A theory of flight flutter testing

A THEORY OF FLIGHT FLUTTER TESTING Erik Mollc)-Christensen -- California Institute of Technology Abstract Finally, we shall look at the accuracy of a flutter prediction, in terms of the precision of the data used in the computation.

Flight flutter testing is considered as a method for finding generalized aerodynamic forces. The co- The Equations of Motion efficients determined from flight flutter tests are used in flutter calculations, using a simple expansion We assume the wing to be perfectly elastic, and in frequency and Mach number. The errors in the assume the motion of the wing to be small, such that procedure are discussed, and expressions for the the aerodynamic loads are proportional to some error in flutter prediction are given. Methods of linear integral-differential transform of the deflec- L_[h*_ procedure _tre discussed.

tions.

This integro-differential dependence ofairloads INTRODUCTION on deflection is proportiohal to dynamic pressure, but may depend upon flight altitude, and depends upon Mach number and frequency of oscillation.

This paper considers flutter testing and flight flutter testing a part of flutter analysis. Very often The equation of motion can then be written: nowadays, tests which were originally intended as z(x, y) = jfW, ngC(X, YlA, _)(Jm(_f. 77)z{_, 72 ÷ proof tests or acceptance tests inadvertently became Area exploration of the unknown. This situation will per- sist until flutter analysis can be used with confidence, F((, 7) + _PU2D,(4, "9," k, M) ffWzngq(:, 77; r, s. _, M) to the extent that the accuracy of a flutter prediction Area can be computed as part of the analysis.

D2(r. s[ _, M) z(r, s)drds}d_d?7 Since this situation exists, one might as well where: consider such tests as links in the flutter analysis, R e [z(x, y)e z°_t] and squeeze out as much information as possible from the test results, rather than rest content with is the deflection of the wing at (x, y) at time t.

say, flutter frequency, speed and Mach number as the only result of a wind tunnel flutter test, which usually 27r_o cannot be repeated using the same model.

is the frequency of vibration.

We shall, therefore, consider the equations of rn(x, y) motion on a wing vibrating in an airstream, examine is the mass per unit wing area at (x, y).

which quantities can be measured, which quantities c(x, y]4, 7) can be found from a simpler test, and attempt to as- is the deflection at (x, y) due to a unit load applied say the accuracy of data obtained from static, ground at(e, 7).

vibration, flight vibration and flutter tests.

Mach number M, the combined equation becomes es- RelF(z, y)e 2c°t] pecially simple, and takes on the form: is the force applied to the wing by shakers, or ground [_/_] = [c_ u] [_@] Is1.] [_,,] b-,_ + supports.

(4)

{%u] Cd_] [_s.] Cz.n] ['o_ n] + D (J::, _; k, M)j_IW_ngq(C, 77; r, s, _, M)I)2(F, s, k, g)z(r, s)drds is the operator which yields the lift per unit area at {%.]{Fu.] + [%.][%._,. M)][z.n][_ pJ;_] ( ,:, _) ) divided by the dynamic pressure for a deflec- tion amplitude distribution z(x, y) at Mach number M where we have included a structural damping term with [d] asthe matrix of damping coefficients, n is and reduced frequency U the test number, so Zun is the deflection amplitude at (x , yu) in the n'th test, _n is the frequency, and At zero airspeed and frequency, this is the equa- tion for a ground static test, for zero airspeed only ¢ P J)n is the dynamic pressure in the n'th test.

it is the equation of a ground vibration test, and for After having performed a set of N tests, where zero impressed force, it is the equation for flutter, all but N columns of one of the matrices in Equation while the whole equation describes a flight vibration (4) are either measured in the tests or known from test.

previous tests or analysis, it is possible to compute the unknown columns. As examples, we shall con- To be able to use the equation, one must rewrite sider a set of static tests, a set of ground vibration it using some kind of approximation. One can use an tests, a set of flutter tests and a set of flight vibra- approximation in natural modes, but that seemspoint- tion tests.

less unless they are known precisely. The alternative is to use an approximation in discrete ordinates, or if one is in a fancy mood, to use station functions, or an A set of static tests should obey the equation: approximation in terms of surface stresses.

[_] : [%_] {_._] We shall use an approximation in discrete or- which can be inverted to yield: dinates, namely the deflections Zv at the points (x., [%_] = [_]{F_] -z y_) where _ refers to the number of the point in some kind of ordered sequence.

