NASA Technical Note D-344
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NASA TN D-344 1 i i' ] oo Z < < Z f f TECHNICAL NOTE D-344 EXPERIMENTAL DETERMINATION OF THE PRESSURE DISTRIBUTION ON A RECTANGULAR WING OSCILLATING IN THE FIRST BENDING MODE FOR MACH NUMBERS FROM 0.24 TO 1.30 By Henry C. Lessing, Jotm L. Troutman, and Gene P. Menees Ames Research Center Moffett Field, Calif. NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON December 1960
Transcript of NASA Technical Note D-344
NASA TN D-344
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TECHNICAL NOTE
D-344
EXPERIMENTAL DETERMINATION OF THE PRESSURE
DISTRIBUTION ON A RECTANGULAR WING
OSCILLATING IN THE FIRST BENDING
MODE FOR MACH NUMBERS FROM
0.24 TO 1.30
By Henry C. Lessing, Jotm L. Troutman,
and Gene P. Menees
Ames Research Center
Moffett Field, Calif.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON December 1960
lit
¥
NATIOHAL AERONAUTICS A}FD SPACE ADMINISTRATION
TECN}TICAL NOTE D-344
L_{PERIMEHTAL DETERMINATION OF TI{E PRESSURE
DISTR]mUTIO_ ON A m_CTm_CULmR WING
0SCILIATING IN THE FIRST BE_{DING
HODE FOR MACH I,YJHBERS FROH
o.24 TO 1.30
By Henry C. Lessing, Jo_m L. Troutman_and Gems P. Menses
SUMMARY
The results of an experimental investigation in a _Tind t_unel to
obtain the aerodynamic pressure distribution on an uns<_:_pt rectangular
wing oscillating in its first symmetrical bending mode arc pros<ntel.
The- wing _<as of aspect ratio 3 with 5-percent-thick biconvex airfoil sec-
tions. Data wer_ _ obtain_d at 0° and 5° angle of attack in th_ _ch
n<_b_r range from 0.24 to 1.30 at Reynolds nm_0ers_ depending om the _[%ch
nmnber_ ranging from 2.2 to 4.6 million per foot. The reduced fr<'quenci_s_
also a function of _eh _mm])er_ ranged from 0.4(i at H = 0.24 to 0.i0
at H : 1.30.
The most important results presented are the chord<else distributions
of the surface pressures generated by the bending oscillations. Similar
data obtained trader static conditions are also presented. The results
sho_,rthat th<o phenomena causing irregularities in the static pressure such
as three-dimensional tip effects_ local shock waves_ and separation effect<:,
_¢ili also produce significant changes in the oscillatory pressures. The
experimental ,data are also compared with the oscillatory pressure distri-
butions comput<,d by means of the most complete linearized theories avail-
able. The comparison shows that subsonic linearized theory is adt,quat_
for predicting the press-0a_es and associated phase anglus at io_,_subsoI_ic
speeds and io_{ angles of attack for this -_ing. Ko<_ever_ the appearance of
local shock _aves and flow separation as the Mach number and anglo of
attack are increased causes significant change s in the experimental data
from that pr<_dicted by the theory. At the low supersonic speeds covered
in the experimental investigation, linearized theory is completely
inadequate, principally because of the detached bo}_ _ave caused by the
_ing thickness.
Some indication of wind-tunnel resonance _as noted; ho_ever_ the
effects on the experimental data appear to be confined to the H : 0.70re sult s.
INTRODUCTION
Flutter is conventionally defined (e.g._ ref. I) as a self-excit<doscillation resulting from a combination of inertial, elastic, oscillatoryaerodynamic dmmpins, and temperature forces. In combination_ these forcescan result in Rustable motion (i.e., flutter) which leads to mild orextremely severe structural failures. It is evident, then_ that the sub-ject or flutter is an extremely complex one and a reliable analysis canbe madeor a specific design only through the use of the most completeinformation about the struct_ire, dynamics; and aerodynamics.
The present paper is concerned _,_ithonly one of these fields - theoscillatory aerodynamics. The basic tool of the flutter analyst as regardsaorodynmnics is theory. Very little has been done v_ith the application of<_xperimentaldata to flutter calc_lations; instead_ exp_:rimental resultshaw been used chiefly to evaluat< th< accuracy of the theoretical methods.Th,_seevaluations have shoT:,a_that the use of t_ro-dimensional theory appliediK a strip analysis_ while yielding good r{_sults for w_ngsof high aspectratio, lea_ to inaccurate results in the case of lo<r aspect ratio surfaces.This fa<rt has led to the developmc_utof three-dimensional theories_ t_roof the most complete beiug the subsonic ke,"n<,l-fu_ction method of Watkins,R_uya_ and Woolston (rer. 2)_ and the supersonic linearized theory ofHi_es (t_<_£.3), an analytically _ct tpJatm_nt api_licable only to certain
A3
>h
The expc_rimental results used to evaluate theories in the past have
teen !.<,rived chiefly from rigid wings ex<_cuting flapping_ pitching, or
pltuK_ing oscillations at subsonic sp<_eds (e.g., refs. 4.. 5, a_<d 6). Moll_-
Christensen (ref. 7) has obtained th,_ oscillatory press<_re amplitudes at
c_rtain points on such vrings at transonic spot,is. To date_ only one
experimental investigation has b<,,u_ r_norted :,_ith a,<¢ing perfomning elastic
defozmlations (ref. $); this investigation was conducted only at low sub-
sonic speeds, however_ and as yet no similar r{k_sults are avaElal01e for
transonic and supersonic speeds.
The pz_loosc _ of the present investigation was to develop a research
t<,cbmique by means of which the oscillatory pressures <_xisting on a
thre_-dimensional wing perfo_n, ing elastic deformations could be obtained
throughout the subsonic, transonic_ and supersonic speed ranges_ and to
compare the exicerimental results with those of the most complete theories
avai lab !e.
NOTATION
The follo_ing definitions are for significant notations which will
be found in the main body of the report. Notations found only in the
appendices are defined therein.
A aspect ratio
b semichor£
Cpu_CpZ static-pressture coefficient on upper and lo',,r<_r surfaces_
respectively,Pu - Poe P_ - Pc,_
q _ q
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Cp
CP_ h
_h
C_@h _Cm<sh
7h,Thm
f
K
h
)
Re( )
k
H
P
P
p!
static lifting-pressure coefficient_ Cp% - Cpu
complex amplitude of oscillatory pressure coefficient per
P
unit _h, q_h
phase angle bet_,,reenoscillatory lifting-pressure coefficient
magnitude and wing-tip bending magnitude, positive for
pressure leading bending displacement _ dog
oscillatory section lift and moment coefficients about
mid-chord per unit _h
phase angle bet_,_een wing-tip bending displacement magnitu_ie
and magnitude of oscillatory lift or moment coefficient_
positive for lift or moment coefficient leading bending
displacement, deg
oscillatory frequency, cycles per unit time
turmel height_ ft
displacement of mid-chord line at any spanwise station_
positiw_ do_¢n
imaginary part of a quantity - that component _¢hich is in
phase _rith the wing-tip velocity
real part of a quantity - that component _,rhich is in phase
with the _.ring-tip displacement
reduced frequency_ _b/V
Mach number
static pressure at wing surface
complex amplitude of oscillatory pressure at any orifice on
_,rin6 surface
complex _aolitude of oscillatory pressure at pressure cell
connected to orifice at which P exists
q free-stream dynamic pressure_ pV£/2
I,
r
R
t
T
V
Y
chord_ise distance from _ing mid-chord to accelerometer root
Reynolds number per ft
time
m_<imumwing thickness
strain-gage supply voltage or free-stream velocity
chor_{ise distance, positive do_,mstream, measured from wing
leading edge
span<_ise distance measured from wing root
IP'lamplitude ratio_
Iml
C_
_h
static or mean value of angle of attack_ deg
magnitude o£ oscillatory angle of attack at wing tip due to
bending_ --_--_ radians
tO
L
(')
f
r
i
free-stremn density
phase angle between
circular frequency of oscillation 5 2_f
T
_ring thickness-chord ratio_ _-_
magnitude of quantity, I( ) 12 = [Re(
d( )
dt
P and P' , positive for P leading P' , deg
)]2 2+ [:h( )]
Additional subscripts which may be attached to preceding quantities
association with forward accelerometer
association with rear accelerometer
component o£ a quantity in phase _{ith bending displacement
or resolver rotor
component of a quantity out of phase with bending displacement
or resolver rotor
T
U
Z
OO
value of a quantity at }rinc ti_
value of a quantity on upper wing stu_face
value el' a quantity on lo:_er _Tinc surfac_
frec-stresm condition
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FXPERI}4F2FfAL I]'_ESTIGATION
Test Facility
The experimental investigations _ere conducted in the Ames by
6-foot supersonic :_ind turmel_ _hich is of the closed-r<,_tumu_ varia]}L -
pressure tk-pe capable of furnishing a continuous Hmch nm_?o,:r range from
0.70 to 2.20 by means of an asymmetric sliding-block throat. This t<uu_l
can also be o oerated at M = 0.24 by halving the available coml'_r<sso±_-
drive po<_Ter. In order to operate at sonic sneed and to obtain more nearly
ur_ifolnn flo_, the test-section floor and ceiling _{ere nerforated.
