## 3 Model Description
### Flexible Model of the System
#### 3.1.2 External Forces
* Once the node \((i-1)\) has left the magnetic area, only node \(i\) senses the magnetic force, so the value of the magnetic function is equal to the total magnetic force. * When node \((i+1)\) is entering the magnetic area, the value of the magnetic force on node \(i\) decreases as already seen for node \((i-1)\) till it gets equal to zero where it leaves the magnetic area. * Once node \(i\) has left the magnetic area, the value of the function of the magnetic force on node \(i\) is equal to zero. The magnetic force is equal to the sum of the contributions of all the nodes. Developing the functions of the magnetic force on the nodes in the way described ensures that the magnetic force is constant and equal to the total magnetic force, for every position z of the carriage. When only one node lies in the area between the sliders, the value of the force is equal to the value of the magnetic force. This condition is true in the area where the function is constant in Figure 5.  
Figure 5: Functions of the magnetic force. The magnetic force is considered only along the Y-axis, in positive direction. Therefore, the resulting magnetic force vector is:  
\[\left\{\mathbf{f_{m}}(z)\right\}\ =\ \ \left\{F_{m,1}(z)\ldots F_{m,3N}(z)\ 0\right\} \tag{6}\]  
where \(F_{m,3i+1}\ =\ F_{m,3i+3}\ =\ 0\); \(F_{m,3i+2}\ =\ M;i\ =\ 0\rightarrow(N-1)\)  
**4. Moving Constraints: Sliding-Guide Force**  
A multi-degree of freedom (mdof) system can be expressed as:  
\[\left[\mathbf{M}\right]\left[\ddot{\mathbf{x}}\right]+\left[\mathbf{K} \right]\left[\mathbf{x}\right]\ =\ \left\{\mathbf{F}\right\}\text{.} \tag{7}\]  
In this particular case, the system in (7) is relative to a free system, in which elastic constraints are not considered. Stiffness matrix \(\mathbf{K}\) is symmetric and has dimension \(h\times h\), with \(h\) equal to the number of degrees of freedom (dof). Constraining the mdof system can be obtained adding to \(\mathbf{K}\) a constraint matrix \(\mathbf{K_{C}}\):  
\[\left[\mathbf{K_{c}}\right]\ =\ \textit{diag}(\textit{c}_{1},\textit{c}_{2}, \ \ldots,\textit{c}_{a}) \tag{8}\]  
\(\mathbf{K_{C}}\) has dimension \(h\times h\) and it is a diagonal matrix. On its main diagonal, the values of the constraints relative to the different dof are present. System (7) becomes:  
\[\left[\mathbf{M}\right]\left\{\ddot{\mathbf{x}}\right\}+(\left[\mathbf{K} \right]+\left[\mathbf{K_{C}}\right])\{\mathbf{x}\}\ =\ \left\{\mathbf{F}\right\}\text{,} \tag{9}\]  
which rearranging is:  
\[\left[\mathbf{M}\right]\left\{\ddot{\mathbf{x}}\right\}+\left[\mathbf{K} \right]\left[\mathbf{x}\right]\ =\ \left\{\mathbf{F}\right\}-\left[\mathbf{K_{C}}\right]\{\mathbf{x}\}\text{.} \tag{10}\]  
This is the solution chosen to represent a constraint as an external force, considering the stiffness matrix of the free system. The constraints due to the sliders were introduced in the model using elastic constraints representing the contact stiffness of the slider along the X and the Y axes. The reaction force is given by the contact stiffness times the displacement \(\mathbf{x}\) of the constrained node. The reaction force is a function of the displacement \(\mathbf{x}\), and its point of application depends on the relative position along the Z-axis between the carriage and the frame. The supports are modeled with discretized contact points. Therefore to correctly model the effect of the supports, it is necessary to ensure:  
* the continuity of the reaction force during its transition between two neighboring nodes;
* that the total stiffness of the constrained nodes is equal to the total stiffness of the slider. In Figure 6, the slider is represented by stiffness connected to the nodes denoted to as constraining nodes. The constraining nodes depend on the Z-position of the carriage; therefore, the sliders were considered as moving in the reference frame of the carriage. The stiffness was defined as function of the Z-position of the carriage. Figure 6: Stiffness of the slider. Lumped-stiffness is associated to each contact node.