# Numerical Analysis of the Kinematic Accuracy of the Hermetic Harmonic Drive in Space Vehicles
## 3 Preparation of Numerical Calculations
Figure 7 presents an ideological way to create a section of flexspline and generator in a CAD program. To improve the readability of the drawing, the circular spline was omitted, which was included in the calculations. In the case of the planar theory of elasticity, the two-dimensional models of the full harmonic drive are analyzed, therefore its calculation using the FEM method and the processing of results takes less time. It is caused by a much smaller number of finite elements needed to describe the model, and through it also a smaller number of formulas are necessary to solve the computing program. In view of device limitations, it is sometimes the only possibility for a precise computer analysis of harmonic drives or for confirming the mating of their components. For the needs of the realized task, the initial project of the hermetic harmonic drive was done by determining its geometric parameters. In addition, check calculations were performed that refer to the interference of the teeth of the mating wheels for each of the four types of generators. Then, on the basis of analytically obtained geometrical parameters, solid models of the main components of the harmonic drive were created. These models were created in the SolidWorks program by defining four assemblies for each of them with the other type of wave generator. On their basis, two-dimensional models were prepared, which are cross sections passing through the middle of the width of the toothed wheel rim of both wheels and the generator. The models prepared in this way were used as a basis for strength calculations using FEM, which were performed in the Abaqus program. A sample model of harmonic drive with a double-roller generator used in the calculations is shown in Figures 8 and 9. In the figure are visible all components of the gear and marks of some boundary conditions introduced during the creation of the computational model. Preparation of numerical calculations of kinematic accuracy started by loading models into the counting program, their settings, and linking them with each other. Then they were discretized using quadrangular finite elements with appropriate mesh density in particularly important areas. The increase in mesh density is defined on the teeth flanks of both gears and on the inner diameter of the flexspline and generator because these surfaces are in contact during operation. Boundary conditions were also defined for all models, as well as displacements and loads, in accordance with the real nature of the operation of the harmonic drive elements. Because the analysis refers to a hermetic harmonic drive in which the flexspline is attached to the casing, in the computational model the possibility of deformation was allowed but the possibility of rotation was blocked. The circular spline and the flexspline were associated with the point positioned on their axis (0) using rigid links. This point was fixed, but the possibility of rotating was allowed to realize the load of the harmonic drive. At this reference point, the angle of rotation of the circular spline was also read after rotating the wave generator about the determined angle \(\varphi\). Figure 7: Essence of creating a two-dimensional model as a cross-section for calculation. Figure 8: Numerical model of harmonic hermetic drive with double-roller generator: 1—flexspline, 2—circular spline, 3—double-roller generator. Figure 9: Characteristic area of computational model of harmonic drive: (**A**) meshed toothed wheel rims (major axis), (**B**) pass each other of teeth (minor axis). Calculations for each variant difference were only the type of generator used and were composed of three steps. In the initial step, a deformation of the flexspline occurred by a wave generator for the tested model. This stage was not taken in the analysis because it had only caused the wheels to mesh, as it is in a real harmonic drive. In the first stage, rotation of the wave generator about angle 1 rad in a positive direction was forced, and the load of harmonic drive by torque was realized.