Test Conditions
The pressure distributions vTere measured over a }_%ch ntmlb<_r raw,c<_
from 0.24 to 1.30. The _ing _as oscillated at its first bending fr<_quency.
f -- 26.5 cycles per second_ at a tip amplitude of approximately 0.2 inch.
and the resultin_ reduced frequency range _ms from 0.i0 to 0.4_ . The
maximum available total _ressure _,_asmaintained at each _izch nmnb _r and
provided Reynolds numbers from 2.2 to 4.6 million per foot. The _ing
angles of attack were 0° and 5° .
Description of Apparatus
Model.- The model consisted of a semispan tms<_ept, rectang:ular _ri:ig
_Tith an effective asoect ratio of 3. (The _<ing _as provided ,,<itha re<rob !
tip _{hich _as not included in computins this aspect ratio.) The airL'oil
consisted of a biconvex circular-arc section, 5 percent thick. _ith :_har:_
leading and trailing edges. A photograph of the model mount<] in tb t_ t
section is sho_n in figure l(a) and a dra_ing of the model giving ]_:rtin _
dimensions is sho_m in figure l(b).
The _<ing ,,_asmachined from forginss of SAN 4130 st< :l _;hich '_a_r
heat treated to approximately Roc}_;ell C-3i_ hat<[mess. The u_u_i_' a_ !o';,r
surfaces _rere machined separately and mated at the chord line to ]<roiuc, _
a monocoque structure :<ith integral spanwisc stiffening. Figur: , l(c) [:_
_j
a l)hOtograph of a representative cross section showing the location ofthe stiffeners and the resulting interior partitio_in_. Figum;es l(d)
and l(e) show the interior sides of the upper and lo_rer surfaces. The
two halves _rere held together by screws through the root section, stiff-
eners, and leading and trailing edges. The shear loads due to _ring deicer-
marion were carried by do_rel pins at the root and tip, together with the
shear blocks set in the soan_rise stiffeners, _Thich ar_ visible in
figum,_e l(e).
_ing-drive mechanism.- The drive mechanism to force the _ring tooscillate in the desired stz_actural deformation mode should be contained
completely within the _ing structure in order not to create flow distur-
Lances _<ith un]eno<_ aerodynamic effects. A unique system satisfying this
requirement _,_asdeveloped for the present investigation and consisted of
two spars extending the full spanwise length of the wing inside the
channels sho_s_ in figures l(d) and l(e). Each spar consisted of an upper
and lower half_ riveted together at the wing tip as sho_n in sketch (a)
and constrained to remain in contact over their length. Contact with the
Tunnel wall -__
F
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Sketch (a)
ring occurred only at the tip and 50-percent semispan station. The lo_er
half of each spar v_s restricted from spanwise movement by being pinned
at the wing root support in the tunnel wall. The upper half of each spar
extended through the tunnel wall and was attached to an electromagnetic
shaker _hich applied a harmonically varying force, F. The resulting
alternating tension and compression in the upper spar half, coupled with
the alternating compression and tension in the lower spar half_ caused the
spar to deflect in a manner analogous to that of a bimetallic strip. The
resulting forces on the two bearing points then forced the wing to
oscillate in its first bending mode, when the frequency was properly
adjusted.
The bearing points were chosen from the results of an analysis which
indicated that not only first bending but second bending and first and
second torsion modes could also be forced by the one drive system. In
practice, ho_ever: it was found that these additional modes could not be
driven with sufficient amplitude to obtain data.
The wing and associated drive mechanism were mounted on the circular
plate sho_ in figure l(a), which allowed the angle of attack to be varied.
An inflatable seal prevented leakage of atmospheric air around the plate
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into the test suction. Tt should b: noted h_r{ that _reat car_ -.raLztSS'_,r
to seal the is,reties of thu wing from the fr_u stream in the: t st s, ction
throuF;h the use of rubb_ r cement on all matins surfacus az_,l asexual th,
pressure-c,_ll mou1_tii_gs. The interior was s_ab.d rrom atmospheric i_r_-
sure at the root by sold__rud lead-throughs rot th, orific,_ lines, pr,_,ssu_:_,-
tight plugs for the cl_ctrical conn,<'ctions_ gask<_ts or rubb r ',_,m<;nt on
all mating surfaces; and flexiblu rubber diaphragm: to _ rmit mowm..nt of
the _,rin[,;-driv<,_ mechanism.
Instrmncntation.- The wing instrumuntation was d, slgJl<d to furnish
two typos of information: the prcssuru distribution ov r th_ _,,_[n_;(_ith.-r
static or dynamic), and tb,,_ d<_f]_ection shape of th<_ o_ciilit_.ug -._kug.
The _¢ing pressure distribution was m_asured on both upp_r anti lower
surfacus approximat_ly at the 90-percent. yO-pcrc nt, and pO-p, re, _t s_mli-
span stations and at th root. T_n O.030-i_._ch-dism=t<r orific ,_ :._r
dJstr_but_ i along the chord at uach of th_'s,_ stations o;_ _:ach rurrsc_.
Th, <;rli'ic{: positions ar_ sho-,m and t_-_bulatod in f;f:R!'> ](b). (Th :rhea;
orific<:s <g_r_ sta6g<_r<,@ aud the root orific_ s <.<trek: eL'f set a_s she<, _] ill t]Yi s
fi{%r_ in or'des %o acco_mnodato the i_str_ncntatlon for ak ai_t_ rn%t_:
pressur_-measuring system at these stat_ol_s. This i_lstrmn, ntation, co_i-
sisting of flat, electrical r_sistanc__ pr_ssur_ cells ]/s i!nch £__1 ,_i@2£l..t i'.
•,,;as installed_ and is visib-L: _._ th_ photographrr o1' fiEw_" : l(a), I(L 0 .
and l(e). The particular pr_s_ur_, _ c_lls ,:Lcslgn_,l !'or thi:_ _;q_ rLm, nt
prow_:d to b_ u',n'_lisbl: iu uar!y tests and :,;,_r_ suL_,,qu ntly a_;'__bn,,_l.
Thu pressure at each orific_ ,_m_ _:_ip_:d :)ut oi' the: :<:in{;through tb
reel l,'a:i-throu_;hs [nsi(]_ _01astic or sta[n]_ ,: -st, _ I t u_J:)in_:
(I.D. = O.O_s in.) _.;hich _,.;asconnected to th_ manifold of a m o:l ! 4 -D
Scm_lvalv<-. This is a co_mmcrcialiy availablu electromechanical pressure
:u,ritch_ capal01< of s{_qu<nrtialAy uxpos[ng 4_} !;rrssure linc_ to a s[r4:k(
r)r<:ssur<-s _llsi]l[;t_<)vi(:c. _,To such UlI_iSS ,'re]'{US,<_d ill OrLb:::" t}]at %]i{:pr_s-
_]l:_<_J rN;s. fr,9]!! COl'rcspo_l,:]._.ii[<;_'ii'ic::_ on %h( l.lp]m_sr ali:i lo?p r LLlrh'ac 's, might
b: f:n,d slut[ '.ts%rl<rou;],ky. ]_h_ pr{:ssur_-s<n_siR S (](:_,,,[c<s used :rer{ coHncr-
c_'a_Lly avaiiabl< dif;.'_ r<;.mt'al pressure c_d_ls of the "_fl_onded straln-gas,:
t}<o< ,. The compl<_t< _ S<OLkciflcatiou Of all oscillatory pressur_ requires fi]u.:
iut_n:L_ation of its two <_omF,ouonts _ and in th_ m_ thod to bc duscrib<_£
pro s<,ntiy .'_',orthat d_%:,rmiriatiou_ it _,,rasus_ ful to r_:vis:_, th_ Sca_ivalv_
<rS.t a_ suppl L{;_lby tb_ manuYacturer :b_ order to acco_m_o,iat<_ %,,re :Ln:i<_cn-
d_nt pressurL, c_lls. As can b,,_- s _cn in f'igurc l(f) th_ mod_[ficatlo'n
r<:sult£!d J_l a_l arraTl(%:9_l<N_t _,Nlich n_:rmitt<_d the _)]'essu_,fL_ to b_' £toJL si]]]u]_-
taneously to th_ face or each of the two pressure= culls hous, d in thu
i01ock. The interior diam<t<rs in the mo<iiY[ J Scaniva!w. arran(_:_.mcnt <_or
held consta_Lt :L:h or<L_r to pr_:v,nt loss "n {if[_!ai striLqli:Nth.
Thu limited rangu or the hish-s_sitivity _or,:ssurc c<_l!s used :_d_n
obtaining th_ ,'ly_usuni<,dats_ mad_ it necessary to _quali_L _ the static _r, s-
sure or_ both sides eL' _:ach eel! diaphra_. For this r,u:rnos a bl _<i-i'ilt, r
consisting of a ,:]-inch i, nNth of sta_nlcs_-sto<'l h)qx<!,::!m_ic t\fi01n/:
(I.D. : 0.OO<% i.u.) was motu_t<_d in the top of the' t_,: an<l _,_-asconk, ct::[ to
the reference-pressure outlet of each c___ll motu_ted in [is block as sho-,nL
in figure l(f). This length of bleed tubing _as found to furnish an
acceptable response time to a representative step change in static pres-
sure •_hile satisfactorily filtering out the oscillatory part of the
pressure signal. When the static-pressure data were obtained, one less
sensitive pressure cell was used in each block_ and it was referenced to
the static pressure at the tunnel wall. This provision is also indicated
diagramatically in figure l(f).
