How can we find the missing equation for solving spatial collisions of bodies?

[Trying2Learn]

Good Morning

May I ask about spatial collisions of bodies?

In undergraduate dynamics, we study that when two particles college, we have two final unknowns: the final velocity of each particle.

We first use the conservation of linear momentum.

However, we supplement the analysis with the coefficient of restitution (and how it relates relative velocities) and can now solve for the final velocities of each body (we have our second equation)

Good. All clear.

Now turn to an analysis for bodies, not particles.

There are 12 final unknowns: the three final linear velocities and angular velocities of each body.

As I look at the undergraduate textbooks (Hibbeler , Beer and Johnson) they all assert the problem is complicated because the coefficient of restitution depends on the nature of the specific problem. I get that.

But may I proceed with this question by assuming we have such a coefficient?

Those same books then do a spatial collision problem where one of the bodies pivots around a hinge. I suppose that simplifies the problem (the pivot, essentially reduces the number of unknowns).

But are the books "chickening" out?

Suppose I do wish to do a spatial collision of two free bodies.

What are the equations?

(Let me assume the 1-axis is directed along the path of collision. And assume a sufficiently simple geometry)

I get we have to use coefficient of restitution in that direction.

But what other "statements" can be made and converted into "equations?"

Yes, we have conservation of linear and angular momentum – I get that.

(BTW: I am not trying to actually solve this. It is not a HW. I am just trying to understand this.)

For particles, we assumed that at the peak of the impact, the two particles move with the same velocity –this helps us once we know the coefficient of restitution.

But what would one do for bodies?

For example: could we say (as we do in particle collisions), that at the extreme peak of impact, the angular momentum of each body is the same?

Surely, we cannot "assert" the angular velocities are the same. Right?

Any advice? I am sure the solution is beyond me. Maybe I will try. I just want to see the assumptions that lead to the additional equations.

 

[Baluncore]

But what other "statements" can be made and converted into "equations?"

Welcome to PF.

As you are considering only linear and angular velocity I assume the bodies you refer to are spheres of radii approaching zero, and that the collisions occur on one axis. An off-axis collision cannot be computed without knowing radii. Radii would also be important for critical simulation timing.

How would you model friction between the surfaces of real spheres during the finite period of contact. As contact pressure increases the surfaces may heat, then lock together to share angular velocity, then release without frictional losses as the spheres separate.

 

[Trying2Learn]

Welcome to PF.

As you are considering only linear and angular velocity I assume the bodies you refer to are spheres of radii approaching zero, and that the collisions occur on one axis. An off-axis collision cannot be computed without knowing radii. Radii would also be important for critical simulation timing.

How would you model friction between the surfaces of real spheres during the finite period of contact. As contact pressure increases the surfaces may heat, then lock together to share angular velocity, then release without frictional losses as the spheres separate.

Thank you.

No, not spheres and not small bodies "approaching the point of becoming particles."

I mean general shaped, large bodies.

But I have since seen the fault in my thinking...

When two bodies collide along one axis, we can make the same assumption that during the peak of collision, the two contacting points become one.

The two contacting points have the same velocity at collisoin peak: this is a kinematic constraint, but an equation, nonetheless (1 equation).

Then there is Euler's equation for each body (3 equations)

Then there is conservation of linear momentum along direction of collision (1 equation).

That is four equations.

How many unknowns?

In general: 12 (three angular velocities and three translational velocities for both bodies, post collision).

We continue and assume no slip (change of velocity) in the directions orthogonal to the collision axis.

This will knock out four unknowns: down to eight now.

We continue and assume no change in rotation about the axis of collision during the peak of collision (smooth, also) -- that knocks out two more (rendering one of Euler's equations -- about the collision axis -- useless).

But there are not enough equations -- we need two more equations.

I suppose one might come from conservation of kinetic energy -- elevating it from an linear solution to a non-linear one. (Also, reducing this from a problem with general coefficient of restitution to one where the coefficient is 1.0).

So let me go with that: perfectly elastic collision

But we are still missing one more equation.

Yes, it will be difficult to solve and set up. That is for another time. Right now, I am only interested in the other equation.