# 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  
In many studies, the calliper pressure has been used to calculate the brake torque, assuming a constant BLCF [35]. It is clear that this assumption is not feasible when one is interested in predicting the actual BLCF since it is influenced by the operating and environmental conditions. For this reason, it is important to take into consideration the phenomena affecting the braking by applying a formulation that captures environment and state dependences. Referring to the state of the art, it is noticeable that a wealth of different models have been implemented for the prediction and estimation of friction properties in automotive brakes under determined working conditions. In [36] an alternative formulation based on the switching between the stick and slip motion is proposed by considering a rate-dependent dumping term for the Stribeck effect and an elastic term for the pre-sliding modelling [37; 38]. In [39], a simple quadratic model in the velocity state, based on experimental tests, is used. In [40], an analytical formulation considering only the friction dependence against the speed is proposed. A very simple analytical formulation based on steady-state experimental tests that correlate the pressure, speed and temperature dependences to the friction and wear is assessed in [41]. In [42] an alternative formulation is put forth that, in addition to the sliding speed, involves thermal effects due to the increase in temperature of the friction materials. In [43] a simplified static LuGre model is used under the assumption that the friction dynamics is considerably faster than the other phenomena involving the vehicle dynamics and therefore negligible for simulation purposes: the friction dynamics is neglected by substituting the state with constants and the formula is augmented with a temperature-dependent term. A dynamic LuGre model is employed in [44] to determine the friction force between the disc and the pad. In addition, the authors in [45] propose a novel static formulation as a function of the brake temperature, brake pressure and braking speed. Experimental data collected from the pin-disc test rig are used to calibrate the model. Finally, in [46], a scaled inertia brake dynamometer is used to investigate the frictional characteristics of three lining materials, based on low-metallic, semi-metallic and non-asbestos inorganic composition, respectively. A dynamic model whose parameters depend on the initial load conditions is developed to measure the frictional dependence against the load, temperature and speed. Unfortunately, the majority of BLCF models have not been subjected to experimental assessment but only to simulation tasks. The listed formulations are not capable of depicting the hysteresis behaviour of BLCF against varying speed and pressure. Hence, they do not constitute reliable solutions for BLCF estimation in automotive brakes and cannot be used for real applications. Moreover, most of the abovementioned models do not account for the combination of thermal and mechanical behaviour of the contact since the temperature does not appear as an influencing variable of the BLCF. It is well known that the extreme temperatures reached during the braking actions have a strong impact on the brake performance and can bring about undesired phenomena such as the fading effect, even for metal pads [4; 5]. The FEM technique has been widely applied for temperature distribution estimation and squeal analysis in disc brakes [47; 48]. However, owing to the high computational burden, it is not possible to use this approach for an online estimation of the brake temperature. For this reason, it is not feasible to use the finite element method, along with a friction model, for prediction purposes. The results presented in the literature refer to off-line simulation analysis, wherein, wrongly, BLCF is assumed to have a constant value [49; 50].