Document
VECTOR PLOTTING AS AN INDICATION OF THE APPROACH TO
FLUTTER
E. G. Broadhent -- Royal Aircraft Establishment,
Farnhorough, England
Abstract.
Because of its success in ground resonance tests the idea has arisen of adapting the technique for A binary flexure-torsion analysis has been made flightflutter testing. It is hoped that from the flight test under continuous excitation the resonances might to check theoretically a method for predicting flutter be obtained in the same way as from a ground test.
which depends on plotting vectorially the amplitudes of response relative to the exciting force and extracting with at the same time estimates ofthe overall damping the relevant damping rate. The results of this calcu- at each resonance frequency. Thus a graph of damping lation are given in the form of graphs both of the rate against airspeed can be obtained from a continuous vector plots themselves and of the estimated damping excitationmethod of flight fluttertesting. In this way _+_ _,_ fnrw_rd _need. The estimated damping itis hoped to obtain the best oftwo worlds; continuous rates are compared with calculated values. The excitation allows more accurate analysis in ii_e p_ _- ence of buffeting than is possible from a decaying method has the advantage that in a flightflutter test oscillation, and at the same time damping can be damping can be estimated from continuous excitation records: the method is an extension of the Kennedy plotted against airspeed; and damping gives a more and Pancu technique used in ground resonance testing. reliable warning of the approach to flutter than does amplitude response. Near the flutter speed, however, the analysis has to deal with adifferent type of equili- brium than in a ground resonance test, because the INTRODUCTION aerodynamic forces are powerful and do not represent a conservative system. In order to see whether this leads to any difficulty in application, a simple flexure- The measurement of normal modes in a ground torsion binary example has been worked out in the resonance test needs an elaborate technique both to present paper and analysed by the Kennedy-Pancu ensure .that the modes are reasonably orthogonal, method at various forward speeds up to the flutter and to ensure that no mode is missed. The presence speed. The dampings are obtained and plotted against of structural damping presents one of the main dif- airspeed and the results are found to agree well with ficulties. Kennedy and Pancu have suggested a method calculated dampings. Some low speed wind-tunnel tests of analysing therecordings taken by plotting vector- carried out by Bristol Aircraft Limited show that the ially the displacements relative to the exciting force.
method can give results with a high degree of repeat- Near circles are obtained for each resonance and ability, even in the presence of buffeting.
practical experience seems to show that this type of plot considerably reduces the likelihood of missing a resonance and also improves the accuracy of deter- THEORY OF THE METHOD mining the resonant frequency. This in itself leads to modes being measured which are a better approxi- mation to the true normal modes than is usually possible from amplitude plots alone. In addition the The basis of the theory is outlined I'ere for structural damping can be estimated directly for each convenience.
resonance.
Hence One Degree of Freedom The equation of motion for one degree of freedom F I - _2
can be written in the form:- (7)
qF 2 _2) 2 a% (i - + g _" ÷ e(1 + _q = _,_t (I) and for a generalized exciting force Fe _t , F -g (B) q_ :-- [ ] 2 2 where a is an inertia coefficient a% (i - _2) + g e is an elastic coefficient As _ is varied the locus of points (qr, qi ) is a smooth curve obtained by eliminating Y_ from these two equa- q is a generalized co-ordinate tions:- g is the phase angle of the restoring force (the qr F damping coefficient).
(9)
+ i 2 2 The steady solution will be motion of the form e __t, % aWo%g so we substitute q = q e __t or Equation (1) now becomes:- qr + q2 + ( )qz = 0 [-_2a + e(1 + _)]_ o F (2) This is the equation of a circle with its diameter lying on the negative imaginary axis andpassing throughthe origin (see Figure 1).
We let % be the natural frequency of the one degree 2 e The Position of Resonance of freedom, i.e., % : -- and we obtain:- a Resonance occurs when _ = 1 andfrom Equation (7) qr=0, i.e., the vector OC on Figure 1 represents (3) the amplitude at resonance. We can obtain a relation between the rate of change of frequency along the curve at resonance and the damping g, so that if the curve itself is obtained from measurements on a structure where Y,_ : ff-_-_ of unknown damping, the damping can be estimated.
