Figure 6.5 Time Histories of Systems with Positive, Neutral, and Negative Dynamic Stability

A system which has positive static stability but no damping at all continues to oscillate without ever decreasing the magnitude or amplitude of the oscillation. It is said to have neutral dynamic stability because over time the system does not get any closer to or farther from equilibrium. The time history of a system with positive static stability but neutral dynamic stability is shown on the left-hand graph of Figure 6.5 (b). On the right side of Figure 6.5 (b) is a time history of a system with neutral static and dynamic stability. When displaced from its intial condition, it is still in equilibrium, like the ball on the flat surface, so it has no tendency to return to the zero-displacement condition.

Figure 6.6 illustrates the longitudinal trim problem for a conventional tail-aft airplane. The aircraft’s center of gravity is marked by the circle with alternating black and white quarters. The lift forces of the wing and horizontal tail are shown acting at their respective aerodynamic centers. The moment about the wing’s aerodynamic center due to the shape of its airfoil is also shown. The upper-case symbols L, L_t, and M_ac are used as in Chapter 4 for wing lift, tail lift, and wing moment respectively to indicate that they are forces and moments produced by three-dimensional surfaces, not airfoils. The horizontal tail is assumed to have a symmetrical airfoil, so that the moment about its aerodynamic center is zero. For consistency with the way two-dimensional airfoil data is presented, the locations of the wing’s aerodynamic center, x_ac, and the whole aircraft’s center of gravity, x_cg, are measured relative to the leading edge of the wing root. The distance of the aerodynamic center of the horizontal tail from the aircraft’s center of gravity is given the symbol l_t.

The moments in (6.1) must sum to zero if the aircraft is trimmed. For steady flight, the forces also sum to zero. Summing in the vertical direction:

Together, (6.1) and (6.2) provide a system of two equations with two unknowns (since the weight is usually known and the moment about the aerodynamic center does not change with lift) which can be solved simultaneously to yield the lift required from each surface for equilibrium. In practice, the elevator attached to the horizontal tail is deflected to provide the necessary lift from the tail so that the sum of the moments is zero when the aircraft is at the angle of attack required to make the sum of the forces zero. Note that for aircraft configurations such as the one shown in Figure 6.6, which have the horizontal tail behind the main wing, trim in level flight normally is achieved for positive values of L_t, so that the horizontal tail contributes to the total lift of the aircraft. Note also that (6.1) and (6.2) are applicable only to the aircraft configuration for which they were derived. Similar relations may be derived for flying wing aircraft, airplanes with canards, etc.

If the aircraft’s geometry and flight conditions are known, then the lift coefficient required from the wing and pitch control surfaces may be determined using L = C_L q S when (6.1) and (6.2) are solved for L and L_t. If any of the required C_L values are greater than C_lmax for their respective surfaces, then the aircraft does not have sufficient control authority to trim in that maneuver for those conditions. To remedy this situation, the aircraft designer must either increase the size of the deficient control surface or add high-lift devices to it to increase its C_lmax . Figure 6.7 shows a McDonnell-Douglas F-4E Phantom II multi-role jet fighter. Note that the stabilators on this aircraft have had leading-edge slots added to them to increase their C_lmax and hence their control authority.