## 2 Dof Static Airfoil Divergence Speed Formula

This is the same answer as one would obtain with an analysis similar to those of Chapter 3.

For looking at flutter, we consider a specific configuration defined by a — —1/5, e — — 1/10, [i — 20, r2 — 6/25, and a — 2/5. The divergence speed for this configuration is Vd — 2.828 (or l! ¡, — 2.828 b oi,, ). Plots of the imaginary and real parts of the roots versus Figure 4.3 Plot of the modal frequency versus V for a = —1/5, e = —1/10, ¡a = 20, r2 = 6/25, and a = 2/5 (steady-flow theory).

V are shown in Figs. 4.3 and 4.4, respectively. The negative of T is the modal damping, and Q is the modal frequency. We consider first the imaginary parts, Q, as shown in Fig. 4.3. When

V — 0, one expects the two dimensionless frequencies to be near unity and a for pitching and plunging oscillations, respectively. Even at V — 0 these modes are lightly coupled because of the nonzero off-diagonal term xg in the mass matrix. As V increases, the frequencies start to approach one another, and their respective mode shapes exhibit increasing coupling between plunge and pitch. Flutter occurs when the two modal frequencies coalesce, at which point the roots become complex conjugate pairs. At this condition, both modes are highly coupled pitch-plunge oscillations. The flutter speed is VF — Up/{bu>g) = 1.843, and the flutter frequency is /twg = 0.5568. The real parts, T, are shown in Fig. 4.4 and remain zero until flutter occurs. When flutter occurs, the real part of one of the roots is positive and the other is negative.

Comparing results from the above analysis with experimental data, one finds that a few elements of realism are at least qualitatively captured. For example, the analysis predicts that flutter occurs at some value of V — V> < V0, which is correct for the specified configuration. Furthermore, it shows a coalescence of the pitching and plunging frequencies as r Figure 4.4 Plot of the modal damping versus V for a = —1/5, e = —1/10, ¡jl = 20, r2 = 6/25, and a = 2/5 (steady-flow theory).

V approaches V>, which is not only correct for the specified configuration but is frequently observed in connection with flutter analysis. However, the above analysis is deficient in its ability to accurately predict the flutter speed. Moreover, the damping of all modes below the flutter speed is predicted to be zero, which is known to be incorrect.

The main reason for these deficiencies is that the aerodynamic theory from Chapter 3 was used. Although the aerodynamic theory has obvious deficiencies, such as its linearity and two-dimensionality, a very significant deficiency as far as flutter analysis is concerned is that it neglects unsteady effects, which are in general very important for flutter problems. The flow is unsteady because of two separate physical reasons. First, because of the wing's unsteady motion relative to the air, the relative wind vector is not fixed in space. Second, the airfoil motion disturbs the flow, shedding a vortex at the trailing edge. The downwash from this vortex, in turn, changes the flow that impinges on the airfoil. Thus, to obtain a more accurate prediction of the flutter speed, it is necessary to include unsteadiness in the aerodynamic theory. This demands a far more sophisticated aerodynamic theory.

Unfortunately, development of unsteady aerodynamic theories is no small undertaking. Unsteady aerodynamic theories can most simply be developed when simple harmonic motion is assumed a priori. Although such limited theories cannot be used in the p method of flutter analysis described in Section 4.1, they can be used in classical flutter analysis, described in the next section. As will be seen, classical flutter analysis can predict the flutter speed and flutter frequency, but it cannot predict values of modal damping and frequency away from the flutter condition. To obtain a reasonable sense of modal damping and frequencies at points other than the flutter condition, a couple of approximate schemes will be discussed in Section 4.4.

If these approximations turn out to be inadequate for predicting modal damping and frequencies, one has no choice but to carry out a flutter analysis that does not assume simple harmonic motion, which in turn requires a still more powerful aerodynamic theory. One such approach that fits easily into the framework of Section 4.1 is the finite-state theory of Peters et al. (1995). Such a theory not only facilitates the calculation of subcritical eigenvalues, but since it is a time-domain model it also can be used in control design.

Hence, in the sections to follow, we first look at classical flutter analysis and the approximate techniques associated therewith and then turn to a more detailed discussion of unsteady aerodynamics, including one theory that assumes simple harmonic motion (the Theodorsen theory) and one that does not (the Peters finite-state theory).

