Inertia coupling tends to occur in aircraft with a
long, slender, high-density
fuselage. A simple, yet accurate mental model describing the aircraft's
mass distribution is a
rhombus of
point masses: one large mass fore and aft, and a small one on each wing. The
inertia tensor that this distribution generates has a large
yaw component and small
pitch and roll components, with the pitch component slightly larger.
Euler's equations govern the rotation of an aircraft. When , the
angular rate of roll, is
controlled by the aircraft, then the other rotations must satisfy \begin{align} I_\text{y}\dot{\omega_\text{y}}&=(I_\text{p}-I_\text{r})\omega_\text{r}\omega_\text{p}+T_y \\ I_\text{p}\dot{\omega_\text{p}}&=-(I_\text{y}-I_\text{r})\omega_\text{r}\omega_\text{y}+T_p \end{align} where y, p, and r indicate yaw, pitch, and roll; is the
moment of inertia along an axis; the external torque from aerodynamic forces along an axis; and
dots indicate
time derivatives. When aerodynamic forces are absent, this 2variable
system is the equation of a
simple harmonic oscillator with frequency : a rolling
Space Shuttle will naturally undergo small oscillations in pitch and yaw. Conversely, when the craft does not roll at all (), the only terms on the right-hand side are the aerodynamic torques, which are (
at small angles) proportional to the craft's angular orientation to the
freestream air. That is: there are natural constants such that an unrolling aircraft experiences \begin{align} I_\text{y}\dot{\omega_\text{y}}&=T_y=-k_\text{y}I_\text{y}\theta_\text{y} \\ I_\text{p}\dot{\omega_\text{p}}&=T_p=-k_\text{p}I_\text{p}\theta_\text{p} \end{align} In the full case of a rolling aircraft, the connection between orientation and angular velocity is not entirely straightforward, because the aircraft is a
rotating reference frame. The roll inherently exchanges yaw for pitch and vice-versa: \begin{align} \dot{\theta_\text{y}}&=\omega_\text{y}+\omega_\text{r}\theta_\text{p} \\ \dot{\theta_\text{p}}&=\omega_\text{p}-\omega_\text{r}\theta_\text{y} \end{align} Assuming nonzero roll,
time can always be rescaled so that . The full equations of the body are then of two
damped, coupled
harmonic oscillators: \begin{align} 0&=\ddot{\theta_\text{y}}-(1+J_\text{y})\dot{\theta_\text{p}}+(k_\text{y}-J_\text{y})\theta_\text{y} \\ 0&=\ddot{\theta_\text{p}}+(1-J_\text{p})\dot{\theta_\text{y}}+(k_\text{p}-J_\text{p})\theta_\text{p} \end{align} where \begin{align} J_\text{y}&=\frac{I_\text{p}-I_\text{r}}{I_\text{y}} \\ J_\text{p}&=\frac{I_\text{y}-I_\text{r}}{I_\text{p}} \end{align} But if in either axis, then the damping is eliminated and the system is
unstable. In dimensional terms (that is, unscaled time), instability requires . Since is small, J_\text{y}J_\text{p}\approx1 In particular, one is at least 1. In thick air, are too large to matter. But in thin air and supersonic speeds, they decrease, and may become comparable to during a rapid roll. Techniques to prevent inertial roll coupling include increased directional stability () and reduced roll rate (). Alternatively, the unstable aircraft dynamics may be
mitigated: the unstable
modes require time to grow, and a sufficiently short-duration roll at limited angle of attack may allow recovery to a controlled state post-roll. ==Early history==