Because of the chordwise rigidity of the wing it was assumed that
any oscillatory motion _as completely determined by the plunging and pitch-
ing oscillations of each of its cross sections, it was further assumed
that the first symmetrical bending and torsion modes of oscillation could
be adequately dete_mined from a description of the motion at four span_ise
stations. Accordingly_ eight cantilever-beam-type accelerometers were
located in cutouts along the extreme fore and aft stiffening spars at the
95-, 75-, 55-, and 35-percent semisp_1 stations, and the oscillating wing
motion was determined from their i1_ependent outputs.
All accelerometers were bolted to the stiffeners of the low_r _ing
panel and are visible in figure l(e). T_o complete strain-gage bridges
were bonded to each tip accelerometer beam, permitting both acceleration
components to be measured simultaneously at that station while only one
such bridge was bonded to each of the inboard accelerometer beams. Thus_
for the inboard accelerometers, the method for component determination
descri%ed below simply required sequential determination of one component
and then the other. The electrical leads from the accelerometers were
run along the wing cavities to cannon plugs in the root section.
All stainless-steel tubing and <_iring <{as bonded to the _{ing interior
_ith PRC 1221 cement which when dry had a rubber-like consistency and thus
could ['lex with the wing without cracking. No fatiguing of either the
bonding or the elastic properties of the cement was encountered dtu_ing the
test.
Electronic circuitry; method for component determination.- The
electronic circuitry was designed to perform a harmonic analysis of the
information provided by each element of instrumentation (whether pressure
cells or accelerometer) and to respond only to the components of the com-
plex amplitude of the fundamental. (The term f_damental as used here
refers to that harmonic term which is of the same frequency as that of the
gage power. In the oscillatory tests this frequency _as the natural
frequency for the first bending mode of the wing while in the static tests
it was zero.)
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2¥
9
The essential features of the electronic circuitry used are sho_m
in sketch (b) for the oscillatory test case. The principle of operation
% Stahc deflectiOnUUUUUHU UH UUv",Uu'--V )
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Sketch (b)
of the circuit is as follows: the output of a 5-kilocycle square-wave
generator is passed through a resolver driven by the alternator supplying
power to the electromagnetic shakers which force the wing motion. After
subsequent demodulation this produces two alternating voltages at the
wing-drive frequency such that one voltage is in phase and the other is
out of phase with the angular position of the resolver rotor. After
amplification, these voltages are used to power the strain-gage bridgesin the various accelerometers and pressure cells in use. (The voltages
were applied sequentially to the single bridge of each accelerometer at
the 35-5 55-_ and 75-percent semispan stations and simultaneously to the
remaining gages.)
When the wing is oscillating the resulting pressures (or accelerations)
impose a time-varyin_ deformation on the various strain gages. If each
gage is powered by an alternating voltage at this same frequency, then its
output is essentially proportional to the product of the voltage signal
and the resistance changes in the bridge associated with the deformations.
Thus_ if the gage is undergoing sinusoidal deformations of the form
B sin (_t + c)
i0
(where c is the phase angle between the gage deformation and the angular
Dosition of the resolver rotor) and is powered by the two voltages
delivered by the resolver in the form
V i : V sin _t
£_ 2'V o = V sin t + _/
then the gage output voltages are, respectively,
v i = K B sin (_t + e) V sin _t
w [cos _ - cos (2_t + c)]=K T
and
Vo = K B sin (_t + e)V sin £_t + 2_
= K _ [_inc + sin (2_t+ c)]
Here K is a general bridge constant depending on the type of bridge in
use, its resistance, and the choice of units of B. Thus, the output of
the strain gage is proportional to one-half the amplitude of the quantity
to be measured, the frequencies are twice the frequency of operation, and
each of the resultant voltages has a time-invariant component.
As sh_n in sketch (b), each output voltage is used to drive a
galvanometer element which has an undamped natural frequency much less
than 2_ so that it is essentially unable to respond to the oscillatory
part of the signal. However_ the galvanometer is capable of responding
_ith a deflection G to the component of the strain-gage signal which is
time-invariant. The result is a static indication of the in-phase and
out-of-phase components of the harmonically varying quantity.
This method is more fully described in reference 9, where it is shown
that such a system is capable of filtering unwanted harmonics from the
data. However, tunnel turbulence and the occurrence of leading-edge sepa-
ration when the wing was set to an angle of attack caused fluctuations in
the strain-gage outputs which reduced the accuracy of the pressure data.
11
Instrument Calibration
Acc_,l<',rom_,_tcrs.- Th_ calibration constants which had to b_: d._t-i_ninod
:'or each acc,_l,_uom_ ,,,_-r eel,re, from apo<.n,lix A, K h _d Krj " t}]U galvamom<.t_:.r
d,zflections p<.r unit lin_:ar acceleration and p<;r re-Lit angular r_c¢'.,;1 _rat[.on
about the acceleromet<:r root. Th, se constant:, _,.rored,_tormin,_d aft,or th,,•
accelorometers had b,,_n instaiffe,i in the: ,,7i:,<iiR order that all _o_rt:_n< nt
L'a,:,.tors, such as root fixity, _,,rer,__ account,x! for.
The mode shape of -the oscillating :zing was d,,-termin_ d <or ou:m}oses
of calculating the accelerome%_r constants by means of a stan[ which held
l.<th microme% rs <rith spring-supported <'onical tips fast_nei to th_ _
snin,:i!.< s _h_ imps orovi,:._,:[ acc02rate po:_itio:_n_t of th,, _ _',unt,,,: b uoh_t an_!.
minim-z,_'A th<_ danger of Rs£,_age to both the ,rir_.g az:d tb, microm_ter. Con-
tact betw_ cn wing and microm(_ter 4as in£icat:::i by a ",'_'atte±_y-Oow- _eR cJ rcuit
_;,rhich _>roduc,:,£ an au_tib].c sidnal in a p_.ir or m_hoz_,<_. The torsio.'_ mOLl,:
,,;as excited by moans of opposed occ,,nttric ,,.rei_Nht's st!ss, cil,,,i to t,]l, ,shaft or
a small <]::-ctric motor n_o_.u_ztod, on the -,,S.n S tip. A p'_zi'e torsional mo_!-:nt
';,_s produced in this mann, r with no accomp:n_};gn6 ['orc< t<_'.din6 to dia-uN_
the ,.ri_,,i< __tL b_nt:1]ng, s::rJJ ::!_tc,: %h_ ,:lastic axir_ kay oR th,k !in,: of r,_s_;,s
c,qttcr::, ]zo b,sh,[i:_{i -,;as [_r_%ts[ i_]. "bJ r _n,: rtia Coupl-_Lg. IX ord.cr shat a]]
<[u.antit, i<.-, mi_J_t %e r f_:rcncek tirectly to tl%: t:[p _m_atitudo, th out_,ut;_
.sf the 9N-p:_rcent semispan s,ccel. rom_.t rs w, lt'._ ca!iLrat, d i]', terms of th<
tip d':fl_'::tion rather than t]%,. :l<-fl_cti:urz of th, 'pS-]',or,cent s:mispnn
;tat i c_".
rh as,:.:, i xrom<:%_r <taiibrotiou showo,d that Kd ,,m:; rx-zj] _g[_l, ".,r_'th
_:'_:m,:ct %o rKh in det<rmin:iz:F: a torsional ,com!_orzent or motion n_t !:h_
first b_,r_,iinf_ i'ruque_cy_ arN. was consequcnt_ly sot equal to ;u:_ro i:_ the
d.ata-rcduction <,quat[on<; us,__<i in the inv_ st'sat:ion. The calibration s]so
show_:<t that no torsio'_a-! mot[o:_ <,:as prr;s<mt <,ri]_;_t th VNk L" <,;<t,_ :i=-iv<:) :[;_
-the first b,):u]i.r_( mo(t.c. Tiz':_ [m:-,ortant z'c.su-]t; .,ras !at<:r 'fout::i h,:;-_,,- Dru:
J_tLring the _.,rin:i-on t,_,::t .ql::,,_. [_his :r,q_,s to b,_ , xo._¢%,_d, cor_si,i_r_:'[k,_ th_
mass s[,M.nmetry o1" the _.rLng _oout the mid-chord line mentioned previous!y_
1%n',lth_, East -i;]_atthe torsional z'_sonant frequency ",.,;as:a!)proxirrlrtt_ly llO
cy,::l,,s _z r s :o_d a_ com_,ared to the first %<,nt]._'_g £r<.quency o:"
apero <imate!V cycles per ......
Pressure c_lls,- The static sensitivities of the pr<_:su_"< c_:lls _¢,.,rc
obtain_!'l by mo_%z_s o_ a precision aaRon{tear. Th< calibratio__1 con_:tants so
obtain<q/ _.<_re used directly in the.. static-Dr< ssure-distribution test. No
basic difference in the static or dynamic ceil sensitivities existo,d sine,
the resonant frequency of the pressure cells <¢as s©vemal orders o£ magni-
tude gr.,at,:r than the test frequency• The dy_amic calibration constants
:l<,notod as Ki and Ko in appendix A differed from the static constants
because the dyp,amic data ",,r_,;rcobtai:_ed by ?owerin 6 the cells with au alt.:r-
hating voltage rather than a constant voltage. The root mean square of
the alternating voltage was maintained at a value equal to the constant
voltage used during calibration; the effect of this change_ coupled _¢ith
12
the factor 1/2 noted in the section on Instrumentation_ resulted in a
d_amic calibration constant equal to _ times the static constant when
expressed in the units of Ki and Ko given in appendix A.