### 4.3 Classical Flutter Analysis

Throughout the aircraft industry most lifting surface flutter analyses performed are based on what is called a "classical flutter analysis." The objective of such an analysis is to determine the flight conditions that correspond to the flutter boundary. It was previously noted that the flutter boundary corresponds to conditions for which one of the modes of motion has a simple harmonic time dependency. Since this is considered to be a stability boundary, it is implied that all modes of motion are convergent (stable) for less critical flight conditions (lower airspeed). Moreover, all modes other than the critical one are convergent at the flutter boundary.

The method of analysis is not based on solving the generalized equations of motion as described in Section 4.1. Rather, it is presumed that the solution involves simple harmonic motion. With such a solution being specified, the equations of motion are then solved for the flight condition(s) that yield such a solution. Whereas in the p method one determines the eigenvalues for a set flight condition, the real parts of which provide the modal damping, it is apparent that classical flutter analysis cannot provide the modal damping for an arbitrary flight condition. Thus, it cannot provide any definitive measure of flutter stability other than the location of the stability boundary. Although this is the primary weakness of such a method, its primary strength is that it needs only the unsteady airloads for simple harmonic motion of the surface, which are more easily and thus more accurately obtained than those for arbitrary motion.

To illustrate classical flutter analysis it is necessary to consider an appropriate representation of unsteady airloads for simple harmonic motion of a lifting surface. Because these oscillatory motions are relatively small in amplitude, it is sufficient to use a linear aerodynamic theory for the computation of these loads. These aerodynamic theories are usually based on linear potential flow theory, which presumes that the motion of the structure is a small perturbation with respect to the freestream speed. For purposes of demonstration it will suffice to consider again the typical section of a two-dimensional lifting surface that is experiencing simultaneous translational and rotational motions, as illustrated in Fig. 4.2. The motion is simple harmonic; thus, h and 0 will be represented as h(t) — h exp(/&>r),

6(t) — 9 exp(/tt»f), where a> is the frequency of the motion. Although the h and 0 motions are of the same frequency, they are not necessarily in phase. This can be taken into account mathematically by representing the amplitude 6 as a real number and h as a complex number. Since a linear aerodynamic theory is to be used the resulting lift, L, and the pitching moment about P, denoted by M where

will also be simple harmonic with frequency &>, so that

The amplitudes of these airloads can be computed as complex, linear functions of the amplitudes of motion as

Here the freestream air density is represented as p^ and the four complex functions contained in the square brackets represent the dimensionless aerodynamic coefficients for the lift and moment resulting from plunging and pitching. These coefficients are in general functions of the two parameters k and M^, where bm k — — (reduced frequency), U

As in the case of steady airloads, compressibility effects are reflected here by the dependence of the coefficients on The reduced frequency parameter k is unique to unsteady flows. This dimensionless frequency parameter is a measure of the unsteadiness of the flow and normally will have a value between zero and unity for conventional flight vehicles. It may also be noted that for any specified values of k and Af«, each of the coefficients can be written as a complex number. As in the case of h relative to 6, the fact that lift and pitching moment are complex quantities reflects their phase relationships with respect to the pitch angle (where we can regard 0 as a real number for convenience). The speed at which flutter occurs corresponds to specific values of k and M^ and must be found by iteration. Examples of how this process can be carried out for one- and two-degree-of-freedom systems are given in this section.

### 4.3.1 One-Degree-of-Freedom Flutter

To illustrate the application of classical flutter analysis a very simple configuration will be treated first. This example is a one-degree-of-freedom aeroelastic system consisting of a rigid two-dimensional wing that is permitted to rotate in pitch about a specified reference point; this is a special case of the typical section configuration in Fig. 4.2 for which the plunge degree of freedom is equal to zero, as depicted in Fig. 4.5. The system equations of motion reduce to one equation that can be written as

To be consistent with classical flutter analysis, the motion of the system will be presumed to be simple harmonic as

The aerodynamic pitching moment, M, in the equation of motion is in response to this simple harmonic pitching displacement. As previously discussed this airload can be described by

where

Substituting these simple harmonic functions into the equation of motion yields an algebraic Figure 4.5 Schematic of the airfoil of a two-dimensional wing that is spring-restrained in pitch.

relation between the coefficients of 0 as kg - arlp = Jtp00b4m2mg(k, Mœ). (4.41)

