## 2 Materials and Methods
### Geometrical Core Concepts
Orthodontic activation cycle period (as effect of one activation): \(\mathrm{t}_{\mathrm{f}}=7\) week
4. The elastomeric chain: \(\mathrm{K}_{\mathrm{u}}=100\) gramsforce (grf); \(\widetilde{\mathrm{u}}_{\mathrm{f}}=0.1\cdot\mathrm{K}_{\mathrm{u}}\)
5. Time constants:
6. \(\mu=\mu_{\mathrm{T}}=\mu_{\mathrm{Tu}}=4;\lambda=\lambda_{\mathrm{T}}=\lambda _{\mathrm{Tu}}=1.5;\mathrm{T}_{1}=\frac{10\cdot\mathrm{t}_{\mathrm{fu}}}{\mu_{ \mathrm{T}}(1+\lambda_{\mathrm{T}})};\mathrm{T}_{2}=\lambda_{\mathrm{T}}\cdot \mathrm{T}_{1};\mathrm{T}_{1\mathrm{u}}=\frac{\mathrm{t}_{\mathrm{fu}}}{\mu_{ \mathrm{Tu}}(1+\lambda_{\mathrm{Tu}})};\) T\({}_{2u}=\lambda_{\mathrm{Tu}}\cdot\mathrm{T}_{1u}\)
7. The fictional axis of the tooth: \(y_{\alpha}^{\prime}=1\) mm; \(y_{\beta}=(0.5\div 0.9)\) mm; \(\gamma_{0}=y_{\alpha}^{\prime};\gamma_{1}=\frac{Y_{\beta}-y_{\alpha}^{\prime}} {\mathrm{s}_{\mathrm{f}}}\)
8. \(u_{0}(t)=\widetilde{u}_{f}+(K_{u}-\widetilde{u}_{f})\cdot(\frac{T_{1\mathrm{ u}}}{T_{1u}-T_{2u}}\cdot e^{-\frac{t}{T_{1u}}}+\frac{T_{2u}}{T_{2u}-T_{1u}} \cdot e^{-\frac{t}{T_{2u}}})\)
9. \(F_{OT}(t)=\frac{1}{T_{1}-T_{2}}\cdot e^{-\frac{t}{T_{1}}}+\frac{1}{T_{2}-T_{1} }\cdot e^{-\frac{t}{T_{2}}}\),
10. \(F_{OS}(s_{0})=y_{\alpha}^{\prime}=\gamma_{0}\)
11. \(s=s_{0},\ldots,s_{f}\)
12. Scaled value for \(\mathrm{K}_{\mathrm{y}}=0.1\) mm/grf  
Due to the positive values of (\(y_{\beta}\)), it results that in this example, we should consider the translation regime. The proportionality coefficient (constant) (\(\mathrm{K}_{\mathrm{y}}\)) makes the connection between the input signal (\(\mathrm{u}_{0}(\mathrm{t})\)) and the output signal \(y_{00}^{\prime}(\mathrm{t},\mathrm{s})\). Knowing the applied force, the \(y_{\alpha}^{\prime}=1\) mm average value is obtained (by observing the clinical evolution of the tooth's movement and according to the data in the orthodontic literature). More exactly, \(y_{\alpha}^{\prime}=1\) mm represents the average of the steady-state values of the \(y_{00}^{\prime}(\mathrm{t},\mathrm{s}_{0})\) movements obtained as effects of the force application. Obviously, for all patients, the same force (having the same Ku = 100 grf coefficient) is used in the treatment procedure, and the treatment objective is to modify the position of the upper central incisor. After obtaining the average value \(y_{\alpha}^{\prime}=1\) mm, we have analyzed the proportionality between it and \(\widetilde{\mathrm{u}}_{\mathrm{f}}=0.1\cdot\mathrm{K}_{\mathrm{u}}\) residual force, resulting in the value \(\mathrm{K}_{\mathrm{y}}=0.1\) mm/grf for the biomedical process proportionality coefficient. For the determined (scaled) value of the proportionality coefficient (\(\mathrm{K}_{\mathrm{y}}=0.1\) mm/grf), the equality between the right and the left member of Equation (5), in steady-state regime, is verified. All the simulations presented in this paper are made in MATLAB/SIMULINK. In Table 1 (under numerical form) and in Figure 5 (under graphical form), the results of the simulation of the (\(\mathrm{u}_{0}(\mathrm{t})\)) input signal (of the applied force in relation to (t)), are presented, using the corresponding parameters defined between the initial data. \begin{table}
\begin{tabular}{c c} \hline
**t[Weeks]** & **u\({}_{0}\)[grf]** \\ \hline
0 & 100 \\ \hline
0.1 & 75.77 \\ \hline
0.2 & 46.22 \\ \hline
0.3 & 27.92 \\ \hline
0.4 & 18.41 \\ \hline
0.5 & 13.83 \\ \hline
0.6 & 11.71 \\ \hline
0.7 & 10.76 \\ \hline
0.8 & 10.33 \\ \hline
0.9 & 10.14 \\ \hline
1 & 10.06 \\ \hline \end{tabular}
\end{table} 
Table 1: The results of the \(\mathrm{u}_{0}(\mathrm{t})\) input signal simulation, in numerical form. From Table 1 and Figure 5, the decreasing evolution of the (u\({}_{0}\)(t)) input signal, in relation to (t), can be observed, between the value of (K\({}_{\rm u}\)) and the value of (\(\widetilde{\rm u}_{\rm f}\)). The dynamics of (u\({}_{0}\)(t)) are determined by the values of (T\({}_{1\rm u}\)) and (T\({}_{2\rm u}\)). The value (t\({}_{\rm fu}=1.2\) weeks) was determined by the technical characteristics of the elastomer.