Eric,

An excellent explanation . . . by Richard Nakka.

**https://www.nakka-rocketry.net/ep_lr7.html**
QUOTE :

**Inertial Roll coupling**
Also known as

*Pitch-Roll coupling* or

* Yaw-Roll coupling*,

*Inertial Roll coupling* is a "resonant divergence in pitch (or yaw) when roll rate equals the lower of the pitch or yaw natural frequencies", according to Reference 1. In other words, this term describes a phenomenon whereby dynamic instability of a rocket (or other flight vehicle) develops under certain flight conditions with potentially catastrophic consequences, if that vehicle has the mass and geometric configuration that makes it susceptible.

The concept of

*roll *and

*pitch *is shown in Figure 6. If a rocket rolls, it will rotate about its own

*principal axis*, the line of least resistance, rather than the flight path (geometric axis), as illustrated in Figure 7. The position of the principal axis is determined by the particular placement of

* items of mass *that make up the rocket. If the angular difference between the principal axis and the geometric axis is sufficiently large, and if the rocket rolls sufficiently quickly, the destabilizing moment from the inertial forces will overcome the stabilizing aerodynamic moment provided by the fins. The centrifugal force due to the roll will cause the nose and tail to try to swing out

*perpendicular *to the rotation axis.The rocket will become directionally unstable, with the pitch angle continually diverging, developing a

*wobble *or

*coning *motion, to the point where the vehicle's structural limit is exceeded, leading to break-up. In order for the rocket's principal axis to be different than the geometric axis (dynamically unbalanced), the distribution of the various components that make up the mass of the rocket would have be uneven, with respect to the centreline of the rocket (geometric axis). Components of the rocket such as the fuselage, fins, nosecone, and motor are generally symmetrical about the rocket's centreline axis, and would not contribute to dynamic imbalance. However, certain items of mass, typically payload items, may have a centre of gravity (CG) that is not in line with the centreline of the rocket. It is these items that offset the principal axis and lead to dynamic imbalance and the potential for inertial roll coupling.

What makes a particular flight vehicle

**susceptible **to inertial roll coupling? The most obvious condition is the presence of roll. For a rocket, roll may be produced by fin tabs, asymmetrically airfoiled fins, or even misaligned fins. Another condition, as mentioned, is the configuration of the vehicle that leads to dynamic imbalance, such as offset items of mass that result in non-coincident principal and geometric axes. A second condition is the existence of a large difference between the

*roll moment of inertia* and

* pitch moment of inertia* for the vehicle. This is typically the case for

*rockets *, which have all the mass contained within the fuselage (low roll inertia) and have long fuselages with heavy motors, payload, etc. (high pitch inertia). From reference [1], this susceptibility can be expressed in terms of a

*coupling inertia ratio* given by

**(Ix-Iy)/Iz** for any flight vehicle, where

**Ix** is the roll moment of inertia about the geometric axis,

** Iy** is the pitch moment of inertia about the geometric axis, and

**Iz **is the yaw moment of inertia about the geometric axis. Coupling tendencies increase as this ratio approaches a value of -1. For most rockets, the pitch and yaw moment of inertia are equal, owing to symmetry, and the roll moment of inertia is usually written as

** IR**. The coupling ratio then reduces to

**IR/IL-1**, where

** IL= Iz = Iy** (longitudinal moment of inertia).

END QUOTE:

Dave F.