NATURE OF THE OPERATORS IN AEROELASTICITY
As an aeroelastic system may be composed of a large number of electrical, hydraulic, mechanical, as well as aerodynamic elements, it is evident that the operators involved are much varied. From the point of view of analysis, it is convenient to classify the operators into two kinds: (1) those relating quantities that are essentially independent of the space coordinates, such as an electric voltage across two terminals, and (2) those relating quantities that are functions of space, such as the elastic displacements of an airplane. In operators of the second kind, space integrals are generally involved. Control-system operators, mechanical, electric,
or hydraulic, are generally of the first kind, while aerodynamic, structural, and inertia operators are generally of the second kind.
For example, a simple resistor and capacitator network as shown in Fig. 11.6 is governed by the following equation:
The interested variables V1 and V2 are essentially independent of the space coordinates. More complicated networks can be built up by such elementary ones. The resulting relations between the input and output can be expressed as ordinary linear differential equations. The analysis of such a system presents little difficulty in principle. The design of a satisfactory system to perform a specified function, (the so-called synthesis problem) of course is more difficult.
On the other hand, the inertia force, aerodynamic force, and elastic deformation, being functions of both space and time, must be governed either by partial differential equations or by integral equations.
In the following sections, the structural and aerodynamic operators occurring in aeroelasticity will be considered. The inertia operator describes inertia forces. In airplane dynamics, it is convenient to use a system of reference coordinates attached to the airplane, and thus moving with respect to an “inertial” frame of reference. The expression of the inertia operator referring to moving axes can be quite complicated, but it has been treated exhaustively in books of theoretical mechanics. f The case of small disturbances from a steady symmetric motion is of particular importance. The inertia operator can be linearized under the assumption that the square and higher-order products of the small disturbances (linear and angular velocities, as well as the elastic displacements) are negligible in comparison with the disturbances themselves.