Friction Formulation in Connectors
The basic concept of Coulomb friction between two contacting bodies is the relation of the maximum allowable frictional (shear) force across an interface to the contact force between the contacting bodies. In the basic form of the Coulomb friction model, two contacting surfaces can carry shear forces, Ft, up to a certain magnitude across their interface before they start sliding relative to one another; this state is known as sticking. The Coulomb friction model defines this critical shear force as μFN, where μ is the coefficient of friction and FN is the contact force. The stick/slip calculations determine when a point transitions from sticking to slipping or from slipping to sticking. Mathematically, the relationship can be formalized as
Frictional stick occurs if Φ<0; and sliding occurs if Φ=0, in which case the friction force is μFN.
Friction in connectors is based on the analogy that contacting surfaces of various parts inside a connector device transmit tangential as well as normal forces across their interfaces. The normal (contact) forces, FN, are typically generated by kinematic constraints or by elastic forces/moments in the connector. Connector friction can be used to model tangential (shear) forces, Ft, in the space spanned by the available components of relative motion for both stick and slip conditions. Figure 1 illustrates the simplest frictional mechanism in connectors, a SLOT connector in a two-dimensional analysis.

The local tangent direction in which frictional sliding occurs is the 1-direction (tangential traction Ft=f1), and the normal force is developed by the kinematic constraint enforcing the SLOT constraint in the 2-direction, FN=f2. The friction model is defined in this case by
which in case of slip predicts a friction force f1=μf2 as expected. In this case the friction model is straightforward to understand as the slip direction is along an intrinsic (1 through 6) component of relative motion and the normal force is given only by the force in one other single component orthogonal to the sliding direction.
In many connectors the definition of the tangential tractions is more complex. For example, friction may develop in a tangent direction that spans two or more available components of relative motion. Consider the case of frictional sliding in a SLIDE-PLANE connection as illustrated in Connector Functions for Coupled Behavior. In this case the friction-generating normal force is given by the constraint force in the 1-direction, FN=f1. However, the magnitude of the tangential tractions is given by
thus including contributions from two components of relative motion. The instantaneous direction of frictional slip in the 2–3 plane is not known a priori.
In many connectors the normal force may have contributions from several connector components. Consider the case of a three-dimensional SLOT as illustrated in Connector Functions for Coupled Behavior. In this case the magnitude of the tangential tractions is given by Ft=f1, but the normal force is generated by constraint forces in both the 2- and 3-directions and can be written as
In the most general case both the tangential tractions and the normal force may have contributions from several components. Further, the component directions may include both translations (forces) and rotations (moments). Thus, friction modeling in connectors is defined in a more general form, as follows. First, the function Φ governing the stick-slip condition is defined as
where f is the collection of forces in the connector; P(f) is the connector potential (see Connector Functions for Coupled Behavior), which represents the magnitude of the frictional tangential tractions in the connector in a direction tangent to the surface on which contact occurs; and FN is the friction-producing normal (contact) force on the same contact surface. Frictional stick occurs if Φ<0; and sliding occurs if Φ=0, in which case the friction force is μFN.
The normal force, FN, is the sum of a magnitude measure of contact force-producing connector forces, FC=g(f), and a self-equilibrated internal contact force (such as from a press-fit assembly), FintC:
The function g(f) is given by a connector derived component definition as illustrated in Connector Functions for Coupled Behavior. Using this formalism, we can easily reconstruct the examples illustrated above:
-
In the two-dimensional SLOT case, P(f)=|f1| and g(f)=f2.
-
In the SLIDE-PLANE case, P(f)=√f22+f23 and g(f)=f1.
-
In the three-dimensional SLOT case, P(f)=|f1| and g(f)=√f22+f23.
See the examples at the end of this section for more complex illustrations of friction definitions in connectors.
If frictional effects are defined for a rotational component of relative motion (such as in a HINGE connector), it is often more convenient to define “tangential” moments and “normal” moments instead of tangential tractions/forces and normal forces. The pseudo-yield function governing the stick/slip behavior is defined in a similar fashion:
where the “normal” moment MN is written as
MintC is the self-equilibrated friction-generating internal “contact” moment (for example, from press fit). See Specifying Friction in a HINGE Connection at the end of this section for an illustration.