-z where [F_] is Equation 1 then becomes: (_) = [c_.] {m.J<J [n] (_u) + [%.] [H] (F.) + !zJ[%_][q.j_, M)]{z a} 2) T'_ is the cofactor of the element f;n in the transpose of [F_] The first order error in where [qua] is the matrix corresponding to the [%u} due to error in the measurement of z_ and linear integro-differential operator which yields the lift distribution. [JJ] is a diagonal matrixof integra- [Fun] can be evaluated as f_llows: tion weights, it has been lumped with the {q.afk, ,v)] -1 in the last term on the right hand side. ACc,_ u] : [A%u] : [Az_] [Fu.] + The equation for flutter states that the determin- ant of (2) must vanish for (F.) = 0 in order to ob- tain a non-trivial solution: o Z _ [Sr_s] Az,. s + _: r [K_.s] AFrs r s 1" $ D _ I [-1] + a_2 [c] [m] [N] +--_pU2[c][q(k, M)]l = O (3) If the errors are given in terms of standard deviations, c_Frs and %_, the standard deviation We shall now proceed to write down the equa- in the element %u is: tions for a set of tests, and to examine the rate of 2 2 2 H 2 change of flutter speed with changes in the elements = O'ZF S of the flutter determinant. The latter will enable us The coefficients /(_'s can be seen to be large to assess the first order error in the flutter predic- when the determinant IFI is small, i.e. when one of tion due to errors in wing parameters and aerody- the columns or rows in the loading matrix is nearly namic coefficients.

a linear combination of the other columns or rows.

The Equation for a Set of Tests The Equation for a Set of Ground Vibration Tests For a set of ground vibration tests, one ob- If one repeats a flight vibration test N times, tains: one obtains N equations like equation (1), which can [%_] : [%u] [_u] Era] [_u_][<o_,] + [%_] [¢u_] + be written as a single equation. If all these tests are performed at the same reduced frequency k and [%u] [d] [z_n] [_.]

Equating real parts: or ['_] " [_v]-l[c_v] -1(Re [z_] - The equation for a set of tests is then Eq. (4), 2 -1 solving for the aerodynamic terms, one obtains: [c_y]Re [Fvn]) [w n] [Reavn ]-1 and we see that if the determinants of [c_.] and [zy n] are small, the first order errors may become large. However, in the flutter equation, [m] only occurs in the combination:

{[_] - [%_] ( C_]CH] C_y_]['_] ÷ , [d] [_] [_y_] [%] +

[c:_y] fay] [_,] [Fuu]) }[(_- pU=)u]-l[Zyu] -1 and therefore only this combination is of interest: This set of equations may be insufficient to de- [cp_] [b_y] [my] = (Re[z_] - [cpv][ReF_n]) [Wn]-l[Re_pn] -1 termine [quy(_, g) ] However, some of the q_y(_, _; are not very important as far as the which shows that these errors tn [z_] and iF, n] flutter speed is concerned, the zero order terms in k are really important. The matrix of first order er- can he determined by wind tunnel tests on stationary rors of the left hand side is [A] , where: but deformed wings (tied down), others can again be -i -I guessed at, at least, from linearized aerodynamic [A] : ([_ez_,] - [%_y] [_ePwn]) [%] [Rez_] theory. The purpose of a flight vibration test or a flutter test is then to determine the remaining aero- + ( freaky] - [c_y] iReFul) ((A[_,_] -1 [Rez_]-l) dynamic coefficients. Without going into a discussion 2 -1 -1 of which aerodynamic coefficients are to be chosen as + [%] A([Rez_n] )) those which neither theory nor wind tunnel static tests The error will therefore be proportional to the in- can yield, we shall consider the precision obtainable verse square of the determinant of [Zez_] ; this in q_,(_, _) when it is determined from tests.

determinant should be maximized by arranging the test such that the columns of the determinant are The term which is most liable to magnify the orthogonal if possible. This means that each test errors is the errors in the inverse of [z_u] . The should be performed at a natural frequency.

value of [z_] is Equation [or a Set of Kitgnt Vtoratton Tests or Ftutter Tests The information obtainable from a set of flight When differentiating to evaluate the error, one obtains vibration tests or flutter tests which cannot be obtained an expression with Izl = in the denominator. To from tests where there are no aerodynamic forces minimize errors, one must try to make lal as large are, of course, the aerodynamic forces.

as possible, i.e., the columns in [z_] should be as different as possible. Vibration is natural modes Since the aerodynamic coefficients depend upon only will go far towards the accomplishment of pre- Mach number, M, and reduced frequency, k, the tests cision.

must either be performed at constant M and k, or Errors in Flutter Prediction due to Errors in Struc- one must somehow approximate this dependence.

tural, Mass and Aerodynamic Parameters One can for example use a Taylor series ex- pansion of [q_.(_. M) ] in k and M about some value Before the obtainable precision in experimental of M, Mref. and zero reduced frequency. One obtains: determination of structural, mass and aerodynamic information can be meaningful in terms of resulting [q:_y(_, /4)] = accuracy in flutter prediction, we have to analyze the sensitivity of a flutter point to such errors.

