Theory of Friction in Rigid Body Contact
The Friction (Rigid Body Contact) feature is used to add frictional forces and losses to the Rigid Body Contact.
The friction force is modeled using a continuous friction law, capable of modeling sliding-sticking phenomena. A strict application of Coulomb’s law involves a discrete transition from sticking to sliding, and vice versa, as dictated by a vanishing relative velocity. A discrete transitions can cause numerical difficulties, and to avoid it the friction force is approximated with a continuous friction law. The magnitude of the friction force is calculated as
Here, Fn is the contact force, Ff,r is the additional sliding resistance, Ff,max is the maximum friction force, is the slip velocity vector and is the characteristic slip velocity.
The characteristic slip velocity should be small compared to the characteristic relative velocities encountered during the simulation. The continuous friction law describes both sliding and sticking behavior, and it replaces Coulomb’s law. Sticking is replaced by creeping between the contacting bodies with a small relative velocity below the characteristic slip velocity.
Friction in the Spherical to Spherical Formulation
In the Spherical to Spherical Formulation, the slip velocity is calculated as
where vc,s and vc,d are the velocities of the source and destination spheres at the contact point. They are defined as
where vsrc and vdst are the velocity vectors of the centers of the source and destination spheres, and the angular velocity vectors of the source and destination spheres are given by and . The distance vectors from the centers of the source and destination, to the contact point, are given by rs and rd. They are defined as
where rs and rd are the radii of the source and destination spheres.
If the inside boundaries of the destination sphere are in contact with the outside boundaries of the source phase, the distance vector from the center of the source sphere to the contact point is replaced by
The friction force vector, applied on the source sphere is defined as
where Ff is the magnitude of the friction force. The friction force vector is applied also on the destination sphere, but with opposite sign.
The frictional moments applied at the center of the source and destination spheres are defined as
In 3D, the quaternion moments corresponding to the physical moments for the geometrically nonlinear case are defined as
where qsrc and qdst are the quaternions of the source and destination spheres.
The energy dissipation rate caused by friction can be written as
Friction is formulated in the same way in the Spherical to Cylindrical Formulation, the Spherical to Planar Formulation, the Cylindrical to Cylindrical Formulation and the Cylindrical to Planar Formulation. The only difference is the computation of the distance vectors from the source and destination centers to the contact point, denoted by rs and rd. In each case, based on the source and destination geometries, they are calculated from the undeformed locations of the source and destination centers and direction vector from the source contact point to the destination contact point.
Friction in Spherical to Arbitrary Formulation
In the Spherical to Arbitrary Formulation, the slip velocity is calculated as
where vc,s is the velocity of the source sphere at the contact point. It is defined as
where vsrc is the velocity vector of the source center and is the angular velocity vector of the source sphere. The distance vector from the center to the contact point for the source is defined as
where rs is the radius of the source sphere. The velocity of the destination boundary, at the contact point, is vc,d.
The friction force vector, applied on the source sphere, is defined as
where Ff is the magnitude of the friction force. The friction force vector applied on the destination sphere has equal magnitude, but points in the opposite direction.
The frictional moment applied at the center of the source is defined as
.
In 3D, the quaternion moment corresponding to the physical moment for the geometrically nonlinear case is defined as
where qsrc is the quaternion of source sphere.
The energy dissipation rate caused by friction can be written as