## 3 Modeling and Simulations of the Assembly Error Effect on the Stability Accuracy
### Modeling and Simulation of the Assembly Error Effects on Stability Accuracy
#### 3.5.2 Modeling and Simulation of Assembly Errors of Run-Outs  
As mentioned, the instantaneous rotation axes are taken as the rotation center lines for error analysis of perpendicularity, rather than the average one. However, in practice, due to bad assembly of the bearings, the rotation center lines can be dynamically unsteady under working state. The maximum deviations between the instantaneous rotation axes and the average ones are defined as the assembly errors of run-outs. Specifically, run-outs include axial shifts \(\Delta s\), radial run-outs \(\overset{\rightarrow}{\Delta}\overset{\rightarrow}{R}_{o}\), and tilting oscillations \(\Delta\overset{\rightarrow}{\gamma}\) (\(\overset{\rightarrow}{\Delta}\overset{\rightarrow}{R}_{o}\) and \(\Delta\overset{\rightarrow}{\gamma}\) can be decomposed along two axes of the coordinate [21]). Taking the azimuth for example, the assembly errors of run-outs are shown in Figure 11. Apparently, displacement errors such as \(\Delta s\) and \(\overset{\rightarrow}{\Delta}\overset{\rightarrow}{R}_{o}\) had no effect on the stability accuracy, and angular tilting oscillations \(\Delta\overset{\rightarrow}{\gamma}\) are what this research focuses on. Figure 10: The signals affected by errors of axis perpendicularities. Figure 9: The axis perpendicularity deviations. The oscillation vector of the azimuth is expressed as \(\triangle\overrightarrow{\gamma}_{a}=[\triangle\alpha_{am},\triangle\beta_{am},0]^{T}\), and that of the pitch is expressed as \(\triangle\overrightarrow{\gamma}_{p}=\left[\triangle\alpha_{pm},0,\triangle \beta_{pm}\right]^{T}\). \(\triangle\alpha_{am}\), \(\triangle\beta_{pm}\), \(\triangle\alpha_{pm}\) are the decomposed components of the maximum tilting oscillations along the coordinates \(\{b\}\) and \(\{a\}\), and they can also be measured [22]. From the perspective of dynamic signal processing, the tilting oscillations are periodic signals, with the angular frequencies of \(\dot{\theta}_{a}\) and \(\theta_{p}\). Taking \(\triangle\alpha_{am}\) for example, its real-time value expression \(\triangle\alpha_{a}(t)\) is a periodic signal with angular frequency of \(\dot{\theta}_{a}\) and period of \(T_{a}=2\pi/\dot{\theta}_{a}\), then \(\triangle\alpha_{am}\) can be expressed as Equation (17) by using Fourier series:  
\[\left\{ \begin{array}{rl}\triangle\alpha_{a}(t)&=\frac{\triangle\alpha_{a0}}{ 2}+\sum_{n=1}^{\infty}\bigl{(}a_{n}\bigl{(}n\dot{\theta}_{a}t\bigr{)}+a_{n} \bigl{(}n\dot{\theta}_{a}t\bigr{)}\bigr{)}\\ &a_{n}=\frac{2}{T_{a}}\int_{-\frac{1}{2}}^{\frac{T_{a}}{2}}\triangle\alpha_{a} (t)c\bigl{(}n\dot{\theta}_{a}t\bigr{)}dt\quad,(n=0,1,2,\ldots),\\ &b_{n}=\frac{2}{T_{a}}\int_{-\frac{1}{2}}^{\frac{T_{a}}{2}}\triangle\alpha_{a} (t)s\bigl{(}n\dot{\theta}_{a}t\bigr{)}dt\end{array} \right.,(n=0,1,2,\ldots), \tag{17}\]  
For simplicity, \(\triangle\alpha_{a}(t)\) is taken as an odd function, and high-frequency signals with \(n>1\) are ignored. Then \(\triangle\alpha_{a}(t)\) can be expressed as Equation (12), and \(\triangle\alpha_{am}\) is considered to be the amplitude of this signal. Similarly, other tilting oscillations are also shown in Equation (18). \[\left\{ \begin{array}{rl}\triangle\alpha_{a}(t)\approx\frac{2}{T_{a}}\int_{- \frac{1}{2}}^{\frac{T_{a}}{2}}\triangle\alpha_{a}(t)s\bigl{(}\dot{\theta}_{a}t \bigr{)}dt\cdot s\bigl{(}\dot{\theta}_{a}t\bigr{)}=\triangle\alpha_{am}s( \theta_{a})\\ \triangle\beta_{a}(t)=\triangle\beta_{am}s(\theta_{a})\\ \triangle\alpha_{p}(t)=\triangle\alpha_{pm}\beta_{pm}\end{array} \right., \tag{18}\]  
Based on the kinematic analysis of axis perpendicularity errors earlier, the rotation transformation matrixes caused by run-outs \(\triangle R_{a}\) and \(\triangle R_{p}\) can be expressed as Equation (19):  
\[