Test Procedure
The data presented for each test condition were obtained in three
distinct stages; the static test, the oscillatory test, and the orifice-
tube calibration. Data from each of these phases were automatically
entered on punched cards and the results appropriate to each phase were
computed electronically.
Static test.- The static-pressure distribution _as obtained by
po_ering one of the cells in each Seanivalve r_uit _ith a constant voltage.
Each cell was referenced to th . static pressure at the _¢all of the test
section, so that its output for each orifice _as directly proportional to
the static-pressure difference between the _ing orifice and the tunnel
wall. Adjustments to free-stream conditions yielded the static-pressureco_d_ficients.
Oscillatory test.- The _in_ _Tas driven in resonance at its first
b_ding frequency in order to obtain the maximum available tip amplitude
,of motion (approximately 0.2 inch for all test conditions). The pressure
coW.ks in the" Scanivalves _ere connected to the bleeds as described in the
section on instrumentation so that no static-pressure difference existed
acro. s the cell diaphragm. The cell outputs, corresponding to the
oscillatory-pressure amplitude, were recorded for each orifice. Simul-
tan_:,ous r:_adinss of the acc_lerometer outputs were also recorded, and
thes,_ data, together _ith the data obtai_ed as described in the next
s,_ction_ yielded the mode shape and oscillatory-pressure distribution when
combined in the equations of appendix A.
Orifice-tube calibration.- The calibration was performed by means of
the fixture shown in figure l(g) which clamped to the wing and simulta-
neously covered all orifices at a given span station. A similar fixture
_¢as used in the calibration of the root orifices. At most test conditions
the oscillating pressures experienced by the pressure cells in the Scani-
valves were not representative of either the magnitude or phase angle of
the pressures existing on the wing surface because of the attenuation and
phase characteristics of the tubing which transmitted the pressures to the
cells. It was experimentally determined that the tube characteristics
were functions primarily of the frequency and static pressure existing in
the tube_ and essentially independent of the amplitude of oscillating
pressure for the values experienced during the investigation.
The tube characteristics could also be affected by the introduction
of foreign particles into the wing orifice and by necessary maintenance
_ork on the Scanivalve. To minimize the possibility of such occurrences
affecting the data, a calibration was made immediately after each
A
354
lj
A
3
54
o_ci!latory test. In th,:>calibration the absolute static prcssur<, as
dcte_nin_d from the static t_st, and the frequency, as detel_nined from
the oscillatory test, _ere matched for each orifice location and an aver-
age value of osciilating-prc.ssure s_mplitude was applied at the orific .
Th<_ calibration fixture consisted of a chamber which appli<d the smn,
conditions to all orifices on the upper or lo_,¢er surfaces of the _¢ing.
S<_al_ng arotu_d each oriL'ice <{as accomplished by clamping on the rubber
grommets visible in the photograph. The static pressure was s_t to tb:
value corresponding to the particular orifice beinL_ calibrat<d. Tb_,
oscillatinc-pr:ssur_, amplitude, supplied by a piston-cy!in_ r arran(_ mo_it
mo_mt_ d on an electromagnetic shaker, '_as indicated by the two pressur_
cells mourLted in each half of the fixtui_e as sho_,n_. It was experimentally
detelmlined that there was no attenuation or phase shift introduced %y th,c
calibration fixture, so that the pressure cells in the fixture gave a tru<
indication o2 th<_ oscillating pressure existing at each orifice.
The data obtained L'rom the calib!'ation were reduced to the fol_ of
amplitude ratio_ Z, and phase angle, 8_ and were used in the data-reduction
equations of appendix A. A plot shovnLng the variation of th_s_ quantities
with static pressure, typical of the orifices at the 90-percent semis,pan
station, is presented in figure 2. Aithouch the investigation was con-
ducted at the ms_cimum turmel pressure available, the decreas< in static
pressure with incr_asing Hach number caused an increasing loss of syst-m
acc_macy because of the resulting attenuation. Because of this _,_fLkct
the ma_:imum Math number at which reasonably accurate data could bc o%taim>£
_,ras 1.30.
RESULTS A}_ DISCUSSION
Static Data
The pressure distributions obtained Lmder static conditions are
present_d in fi_ure 3 for s_Igles of attack of 0° and 5° . (These angle of
attack values are nominal in that the ttmnel stream angle varied across
the span, generally %ecominc more neEative toward the root as can be s< m
from the zero angle of attack data.) The data are given in the form of
th_ chord<<i.s_ distributions of the s_rrface pressure coeffici<_nts (upper
and lower) and the lifting pressure coefficient at the four instrumented
_ao=omo. In {>eneral, the static data agree with those of other biconvex
airfoil and rectangular wing studies (e.g., refs. i0 and ii) and require
little comment.
For the transonic soeed ran{p (H = 0.9 - l.lO)_ the local surface
Hath ntunb :rs w_rc com_uted from equation 11.4 of reference 12 and the
result:', are presented in figum_e 4. These data sho-_ that for Hach numb,:r_
l.O and i.l, th_: _L%ch number frcczc condition (see, o._._ ref. 9) exist,:_l
14
over the entire surfaces at both angles of attack, while at 0.9 Mach
number a significant freeze occurred only at _o angle of attack and then
was confined to the leading edge of the upper surface.
Unsteady static pressures occurred at the root station over both
surfaces and were most significant near the leading edge. These were
attributed to the effects of a turbulent wall boundary layer. In addi-
tion, certain unsteady pressure phenomena which were encountered during
the oscillatory tests are believed to have their origin in the static
environment of the wing_ and the relevant features will be pointed outhere.
First, at Mach number 0.9, the presence and location of the shock
wave terminating the region of local supersonic flow should be noted as
seen in figures 3(g) and 3(h). Although it is not sho_cn in figures 4(e)through 4(h)_ there were also shock waves terminating a small region of
supersonic flow near the leading edge of the upper surface at H = 0.7
for _ = 9°. Small changes in the location of these shock waves were
constantly occurring as a result of slight changes in the flow conditions.
Second, although the effective semiwedge angle of the airfoil at the
leading edge was approximately 7°_ and thus the upper surface of the _ing
leading edge was at approximately a 2° negative angle of attack _¢ith
respect to the free stream_ the _ = 5 ° data of figure 3 indicate that
the induced upwash raised the effective flow angle sufficiently to require
the flow to expand over the upper leading-edge surface. Because of the
sha_pness of the leading edge, a small region of separated flow probably
existed for most test conditions. However_ the data of figure 3 indicate
the possible separated region to be confined to the region ahead of the
5-percent chord under static conditions.
Finally, at 5° , for Mach numbers 0.9_ 1.0, and i.i, the local Mach
number plots (fig. 4(e) through 4(h)) sho_¢ that the flo_¢ over the uoper
surface of the wing _ras almost entirely supersonic_ therefore, the wing
tip may be expected to have a domain of influence on the upper surface
similar to that of a purely supersonic wing, and the associated abrupt
changes in the pressure at the boundary of this region. The approximate
location of this cu_ed boundary line for M = i.i was computed from the
local Mach number data of figure 4, and is shown in figure 5; as a result
of the transonic Mach number freeze over the tip portion of the upper
surface, this boundary-line position is also applicable for M = 0.9
and 1.0. The correspondence of its location with the position of the
slight peaks nearest the leading edge in the static-pressture distributions
of figures 3(h), 3(k), and 3(n) at the three outboard stations shows that
the peaks mark the boundary of the tip domain of influence. The boundary-
line location for M = 1.3 was computed in a similar manner and is also
shown in figure 5- However, for this Mach number, the tip effect, in
terms of peaks in the pressure distribution_ was so small that it was
negligible statically and unnoticeable dynamically.
A
3
54
15
Dynamic Data
A
3
54
Mode shape.- The mode shape, normalized with r_spect to th_ tip
amplitude_ is shown in figure 6 for both wind-off and _¢ind-on conditions.
The wind-off curve was obtained by means of depth micrometers as described
previously. The wind-on data were obtained through use of the acceier-
ometer equation (A4) of appendix A. Data are missing from the 35-pc_reent-
station accelerometers because of an electrical failure at this station
in the early stages of the investigation.
Chordwise pressure distributions; _ : 0°.- Figure 7 presents the
result of applying equations (All) to the data for M = 0.24 to give the_
components of the chor_rise surface pressure distributions her unit tip
angle of attack, in phase and out of phase with the wing bending deflec-
tion. Application of equations (AI2) to these data yields the in-phase
and out-of-phase components of the chef,rise distribution of lifting-
pressure coefficient as sho<m in figure $. Finally_ equations (AI3) yield
the chordwise distribution of the lifting-pressure coefficient amplitude
and phase angle shown in figure 9, the form in which most of the d_u_uic
results of this investigation are presented.
