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Figure 2.45 Approximate fundamental frequency for a clamped-free beam with a particle of mass mi attached at .v = ri.

22. Consider a clamped-free beam to which is attached at spanwise location x = tr a particle of mass unit. Using a two-term Ritz approximation based on the functions in Eq. (2.311), plot the approximate value for the fundamental natural frequency as a function of r for // = 1.

Static Aero elasticity

I discovered that with increasing load, the angle of incidence at the wing tips increased perceptibly. It suddenly dawned on me that this increasing angle of incidence was the cause of the wing's collapse, as logically the load resulting from the air pressure in a steep dive would increase faster at the wing tips than at the middle. The resulting torsion caused the wings to collapse under the strain of combat maneuvers. A. H. G. Fokker in The Flying Dutchman (Henry Holt and Company, 1931)

The field of static aeroelasticity is the study of flight vehicle phenomena associated with the interaction of aerodynamic loading induced by steady flow and the resulting elastic deformation of the lifting surface structure. These phenomena are characterized as being insensitive to the rates and accelerations of the structural deflections. There are two classes of design problems that are encountered in this area. The first and most common to all flight vehicles is the effect of elastic deformation on the airloads associated with normal operating conditions. These effects can have a profound influence on the performance, handling qualities, flight stability, and structural load distribution. The second class of problems involves the potential for static instability of the structure that will result in a catastrophic failure. This instability is often termed "divergence" and can impose a limit on the flight envelope.

The material presented in this chapter provides an introduction to some of these static aeroelastic phenomena. To illustrate the physical mechanics of these problems and maintain a low level of mathematical complexity, relatively simple configurations are considered. The first items treated are rigid aerodynamic models that are elastically mounted in a wind tunnel test section. Such elastic mounting is characteristic of most load measurement systems. The second aeroelastic configuration to be treated is a uniform elastic lifting surface of finite span. Its static aeroelastic properties are quite similar to most lifting surfaces on conventional flight vehicles.

### 3.1 Wind Tunnel Models

In this section we consider three types of mounting for wind tunnel models: wall-mounted, sting-mounted, and strut-mounted. Expressions for the aeroelastic pitch deflections are developed for these simple models that, in turn, lead us to a cursory understanding of the divergence instability. Finally, we will return to the wall-mounted model briefly in this section to consider the qualitatively different phenomenon of aileron reversal.

### 3.1.1 Wall-Mounted Model

Consider a rigid, spanwise-uniform model of a wing that is mounted to the side walls of a wind tunnel in such a way as to allow the wing to pitch about the support axis, as illustrated in Fig. 3.1. The support is flexible in torsion, which means that it restricts the pitch rotation of the wing in the same way as a rotational spring would. We denote the rotational stiffness of the support by k; see Fig. 3.2. If we assume the body to be pivoted

Figure 3.1 Planform view of a wind tunnel model on a torsionally elastic support.

about its support O, located at a distance x0 from the leading edge, moment equilibrium requires that the sum of all moments about O must equal zero. In anticipation of using linear aerodynamics, we assume the angle of attack, a, to be a small angle, such that sin(a) = a and cos(a) = 1. Thus,

If the support were rigid, the angle of attack would be ar, positive nose-up. The elastic part of the pitch angle is denoted by 0, which is also positive nose-up. The wing angle of attack is then a = a1■ + 0. For linear aerodynamics, the lift for a rigid support is simply

¿rigid = qSCLar, whereas the lift for an elastic support is L =qSCL.(aI + d),

where q — \p^U2 is the freestream dynamic pressure (i.e., in the far field - often denoted by <?oo)- U is the freestream air speed, p^ is the freestream air density, S is the planform area,

and Cl„ is the wing lift-curve slope. Note that L / ¿rigid; and, for positive 0, L > Ln„l(|. We can express the moment of aerodynamic forces about the aerodynamic center as

If the angle of attack is small, Cm&c can be regarded as a constant. It should be noted here that linear aerodynamics implies that the lift-curve slope C /_„ is a constant. A further simplification may be that Cia — 2tt in accordance with two-dimensional thin-airfoil theory. If experimental data or results from computational fluid dynamics provide an alternative value, then it should be used.

Using Eqs. (3.3) and (3.4), the equilibrium equation, Eq. (3.1), can be expanded as qScCMac + qSCLa(uv + 0) (x0 - xac) - W(x0 - JcCg) = k6. (3.5)

Solving Eq. (3.5) for the elastic deflection, one obtains qScC/M ac + qSCLaarU'o - xac) - W(x0 ~ Xcg)

When and q are specified the total lift can be determined.

When the support point O is aft of the aerodynamic center, so that x0 > xac, the denominator can vanish, which implies that 0 blows up. This behavior is a static aeroelastic instability called "divergence." The dynamic pressure at which divergence occurs for this case is k

From this, the air speed at which divergence occurs can be found as

It is evident that when the aerodynamic center is coincident with the pivot, so that x0 — xac, the divergence dynamic pressure becomes infinite. Also, when the aerodynamic center is aft of the pivot, so that x0 < xac, the divergence dynamic pressure becomes negative. In either case divergence is impossible.

To pursue the character of this instability a bit further, consider the case of a symmetric airfoil (CMilC — 0). Furthermore, let Xq — -^-cg SO that the weight term drops out of the equation for 0. From Eq. (3.7) we can let k — qDSCLa U'o — xac), and so 0 can be written simply as ar

The lift is proportional to ar + 0. Thus, the change in lift divided by the rigid lift is given by

Jrigid

Both 0 and AL/LngKi clearly approach infinity as q qD. Indeed, a plot of the latter is given in Fig. 3.3 and shows the large change in lift caused by the aeroelastic effect. The lift evidently starts from its "rigid" value, that is, the value it would have were the support rigid, and increases to infinity as q —qn ■ However, keep in mind that there are limitations on the validity of both expressions. Namely, the lift will not continue to increase as stall is

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