Forces transmitted through a contact can include both normal forces as well as tangential forces,
due to friction.
To understand the friction force, imagine pulling a block with a spring.
To cancel the downward gravitational force on the block, the floor pushes upward with
a normal force f_n.
The force applied by the spring is f, which is opposed by the friction force f_t applied
by the floor.
At first, the pulling force f is too small to move the block.
As we extend the spring, the force f gets larger, as does the resisting friction force
f_t.
When f grows large enough, the block begins to slide.
If the spring is pulled with a constant velocity, the block matches the velocity, and f and
f_t are equal and opposite.
Let's plot the friction force f_t as a function of the block's sliding velocity v. When the
velocity is zero, the friction force f_t could be anywhere in the range minus mu f_n to mu
f_n, where mu is called the friction coefficient.
When the sliding velocity is not zero, the magnitude of the friction force is mu f_n,
and it acts in the direction opposite the sliding velocity.
By this model, the friction force depends only on the direction of sliding, not the
speed of sliding.
This empirical, approximate model of dry friction is called Coulomb friction.
According to this approximate model, if the sliding velocity is zero, then the magnitude
of the tangential friction force is less than or equal to mu times the normal force, which
is nonnegative.
The frictional force could act in any direction.
If the sliding velocity is nonzero, then the friction force magnitude is mu f_n, and it
acts in the direction opposite the sliding direction.
If the velocity is zero, but the acceleration a is nonzero, then slip is about to occur,
and the same equation applies, substituting a for v.
The Coulomb friction model is just a rough approximation for the micromechanics of contact,
and there are many more detailed models.
One common enhancement to the model is to define two friction coefficients, a static
friction coefficient mu_s and a kinetic friction coefficient mu_k, where the static coefficient
is larger than the kinetic coefficient.
This friction law can be visualized as shown here.
Larger friction forces are available to resist initial sliding, but once sliding is initiated,
the friction coefficient drops.
In the rest of this chapter, though, we will use the basic Coulomb friction law with a
single friction coefficient.
This model is attractive for its simplicity, and because it approximately captures the
behavior of many dry surfaces in contact.
The friction coefficient mu depends on both materials in contact, and typically ranges
from values close to zero, when one of the materials is teflon or ice, to values around
1 when one of the materials is rubber.
The set of all forces that can be transmitted through a Coulomb friction contact can be
visualized as a friction cone.
For a frame defined at the contact, the normal force f_z must be nonnegative, and the tangential
force magnitude must be less than or equal to mu f_z.
Looking at this cone from the side, we define the friction angle alpha, which is the inverse
tangent of mu.
This figure also represents the friction cone for a planar contact, and we can define f_1
and f_2 to be vectors along the friction cone edges.
Then the set of all forces that can be transmitted through a planar contact is the positive span
of f_1 and f_2.
Unlike a planar friction cone, which can be represented as the positive span of two forces,
a spatial friction cone cannot be represented as the positive span of a finite number of
forces.
For computational purposes, though, it's common to approximate a quadratic cone as a polyhedral
cone defined as the positive span of four forces, where the z component of each force
is 1 and the x or y component is mu or minus mu.
The polyhedral cone is an underapproximation of the friction cone.
To more closely approximate the quadratic cone, one could use more cone edges.
Contact forces create moments about coordinate frames not at the contact point.
To represent these moments, we can define a wrench cone that corresponds to the friction
cone.
For the coordinate frame shown here, and a planar friction cone which is the positive
span of f_1 and f_2, the wrench cone includes the moments p cross f, where p is the contact
point in the coordinate frame.
The planar friction cone can be plotted as a wrench cone in the three-dimensional wrench
space.
The wrench cone is the positive span of the wrenches from the friction cone edges.
The linear components of the wrench cone edges are mostly in the f_y direction, and the moments
about the z-axis are negative.
Adding another friction cone creates a corresponding wrench cone with positive moments.
The set of all wrenches that can be transmitted through the two contacts is the positive span
of the four wrenches at the edges of the friction cone.
We call this a composite wrench cone, composed of wrenches due to multiple contacts.
In the next video I'll introduce a convenient graphical representation of planar wrench
cones, analogous to our graphical representation of planar twist cones.