustic field derived by White [18]. Vibration response spectra were computed with the aid of a modal analysis computer program [21]. This computer program was developed for the prediction of plate and shell responses to in-flight fluctuating pressures, rocket noise, reverberant acoustic fields and grazing incidence and normal incidence acoustic waves. 3.2. PREDICTEDSHROUDVIBRATIONSPECTRA Normalized shroud acceleration spectra for the various fluctuating pressure environments at Mach 0.7 are shown in Figure 9. Each spectrum represents the space-average normalized acceleration response, in g2/(N/m2)2, of the whole cylindrical section of the shroud when exposed to a single in-flight fluctuating pressure environment concentrated over a particular zone. Results for the conical section of the shroud have not been shown, since this region of the shroud sees only attached flow throughout the flight Mach number range, and the vibration amplitudes were found to be significantly lower. At low frequencies, the pressure fluctuations beneath the separated flow in zone 1 are observed to be relatively inefficient in terms of forcing the overall shroud, whereas at high frequencies the converse is true. Below about 600 Hz, the

416

J. A. C O C K B U R N A N D J. E. ROBERTSON

response spectra are distinctly modal in character, and peak responses occur at the ring frequency, corresponding to the fundamental "breathing" mode of the shroud at approximately 630 Hz. In the vicinity of the ring frequency, the modal density of the shroud is a maximum. Above the ring frequency, the modal density becomes essentially constant and the acceleration levels are controlled primarily by hydrodynamic coincidence effects. The relative levels of the three acceleration spectra shown in Figure 9 are governed by two important A (M

10_3

i

I

i

i

I

I

Z

n-

co-'

@ j o -~

/I 4

~

_

8

_

.~t:3

i~'

\ \

/

I I J./Coincidence frequencylimits //I IV for excitation over zones2ondI

I

-

l

V

t~

J

I~

"1

Coincidence frequency limits for excHolion overzone I

3

_

-

o

/ z

10-9 I0

1130

I000

10000

Frequency (Hz)

Figure 9. Predicted normalized acceleration spectra for M| = 0.7. - - - , Acceleration induced by separated flow over zone l; . . . . . , acceleration induced by modified attached flow over zone 2; - - - , acceleration induced by modified attached flow over zone 3. phenomena: the first of these is the fraction of the shroud area over which the particular excitation is applied, and the second is the spatial correlation of these fluctuating pressures. For frequencies up to about 2000 Hz, the modified attached flows in zones 2 and 3 are the major contributors to the shroud vibration levels. Above 2000 Hz, the separated flow in zone 1 becomes the major contributor to the vibration levels. Several frequencies have been identified on the abscissa of Figure 9 for the purpose of clarification:f12 denotes the lowest shroud mode (m = 1, n = 2) at 120 Hz,f, denotes the ring frequency (at 632 Hz), andf~ denotes the acoustic criticalfrequency for the shroud at approximately 5220 Hz. The acoustic critical frequency is the lower limit for acoustic coincidence, corresponding to coincidence for grazing incidence acoustic waves [22]: i.e., wavelength matching between the pressure wavelength and the modal wavelength of the structural mode. The acoustic critical frequency is defined by the relation

c~ /p

f~ = 6"23 rq 1)='

(13)

where Co is the speed of sound in m/s, p is the mass per un;t area of the shroud surface in kg/m 2, and D= is the fiexural rigidity for axial bending in Nm. Similarly the hydrodynamic

VIBRATION

OF SPACECRAFT

SHROUDS

417

coincidencefrequency is defined by

u? / f"c = g:i3 d

(14)

where Uc is the convection velocity in m/s and the other symbols are as defined above. Since each frequency is associated with a unique convection velocity, however, f . c is not a lower limit for coincidence. The effects o f coincidence, including its dependence upon convection velocity, will be discussed later. I0

I

I

II

I

I

I I I I

I

I

!

l,l l 1000

i

J

[/"\ > c . 0i O

r

.= E

i 0 -~ _

~J

e

0

I

id 3 -m

o

s 0-4

10

!,v Jf 100

I

I

i

10000

Frequency(Hz)

