CHAPTER 4 FRICTION REDUCTION BY TANGENTIAL VIBRATIONS
4.3 Sliding with Tangential Compliance
(m
K rx
ζ . A significant friction reduction effect is observed whenever the macroscopic velocity is smaller than the velocity amplitude of the vibration component. For velocity ratio 95ζ <0. , the vibration parallel to the direction of the macroscopic velocity (θ =0) exerts the greatest effect on the friction reduction. For velocity ratio ζ >1, the vibration perpendicular to the direction of the macroscopic velocity (θ =π 2) has a larger effect on the friction reduction, but the amount of the friction reduction is limited.
4.3 Sliding with Tangential Compliance
Surfaces are very irregular at the microscopic level. Therefore two surfaces contact at a
number of asperities. When a tangential force is applied, the asperities will deflect like springs giving rise to the friction force. If the force is sufficiently large, some of the asperities deflect so much that they will slip. The average behavior of the asperities can be represented by the physical analogy depicted in Fig. 4.5. Here, the sliding body experiences a friction force due to the deformation of a single lumped asperity contact.
The deflection z of the lumped asperity is defined as the horizontal distance between points P and T. The deflection z can be modeled by the extension of Dahl model. The Dahl model has the general form
i
where F denotes the friction force, x represents the displacement of the sliding body, σ0 is the stiffness of the asperity, Fc denotes the Coulomb friction force and i represents a parameter that determines the shape of the friction-displacement curve. The value i is most commonly used. Higher values will gives a friction-displacement curve with a sharper bend, as shown in Fig. 4.6. Notably, in this model the friction force is only a function of the displacement and the sign of the velocity. This so-called rate independence is an important property of the model.
=1
To introduce the deflection z into the model, the friction force is defined as z
F~=σ0 , (4.13)
then the model can be written as
i
Equation (4.14) claims that during the unidirectional sliding the deflection z approaches the magnitude
σc0 ss
z = F , (4.15)
which is the steady state deflection of the asperity. Thus Eq. (4.14) can be written as
i
ss
z v v z dt
dz
−
= 1 sgn( ) . (4.16)
The hypothesis of Dahl model, including most friction models, is that the friction force is parallel to the velocity of the sliding body. Some difficulties arise in modeling the behavior of the asperity in the friction system shown in Fig. 4.1, where the instantaneous friction force may not be parallel to the velocity of the sliding body. In this friction system, the velocity of the sliding body frequently changes direction in accordance with the vibrations. The vibration direction must be parallel to the direction of the macroscopic velocity (i.e. θ =0) for the instantaneous friction force to parallel to the velocity of the sliding body, and the Dahl model can be applied without difficulty. However, the behavior of the lumped asperity becomes more complicated when the direction of vibrations is not parallel to the direction of the macroscopic velocity (i.e. θ ≠0).
Figure. Fig. 4.7 shows the behavior of the lumped asperity when the sliding body moves along a curve. This figure is the top view of Fig. 4.5 and only the points P and T are shown.
The trajectory of point P of the sliding body is known and the trace of the point T of the asperity needs to be determined to calculate the friction force. The trace of the point T can be approximated by the following procedure. At time t, point P of the sliding body is in position P(t) and point T of the asperity is in position T(t). At time (t+∆ t), point P moves to the position P(t+ ∆ t). If the time increment ∆ t is small, the asperity is pulled approximately along line T(t)P(t+∆t) to a new position T(t+∆ t). The length of line
) ( )
(t t P t t
T +∆ +∆ , namely the new deflection of the asperity, depends on the friction force and the elasticity of the asperity, which are discussed below. Following this scheme and
using a small time change ∆t can obtain the trace of the point T, as shown in Fig. 4.8.
4.3.1 Asperity Slip without a Stiction Phase (Dahl Model with i=1)
Referring to Dahl model to Eq. (4.14), the deflection change after a small time increase t can be expressed as approaches the steady state deflection
) be approximated by the mean velocity of the point P along line
)
According to Eq. (4.17), the deflection of the asperity at time (t+∆ t) is written as
t
which can be obtained by inserting Eqs. (4.18) and (4.19). Once the new deflection of the asperity at time (t+∆ t) is obtained, the new position of point T, T(t+ ∆ t), is given by
and ( ( ) ( ))
The friction force depends on the deflection and the direction of the asperity. The friction force at time (t+∆ t) therefore can be expressed as (refer to Eq. (4.13))
))
which is the component form of the friction force. Following Eqs. (4.18)~(4.24), the friction force at time (t+2 t) can be obtained. Continuing this process can obtain the friction force during sliding with tangential vibrations.
∆
4.3.2 Asperity Slip with a Stiction Phase (Dahl Model with i=0) Equation (4.14) shows that in the Dahl model with i≠0
0
the asperity can slip (i.e.
) even when the deflection is very small. Thus, when an oscillatory applied force that is far smaller than the Coulomb friction F
dt
c applies to the sliding body, the position of the sliding body drifts. To minimize the drift, Dupont et al. (2000, 2002) proposed an elasto-plastic friction model that possesses a stiction phase. The asperity sticks (i.e.
) when its deflection is smaller than a breakaway deflection. Consequently, it is reasonable to assume that the asperity as shown in Fig. 4.7 sticks when its deflection is smaller than the steady state deflection. Here, the value i= is used in the Dahl model to render stiction. The Dahl model then reduces to
ss
which is essentially an elastic Coulomb friction model. The asperity is modeled as a linear spring. When an increasing tangential force is applied, the asperity does not slip until the force increases to the size of the Coulomb friction Fc. Before slippage, the deflection of the asperity equals the displacement of the sliding body. Thus, the deflection of the asperity at time (t+ t) in Fig. 4.7 can be written as ∆
Replacing Eq. (4.20) with Eq. (4.27) and following Eqs. (4.18)~(4.24) can yield the friction force during sliding with tangential vibrations.