The pressure distribution for M = 0.24 (fig. 9(a)) appears smooth_
but as the Mach number increases, various irregularities in th<_ data occur_
as is evident in the succeeding parts of figure 9. The general deteriora-
tion in smoothness can be attributed to the inherent loss in system
accuracy with increasing Mach number discussed previously. However_ cer-
tain of these irregularities are isolated enough to demand an explanation
based on the physical characteristics of the flow.
The first of the isolated irregularities is the sharp break in the
phase-amgle distribution at M = 0.70 (fig. 9(b)), particularly at the
two outboard stations. Because of the associated small pressure amplitudes,
these jumps of nearly 180 ° are probably partially accounted for by the
inherent inaccuracies in the phase-angle computations for small pressu1_e
amplitudes, but their consistent nature may be due to possible tm_lel
resonance effects which will be considered later.
In the M = 0.9 data (fig. 9(c))_ this jump is still present and is
accompanied by sharp peaks in the pressure amnlitude distr!bution over
the aft portion of the chord. The static-pressure data show that the term-
inating shock wave is also in this vicinity, and the pressure peaks are
attributed to the movement of the shock with the wing motion. The ampli-
tude of the peak is roughly a measure of the harmonic content of the
oscillations in static pressure associated _ith this shock motion_ which
is s_nerposed on that arising from the oscillatory angle of attack. T_o
sets of data are also shown for this Mach number_ and the poor repeata-
bility in the neighborhood of the shock is further evidence of the
sensitivity of the shock location to slight changes in the flow conditions.
16
Chord_¢ise _ressure distribution_ _ = 5o. - The components of the
chord_,_ise pressure distribution on the upper and lower surfaces are shown
in figure i0 for H = 0.24; also sho_n for comparison are the correspond-
im6 data for _ = 0° from figure 7- The radical change in the pressttre
distribution at the leading edge of the upper surface is apparently the
r_,_sult of th_ ttnsteady pressures associated with the changing chordwise
extent of the leading-edge separation mentioned previously, caused by
the additional increment in angle of attack furnished by the wing oscilla-
tions. As shown in figure ll(a), the phase angles were relatively
_uuaffected by the separation.
At a Hach number of 0.70 (fig. ll(b)), both the amplitude and phase
anglc_ of the pressure coefficient exhibit leading-edge irregularities_
but th__'s,_,may not result entirely from flo_ separation. As mentioned pre-
vious! K. local supersonic flow occurred over the leading edge at }4 = 0.70
_rhen at angle of attack, and movement of the shock terminating this region
could have caused increased pressui_e amplitudes in the same manner as that
,described for H = 0.90 at 0o. Because of the reduced angle of attack
associated with bending oscillations (as compared with that for M = 0.24)
sild also the increased Reynolds number_ it is doubtful if significant
l_ading-edEe separation occurred except perhaps near the tip. This appears
to be the case for the data of Hach numbers 0.90_ 1.0, and l.l (figs. ll(c),
(i), and (e)) which show a large pressure-amplitude peak at the 90-percent
s_mispan station. The data also show pressure amplitude peaks at the
three-tenths and mid-chord points of the 70 - and 50-percent semispan sta-
tions, respectively. The similarity of the data for these three Bach num-
bers is evidence of the Mach number freeze sho_n in figure _. As discussed
_,r_viously, the freeze held only over the for_rard portion of the chord at
H = 0.90. Altho_h not presented herein_ the sel_arate surface data show
that the peaks occ_rrcd on the upper surface only; the coincidence of their
locations _ith those of the abrupt changes in static pressure which were
previously sho_m to define the boundaries of the tip domain of influence
suggests that the peaks are the result of the movement of this boundary
line with the wing motion. As can be seen_ relatively small (but well
defined) changes in static pressure at the boundary may produce a surpris-
inEly large pressure-amplitude peak in the immediate neighborhood of the
botu_dary. However, the diminishing of the peak at the 50-percent station
_hen the flo_¢ is accelerated from H = 1.0 to H = i.i, attests that this
neighborhood is extremely small_ and the peak might not be detected unless
there _ere an orifice located there.
Section lift and moment distributions; _ = 0°.- The oscillatory
aerodynamic section lift and moment coefficients were obtained by numeri-
cally integrating the lifting pressure coefficient distributions for each
l_ch number corresponding to those of figures 8(a) through 8(d) for
M = 0.24. It -_as assumed that these pressure distributions varied linearly
bet_een adjacent chord points, and the necessary leading- and trailing-
<d[]_::values were selected by inspection for each spanwise station. A
trapezoidal scheme of integration, consistent with these assumptions; was
th_n utilized to obtain the in-phase and out-of-phase components of both
A
3
54
8¥
17
A
354
the section lift and the section moment coefficients per unit effective
tip angle of attack. From this information, the magnitude and phase angle
of the oscillatory section coefficients were computed and the results are
show_ in figure 12. Of particular interest are the rerun results of
figures 12(c) and 12(f) which show remarkable agreement considering the
approximate nature of the integrations and the general erratic character
of the original pressure distributions.
Although there is some question as to the qu_utitative accuracy of
the results, definite trends with increasing Hach number are in evidence.
Among these are the decrease in the section lift phase angle from approxi-
mately 120 ° at H = 0.24 to 55 ° at H = 1.3, although at each Hach number
this phase angle is relatively constant across the span. One of the most
remarkable features of these data is the gradual flattening of th_ distri-
bution of the section lift coefficient amplitude, until at H = 1.3
(fig. 12(f)) it is almost elliptical. It might be expected that at least
the out-of-phase component _,_ould retain some evideuee of the mo_e shape
and have its m_ximum somewhere in the vicinity of the wing tip as it _ou.l!
theoretically for a parabo!ically twisted wing in st_a:ly linearizod super-sonic flow at M = 1.3; but figure 12(f) shows that the m_im<_ oscillatory
lift is produced at the root section where the wing deflection is zero.
Similarly, for the section moment distribution, the most striking effect
of compressibility is the decreasing of the root phase _gle from 75 ° at
M = 0.24 to -_15° at M = 1.3_ _hile the phase m_gles for the thre< outboar<{
stations are relatively unaffected.
Because of the general irregularity of the pressure distributions for
: 5° , no attempt _as made to integrate them and obtain the associated
section lift and moment coefficients.
Comparison With Theory
Of the many oscillating wing theories developed during tb:_ last ten
years (see ref. 13 for a fairly complete survey) only two furnish the
lifting-pressure distribution on a defonning wing of moderate aspect ratio_and both of these assume the conditions for linearized flow. For subsonic
flow, there is the approximate kernel function approach of references 2_
14, and 19 which has recently culminated in a usable program for the IBM 704
electronic computer_ and for pure supersonic flow there is the analytically
exact potential obtained %y Miles (ref. 3). The latter has also been
developed into a usable program for this report (appendix B). Mollj-
Christensen (ref. 7) and Miles (ref. 16) have investigated the necessary
conditions under which unsteady linearized flow theory is valid, and
reduced them to the following:
18
_¢here A
1. A_ MA_ kA, k_ << 1
2. A-_ >>SmJs /
3. I}_ - iI >>a213
4. k >> A 2/3
any one or mor,_:of these
is the thickness ratio of the wing.
From the values of the appropriate test parameters, it is seen that
although condition i was a_¢ays met, conditions 2 and 4 were never met,
while condition 3 would require either M >> 1.O65, or M << 0.930. There-
fore, in general, it }_ould not be expected that the results of these linear-
ized theories should agree well with those obtained experimentally, except
possibly at H = 0.24, 0.70, and i.30. With the reservations dictated by
these considerations, the comparison between experimental results and thoseof the above linearized theories _ill be examined in more detail.
For the subsonic speeds, the computing progrs_n r_quires the selection
of control ooints on the plan form at _¢hich the dowY_wash (as determined by
the mode shape) is to be specified. 9_elve control points were chosen;
the 1/4-, 1/2-, and 3/4-chord points at the 20-, LO-, 60-, and SO-percents_m_ span stations.
At H = 0.24_ c_ = O, the agreemen± bet_,¢een the theoretical and
_xperimental pressure distributions is good (figs. 7, 8, and 9(a)). The
data were obtained at the greatest static pressure of any data of the
inv,sstigation, so that the phase lag and attenuation of the, orifice tubes
were at their minimum values. There ',¢asalso no evidence of stream fluc-
tuation, so that these data are the most reliable of all that were obtain,_d
in the oscillatory testing. The good agreement bet_¢een theory and experi-
ment _,ho_¢s that linearizecl theory is adequat_ to d_scrib_ _ -_,hc flo,,rat this
blach humbler. The comparison bet_,r,;_,enth,_ory an i oxperfmeT_t in tht_ fo_n of
section lift and moment coefficierlts (fig. l,_(a)) sho_¢s considerably .Ynore
discrepancy than would be expected on the basis of the pressure distribu-
tion comparison. The primary cause of the discrepmncy has been traced to
the oressures assumed to exist at the leading edge; _¢hereas the exp<_rimen-
tal data were extrapolated to pressures considered reasonable on a physical
basis, the theory, of course, assumes the usual subsonic singularity.
Therefore, if a more realistic handling of the leading-edge pressures could
b(_ incorporated into the theory_ better agreement with the experimental
values of lift and moment would undoubtedly result.