(JO Consider the point D in Figure 1 when the fre- quency is% ÷ _,. At D For the purpose of vector plotting _ is written the form:- qr
: ta_- (il)
% 2
(4)
: qr + _'C]7, _L For any exciting frequency, _, the quantities qr and qi can now be calculated and plotted on an Argand diagram to give the response vector at that frequency relative to the exciting force; i.e., F is taken to lie along the real axis.
Substituting Equation (4) in Equation (3)and equating real and imaginary parts leads to:-
(5)
awo[qr( I _ _2) _ qzg] = F _e÷$1 _o and Figure 1. Vector Diagram for One Degree of
(6)
aWo [qrg + q_ (l - _)] = 0 Freedom -- Hysteresis Damping _2 i - co D Comparing this with equation (1) (12) g dq=eigq (18) from E.iuations (7) and (8).
Hence and substituting q = _ e i_vt g :-- C2 *--) cot-- (13) co o o) 0 2 _ = _e_ (19) bw Hence It can be seen from Equation (13) that if _ is small, equal angles will be subtended by equal fre- _d quency increments on either side of the resonance.
: (2o)
¢ In the particular case when _ = 2-we have:- c c But d = 2 -_c "/'_ where _'c is the fraction of critical damping: - _v4 4,2 Hence COO and when o : (14) co : 2_--- IT; (21) m/3 _2 0 C %, = !-g =-- 6) o d so that at resonance ¢ = 2_ : _ (22) C Hence It should be noted that if the damping is of the form given by Equation (17) the locus of points (qr, qi) is no longer a circle; the steady solution will be 2g = -- motion of the form e zcvt, and substitutingq ='_ ¢ z_t the % (15) equation becomes:- and (- aJ + d,w + e)_ = Y (23) Proceeding as before we obtain:- Whence F I - _2 qr =_ (24) 2 2 aoJ o (l - _) _ + _o_ _ g : __ (16) and _A + cod q_ _-- (25) a% (i - _2;2 . cj_2 2w o It is common practice in this country to express the so that the two systems represented by Here _ =_e damping as a percentage of the critical damping. As long as the damping is small, g canbe directly related Equations (1) and (23) will have the same properties to the percentage of critical damping which is derived at resonance if g =_. The vector q defined by Equa- from the concept of velocity damping: i.e., the ap- tions (24) and (25) now describes a quartic curve propriate differential equation is:- / point{ F--'-7 , 0) when _ =0 and finish- at the starting \ aO9 0 } ing at the origin when co--,-_ ; any other branches are a_ + d_ ÷ e¢ _ ye'_t (17) for unreal frequencies. In practice for small values of g the curve is indistinguishable from a circle ex- lating the response of a simple binary example at cept at low frequencies; this is shown in Figure 2 where various speeds up to the flutter speed.
the circle of Equations (7) and (8)is compared with the quartic of Equations (24) and (25).
BINARY EXAMPLE Basic Data: Geometry For simplicity a 2-dimensional rigid wing, re- strained by springs in vertical translation and pitch was considered. The two degrees of freedom are: Vertical translation: z = cql (representing wing flexure) a_.e 0"_)_ = r- _'t Pitch: _ = q2 (representing wing a _ , cL$-_e_ =rq ''t --- torsion) Figure 2. Vector Diagram for One Degree of in general z = cql + xq2 Freedom -- Comparison Between Hysteresis and Velocity Damping The axis of pitch is at the half chord.
The axis of centre of gravity is at the half chord.
