Yes, moonburners can be a source of pitch oscillations of significant magnitude that may lead to dynamic instability and substantial loss of altitude or structural failure.
As you point out, the offset core in the motor means that the center of mass (CoM) does not lie on the rocket’s axis of symmetry. It lies somewhere to the side. This causes two issues:
- Because the thrust is directed along the rocket’s symmetry axis while the CoM is off-axis, a pitching moment will be generated, as you point out. In the absence of roll, this will induce a steady pitch rate in one direction. Now, if you superimpose a bit of roll rate to this scenario, you end up with a “coning” motion. Drag will increase because the rocket is now flying at an angle of attack, and the rocket will likely reach an altitude lower than predicted (assuming those effects are not accounted for in your aerodynamic simulations).
- The second, and more problematic consequence, is that the offset CoM means that the principal axes of inertia are not aligned with the geometric axes of the rocket (the axes that visually correspond to roll, pitch, and yaw). The principal axes of inertia are found by diagonalizing (taking an eigendecomposition) of the inertia tensor — the principal axes are the eigenvectors. The inertia tensor describes how mass is distributed about the three axes, as well as cross terms (“products of inertia") that account for the off-axis imbalances. An “ideal” scenario is one in which the inertia tensor is diagonal when viewed in the coordinate system aligned with the geometric axes — a roll about the geometric axes also corresponds to a roll about the principal axes. People go to great lengths to ensure that this is the case, especially for high performance rotating machinery such as pumps and turbines. More on this in a bit.
When the principal axes of inertia are not aligned with the geometric axes, trouble ensues. The angular momentum equation:
H=
Iw (where
w is omega, the angular velocity vector, and
H is the angular momentum vector), in three dimensions, is expressed as a matrix equation transforming the vector
w (in 3 space) to the angular momentum vector
H (also 3D). Consider for a moment the “ideal case”: the geometric axes are aligned with the principal axes. If a roll rate is induced about the symmetry axis of the rocket, the body will acquire an angular momentum vector omega that is parallel to this axis, which is also a principal axis. In this case, the action of the matrix
I in the equation
H=
Iw is only to stretch the vector
w, not to change its direction (in mathematical terms, you can say that
w is then an eigenvector of
I, and the corresponding diagonal term in the inertia tensor is the eigenvalue). Thus,
w is aligned with
H, and the expected motion, a simple roll, results.
When products of inertia exist (cross terms in the inertia tensor, from imbalances in the mass distribution), the inertia tensor will no longer be diagonal, and the action of the matrix
I on the angular velocity vector
w (aligned with the body axes) will be to transform the magnitude and direction of the resultant angular momentum vector. Thus, the angular momentum vector and angular velocity vector will be pointing in different directions.
This poses serious problems for a rocket’s flight dynamics (or that of any aerospace vehicle, for that matter). Assume for a minute that the body is free of external moments (other than the one that created the initial angular velocity). The conservation of angular momentum means that the angular momentum vector is fixed in space, while the angular velocity vector is offset. See the image below, which illustrates a case of an aircraft attempting a roll about the geometric (body) axis which is not a principal axis of inertia:
This is a special case of the Euler equations for rigid body dynamics in the case of absence of external moments, for which an analytical solution exists. Due to conservation of angular momentum, the vector
H remains fixed in space while the angular velocity vector
w “orbits” or precesses
H. This orbiting causes the vehicle to roll as well as pitch at an angle theta as it precesses. In the example above, the pilot may have thought that they were executing a simple roll of the aircraft, but they got a pitch as well because of the offset principal axes. The same is true for rockets.
In this way, a connection between the roll rate and precession rate is established. Now, what might cause a rocket to roll? Most amateur rockets are passively stabilized and unguided, so they have no way to counteract unwanted body rates as they may arise in flight. These body rates may arise from manufacturing imperfections, that while small, may impart significant torques on the rocket. Among many examples, imperfections in the alignment of the fins can lead to small (usually visually unnoticeable) deviations of the position and cant angle of the fin on the airframe. Taken together, the misalignment across all the fins can induce a net rolling moment about the body axes of the rocket. Of course, now that an angular velocity has been introduced, the analysis above now takes effect, and you get a coning motion if the body axes are not aligned with the principal axes of inertia.
This effect matters more for higher performing rockets (for which moonburners are commonly used). The roll moment is a function of both the magnitude of fin misalignment and the dynamic pressure. Higher performing rockets can hit very high dynamic pressures (hundreds of kPa) and are thus more susceptible to higher induced roll rates. They also have relatively high propellant mass fractions, so the CoM of the motor contributes significantly to the CoM of the total vehicle. You can imagine how taken together, a high roll rate combined with the mass imbalance issue can lead to substantial precession rates and pitch angles.
It is quite possible that if the rocket does survive the boost, the loss of altitude due to incident drag from flying at an angle of attack more than detracts from the theoretical performance gain of using the moonburner.
It is important to note that this “coning” motion is a completely separate phenomenon from the roll-pitch coupling scenario (also sometimes referred to as “coning”). The scenario discussed above is a consequence of the “free body” (inhomogeneous) solution to the Euler equations. Roll-pitch coupling can also be derived from the Euler equations, but is an inhomogeneous solution under the condition at which the roll rate (frequency) is equal to the natural pitch frequency. Of course, the roll rate is induced in the same manner as above, forced from the misalignment of the fins. The result is a pitch angle that manifests itself in a similar way as the precession above (i.e. looks the same from the ground).
Still other sources of the “coning” motion exist, including the decay of rotational kinetic energy due to dissipative loss mechanisms (departure from the rigid body assumption). Slop in joints, slosh in tanks, or the motion of improperly constrained payload or recovery components are examples of these loss mechanisms. A body that is initially spinning about a minimum axis of inertia, in the presence of these loss mechanisms, will always decay to a spin about a major axis of inertia. This can be seen by examining both the equations for rotational energy and angular momentum:
Energy: E = Iw^2 (scalar equation)
Momentum:
H =
Iw (vector equation)
Imagine a long slender satellite in space (or an amateur rocket, as many of us like to imagine), spinning about its roll axis and free from external torques. If internal energy loss mechanisms are present, they will reduce the rotational kinetic energy of the system. This forces the angular velocity,
w, to decrease. However, since the system is free from external torques, it must obey the conservation of angular momentum, thus
H must remain constant. The only way for
H to remain constant (in both magnitude and direction, remember that
H=
Iw is a vector equation in 3D), is for the moment of inertia to increase (more precisely, the components of
w in the direction of the larger diagonal terms in
I must increase). This can happen if the angular velocity vector becomes aligned with a principal axis of inertia that is greater than that about which the rotation was initialized. By the end of this process, the body will always end up rotating about the largest (major) axis of inertia in the absence of controlling/restorative torques, in the presence of these loss mechanisms.
This simple thought experiment has enormous practical consequences for spinning objects, especially long slender ones like rockets and spacecraft. Because the roll axis is that of least inertia, this energy dissipation will result in the object going into a flat spin. In fact, the first US satellite, Explorer 1, encountered this phenomenon: it was spin stabilized during launch, but flexible antennas deployed after launch led to energy dissipation that sent it into an eventual flat spin:
https://en.wikipedia.org/wiki/Explorer_1.
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