## 2 Materials and Methods
### Description of the Model
#### 2.1.2 Equations of Motion  
The system of equations that govern the wheel is directly presented in modal coordinates, where the modal base is calculated by finite elements, with \(n\) modes and one physical degree of freedom corresponding to the contact point [6], as follows:  
\[\mu\ddot{\mathbf{Q}}(t)+c\dot{\mathbf{Q}}(t)+\gamma\mathbf{Q}(t)=\mathbf{ \Phi}^{T}N\mu(s,N) \tag{2}\]  
\(\mu,c,\gamma\in\mathbb{R}^{n\times n}\) are the generalized mass, damping and stiffness matrices of the wheel, respectively, and proportional damping is assumed, so that matrix \(c\) is diagonal. \(\mu(s,N)\) stands for the creepage and normal force-dependent friction coefficient, according to the friction law explained previously. The modal base \(\mathbf{\Phi}\in\mathbb{R}^{1\times n}\), as well as the modal contributions vector \(\mathbf{Q}(t)\in\mathbb{R}^{n\times 1}\) can be written as follows (component \(\Phi_{i}\) of \(\mathbf{\Phi}\) represents the modal deformation of mode \(i\) in the tangential direction):  
\[\mathbf{\Phi} =[\Phi_{1}\;\Phi_{2}\;\cdots\;\Phi_{i}\;\cdots\;\Phi_{n}] \tag{3}\] \[\mathbf{Q} =[Q_{1}\;Q_{2}\;\cdots\;Q_{i}\;\cdots\;Q_{n}]^{T} \tag{4}\]  
For the study of friction-destabilized systems, it is convenient to express the equations relative to the quasi-static equilibrium, by eliminating the steady-state component of the displacement and force. This way, \(u(t)=U(t)-U_{st}\) corresponds to the wheel's instantaneous displacement relative to the quasi-static equilibrium \(U_{st}\) and the derivatives of \(u\) are \(\dot{u}(t)=\dot{U}(t)\) and \(\ddot{u}(t)=\ddot{U}(t)\).  
This procedure can be extended to the modal contributions vector in a similar manner by stating \(\mathbf{q}(t)=\mathbf{Q}(t)-\mathbf{Q}_{st}\) and \(\mathbf{\dot{q}}(t)=\mathbf{\dot{Q}}(t)\), \(\mathbf{\ddot{q}}(t)=\mathbf{\ddot{Q}}(t)\), for its derivatives. Assuming the system oscillates stably around \(U_{st}\), it is possible to obtain \(\mathbf{Q}_{st}\) from Equation (2), by setting \(\mathbf{\dot{Q}}\) and \(\mathbf{\ddot{Q}}\) to zero, as follows:  
\[\mathbf{Q}_{st}=\gamma^{-1}\mathbf{\Phi}^{T}N\mu_{st} \tag{5}\]where \(\mu_{st}=\mu(\alpha,N)\) stands for the friction coefficient evaluated at the quasi-static conditions. Equation (2), in relative terms, becomes:  
\[\mathbf{\mu}\ddot{\mathbf{q}}(t)+\mathbf{c}\dot{\mathbf{q}}(t)+\mathbf{\gamma}\mathbf{q}(t)= \mathbf{\Phi}^{T}N(\mu(s,N)-\mu_{st}) \tag{6}\]  
Differential Equation (6) is solved numerically for \(\mathbf{q}\) and using each one of the initial conditions explained in Section 2.3. [PERSON]'s method [29] for the nonlinear case is chosen for this purpose and implemented in MATLAB, as it is particularly well adapted to second-order differential equations and therefore for structural dynamic problems, due to its good numerical stability and energy conservation properties. An average constant acceleration algorithm (for which \(\gamma=1/2,\ \beta=1/4\)) was implemented, as it is the best unconditionally stable scheme [30] and the time step \(\Delta t\) was carefully tuned at \(1\times 10^{-6}\) s, by performing a power conservation verification.  
The response of the wheel \(u(t)\) at the contact point can be easily found by premultiplying the just obtained modal contribution vector \(\mathbf{q}(t)\) by the modal base \(\mathbf{\Phi}\):  
\[u(t) =\sum_{i=1}^{n}q_{i}(t)\mathbf{\Phi}_{i} \tag{7}\] \[u(t) =\mathbf{\Phi}\mathbf{q}(t) \tag{8}\]  
Likewise, the derivatives of \(u(t)\) are calculated as follows:  
\[\dot{u}(t) =\mathbf{\Phi}\dot{\mathbf{q}}(t) \tag{9}\] \[\ddot{u}(t) =\mathbf{\Phi}\dot{\mathbf{q}}(t) \tag{10}\]  
A tool that is useful to analyze transient nonlinear response is the modal projection, as it allows us to better understand the evolution in time of each of the modes of the structure. However, instead of plotting the evolution of the modal contributions to the velocity \(\dot{q}_{i}\), the product \(\dot{q}_{i}\Phi_{i}\) provides a more physical comprehension.