At H = 0.70, _ = 0 (figs. 9(b) and 12(b)), although the pressure
_aplitudes are in fair agreement, the phase angles are not, particularly
near the three-quarter chord line_ while the lift amplitudes show a
noticeable decrease from their values at H = 0.24. The latter circum-
stance suggests the possibility of wind-tunnel resonance rather than
violation of the limits for application of linearized theory. If the
19
resonance induced by a bending wing in a rectangular test section is
considered to be primarily a two-dimensional problem_ the theory of ref-
erence 16 is applicable for predicting the combination of Mach number and
reduced frequency at which resonance could occur. The slotted floor and
ceiling construction of the test facility necessitate considering an
effective tunnel height somewhere between 6 and 18 feet. A plot of the
appropriate test parameters is given in figure 13, with critical resonance
curves from reference 17, based on tunnel heights of 6, 12, and 18 feet.
As can be seen from the figure, resonance at H = 0.7 could indeed occur,
for an effective t_el height of approximately 13 feet_ and this is
believed to furnish a partial explanation for the above-mentioned
discrepancies.
At H = 0.9 (fig. 9(c)),the agreement is poor_ particularly in the
vicinity of the shock waves over the afterportion of the chord. This is
to be expected, since linearized theory is unable to account for the
pr_sence or the effects of these shock waves. The poor agreement is also
evident in the lift and moment plots of fig_re 12(c).
For H = 1.0, the theoretical subsonic coefficients evaluated at
H = 0.99 have been used for comparison. Although the amplitudes of the
pressure coefficients are in better agreement than they were at M = 0.90,
the corresponding phase angles are not (fig. 9(d)), nor are the section-
lift and section-moment coefficients (fig. 12(d)). The requirement for
the application of linearized theory is clearly invalidated at this Hach
number and good agreement could not be expected.
For H = 1.10 and 1.30 (figs. 9(e), 9(f), 12(e), and 12(f)), there is
little or no resemblance between the results of the experiment and those
predicted by Hiles linearized theory for parabolic bending_ even in the
plan-form regimes in which the latter is formally applicable (see appen-dix B). The discrepancies here are primarily accounted for by the non-linear effects (detached bow wave) associated with the thickness of the
wing_ and partially by the over-all loss in system accuracy discussed
earlier in the text. For M = 1.30_ it is evident that the condition for
linearized flow theory (M >> 1.065) remains unsatisfied, while the general
r_peatability evidenced in the rerun data provides a measure of the
reliability of these data.
CONCLUSIONS
An experimental investigation was made in which the pressure
distributions on an unswept rectangular wing of aspect ratio 3 oscillatingin its first symmetrical bending mode were obtained over the Hach number
range 0.24 to 1.30. From the results of the investigation, the followingconclusions can be made:
2O
i. Large effects on the oscillatory pressure distribution can becaused by the presence of static phenomena_such as local shock waves,flo_ separation, and finite span effects.
2. The results of linearized theory comparedfavorably with thoseobtained experimentally for M = 0.24 at _ = 0° where the flow was essen-tially incompressible and inviscid. However, the agreement deterioratedat transonic speeds and at angle of attack where the important effects ofthickness_ local shock waves, and separation could not be adequatelydescribed by linearized theory.
3- The comparison of the experimental and theoretical results tendsto confirm the Miles and Moll_-Christensen criteria for the application oflinearized flo_ theory.
4. Someevidence of _ind-tumnel resonance _as noted; however, theeffects on the experimental data appeared to be confined to the M = 0.70results.
_. The experimental method for obtaining the oscillatory pressuredistributions developed in this investigation in general gave satisfac-tory results. The upper range of Machnumbers_as limited by theincreasingly poor frequency response characteristics of the pressure-measuring equipment.
AmesResearch CenterNational Aeronautics and SpaceAdministration
Moffett Field, Calif._ July 26, 1960
A354
21
APPF_'@ IX A
OSCILLATORY TEST DATA REDUCTION EQUATIONS
Acceleromet er Equations
A
3
54
In the follo_ing equations, the outputs of the accelerometer strain-
gage bridges, which in reality are alternating voltages, are given the
symbol G which, from the section on Instr_aentation, indicates a static
galvanometer deflection. No confusion should be caused, however, since
it was sho_ that the static deflection is directly proportional to the
amplitude of the alternating quantity, after scaling by the appropriate
constants. All constants are taken to be included in the factors K h
and K@.
8
h
Sketch (c)
Consider the accelerometer geometry shown in sketch (c). Assume
harmonic wing motion of bending and torsion. As a result of the chord-
wise rigidity, the motion of each cross section is completely specified
by
h : lhl sin et
@ = I@I sin (tot + N)
For each of these motions, by virtue of the high natural frequency of the
accelerometers, the total output of the strain-gage bridges is directly
proportional to the amplitude of the motion in question and may be taken
to be in phase with it.
22
These outputs are as follows:
a. due to bending b. due to torsion
forward accelerometer Khfl h 1_o2 (rKhf + Kgf) lel_2
rear accelerometer Khrl h l_2 (rKh r + Ker )l@lw 2
_here
Khf fo_ard accelerometer galvanometer deflection per unit linear
acceleration
Kh r rear accelerometer galvanometer deflection per unit linear
acceleration
K_f forward accelerometer galvanometer deflection per unit angular
acceleration about forward accelerometer root
K@ r rear accelerometer galvanometer deflection per unit angular
acceleration about rear accelerometer root
For the combined motion_ the rotating vector diagrams for the
fo_ard and rear accelerometers are shown in sketches (d) and (e):
Forward /accelerometer L'/_ ..._flh I
Khflhlw=
%f I --./_-7-L_-_ I_L
_-_'- "G if -- rotor
(rKhf + KSf)181 '_z I
A
354
• Sketch (d.)
23
A
35
/Rear /
accelerometer
G°r F Kh---r+iOr]ol _Jt _ S_ \ _,h I
lIj tr__ __ i__ _iResolve r
rotor
Sketch (e)
For the foz_¢ard accelerometer_
Oil = Khfl hle%os ;_h- (rKhf + K@E) @ leacos F_6
Gof = Khfl hleasin 2 h - (rKhf + K@f) _ i_2sin f_a
(AZ)
For the rear accelerometer_
Oi r = Khrl hl_2cos nh + (rKh r + K_ r)
Go r : Khrlhlm2sin 2h + (rKh r + K_ r) 0 le2sin N0
(A2)
_h,:,r<: G i and G O are the galvanometer deflections proportional to the
strain-6age signals in phase and out of phase _,¢ith the resolver rotor.
In actual practice in-phase and out-of-phase components were indicated
by a separate strain-cage bridge or else the t_¢o components _ere read
consecutively; with either procedure there _as the possibility of diff-
,_rent sensitivities Kh and K e for the t_o components. To indicate the
_possioility of different sensitivities for the in-phase and out-of-phase
eom_)orle_ts_ eq-,ations (A1) and_ (A2) are rewritten as
24
Gif = Khfilhl _ac°s g_h - (rKhfi + K_=i_')l@Iw 2cos f_@
= hhl wasin _h - (rKhfo + _±,o )l@l_2sin g@Gof Khfo
= Ihlw2c°s _h + (rKhri + K@ri) l@l_2c°s g@Gi r Khri
Go r = Khrolhl _2sin gh + (rKhro + Kero) l@l_2sin _@
(A3)
Equations (A3) can be solved for the following quantities at each of the
four accelerometer stations:
(a) Bending amplitude
m(rKhr + K8 r )Gif + (rKhf + Kggf )Girt
l I __i___i ..... S____L Ilhl : _ b 2rKhfiKhri + KhfiK@ri + KhriKSfi ]
(rKhf O + KOfo )GOr + (rKhr O + Kero )GOf._2 }i/2+ [ 2rK----hfo-Khr--_+ K_fo--K_r_ + "Kh%o-KOf% j (A4)
A
354
(b) Phase angle between resolver rotor and bending amplitude
I rKhf____o_O i K@fO)f or i (rK___hr_O i KS rO)______GO 7_o_ro +_o_ro+_hro_oJ_h = tan'm
+ _+ +_2rKhfiKhri + KhfiKSri + KhriK@fi J
(AS)
(c) Torsion amplitude
I m { KhfiGir _ KhriGi f )218' = _-_ b_ 2rKhfi Khr_" + K--hfiKor-_m + _KhriKSfi
+_2rKhfoKhrN + K hfoK8 r----_+ K hroK8 fo
(A6)
4¥
25
A
35
(d) Phase angle between resolver rotor and torsion amplitude
_? = tan -1 L2rKhfoKhro + KhfoK0ro + KhroK0 f
__ KhfiGir - KhriGif
(e) Phase angle be%¢een bending and torsion
(AT)
fi : f18 - 2h (A_%)
Pressure-Measurement Equations
Consider the harmonic pressure variation of amplitude IPI and ohase
angle hh occurring on the surface of the wing at a particular orifice_
where hh is measured with respect to the wing-tip position. Because of
the attenuation and phase characteristics of the pressure line conmectin_
the orifice at the wing surface with the pressure cells outside the tmunei_
the pressure cells experience a pressure of amplitude IP'l which lags
the pressure at the orifice by the phase angle 5. As mentioned previ-
ously in the text_ all aerodynamic quantities are made dimensionless _rith
respect to the wing-tip angle of attack amplitude. The necessary geometry
is indicated in the rotating-vector diagram of sketch (f).
