Transient Response to Aileron and Rudder
We have seen that useful lateral steady states are produced only by certain definite combinations of the control deflections. It is evident then that our interest in the response to a single lateral control should be focused primarily on the initial behavior. The equations of motion provide some insight on this question directly. Following a step input of one of the two controls the state variables at t = 0+ are all still zero, and from (4.9,19) we can deduce that their initial rates of change are related to the control angles by
v = ^sA
р = ад + ад (7.11,1)
r = NSa8a + M SA
The initial sideslip rate v is thus seen to be governed solely by the rudder and, since °}JSr > 0, is seen to be positive (slip to the right) when Sr is positive (left rudder). Of somewhat more interest is the rotation generated. The initial angular acceleration is the vector
d) = ip + kr (7.11,2)
The direction of this vector is the initial axis of rotation, and this is of interest. It lies in the xz plane, the plane of symmetry of the airplane, as illustrated in Fig. 7.28a. The angle £ it makes with the x axis is, of course,
£ = tan-1 — (7.11,3)
Let us consider the case of “pure” controls, that is, those with no aerodynamic crosscoupling, so that LSp = NSa = 0. The ailerons then produce pure rolling moment and
Figure 7.28 Initial response to lateral control, (a) General. (b) Example jet transport.
the rudder produces pure yawing moment. In that case we get for 8n — 0 the angle gR for response to rudder from
and similarly for response to aileron:
tan i, = IJI. (7.11,5)
The angles are seen to depend very much on the product of inertia Lx. When it is zero, the result is as intuitively expected, the rotation that develops is about either the x axis (aileron deflected) or the z axis (rudder deflected). For a vehicle such as the jet transport of previous examples, with IXp = 0.4IZp, the values of Ix, lz, /., given by (4.5,11) yield the results shown in Fig. 7.29. The relations are also shown to scale in Fig. 7.28b for є = 20° (high angle of attack). It can be seen that there is a tendency for the vehicle to rotate about the principal x axis, rather than about the axis of the aerodynamic moment. This is simply because IJ1Z is appreciably less than unity. Now the jet transport of our example is by no means “slender,” in that it is of large span and has wing-mounted engines. For an SST or a slender missile, the trend shown is much accentuated, until in the limit as aspect ratio —* 0, both tan and tan tend to tan e, and the vehicle rotates initially about the xp axis no matter what control is used!