Introducing the natural frequency of the system at zero airspeed,

and rearranging the algebraic relation, one obtains the final equation to be solved for the flight condition at the flutter boundary as

TtPoob4

To solve this equation it will be presumed that the configuration parameters IP, <>>,,, and b are known. The unknown parameters that describe the motion and flight condition are co. Poo, k, and These four unknowns must be determined from the single algebraic equation, Eq. (4.43). Since the aerodynamic coefficient, mg(k, M^), is complex, it can be written as mg(k, Moo) = m, Moo) + Mqo). (4.44)

As a consequence, both the real and imaginary parts of the algebraic relation must be zero, thus providing two real equations to determine the four unknowns. Therefore, two of the unknown parameters should be specified. A fixed altitude will be chosen that specifies the freestream atmospheric density, p^. The second parameter to be fixed will be the Mach number, which can be given a temporary value of zero. This, of course, implies that the flow is incompressible and the aerodynamic moment coefficient is then only a function of the reduced frequency. The governing algebraic equation can now be written as

Ttprvb4

Equating the imaginary part of the left-hand side to zero gives a relation that can be solved for the reduced frequency, kF, at the flutter boundary,

With k f known :H(A>. 0) can be numerically evaluated. Equating the real part of the left-hand side to zero now enables the frequency, coF, to be determined from i^)2 = i + 0) (4 4?) \coF' Ip

Now that kF and coF have been determined, it is possible to compute the flutter speed as bcoF

The flutter speed determined by the above procedure corresponds to the originally specified altitude and is based on an incompressible representation of the airloads. After this speed has been determined, the speed of sound, c^, at the specified altitude can be used to find the flutter Mach number as

If this flutter Mach number is sufficiently small to justify the use of incompressible aerodynamics coefficients, then the altitude-speed combination obtained is a point on the flutter boundary. If the flutter Mach number is too high to validate the incompressible approximation, then the entire procedure should be repeated using aerodynamic coefficients that are based on the initially computed flutter Mach number. Using the standard atmospheric model, which relates density and the speed of sound, this iterative scheme will converge to a flight condition on the flutter boundary.

### 4.3.2 Two-Degree-of-Freedom Flutter

The analysis of multi-degree-of-freedom systems for determination of the flutter boundary can be adequately demonstrated by the simple two-degree-of-freedom configuration of Fig. 4.2. The equations of motion, already derived as Eqs. (4.24), are m(h + bxgO) + ki,h — —L,

The next step in classical flutter analysis is to presume that the motion is simple harmonic as represented by h — h exp(icot),

0—0 exp(/&)f). The corresponding lift and moment can be written as L — L exp (icot),

Substituting these time-dependent functions into the equations of motion, one obtains a pair of algebraic equations for the amplitudes of h and 0 of the form

-co2mbxgh - ccrlpO + IpcojO — M, where we recall that

Substituting these lift and moment amplitudes into Eq. (4.54) and rearranging, one obtains a pair of homogeneous, linear, algebraic equations for h and 0, given by

Ttprvb2

_7T Prvb2

mxg npoob2

Jtpo^b4

The coefficients in these equations that involve the inertia terms will be symbolically simplified by defining the dimensionless parameters used earlier, namely,

Ttp^b2

Ip mb2

Using these parameters allows us to rewrite the above two homogeneous equations in a somewhat simpler way:

The third step in the flutter analysis is to solve these algebraic equations for the flight condition(s) for which the presumed simple harmonic motion is valid. This result will correspond to the flutter boundary. If it is presumed that the configuration parameters m, e, a, Ip, ft)/,, ox,, and b are known, then the unknown quantities h, 0, ft), px, M0Q, and k describe the motion and flight condition. Because Eqs. (4.58) are linear and homogeneous in h/b and 0, the determinant of their coefficients must be zero for a nontrivial solution for the motion to exist. This condition can be written as fix g + m h fir fix g + ig

The determinant in this relation is called the "flutter determinant." It should be noted that the parameter a — ft)/, /o),, has been introduced, so that a common term explicit in &) is available, namely, cog /a>. Thus, expansion of the determinant will yield a quadratic polynomial in the unknown (a>g /ft))2.

