The Arm Stall is when the arm stops moving.
Once the arm has stopped moving, it no longer has any kinetic energy. That means any kinetic energy that was in the arm has been transfered, usually to the projectile (and sling) or back to the counterweight.
Background: Energy Transfers
Trebuchets, before they throw, they are cocked. A cocked trebuchet has no kinetic energy (nothing is moving), but it does have lots of potential energy stored in the counterweight's potential drop height. During the throw energy is preserved (conservation of energy). Once the trigger is pulled, the potential energy of the counterweight begins to be transformed into kinetic also stored in the counterweight. As the counterweight falls it is slowed by the arm. Thusly, it transfers the kinetic energy to the arm. If all the kinetic energy is transfered, the counterweight will no longer be moving (no kinetic energy) and is considered stalled. (Counterweight Stall)
Just as the counterweight is slowed by its transfer of energy to the arm, the arm is slowed by its transfer through the sling to the projectile. If all the kinetic energy from the arm is transfered to the projectile, the arm will no longer be moving (no kinetic energy) and is considered stalled. (Arm Stall)
Sometimes energy is stored in other ways during a throw such as the kinetic energy of the frame or floating axle. This adds the concepts of frame and axle stall to some types of trebuchets.
Arm to Projectile Energy Transfer
A few things can be easily determined about this energy transfer.First, if the arm is rotating, and the angle between the back (trailing side) of the arm and the sling is between 0 and 180 degrees (0 to pi radians) then the projectile will be accelerating, and the arm decelerating. This can be explained by the fact that in this case, the arm tip is moving away from the projectile and thus doing work on it through the tensioned sling. The work (transfer of energy) is force times distance. The arm opposes the force of the sling with an equal opposing force supplied by the inertia of the arm. If the arm moves away from the sling, the inertia of the arm is reduced and work is done on the sling (which accelerates it). If the arm moves toward the sling, work is done on the arm by the sling which accelerates the arm and decelerates the sling. During this transfer, only the portion of the sling tension that is tangential to the path of the arm tip matters, the rest (the normal component) is balanced by a load on the axle through the arm. The amplitude of the energy transfer can be determined by computing the sling tension's tangential component, and the velocity of the arm tip, and multiplying them together.
The Simplified Case
To analyze the arm to projectile energy transfer (through the sling), the situation must first be defined. To avoid all possible complexity, the axle will be assumed to be fixed, and the counterweight to be no longer effecting the arm. The sling is assumed to be massless, and all parts are assumed to be frictionless and inelastic. Basically, this simplified case consists of a spinning arm with a sling and projectile. Also, gravity is ignored and all angles are in radians.
There are several factors involved in the arm to projectile energy transfer.
Mass of the projectile.
Distance from arm tip to center of projectile.
Arm's moment of inertia
The moment of inertia of the arm about the axle.
Distance from the center of the axle to the arm tip (attachment of the sling).
Choose units where the arm length is one, and the arm moment of inertia is also one. The arm can be assumed to consist of a point mass at the end of the arm, and with the above values, that point mass will be considered one unit of mass.
Remaining adjustable constants:
- Sling length (in arm lengths)
- Projectile mass (in arm point masses)
- Initial conditions for some of the below variables
- Sling angle
- Projectile Velocity
- Arm Velocity
Angle between the sling and the arm measured from the arm to the sling in the initial rotational direction of the arm. (Initial=Supplied)
The tension force in the sling. (Initial=Computed)
The rate of change of the projectile's velocity (vector). Equal to the sling tension (vector) divided by the projectile mass. (Initial=Computed)
The speed and direction of the projectile's movement (vector). The integral of Projectile Acceleration. (Initial=Supplied)
The angle of the arm measured from the arm's starting location to the arm in the initial rotational direction of the arm. (Initial=0)
Arm Acceleration (Angular)
The rate of change of the arm's velocity (angular). Equal to the sling tension's normal component (sin(sling angle)*sling tension)divided by the arm's point mass (one) times the arm tip's speed (arm velocity in radians*arm length of one) which is simply the arm velocity. Thus, equal to: sin(sling angle)*sling tension*arm velocity (Initial=Computed)
The speed of the arm's rotation. The integral of Projectile Acceleration. (Initial=Supplied)
Further mathematical analysis coming soon.