Document
ON THE PREDICTION OF CRITICAL FLUTTER CONDITIONS FROM
SUBCRITICAL RESPONSE DATA AND SOME RELATED
WIND-TUNNEL EXPERIENCE
J. C. Houholt and A. G. Rainey -- NACA, Langley Laboratory,
Langley Field, Virginia
Abstract ocity enters. Actually, the work startedwhenwe were considering the application of ideas suggested by Methods of interpreting response measurements Professor Moll_-Christensen. The present work which could be amenable to flight flutter testing pro- evolved as a special consideration, and we thought it cedures are being studied analytically and in the wind to be of enough interest to merit separate attention.
tunnel. One suggested scheme, which requires evalu- ation, is an iterative technique in which derivatives In the first part of the paper an elementary but obtained from subcritical response data are used to rational analysis is given to show how the response indicate the approach to flutter. This paper considers of a wing system might be expected to depend on air a simplification of this procedure by examining the density, for both the cases of sinusoidal and random manner in which a single characteristic of the sut)- torce input. A theur_i_i ,nodel i11u_trating thc critical response behaves in relation to variations of technique of extrapolation to the flutter condition is the density or dynamic pressure in the approach to then considered. Then, inthe second part of the paper, flutter. The use of this single parameter scheme is attentionis focused on the experimental testing of the examined for random excitation as well as for sin- approach by application to some wind-tunnel studies.
usoidal forcing. The feasibility of the method is illustrated by several examples and the relative merits of random and sinusoidal excitation are discussed: ANALYTICAL TREATMENT INTRODUC TION Derivation of Extrapolation Equations In this paper certain new slants are given on the Let us consider an aeroelastic system which is prediction of critical flutter condition from subcritical being excited into motion by either a sinusoidal shaker response data. Specifically, the technique considered or a sinusoidal gust, and then proceed to investigate herein deals with the manner in which the forced how the amplitude of the response, such as deflection, response behavior of an aeroelastic system varies is dependent on the density of the air flow. To do this, with changes in air density, while velocity is being introduce the equation governing the motion of the held essentially fixed. The impression is not to be system as follows given that density considerations are necessarilynew, but rather the point of view is held that a further Dw = p = J mw + p V2DL TM + F s + z Fg (1) examination of density effects may lead to a simple index which may be useful in the prediction of flutter.
The motivation stems from the fact that density appears where the equation may be interpreted either in dif- in a rather clean-cut fashion in the equations for ferential operator form or in matrix notation. The flutter, in contrast to the complex way in which vel- operator D on the left hand side converts the surface deflection w into the total surface loading composed By inverting this equation and at the same time sep- of the inertia, aerodynamic, and applied loadings on arating the effects of the shaker and gust terms, we the right hand side. The operator D L is complex and arrive at the final two equations which indicate how the is a function of Mach number and reduced frequency, amplitude of wing deflection varies with density and when operating on the deflection, leads to the aero- dynamic loading; the shaker force F s (considered to
1 ,')IA.,<I
shaker only (7a) be distributed over a small area to give an intensity)
1%'_7 = Io_1 rp.f-p_
and the gust loading p Fg are treated together for convenience, and will be separated later. It is re- marked that the sinusoidal gust condition is introduced
Ps_}l_sl _
gust only (Tb) because this condition yields a necessary part -- the
I% I t°el P Ps
transfer function -- of the solution for response when random inputs are involved; the density p is shown These two equations suggest the basic linear extra- specifically as an ingredient of the gust loading so as polation procedure of this paper. Thus, assume that to keep the density in an explicit sense throughout in-flight measurements of response are made accord- the analysis.
ing to the following plan: we fly at a velocity near the expected flutter speed (or at a velocity for which we We now choose to make an approximate solution want to prove the aircraft safe), but take care to first of equation (1), since our essential result is arrived fly at a high altitude where the density is low. Then, at rather quickly, and will leave a more rigorous, repeat the tests at successively lower altitudes. Then, but lengthier, treatment which leads to the same re- for tests utilizing a sinusoidal shaker input, we might sult to an appendix. The approximate solution is of expect a plot of the reciprocal of the amplitude versus the Galerkin type and is made by assuming that the density to form a straight line, which when extra- deflection is expressed in terms of the modal shape I which occurs at flutter, thus polated to _ = o yields the density that ought to pro- ,-11 duce flutter. For the case of agust input,X'_-'_is plotted w = alw f (2) against p for an expected linear relationship. In the actual testing in a random force input environment, the where a 1 is a coefficient to be determined and wf is output spectrum of response will be found. But since the flutter deflection shape which satisfies the equation this spectrum is proportional to the square of the frequency response function for sinusoidal gust input, we see that the reciprocal of the square root of the Dwf = J/ mwf +p/vf2DL?f (3) output spectrum should be plotted against-- I , to arrive P which is simply equation (1) with the forcing terms at a condition consistent with that indicated by equation suppressed. Substitute equation (2) into (1), use equa- (7b).
