# Numerical Analysis of the Kinematic Accuracy of the Hermetic Harmonic Drive in Space Vehicles
## 3 Preparation of Numerical Calculations  
The issues related to the cooperation of gears in the harmonic drive are complicated and require a large number of operations. If the calculations are made in an analytical method, it is necessary to transform and solve many geometrical dependencies. The results obtained this way, through the simplifications contained therein, do not provide solutions equivalent to the real ones. Also useful are any numerical tools and computer programs that can improve this calculation process. A good universal calculation tool  
Figure 6: The trajectory of A\({}_{1}\) point of the tooth of flexspline. that supports engineering design is the finite element method (FEM). There are a lot of computer programs based on the numerical FEM method, which definitely facilitates calculations concerning harmonic drives. Many of them are integrated with programs for computer-aided design (CAD). Moreover, the possibilities of computer modeling and numerical calculations were used during the preparation and calculation of the hermetic harmonic drive. The considered issue concerning the influence of the generator type on the kinematic accuracy of the hermetic harmonic drive is a very complicated task. In the numerical calculation, it is necessary to include three cooperating elements of the harmonic drive, namely flexspline, circular spline, and wave generator. Moreover, it is necessary to define the surface of contact between pairs of these elements and their continuous control because conditions of contact are changed at different stages of calculation. The next problem is the elastic deformation of the flexspline, which should be defined correctly. A much more complicated problem is the forced rotation of the circular spline relative to the flexspline. This motion of the circular spline is the result of the deformation of the flexspline by the wave generator with simultaneous rotation of the generator about the axis of the gear. Analysis of the accuracy of the cooperation of gears in the harmonic drive requires a precise definition of the boundary conditions for the rotation of the wave generator. Numerical analysis FEM using full model hermetic harmonic drives would require large memory operation resources of the working station and would take a very long time. However, for special causes, it is necessary to find the exact way to solve construction tasks. For the vast majority of numerical calculations, it is enough to use only simplified models. The calculation models can be fragments of real components or have changes in geometry to simplify their build. In many analyses, gears can be successfully used with two-dimensional models being their cross sections. Such a simplification is also possible in the case of the analyzed hermetic harmonic drive. The flexspline had a toothed wheel rim in the middle of its length and straight teeth with a constant contour. The dimension of the width of the toothed wheel rim compared to the height and width of a single tooth is large. Moreover, it is necessary to mention that all cross-sections of a single tooth have the same profile and are between non-movable end walls, which are impossible to relocate toward the main axis of the gear. In addition, it was assumed that the load of the circular spline is evenly distributed across the width of the toothed flexspline wheel rim. With such assumptions, each single tooth cross-section is under the same conditions and is equally loaded. If the assumptions presented are met, it can simplify the spatial problem of the wheel rim to the two-dimensional problem in a plane state of strain. Since each cross-section is the same, when carrying out numerical analyses, it is enough to consider only the middle layer of the toothed wheel rim with a unit thickness, i.e., an adequate cross-section of the flexspline passing through the center of the toothed wheel rim. In this way, two-dimensional models for numerical calculation FEM realized in the Abaqus program were made.