Figure 10. Predicted mean square acceleration levels in one-third octave bands for M.~ = 0-7. - - - - - , Acceleration induced by separated flow over zone I ; , acceleration induced by modified attached flow over zone 2; . . . . , acceleration induced by modified attached flow over zone 3; . . . . . , total mean square acceleration at 3,/,0 = 0.7. Corresponding mean square acceleration levels in one-third octave bands are shown in Figure 10 for the shroud at Mach 0.7. These results were obtained by multiplying the normalized spectra in Figure 9 by the spectrum levels for the appropriate in-flight environments, and then summing over one-third octave bands. Again the mean square acceleration levels are shown for each individual fluctuating pressure environment. The total mean square acceleration levels o f the shroud, obtained by adding individual mean square levels in one-third octave bands, are also shown. It may be seen that below 2000 Hz, the shroud vibrations are caused exclusively by the modified attached flows in zones 2 and 3, while above this frequency the shroud is driven largely by the separated flow in zone 1. Normalized acceleration spectra corresponding to Mach 0.8 are shown in Figure 11. Again each spectrum represents the space-average normalized acceleration of the whole shroud when driven separately by the various fluctuating pressure environments. The relative efficiencies o f the separated and attached flows are similar to the previous Mach number.

418

J. A. COCKBURN

A N D J . E. R O B E R T S O N

The acceleration spectra are again modal in character at low frequencies, reaching peak levels in the vicinity of the ring frequency and show coincidence effects at higher frequencies. The normalized spectrum for excitation by the shock wave oscillation suggests that this environment is highly efficient in terms of driving the structure. However, inspection of Figure 12, which describes the mean square acceleration levels in one-third octave bands, shows that

~,

I0-'

~

,'

"-'~\\

Z

/ ~J/~v ~.

"\\

~o-~

3

-~

All~

Coincidence frequency limits excitotiOnover forzone I

/

io-~

o

9

I 0-r

/

/

g ~6

I0 e

/

/

l!

II ,t |

/ //

[/ u !J/

] Ai t ./V

Coincidence frequency limits forexcitotionoverzone3

/v/

/A

/

,

,

p

-'1

o

Coincidence frequency limits for excitation over zone 2

i i Z

/ II

r I

I0

i

I

i

I00

1t I000

t

I I 0 000

Frequency (Hz)

Figure I l. Predicted normalized acceleration spectra for Moo---0.8. - - . . . . , Acceleration induced by attached flow over zone 1; . . . . , acceleration induced by separated flow over zone 2; , acceleration induced by shock wave oscillation over zone 2; . . . . . . , acceleration induced by modified attached flow over zone 3.

because of the primarily low frequency spectral content of the shock oscillation, it contributes little to the response of the overall shroud. At Mach 0.8 the shroud is again driven almost exclusively by the modified attached flow in zone 3 for frequencies up to about 2000 Hz. Above this frequency the shroud is driven largely by the separated flow in zone 2. The normalized acceleration spectrum corresponding to Mach 2.0 is shown in Figure 13; for this condition the flow is attached over the entire length of the shroud. The characteristics of this spectrum are similar to the previous spectra for excitation by attached flows. However this spectrum is quite fiat at higher frequencies because of coincidence effects and does not begin to roll off until about 7000 Hz. The corresponding mean square acceleration levels in one-third octave bands, obtained by multiplying the normalized spectrum by the pressure spectral density and summing over one-third octave bands, are shown in Figure 14. Also shown in Figure 14 are the mean square acceleration levels in the conical section of the shroud for the purposes of comparison.

VIBRATION OF SPACECRAFT SHROUDS

419

A

%

R 10_1

\ \ g

10-2

\

7o

r/

I

'

! J

c 0

i

I0

o

"

~

I00

\ I000

I0000

"

Frequency(Hz) Figure 12. Predicted mean square acceleration levels in one-third octave bands for Moo = 0.8. - - . . . . . . . , Acceleration induced by attached flow over zone 1 ; . . . . . , acceleration induced by separated flow over zone 2; - - - - - , acceleration induced by shock wave oscillation over zone 2; . . . . , acceleration induced by modified attached flow over zone 3; , total mean square acceleration at M| = 0"8.

Z

I 0 -]

~

,o~

~

I o~

~.

l0 6

.~

I 0 -r

I

I I I

I

t

I

I I

I

!

I

I

I I ~

I

I

I

I I 1000

I

I

I

I-

l O.s

Z

/

1 0 -9 I0

I00

I0 0 0 0

Frequency( Hz} Figure 13. Predicted normalized acceleration spectrum for Moo = 2.0; attached flow over whole shroud.

