Precession of an air rifle pellet in flight

[TCups]

Unlike firearm projectiles, air rifle pellets have evolved to a generally typical diabolo configuration. For reference, see: http://www.photosbykev.com/wordpress/2009/01/20/air-rifle-pellet-database/

The initial shape was described as like that of a badminton shuttlecock, and "skirt" at the bottom of the pellet acted as a stabilizing fin for projectiles of this type when fired from smooth, un-rifled barrels. Like firearms, modern air rifles have rifled barrels and use gyroscopic spin to stabilize the projectile in flight.

The skirt's second function, being lighter and thinner than the head of the pellet, is to balloon, like Marilyn's skirt over a subway vent, expanding to seal the back side of the pellet, forming to the grooves and lands of the rifled barrel, while the narrower head rides the surface of the rifling.

To a variable extent of depth and diameter, the underside of the skirt is hollow.

The pellet exits the rifle in the axis of the bore with a uniform rate of spin (angular velocity) as well as forward velocity.

Consider, as a generic model, a solid sphere with an attached, hollow truncated cone. The solid sphere, or head of the pellet, has a different angular momentum and rotational kinetic energy than the cylindrical (ring-like) skirt, but their initial angular velocity is identical.

As the initial velocity and stored energy of the pellet decreases traveling down range, due to air resistance (both its linear and rotational kinetic energy), do the inertia of the rounded head and hollow skirt, both on a fixed axis, induce a rotational precession or wobble due to the linear differential of inertia from head to skirt of the pellet, independent of any other force acting on the pellet? Is gyroscopic stabilization of this type of pellet inherently unstable compared to a typical bullet shape due to differential in angular momentum of head and skirt?

I hope I am asking clearly and in correct terms, as it's been a long time since I had any formal physics courses. If not, I hope you still understand the question.

Thanks.

 

[A.T.]

Is gyroscopic stabilization of this type of pellet inherently unstable compared to a typical bullet shape due to differential in angular momentum of head and skirt?

I think the reason is more generally related to combining a fin and rotation as methods of stabilization. The aerodynamic force on the skirt creates a torque that tries to align the projectile axis with the relative airflow, but the rotation causes it to rotate around an axis at 90° to that external torque.

 

[TCups]

I offered up this hypothesis (speculation) on an air rifle forum, but I don't know for sure

[quote someone else]
There is no difference in angular momentum between the head and skirt of a pellet, unless the pellet had a swivel at its waist! There is one and only one angular momentum vector of a pellet. Sure, a lead atom in the skirt contributes more to the angular momentum than an atom in the center of the head (after being shot, usually) but that doesn't change the fact that there is only one commonly accepted meaning of "angular momentum of a rigid body".[/quote]

Agreed. Only one angular momentum for a rigid body. But if the mass is not evenly distributed along the axis of a rotating rigid body (i.e., the rigid body is not a sphere) and the mass is rotating, and if the distribution of the mass on one end of the axis is spherical and on the other end is cylindrical . . .



Then, if an axel had a disk wheel on one end and a rimmed wheel on the other, or say a spherical wheel on one end and cylindrical wheel on the other, and I put it in a vacuum with no external forces acting on it (i.e.,no wind resistance gravity, etc) and spin it, it spins on its axis and stays centered. But if I apply a force to one end or the other, is the resulting perturbation the same?

Think of it this way -- you and a friend are standing on the right and left side of a road. A road - sized barbell is rolling toward you and will hit you both at the same time. On the left end of the barbell, the wheel is a solid lead ball, diameter 3 ft. On the right end of the barbell is a spoked cylindrical rimmed wheel, diameter 3 ft. Hypothetically,both the sphere and the cylinder, as you face them, have roughly the same presenting, visible surface area. Irrespective of the center of gravity of the rolling (cylinder)--------(sphere) about to smack both of you -- the COG won't be the centerline of the road, but far closer to the more massive sphere, of course -- which side of the road would you prefer to be standing on when this thing hits both of you? And if you and your buddy, say, have equal mass, the impact is not only going to perturb both of you, but also perturb the rolling (cylinder)----(sphere), then which end of the (cylinder)-----(sphere) has the greater momentum and is less affected?

And now, imagine the rolling (cylinder) ---- (sphere) is a JSB diabolo pellet proceeding down range at -- pick a velocity -- 800 fps in a 90-degree crosswind from the left at 10 mph. Are the force of the wind resistance on the leading face of the sphere and the drag resistance on the tailing edge of the pellet equal? I think not. Is the force of the cross wind on the side of the pellet designed, roughly we will say, to have the same surface area of head and skirt presenting to the crosswind? Is the crosswind force an equal vector force (lb/inch*inch) applied to the presenting surface areas of the head and skirt (roughly equal), and therefore, an asymmetrically applied force relative to the center of mass (much closer to the pellet head)? If angular momentum must be conserved for the rigid body, how does the asymmetric force perturbation affect the motion of the pellet? Does it move straight sideways, or does the force acting on the skirt push it askew more than the head, rotating the pellet on it's center of mass? And if begins to precess, such that the less massive skirt is now swinging in a wider arc than the more massive head, then wouldn't angular momentum still be conserved, even though the pellet is no longer spinning on the same axis as the trajectory? Do JSB's with their shallow-holed underskirt and more sphere-like head fare better than pellets with deeper holes? Does the secant ogive of the JSB fare better than the tangent ogive of Crossman Premier where drag and BC are concerned, analogous to the ogive's effect on a rifle bullet? Inquiring minds want to know.

And I don't know -- I am only speculating and asking questions, but I would bet 50¢ that diabolo pellets are inherently dynamically unstable when spun, and that conservation of energy and momentum, about the center of mass far forward of the skirt vs. asymmetry of typical ballistic external forces affects head and skirt much differently; that diabolo pellets vs. bullets are much more likely to be affected in odd ways by external forces -- wind, air resistance, drag, and maybe even gravity, compared to a more uniform cylindrical solid. I think that diabolo pellets must have to pay a significant price in rotational stability (in the ballistic sense) for having to have a skirt that is necessary for a proper seal in an air rifle and drag in excess of the head. But again, I've been wrong before and may be wrong now.

 

[A.T.]

You are over complicating this. It can be explained using solely the total angular momentum vector and the external torque vector from asymmetrical drag on the skirt.