# Optomechanical Analysis and Design of Polygon Mirror-Based Laser Scanners
## 5 Finite Element Analysis (FEA) of PMs  
In the context of the high rotational velocities \(\omega\) that are necessary to obtain a scan frequency \(f_{\rm s}\) of PMs as high as possible, as required by most applications, an FEA must be performed to determine the amount of mechanical stress produced by centrifugal forces within the PM. While such an FEA is specific to each design and set of materials of the PM assembly (including shaft and other parts), the multi-parameter analysis in this section is made for PMs with a common configuration. Several constructive parameters of the PMsare considered for an \(\omega\) of up to 120 krpm (but also for lower, common values of \(\omega\) of up to 60 krpm). An Al alloy 5052 is considered for the PM material, with a tensile yield strength of 214 MPa and an ultimate tensile strength of 268 MPa. While such a material is the most common and appropriate one to be utilized, steel and plastic could be employed for small values of \(\omega\). However, if Al alloys fail the FEA, beryllium alloys must be considered for high values of \(\omega\); however, they are expensive and toxic during manufacturing. The main scope of an FEA for a PM, as addressed in the following, is to verify its structural integrity. A secondary scope would be to determine the level of deformations, which can be the start of another analysis, considering manufacturing and tolerances, as well. Such an analysis is the subject of future work, as in this study, we consider no manufacturing or mounting errors. This is a necessary simplifying hypothesis to carry out the proposed analysis. Another direction of future work is the validation of such an FEA by experiments--to be carried out for a number of materials, dimensions, and PM designs. The main parameters of the PM for the FEA are the number of facets \(n\), the apother \(R\), the PM thickness \(b\), and the rotational velocity \(\omega\)--Figure 2a. However, several other constructive parameters must be considered as well (Figure 11). They include the radius \(r\) of the central hole (to mount the PM shaft), the number \(n_{\text{h}}\) of supplementary (identical) holes (to mount the PM on a rotational disk, as usually utilized in such an assembly), and each of these holes has a \(\delta\) diameter and placed with its center on a circle of \(R_{\text{h}}\) radius. The number \(n_{\text{h}}\) of these supplementary holes must be chosen without affecting the structural integrity of the PM, i.e., the effective section at the radius \(R_{\text{h}}\). They must be considered in the direction of the apex of the PM to obtain a symmetrical deformation of the facets. When this is not possible because there are too many facets, a limited number of holes (three to six) should be used [1]. In the following FEA, a quarter of each PM plate is considered, as shown in Figure 11. The analysis is carried out using ABAQUS (Dassault Systemes, Paris, France). In Figure 12, three PMs are considered, with \(n\) equal to four, six, and eight facets (on each of the three rows, respectively), for four steps of the rotational velocity \(\omega\): (a) 60 krpm, (b) 80 krpm, (c) 100 krpm, and (d) 120 krpm, on each of the four columns, respectively. The PM parameters are \(R\) = 25 mm, \(r\) = 5 mm, \(b\) = 10 mm, \(R_{\text{h}}\) = 13 mm, and \(\delta\) = 2 mm. The number of mounting holes is \(n_{\text{h}}=n\) in each case, and they are placed in the direction of the PM apex. The color code for the mechanical stress \(\sigma\), preserved in this entire FEA, is presented in Figure 12(a): The maximum value (red) was set at 200 MPa, bellow the tensile strength of the material (218 MPa), as a safety margin.