Two Degrees of Freedom Since the modes are uncoupled at zero flight speed they are normal modes and the frequency ratio is Kennedy and Pancu suggest that with N degrees _%:%_: : 0.4676:1.
of freedom there will be N near circles. For any particular resonance, the best circle is put through the points and the resonance is given by the minimum Structural damping at a value of g = 0.02 is b_o assumed to be present in each degree of freedom. It , where s represents distance along the curve. If <s,s is assumed that displacements to be recorded in flight tests are linear displacements at the half chord, equal increments of c_ are taken the greatest change quarter chord and leading edge and the angle of pitch.
of phase gives the resonance. The damping (g) can Thus the first and last of these 'pickups' give meas- then be extracted as for one degree of freedom.
urements proportional to the generalized co-ordinates ql and q2 respectively. Finally it is assumed that Because this method appears to be the best way of estimating damping in ground resonance tests, it the excitation is linear vertical excitation applied at the quarter chord.
has been suggested that it might well be extended to the estimation of damping in a flight flutter test, where continuous excitation is being employed. The method may be difficult when the dampings are high at medium Wing Flutter flight speeds, but should improve againfor low damping near the flutter speed. The difference between the The aerodynamic derivatives are'assumed to be flight condition near the flutter speed and the ground constant both with the frequency parameter and for- condition, where the damping is low in each case, is ward speed, i.e., any Mach number effect is neglected.
that in flight there will be large asymmetric couplings arising from the aerodynamic forces. It was decided The equations for free oscillation can be written in the form:- to see how important these were in practice by calcu- -14.0492 + 1.9_ _'u_7. + (1 * O.O2_)y o O.Ppb:,_ * p.2r*S :
: o (20)
- .4RUM -0.n908S + 0.24vp_ - .565v '_" +.20 (i + O.OSzJyo where V c = flutter speed COg
To'i
V c o_e o.8ts g 0.77_ 0"9 gc 0.75_ \\_ , o._S Eaz
-,% i'_ --_-
, "</i'_ _--
YO = _ 2 _ PVcSC 0-'; / \\ / "_ 0-57, \ / J _, c is the wing chord o,Z ', s is the wing span The equations were solved for Yo with v = 1 (cor- responding to the critical flutter speed), and gave Yo = 2.92 and y = 0.666.
From a knowledge of Yo it is possible to relate any known Ell (the spring restraint against vertical translation) to an actual flutter speed (Vc) , knowing the dimensions. Here, however, we are only interested 0.41; I 0-45 o. 43"75 in the relative speeds, i.e., v, the fraction of V c.
!
Response Calculations J -2.0 With the excitation at the quarter chord and Figure 3. Vector Diagram for Binary Example: after the substitution for Yo = 2.92, Equation (26) v = 0.75, displacement 1 becomes:- (-14.04Y _ + 2.92) + (l.98vv + 0.0594)_ 2.2"tv 2 + O._SvvZ -0.25 (27) (-.9908_ _ - 0.585v 2 + 0.q4894 q - .4Pvw_ + (O.94vy * 0. 0189.98) _ which gives a direct measure of the first co-ordinate where F is an arbitrary force level. For simplicity in the calculation. At zero speed the co-ordinates are F is taken to be unity in the calculation which follows.
normal co-ordinates so that the vector diagram re- Values of v = 0, 0.25, 0.5, 0.75, 0.9 and 1.0 were sults in a single pure circle with a resonance fre- chosen, and in each Case ql and q2 were calculated for a set of increments in vo. Assuming perfect quency given by _o = 0.456. As speed is increased the size of the circle reduces (the same scale has been accuracy of recording the measurements taken in flight kept throughout each of Figures 4 to 7, although of from the four 'pickups' (half chord, quarter chord, course different scales were used to estimate fre- leading edge, pitching angle) would be ql, ql-1/4q2 , quency rates of decay in practice) and a small sec- ql-1/2q2' q2" ondary circle starts to appear near the origin. This second circle occurs at the frequency of the pitching These quantities were plottedvectorially andthe mode which is now beginning to couple slightlywith frequencies and rates of decay were estimated from the near circles; a typical example is shown in Figure the bending mode due to the presence of the aero- 3 for pickup 1 at 3/4 of the flutter speed. dynamic forces. The new circle continues to increase in size until at a speed of nine tenths of the flutter Comments on Figures speed it is the greater of the two. The last diagram in this series is drawn for the flutterspeed itselfat which one of the circles must have increased indefi- The change in character of each vector diagram as the forward speed is increased is indicated in Fig- nitely in size. This is in fact the new circle cor- ures 4 to 7. Consider first Figure 4 for displacement responding to the higher frequency.
