Theory for Friction in Joints
The Friction feature is used to add frictional loss to a joint. It is available for Prismatic Joint, Hinge Joint, Cylindrical Joint, Screw Joint, Planar Joint, and Ball Joint.
The friction force is modeled using a continuous friction law, which is capable of modeling sliding-sticking phenomena. A strict application of Coulomb’s law involves discrete transition from sticking to sliding and vice versa, as dictated by a vanishing relative velocity. These discrete transitions cause numerical difficulties and to avoid them, the friction force is approximated with a continuous friction law:
where Ff is the friction force, μ is the frictional coefficient, N is the normal force, v is the slip velocity, and v0 is the characteristic slip velocity. The term
is called the regularization factor.
The regularization factor smooths the friction force discontinuity. The characteristic slip velocity should be chosen to be small in comparison to the characteristic relative velocities encountered during the simulation. The continuous friction law describes both sliding and sticking behavior, that is, it completely replaces Coulomb’s law. Sticking is replaced by creeping between the contacting bodies with a small relative velocity.
The exact interpretation of the friction model for each joint type is described in the following. In general, an expression for the friction force such as
is used. Here Ff,r is the additional sliding resistance, Ff,max is the maximum friction force, and is the relative velocity in the joint. Since the joints are abstractions, there is no unique way to define the normal force N, to which the friction force is proportional. For each joint there are several available methods for computing N and you must select the most representative one.
Prismatic Joint
Friction Force
The total friction force is defined as
The total normal force can be computed from joint forces, interference fit, or the combination of these two.
Normal Force Case 1: Contribution from Joint Forces
Case-1.1:
Normal force computed from attachment reaction forces:
where Fa are the reaction forces on the attachment boundaries, and n is the surface normal in the deformed state.
Case-1.2:
Normal force computed from joint forces and moments:
where Fj are the joint forces, e is the joint axis, Mjli are the components of the joint moment in joint local coordinate system, and αli are contribution factors given by the user.
Case-1.3:
Normal force computed through joint force in a specified direction.
where en0 is a direction given by the user. Rsrc represents the rotation matrix for the rotation of the source attachment.
Normal Force Case 2: Contribution from Interference Fit
Case-2.1:
Normal force defined by pressure and contact area,
where p is the contact pressure, and A is the contact area.
Case-2.2:
Normal force defined by total force
where Fn is the total normal force.
Virtual Work and Power Loss
The virtual work contribution from the total friction force is obtained through a multiplication by the variation of the relative displacement between the two parts:
The energy dissipation rate caused by friction can be written as
Hinge Joint
Most of the theory is the same as for prismatic joints. Any differences are described here.
Friction Moment
The total friction moment is defined as:
where is the relative angular velocity in the joint, and r is the hinge radius, supplied by the user.
Normal Force
Case-1.2:
Normal force computed through joint forces and moments:
Virtual Work and Power Loss
The virtual work contribution from the total friction moment is obtained through a multiplication by variation of the relative rotation between the two parts
The energy dissipation rate caused by the friction moment can be written as
Cylindrical Joint
Most of the theory is the same as for the hinge joint. Any differences are described here.
Friction Force and Moment
The total friction force and total friction moment are defined as:
where r is the entered cylinder radius.
Virtual Work and Power Loss
The virtual work contributions of the total friction force and the total friction moment are obtained through a multiplication by the variations of the relative displacement and relative rotation between the two parts respectively.
The energy dissipation rate caused by friction can be written as
Screw Joint
Most of the theory is same as for cylindrical joints. Any differences are described here.
Friction Force and Moment
The total friction force and total friction moment are defined as:
where n is the number of starts, p is the pitch, and r is the screw radius.
Normal Force
Case-1.1:
The normal force computed from attachment reaction forces is:
Case-1.2:
The normal force computed through joint forces and moments is:
Planar Joint
Most of the theory is same as for prismatic joints. Any differences are described here.
There are two ways to model the friction force in a planar joint. In the first approach, the total friction force is applied at the center of joint. This is similar to other joint types, but in this approach, no friction moment is caused by the relative rotation. In the second approach, the distributed friction forces are applied on the selected boundaries, and hence there is an effective friction moment for relative rotation.
Friction Force: Total Force at Center of Joint
The total friction force is defined as:
where is the relative velocity vector in the joint.
Friction Force: Distributed Force on Selected Boundaries
The distributed friction force is defined as:
where X is the material coordinates on the selected boundaries, Xc is the center of joint, and A is the total contact area.
Normal Force
Case-1.2:
The normal force computed through joint forces and moments is:
Virtual Work and Power Loss: Total Force at Center of Joint
The virtual work contribution from the total friction force is obtained through a multiplication by the variation of the relative displacement vector between the two parts:
The energy dissipation rate caused by friction can be written as
Virtual Work and Power Loss: Distributed Force on Selected Boundaries
The virtual work contribution of the distributed friction force is obtained through a multiplication by the variation of the relative displacement vector between the two parts on the selected boundaries.
The energy dissipation rate caused by friction can be written as:
Ball Joint
Most of the theory is same as in for hinge joints. Any differences are described here.
Friction Moment
The total friction moment is defined as
where is the relative angular velocity vector at joint and r is the ball radius. The factor πr/ 4 is the effective ball radius.
Normal Force
Case-1.2:
The normal force computed through joint forces is:
Virtual Work and Power Loss
The virtual work contribution from the total friction moment is obtained through a multiplication by the variation of the relative rotation between the two parts,
where {ar, br} is the relative quaternion in the joint.
The energy dissipation rate caused by friction can be written as