Short-Period Oscillation and Phugoid Motion
The diagrams in Figure 12.12 show an exaggerated aircraft flight path (i. e., altitude changes in the pitch plane). In the pitch plane, there are two different types of aircraft dynamics that result from the damping experienced when an aircraft has a small perturbation. The two longitudinal modes of motion are as follows:
1. Short-period oscillation (SPO) is associated with pitch change (a change) in which the H-tail plane acts as a powerful damper (see Figure 12.1). If a disturbance (e. g., a sharp flick of the elevator and return) causes the aircraft to enter this mode, then recovery is also quick for a stable aircraft. The H-tail acts
like an aerodynamic spring that naturally returns to equilibrium. The restoring moment comes from the force imbalance generated by the angle of attack, a, created by the disturbance. Damping (i. e., resistance to change) comes as a force generated by the tail plane, and the stiffness (i. e., force required) comes from the stability margin. The heavy damping of the H-tail resists changes to make a quick recovery.
The bottom diagram of a short period in Figure 12.12 plots the variation of the angle of attack, a, with time. All aircraft have a short-period mode and it is not problematic for pilots. A well-designed aircraft oscillatory motion is almost unnoticeable because it damps out in about one cycle. Although aircraft velocity is only slightly affected, the angle of attack, a, and the vertical height are related. Minimum a occurs at maximum vertical displacement and maximum a occurs at about the original equilibrium height. The damping action offered by the H-tail quickly smooths out the oscillation; that is, one oscillation takes a few seconds (typically, 1 to 5 s). The exact magnitude of the period depends on the size of the aircraft and its static margin. If the H-tail plane area is small, then damping is minimal and the aircraft requires more oscillations to recover.
2. Phugoid motion is the slow oscillatory aircraft motion in the pitch plane, as shown in the bottom diagram in Figure 12.12. It is known as the long-period oscillation (LPO) – the period can last from 30 s to more than 1 min. Typically, a pilot causes the LPO by a slow up and down movement of the elevator. In this case, the angle of attack, a, remains almost unchanged while in the oscillatory motion. The aircraft exchanges altitude gain (i. e., increases in potential energy [PE]) for decreases in velocity (i. e., decreases in kinetic energy [KE]). The phugoid motion has a long period, during which time the KE and PE exchange. Because there is practically no change in the angle of attack, a, the H-tail is insignificant in the spring-mass system. Here, another set of spring – mass is activated but is not shown in schematic form (it results from the aircraft configuration and inertia distribution – typically, it has low damping characteristics). These oscillations can continue for a considerable time and fade out comparatively slowly.
The frequency of a phugoid oscillation is inversely proportional to an aircraft’s speed. Its damping also is inversely proportional to the aircraft L/D ratio. A high L/D ratio is a measure of aircraft performance efficiency. Reducing the L/D ratio to increase damping is not preferred; modern designs with a high L/D ratio incorporate automatic active control (e. g., FBW) dampers to minimize a pilot’s workload. Conventional designs may have a dedicated automatic damper at a low cost. Automatic active control dampers are essential if the phugoid motion has undamped characteristics.
All aircraft have an inherent phugoid motion. In general, the slow motion does not bother a pilot – it is easily controlled by attending to it early. The initial onset, because it is in slow motion, sometimes can escape a pilot’s attention (particularly when instrument-flying), which requires corrective action and contributes to pilot fatigue.
directional stability = Cnp lateral stability =
12.6.2 Directional and Lateral Modes of Motion
Aircraft motion in the directional (i. e., yaw) and the lateral (i. e., roll) planes is coupled with sideslip and roll; therefore, it is convenient to address the lateral and directional stability together. These modes of motion are relatively complex in nature. FAR 23, Sections 23-143 to 23.181, address airworthiness aspects of these modes of motion. Spinning is perceived as a post-stall phenomenon and is discussed separately in Section 12.7.
The four typical modes of motion are (1) directional divergence, (2) spiral, (3) Dutch roll, and (4) roll subsidence. The limiting situation of directional and lateral stability produces two types of motion. When yaw stability is less than roll stability, the aircraft can enter directional divergence. When roll stability is less than yaw stability, the aircraft can enter spiral divergence. Figure 12.13 shows the two extremes of directional and spiral divergence. The Dutch roll occurs along the straight initial path, as shown in Figure 12.14.
The wing acts as a strong damper to the roll motion; its extent depends on the wing aspect ratio. A large V-tail is a strong damper to the yaw motion. It is important to understand the role of damping in stability. When configuring an aircraft, designers need to optimize the relationship between the wing and V-tail geometries. The four modes of motion are as follows:
1. Directional Divergence. This results from directional (i. e, yaw) instability. The fuselage is a destabilizing body, and if an aircraft does not have a sufficiently large V-tail to provide stability, then sideslip increases accompanied by some roll, with the extent depending on the roll stability. The condition can continue until the aircraft is broadside to the relative wind, as shown in Figure 12.13.
