A step that is always involved in an aeroelastic analysis is the determin­ation of elastic deformation from exterior forces. This can be expressed in operational form by an equation:

= Ostru {Paero + Oiner – f F„} (1)

where % is a vector describing the elastic deformation, Ostru is a structural operator giving % for any system of exterior forces. Oaer0 and 0lner are, respectively, the aerodynamic and inertia operators winch give the exterior forces caused by a given elastic deformation, and F„ is the acting exterior force that is independent of the elastic deformation. The operators Ostru, Oaero can be expressed as integrals with proper kernels.

When % represents the true elastic displacement in the body, the struc­tural operator (or rather the kernel of the integral representing the structural operator) is symmetric according to Maxwell’s reciproca relation, which states that the elastic displacement % at a point A causer by a force F acting at a point В is equal to the displacement (in the direc tion of F) at В caused by a force (acting in the direction of?) at A. Bu the kernel of the aerodynamic operator is in general unsymmetric, becaus the aerodynamic force at A caused by a displacement at В is, in general, different from that at В caused by a “corresponding” displacement at A. Thus Eq. 1, as an integral equation, has an unsymmetrical kernel.

The unsymmetry of the kernel of the basic integral equation 1 may also arise from the structural operator when % represents the displacement in generalized coordinates. For example, the deformation of a slender wing may be described by a deflection w perpendicular to the plane of the wing and a rotation в about a reference axis. Then f may be considered as having two components w and в. Correspondingly the exterior forces are generalized forces: a lift corresponding to w and a moment corres­ponding to в. In this case the structural operator, which consists of integrals of the products of the exterior forces with proper influence functions, becomes unsymmetric in general, because the deflection at a point A due to a couple acting at a point В is in general unequal to the deflection at В due to a couple at A. In other words, the influence function connecting w with the moment about the elastic axis is unsym­metric.

An integral equation with a real symmetric kernel possesses many nice properties that are lost when the symmetry of the kernel is lost. The main mathematical difficulty in aeroelasticity lies in the unsymmetry of the kernels of the governing integral equations.

Symmetric cases are exceptions rather than the rule in aeroelasticity. It is interesting to consider the conditions under which the operators become symmetric. For concreteness let us consider a slender wing. The structural operator becomes symmetric when bending and torsion are “separated,” i. e., when a torsional moment about the elastic axis induces no deflection of that axis and vice-versa. This occurs for an unswept cantilever wing with a straight elastic axis. The aerodynamic operator is simplified when the strip assumption is introduced, according to which the aerodynamic forces and moments at any section of the wing simply depend on the local % and wing chord. The inertia operator is symmetric if linearized. Thus the kernel of Eq. 1, as an integral equation, may become symmetric when bending and torsion of the wing are elastically uncoupled and when the strip assumption for aerodynamic forces is introduced. The problems of torsional divergence and aileron reversal of a normal wing as given in § 3.2 and § 4.2 are examples of this case. The divergence and reversal of a swept wing (§ 4.5) are examples of the unsymmetric case. In the flutter problem, the aerodynamic forces are complex functions of the reduced frequency; the kernel is no longer real valued.

Most problems in aeroelasticity can also be formulated as boundary – value problems in differential equations, which are connected with the integral equations by proper Green’s functions (i. e., the influence func­tions). It is well known that the Green’s functions are symmetric if the boundary-value problem defined by the differential equation is “self – adjoint” (see Collatz).11® The unsymmetry of the aeroelastic operators is associated with the non-self-adjointness of the boundary-value problem.

The stability problems in aeroelasticity are eigenvalue problems. Under certain conditions, among which the Hermitian self-adjointness or the Hermitian symmetry of the kernel is the most important, it can be shown that eigenvalues always exist and are real valued, that the eigenfunctions form a “complete” set of functions, that the iteration procedure for the calculation of the eigenvalues and eigenfunctions is valid, and that the bounds to the eigenvalues can be estimated. The same is not all true with regard to non-self-adjoint problems. The eigenvalues are in general complex and may not always exist. The completeness of the eigenfunctions is questionable, and, consequently, ordinary proofs of the convergence of the iteration procedure requires re-examination. A simple estimation of the bounds of the eigenvalues is yet unknown.

It is beyond the scope of this book to discuss the mathematical problems connected with the non-self-adjoint equations in aeroelasticity. There exists an extensive mathematical literature, but simple and decisive theorems useful for practical calculations are few. It may be pointed out, however, that recent studies initiated by the flutter research have already yielded many significant results. Of greatest importance is Wielandt’s proof1124 that the classical iteration procedure can be used to find the eigenvalues (if they exist at all) and eigenfunctions of non-self-adjoint equations. Of practical methods of calculation, Lanczos’s “minimized iterations” method1113,11-14 is powerful and labor-saving, particularly when several eigenvalues and eigenfunctions are desired. Wielandt’s “iterative transformation” procedure11-24-11-26 is applicable to flutter and similar eigenvalue problems. (A partial but much more readable account of Wielandt’s method is given by Gossard11-9.) By a simple extension, Wi. elandt also gives a “broken (gebrochene) iteration” procedure11-24 which can be used to correct a given approximation for any arbitrary higher eigenvalue and the corresponding eigenfunctions without the knowledge of the preceding eigenvalues. Both Wielandt’s and Lanczos’s methods are applicable to algebraic (matrix), differential, or integral operators, and are of importance in studying the fundamental questions in aeroelasticity and in checking approximate solutions. On the other hand, solution of nonhomogeneous equations on the basis of expanding an arbitrary function in series of biorthogonal functions, the concept of “adjoint energy function,” and a variational principle which leads to a procedure of the Rayleigh-Ritz type have been introduced by Flax.7-1 A

method of calculating the eigenvalues of complex matrices is given by Wielandt,11-24 who gives also a simple algorithm to determine whether some of the eigenvalues have a positive imaginary part. A different form of the last mentioned generalization of Routh’s rules for discriminating the pseudo-negative roots of a polynomial with complex coefficients is given independently by Sherman, DiPaola, and Frissell.11-20,11-21

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