Resolverrotor
Sketch (f)
The amplitude and phase angle of the tip motion are indicated by the
subscript T.
_6
Demote by the symbol Z the ratio of the pressure amplitude
experienced by the pressure cells to the pressure amplitude IPI
on the wing surface. Then
IP'I
exi st ing
and
Now
or
IP'IIPI = Z
hh = A + _ - _hT
Pi = IPlcos hh
= IPloos(A+ _ - _hT)
Po'I P' I Pi' cos _)) +-
= T IP'l (_ - IP'l sin(_hT - _)]
Pi = _ Pi' c°s(_h T - b) + Po' sin(_h T - _)
A
353
and similarly,
Po:_ Po' cos(a_-_) -Pi' si_(_-_)
Now
Pi' = KiGi Po' = KoGo
where
Ki
}[o
in-phase component of pressure amplitude per unit galvanometer
deflection
out-of-phase component of pressure amplitude per unit galvanometerdeflection
27
T]lerl
Pi =
Po =
[XiGicos(_hT _) + Xo_osin(_,_ _)_J } (A9)
A
3
4
The amplitude of the wing-tip angle of attack is given by
lil T eb hi T- (ilO)_h - V V
Dividing equations (Ag) by (AIO) and by the dynamic pressure q
gives the component of the surface pressure coefficient in phase and outof phase (hereafter denoted by Re( ) and Im()) _rith the bending
amp iitude.
}_e( cp_ h) -
m(cp_h) -
V IKiGicos (ShT -
V [KoGocos (_hT _z_qlhLT
8) - KiGisin(i'_hT - 3)
(All)
where Z and _ are found as indicated in the text.
Equations (All) were developed for a general orifice on either the
upper or lo_¢er _ing surface. The data for M = 0.24 obtained by means of
these equations are show_ in figure 7 as Re Cp_ ] , Re ' and\ h/u
Im p_lu, Im , the subscripts u and Z denoting upper and
lower wing surface.
The real and imaginary components of the lifting-pressure amplitude
are given by
Re p_h/l = Re - Re p_u
Im C- Im p@_P_nl P_ _ u
(AI2)
and are shown for M = 0.24 in figure 8.
28
The magnitude and phase angle with respect to bending _nplitude of
the lifting-pressure coefficient are given by
icp_ht: {[Re(Cp%)]_ + [I_(Cp%)]_}_j_
{h : tan-l
Re(Cp%)
(A13)
The data are presented in this form for all Mach numbers in figure 9-
Integration of equations (AI2) with respect to dimensionless chordwise
coordinates gives
SO I
I_(Cp%) a{
(AI_)
A
354
where
_(Cr_h) : I_(Cmh) (_' - _)a_
= x/2b and [ is the center of moments ([ = 0.5) from _,rhich
I c_%L : {IRe(Or%)] _ + [I_(Ct%)_} _j_
7h = tan-i
Ic_l : {[Re(C_)] _ + [m(C_p]_} _j_
7hm = tan -lRe(C_h)
(AIS)
29
Figure 12 giv<-s tho r,:sults for all Hach numbers at zero angle of
attack in this form comouted from numerical integrations correspon:lingto <quations (AI4).
A
3
4
3O
APPENDIX B
APPLICATION OF MILES' THEORY FOR SUPERSONIC
RECTANG_ OSCILLATING WINGS
X,p X I
Sketch (g)
In reference 3 Miles considers
the linearized problem of an infini-
tesimally thin quarter-infinite
lifting surface having one straight
edge parallel and another normal to
_--Y,Y,the stream_ undergoing harmonic
oscillations of frequency _. For
ease of reference, the surface and
an appropriate coordinate system is
shown in sketch (g).
By use of the laplace trans-
formation, he obtained the exact
linearized expression for the com-
plex amplitude of the perturbation
velocity potential on this surface.
With suitable identifications, and
with the signs of exponentialsreversed for more conventional
phase interpretation_ this expres-sion is as follows:
A
354
kiZ _oxl Xle-iMk]_u_
k \_ VA
(BI)where
2kM xk I = -- X I =
_a 2b
Yl = i 9 = i - yl
_a = Ma - i J1 =
first-order
Bessel
function$ = _e -i_t ml = 2
31
and where Z and 2b are reference lengths in the span_ise and streaaT_isedirections, respectively; _(xl,yl,k) is the downwashamplitude resultingfrom a wing deformation prescribed by
H(x,y,t) = ZH(xm,Yl) e i_°t (postive do_) (B3)
A
3
54
and related to it through the equation
_(xl'y!_k) = - VA2 [_ (x1'Yl)+ 2ikH(x1'Yl)l(m_)
The superposition and lift cancellation techniques of supersonic steady
flow can be used to obtain from (BI) the potential at all points on a
finite rectangular plan-form lifting surface performing either symmetrical
or antisymmetric oscillations, such as that indicated by the dashed lines
in sketch (g)_ providing the Mach line from one wing tip does not intersect
the side edge of the opposite wing panel (i.e., _A _ i).
From this potential, with
Z_p =_peiWt = 2p _-_ + V _)
defining the lifting pressure_ the expressions for that pressure_ the
section lift, and the section moment distributions are obtained (after
some integration by parts) as follows:
(Bs)
_(y_k) = [#- _ VT (1,y_)
+ 2ik _o I _ (_l,yl k)_xl] (B6)
D_
_ma_ (Y_,k) = •4qb 2 VZ
_,1 ¥ (xl,yz ,k) 4xz- 4iki x_ v-£
"- O
(B7)
where the vanishing of the potential at the wing leading edge has been
used, and xl = al is the moment axis.
Even for elementary deformation modes, the analytical integration
of equations (BI) and (BS) through (BT) has not been accomplished as yet,
and in order to facilitate numerical evaluation, it is convenient to
consider the following classes of mode shapes.
For s_mmetric modes:
H(xm,yl) = Hn(S)(xm,yl) = + lymln(Anxl + Bn)
for all Yl, n = O, i, 2
A
354
For antisymmetric (rolling) modes:
H(xl,yl) = Hn(A)(xl,y!) = + lylln-l(Anxl + Bn)yl
n : 1, 2 (BS)
It is seen that this choice of mode shapes limits subsequent
application to wings undergoing chordwise rigid deformations for which
the spanwise variation is expressible in a power series in Yl = }-
Upon substitution of equations (B$) and (BI) into equations (BS)
through (BT) it is possible (after some interchanging of the orders of
integration) to carry out all except the final two integrations exactly,
but numerical methods are necessary in order to complete these. Series
methods are applicable to the inner integrals since their integrands con-
sist of certain products of Bessel, trigonometric, and algebraic functions,
and as the resulting integrated functions are well behaved, Gauss' method
is employed to effect the outer integration.
This work and the subsequent combinations necessary to obtain the
final answers was programmed for the IBM 704, and this programwas used
33
in obtaining the supersonic theoretical results presented in this reoort.
The input information required is the specification of the Mach number H,
eb Z the momentthe true reduced frequency k = -_- _ the aspect ratio A : _ ,
axis al, and the mode shape as reflected in its general classification
(symmetric or antisymmetric)_ the choice of n_ and the coefficients An
and Bn. In its present form_ the program is only applicable to the
aerodynamic situations for _<hich _A k 2_ i.e._ those for which each tip
}hch line is entirely contained on its wing panel.
The program furnishes the pressure distribution at any specified
points on the plan form_ and the section lift and pitching moment for any
designated spanwise stations. Since the various integrands are essentially
nonsingular_ the required series expressions and integrations are straight-
fo_£ard and it is believed that the final answers are in error by no more
than i percent.
34
REFERENCES
i. I,_ACA Subcommittee on Vibration and Flutter: A Survey and Evaluation
of Flutter Research and Engineering. NACA RM 56112, 1956.
o
,
,
f
ril.
Watkins, Charles E., Runyan, Harry L., and Woolston, Donald S.: On
the Ker_lel Fuslction of the Integral Equation Relating the Lift and
Dovrnwash Distributions of Oscillating Finite Wings in Subsonic Flow.
NACA Rep. 1234, 1955. (Supersedes NACA _ 3131)
•
i0.
Miles, John W. : A General Solution for the Rectangular Airfoil in
Supersonic Flow. Quart. Appl. Hath., vol. ii, no. i, 1953.
Woolsto1_, Donald S., Clevenson, Sherman A., and Leadbetter, Sumner A.:
Analytical and Experimental Investigation of Aerodynamic Forces and
Moments on Low-Aspect-Ratio Wings Undergoing Flapping Oscillations.
I_ACA T}_ 4302. 1955.
ii.
Wilmeyer, Edward, Jr., Clevenson, Sherman A., and Leadbetter,
S_mner A.: Some Measurements of Aerodynamic Forces and Moments
at S_sonic Speeds on a Rectangular Wing of Aspect Ratio 2 Oscil-
lating About the _dchord. NACA TN 4_40, 1958.
Laidlaw, W. R.: Theoretical and Experimental Pressure Distributions
om Lov_-Aspect-Ratio Wings Oscillating in an Incompressible Flow.
M.I.T. Aeroelastic and Structures Res. Lab. Rep. 51-2 (Contract
HO a(s) )2-5_6-6), Sept. 1954.
