Consider a single-degree-of-freedom system such as a mass-spring – dashpot model. An idealized equation of motion is (§ 1.8)

mx + fix + Kx = X (1)

where the constants m, /?, К are such that the kinetic energy, potential energy, and the dissipation function of the system are given, respectively, by

bmx2, Kx%, I fix2

The dissipation function has the dimensions of energy per unit time, f) is interpreted as viscous damping factor. A is a forcing function.

In § 2.4, p. 69, it is shown that Eq. 1 can be conveniently written as

x – f 2уш0х + a>02x — — X (2)


where a)02 = Kjm, and у is the ratio of actual damping /? to the critical damping factor (>„ = 2mco0.

If the forcing function is a harmonic function, the solution of Eqs. 1 and 2 has already been given in § 1.8.

It is instructive to compare the form of damping assumed in Eq. 1 with that commonly used in flutter analysis. Following § 6.9, we write

X + (1 + ig)u>fa = — X (3)


which is applied to harmonic oscillations only. But if a motion is har­monic so that x = x0eimi, Eq. 2 may be written as

x – r (1 + 2 iy — I a>02x = — X (4)

a>J m

This is identical in form with Eq. 3 if, and only if,

ш = 0,o. g = 2У (5)

It is therefore clear that the mechanisms of damping suggested by Eqs. 1 and 3 are entirely different if ш Ф ш0, and if the motion is not a harmonic one.

As is discussed in § 11.4, the detailed mechanism of damping in structures concerned with in aeroelasticity is yet unknown. Hence, a choice of the particular form of damping is open to question. For the same reason, the value of g is rarely accurately determined. It is known, however, that g is small for metal airplanes.

When g is small, a free oscillation of a system described in Eq. 2 follow­ing an initial disturbance will be almost sinusoidal with a frequency close to co0. Assuming that Eq. 3 is applicable also to this case, then we have approximately

g = 2У

as in Eq. 5. It is then easily derived from the solution given in § 1.8 that, when g<^l, y<^l, the logarithmic decrement 6 in a free oscillation following an initial disturbance is

<5 = ng = 2 ny

More generally, g can be deduced, approximately, from the rate of decay of free oscillations,

[1] Some authors define elastic axis as the locus of flexural centers or as a flexural line. This is not the sense to be used in this book.

[2] We shall assume tacitly that the structure is rigidly supported in a specific manner when the influence functions are measured. However, this does not prevent their usefulness in applications to an airplane in free flight, because all that is necessary is to measure the elastic displacements (or influence functions) of the airplane structures with respect to a set of rectangular coordinates attached to the airplane, with the origin located at a convenient point, e. g., a point on the center line of the main spar, or the air­plane center of gravity. The structure may be regarded as clamped with respect to the coordinate system at its origin. The change in direction of the coordinates and the motion of the origin can then be determined by the free-body motion of the airplane.

[3] The “local” lift coefficient at any point (x, y) on a lifting surface is defined as the limit of dL/qdS as dS -*■ 0, where dS is a surface element enclosing the point (x, y) and dL is the lift force acting on dS. The local lift coefficient Ct(y) used in Eq. 18 is defined over a chordwise section, so that dS = cdy. We shall use lower case subscript to indicate a local aerodynamic coefficient such as Cu whereas a capital letter subscript will be used for coefficients referred to the entire wing.

[4] A prime over the summation sign is introduced to indicate that the term j = і must be deleted from the sum.

[5] It is necessary only to satisfy the boundary conditions imposed by constraints against displacement. Such boundary conditions are called the “rigid” boundary conditions. The boundary conditions at an end where no constraint is imposed, such as those at the free end of the beam, are called the “natural” boundary conditions. The natural boundary conditions will be satisfied in the limit by the method of successive approximations.

[6] This type of oscillation, due to periodic shedding of vortices, may be called the oscillations of the Aeolian harp type. The principle of playing strings by the wind was recognized in ancient times. King David, according to Rabbinic records, used to hang his kinnor (Ki“ ra) over his bed at night, where it sounded in the midnight breeze.

[7] In some books, the Strouhal number is defined by replacing со by the number of cycles per second. For uniformity in notation, we shall not distinguish the reduced frequency and the Strouhal number in this book.

t There are some quantitative differences between Kovasznay’s and Relf and Simmons’ results, probably reflecting the difference between the wind tunnels they used.

[8] We shall assume that the Reynolds number is in the range of vortex shedding with a clearly defined predominant frequency, 40 < R < З X 106.

[9] We assume that the total angle of attack remain so small that Cj is linearly pro­portional to the angle of attack. This requires that the initial angle of attack do not approach the stalling angle. If the initial angle of attack is large, the additional elastic twist may cause the wing to stall. Divergence near stalling angle is an interesting nonlinear problem, which is closely related to stall flutter and buffeting.

[10] q&lv /1 _ 1 a qaiJ’

plotted against 1 lq will be a straight line. The divergence dynamic pressure can be found either from the slope of this line, ^div/“> or from its intercept on the Щ axis, l/?dtv – This relation can be used for an experimental determination of the divergence speed.

