12.3 Static and Dynamic Stability

in 3D space. They are decomposed into the three planes; each exhibits its own stability characteristics, as listed herein. The sign conventions associated with the pitch, yaw, and roll stabilities need to be learned (they follow the right-handed rule). The brief discussion of the topic herein is only for what is necessary in this chapter. The early stages of stability analyses are confined to small perturbations – that is, small changes in all flight parameters.

1.Longitudinal Stability in the Pitch Plane. The pitch plane is the XZ plane of aircraft symmetry. The linear velocities are u along the X-axis and w along the Z-axis. Angular velocity about the Y-axis is q, known as pitching (+ve nose up). Pilot-induced activation of the elevator changes the aircraft pitch. In the plane of symmetry, the aircraft motion is uncoupled; that is, motion is limited only to the pitch plane.

2.Directional Stability in the Yaw Plane. The yaw plane is the XY plane and is not in the aircraft plane of symmetry. Directional stability is also known as weathercock stability because of the parallel to a weathercock. The linear velocities are u along the X-axis and v along the Y-axis. Angular velocity about the Z-axis is r, known as yawing (+ve nose to the left). Yaw can be initiated by the rudder; however, pure yaw by the rudder alone is not possible because yaw is not in the plane of symmetry. Aircraft motion is coupled with motion in the other plane, the YZ plane.

3.Lateral Stability in the Roll Plane. The roll plane is the YZ plane and also is not in the aircraft plane of symmetry. The linear velocities are v along the Y-axis and w along the Z-axis. Angular velocity about the X-axis is p, known as rolling (+ve when right wing drops). Rolling can be initiated by the aileron but a pure roll by the aileron alone is not possible because roll is associated with yaw. To have a pure rolling motion in the plane, the pilot must activate both the yaw and roll controls.

It is convenient now to explain the static and dynamic stability in the pitch plane using diagrams. The pitching motion of an aircraft is in the plane of symmetry and is uncoupled; that is, motion is limited only to the pitch plane. The static and dynamic behavior in the other two planes has similar characteristics, but it is difficult to depict the coupled motion of yaw and roll. These are discussed separately in Sections 12.3.2 and 12.3.4.

Pitch-plane stability may be compared to a spring-mass system, as shown in Figure 12.1a. The oscillating characteristics are represented by the spring-mass system, with the resistance to the rate of oscillation as the damping force (i.e., proportional to pitch rate, q) and the spring compression proportional to pitch angle θ . Figure 12.1b shows the various possibilities of the vibration modes. Stiffness is represented by the stability margin, which is the distance between the CG and the neutral point (NP). The higher the force required for deforming, the more is the stiffness. Damping results from the rate of change and is a measure of resistance (i.e., how fast the oscillation fades out); the higher the H-tail area, the more is the damping effect. An aircraft only requires adequate stability; making it more stable than what is required poses other difficulties in the overall design.

Figure 12.2 depicts the stability characteristics of an aircraft in the pitch plane, which provide the time history of aircraft motion after it is disturbed from an initial

Figure 12.3. Longitudinal static stability

moment. It remains where it was after the disturbance and requires an applied pilot effort to force it to return to the original state. The tendency of an aircraft to return to the original state is a good indication of what could happen in time: Static stability makes it possible but does not guarantee that an aircraft will return to the equilibrium state.

As an example of dynamic stability, Figure 12.2 also shows the time history for when an aircraft returns to its original state after a few oscillations. The time taken to return to its original state is a measure of the aircraft’s damping characteristics – the higher the damping, the faster the oscillations fade out. A statically stable aircraft showing a tendency to return to its original state can be dynamically unstable if the oscillation amplitude continues to increase, as shown in the last diagram of Figure 12.2. When the oscillations remain invariant to time, the aircraft is statically stable but dynamically neutral – it requires an application of force to return to the original state.

12.3.1 Longitudinal Stability: Pitch Plane (Pitch Moment, M)

Figure 12.3 depicts the conditions for aircraft longitudinal static stability. In the pitch plane, by definition, the angle of attack, α, is positive when an aircraft nose is above the direction of free-stream velocity. A nose-up pitching moment is considered a positive. Static-stability criteria require that the pitching-moment curve exhibit a negative slope, so that an increase in the angle of attack, α, causes a restoring negative (i.e., nose-down) pitching moment. At equilibrium, the pitching moment is equal to zero (Cm = 0) when it is in trimmed condition (αtrim). The higher the static margin (see Figure 12.11), the greater is the slope of the curve (i.e., the greater is the restoring moment). Using the spring analogy, the stiffness is higher for the response.