Rigid Body Dynamics - 4th Equation

[Abastion]

Let's say there are two rigid bodies in space, each has an initial translational velocity as well as an initial angular velocity. Eventually, these two bodies collide. The collision causes the translational and angular velocities of both bodies to change, resulting in 4 unknowns (each unknown being a vector). All of the initial conditions are known, as well as all information about the collision point.

I need four equations to solve the problem. So far, I have three: conservation of linear momentum of the system, conservation of angular momentum of the system, and the coefficient of restitution equation at the collision point. I can't figure out the last equation to give me my last unknown. Any ideas?

 

[AlephZero]

So far, I have three: conservation of linear momentum of the system, conservation of angular momentum of the system, and the coefficient of restitution equation at the collision point.

Those are three good physical principles to use, but you can get more than three equations from them. (Hint: vectors).

 

[D H]

So far, I have three: conservation of linear momentum of the system, conservation of angular momentum of the system, and the coefficient of restitution equation at the collision point. I can't figure out the last equation to give me my last unknown. Any ideas?

You're missing more than just one equation. You need to know something about the geometry of the collision. You also need to know to some extent how the objects react during the collision. The coefficient of restitution is just part of the picture.

Make some simplifying assumptions such as having the objects being hard spheres. The collision between two lumpy, deformable objects would make for a nice simulation problem, but not for an analytic problem. You can make some general statements of the post-collision situation in terms of scattering angles and such, but you are not going to be able to come up with an analytic solution to the problem of the collision between lumpy, deformable objects.

You could also look at a simplified version of the problem. For example, start with two pool balls colliding on a frictionless pool table, and try to generalize from that simplified problem.



What, exactly, is the problem you are trying to solve?

 

[Abastion]

Well, I'm writing some code for a physics simulation. I have two 3-dimensional rigid (non-deformable) bodies moving around in a 3-D environment, and occasionally, they collide with each other. To account for the general case, I have to assume they both have initial translational and rotational velocities (both vectors in 3D) just before they collide. Sometime mid-frame, the objects collide and I calculate the time of the collision as well as the collision point in space. I also have a coefficient of restitution for the collision. From that information, I need to calculate the final translational and rotational velocities of the two bodies.

Arbitrarily, the rigid bodies are rectangular boxes.

 

[haruspex]

Let's restrict it to uniform spheres for now, as DH suggests.
You can normalise the co-ordinate system to reduce it to a 2D problem.
Your 4 unknown vectors constitute 6 degrees of freedom (the spins now only being a scalar each). The conservation laws pin down 3.

At impact, the relative motion will not in general be along the line of centres. This means they experience a glancing blow, and friction becomes significant. If they're smooth, no spin will be imparted. If sufficiently frictional there is instantaneous rolling contact in the split second before they bounce apart again. If you look at the direction of motion of the points of contact immediately after bounce, their relative motion must be directly away from each other. So in either extreme, you get one more degree of freedom pinned down. In between the extremes, there is sliding but frictional contact. This will apply a tangential impulse equal to the dynamic coefficient of friction multiplied by the normal impulse.

The angle of contact (angle between line of centres and initial relative motion) also feeds into subsequent linear motion. That's 5. Restitution gives you the sixth.