[11] See P. D. Grout, Marchant Calculating Machine Co. Bulletin ММ-182, MM-183 (Sept. 1941). Also Trans. AIEE, 60 (1941).

P. S. Dwyer, Psychometrika 6, 101-129 (1941).

W. E. Milne, Numerical Calculus, Princeton Univ. Press, Chapter 1 (1949).

Note that, according to Eq. 17, the &th column of A-1 can be obtained by solving Eqs. 16 for a column matrix В for which every element is zero except the Arth element which is 1. Thus the calculation of the inverse of a matrix is equivalent to the numerical solution of a system of linear equations.

[12] See for example, С. B. Millikan, Aerodynamics of the Airplane, Chapter 4, John Wiley & Sons (1941). C. D. Perkins, and R. E. Hage. Airplane Performance, Stability and Control, Chapters 5 and 6, John Wiley & Sons (1949).

t The characteristic wing-twisting angle 6W and the spanwise location of the wing reference section rjw will be defined by an equation analogous with Eq. 5 of § 4.6.

[13] In order to distinguish “flutter” from “stall flutter,” some authors use the term “classical flutter.”

f From the results of § 1.9, it is natural to expect that spanwise shift in phase angle exists at the critical flutter condition, because the aerodynamic force contributes a term proportional to the velocity of movement, but such spanwise shift in phase is relatively small.

[14] Flexural-stiffness criterion:


Fig. 6.1. Steady flow over a two-dimensional airfoil.

[16] It should be pointed out that the method used in these sections can be applied whenever the aerodynamic forces are expressed in the form of the so-cdlled “classical derivative coefficients” (see p. 228), and are not restricted to quasi-steady coefficients.

[17] The assumed functions f(y) and Ф(у) must satisfy the “rigid” boundary conditions at у = 0 (Eqs. 4, § 6.3). See footnote on p. 54.

[18] x, x • • • „ dR d*R d3R „ „

* Note that it is impossible to have R — — = —— — —; = 0 while —— ф 0.

‘ dU dUi dU3 dU1

t The validity of these relations is restricted by the quasi-steady assumption made in their derivations. The fluid is incompressible. The wing is unswept,, and the aileron is locked. For a more exact survey, see, for example, Ref. 6.23.

[19] The expression 8a gives somewhat better approximation for к < 0.5, whereas 86 is better for к > 0.5. The real part F is well approximated by both 8a and 86, the maximum percentage errors based on the exact values are + 2.6 per cent, — 2.1 per cent throughout the range (0, oo). The error of the absolute value of the imaginary part, — G, is much larger. The maximum percentage errors based on the exact values are as follows: For 8a (R. T. Jones), + 8.5 per cent, — 13.5 per cent. For 86 (W. P. Jones), + 10 per cent, — 11 per cent. A detailed numerical comparison can be found in Ref. 15.80.

[20] This is not true in all cases. For flutter involving ailerons, for instance, the effect of structural damping can be exceedingly large. See NACA Rept. 741.

[21] The (5 function can be defined as a limit of some well-defined functions. See Chapter V of van de Pol and Bremmer.8-2

[22] This curve is given by P. MacCready.8-52

[23] If the mean value does not vanish, we may first subtract from »(/) the mean value. In other words, we may consider only the deviation from the mean value.


p(co) = — І у>(т) cos сот dT rr Jo

[25] It must be observed that the possibility of such a separation into two independent problems depends on the assumption of small disturbances, so that the hydrodynamic equations may be linearized. Hence, the analysis is theoretically valid only for turbu­lences of small intensity.

[26] Within a certain region the total head of the flow is smaller than that of the un­disturbed stream. Although the transition to the normal value takes place gradually, the boundaries of this “total-head wake” are quite well defined. It is not clear how to interpret the total-head wake in a turbulent flow. However, the main indication of the experiment is that the buffeting intensity is related to the turbulences in the flow.

t A proper fillet, i. e., a smooth fairing at a wing-fuselage junction, is most effective in reducing flow separation caused by wing-fuselage interference. Von Karman, in Aerodynamics, Selected Topics in the Light of Their Historical Development, p. 151 (Cornell Univ. Press, 1954), relates an interesting story about his reporting of the effec­tiveness of a fillet in preventing tail buffeting in a lecture in Paris in 1932. Some French engineers were apparently troubled by the same difficulty and immediately tried K&rman’s fillets. Henceforth, the “fillet” was known in France as “karman,” and expressions like “large karman” and “small karman” are used. Von Kdrman himself attributes the development of fillet to the team work of himself, Clark Millikan, and Arthur Klein at the California Institute of Technology. See Karmdm, Th. von.: Quelques problemes actuels de 1’aerodynamique. ./. techniques internationales de Гaeronaut, 1-26 (Paris, 1933), Klein, A. L.: Effect of Fillets on Wing-Fuselage Inter­ference. Trans. A. S.M. E. 56, 1-7, AER-56-1 (1934).