2. Spiral. However, if the aircraft has a large V-tail with a high degree of directional (i. e., yaw) stability but is not very stable laterally (i. e., roll) (e. g., a low – wing aircraft with no dihedral or sweep), then the aircraft banks as a result of rolling while sideslipping.
Figure 12.14. Dutch roll motion
This is a nonoscillatory motion with characteristics that are determined by the balance of directional and lateral stability. In this case, when an aircraft is in a bank and sideslipping, the side force tends to turn the plane into the relative wind. However, the outer wing is traveling faster, generating more lift, and the aircraft rolls to a still higher bank angle. If poor lateral stability is available to negate the roll, the bank angle increases and the aircraft continues to turn into the sideslip in an ever-increasing (i. e., tighter) steeper spiral, which is spiral divergence (see Figure 12.13). In other words, spiral divergence is strongly affected by Clr.
The initiation of a spiral is typically very slow and is known as a slow spiral. The time taken to double the amplitude from the initial state is long – 20 s or more. The slow buildup of a spiral-mode motion can cause high bank angles before a pilot notices an increase in the g-force. If a pilot does not notice the change in horizon, this motion may become dangerous. Night-flying without proper experience in instrument-flying has cost many lives due to spiral divergence. Trained pilots should not experience the spiral mode as dangerous – they would have adequate time to initiate recovery actions. A 747 has a nonoscillatory spiral mode that damps to half amplitude in 95 s under typical conditions; many other aircraft have unstable spiral modes that require occassional pilot input to maintain a proper heading.
3. Dutch Roll. A dutch roll is a combination of yawing and rolling motions, as shown in Figure 12.14. It can happen at any speed, developing from the use of the stick (i. e., aileron) and rudder, which generate a rolling action when in yaw. If a sideslip disturbance occurs, the aircraft yaws in one direction and, with strong roll stability, then rolls away in a countermotion. The aircraft “wags its tail” from side to side, so to speak. The term Dutch roll derives from the rhythmic motion of Dutch iceskaters swinging their arms and bodies from side to side as they skate over wide frozen areas.
When an aircraft is disturbed in yaw, the V-tail performs a role analogous to the H-tail in SPO; that is, it generates both a restoring moment proportional to the yaw angle and a resisting, damping moment proportional to the rate of yaw. Thus, one component of the Dutch roll is a damped oscillation in yaw. However, lateral stability responds to the yaw angle and the yaw rate by rolling the wings of the aircraft. Hence, the second component of a Dutch roll is an
oscillation in a roll. The Dutch-roll period is short – on the order of a few seconds.
In other words, the main contributors to the Dutch roll are two forms of static stability: the directional stability provided by the V-tail and the lateral stability provided by the effective dihedral and sweep of the wings – both forms offer damping. In response to an initial disturbance in a roll or yaw, the motion consists of a combined lateral-directional oscillation. The rolling and yawing frequencies are equal but slightly out of phase, with the roll motion leading the yawing motion.
Snaking is a pilot term for a Dutch roll, used particularly at approach and landing when a pilot has difficulty aligning with the runway using the rudder and ailerons. Automatic control using yaw dampers is useful in avoiding the snaking/Dutch roll. Today, all modern transport aircraft have some form of yaw damper. The FBW control architecture serves the purpose well.
All aircraft experience the Dutch-roll mode when the ratio of static directional stability and dihedral effect (i. e., roll stability) lies between the limiting conditions for spiral and directional divergences. A Dutch roll is acceptable as long as the damping is high; otherwise, it becomes undesirable. The characteristics of a Dutch roll and the slow spiral are both determined by the effects of directional and lateral stability; a compromise is usually required. Because the slow-spiral mode can be controlled relatively easily, slow-spiral stability is typically sacrificed to obtain satisfactory Dutch-roll characteristics.
High directional stability (C„p) tends to stabilize the Dutch-roll mode but reduces the stability of the slow-spiral mode. Conversely, a large, effective dihedral (rolling moment due to sideslip, Cy) stabilizes the spiral mode but destabilizes the Dutch-roll motion. Because sweep produces an effective dihedral and because low-wing aircraft often have excessive dihedral to improve ground clearance, Dutch-roll motions often are poorly damped on swept-wing aircraft.
4. Roll Subsidence. The fourth lateral mode is also nonoscillatory. A pilot commands the roll rate by application of the aileron. Deflection of the ailerons generates a rolling moment, but the aircraft has a roll inertia and the roll rate builds up. Very quickly, a steady roll rate is achieved when the rolling moment generated by the ailerons is balanced by an equal and opposite moment proportional to the roll rate. When a pilot has achieved the desired bank angle, the ailerons are neutralized and the resisting rolling moment very rapidly damps out the roll rate. The damping effect of the wings is called roll subsidence.