## 3 Model Description
### Flexible Model of the System
#### 3.1.2 External Forces  
The forces acting on the carriage are:  
* the weight of the carriage \(\mathbf{f_{p}}\);
* the cutting force with components \(\mathbf{f_{c}}\) and \(\mathbf{f_{a}}\);
* the magnetic force \(\mathbf{f_{m}}\) generated by the linear motor. In the proposed method, the constraints imposed by the sliders of the carriage have been considered as moving reactions, called slider-guide forces \(\mathbf{f_{v}}\). It will be shown in Section 5 that replacing the physical constraints with the slider-guide forces does not alter the dynamic behavior of the carriage. The carriage was considered to have uniformly distributed mass, so its weight \(W\) was equally distributed among the N master nodes. The force is exerted on the vertical axis, downwards, which means that the relative force vector \(\mathbf{f_{p}}\) must be filled only in the rows relative to the Y-axis, with negative sign:  
\[\left\{\mathbf{f_{p}}\right\}\ =\ \left\{W_{1}\ldots\ W_{3N}\ 0\right\} \tag{5}\]  
with \(W_{3(n-1)+1}=W_{3(n-1)+3}\ =\ 0;\ W_{3(n-1)+2}\ =\ -\frac{W}{N};\ n\ =\ 1\to N\)  
The cutting force depends on the machining operation. In a generalized form, it is represented as a sinusoidal function. It is applied only on the node representing the position of the spindle, in vertical direction, when the carriage approaches the position of the workpiece. Its transversal component is \(\mathbf{f_{c}}=F_{c}\ sin(\omega t)\), with \(\omega\) the spindle speed, in rad/s, and \(F_{c}\) the amplitude of the cutting force. Their value is reported in Section 6. The magnetic force is exerted in the area, denoted to as magnetic area, located between the middle points of the sliders. The magnetic force on the nodes was modeled with a trapezoidal shape as a function of the position \(\mathbf{z_{c}}\) of the carriage. The total magnetic force is given by the sum of all the magnetic forces on the nodes. With a proper choice of the trapezoidal shape, the total magnetic force remains constant when the nodes enter or leave the magnetic area. Nodes outside the magnetic area undergo a null magnetic force. Figure 5 plots the magnetic force perceived by 4 adjacent nodes as a function of the normalized carriage position \(z_{o}\)/\(z_{c,max}\). For a simplicity's sake, only two nodes at most were considered to lie in the magnetic area. Considering the magnetic force acting on node 2, the trapezoidal function works as follows (see Figure 5):* in the area between 0 and A, both node 1 and node 2 are in the magnetic area. Node 2 is entering in the area in position 0 and node 1 is leaving the area in position A. The value of the magnetic force of node 1 is decreasing from the maximum value of the total magnetic force to zero. The value of the magnetic force on node 2, instead, is increasing from zero to the maximum force. The sum of the two force contributions is constant and equal to the value of the total magnetic force; this is evident in position D where both the forces are equal to half of the total magnetic force. * In the area between A and B, only node 2 is in the magnetic area. The trapezoidal function is constant and equals the total magnetic force. * In the area between B and C, node 3 enters the magnetic area: the magnetic force on node 3 is increasing and the magnetic force in node 2 is decreasing. Generalizing, the magnetic force for a master node is developed as follows:  
* before entering the magnetic area, a node \(i\) does not sense the influence of the magnetic force, so the value of the magnetic function on node \(i\) is equal to zero. * When node \(i\) enters the magnetic area, a node \((i-1)\) is present in the magnetic area; the value of the force on the node \(i\) increases from zero in a straight line, and the value of the force on node \((i-1)\) decreases, following a straight line having the same slope as the previous one, but with opposite sign. This ensures that the sum of the force on nodes is constant and equals the value of the total magnetic force.