# Survey on Modelling and Techniques for Friction Estimation in Automotive Brakes
## 2 Techniques for the Estimation of BLCF in Automotive Disc Brakes
### Static and Dynamic Model-Based Approaches for the Estimation of BLCF
Regardless, in the following rows, two simple models that try to capture the temperature influence on the brake effectiveness are reported. In [51], [PERSON] introduces a static model that takes into account the temperature dependence of the BLCF for different pad compositions. It is possible to further understand the capabilities of this approach by explicating the proposed formulation. The dependence of BLCF \(f\) on the temperature \(T\), in accordance with the method proposed in [51], has the form:  
\[f(T)=f_{1}+\frac{f_{2}}{\left(f_{3}\Big{(}T-T_{f}\Big{)}\right)^{2}+1}. \tag{1}\]  
The coefficients \(f_{1},\ f_{2},\ f_{3}\) and \(T_{f}\) are defined experimentally and depend on the pad chemical composition. The main drawback of this model is that it does not account for the static friction, giving no analytical formulation for the stick regime, and does not consider the dynamics of BLCF itself. Above all, this model does not reproduce the Stribeck effect and consequently the hysteresis against the speed. In [17], [PERSON] proposes a dynamic model for the prediction of BLCF based on two differential equations: the former estimates BLCF; the latter represents the temperature state. This dynamic model tries to capture the dynamical behaviour of BLCF and its dependence on temperature, thus explaining the fading effect of the brake system. The frame of the model rests on the assumption that BLCF is the result of the equilibrium between the flow of birth and death of contact patches. It is worth noticing that the destruction of the plateaus depends on the wear of the neighbouring plateaus, their temperature and the local mechanical stress. The described process is responsible for the equilibrium of the patch flow and thereby determines the friction and wear behaviour of the contact. The patches are responsible for the transformation of the kinetic energy into friction. In particular, both the number and size of the contact patches are tied to the heat and wear phenomena: indeed, the wear causes a particle flow that increases the size of the contact patches; the heat modulates the wear and thus the destruction of the contact patches [17]. Heat and wear are in turn induced by the sliding resistance. A schematic representation of this process is reported in Figure 2.  
Along with the friction modelling, this approach also embeds the wear effect on the brake dynamics; a reduction of the pad thickness affects the temperature space distribution and, above all, the wear is responsible for the destruction of the contact plateaus. This model is able to capture the fading effect thanks to the inclusion of the temperature dependence, but, as shown later, is unable to  
Figure 2: Energetic balance involving friction phenomena, as proposed in [17]. Reprinted with permission from [17]. Copyright Elsevier, 2003.  
render the hysteresis phenomenon against the sliding speed and the pressure. The typical formulation dwells on the following set of differential equations:  
\[\dot{\mu}=-[b\times(Nv(t)+\varepsilon)\mu-a\times T] \tag{2}\]  
\[\dot{T}=-c\times[(T-T_{0})-\gamma\times Nv(t)] \tag{3}\]  
The first equation depicts the dynamics of the BLCF \(\mu\); the latter represents the dynamics of the brake temperature \(T\) in the contact region. The group \(Nv(t)\) represents the magnitude of the combination between the calliper normal load and the brake disc speed; \(a,\ b,\ c,\ \gamma,\varepsilon\) are constant parameters linked to the pad chemical composition. In particular, the parameter \(a\) is responsible for the growing behaviour of the patches against the temperature and thus of the increase of the BLCF; \(b\) is instead responsible for the destructive behaviour of the mechanical grinding action; the parameter \(\varepsilon\) is linked to the real area of contact patches. The term in parentheses in Equation (2) describes the destructive effect linked to the friction power and the extent of the patches area, supposedly proportional to BLCF.