SpinBot Physics
What happens when a bot with an "angular momentum weapon" collides with another bot?
To investigate this situation, I'm going to consider a stationary bot with its weapon spinning at some angular velocity. A "worst case" collision will be simulated by introducing an immovable object into the path of the spinning shell. Unfortunately I can't seem to find a decent program to use to draw these diagrams so I'll do my best using 3dsMax.
First, we have our robot, stationary with its spinning weapon at full speed:
Next, the shell collides with some immovable object (just to keep things a little simpler). The forces involved in are going to be proportional to the change in the angular momentum of our robot and happen over a very short time. In this case, we use the mathematical concept of an impulse which is an instantaneous change in momentum. The collision impulse will be tangential to the shell an pointing opposite the direction the shell was spinning. To be clear, this is a large linear force, applied at the point of contact, tangential to the shell acting over a very short period of time.
Now, how does this impulse affect the robot? In order to compute the effect that an impulse has on an object; you calculate the equivalent impulse on the CM. To paraphase from David Baraff: If we apply impulse J to a rigid body with mass M, then the change in linear velocity is: delta_v = J / M. If the impulse acts at a point p, then just as a force produces a torque, J produces an impulsive torque of:
( http://www.cs.cmu.edu/~baraff/sigcourse/notesd2.pdf page 14).
But where does the Torque go?
So far we've determined that an impulsive force and torque is applied to the weapon. The next step is to determine how this affects the rest of the bot. In order to do this we need some free-body diagrams. Clearly, the linear impulse is going to accelerate the entire body of the robot in the direction of the impulse. The frame of the robot must be designed to withstand this force. The subject of great debate on the battlebots forum (which I participated in loudly on the wrong side for quite a while!) is what happens to the torsional impulse?
Below is an image of the state of a horizontal spinning weapon drive-train right at the instant of impact. The relevant quantities are the radii of the disk and two sprockets, and the angular momentum in the weapon and motor.
In this analysis, we're going to assume the worst case: the impact causes the weapon and motor to completely stop. We'll start with the free-body diagram for the weapon. Since the collision is going to completely stop the weapon; the torsional impulse from the collision and the torsional impulse applied to the weapon's sprocket must sum to the angular momentum of the weapon.
From the following free-body diagram of the motor, we can see that the torsional impulse carried through the chain is exactly equal to the angular momentum in the motor.
So the torsional impulse from the collision is divided between the weapon and the motor and the linear impulse carried through the chain/belt is proportional to the angular momentum in the motor (where the constant is the radius of the sprocket on the motor). There is some minorly interesting stuff happening related to the radii of the two sprockets used but other than that the whole thing is quite simple and obvious to me now. I must confess that for a long time, I mistakely thought that the entire torsional impulse from the collision would be carried through the chain to the motor. To Brian Nave ( http://www.TeamLOGICOM.com ) Hopefully this web page is clear enough that other clutch lovers can see the light as I have :-)
EXCELLENT PAGE! Spinning Disk Weapons: http://homepages.which.net/~paul.hills/Spinningdisks/Disks.html
Vector Mechanics for Engineers, by Ferdinand P. Beer and E. Russell Johnston Jr.
http://www.cs.cmu.edu/~baraff/sigcourse/notesd1.pdf
http://www.cs.cmu.edu/~baraff/sigcourse/notesd2.pdf