SOLUTION FOR LARGE ANGLES

The preceding analysis shows how a lateral response starts, but not how it continues. For that we need solutions to the governing differential equations. As remarked ear­
lier, at the beginning of this chapter, control responses can rapidly build up large val­ues of some variables, invalidating the linear equations that we have used so far. There is a compromise available that includes only some nonlinear effects that is use­ful for transport and general aviation airplanes, which are not subjected to violent maneuvers. The compromise is to retain a linear representation of the inertia and aerodynamic effects, but to put in an exact representation of the gravity forces. This allows the angles ф, 9, and ф to take on any values. As we shall see in the following example, the solution obtained is then limited by the airplane speed growing beyond the range of linear validity, that is, it is an aerodynamic nonlinearity that then con­trols the useful range of the solution. When the procedure that led to (4.9,18 and 19) is repeated without the small angle approximations we get the following for 90 = 0 (see Exercise 7.8):

The data for the B747 jet transport previously used was incorporated into the preced­ing equations. A step aileron input of —15° was applied at time zero, the other con­trols being kept fixed, and the solution was calculated using a fourth-order Runge – Kutta algorithm. The results are shown in Fig. 7.30. The main feature is the rapid acquisition of roll rate, shown in Fig. 130b, and its integration into a steadily grow­ing angle of bank (Fig. 7.30c) that reaches almost 90° in half a minute. Sideslip, yaw rate, and yaw angle all remain small throughout the time span shown. As the airplane rolls, with its lift remaining approximately equal to its weight, the vertical component of aerodynamic force rapidly diminishes, and a downward net force leads to negative 9 and an increase in speed. After 30 seconds, the speed has increased by about 10% of m0, and the linear aerodynamics becomes increasingly inaccurate. The maxi­mum rotation rate is p = .05 rad/s, which corresponds to p = 0.01. This is small enough that the neglect of the nonlinear inertia terms in the equations of motion is justified.

(e) Attitude angles

Figure 7.30 Response of jet transport to aileron angle; Sa = —15°. (a) Velocity components, (b) Angular velocity components, (c) Attitude angles.

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