# Numerical Analysis of the Kinematic Accuracy of the Hermetic Harmonic Drive in Space Vehicles
## 2 Impact of Type of Generator on Flexspline Deformation Form
On the major axis of the flexspline, the teeth are fully in mating, while in the minor axis of the generator, the teeth of the two gears pass each other, as shown in Figure 5.  
Point \(A\) on the tip of the flexspline teeth determines a trajectory with a shape depending on the type of generator, parameters of the rim of the toothed wheel rim, and which  
Figure 5: Position of teeth in moment \(t=0\): \(y\), \(z\)—the coordinate system in which the relative movement of the tooth is considered, R—radius of the neutral layer of not deformed flexspline, \(r_{a1,r2}\)—tip radii, \(r_{f1}\),\(r_{f2}\)—root radii. of the gears is stationary. Determining the relative path of the teeth takes place in the \(y\),\(z\) coordinate system associated with a fixed circular spine. In the situation described above, the beginning of this system (0) is at the intersection of the undeformed layer of the flexspline with radius \(R\) with a minor axis of the generator. To define the location unambiguously, the tooth in this coordinate system should be determined by the position of its two characteristic points. In an undistorted flexspline, these points are \(A_{0}\)--at the intersection of the tooth axis with the tip circle and \(F_{0}\)--at the intersection of the tooth axis with the root circle. The coordinates of the points \(A_{2}\) and \(F_{2}\) of the teeth of the circular spine in the \(y\),\(z\) coordinates system on the \(z\)-axis are presented in Formulas (1) and (2) [22; 35; 36]:  
\[w_{a2}=r_{a2}-R, \tag{1}\]  
\[w_{f2}=r_{f2}-R, \tag{2}\]  
The coordinates of \(v_{a2}\) and \(v_{f2}\) are equal zero because these points laying on \(z\) axis (3) and (4):  
\[v_{a2}=0, \tag{3}\]  
\[v_{f2}=0, \tag{4}\]  
The position of the tooth of flexspline in the \(y\), \(z\) coordinate system is variable and depends on the angle \(\varphi\) of the generator rotation. At the initial moment \(t=0\), the major axis of the generator is parallel to \(y\)-axis, and after time \(t>0\) will turn about angle \(\omega_{g}t\). The current location of the major axis of the generator relative to the starting position is determined by angle \(\varphi\) (5) [22; 34; 35]:  
\[\varphi=\frac{\pi}{2}-\omega_{g}t, \tag{5}\]  
where: \(\omega_{g}\)--angular velocity of the wave generator, \(t\)--time. Depending on the position of the major axis of the generator, the angle of the rotation of flexspline \(\varphi_{2}\) has values according to the dependency [22]:  
\[\varphi_{2}=\frac{\left(\frac{\pi}{2}-\varphi\right)}{i}, \tag{6}\]  
where: \(i\)--gear ratio. The coordinates of the points \(A_{1}\) and \(F_{1}\) of the adequate flexspline tooth on the \(z\)-axis (\(w_{a1}\), \(w_{f1}\)) and on the \(y\)-axis (\(v_{a1}\), \(v_{f1}\)) are the same as in Formulas (7)-(10) [22; 34; 35]:  
Figure 4:0  
Figure 4:1  
Figure 4:2  
Figure 4:3  
where: \(\theta\)--the angle of rotation of the normal exposure to the neutral layer of the flexspline by angle \(\varphi\), \(w\)--radial displacement of the neutral layer point of flexspline, \(v\)--circumferential displacement of the neutral layer point of the flexspline. Figure 6 shows the trajectory of the \(A_{1}\) point of a single tooth lying on the intersection of the axis of symmetry with the tip radius of the flexspline. The displacement of this point is determined in the coordinate system \(y\),\(z\) and combined with a fixed circular spline and calculated based on dependences (7) and (9) for the values of the rotation angle of generator in range \(0\leq\varphi\leq\
icefrac{{11}}{{2}}\). In the flexspline initial moment (\(\varphi=0^{\circ}\)), the tooth enters maximally in the space of the circular spline. If that occurs, then the full penetration \(h_{iw}\) is predicted for the harmonic drive.