tion (3), multiply by wf and integrate over the wing surface; the result leads to the following _olution for In applying equations (Ta) and (7b),itis implied a I that the frequency of flutter is known. This is, of course, not so; therefore the procedure to follow is to Qs + P Qg observe the amplitude-density behavior at several frequencies until it becomes clear from the frequency J - J M -p a 1 - vf2A f 2A (4) f + P/ v response plots what frequency is emerging as the flutter frequency.
where Qs and Qg are in the nature of generalized Example of Calculated Results forces As a test of the possible range of applicability Qs = f WfFsdS ' Qg = j WfFgdS of equations (7a) and (7b), response calculations were made for a rectangular cantilever wing, and inter- and preted in accordance with these equations. The re- sponse analysis was limited to two degrees of freedom, M = f mwf2dS, Af = f WfDLfWfdS , A = f WfDLWfdS (5) one bending and one torsion, and employed the aero- dynamic coefficients for M = 0.8 in a strip fashion.
In general, all of these generalized coefficients are The frequency response functions obtained for ampli- complex. At a velocity and frequency equal to the tude of torsional displacement at the wing tip are values at flutter but at a subcritical value for density, shown in Figure 1, where the curves at the left are the value of a 1 is particularly significant and is for a sinusoidal gust input, whereas the curves at the right are for a sinusoidal shaker input located at the Qs + p Qg tip and at 10 percent chord position. The parameter a I = (6) is a ratio of structural mass to air mass, and vf2Af _Pf - p,,
FREQUENCY RESPONSE
RANDOM SINUSOIDAL
I0
I st Ist
BEND. TOR.
I
REL I
,oo
I00 I
AMP 4 !
0 0 200 400
200 400
FREQ, W
Fig_are 1. Frequency Response therefore may be regarded as inversely proportional Figure 2. Extrapolation of the curves to_-_L= 0 indi- to air density. It is seen that as the air density cates a flutter density ( # = 89) which agrees identi- increases ( _ decreasing) an ever _ruwh_ ,_iid s,hr.rper cally with that given by a conventional flutter anaiysls.
peak develops at a frequency of 158 cps, thus suggest- The very pronounced range of linearity is also to be ing a frequency of flutter.
noted; in fact, using only the data at densities of 45 and 75 percent of the flutter density would give a Application of equations (7) to the amplitude flutter prediction erring by only a few percent. It is values at this frequency gives the curves shown in
EXTRAPOLATION TECHN IQUE
SINUSOIDAL
RANDOM
1.6
1.2
I
/_
--.8
b.4
FLUTTER
_U R
.4
i i _/'= ,I l I l I
0 40 80 120 160 ?00 _)_0 0 .004 .008 .012 .016
' (_P/
g'
Figure 2. Extrapolation Technique unimportant in these instances, and this is actually significant to note also that the data point correspond- what the experiment shows. Thus, any flight investi- ing to the 45 percent of critical density condition is gation should keep this possibility in mind.
not a major peak in the frequency response curve for this density. Thus, subcritical response data which have not yet indicated peaks may still be useful.
The single data point and dashed curve shown for EXPERIMENTAL RESULTS densities above the critical value are shown simply as a matter of interest to indicate that the theoretical response calculations based on sinusoidal conditions The previous section concerned the analytical show a branch above the flutter condition as well as background which has formed a guide to some wind- below.
tunnel experiments discussed in this section.
The main conclusion to be drawn from this The linear extrapolation technique has been example is that the present technique for predicting examined experimentally for six cases involving ran- flutter appears quite promising. In the secondpart of dom excitation and for one case of sinusoidal excita- the paper we shall see how well itworks when applied tion. These various cases are illustrated in Figure to wind-tunnel studies.