Z qlzy(/_, g r=o s:o -aM r _k s k = o r! s!

Flutter occurs whenever the determinant (Eq.

(3)) vanishes: M = Mre f D(_, g,_-}.-pU _, m ...... mf¢, d ...... d w Instead of expanding in power of (M-Mref), one can expand in powers of (M2-1) for transonic Mach num- q,_ ...... qNN' c ..... , CNN) = O bers and (M2-1)-h for supersonic Mach numbers.

Vary one of the parameters, which we shall call P.

Both the real and imaginary parts of the flutter de- As an engineering approximation one would only use the first and zero order terms.

terminant will then change, and k and _-pj must then be changed to compensate, such as to maintain lined. It is realized that only the practising flutter the value of the flutter determinant at zeroat constant analyst can choose the method of analysis and the tests to be performed, knowing the limitations of his M. Instead of changing kland -_-Q]7I , k and M can be changed, at constant -_-Qf , or i _ and k can be facilities and his personnel.

changed only, at constant M and U.

The method which has been outlined is clearly i impractical; however, if some of its elements are used, We shall only consider changes in k and -2P Lf at constant M. or if nothing else, its viewpoint is adopted, the paper will have accomplished its purpose.

To maintain flutter for a change in P, one must have: Bibliography Ref:hUJ) = Re('f'_p ZU') + 2e('_JiK&(-pL )+ Re(--J._k = 0 _O _i) l 2 5D Im/a_)}= Im(--.A£ I + Im{--J _.(_pL' ) + ]m(_kJ _Z_ = 0 1.

Molineaux, W. G., The Determination of Aero- dynamic Coefficients from Flutter Test Data, unpublished Ministry of Supply Report, U.K.

Solving for A -_pL, '_} and _k, one obtains: 2.

Mollo-Christensen, E., and Martuccelli, J. R., M = const.

A Study of the Feasibility of Experimental De- termination of Aerodynamic Forces for use in Flutter Calculation, M.I.T. Aeroelastic and Struc- tures Research Lab., Technical Reports 66-1, 66-2, 66-3, June, 1957.

3.

Bisplinghoff, R. L., Ashley, H., and Halfman, R. L., Aeroelasticity, Addison Wesley, Cam- bridge, 1955.

, a '_ D/} } ,_J' _ 1 ,> 4.

Garrick, I. E. and Rubinow, S. I., Flutter and ( XA" ; )/ = ,? _ Oscillating Air-Force Calculations for an Airfoil bU _1) in Two-dimensional Supersonic Flow, NACA Re-

:,,, t,_ ) --Sy _

_: it){'' ) port 846, 1946.

2_ where the bars denote the complex conjugate and the 5. Ashley, H., and Zartarian, G., Piston Theory-- derivative with respect to k is taken at constant M A New Aerodynamic Tool for the Aeroelastician, and and the derivative with respect to 1 Jour. Aero. Sci., Vol. 23, No. 12, pp. 1109-1118, ' _pl,: December, 1956.

is taken at constant k and M.

6.

Goland, M., The Flutter of a Uniform Cantilever It is, of course, complicated to evaluate these Wing, Journal of Applied Mechanics, Vol. 12, derivatives, but it seems to be necessary for finding No. 4, December, 1945.

the sensitivity of a flutter point to parameter changes.

With modern computers it may, however, be possible. 7.

Buxton, G. H. L., and Minhinnick, I. T., Expres- sions for the Rates of Change of Critical Flutter A rough knowledge of the precision of a flutter Speeds and Frequencies with Inertial, Aerody- prediction will always be useful; one must keep firmly namics and Elastic Coefficients, BR. A.R.C., in mind, however, that the estimate of precision is in R. and M. 2444, September, 1945.

terms of a given numerical approximation, and can give no information about the remainder term of the 8.

Baker, R. C., Effect of Errors and Changes in numerical approximation.

System Parameters on the Flutter Point, S.M.

Thesis, M.I.T., 1956.

In practice, when a flutter point proves very in- sensitive to parameter changes, it should not be al- 9.

Crout, P. D., A Short Method of Evaluating De- lowed to cause unalleviated elation, since then it will terminants and Solving Systems of Linear Equa- take a major design change to move the flutter point tions with Real or Complex Coefficients, Trans.

out of the flight envelope of the airplane, for example.

A.I.E.E., Vol. 60, pp. 1235-1240, 1941.

Conclusion 10.

Cramer, H., Mathematical Methods of Statistics, p. 213, Princeton University Press, 1954.

A viewpoint and a method of approach to flight flutter testing and to flutter in general has been out-