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The various classes of problems, and the corresponding techniques for solution of the equations, can be briefly previewed as follows. Symmetry If the airplane is symmetrical, it is conceptually legitimate to consider pure longitudinal motions. These may involve changes of forward velocity, angle of attack, and pitch attitude; but always in straight, wings-level flight at zero sideslip. For these motions and flight conditions, the symmetry reduces every rolling moment, yawing moment, and side force identically to zero and the pure longitudinal motions are described by three equations for drag and lift forces and for pitching moments.
In diagrams it is represented by the symbol > . Input (error) >\ ^ Output The analyst needs to know only a few things about these amplifiers. One is that they draw practically no error current; another, that their voltage gain is very large, in fact almost infinity; and that their dynamic response is essentially perfect to frequencies higher than those in his problem. Because of the almost infinite voltage gain, for any ordinary 7 Inputs '» Ί ee 29 —^w— r . terror Output r > 3'« FIG. 7. Amplifier used for summing.
In this example lX-PM-X> + Cl =0 (12b) where Pi=-p2=jVC0 (12c) *i = 0 (12d) Then plot the poles and zeros on the complex plane (Fig. 1). Consider a trial value for λ, shown as the X in Fig. 1. The vector (λ — Pi) (λ — />2)/(λ — #i) is the product of the two solid vectors divided by the dashed one. Its phase is the sum of the solid vector phases minus that of the dotted one. It must equal 180°. The trial point in the diagram 30 I I . MATHEMATICAL TECHNIQUES FOR LINEAR DIFFERENTIAL EQUATIONS Imaginary axis Real axis Σφρ - Σφζ = 180 λ -/> FIG.