3, where a typical flutter boundary is used to illustrate the manner in which the flutter condition was ap- Before looking at the experimental results, we proached. Geometric properties of the four semi- might make a few comments on the general applica- span, cantilever mounted models are listed in Table bility of the density extrapolation technique. As with I. Model A was used to obtain three sets of sub- other flutter extrapolation techniques, there will un- critical response data -- Case I and Case II at two doubtedly be cases where this scheme breaks down.
different stagnation pressures, but increasing velocity, One possible example is that associated with wing and Case 1TI at constant velocity but increasing systems which are capable of a single degree of density. Models B and C were tested at constant stag- freedom type flutter. Interestingly enough, equation nation pressure and increasing velocity. ModelDwas (7) can be used to demonstrate why. Up to now we equipped with an electro-hydraulic shaker housed in a have tacitly assumed that unbounded response (al-----_ tip tank. This model was examined for two cases -- oo ) occurs when _/ - p becomes zero. It, of course, Case I, random excitation at constant stagnation pres- also is possible for the response to become infinite sure, and Case II, sinusoidal excitation at constant when A vanishes, and this may occur either in a velocity. In all of the cases examined the type of classical way for attached flow, or what is more flutter encountered was classical bending torsion likely, when the flow becomes separated, such as in involving the coupling of well separated modes.
stall flutter. The equation indicates that density is TYPICAL FLUTTER BOUNDARY SHOWING MANNER OF APPROACH FOR VARIOUS CASES MODEL A B I!
# DYNAMIC C PRESSURE /, CASE I D IT 'm- CASE I I I I I .I I 0 .2 .4 .6 .8 1.0 1.2 MACH NUMBER Figure 3. Typical Flutter Boundary Showing Manner of Approach for Various Cases TABLE I GEOMETRIC PROPERTIES OF MODELS TESTED Aspect Taper Sweep Airfoil Model Ratio Ratio at 1/4 C Section A 5 1.0 0 ° 6 percent Cir- cular Arc B 6 1.0 45 ° Flat Plate C 3 1/7 45 ° NACA 65A004 D 3 1.0 0 ° NACA 65A010 Random Excitation response. These results are illustrated in Figure 4 where the response magnitudes are shown as functions The subcritical response data for Models A, B, of the ratio of the dynamic pressure at flutter to the and C were obtained by recording the output of re- dynamic pressure associated with each point.
sistance wire strain gage bridges mounted near the root of the model, while the model was responding to It should be pointed out that this form of pre- the normal turbulence in the wind-tunnel airstream.
sentation is not identical to that suggested by the The response data were recorded on magnetic tape analysis. Some of the experiments were completed using frequency modulation amplifiers (ref. 1). After before the analysis was available, and the form of completing the tunnel runs, thirty-second samples of presentation chosen was such that all of the experi- the tape records were analyzed using analog data ments would be consistent within themselves. For reduction equipment described in reference 1. The example, the velocity squared term has been combined peak values in the power spectra of strain response with the density to form the dynamic pressure. This were operated on to yield numbers proportional to is a necessary step in that some of the experiments the reciprocal of the absolute magnitude of the strain involved an approach to the flutter condition primarily
EXTRAPOLATION TO FI lITTER CONDITION FROM RANDOM EXCITATION
_'GAGES)
_" GAG:)
1 o
[
[?o o o
I I I I l I [ i = I I _1
1.0 1.4 1.0 1.4 1.8 2.2 1.0 1.4 1.8 2.2
I
E
CASE -r CASETr
o CASE TrT
jl I I I I
J,,°
I i I I I I . 1 I
1.0 1.4 1.8 2.2
1.0 [4 1.8 2.2
ID [4 1.8
q FLUTTER
q
Figure 4. Extrapolation to Flutter Condition from Random Excitation through increases in velocity. These variations in shown in Figure 5. The data in the left hand part of velocity require the statement of additional qualifi- the figure were obtained in the same manner as the cations to those already mentioned ifone is to expect data of the previous figure except that the angular a linear extrapolation of the response data. Perhaps motion of the tip of the model was deduced from the combined output of two linear accelerometers mounted the most important of these additional assumptions is that near the flutter condition, the air forces in the tip tank.
associated with flutter do not vary rapidly with the The data shown in the right hand part of Figure reduced frequency and Mach number.
5 were obtained by measuring the amplitude of re- An idea of the usefulness of these extrapolation sponse at the two accelerometer stations due to a sinusoidal applied force. The amplitudes were meas- methods can be gained by examining Figure 4. A ured after the shaker had been tuned to the frequency reasonable degree of linearity of the response data is of maximum response which, in this case, appearedto indicated for all of the cases, when the dynamic pres- be associated with the torsional mode. Although some sure is within about 20 percent of the critical value response due to turbulence was present during the and the extrapolation gives a good indication of the flutter condition. The least encouraging results were shaker tests, the phase sensitive instrumentation used effectively eliminated its effects.
obtained for Model B which was poorly instrumented.
