I have decided that friction is a drag. It's almost as easy to understand as gravity. We deal with it every day. Friction keeps me from sliding completely under my desk when I slouch in my chair. It keeps my car from spinning out of control as I turn corners with reckless abandon.
This experience with friction begins when as babies we attempt to scoot across the floor and find the carpet difficult and the linoleum floor relatively easy. We build upon our experience until as elementary-age children we are able to pick up our video console controller and expertly proclaim, "This game looks so fake - the cars are sliding all over the place. The physics in this game bites!"
That is the challenge game developers face. The physical world is so familiar to everyone in your potential audience, any departure from realism can be glaring. However, realistically simulating these simple physical properties is quite challenging. This month, I'm going to discuss simulation of friction in real-time 3D applications, otherwise known as the field of tribology.
Why Is It Such A Drag?
Let's take a look at what makes up the experience we term "friction." Grab your trusty copy of Computer Graphics: Principles and Practice and set it on the table. Give the book a push with a small horizontal force. Notice that if the force is small, the book will not move. As you increase the force, you will reach a point where the book will start moving. Once it's moving, you may notice that it takes a little less force to keep it moving.
Table 1. A summary of notations
used in this article.
How is it possible for a smooth book on a smooth table to create a force that resists your efforts to push it? Well, it turns out that even relatively smooth surfaces are actually pretty rough if you look closely enough. It's this roughness that opposes your efforts. But even more interesting is the fact that on a smaller scale, when objects rest against each other, atomic bonds tend to form between them. These bonds form a kind of glue that makes it necessary to apply extra force to get an object moving.
It's possible to measure the effect of this roughness. In fact, this is exactly what Charles Coulomb did in the late eighteenth century. He established a theory of dry friction (since called Coulomb friction) that predicts the maximum friction forces that are exerted on an object in contact with a dry surface before that object moves and becomes dynamic. The theory also predicts the friction forces that the surfaces exert when they are in motion relative to each other.
Don't Give Me No Static
When you are applying force on the book, the friction force opposes your efforts. Let's take a look at a diagram of this situation. Figure 1 shows a free body diagram of the book in static equilibrium, meaning that the book is not moving.
Figure 1. A book in a state
of static equilibrium.
Since the book is in static equilibrium, we can determine a number of things via the principles of statics. The normal force, N, to the collision of the book with the surface is equal in magnitude to the weight of the book, W. Also, the friction force, f, must also be equal in magnitude to the force being applied on the book, F.
(Coulomb Static Friction)
The Coulomb static friction model states that the magnitude of the friction force is less than or equal to the normal force, N, multiplied by a constant coefficient of static friction, ms. This coefficient describes the degree of smoothness between the two surfaces and generally depends on the material composition of the contacting objects. This value typically varies from 0 (which would be a perfectly smooth, frictionless surface) to 1 (for a very rough surface). Some examples of coefficients of static friction can be seen in Table 2.
Table 2. Some coefficients
of static friction.
There are some circumstances where ms can actually be greater than 1. Drag racing tires, for example, are designed to be sticky so that the friction force they exert is greater then the normal force exerted by the road.
When the force you are applying on the book causes the book to be on the verge of sliding, the friction force that opposes your efforts is at its maximum. At this point, slip is said to be impending. Through statics you can calculate the magnitude of the force necessary to cause this slip.
(Coulomb static friction model)
(Objects are in static equilibrium)
(The maximum F before a slip occurs)
Therefore, the maximum force that can be applied on the book before it begins to slip is µsN. What is interesting, and complicated, about static friction is the fact that the friction force increases to equal the applied force until this threshold has been reached.
What Happens Then?
Once the applied force is greater than the slip threshold, the object starts moving. We now leave the world of statics and enter the world of dynamics. It's actually very similar to static friction. The magnitude of the friction force between two dry contacting surfaces that are sliding relative to each other is
where µk is the coefficient of kinetic friction. This force resists the motion of the two bodies. Its direction is opposite the vector of relative velocity between the objects. In general, the value of µk is smaller than ms. However, this does not always have to be the case.
That covers the Coulomb dry friction model in both static and dynamic situations. By simply implementing these two methods, you can create a world represented by interesting physical properties.