1, i.e. the displacement of the first pickup (see above) "IJr - o. Z5 "It - o.s "u'- 0-7S "U'_ O- 9 V=I'O 2/'-0 $_, _L 0 5 -5 $_ '_' ST :I" I- I_I" -s ' _" -/-5 -I0 Figure 4. Vector Diagram for Binary Example: Displacement 1, Varying Speed 2O _J'- I'O '_)', 0-5 "0". 0-7._ "tr,O 5L -Z 2_ -5 -- %
- '(
S $ 5 Figure 5. Vector Diagram for Binary Example: Displacement 2, Varying Speed -_'o -Lr= o._j "U'= 1.0 "q : 0-'16 '_, 0-o # <1, i. "t.r, O. 5 "V, 0?.5 $_.
SC _.L.
S ?_ Z _ .p ;, -2 2 io S g '5 Displacement 3, Varying Speed igure 6. Vector Diagram for Binary Example: "tJ"• 0-?$ "lLr • 0.5 "V" =0.'/$ "tr = o-g "IT= I'0 o,,_.
,¢ti -5 -_ 2 -Z Z - /I ' "q,r
/
$ [ -5 .5 Figure 7. Vector Diagram for Binary Example: Displacement 4, Varying Speed Figure 5 gives the diagrams for displacement 2, the quarter chord, which shows two circles even at I0 zero speed; neither of these circles are perfect al- though the error is not detectable on the scale shown.
Both circles reduce with increasing airspeed for a time and the smaller (corresponding to the higher fre- quency) changes its position relative to the origin.
Ultimately, as before, the higher frequency circle o{ increases in size to an indefinite extent at the flutter speed. Similar sequences are shown for the other pickups in Figures 6 and 7, although in the last figure the hi_her frpn._ney. .... circle ."cmaln-,o the l_-geL- throughout.
Estimation of Damping in Flight and Conclusion As outlined in paragraph 2 we estimate the damping _c from the circles. Near each resonance suitable equal increments in frequency are chosen, and these are marked on the curves of Figure 3. The 0'4 actual resonance is picked out from the figures by using a pair of dividers to get the maximum phase change, In this example there was never any difficulty in putting a circle through the points (a typical circle is shown in Figure 3) and the damping was estimated from convenient increments of frequency as can be seen from the construction on Figure 3.
0._ The damping as obtained from each pickup was then plotted against forward speed, and the results are shown in Figure 8. Since our example is completely specified mathematically, the dampings can also be calculated exactly. In Figures 9 and 10 the calculated roots are plotted and compared with the estimates 0 0-25 from each of the four 'pickups'. Figure 9A, shows the change in frequency of the lower frequency with for- Figure 8. Damping Estimates from the Vector ward speed and Figure 9B, shows the change in damp- Diagrams Against For_vard Speed ing: Figures 10A and B give the corresponding re- sults for the higher frequency root, which is the one that leads to flutter at v = 1.0. The agreement in ,-o _.
O.f 0"3 I_lSON A.NC ! FA[QUI_Y O-i O-I O'_ ,o --_ v O._S 0.5 0._$ C-Z5 Q.S 0.'7, I.O _"
(a)
@
EXACT CA_,C_LATION .......
tXACT CALCUL&TION ....
OI_Pt._II M1.NT I D_S_. ACF'hIENT ,,_ ......
DI_PLACIM_NV I DiS_. ACEHENT 4 . . . . . . . . .
01SP_ACI_KMT 3 DISPL_CEMI_T 4 .........
