# On the Effect of Friction on Tibiofemoral Joint Kinematics
## 8 Results and Discussion
### Coulomb and Stribeck Friction Models  
In this section, an in-detail comparison between the two well-known friction models, Coulomb and Stribeck friction laws, is conducted. The effect of friction coefficient on linear and volumetric wear rates is investigated while using Coulomb friction law and acquired outcomes are listed in Table 6 using Method _iii_[77]. It can be observed that the maximum linear wear rates are obtained in the case study with \(\mu_{c}\) = 0.1. The lateral linear wear with friction coefficient 0.2 is lower than the others, which means it is not a direct relationship between the wear values and friction coefficient. The interesting outcome is that the maximum volumetric wear rate occurs for the case study with \(\mu_{c}\) = 0.05 while the minimum volumetric and linear wear rates are associated with the frictionless joint and the one with a friction coefficient of 0.2. When the friction is higher, greater linear wear can be expected as the worm area decreases due to the changes in the trajectory of the femoral part. However, there is another important contributing factor to wear occurrence, which can prevent the wear to increase. When UHMWPE slides against a metallic counterface, molecular chains preferentially become oriented and hardened along the direction of the PMO, resulting in higher wear resistance of polyethylene [14; 104]. Figure 7: FFT analysis of tibiofemoral joint using different friction models and friction coefficients: (a) Coulomb law: \(\mu_{c}\) = 0.05; (b) Stribeck model: \(\mu_{s}/\mu_{c}\) = 0.05/0.0325; (c) model A\({}_{1}\); (d) model B\({}_{1}\); (e) model A\({}_{2}\); (f) model C\({}_{1}\). The effect of friction coefficients in the Coulomb and Stribeck friction models on the loci of the tibofemoral joint on the transverse plane is also investigated and illustrated in Figure 8. It can be seen that increasing the friction coefficient leads to a significant change in the trajectory of the femur center in the lateral-medial direction in particular. However, the friction influence is smaller in the posterior-anterior direction, although some variations have been observed for the case study with friction coefficient 0.05 and 0.2 for Coulomb friction law and 0.2/0.125 for Stribeck model, in particular, compared to others. This is also observed in that the trajectory loops of all case studies are closed. Although friction and viscous contact lead to energy loss in the system, energy is continuously introduced in the system due to the rheonomically constrained flexion-extension rotation and boundary conditions, e.g., loads and moments imposed on the femoral bone from muscles, hip joints, among others, which is why the femur center motion converges to a steady-state periodic motion. It can be inferred that the trajectory obtained while using the Stribeck friction model is less influenced by friction force due to the difference existing between the dynamic and static friction coefficient and the velocity-dependent characteristic of friction once the Stribeck model is utilized. The damage map and distribution using the Coulomb friction model with four different friction coefficients after five million cycles in service are considered and illustrated in Figure 9. It can be observed that the distribution of damage in both condyles expands in the anterior-posterior direction once the friction coefficient increases. Moreover, increasing friction gives rise to damage values on both condyles as can be observed in Figure 9 and the more intense color of the damage map becoming red. It is worth mentioning that the damaged area on the lateral condyle of the knee joint is seen to increase significantly with increasing the friction coefficient. The other finding is the maximum damage on the lateral side of the joint occurs close to the middle plateau of the tibia when \(\mu_{c}=0.05\), while it is shifted to the anterior part of the lateral condyle with increasing friction as is observable in