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Aero elastic Flutter

The pilot of the airplane ... succeeded in landing with roughly two-thirds of his horizontal tail surface out of action; some others have, unfortunately, not been so lucky The flutter problem is now generally accepted as a problem of primary concern in the design of current aircraft structures. Stiffness criteria based on flutter requirements are, in many instances, the critical design criteria. ... There is no evidence that flutter will have any less influence on the design of aerodynamically controlled booster vehicles and re-entry gliders than it has, for instance, on manned bombers.

R. L. Bisplinghoff and H. Ashley in Principles of Aeroelasticity ( John Wiley and Sons, Inc., 1962.)

Chapter 2 dealt with the subject of structural dynamics, which is the study of phenomena associated with the interaction of inertial and elastic forces in mechanical systems. In particular, the mechanical systems considered were one-dimensional continuous configurations that exhibit the general structural dynamic behavior of flight vehicles. If in the analysis of these structural dynamic systems aerodynamic loading is included, then the resulting dynamic phenomena may be classified as aeroelastic. As has been observed in Chapter 3, aeroelastic phenomena can have a significant influence on the design of flight vehicles. Indeed, these effects can greatly alter the design requirements that are specified for the disciplines of performance, structural loads, flight stability and control, and even propulsion. In addition, aeroelastic phenomena can introduce catastrophic instabilities of the structure that are unique to aeroelastic interactions and can limit the flight envelope.

Recalling the diagram in Fig. 1.1, one can classify aeroelastic phenomena as either static or dynamic. Whereas Chapter 3 dealt only with static aeroelasticity, in the present chapter we examine dynamic aeroelasticity. Although there are many other dynamic aeroelastic phenomena that could be treated, we focus in this chapter entirely on the instability called flutter, which generally leads to a catastrophic structural failure of the flight vehicle. A formal definition of aeroelastic flutter may be given as: A dynamic instability of a flight vehicle associated with the interaction of aerodynamic, elastic, and inertial forces. From this definition it is apparent that any investigation of flutter stability requires an adequate knowledge of the system's structural dynamic and aerodynamic properties.

Of the various phenomena that are categorized as aeroelastic flutter, lifting surface flutter is the one that is most often encountered and most likely to result in a catastrophic structural failure. As a result, it is required that all flight vehicle lifting surfaces be analyzed and tested to assure that this dynamic instability will not occur for any condition within the vehicle's flight envelope. If the airflow about the surface becomes separated during any portion of the elastic oscillation, the instability is called stall flutter and the governing equations become nonlinear. This type of instability most commonly occurs on turbojet compressor and helicopter rotor blades. Other phenomena that result in nonlinear behavior include large deflections, mechanical slop, and nonlinear control systems. Nonlinear phenomena will not be considered in the present treatment.

Even with this obvious paring down of the problem, one still finds that linear flutter analysis of clean lifting surfaces is complicated. We can only offer a simplified discussion of the theory of flutter. The reader is urged to consult the bibliography for additional reading on the subject.

This chapter begins by using the modal representation to set up a lifting surface flutter analysis as a linear set of ordinary differential equations. These are transformed into an eigenvalue problem, and the stability characteristics are then discussed in terms of the eigenvalues. Then, as an example of this methodology, a two-degree-of-freedom "typical section" analysis is formulated using the simple steady-flow aerodynamic model used in Chapter 3. The main shortcoming of this simple analysis is the neglect of unsteady effects in the aerodynamic model. Motivated by the need to consider unsteady aerodynamics in a meaningful but simple way, we then introduce classical flutter analysis. Engineering solutions that partially overcome the shortcomings of classical flutter analysis are then presented. To complete the set of analytical tools needed for flutter analysis, two very different unsteady aerodynamic theories are outlined, one suitable for use with classical flutter analysis and its derivatives, and the other suitable for eigenvalue-based flutter analysis. After illustrating how to approach the flutter analysis of a flexible wing using the assumed modes method, the chapter concludes with a discussion of flutter boundary characteristics.

### 4.1 Stability Characteristics

The lifting surface flutter of immediate concern can be described by a linear set of structural dynamic equations that include a linear representation of the unsteady airloads in terms of the elastic deformations. The surface could correspond to a wing or stabilizer either with or without control surfaces. Analytical simulation of the surface is sometimes made more difficult by the presence of external stores, engine nacelles, landing gear, or internal fuel tanks. Although such complexities complicate the analysis, they do not significantly alter the physical character of the flutter instability. For this reason the following discussion will be limited to a "clean" lifting surface.

When idealized for linear analysis, the nature of flutter is such that the flow over the lifting surface not only creates steady components of lift and pitching moment but also creates dynamic forces in response to small perturbations of the lifting surface motion, pitch and plunge motions in particular. Recall that the pitching motion of an airfoil may arise from torsional deformation, and the plunging of the airfoil may arise from bending deformation. When a lifting surface that is statically stable below its flutter speed is disturbed, the oscillatory motions caused by those disturbances will die out in time with exponentially decreasing amplitudes. That is, one could say that the air is providing damping for all such motions. Above the flutter speed, however, rather than damping out the motions caused by small perturbations in the configuration, the air can be said to be providing negative damping. Thus, these oscillatory motions grow with exponentially increasing amplitudes. This qualitative description of flutter can be observed in a general discussion of stability characteristics.

