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Aerodynamic Forces and Moments
The most difficult problem in atmospheric flight mechanics is the mathematical modeling of the aerodynamic forces in a form that can be analyzed and evaluated quantitatively. Because the functional form is not known, the aerodynamic force functions are expanded in Taylor series in terms of the state variables relative to a reference flight. Even for digital computer simulations, restrictions for storage and computer time require that the number of independent variables in the aerodynamic tables be kept to a minimum. The dependency on the other variables then is expressed analytically by Taylor-series expansions.
For analytical studies a complete expansion is carried out for all state variables. There are two requirements that must be met. First, the partial derivatives of the expansions must be continuous—a condition that is usually satisfied; and second, the expansion variables must be small. In generating the aerodynamic forces three frames are involved: the atmosphere-fixed frame A, the body frame B, and the relative wind frame W. If the air is in uniform rectilinear motion relative to an inertial frame, A itself is an inertial frame. The wind frame has the c. m. of the vehicle as one of its points.
Usually it is postulated that the aerodynamic forces depend on external shape and size (represented by length /), atmospheric density p, and pressure p, the linear velocity of the airframe, c. m. relative to the atmosphere v д, the angular velocity of the body relative to atmosphere шВА, the acceleration of the c. m. wrt the atmosphere DAv g and, finally, the control surface deflections rj. In summary, the functional form is
fa = //(Л P, P, «BA, DAvj, V) (7.40)
The same functional relationship holds for the aerodynamic moment.
ma = /т(/, p, p, vAB, uBA, DAvA, rj) (7.41)
The expansions, called force expansion according to Hopkin,2 are carried out in the form of Eqs. (7.40) and (7.41). Variables that remain small throughout the perturbed flight must be identified. If the body frame does not yield these variables, the dynamic frame of the preceding section is introduced. As an example, a spinning missile requires a nonrotating body frame as dynamic frame.
