## 3 Model Description  
The motion of the carriage is given by the superposition of a rigid body motion (actuated by the linear motors) and vibrations due to its flexibility. These two motions are studied separately and then coupled. Figure 2 shows the block diagrams of the rigid and flexible models. The actuation of the carriage is modeled in Matlab (R2019a, MathWorks, Natick, MA, USA, 2019) considering the electric linear motor. The flexible motion is modeled in the state-space (S.S.) representation. The system is fed with the forces \(f\) while the outputs are the position \(x\) and velocity \(\dot{x}\) of the tool.  
Figure 1: Z-axis carriage.  
The force vector \(f\) is composed by four types of forces acting on the carriage:  
* the weight force \(\mathbf{f_{p}}\) constant in time;
* the magnetic force with components \(\mathbf{f_{m}}\) and \(\mathbf{f_{r}}\), due to the linear motor and depending on the position of the carriage according to the action of the control;
* the cutting force \(\mathbf{f_{c}}\) between the spindle of the carriage and the workpiece, variable in time according to the machining process, with a possible axial component \(\mathbf{f_{a}}\);
* the slider-guide force \(\mathbf{f_{v}}\), depending both on the Z-axis position of the carriage and on the position \(\mathbf{x}\) of the nodes currently in contact with the slider.  
Figure 3 shows the Z-carriage in two different configurations. The numbered nodes were selected as master nodes for the Craig-Bampton reduction of the full FE model of the carriage. This model was developed with a commercial FE software. The Craig-Bampton reduction was performed using the first 20 modes and 11 master nodes. This kind of reduction gives an accurate dynamic behavior of the carriage in the frequency range chosen for the reduction. Since the first modes are the most influencing in dynamic systems, considering 20 modes provides a good approximation. At the same time, mass and stiffness matrices (denoted as \(\mathbf{M}\) and \(\mathbf{K}\), respectively) are small enough to allow fast simulations with an in-house code. Further details on the reduction are reported in [22].  
The reduced model of the carriage was compared with the full FE model in the situation as depicted in Figure 3.  
Figure 3: Forces on the nodes changing with time.  
Figure 2: Schematic representation of the model.  
As shown in Figure 4, four nodes were constrained along X-, Y- and Z-directions. In Matlab environment, this is expressed adding stiffness in stiffness matrix. The values of the stiffness were assumed \(1.58\times 10^{20}\) N/m along X- and Z-directions and \(1.50\times 10^{20}\). N/m along Y-direction. A force equal to 1000 N along the negative direction of the Y-axis is applied on a target node, whose displacement is the element of comparison between the two approaches.  
Table 1 shows the results of the comparison in terms of displacements.  
When the carriage moves from an initial configuration at time t = 0, in which the Z-coordinate is \(z_{c}=0\), to the next configuration at t = t\({}_{1}\) with \(z_{c}=z_{c1}\), the relative position of the corresponding nodes on the carriage with the frame changes. Therefore, the magnetic force (in yellow in both Figures 1 and 4) and the constraint due to the sliders (in red in both Figures 1 and 4) are applied to a different set of nodes on the carriage.