Dry friction

Most of the textbooks on engineering mechanics recognize just two types of friction opposing relative motion of bodies in mechanical contact: fluid or viscous friction, and dry or Coulomb friction. We have seen in modules on translational and rotational systems already that modeling of fluid friction of a constant damping factor is quite easy.

Dry friction, however, is often neglected just because it is poorly understood and considered as difficult to model. Yet dry friction may effect machine behavior. in a way deteriorating or even disabling machine control. Dry friction-induced vibrations, often referred to as stick-slip, are causing such industry plaguing effects like break and bearing squeals or chatter of machine tools. Dry friction modeling is not an unattainable goal, however.

Dry friction between two rigid bodies in mechanical contact is commonly assumed to be caused by a dry friction force of two components - the kinetic friction force and the static friction force. Both these reaction forces are developing in the direction tangential to the plane of contact and both are opposing the relative motion of the contact surfaces.

The kinetic friction force F_fk can be specified by the expression

(4.1)

where μ_k is a constant called the coefficient of kinetic friction, F_n is the normal force driving the two surfaces into contact, and is the relative velocity of the two surfaces. The signum function is defined as

(4.2)

At zero relative velocity , it is the static friction force that tends to prevent motion. It is usually defined as

(4.3)

where μ_s is the constant coefficient of static friction. In most cases, μ_s > μ_k.

The two dimensionless coefficients of friction can thus be defined as

(4.4)

Values of both the coefficients are assumed to be independent of the geometric area of the mechanical contact.

Fig. 4.3 displays two different characteristics of the dry friction force defined above. In Fig. 4.3a the friction force F_f is plotted with respect to the relative velocity of the contact surfaces whereas Fig. 4.3b shows the way in which the dry friction force F_f is related to the tangential force F applied to the contact surfaces. It is important to note in this figure that a part of the displayed characteristic for increasing F differs from that for decreasing F.

From the characteristics in Fig. 4.3 we can observe that three different situations may occur when two surfaces are in mechanical contact:

1. The surfaces are in mutual rest and the applied force F is smaller (in absolute value) than F_s. F is thus not large enough to set the surfaces in motion as the surface reaction, the friction force F_f, develops to be in equilibrium with F. The surfaces remain in rest up to the point when the applied force F reaches the value F_s. Then the surfaces are just about to slide along each other — their motion is impending.
2. The applied force F has exceeded the value F_s. Then the friction force F_f cannot balance the applied force F any more and the surfaces start sliding. As soon as they were set into motion, the magnitude of F_f drops into a lower value F_k. From there on the surfaces keep sliding while the friction force remains constant at the value F_f = F_k as far as the force F provided that it does not drop below the value F_k.
3. The applied force F has dropped below the value F_k. The friction force F_f gets into balance with F and the sliding motion stops.

As the dry friction description given above was formulated rather early (Amontos in 1699, Coulomb in 1755) it has been termed as the ‘law of dry friction’, and the coefficients of friction are treated as intrinsic material constants. Despite this, the expressions (4.1) and (4.3) should be regarded only as a specification of a rough empirical model of limited validity.

And more than that, this model is inconvenient for numerical computations because of its discontinuities at zero velocity. Besides functions specifying any nonlinearities, the algorithms for numerical integration of differential equations operate also with the derivatives of these functions. Most of the algorithms will just collapse when encountering a characteristic of dry friction given above. The robust algorithms will have to shorten the integration steplength to such an extent that the computations will slow down very significantly.