The strain gage bridges were mounted very near the It is noted that both sets of response data indi- root and were about equally sensitive to bending or torsional motions. The response data for the other cate an equally good extrapolation to the flutter con- dition. If it is assumed that random excitation and cases were taken from strain gages arranged such sinusoidal excitation will yield equally adequate extra- that they were sensitive primarily to torsional strains.
polation results, the question of relative cost or dif- It might be mentioned that the results shown for the third case of Model A indicate a linear relation to ficulty of the two methods is of interest. It was mentioned earlier that six cases of random excitation lower values of dynamic pressure than most of the other cases. This result may be associated with as opposed to one case of sinusoidal excitation have been examined. In the wind tunnel, at least, it is the constant velocity method of obtaining the response believed that this six-to-one ratio is a fair estimate data in this case.
of the relative difficulty of the two methods. This is Sinusoidal Excitation due, primarily, to the fact that the turbulence is always available while the shaker must be constructed and In order to gain some insight regarding the installed. Although turbulence also exists in the relative merits of sinusoidal excitation as opposed to atmosphere, the problem of finding it during a flight test and determining enough of its properties to permit random excitation, two cases have been examined for its use might improve the relative attractiveness of a a model equipped with an electro-hydraulic shaker sinusoidal shaker as a source of excitation.
contained in a tip tank (Model D). These results are COMPARISON OF EXTRAPOLATION FROM RANDOM lAND SINUSOIDAL EXCITATION RANDOM SI NUSOIDAL F [] alo
/
Iol
o I i 2.2 0 .4 .8 1.2 0 1.0 1.4 1.8 q q FLUTTER q FLUTTER q Figure 5. Comparison of Extrapolation from Random and Sir.u. Adal Excitation APPENDIX We may now proceed to solve equation (1) by expressing the deflection by the following series ex- pansion involving w n The Response-Density Relationship A more rigorous development of equation (7)can w = alw 1 + a2w 2 + a3w 3 + ... (A5) be made along the following lines. Introduce the two equations where the an'S are unknown coefficients to be deter- (D - _2m)w = p V2DL TM (Ala) mined. Substitute into equation (1), use equation (A2a), multiply by Zm, integrate over the surface and then apply equation (A4); the result is an independent solu- (Alb) (D - _2m)z = pV2DL'Z tion for a n as follows where the first is simply the statement of flutter, i.e., f ZmFsdS + p J- ZmFgdS equation (1) with forcing terms suppressed, and the a = (A6) second is what we shall term the transposed mate of n (p. _ p) v2A n equation (Ala). For fixed v and _, these equations may be regarded as eigenvalue statements of p ; they Now, if w, v, and Pl are chosento represent an actual may be shown to have the same eigenvalues P, (which flutter condition (w = _f, v = vf, p 1 = p/), then w 1 will in general may be complex), and hence may be written represent the associated flutter mode shape, and the solution for a 1 becomes Bw n = Pn V2DLWn (A2a) f ZlFsdS + pf ZlFgdS a 1 = (A7) Bz m =Pm _V2DL'Zm (A2b) ,(pf - p) vf2A1 where B = D -_ m. Considered jointly, some signifi- cant relations between w n and z m may be found. Thus, This solution thus confirms the validity of equation multiply equation (A2a) by Zm, equation (A2b) by Wn, (7) presented in the body of the paper. The form of the integrate both over the wing surface, then subtract the equations is the same, but it is of interest to note re_ultin_r ex__ressions and make use of the fact that the more rational analysis presented here indi- cates that the generaltzeci iorces are associated wi[h the work done by the applied forces in moving through that f ZmBWndS = f WnBZmdS and f ZmDLWndS = thecnodal displacements of the transposed system.
f WnD L'zmdS; there results the relation (A3) (Pro - Pn) f ZmDLWndS REFERENCE From this equation we arrive at the basic orthogonality properties of w n and z m as given by the following 1. Smith, Francis B.: Analog Equipment For Pro- equation cessing Randomly Fluctuating Data. Aero. Egnr.
Review, Vol. 14, No. 5, pp. 112-119, May 1955.
f ZmDLWndS = O m f n (A4a) A m = n (A4b) n