# the effect of aerodynamic forces

Aeroelasticity is the study of the effect of aerodynamic forces on elastic bodies.

The classical theory of elasticity deals with the stress and deformation of an elastic body under prescribed external forces or displacements. The external loading acting on the body is, in general, independent of the deformation of the body. It is usually assumed that the deformation is small and does not substantially affect the action of external forces. In such a case we often neglect the changes in dimensions of the body and base our calculations on the initial shape. Even in problems of bending and buckling of columns, plates, or shells, either the external loading or the boundary constraints are considered as prescribed. The situation is different, however, in most significant problems of aeroelasticity. The aerodynamic forces depend critically on the attitude of the body relative to the flow. The elastic deformation plays an important role in determining the external loading itself. The magnitude of the aerodynamic force is not known until the elastic deformation is determined. In general, therefore, the external load is not known until the problem is solved.

One of the interesting problems in aeroelasticity is the stability (or rather instability) of a structure in wind. Since, for a given configuration of the elastic body, the aerodynamic force increases rapidly with the wind speed, while the elastic stiffness is independent of the wind, there may exist a critical wind speed at which the structure becomes unstable. Such instability may cause excessive deformations, and may lead to the destruction of the structure.

A major problem is the flutter of structures such as airplanes or suspension bridges, when small disturbances of an incidental nature induce more or less violent oscillations. It is characterized by the interplay of aerodynamic, elastic, and inertia forces, and is called a problem of dynamic aeroelastic instability. The particular case of an oscillation with zero frequency, in which in general the inertia force may be neglected, is called the steady-state, or static, aeroelastic instability.

Quite different from the above are the response problems in which the response of an aeroelastic system to an externally applied load is to be found. The external load may be caused by a deformation of the elastic body, such as a displacement of the control surfaces of an airplane, or by disturbances such as gusts, landing impacts, or turbulences in the flow.

The response to be found may be the displacement, the motion, or the stress state induced in the elastic body. Again the response problems may be classified into the steady-state or static problems, in which the inertia forces may be neglected, and the dynamic problems, in which the aerodynamic, elastic, and inertia forces all enter into the picture.

There is a close relationship between the stability problems and the response problems. Mathematically, most stability problems can be described by a system of homogeneous equations, which are satisfied by a trivial solution of zero displacement (or zero motion), meaning that nothing happens at all. On the other hand, a response problem is represented by a nonhomogeneous system; i. e., the initial conditions and the external forces are such as to cause the governing equations to be nonhomogeneous, and to admit a solution not vanishing identically. A response problem generally associates with a stability problem. As an example, consider the response of an airplane wing to atmospheric turbulences. We can formulate the problem of flutter by asking the following questions: Is there a critical speed of flight at which the airplane structure becomes exceedingly sensitive to the atmospheric turbulence; i. e., does there exist a speed at which the structure may have a motion of finite amplitude, even in the limiting case of an atmospheric turbulence of zero intensity? This is equivalent to the following formulation which is usually made in flutter analysis: Is there a critical speed at which the aeroelastic system becomes neutrally stable, at which motion of the structure is possible without any external excitation? Thus the response of an airplane structure to atmospheric turbulence and the flutter problem are linked together. When the response of the structure to a finite disturbance is finite, the structure is stable, and flutter will not occur. When the structure flutters at a critical speed of flow, its response to a finite disturbance becomes indefinite.

This alternative theorem is true in practically all corresponding response and stability problems. Either the homogeneous system has a nontrivial solution while the corresponding nonhomogeneous system has no solution, or the nonhomogeneous system has a solution while the corresponding homogeneous system has no solution other than the trivial one.* It is thus proper to discuss the response and stability problems together as two phases of the same phenomenon.

There exists, however, a very important distinction between the response and stability problems, in regard to the justification of the linearization process often used in the mathematical formulation of a physical problem.

* In exceptional cases both the nonhomogeneous and the corresponding homogeneous system may have a solution. But such exceptions ordinarily have little engineering significance.

In the stability problems, the amplitude of the elastic deformation is indeterminate, and only the modes of deformation (not their absolute magnitude) are of interest; hence, it is logical to consider the elastic deformation as infinitesimal in the neighborhood of an equilibrium state. Therefore the small deflection theory in elasticity and aerodynamics is applicable, and linearization of the governing equations can be justified. On the other hand* the absolute magnitudes of the deformation and stress in a structure are of primary interest in the response problems. Hence, it is necessary to consider finite deformations. As the fundamental equations of fluid and solid mechanics are often nonlinear, it is necessary to consider the effects of nonlinearity, whenever the response reaches a finite amplitude. Thus the justification of linearization of the fundamental equations is always open to question.

Of course it is desirable to treat the nonlinear equations per se, but the mathematical difficulties are generally insurmountable. Generally we are forced to linearize, in order to reach a practical solution. Then it must always be remembered that the justification of the linearization remains to be shown.

In this book, attention will be directed mainly to the stability problems, not because the response problems are less important, but because they are well-known in engineering philosophy. On the other hand, the stability aspect of aeroelasticity is novel.

Generally speaking, aeroelasticity includes the study of all structures in a flow. But those problems in which the elastic deformation plays no significant role in the determination of the external loading will not be discussed in this book. For example, the problem of the distribution of wind load on a building will be excluded.

A survey of the field of aeroelasticity is given in Chapters 1 through 11. Important problems are discussed from the physical point of view. The chief aim is to provide an elementary treatment of the basic problems and to point out the essential parameters involved in their solution. The aerodynamic problems are discussed in greater details in Chapters 12 through 15.