Static and Dynamic Stability
It is pertinent here to briefly review the terms static and dynamic stability. Stability analyses examine what happens to an aircraft when it is subjected to forces and moments applied by a pilot and/or induced by external atmospheric disturbances. There are two types of stability, as follows:
1. Static Stability. This is concerned with the instantaneous tendency of an aircraft when disturbed during equilibrium flight. The aircraft is statically stable if it has restoring moments when disturbed; that is, it shows a tendency to return to the original equilibrium state. However, this does not cover what happens in the due course of time. The recovery motion can overshoot into oscillation, which may not return to the original equilibrium flight.
2. Dynamic Stability. This is the time history of an aircraft response after it has been disturbed, which is a more complete picture of aircraft behavior. A statically stable aircraft may not be dynamically stable, as explained in subsequent discussions. However, it is clear that a statically unstable aircraft also is dynamically unstable. Establishing static stability before dynamic stability is for procedural convenience.
The aircraft motion in 3D space is represented in the three planes of the Cartesian coordinate system (see Section 3.4). Aircraft have six degrees of freedom of motion 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 в. 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 12.1. Aircraft stability compared to a spring-mass system equilibrium. It shows that aircraft motion is in an equilibrium level flight – here, motion is invariant with time. Readers may examine what occurs when forces and moments are applied.
A statically and dynamically stable aircraft tends to return to its original state even when it oscillates about the original state. An aircraft becomes statically and dynamically unstable if the pitching motion diverges outward – it neither oscillates nor returns to the original state. The third diagram of Figure 12.2 provides an example of neutral static stability – in this case, the aircraft does not have a restoring
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.