Moll_-Christensen, Erik L.: An Experimental and Theoretical Inves-
tigation of Unsteady Transonic Flo_. Sc.D. Thesis, M.I.T., 1954.
Epperson_ Thomas B.. Pengelley, C. Desmond_ Ransleben_ Guido E., Jr.,
a_d Younger_ Dewey G.. Jr.: (Unclassified Title) Nonstationary
Air!oad Distributions on a Straight Flexible Wing Oscillating in
a Subsonic Wind Stream. WADC TR 5_-_23, Jan. 1956 (Confidential
Report) .
Lessihg_ Henry C., Fryer, Thomas B., and Head, Merrill H.: A System
for Measuring the Dynamic Lateral Stability Derivatives in High-
Speed Wind T_inels. NACA TN 3345, Dec. 1954.
Knochtel, Earl D.: Experimental Investigation at Transonic Speeds
of Pressure Distributions Over Wedge and Circular-Arc AirfoilS_ctions and Evaluation of Perforated-Wall Interference. NASA
TN D-15, 1959.
Croom, Delwin R.: Low-Subsonic Investigation to Determine the
Chordwise Pressure Distribution and Effectiveness of Spoilers on
a Thin, Lo_-Aspect-Ratio, Unswept, Untapered_ Semispan Wing and on
the Wing With Leading- and Trailing-Edge Flaps. NACA RHL58B05,
1958.
3_
12.
13.
14.
15.
17.
Liepmann_ H. W. and Roshko_ A.: Elements of Gasdynamics. John Wiley
and Sons, Inc., 1957_ Pp.342-346.
Garner, H. C._ and Acum, W. E. A.: Note on Some Recent P_ers on
Oscillating Wings. British ARC 18952, January, 1957. Title
Unclassified_ Contents Confidential.
Runyan, Harry L._and Woolston_ Donald S.: Method for Calculating
the Aerodynamic Loading on an Oscillating Finite Wing in Subsonic
and Sonic Flow. NACA Rep. 1322, 1957. (Supersedes NACA TN 3694)
Watkins, Charles E._ Woolston, Donald S._ and Curmingham, Herbert J.:
A Systematic Kernel Function Procedure for Determining Aerodynamic
Forces on Oscillating or Steady Finite Wings at Subsonic Speeds.
NASA TR R-48_ 1959.
Miles, John W.: The Potential Theory of Unsteady Supersonic Flo_.
Cambridge University Press_ 1959.
Runyan_ Harry L.,and Watkins_ Charles E.: Considerations on the
Effect of Wind-Tunnel Walls on Oscillating Air Forces for Two-
Dimensional Subsonic Compressible Flow. NACA Rep. 1150, 1953.
(Supersedes NACA TN 2552)
36
A
3
54
37
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54
(d) Interior of upper surrrce of wing.A-235t0
Acceleromet,
(e) Interior of lower surface of wing.
Figure i.- Continued.
A-23509
6¥
41
A
354
Bleed filter
To wing orifice tube--_--
I
L
jfJ
fJ
fJf/
fJ
//
ft
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Fj
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o'orMode148-D
Scanivalve body
ToP.(for static testing only)
Pressure cell
Sconivolve manifold(stator)
(f) Details of Scanivalve modification.
Figure i.- Continued.
42
L_
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orlo
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_o ,-i._i
b_0
A
3
4
43
A
3
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20
O
!ili:i!il!ii_i_i:_i!ii!iiii
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24
Cp uor
CPt
-.2
+-.2
0
+-2
0
-+.2
,20 .2 .4 .6 .8 1.0
xl2b
(a) Pressures on upper and lower surfaces; M = 0.24, _ = 0°.
Figure 3.- Static-pressure distributions.
A
3
54
Cpu
or
CPl
-.5
0
+.5
0
±.5
0
+-.5
.50 .2 .4 .6 .8 1.0
x/2b
Y/Ab=90
Y/Ab=2'0
Y/Ab=.50
Y/Ab:0
(b) Pressures on upper and lower surfaces; M = 0.24_ _ = 5°.
45
Figure 3-- Continued.
46
o
0
Cp
o
0
R=2.1x I06
.2 .4 .6 .8x/2b
(c) Lifting pressures; M = 0.24, _ = _o.
Figure 3.- Continued.
Y/Ab=.90
Y/Ab=.70
Y/Ab-50
Y/Ab=O
-2
47
0 Y/Ab--.90
-+.2
Gpuor
Cpz
0 Y/Ab:.70
+-.2
Y/Ab=.50
+--.2
0 :Y/Ab=O
.20 ,2 .4 ,6 .8 1.0
x/2b
(d) Pressures on upper a_d lower surfaces; M = 0.70_ o_ = 0 °.
Figure 3.- Continued.
48
Cpuor
-I.O,
-.5
0
+--.5
0
0
0
.50 .2 .4 .6 .8 1.0
xl2b
Y/Ab=.90
Y/A b=.70
Y/Ab=.50
Y/Ab=O
(e) Pressures on upper and lower surfaces; M = 0.70, _ = _o.
Figure 3.- Continued.
'¥
15
49
I0
5
Y/Ab=.90
¥5
Cp 0 Y/Ab=.70
_.5
0 Y/Ab=.50
¥-5
0 Y/Ab=O
-.5 0 .2 .4 .6 .8 1.0x/2b
(f) Lifting pressures; M = 0.70, _ = _o.
Figure 3.- Continued. I
5o
Y/Ab =.90
Y/Ab=.70
A
354
.20 .2 .4 .6 .8 1.0
x/?_b
(g) Pressures on upper and lower suI_faces; M = 0.90_ _ = 0 °.
Figure 3.- Continued.
--I0
51
--5
O YlAb=.90
A
3B4
CPuor
cpt
+-.5
0
±.5
Y/Ab =.70
0 Y/Ab=.50
---.5
0 Y/Ab: 0
'50 .2 .4 .6 .8 I.Ox/P_b
(h) Pressures on upper and lower surfaces; M = 0.90_ _ = 5°.
Figure 3.- Continued.
52
1.5
0 Y/Ab=.90
Cp 0 Y/Ab-70
0 Y/Ab=.50
0 Y/Ab=O
•2 .4 .6 .8 1.0x/2b
(i) Lifting pressures; M = 0.90_ _ = 5°.
Figure 3.- Continued.
53
A
354
O Y/Ab=.90
Cpuor
cpz
0 Y/Ab=.70
0 Y/Ab=.50
0 Y/Ab=O
40 .2 .4 .6 .8 1.0x/2b
(j) Pressures on upper and lower surfaces; M = 1.00, _ = 0°.
Figure 3.- Continued.
54
GPu
or
cpz
- 1.0
-.5
+.5
0
-+.5
0
-.5
.50 .2 .4 .6 .8 1.0
x/2b
Y/Ab =90
Y/Ab=.70
Y/Ab=.50
Y/Ab=O
(k) Pressures on upper and lower surfaces; M = 1.00, _ : 9° .
A
3
54
Figure 3.- Continued.
55
5
Y/Ab=.90
A
3
54
O
Cp
Y/Ab=.70
0 Y/Ab=.50
O Y/Ab=0
.2 .4 .6 .8 I.Ox/2b
(2) Lifting pressures; M = 1.00_ _ = 5 ° •
Figure 3-- Continued.
56
-4
-2
GPuor
cpz
0
+.2
0
-+-.2
0
-+.2
0
.2
.40 .2 .4 .6 .8 1.0
x/2b
Y/Ab=.cjo
y/.,ed)=.70
Y/Ab:.50
Y/Ab=O
(m) Pressures on upper and lo_el• surfaces; M = i.i0, _ = 0°
A
3
54
Figure 3.- Continued.
8Y
A
3
54
Cpuor
c%
-5
+.5
+-.5
0
+-.5
0
iiii
i iii R=46x I06
L.._O CPuD c%
L_LL
i i i,_.
_+-+-+-+-
IIJI-LLL[
- _ 4- F-I--
_ +_
_-U!1-TFT !
!
58
Cp
lO
0
¥.5
(o) Lifting pressures; M : !.i0, _ = 9°.
Y/Ab =.90
Y/Ab=.70
Y/Ab=.50
Y/Ab= 0
A
354
Figure 3-- Continued.
59
Y/Ab =.90
A
354
CPuor
CPl
Y/Ab=.70
Y/Ab :50
0 Y/Ab=O
i.4
0 2 4 .6 .8 1.0x/2b
(p) Pressures on uiope:__ :_nd lowez surfaces; H = 1.30, _ = 0 °.
FLgure 3.- Continued.
6o
--.5
Cp uOf"
cpz
0
-+.5
±.5
-+.5
0
.50 .2 .4 . 6 .8 1.0
x/2b
Y/Ab=.90
Y/Ab=.70
Y/Ab=.50
Y/Ab=O
(q) Pressures on upper and lower surfaces; M _- !-30, e = _o.
A
3
54
Fig u_e 3.- Continued.
3
4
IO
¥5
-T-.5
0
0
6
t
T÷
i
T
i
4+
L
L
•2 .4 .6 .8 1.0x/2b
(r) Lifting pressures; M : 1.30_ _ : 9o.
YYAb:.90
Y/Ab=70
_Ab:.50
Y/Ab=O
61
Figure 3.- Concluded.
62
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