Aircraft dynamic stability focuses on the time history of aircraft motion after the aircraft is disturbed from an equilibrium or trim condition. This motion may be first order (exponential response) or second order (oscillatory response), and will have either positive dynamic stability (aircraft returns to the trim condition as time goes to infinity), neutral dynamic stability (aircraft neither returns to trim nor diverges further from the disturbed condition), or dynamic instability (aircraft diverges from the trim condition and the disturbed condition as time goes to infinity). The study of dynamic stability is important to understanding aircraft handling qualities and the design features that make an airplane fly well or not as well while performing specific mission tasks. The differential equations that define the aircraft equations of motion (EOM) form the starting point for the study of dynamic stability.

7.1 Mass Spring Damper System and Classical Solutions of Ordinary Differential Equations

The mass spring damper system illustrated in Fig. 7.1 provides a starting point for analysis of system dynamics and aircraft dynamic stability. This is an excellent model to begin the understanding of dynamic response.

Figure 7.1: Mass spring damper system.

We will first develop an expression for the sum of forces in the vertical direction. Notice that x( t) is defined as positive for an upward displacement and that the zero position is chosen as the point where the system is initially at rest or at equilibrium. We know that

(7.1)

There are two forces acting on the mass, the damping force, and the...