To complete the solution for the flight condition at the flutter boundary it must be recognized that four unknowns remain: cog/ft), fi — m/(np^b2), Mm, and k — boj/U. The one equation available for their solution is the second-degree polynomial equation from the determinant. However, because the aerodynamic coefficients are complex quantities, this complex equation represents two real equations, wherein both the real and imaginary parts must be identically zero for a solution to be obtained. This means that two of the four unknowns must be specified. A procedure to solve for and map out the flutter boundary is outlined as follows:

1. Specify an altitude, which fixes the parameter //.

3. Specify a set of trial k values, say from 0.001 to 1.0.

4. For each value of k (and the specified value of Af«,) calculate the functions ('/,, lg, /«/,, and mg.

5. Solve the flutter determinant, which is a quadratic equation with complex coefficients, for the values of (wg /a»)2 that correspond to each of the selected values of k. Note that these roots will be complex in general, the real part being an approximation of (&>e/&>)2, and the imaginary part being related to the damping of the mode.

6. Interpolate to find the value of k at which the imaginary part of one of the roots becomes zero. This can be done approximately by plotting the imaginary parts of both roots versus k, so that the value of k at which one of the imaginary parts crosses the zero axis can be determined. This value of k then has a corresponding real value of (ong/ca)2, which provides the value of &>.

7. Determine U — bai/k and Af«, — U/c<x>.

8. Repeat steps 3-7 with the value of Mobtained in step 7 until converged values are obtained for MXf, kF, and CV for flutter at a given //.

9. Repeat the whole procedure for various values of // (an indication of the altitude for a given aircraft) to determine the flutter boundary in terms of, say, altitude versus MOOF, kF, and Ur.

Step 6 above can also be carried out easily with computerized symbolic manipulation software such as Mathematica or Maple. One simply finds the value of k that makes the imaginary part of one of the two roots of the flutter determinant vanish.

### 4.4 Engineering Solutions for Flutter

It has been noted in the preceding section that the presumption of simple harmonic motion in classical flutter analysis has both advantages and disadvantages. The prime argument for specification of simple harmonic time dependency is, of course, its correspondence to the stability boundary. Identification of the flight conditions along this boundary requires the execution of a tedious, iterative process such as the one outlined above. This type of solution can be attributed to Theodorsen (1934), who presented the first comprehensive flutter analysis with his development of the unsteady airloads on a two-dimensional wing in an incompressible potential flow.

Although unsteady aerodynamics analyses for simple harmonic motion are not simple to formulate and execute, they are far more tractable than those for oscillatory motions with varying amplitude. Over the years since the work of Theodorsen, numerous unsteady aerodynamic formulations have been developed for simple harmonic motion of lifting surfaces. These techniques have proven to be quite adequate for compressible flows in both the subsonic and supersonic regimes. They have also been developed for three-dimensional surfaces and in some cases with surface-to-surface interaction. This availability of relatively accurate unsteady aerodynamic theories for simple harmonic motion has been the stimulus for further development of flutter analyses beyond that of the classical flutter analysis described in Section 4.3.

There are two other very important considerations of the practicing engineer. The first is to obtain an understanding of the margin of stability at flight conditions in the vicinity of the flutter boundary. The second, and possibly the more important, is to obtain an understanding of the physical mechanism that causes the instability. With these two pieces of information, the engineer can propose design variations that may alleviate or even eliminate the instability. When a suitable unsteady aerodynamic theory is available, the p method can address these considerations. In this section we look at alternative ways that engineers have addressed these problems when unsteady aerodynamic theories that assume simple harmonic motion must be used.

### 4.4.1 The k Method

Subsequent to Theodorsen's analysis of the flutter problem, numerous schemes were devised to extract the roots of the flutter determinant and thus identify the stability boundary. Scanlan and Rosenbaum (1951) presented a brief overview of these techniques as they were offered during the 1940s. It was fairly common to include in the flutter analysis a parameter that simulated the effect of structural damping. Observations at that time indicated that the energy removed per cycle during a simple harmonic oscillation was nearly proportional to the square of the amplitude but independent of the frequency. This behavior can be characterized by a damping force that is proportional to the displacement but in phase with the velocity.

To incorporate this form of structural damping into the analysis of Section 4.3.2, Eqs. (4.50) can be written as m [h + bxBd) + ki,h — —L + D/,, I pi) + mbxgh + kg6 — M + Dg, where the dissipative structural damping terms are Dh — Dh exp{icot)

— -igglpcoje exp(/£Wf). Proceeding as before, Eqs. (4.58) become