# A Predictive Multibody Model of Paper Applied to Cash Recycling
## 2 Modeling the Banknote
### Methodology
#### 2.2.1 Tuning Procedure  
Considering the banknote placed on the \(xy\) plane, springs with \(k_{x}\) stiffness connect two boxes along the X direction and springs with \(k_{y}\) stiffness connect the boxes along the Y direction. A schematic is shown in Figure 4.  
A translational \(k_{z}\) spring connects adjacent boxes to resist out-of-plane motion. In the center-to-center connection, one spring per typology connects the mass centers, while in the edge-to-edge connection two springs per typology are necessary. Given the small thickness of the banknotes (about 0.1 mm), it is sufficient to use only one layer. In addition to the translation springs, it is necessary to include rotational springs with stiffnesses \(k_{tx}\), \(k_{yy}\), and \(k_{tz}\), which act with a resisting torque when a relative rotation is created between the boxes, as shown in Figure 4. In practice, the connection between rigid bodies in the MotionSolve(r) software is made using the 'Bushing' entity, which allows translation and rotation stiffnesses, and the associated damping, to be entered. A value must be assigned to each of the above stiffnesses. The specific values depend on the chosen discretization and on the configuration (i.e., number of springs of the center-to-center vs. edge-to-edge layout). As an example, let us consider a EUR 10 banknote (67 mm \(\times\) 127 mm \(\times\) 0.1 mm), shown in Figure 5, discretized in \(n_{x}\) = 10 boxes along the X direction and \(n_{y}\) = 20 boxes along the Y direction. Starting from the mass and dimension of the banknote and the chosen discretization (variables \(n_{x}\) and \(n_{y}\)), it is possible to define the following quantities:  
* The dimension of the boxes \[ \begin{split} d_{x}&=\frac{L_{x}}{n_{x}-1},\\ d_{y}&=\frac{L_{y}}{n_{y}-1},\end{split} \] (1)
* The mass of each box  
Figure 4: Center-to-center connection to represent the banknote focus on translational springs (**left**) and edge-to-edge connection focus on rotational springs (**right**). Figure 5: EUR 10 banknote discretized into 200 boxes. \[m=\frac{M}{n_{x}\cdot n_{y}}, \tag{2}\]
* The section of each box normal to directions X and Y \[A_{x}=d_{y}\cdot s,\] (3) \[A_{y}=d_{x}\cdot s.\] All relevant quantities are reported in Table 1.  
The values of the translation stiffnesses \(k_{x}\) and \(k_{y}\) are obtained from the stress-strain curves of the paper. In detail, with reference to the chosen orientation of the banknote (see Figure 5), \(k_{x}\) may be obtained from the stress-strain curve obtained in machine direction MD, while \(k_{y}\) may be obtained from the tensile test performed in cross direction CD (see Figure 1). The data presented here are taken from [13]. Considering the machine direction MD, i.e., X direction, and the center-to-center configuration, the stress-strain relation can be expressed as follows:  
\[\sigma=E_{MD}\cdot\epsilon=\frac{k_{x}\cdot\Delta L}{A_{x}}, \tag{4}\]  
where the strain is defined as \(\epsilon=\Delta L/L_{0}=\Delta L/dx\). By substituting the definition of \(\epsilon\) into Equation (4) as follows:  
\[k_{x}=\frac{E_{MD}\cdot A_{x}}{d_{x}}, \tag{5}\]  
and similarly, one can define \(k_{y}\) as follows:  
\[k_{y}=\frac{E_{CD}\cdot A_{y}}{d_{y}}. \tag{6}\]  
In the simulation, one may chose to use as Young's Modulus along the different directions (\(E_{MD}\) and \(E_{CD}\)), a mean value valid for all strains, or feed MotionSolve(r) with the nonlinear stress-strain curves. In the edge-to-edge configuration, the single \(k_{x}\) and \(k_{y}\) values will be 50% of their center-to-center counterpart; however, the global stiffness will be the same. In detail, considering the X direction as an example, the connections on the outer edges of the banknote will be linked by a \(k_{x}\) spring, while the inner ones will be linked by a \(2k_{x}\) spring. The value of the \(k_{x}\) spring is chosen to be one order of magnitude higher than that of \(k_{x}\) and \(k_{y}\) to prevent limiting out-of-plane motion between neighboring boxes.