## 3 Results
From Figure 10, it results that until the moment \(t1=t_{\rm a}\), the value of the rotation center (\(s_{\rm c}\)) is higher than 20 mm; more exactly, its position is higher than the tooth's apex position, outside of the tooth's volume. This aspect is due to the positive values of the \(y_{\rm\beta}(t)\) signal (Figure 9) between the time interval [0; \(t_{\rm a}\)] weeks (in this time interval, from a mathematical perspective, the tooth behavior is similar to the case of the translation regime). At the moment \(t_{\rm 1}=t_{\rm a}=1.35\) weeks, the rotation center has the position \(s_{\rm c}=20\) mm, which corresponds to the tooth's apex position. Practically, when \(s_{\rm c}=20\) mm, then \(y_{\rm\beta}(t)=0\) mm; the tooth's apex reaches again the initial position in its movement to negative values. After \(t_{\rm 1}=t_{\rm a}=1.35\) weeks, the value of (\(s_{\rm c}\)) becomes lower than 20 mm, highlighting positions inside the tooth's volume. This aspect, according to Figure 8, proves the tooth's rotation regime during the orthodontic treatment (for values of (\(s_{\rm c}\)) smaller than 20 mm, on Figure 10, the signal \(y_{\rm\beta}(t)\) has negative values). After the moment (\(t_{\rm b}\)), the (\(s_{\rm c}\)) enters the steady state regime (its steady state value being \(s_{\rm c}=10.15\) mm), which signifies the end of the tooth's movement (in the case of the applied orthodontic treatment procedure). A very important and interesting study is represented by the analysis of the tooth's movement dynamics in relation to the value of the tissue resistance weighting coefficient (\(\sigma\)). In Figure 11, the function (\(s_{\rm c}(\sigma)\)) is represented, for different values of time (t). The curves from Figure 11 highlight two important aspects regarding the tooth's dynamics during the orthodontic treatment. First, analyzing Figure 11, it can be observed that for higher values of the time independent variable (t), we obtain lower values of the rotation center position (\(s_{\rm c}\)). Consequently, on the entire range of time (t) values (on the  
Figure 11: The evolution of the rotation center (\(s_{\rm c}\)) in relation to the weighting coefficient (\(\sigma\)), for different values of the time (\(t\)) independent variable. entire period of the orthodontic activation cycle) the evolution of the (s\({}_{\rm c}\)(t)) function is a monotone, decreasing one. This remark is proved by both the simulations presented in Figures 10 and 11. It is obvious that immediately after the treatment starts, the position of the rotation center is far from the tooth's top; after that, during the treatment, the position of the rotation center becomes closer to the tooth's top and, in the final part of the treatment, the position of the rotation center is the closest in relation to the tooth's top. Secondly, analyzing Figure 11, the decreasing evolution of the (s\({}_{\rm c}\)(\(\sigma\))) function, on the entire range of (\(\sigma\)) values and for any value of the time independent variable (t), can be remarked. For any value of (t), the lower the value of the (\(\sigma\)) weighting coefficient, the higher the value of the tissue resistance and, implicitly, the higher the value of the (s\({}_{\rm c}\)(\(\sigma\))) function. This phenomenon is physically explained by the fact that the higher the tissue's resistance, the translation movement during the treatment has a more pronounced character, the differences between the values of the movements \(y^{\prime}_{\alpha}(t)\) and \(y_{\beta}(t)\) being a smaller one and implicitly the position of the rotation center (s\({}_{\rm c}\)(t,\(\sigma\))) being more distant in relation to the tooth's apex. For any value of (t), the higher the value of (\(\sigma\)) weighting coefficient, the lower the value of the tissue resistance, and, implicitly, the lower the value of the (s\({}_{\rm c}\)(\(\sigma\))) function.