420

J. A. COCKBURNAND J. E. ROBERTSON I

ion v

I

I I i

t

I

I I i

I

I

I

/,,Y"

I 0 -~

~#/

g

-2

/

Id2

w E

l~ I

Id3

o

id 4

O

I 0 "~

I 10

I

I II

I

I

I II

Ioo Iooo Frequency ( Hz }

l

I

,

IO0oo

Figure 14. Predicted mean square acceleration levels in one-third octave bands for M| = 2.0. , Total mean square acceleration in cylindrical section of shroud; . . . . , total mean squai'e acceleration in conical section of shroud. 4. DISCUSSION The vibration spectra presented in Figures 9 through 14 are governed primarily by hydrodynamic coincidence effects at high frequencies. At coincidence, the axial joint acceptance terms, j~(og), reach maximum values. It was pointed out in the previous section that because each frequency is associated with a unique convection velocity, the value of the hydrodynamic coincidence frequency f , c given by equation (14) does not represent a lower limit for coincidence in the same manner as the acoustic critical frequency. To illustrate this point further, a curve describing the theoretical hydrodynamic coincidence frequencies as a function of convection velocity is shown in Figure 15 for the Atlas-Agena shroud. Also shown in this figure are the upper and lower limits of the convection velocity in each zone of fluctuating pressure at the three flight Mach numbers considered during this study. The upper and lower limits for the convection velocities have been inserted directly onto the theoretical curve shown in Figure 15. The ranges of convection velocity were based upon Bies' results [12] for attached flows, and Coe and Rechtien's results [9] for separated flows. For attached flows, Uc varies approximately from 0.83 Ut at (o96dU~)= 0.8 to 0.58 U~ at (o~6dUt) =.80.0. For separated flows, Uc varies approximately from 0.3 U~ at (~6dUt) = 0.63 to 0.8 Ut at (w6~/U~)= 6"3. Thus from Figure 15 it may be inferred, for example, that the limiting frequencies over which hydrodynamic coincidence may occur extend from about 430 Hz to 3100 Hz for excitation by the separated flow in zone 1 at M~ = 0.7. Similarly the limiting frequencies extend from about 5300 Hz to 11 300 Hz for excitation by the attached flow at M~o= 2.0. The frequency ranges corresponding to possible hydrodynamic coincidence frequencies, identified in Figure 15, have also been shown in Figures 9, 11 and 13 which describe the normalized acceleration spectra for the shroud at M~ = 0.7, Moo = 0.8 and M~ = 2.0, respectively. The relationship between the high frequency roll-off exhibited by individual spectra and the corresponding range of possible coincidence frequencies is clearly evident.

421

VIBRATION OF SPACECRAFT SHROUDS

Thus far, the discussion has been concerned with the hydrodynamic coincidence frequency and its dependence upon the convection velocity. The equation forf~,c simply defines the theoretical hydrodynamic coincidence frequency for a particular value of Uo based upon the condition of matching of the bending wavespeed in the structure and the convection speed of the fluctuating pressure field. It is also necessary to examine modal frequencies and flexural

20 0 0 0

I

I

i

I

J

I

i

l

i

I

i

l

I 400

l

I 500

f

I0000 A N Z

/'

1000

/

8

~ E

Ioo

Equation1141

IO 0

I I00

T

I 200

i

I 300

600

Convectionvelocity tlc (m/s1

Figure 15. Hydrodynamic coincidence frequency t'ersusconvection velocity for the Atlas-Agena shroud. o, Limits of U, in zone 1 for M,o = 0.7;., limits of Uc in zones 2 and 3 for M| = 0.7; o, limits of Ucin zone I for ~./~ = 0'8; a, limits of Uc in zone 2 for M.o = 0.8; 13, limits of Uc in zone 3 for M~ = 0.8; <>,limits of Uc for M,o = 2"0.

wavelengths relative to hydrodynamic coincidence. In each of the coincidence frequency ranges shown in Figures 9, 11 and 13, the structural modes which are excited are classified according to whether or not there is exact matching between the flexural wavelength and the pressure wavelength. These modes are defined as hydrodynamically fast (HF), hydrodynamically coincident (HC) and hydrodynamically slow (HS) according to whether the axial component of the bending wavespeed, Cxm,, of the mn mode at frequency fro, is greater than, equal to, or less than the convection velocity, Uc, at this frequency. Thus at coincidence, HC modes are defined by the condition Cxmn2efmn

Uc

- - -

2vfttc

fmn

~tc

= 1,

(15)

where 2e and 2p are the flexural and pressure wavelengths, respectively. HF modes and HS modes are defined by the conditionsf,~,/f,c > 1, andf,~,/f,c < I, respectively. Also equating the flexural and pressure wavelengths at coincidence and settingfnc =fro,,, gives

ntU~ fnc --f,,n = 2L'-'--~'

at coincidence,

where m is the axial mode number corresponding to coincidence.

