# On the Effect of Friction on Tibiofemoral Joint Kinematics
## 3 Contact Solver: Tangential Friction and Normal Contact Forces
[30], in which \(v_{F}\) is the velocity of the femoral part at the contact point, which is calculated according to the radius of the functional flexion-extension axis, multiplied by the respective angular velocity and \(v_{T}\), the speed of the tibial insert at the tibial plateau surface. The latter is computed as the summation of the anterior-posterior linear velocity and the multiplication of the radial distance from the tibial axis to the bottom of the tibial plateau and the internal-external angular velocity. For further information on how to determine these velocity terms, the reader is referred to the reference [28]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Main Friction Model Structure** & **Friction Coefficient** & **Model No. ** \\ \hline A. Modified Coulomb friction model: & \(\mu_{c}=constant\) & A\({}_{1}\) \\ \hline B. Stribeck friction model: & \(\mu\left(\|\mathbf{u}_{t}^{kj}\|,\sigma_{kj}\right)=\left\{ \begin{array}{c} \mu\left(\|\mathbf{u}_{t}^{kj}\|,\sigma_{kj}\right)=\\ \left\|\mathbf{u}_{t}^{kj}\right\|-v_{0}\end{array} \right.\) & \(\|\mathbf{u}_{t}^{kj}\|\leq v_{0}\) \\ \hline B. Stribeck friction model: & \(\mu\left(\|\mathbf{u}_{t}^{kj}\|,\sigma_{kj}\right)=\mu\left(\|\mathbf{u}_{t}^{kj }\|,\sigma_{kj}\right)=\mu\left(\|\mathbf{u}_{t}^{kj}\|-v_{0}\right)^{2}+\mu _{s}\left(\|\mathbf{u}_{t}^{kj}\|-v_{0}\right)^{2}+\mu_{t}\left(\|\mathbf{u}_{ t  
At this stage, a short description of the joint kinematic is presented to determine the velocity of master nodes, including their tangential speed. The relative velocity of a master node like \(Q_{F}^{k,j}\) with respect to its corresponding slave node \(Q_{T}^{k,j}\) locating on body 2, i.e., the tibial insert, can be determined as follows:  
\[\mathbf{U}^{k,j}=\dot{\mathbf{X}}^{t_{n}}+\mathbf{\Omega}_{F}\times\left( \mathbf{Q}_{F}^{k,j}-\mathbf{X}^{t_{n}}\right) \tag{16}\]  
\[\mathbf{u}_{n}^{k,j}=\left(\mathbf{U}^{k,j,\,tn}\mathbf{n}_{F}^{k,j}\right)^{ tn}\mathbf{n}_{F}^{k,j} \tag{17}\]  
\[\mathbf{u}_{t}^{k,j}=\mathbf{U}^{k,j}-\mathbf{u}_{n}^{k,j} \tag{18}\]  
where \(\mathbf{U}^{k,j}\) is the velocity vector of the point \(Q_{F}^{k,j}\) with respect to \(Q_{T}^{k,j}\), \(\dot{\mathbf{X}}^{t_{n}}\) and \(\mathbf{\Omega}_{F}\) depict the translational velocity of the local coordinate system and the angular velocity of body 1, respectively, \(\mathbf{Q}_{F}^{k,j}\) depicts the coordinate vector of the point \(Q_{F}^{k,j}\), and \(\mathbf{X}^{t_{n}}\) is the location of the origin of the local coordinate of body 1, i.e., the femoral body, at time \(t_{n}\). \({}^{tn}\mathbf{n}_{F}^{k,j}\) is the vector normal to the femoral body at the point \(Q_{F}^{k,j}\) at time \(t_{n}\). and \(\mathbf{u}_{n}^{k,j}\) and \(\mathbf{u}_{t}^{k,j}\) also are the normal and tangential relative velocities of the point \(Q_{F}^{k,j}\). As body 2 is assumed stationary, the components of the angular velocity vector resolved in the fixed basis could be evaluated based on Euler angles as follows [68]:  
\[\mathbf{\Omega}_{F}=\mathbf{H}_{z-x-y}\left\{ \begin{array}{c}\dot{\phi}\\ \dot{\theta}\\ \dot{\psi}\end{array} \right\} \tag{19}\]  
where  
\[\mathbf{H}_{z-x-y}=\left[ \begin{array}{ccc}0&\cos\phi&-\sin\phi\cos\theta\\ 0&\sin\phi&\cos\phi\cos\theta\\ 1&0&\sin\theta\end{array} \right] \tag{20}\]  
in which \(\left\{\phi,\,\theta,\psi\right\}\) are Euler angles and \(\left\{\dot{\phi},\,\dot{\theta},\dot{\psi}\right\}\) depict their time derivatives. Eventually, the friction force at each element on the femoral components is evaluated knowing the tangential sliding velocities, Equation (18) at all element nodes, the normal load applied to the element, and contact pressure, which are addressed later in this paper.