The evaluation of static stability provides some measure of the airplane dynamics, but only a rather crude one. Of greater relevance, especially for lateral motion, is the dynamic response of the aircraft. As seen below, it is possible for an airplane to be statically stable, yet dynamically unstable, resulting in unacceptable characteristics.
Just what constitutes acceptable characteristics is often not obvious, and several attempts have been made to quantify pilot opinion on acceptable handling qualtities. Subjective flying qualities evaluations such as Cooper-Harper ratings are used to distinguish between "good-flying" and difficult-to-fly aircraft. New aircraft designs can be simulated to determine whether they are acceptable. Such real-time, pilot-in-the-loop simulations are expensive and require a great deal of information about the aircraft. Earlier in the design process, flying qualities estimate may be made on the basis of various dynamic characteristics. One can correlate pilot ratings to the frequencies and damping ratios of certain types of motion as in done in the U.S. Military Specifications governing airplane flying qualities. The figure below shows how the short-frequency longitudinal motion of an airplane and the load factor per radian of angle of attack are used to establish a flying qualities estimate. In Mil Spec 8785C, level 1 handling is considered "clearly adequate" while level 3 suggests that the airplane can be safely controlled, but that the pilot workload is excessive or the mission effectiveness is inadequate.
Rather than solve the relevant equations of motion, we describe here some of the simplified results obtained when this is done using linearized equations of motion.
When the motions are small and the aerodynamics can be assumed linear, many useful, simple results can be derived from the 6 degree-of-freedom equations of motion. The first simplification is the decoupling between symmetric, longitudinal motion, and lateral motion. (This requires that the airplane be left/right symmetric, a situation that is often very closely achieved.) Other decoupling is also observed, with 5 decoupled modes required to describe the general motion. The stability of each of these modes is often used to describe the airplane dynamic stability.
Modes are often described by their characteristic frequency and damping ratio. If the motion is of the form: x = A e (n + i w) t, then the period, T, is given by: T = 2p / w, while the time to double or halve the amplitude of a disturbance is: tdouble or thalf = 0.693 / |n|. Other parameters that are often used to describe these modes are the undamped circular frequency: wn = (w2 + n2)1/2 and the damping ratio, z = -n / wn.
When the aircraft is not perturbed about the roll or yaw axis, only the longitudinal modes are required to describe the motion. These modes usually are divided into two distinct types of motion.
The first, short period, motion involves rapid changes to the angle of attack and pitch attitude at roughly constant airspeed. This mode is usually highly damped; its frequency and damping are very important in the assessment of aircraft handling. For a 747, the frequency of the short-period mode is about 7 seconds, while the time to halve the amplitude of a disturbance is only 1.86 seconds. The short period frequency is strongly related to the airplane's static margin, in the simple case of straight line motion, the frequency is proportional to the square root of Cma / CL.
The long-perioid of phugoid mode involves a trade between kinetic and potential energy. In this mode, the aircraft, at nearly constant angle of attack, climbs and slows, then dives, losing altitude while picking up speed. The motion is usually of such a long period (about 93 seconds for a 747) that it need not be highly damped for piloted aircraft. This mode was studied (and named) by Lanchester in 1908. He showed that if one assumed constant angle of attack and thrust=drag, the period of the phugoid could be written as: T = p V2 U/g = 0.138 U. That is, the period is independent of the airplane characteristics and altitude, and depends only on the trimmed airspeed. With similarly strong assumtions, it can be shown that the damping varies as z = 1 / (V2 L/D).
Three dynamic modes describe the lateral motion of aircraft. These include the relatively uninteresting roll subsidence mode, the Dutch-roll mode, and the spiral mode.
The roll mode consists of almost pure rolling motion and is generally a non-oscillatory motion showing how rolling motion is damped.
Of somewhat greater interest is the spiral mode. Like the phugoid motion, the spiral mode is usually very slow and often not of critical importance for piloted aircraft. A 747 has a nonoscillatory spiral mode that damps to half amplitude in 95 seconds under typical conditions, while many airplanes have unstable spiral modes that require pilot input from time to time to maintain heading.
The Dutch-roll mode is a coupled roll and yaw motion that is often not sufficiently damped for good handling. Transport aircraft often require active yaw dampers to suppress this motion.
High directional stability (Cnb) tends to stabilize the Dutch-roll mode while reducung the stability of the spiral mode. Conversely large effective dihedral (rolling moment due to sideslip, Clb) stabilizes the spiral mode while destabilizing the Dutch-roll motion. Because sweep produces effective dihedral and because low wing airplanes often have excessive dihedral to improve ground clearance, Dutch-roll motions are often poorly damped on swept-wing aircraft.