[27] This is a Tauberian theorem. A sufficient condition is that F(s) be the Laplace transform of some function F(i) for all R1 у > 0, F (j) -* К as ^ -* + 0, and F(t) is of order oO/O as r-> oo. To apply this theorem to our example, we should consider the unit-step function 1(0 as the limit of e – A< 1(f) as Л —>■ 0.

[28] The argument of z, written as arg z, is the angle between the radius vector z = x + iy and the «axis; i. e., arg z = arc tan yjx. Thus arg/(z) = arc tan у>/ф. The argu­ment off(z) is increased by 2ir if the end point of the radius vector f(z) encircles the origin once in the positive direction; it is decreased by 2v if the radius vector encircles the origin once in the negative direction. Thus in Fig. 10.15, if Дг) traces C in the positive direction, arg/(z) is increased by 4- as г goes once around C in the positive direction.

harmonic excitations. Therefore the explicit form of the functions Д(г), ДиСг), etc., can be obtained from Eqs. 11 of § 6.9, by replacing ik by s.

f This conclusion is based on an important theorem by A. Erdelyi (Lecture Notes on Laplace Transformation, 1947, California Institute of Technology). Let S_д denote a region that consists of all points satisfying the condition |arg (s — j0)| <i (іг/2) + Д, where 0 < Д < ir/2. Then the theorem states that under the assumptions:

[30] The function /0) is analytic, and regular in S_A (but not necessarily at j0 or at 00).

[31] f(s) -> 0, uniformly in S„A, as j -> со.

[32] f(s) has the following asymptotic representation in the sense of Poincare:


fG) ~ 2 cn(s — ■*o)An> (— 1 < A0 < < • • ■ < A. v)

n== 0

uniformly in д, as s -> s0.

One concludes that the inverse Laplace transform of f(s) has the asymptotic repre­sentation

[33] There are numerous books and papers on automatic-control systems. See, for example, a book by H. M. James, N. B. Nichols, R. S. Phillips,*-81 and a paper by Bollay,10-10 from which further references can be found.

t For example, A. G. Webster, The Dynamics of Particles and of Rigid, Elastic, and Fluid Bodies, §§ 76-79, 91, and 102, reprinted by G. E. Stechert & Co., New York (1942). For the linearization of the inertia operator, see Refs. 10.10-10.13.

[34] The condition of tangency does not suffice to determine uniquely the flow around the airfoil. In addition, the Kutta-Joukowsky condition (§ 12.1) at the trailing edge must be satisfied. This condition determines the circulation around the airfoil.

The leading edge of an actual airfoil is rounded, and no special edge condition is needed. In a linearized theory, the airfoil being regarded as infinitely thin, the singu­larity at the leading edge may be regarded as the limiting form of such a rounded edge. In the small perturbation theory it suffices to require that the total integrated force be finite, and that the infinity in the pressure distribution at the leading edge be of a proper order, in analogy to the steady-flow theory.

[35] The following method is given by M. A. Biot.13,1

[36] 6 ~ v at the leading edge, б = 0 at the trailing edge, t Schwarz gives a second derivation of this equation in Ref. 13.29.

a + {тта + l)k + 277k2

[38] x0 is the point about which the moment is taken.

[39] The pressure perturbation p is continuous outside the airfoil, is antisymmetric in y, and hence vanishes behind the trailing edge when у — 0. Therefore, the lift, which is the difference of pressure across the airfoil, vanishes at the trailing edge. It does not vanish at the leading edge because the leading edge is a singular point for L.

t A shorter and more general derivation is given by Kiissner16,12 on the basis of Lorentz transformation of the wave equation. Kiissner’s integral equation holds for three-dimensional flow in both supersonic and subsonic cases. Specialization – into Possio’s integral equation in the subsonic case, into Birnbaum’s equation in the incom­pressible case, and into Prandtl’s equation in the steady-state case are demonstrated. But the form of Kussner’s integral equation is simple only if it is stated in terms of divergent integrals. In practice, an involved limiting process is required to trans­form such divergent integrals into those for which Cauchy’s principal value can be formed.

t See M. Muskhlishvili: Singular Integral Equations, in Russian, translated by Radok and Woolnough, Ministry of Supply, Australia. Published by P. Noordhoff, Groningen, Holland (1953).

[40] The notation is as follows: If a table is given for a range of the argument from 0.10 to 0.50 at intervals of 0.05; this is indicated by writing 0.10 (0.05) 0.50. The notation (5 fig.), (6 dec.), etc., implies that most of the figures in the table in question are given to the apparent accuracy of 5 figures or 6 decimals, respectively.

[41] Most of the experimental investigations are restricted to the determination of the forces normal to the plane of the airfoil, i. e., lift, moment, and pressure distribution. Little is known about the unsteady drag force, which plays only a minor role in aeroelastic problems. Only the normal forces will be considered here.

[42] Timoshenko, Theory of Elasticity, p. 301, McGraw-Hill (1934). t Trefftz, op. cit.