(16)

422

J.A. COCKBURNAND J. E. ROBERTSON

Substituting a nominal value for Uc of 0.6 Ut in equation (16) together with the shroud length, Lx, yields the following approximate conditions for hydrodynamic coincidence for the various fluctuating pressure environments: M~=0.7 zone 1 (separated flow): zone 2 (modified attached flow): zone 3 (modified attached flow): M= = 0 . 8 zone 1 (attached flow): zone 2 (separated flow): zone 2 (shockwave oscillation): zone 3 (modified attached flow): M~ = 2.0 attached flow:

f n c 30 m, f n c - 21 m, f u c - - 21 m; - -

fuc fuc fuc fnc

-

39 32 38 24

m, m, m, m;

f n c - 52 m.

By inspecting Table 3, which contains a partial listing of the resonant frequencies of the shroud, in conjunction with the above approximations, the structural modes may be divided into HF, H C and H S modes accordingly. For example, at M~ = 0.7, w h e r e f n c - 30 m for the separated flow concentrated over zone I, the hydrodynamic coincidence frequency is less than the structural resonance frequency, f,~,, for all m < 24, thus the structural modes are primarily H F m o d e s with some near-coincident modes and H S modes for values o f m > 24. For the attached flows over zones 2 and 3, wherefuc - 21 m, all structural modes are H F modes. At Moo = 0.8, wherefuc - 39 m for the attached flow concentrated over zone 1, the structural modes are divided into H F modes for all m < 18 and H S modes for values of m > 18, with some near-coincident modes. TABLE 3

Partial listing o f resonant frequencies f,~n (Hz) o f the A tlas Agena shroud

m•

0

1

2

3

4

6

8

1 2 3 4 5

634 634 633 633 633

241 450 536 575 595

118 255 378 459 511

225 261 330 402 462

415 425 450 486 526

968 970 975 985 998

1741 1743 1746 1750 1758

6 7 8

633 633 633

606 613 618

545 569 586

509 546 574

567 603 634

1014 1033 I053

1766 1777 1789

9 10 12 14 16 18 20 22 24 26 28

633 633 635 637 641 648 657 670 686 708 734

622 625 630 636 642 651 662 677 696 729 747

600 609 625 639 653 668 685 706 730 758 791

597 615 645 670 694 718 744 772 803 838 877

661 684 725 759 792 824 857 893 931 973 1018

1073 1093 1133 1173 1212 1252 1294 1338 1385 1435 1487

1803 1818 1850 1886 1924 1964 2008 2054 2103 2155 2211

423

VIBRATION OF SPACECRAFT SttROUDS

At M= = 2.0, wherefnc - 52 m for the attached flow over the entire shroud, the structural modes are divided into H F m o d e s for all m < 14 and HS modes for values of m > 14, with more near-coincident modes. These results show that over the frequency range considered, a larger number of nearcoincident modes will occur at M~o = 2.0; this is borne out by the acceleration spectrum shown in Figure 13 which exhibits high response levels above the ring frequency, and does not begin to roll off until about 7000 Hz. Conversely, mostlyHFmodes (and no near-coincident modes) are anticipated at M= = 0.7 where the excitation is concentrated over zones 2 and 3; this is also borne out by the acceleration spectra shown in Figure 9, where the normalized spectra corresponding to the attached flows begin to roll off at frequencies just above the ring frequency. Finally, the relative severity of s various in-flight fluctuating pressures is illustrated in Figure 16, which compares the total mean square acceleration levels of the I0 :1

~

o

~

-I0

I

I I

I

I I

i~

i

t I

I

- ,I., , ,\11 ,

I I

i

I

I

I

1 I I

I l

1

-

I/X' II v - ' l f/ \~ 0

'/ V/,,/

oz0

0 "U

~

ii

I

-30

I

=-

I

-

. -

I II 31

63

.:

it,/ V

-40

I~

.:

I I I I I I I I I I I I I IZ5

250

500

I000

ZOO0

I I I I~ 4000

8000

One- !hird octave band center "frequency (Hz)

