Coupling in Aircraft — A Complex Subject Made Simple
What is a simple explanation for inertia coupling?
Airplanes move above the Earth and have on occasion displayed large complex motions in three-dimensional atmospheric space. To conduct analyses, to bring understanding, and order out of chaos we engineers assign what might seem to be an arbitrary airplane reference coordinate axis system. This axis system has its origin fixed at the airplane’s center of gravity (cg). We then assign a clever sign convention with directions that are: forward (+X), right
(+Y), and down (+Z) or backward (-X), left (-Y), up (-Z). We call this an airplane body axis and we align our pilot’s seat and sensors such as gyros, and accelerometers with respect to this axis for his comfort and data acquisition purposes. In addition, we call the X-axis the roll-axis, the Y-axis the pitch-axis, and the Z-axis the yaw-axis. The center of gravity (cg) of the airplane is located at about at the 25-percenct of the wing’s mean aerodynamic chord1
The modern hypothetical airplane configuration I will consider as an example that represents one designed for high speed flight. Its physical characteristics of size, mass, mass distribution and aerodynamics will meet this demand.
The fuselage is long and slender with its high density mass distributed along the X-axis. The wings are short and thin and may be swept. Typical vertical tail and horizontal tailplane complete our mental image. This results in an aircraft having relatively large moments of inertia2 in pitch and yaw and because most of the mass is close to the X or roll axis, a small roll inertia by comparison. The example airplane’s mass-distribution is not uniform about the body axes; is not symmetric in the XZ plane (see the footnote) due to the vertical tail surface(s). Maneuvering about one axis, roll for example may induce motion about the yaw and pitch axis; this is what we would call coupling.
What is coupling?
In general ‘Coupling’ results when some disturbance about one airplane axis causes a disturbance about another axis.
To keep the discussion relatively simple ‘some disturbance’ is replaced by pilot command and we will consider roll coupling to illustrate the problem.
What is roll coupling?
Roll ‘Coupling’ results when a roll angular acceleration is induced by a pilot lateral stick command to achieve some result like bank angle to turn or rapid tactical maneuvering about the roll (X) airplane axis that in turn causes an angular disturbance about the pitch (Y) and yaw (Z) axis.
Roll coupling includes pitching and yawing motions, the longitudinal and directional stability are also important in determining the overall characteristics of the coupled motion and the end result can be mild or violent.
Without going into the complex formulation of the airplane’s equations of motion that include the mass properties, coupling terms, aerodynamic characteristics and esoteric concepts such as inertia and wind axes; we can be bold enough to say that inertial coupling is the tendency for inertial forces in flight maneuvers to overcome stabilizing aerodynamic forces. Rapid roll may result in a violent divergence or oscillations in pitch about the principal inertial axis or complete loss of control; this tendency is usually increasingly marked with increasing altitude owing to possible divergence of this axis from the relative wind.
The actual in-flight motion of the airplane will be the result of the complex combination of the aerodynamic and inertia coupling. Indeed most airplanes exhibit some form of aerodynamic and inertial coupling. Roll coupling causes little problem when aerodynamic restoring moments can cope with the inertia couple.
Modern digital flight control system designs should be capable of precluding any and all adverse inertial and aerodynamic coupling. What then could possibly go wrong . . .go wrong . . . go wrong . . . go wrong, etc.?
Robert W. Kempel, Senior Flight Research Engineer, (Ret.)
13 December 2013
Footnotes:
1 In practice this is usually approximately close to the wing’s mean chord.
2 The moment of inertia of a body with respect to any axis is the sum of the products obtained by multiplying each elementary mass by the square of its distance [I (mass-length2)] from the axis. One might say this is the measure of the mass-distribution of a body about a particular
axis. Roll, pitch, and yaw moments of inertia are Ixx, Iyy, Izz respectively. Metric units are kg-m2 and English units slug-ft 2. Due to the vertical tail