# On the Effect of Friction on Tibiofemoral Joint Kinematics
## 8 Results and Discussion
### Dynamics and Tribology of the Joint
** \\ \hline A\({}_{1}\) & 0.100 \({}^{4}\)/0.086 \({}^{5}\) & 8.79 & 0.045 \({}^{4}\)/0.059 \({}^{5}\) & 4.97 & 0.056 \({}^{4}\)/0.052 \({}^{5}\) & 6.76 \\ A\({}_{2}\) & 0.098/0.095 & 9.21 & 0.034/0.052 & 4.54 & 0.047/0.043 & 6.62 \\ A\({}_{3}\) & 0.087/0.084 & 9.54 & 0.031/0.050 & 4.37 & 0.049/0.040 & 6.72 \\ B\({}_{1}\) & 0.092/0.090 & 9.07 & 0.035/0.055 & 4.97 & 0.050/0.044 & 6.86 \\ B\({}  
The loci of the motion of the femur center with respect to the tibia on the transverse plane for the following five friction models, i.e., A\({}_{1}\), B\({}_{1}\), B\({}_{2}\), B\({}_{3}\), and C\({}_{1}\), are illustrated in Figure 5. Static and dynamic friction coefficients are 0.1 and 0.085 while \(v_{0}\) and \(\xi\) are equal to 1 mm/s and 10, respectively. To estimate such friction coefficients for the sake of computational analysis, the average magnitude of friction coefficient from empirical models, i.e., Saikko model and [PERSON] et al. 's, are obtained over the range of contact pressure the knee joint undergoes over a walking gait cycle. The trajectories show the same trends to some extent, although some variations are observed; for example, the one corresponding to model A\({}_{1}\) is more influenced by friction than that acquired from model C\({}_{1}\) as can be observed in the left-hand side plot of Figure 5. The external-internal rotation angle is also plotted against the abduction-adduction angle over a gait cycle in Figure 5. The models A\({}_{1}\) and B\({}_{1}\) lead to the very similar shapes of the rotational path, while the same can be observed in the case of models B\({}_{2}\) and C\({}_{1}\). The ranges of variation of both angles when using model B\({}_{3}\) decrease. Phase portrait diagrams of the dynamic motion of the tibofemoral joint are also presented in Figure 6 for the friction models A\({}_{1}\), B\({}_{1}\), and C\({}_{1}\). It can be seen that the range of velocity increases once model C\({}_{1}\) is utilized, while model A\({}_{1}\) produces the highest acceleration magnitudes and the lowest maximum velocities among others. The norm of position vector varies in the range of 44.26-46.30 mm. Moreover, the highest speed occurs during the swing phase at around 70% of the walking gait cycle in Figure 6a,c,e. Phase portrait diagrams can help one study the type of dynamic response observed in the knee joint while employing different friction models. Figure 6b,d,f show chaotic behaviors of the tibofemoral motion, which can be considered sensitive to the used friction model, and densely filled in the phase portrait diagram [103]. The chaotic response suggests that the vibration can occur due to friction, the so-called friction-induced vibration, and impact/contact between the femoral part and the tibial insert. This finding is in agreement  
Figure 4: The effect of choosing friction model on linear wear prediction (m) of TKA using friction models (a) \(A_{1}\); (b) \(A_{3}\); (c) B\({}_{1}\); (d) C\({}_{1}\). with the acceleration level found for the frequency at which the joint is likely to vibrate, as is shown in Figure 7, where the highest acceleration is obtained once the model A\({}_{1}\) is used. Figure 5: The effect of the used friction model on trajectory of TKA employing friction models (_a_) A\({}_{1}\); (_b_) B\({}_{1}\); (_c_) B\({}_{2}\); (_d_) B\({}_{3}\); (_e_) C\({}_{1}\). Figure 6: Phase space portraits for different friction models: (**a**,**b**) A\({}_{1}\); (**c**,**d**) B\({}_{1}\); (**e**,**f**) C\({}_{1}\), which presents the acceleration magnitude of the femoral part versus velocity and the femur velocity magnitude versus its position vector norm. A FFT frequency analysis is performed to characterize the friction-induced vibration of the tibiofemoral joint. It is found that the tibiofemoral joint does not vibrate when the friction is below a critical value as the joint does not oscillate using Coulomb or Stribeck friction models with friction coefficient 0.05. It has also tested that the frictionless joint does not undergo oscillation.