Figure 16. Comparison between predicted one-third octave band acceleration spectra for the Atlas-Agena shroud at transonic and q,aaxflight conditions; - - - - - - , M| = 0.7; , M= = 0-8; - - - - - , M= = 2"0 (q,,~,). shroud in one-third octave bands for the three flight Mach numbers considered. The severity of the transonic flight regime at frequencies below 2500 Hz may be observed immediately, particularly with respect to the low order shroud modes between 100 Hz and 250 Hz. The shroud vibration levels corresponding to M= = 0-7 are higher than those corresponding to M= = 0.8 up to about 1600 Hz. Above this frequency the vibration levels are essentially the same for these two Mach numbers. At maximum dynamic pressure (i.e., M= = 2.0) the shroud vibration levels are generally lower than for the transonic flight regime, except at higher frequencies, above 2500 Hz. At these highe r frequencies, coincidence effects contribute significantly to the shroud vibration spectrum. 5. CONCLUDING REMARKS Little reliable in-flight vibration data for spacecraft shrouds is available. Generally, inflight measurements have consisted of external fluctuating pressure measurements, internal acoustic measurements within the shroud, or vibration measurements within the shroud in the region of the spacecraft adapter [I, 13]. The most relevant in-flight vibration measurements are those obtained on the Spacecraft Lunar Module Adapter (SLA)during flight of Saturn

J. A. COCKBURN AND J. E. RoBERTSON

:4

I vehicles AS201 and AS202 [23]. Some of these measurements are shown in Figure 17 r transonic and supersonic flight conditions. The exact flight Mach numbers are uncertain, though the supersonic data were acquired after qm~x(M=- 1.4) but just prior to M~ = 2.0, ,d data above 250 Hz were unavailable because of telemetry cut-off. The SLA structure nsists of a 4-3 • 10-2 m (1.7 in) thick aluminum honeycomb shell having a ring frequency approximately 175 Hz. Qualitatively, the in-flight vibration data shown in Figure 17 hibit similar trends to the theoretical spectra shown in Figure 16, particularly with regard the higher vibration levels in the lower third octave bands at transonic Mach numbers. afortunately, because of telemetry cut-off, response measurements above the ring frequency .~not available for even a qualitative comparison with the theoretical results. 20

I I I I I

I

II

III

III

I I

|

|

I I

I

I

I

I

I I

I I

I

IO

un'~

Tronsonic I

Supersonic

-IO

0

-20 8o

I I I I 16

31

63

125

250

I I II 500

! II II II II I 1(300 2000 4000 8000_

One-'third octave bondcenter frequency (Hz)

:igure 17. One-third octave band acceleration levels of the SLA structure during transonic and supersonic ht. Data from Saturn 1B vehicles AS201 and AS202 [23].

l'he results presented in this paper have shown that shock-wave oscillation contributes very le to the overall space-average acceleration of the shroud except at very low frequencies. .wever, the lowest structural mode of the shroud considered during the present study was ~roximately 120 Hz. Because of the predominant low frequency spectral content of the ,ck wave oscillation, this fluctuating pressure environment may be more severe for shroud vehicle structures having fundamental structural modes at much lower frequencies. )he immediate application of the present results is in the area of structural qualification :ing, acceptance testing, or fatigue testing of spacecraft structures. Because of the pressure relation differences between laboratory acoustic fields and in-flight fluctuating pressure :Is, direct simulation of the in-flight acoustic environment in a laboratory will not essarily induce the same vibration response levels. It is therefore necessary to adjust the icipated in-flight acoustic spectrum so as to simulate the vibration response levels in the aratory. This procedure avoids over-testing or under-testing spacecraft and vehicle tctures. Hence theoretical prediction of vibration levels for a range of fluctuating pressure :Is (including the laboratory environments) allows the derivation of equivalentlaboratory ustic spectra [1, 23] which induce vibration levels in the laboratory that are compatible h those vibration levels which would be experienced during flight. ACKNOWLEDGMENTS 'he authors wish to thank D. M. Lister for his programming work during this study. The .~stigation was partially supported under Contract NAS5-21203 by the National Aerotics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland.

