Modeling Deformation due to collisions

[Captain Newbi]

I am taking a physics modeling class and for our projects we need to add in some element to our physics engine that is unique and different. I choose to model the deformation of objects as a result of collisions.

For clarification, When things bounce, I want to know by how much do they deform and to model the impulse and energy lose.

My question: I Would like to know any references to equations and or the equations that represent the energy loss and by how much the object deforms so i can have an idea of what parameters i would need to give my objects.

I am currently looking into youngs modulus, coefficient of restitution, stress, elastic collisions and more, but i am having trouble finding the correct equation at the moment.

And insight and help is always appreciated, thanks for your time and patience.

 

[Chestermiller]

The most important thing to remember in doing modeling is to START SIMPLE. Why? If you can't solve a simple version of the model, you certainly won't be able to solve a more complicated version. Plus, once you solve the simple model, you will have some results under your belt, and these results can then be compared with those for more complicated versions (if you get that far). In using modeling to get an understanding of collisions, it's important not to start by looking at collisions of balls or even more complicated shapes. The kinematics of such deformations are too complicated, and the problems in dealing with the changing contact patch are daunting. I recommend starting with the head-on collision of two identical circular cylinders, in which the axes of the cylinders coincide. This is a hard enough problem.

The first step in developing a model for a physical system is to articulate in your own words what is happening physically, from a fundamental mechanistic point of view. Once you are able to do this, you can then translate this mechanistic description into the language of mathematics (i.e., equations). So, what is your fundamental mechanistic description of what happens physically when two cylinders collide head-on, in terms of the elastic deformations that occur and the changes in velocity of the various parts of the cylinders?

 

[Captain Newbi]

We do already have ball to ball collisions and ball to rectangle collisions being handled.

Currently, I have deduced a simple model treating the deformation of the ball being represented by a spring force where the delta x is the distance from the center to edge and maximum deformation is when (position - edge)=0.
the spring constant will be represented by the correlation coefficient which we are using for our momentum/energy conversation approximation since we are working in 2D. The increase in the force resistance as the ball increases its deformation will come from the force. The balls dimensions will be circle/oval that is bounded in a rectangle such that the height and width could be adjusted giving the illusion of the squish.

I think this is a good start and will provide a believable first step.
I forgot to mention, we don't need real world exact answers, but I'm trying to use the equations to create a believable looking situation based on physics principles.
(for example gravity in my engine has a magnitude of 1 since a magnitude of 9.8 would make the particle move way to fast on screen, i suppose I could use 9.8 for gravity and just slow my time steps down (maybe 10x slower), but this would yield an equivalent result)

 

[Chestermiller]

I have an additional suggestion for you to consider. Including gravity is something that doesn't really influence the fundamentals of what is happening in a collision. So, rather than including gravity, I would give the body an initial horizontal velocity and let it hit a vertical rigid wall. Consider modeling the body as a series of springs and masses, with the masses sandwiched between the (massless) springs. The springs would simulate the elastic nature of the body, and the masses would represent the distributed mass of the body. Just a thought.

In an elastic collision, what happens is the leading edge of the body hits the wall first, and is stopped by the wall, while the remainder of the body continues to move at the original velocity, unaware that anything has happened yet. The material at the leading edge of the body begins to compress, and this compression translates into a force which begins to slow down the part of mass near the leading edge. A compression wave forms that propagates at the speed of sound along the body. Within the compression zone, the material has stopped moving, while beyond the compression zone, the velocity is still the original velocity. Eventually, the compression wave propagates to the far end of the body, at which point, the body is fully stopped. The compression is then relieved at the far end of the body, and the velocity within the relief zone becomes equal to the original velocity before the collision, but in the opposite direction. The relief zone propagates back along the body at the speed of sound, at which point the entire body is moving away from the wall at the original velocity. At this point, the body loses contact with the wall. The mass-spring sequence model is an ideal 1D model for approximating this.

Chet