Model behaviour

In this section a discussion on the characteristic features of the model both in the pre-sliding and gross sliding regimes of friction is presented. Thus highlighting the peculiarities of the new model and its similarities to some existing models.

Mechanism of the pre-sliding friction function

In the pre-sliding regime friction exhibits a non-local memory characteristic effect as de- scribed previously in chapter 2, section 2.2. This hysteresis is captured by the function in equations 3.1, and 3.2. Figure 2.4 of chapter 2 is also used for the analysis here. Prior to application of force at point a in figure 2.4a, the values of the zr and fr reflect respectively

the initial displacement and force the bristle is subjected to. For surface under tension this is often non-zero while for non-tensed surfaces it is regarded as zero. For the analysis the initial values of these two variables are 0. zt is the target displacement at which maximum force

(see section 2.2), and the target force is the stiction force Fs. Thus the virgin curve could be

said to be the path the friction force traced from rest position towards the saturation force against the displacement. Hence, as the applied force rises from a − b, the branch friction force given as f_b(z) starts from zero and rises non-linearly with the externally applied force until either there is a reversal of the direction of motion (as is the case here) or saturation of the friction force. The friction force then traces (a′− b′).

Direction reversal:

When the external force changes direction at b resulting in direction change from b − c before saturation, an inner loop results. For this new branch b − c the variable zr assumes the value

of the displacement at the beginning of this new branch, that is the value of displacement at b, while the displacement variable z takes on the displacement value at point b also, hence the function f_b(z) starts again from zero rising as the path (b′− c′) is traced. The value of fr is given as the force of the (a′− b′) branch at point b′. The new target displacement zt

and the new target force ft are recalculated by an algorithm. Internal or inner loops result

when several direction reversals occur without the saturation of the friction force or when the displacement is less than the breakaway value which is the case here. As the force reverses direction again at c, f_b(z) restarts again from zero, while the value of the variables zr and

fr are updated to the respective values of the displacement and force at the reversal point.

The values of the target displacement (zt) and force ( ft) recalculated again reflecting current

scenarios. The same happens for the path c − d and d − e. Inner loop are never closed unless the current loop displacement z is larger or equal to the any previous loops. However when such conditions exist inner loops are closed from inside towards outside with inner ones closing before the outer ones. In such a scenario which results in this case as the applied force traces the path (e′− f′− g′), on reaching f′towards (g′), the displacement z for the current branch becomes larger than the previous immediate displacement in the same direction thus the loop (d′− e′− f′) will be closed since d′ and f′ are of the same displacement value. The current branch takes the trajectory of the previous branch in the same direction, branch (c′− d′) for this case and this action wipes out the inner loop. It achieves this by taking on the values of zt and ft of the previous branch and thus extending the branch as if it never

reversed. This behaviour is known as non-local memory hysteresis. Friction force saturation:

At any time when the displacement z reaches the breakaway value Z_b, the friction force becomes a maximum value and saturates (reaches breakaway value), it therefore becomes independent of the displacement, however dependent on the direction of the motion, this is represented as the stiction force Fs since z being the deflection of bristles is assumed to

gross-sliding.

The pre-sliding friction force at any point in the process is always given as the sum of f_b(z) and fr represented as Fhyst(z) = fb(z) + fr.

Motion dynamics

Pre-slide and gross-slide features:

In general consider the model representation of equations ( 3.4), ( 3.5); During the pre-sliding regime

|F_hyst(z) + σ v| ≤ |γ| (3.6) the deflection rate of the bristle is the same as the pre-slide velocity, that is

˙z = v (3.7)

and the friction force becomes

F_f = Fhyst(z) + σ ˙z + fvv (3.8)

putting (3.7) into (3.8)

F_f = F_hyst(z) + (σ + fv)v (3.9)

Given that the micro-damping parameter (σ ) is often chosen much greater than viscous damping coefficient ( fv), implies the micro damping dominates the viscous damping in the

pre-sliding regime. One can also approximate this pre-sliding force as

F_f = F_hyst(z) + σ v (3.10) since σ ≫ fv

During the gross-sliding regime

˙z =γ − Fhyst(z)

σ (3.11)

and

F_f = γ + fvv (3.12)

Thus (3.12) shows that the micro-damping parameter does not appear in this regime but the macro damping parameter fv. So during gross-sliding the macro-damping dominates.

Steady state features:

The steady state features of the proposed non-drift dynamic friction model is investigated considering the constant zero velocity (pre-sliding case) and the constant non-zero velocity

(gross-sliding case):

The first case with ˙z = 0, v = 0

F_hyst(z) = sat(γ, Fhyst(z) + σ v) (3.13)

so that

F_hyst(z) = F_hyst(z) (3.14)

and

F_f = Fhyst(z) (3.15)

This shows that during pre-sliding regime the slowly varying displacement is ˙z ≈ 0 the friction force is described by the pre-sliding friction force.

For the second case with ˙z = 0, and v ̸= 0

0 = sat(γ, F_hyst(z) + σ v) − Fhyst(z) (3.16)

which implies

γ = Fhyst(z) (3.17)

and

Ff = γ + fvv (3.18)

This also shows that during the constant velocity regime (gross-sliding) the friction force is simply a lagged version of the Stribeck function and the viscous force. Note the equivalence of equations 3.18 and 3.12 suggesting that the influence of the micro-damping is limited to the pre-sliding regime.