VIBRATION OF SPACECRAFT SHROUDS

425

REFERENCES 1. J. A. COCKBURN 1971 IVyle Laboratories Research Staff IVR 71-7. Evaluation of acoustic testing techniques for spacecraft systems. 2. J. E. ROBERTSON 1967 AEDC-TR-66-266, ArnoM Enghteering Developntent Center, Air Force Systems Command, Arnold Air Force Station, Tennessee. Wind tunnel investigation of the effects of Reynolds number and model size on the steady and fluctuating pressures experienced by conecylinder missile configurations at transonic speeds. 3. LEWIS RESEARCHCENTER, Cleveland, Ohio 1969 NASA TI~IX-1768. Atlas-Centaur AC-11 flight performance for Surveyor IV. 4. C. F. CUE 1969 Basic Aerodynamic Noise Research Conference Proceedings NASA SP-207, IVashhtgton D.C., 409--424. Surface pressure fluctuations associated with aerodynamic noise. 5. M. V. LowsoN 1967 tVyle Laboratories Research Staff IVR 67-15. Prediction of boundary layer pressure fluctuations. 6. J. E. ROBERTSON 1971 tVyle Laboratories Research Staff tVR 71-10. Prediction of in-flight fluctuating pressure environments including protuberance induced flow. 7. W . V . SPEAKERand C. M. AILMAN 1966 NASA CR-486. Spectra and space-time correlations of the fluctuating pressures at a wall beneath a supersonic turbulent boundary layer perturbed by steps and shock waves. 8. W. J. CHYu and R. D. HANLY 1968 American htstitute ofAeronalttics attd Astronautics Preprint No. 68-77. Power and cross-spectra and space time correlation of surface fluctuating pressures at Mach numbers between 1"6 and 2.5. 9. C. F. CUE and R. D. RECHTIEN 1969 Paper presented at the American Institute of Aeronautics and

Astronautics Structural Dynamics and Aeroelasticity Specialist Conference, New Orleans, Louisiana. Scaling and spatial correlation of surface pressure fluctuations in separated flow at supersonic Mach numbers. 10. R. D. RECttTIEN 1970 University of Missouri-Rolla UMR Research Report. A study of the fluctuating pressure field in regions of induced flow separation at supersonic speeds. 11. J. E. ROBERTSON 1989 IVyle Laboratories Research Staff tVR 69-3. Characteristics of the static and fluctuating pressure environments induced by three-dimensional protuberances at transonic Mach numbers. 12. D. A. BIES 1966 NASA CR-626. A review of flight and wind tunnel measurements of boundary layer pressure fluctuations and induced response. " 13. L. A. WILLIAMS and W. B. TERENIAK 1967 Shock and Vibration Balletht 36, 89-102. Naval Research Laboratory, IVashhtgton D.C. Noise level measurements for improved Delta, Atlas/ Agena-D, and TAT/Agena-D launch vehicles. 14. A. POWELL 1958 Journal of the Acoustical Society of America 30, 1130-1135. On the fatigue failure of structures due to vibrations excited by random pressure fields. 15. A. POWELL 1958 Journal of the Acoustical Society of America 30, I 136-I 139. On the approximation to the infinite solution by the method of normal modes for random vibrations. 16. M. K. BULL, J. F. WILBY and D. R. BLACKMAN 1963 University of Southampton A A S U Report 243. Wall pressure fluctuations in boundary layer flow and response of simple structures to random pressure fields. 17. M . J . CROCKER and R. W. WHITE 1966 Wile Laboratories Research Staff tVCR 66-I 1. Response of an aircraft fuselage to turbulence and to reverberant n o i s e . . 18. R. W. WHITE 1967 Wile Laboratories Research Staff tVR 67-4. Predicted vibration responses of Apollo structure and effects of pressure correlation lengths on response. 19. J. A. COCKBURN and A. C. JOLLY 1967 Wile Laboratories Research Staff )VR 67-16. Structuralacoustic response, noise transmission losses and interior noise levels of an aircraft fuselage excited by random pressure fields. 20. D. J. BozicH and R. W. WHITE 1970 NASA CR-1515. A study of the vibration responses of shells and plates to fluctuating pressure environments. 21. D. M. LISTER 1970 Wile Laboratories Program Description 70/002S-1. Computer program ACCRES.

22. E. J. RICHARDS and D. J. MEAD 1968 Noise andAcoustie Fatigue hz Aeronautics. London: John Wiley and Sons Ltd. See Chapter 23. 23. W . D . DORLAND,R. J. WREN and K. M c K . ELDRED 1968 Shock and Vibration Bulletin 37,139-152. Naval Research Laboratory, Washhrgton D.C. Development of acoustic test conditions for Apollo Lunar Module flight certification.