Before attempting to conduct an analysis of flutter, it is instructive to first examine the possible solutions to a structural dynamic representation in the presence of airloads. We will presume that the flight vehicle can be represented in terms of its normal modes of vibration. We illustrate this with the lifting surface modeled as a plate rather than a beam. This is somewhat more realistic for low-aspect-ratio wings, but in the present framework this increased realism presents very little increase in complexity because of the modal representation. For displacements w(x. y. t ) in the z direction normal to the plane of the planform (the x-y plane), the normal mode shapes can be represented by </>,(.r. v) and the associated natural frequencies by &>,. A typical displacement of the structure can be written as n

where §,-(0 is the generalized coordinate of the z'th mode. It should be noted that the summation includes the index value i — 0, which symbolically denotes that rigid-body degrees of freedom have been included. The set of generalized equations of motion for the flight vehicle can be written as

where M, is the generalized mass associated with the mass distribution, m(x. y), and can be determined as

planform

The generalized force, S,(/), associated with the external loading, F(x. y, t), can be evaluated as

planform

To examine the stability properties of the flight vehicle, the only external loading to be considered is from the aerodynamic forces, which can be represented as a linear function of w(x, y, 0 and its time derivatives. It will be presumed that all other external disturbances have been eliminated. Such external disturbances would normally include atmospheric gusts, store ejection reactions, etc. Recalling that the displacement can be represented as a summation of the modal contributions, the induced pressure distribution, A p(x, y, t), can be described as a linear function of all the generalized coordinates and their derivatives. Such a relationship can be written as n

Ap(x, y, 0 - }j«/( v. y)i/C) + bj(x, y)kj(t) + Cj(x, y)|,■(/)]. (4.5)

The corresponding generalized force of the z'th mode can now be determined from

planforn

planform cij (jc , y )<pi (x, y ) dx dy bj (x, y )</>; (x, y ) dx dy (4.6)

planform

Following the convention in some published work, we have factored out the freestream air density p^ from the aerodynamic generalized force expression. Although not necessary, this step does enable the analyst to identify altitude effects more readily. It also shows explicitly that all aerodynamic effects vanish in a vacuum where poo vanishes. Moreover, the normalization involving powers of b/U, where b is a reference semi-chord of the lifting surface, allows the matrices [o], [/?], and [c] to have the same units. Any nonhomogeneous terms in the generalized forces can be eliminated by redefinition of the generalized coordinates so that they are measured with respect to a different reference configuration. Thus, the generalized equations of motion can be written as a homogeneous set of differential equations when this form of the generalized force is included. They are

The general solution to this set of second-order, linear, ordinary differential equations can be described as a simple exponential function of time, because they are homogeneous. The form of this solution will be taken as

Substitution of this expression into Eqs. (4.7) yields n + 1 simultaneous linear, homogeneous, algebraic equations for the §,-s since each term will contain an exp(vf). Thus,

, , - v^ (b2v2 bv \ — Mi (v2 + w2) Hi ~ Poo I -jjrCij + —bij + au 1^=0 (I = 0, 1 n).

For a nontrivial solution of the generalized coordinate amplitudes, the determinant of the array formed by the coefficients of Hi must be zero. It is apparent that this determinant is a polynomial of degree 2 (n + 1) in v. Subsequent solution of this polynomial equation for v will typically yield n + 1 complex conjugate pairs, represented as vk = rk±iQk (A: = 0,1 n). (4.10)

For each v* there is a corresponding complex column matrix , j = 0, 1,..., n. Thus, the solution of the generalized equations of motion with aerodynamic coupling can be written as n w(x, y, t) = y) exp [(TA. + iQk)t] + wk(x, y)exp [(rA. - iQk)t]}, k=0

where uJk is the complex conjugate of wk. This expression for w(x, v. t) turns out to be real, as expected. Each wk represents a unique linear combination of the mode shapes of the structure; that is, n wt(x, y) = y) (A: = 0,1 n). (4.12)

Note that only the relative values of Hi can be determined unless the initial displacement and rate of displacement are specified.

It is apparent from the general solution for w(x. y. t), Eq. (4.11), that the Ath component of the summation represents a simple harmonic oscillation that is modified by an exponential function. The nature of this dynamic response to any specified initial condition is strongly dependent on the sign of each F/,. Typical response behavior is illustrated in Fig. 4.1 for positive, zero, and negative values of when Q/, is nonzero. We note that the negative of r*. is sometimes called the modal damping of the At h mode, and Q/, is called the modal frequency. It is also possible to classify these motions from the standpoint of stability. The convergent oscillations when Ta < 0 are termed dynamically stable and the divergent oscillations for r* > 0 are dynamically unstable. The case of Ta = 0 represents the boundary between the two and is often called the "stability boundary." If these solutions are for an aeroelastic system, the dynamically unstable condition is called flutter, and the stability boundary corresponding to simple harmonic motion is called the flutter boundary.

r* |
Qk |
Type of Motion |
Stability Characteristic |

< 0 |

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