O.G OAMPI_ O'l O.4 o.1
f
O._5 0'_ O.1_ I'0 'v-
0'_,5 O'S O.'/S 1,0 -_ep "U" (b)
Figure 9. Comparison Between Estimates of Damping Figure 10. Comparison Between Estimates of Damping and Frequency, and Exact Calculation, and Frequency, and Exact Calculation, Bending Mode Torsion Mode general between the different estimaes andthe calcu- be however, that with many degrees of freedom pres- lated values is very good. The only serious error in ent, as on real aircraft, the choice of pickup position the lower frequency root is obtained from the rota- is more important than in the binary example. In tional 'pickup'; this seems to give the wrong trend of general the flight analysis would be carried out for frequency with speed when the damping exceeds 10%of two or three pickups as a normal safety precaution.
critical -- a condition which would in any case be unimportant in practice. For the higher frequency root the accuracy is good throughout, and best for this same rotational pickup, as might be expected on RESULTS FROM A LOW SPEED WIND-TUNNEL MODEL qualitative grounds. Any of the pickups, however, would give a good prediction of flutter speed (see Figure 10B) provided the speed increments chosen The method outlined above has been applied by Bristol Aircraft Limited to a wind-tunnel model de- were not too large.
signed to investigate flutter of a T-tail configuration.
From flight measurements in practice one could Figure 11 shows a typical vector diagram at a for- scarcely hope to get such a consistent set of results ward speed that is about 83% of the extrapolated as has been obtained from the estimates in this flutter speed. The diagram is for the mode which simple binary example. On the other hand the example starts at zero speed as tailplane fundamental sym- does suggest that the method is sound in principle so metric torsion, and which provides the main pointer that if there are practical arguments which favour to the critical flutter condition as did wing pitch in recording from continuous excitation rather than the theoretical example of section 3. The experimental decaying oscillations the Kennedy and Pancu type of results are consistent and define a very good circle.
analysis is likely to provide good results. It may well Figure 12 shows the variation in frequency anddamp- ing with airspeed of the fundamental bending mode of the tailplane and Figure 13 gives the corresponding results for the fundamental torsion mode*. The graph of Figure 13 can be extrapolated to the flutter speed.
It is not the purpose of this paper to deal with the experimental technique involved but one or two points should be made. It is necessary to have a phase meter available that gives accurate readings in the presence of buffeting. The instrument used by Bristols measures in-phase and quadrature compo- nents, and is arranged to descriminate against noise (as in a wattmeter type of phasemeter). It can give an accuracy of about 5% even with a signal to noise ratio as low as unity. The rate of sweep of the ex- citer (in terms of frequency) is determined by trial and error, and a satisfactory rate will depend on the damping in each case. The frequency control of the exciter must be accurate, i.e., high short term stabil- ity is required, and in practice at low dampings the frequency increments may need to be as small as 0.4_ in order to get a reliable measure of the damping.
*These terms are used for descriptive purposes only: in practice, of course, the modes change shape under the aerodynamic forces.
Figure 11. Example of Phase Against Amplitude Plot with Damping Analysis c g 4o O-_.O 3o O.IS O.lO 0.05 IO ioo _ 300 o V $.p.s.
Figure 12. Tailplane Fundamental Symmetric Bending Resonant Frequencies and Damping Against Airspeed
c._
O.10 o. o75 5O 0.05 Z0
/
O.OZS I0
/
o ioo &oo v _.p.5, 300 Figure 13. Tailplane Fundamental Symmetric Torsion Resonant Frequencies and Damping Against Airspeed List of Symbols (cont) ACKNOWLEDGEMENT The author wishes to express his thanks to is a frequency parameter % Bristol Aircraft Limited for making available the Vc c is the wing chord results of their wind-tunnel tests, and to Miss E. V.
Hartley for carrying out the binary calculations.
s is the wing span p is the air density Eli is the spring restraint against vertical trans- LIST OF SYMBOLS lation Ell a is an inertia coefficient YO - PFcS c 2 d is a damping coefficient e is an elastic coefficient z is vertical displacement g is the phase angle of the restoring force (a a is the angle of pitch damping coefficient} q is a generalized co-ordinate F is a generalized exciting force REFERENCE % is the natural frequency of one degree of freedom 0) is the exciting frequency Ref. No. Author Title, etc.
_2 °)2 Use of vectors in vi- 1 Kennedy, C.C.
bration measurement Pancu, C.D.P.
tOo and analysis.
V c is the flutter speed Journal of the Aero- nautical S c i e n c e s.
V is the forward speed Vol. 14, No. 11.
V November, 1947.
v - Vc