# Optomechanical Analysis and Design of Polygon Mirror-Based Laser Scanners
## 5 Finite Element Analysis (FEA) of PMs
The color code ranges from blue (low stress, the minimum set at zero) to green (medium stress) and red; the values of stress that exceed 200 MPa appear in white (see, for example, the inner part of the central hole at the upper  
Figure 11: Selection of a quarter of a (hexagonal) PM (**a**) for an FEA (**b**), highlighting the three constructive (mounting) parameters \(r\), \(\delta\), and \(R_{\text{h}}\). limit of \(\omega\) in Figure 12). One can observe that the maximum stress \(\sigma_{\max}\) is reached in the inner part of the PM and in its central plane, and increases consistently with \(\omega\), but not with \(n\). Thus, \(\sigma_{\max}\) is almost equal to the yield value for \(n=6\) (i.e., 217 MPa), while it is lower for \(n=4\) (i.e., 185 MPa), as well as for \(n=8\) (for which \(\sigma_{\max}=210\) MPa, therefore closer to the yield limit). This conclusion may lead to exploring other designs, which may avoid mounting the PM on a distinct rotational shaft. The minimum level of the stress is reached in the exterior of the apex of the PM. It increases with \(n\) for \(\omega=120\) krpm, \(\sigma_{\min}\) (\(n=4\)) = 8.53 MPa \(<\sigma_{\min}\) (\(n=6\)) = 13.96 MPa \(<\sigma_{\min}\) (\(n=8\)) = 21.01 MPa. In Figure 13a, a study with regard to the ap problem \(R\) is carried out, for a hexagonal PM and two values of \(\omega\): A common one of 60 krpm (on the first row) and an extreme one of 120 krpm (on the second row). The ap  
Figure 12: Extreme values of the stress (\(\sigma_{\max}\) and \(\sigma_{\min}\)) for rotational PMs with (**a**) \(n=4\), (**b**) \(n=6\), and (**c**) \(n=8\), for four levels of the rotational speed \(\omega\): (**1**) 60 krpm; (**2**) 80 krpm; (**3**) 100 krpm; (**4**) 120 krpm. (a) 20 mm, (b) 25 mm, and (c) 30 mm. At \(\omega\) = 120 krpm, \(\sigma_{\rm max}\) becomes slightly higher than the yield tension for \(R\) = 25 mm and reaches 338.1 MPa for \(R\) = 30 mm, in the latter case with \(\sigma_{\rm max}\) in the small mounting hole. For the maximum value of \(\omega\), the dimension \(R\) can be thus obtained (from the point of view of the FEA) for a certain set of parameters. To complete the design process, the other PM parameters must be explored (and exploited) as well. Thus, in Figure 14, the contrast between two levels of two of these parameters is considered, with \(b\) = 5 mm versus \(b\) = 10 mm in Figure 14a versus Figure 14b, respectively, and \(r\) = 2.5 mm versus \(r\) = 5 mm in Figure 14c versus Figure 14b, respectively. The maximum stress \(\sigma_{\rm max}\) decreases slightly when \(b\) increases (from 219.3 MPa for \(b\) = 5 mm to 217.2 MPa for \(b\) = 10 mm) and \(r\) decreases (from 217.2 MPa for \(r\) = 5 mm to 200.2 MPa for \(r\) = 2.5 mm). In Figure 15, the influence of the constructive parameters is explored. A variation of the radius \(R_{\rm h}\) from 13 mm to 14 mm is considered in Figure 15a,b, respectively. In consequence, \(\sigma_{\rm max}\) decreases from 217.2 MPa to 212.2 MPa, just lower than the yield limit. A variation of the radius \(\delta\) of the supplementary holes is considered from 2 to 3 mm from Figure 15a,c,  
Figure 13: Variation of \(\sigma_{\rm max}\) and \(\sigma_{\rm min}\) of hexagonal rotational PMs (with \(b\) = 10 mm and \(r\) = 5 mm) for two levels of \(\omega\), (**1**) 60 krpm and (**2**) 120 krpm, for three values of \(R\): (**a**) 20 mm, (**b**) 25 mm, and (**c**) 30 mm. while all parameters remain the same. In consequence, \(\sigma_{\max}\) decreases from 217.2 MPa to 192.7 MPa. In Figure 15d, the six mounting holes are moved from the symmetry axes oriented towards the PM apex (as in Figure 15a) to positions oriented towards the middle of the PM facets. In consequence, for the configuration in Figure 15d, \(\sigma_{\max}\) = 207.7 MPa, lower than \(\sigma_{\max}\) for the configuration in Figure 15a.