# Aeroelastic 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 3 addressed 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 observed in Chapter 4, 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, we can classify aeroelastic phenomena as either static or dynamic. Whereas Chapter 4 addressed only static aeroelasticity, in this chapter, we examine dynamic aeroelasticity. Although there are many other dynamic aeroelastic phenomena that could be treated, we focus entirely on the instability called “flutter,” which generally leads to a catastrophic structural failure of a flight vehicle. A formal definition of aeroelastic flutter is as follows: 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. To further elaborate, flutter is a self-excited and potentially destructive

oscillatory instability in which aerodynamic forces on a flexible body couple with its natural modes of vibration to produce oscillatory motions with increasing amplitude. In such cases, the level of vibration will increase, resulting in oscillatory motion with amplitude sufficiently large to cause structural failure.

Because of this, structures exposed to aerodynamic forces—including wings and airfoils but also chimneys and bridges—must be carefully designed to avoid flutter. In complex systems in which neither the aerodynamics nor the mechanical properties are fully understood, the elimination of flutter can be guaranteed only by through testing. Of the various phenomena that are categorized as aeroelastic flutter, lifting – surface flutter is most often encountered and most likely to result in a catastrophic structural failure. As a result, it is required that lifting surfaces of all flight vehicles be analyzed and tested to ensure that this dynamic instability will not occur for any condition within the vehicle’s flight envelope.

If the airflow about the lifting surface becomes separated during any portion of an unstable oscillatory cycle of the angle of attack, the governing equations become nonlinear and the instability is referred to as “stall flutter.” Stall flutter 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 are not considered in the present treatment. Even with this obvious paring down of the problem, however, we still find that linear-flutter analysis of clean lifting surfaces is complicated. Thus, we can offer only a simplified discussion of the theory of flutter. Readers are urged to consult the references for additional information 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 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 4. 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 follow. To complete the set of analytical tools needed for flutter analysis, two different unsteady-aerodynamic theories are outlined: one suitable for use with classical flutter analysis and its derivatives; 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.

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