# A Predictive Multibody Model of Paper Applied to Cash Recycling
## 2 Modeling the Banknote
### Methodology
#### 2.2.1 Tuning Procedure
Dampings \(c_{x}\) and \(c_{y}\) are derived from the mass and stiffness information, applying the single mass-spring-damper system as follows: \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Symbol** & **Value** \\ \hline Banknote mass (g) & \(M\) & 0.72 \\ Banknote length short side (mm) & \(L_{x}\) & 67 \\ Banknote length long side (mm) & \(L_{y}\) & 127 \\ Banknote thickness (mm) & \(s\) & 0.1 \\ Number of boxes along X & \(n_{x}\) & 10 \\ Number of boxes along Y & \(n_{y}\) & 20 \\ Box length along X & \(d_{x}\) & 7.44 \\ Box length along Y & \(d_{y}\) & 6.68 \\ Box mass (g) & \(m\) & 0.0036 \\ \hline \end{tabular}
\end{table} 
Table 1: Relevant quantities of banknote model. \[c_{x} =2\xi\sqrt{k_{x}\cdot m}, \tag{7}\] \[c_{y} =2\xi\sqrt{k_{y}\cdot m},\]  
where \(\xi\) is the relative damping coefficient and depends on the material, \(k_{x}\) is the stiffness, and \(m\) is the mass of a box. The damping coefficient should be derived from experimental measurements and by the logarithmic decrement method. Since such tests are not available, relative damping \(\xi=12\%\) was chosen, assuming that paper has greater damping than steel, for which a coefficient of 3-5% is typically used. In addition, coefficients of 15-20% are assumed for wooden structures, and paper is a material that is produced from it, so the value of 12% is reasonable. The rotational springs were calibrated by reproducing in simulation the experimental tests described below. To calibrate \(k_{tx}\), a test was performed by keeping the short side fixed and letting the banknote drop by gravity, as shown in Figure 6. The displacement of the free side is measured with a probe. The displacement of the free side measured experimentally is 60.76 mm. The MB model that reproduces the experimental condition consists of the banknote, decomposed into 200 boxes, with fixed joints on one of the short sides. The rotational springs \(k_{tx}\) are the only ones activated in this condition. The simulation was run iteratively with different tentative values of \(k_{tx}\), and the resulting contour plot (lower portion of Figure 6) was compared with the experimentally measured displacement. In this context, one must then consider errors in the experimental measurement and the fact that the simulation may not perfectly represent the experimental boundary conditions. In fact, the size of the boxes limits the precision with which one can emulate the portion of the banknote which is fixed. Assigning the value \(k_{tx}=0.27\) N mm/rad for the center-to-center configuration yields a simulated free side displacement of 60.26 mm; in light of the considerations just made this can be considered acceptable. The relative damping coefficient used is once again used \(\xi=12\%\) together with Equation (7) to define \(c_{tx}\). A further confirmation of the adequacy of the \(\xi\) estimate is the fact that the number of oscillations observed in the contour plot matches those observed experimentally. A similar procedure was performed to estimate \(k_{ty}\) and \(c_{ty}\). The procedure was repeated with the edge-to-edge configuration, yielding consistent results. Figure 6: Experimental set-up of banknote bending measurement while keeping the short side fixed (**a**). Experiment reproduction in MotionView® (**b**). Contour diagram of the deformable surface (**c**). Displacement along the Z direction of point \(\Gamma\) on the short free side (**d**). The values of the translation and rotation stiffness values with associated damping are summarized in Table 2 for the two configurations. It is emphasized that the rotational stiffness \(k_{tz}\) was not calibrated based on experimental tests but was assumed to be high enough to prevent the rotation of two adjacent boxes around the Z axis.