# On the Effect of Friction on Tibiofemoral Joint Kinematics
## 3 Contact Solver: Tangential Friction and Normal Contact Forces  
As was discussed before, when two surfaces enter into a contact phase and tend to slide against each other, friction develops and acts as a resistance to the relative motion. In this study, the friction forces are taken into account employing several friction models, listed in Table 1, that are: (A\({}_{1}\)) a modified Coulomb friction model with a constant friction coefficient; (A\({}_{2}\)) a modified Coulomb friction model with a pressure-dependent friction coefficient extracted from [PERSON] et al. 's formulation; (A\({}_{3}\)) a modified Coulomb friction model with a pressure-dependent friction coefficient extracted from [PERSON]'s formulation; (B\({}_{1}\)) a Stribeck friction approach with constant friction coefficients; (B\({}_{2}\)) a Stribeck friction approach with pressure-dependent friction coefficients extracted from [PERSON] et al. 's formulation; (B\({}_{3}\)) a Stribeck friction approach with pressure-dependent friction coefficients extracted from [PERSON]'s formulation; and (C\({}_{1}\)) [PERSON] et al. 's empirical friction model. The tangential friction force at a node of the femoral part, e.g., \(Q_{F}^{k,j}\), is computed using the following formulation:  
\[\mathbf{F}^{k,j}=-\mu\left(\|\mathbf{u}_{t}^{k,j}\|,\sigma_{k,j}\right)\| \mathbf{F}_{n}^{k,j}\|\frac{\mathbf{u}_{t}^{k,j}}{\|\mathbf{u}_{t}^{k,j}\|} \tag{15}\]  
where \(\mathbf{u}_{t}^{k,j}\) is the tangential relative velocity of the node \(Q_{F}^{k,j}\) in contact as the tibial insert is assumed stationary, and \(\mu\left(\|\mathbf{u}_{t}^{k,j}\|,\sigma_{k,j}\right)\) is the friction coefficient at the contacting node, which can be considered as a function of either relative velocity or contact pressure and can be substituted from each of the modified Coulomb friction and Stribeck friction models, outlined in Table 1. Moreover, static and dynamic friction coefficients can be dependent upon the contact stress based on empirical models reported by [PERSON] et al. [26] and [PERSON] [27]. Eventually, seven case scenarios are defined in Table 1, each of which is considered in this study to study kinematics and tribology of TKAs. In Table 1, \(\chi\) is a constant value that takes a value greater than one. Moreover, the pressure-dependent friction coefficient is used in the models A\({}_{2}\) and B\({}_{2}\), illustrated in Figure 2, to show the difference between such models. As can be seen, the friction coefficient is a function of both relative velocity and pressure when model B\({}_{2}\) is employed, while it varies merely with contact pressure in model A\({}_{2}\). It is worth adding that both models use the pressure-dependent friction relationship [PERSON] et al., proposed in [26]. However, such a relationship is integrated into the Coulomb friction formula for model A\({}_{2}\) and the Stribeck one for model B\({}_{2}\). It is worth noting that the empirical model [PERSON] et al. [26] suggested does not address the friction coefficient once the contact stress approaches zero. Therefore, the strategy considered here is to prevent the friction coefficient to exceed the value 0.25 that corresponds to the contact stress of 0.37 MPa. In addition, the models A\({}_{3}\) and B\({}_{3}\) are written based on the relationship recommended by [PERSON]. His empirical mathematical formulation is valid for the contact stress equal or greater than 1.1 MPa. According to the plot that shows the relationship between the contact stress and friction coefficient in his paper, it can be concluded that the friction coefficient decreases once the contact stress diminishes below 1.1 MPa until it gains an average value of 0.25 once the contact stress reaches zero value. Therefore, a quadratic function is fitted to the data while outliers are removed to address the value of friction once the contact stress gains a value less than 1.1 MPa. The model C\({}_{1}\) is the one developed by [PERSON] et al.