# Dynamic Measuring Method of Laser Beam Incident Angle for Laser Doppler Vibrometer
## 5 Error Analysis and Discussion
### Rotating Disk System Error
#### 5.3.1 Roll Angle \(\alpha_{y}\)  
When \(\theta_{y}=\theta\) and \(\varphi_{y}=\varphi\), the roll angle \(\alpha_{y}\) may reach a maximum value, which will have a more significant impact on the measured value of LDV. In this case, the axes Y, Y\({}_{1}\), and Y\({}_{2}\) are coincident, and the axes Z, Z\({}_{1}\), and Z\({}_{2}\) are coplanar. See Figure 12 for the positional relationship between the rotating disk and the LDV laser beam. Since Z and Z\({}_{1}\) do not coincide, the rotating disk will periodically 'vibrate' around the Z-axis. From the perspective of \(-\)Y\({}_{2}\), the rotating disk is reciprocating, as shown in Figure 13. Assume that the positions indicated by \(t_{1}\) and \(t_{2}\) are the upper and lower limits of the rotating disk, where \(t_{2}-t_{1}=(n+0.5)\cdot T\), \(n=1,2,3,\ldots\), Parameter \(T\) represents the rotation period of the rotating disk, \(T=2\pi/\omega\). Without a loss of generality, let \(n\) equal 0, then \(t_{2}-t_{1}=0.5\cdot T\). Its physical meaning means starting from time \(t_{1}\), after \(0.5\cdot T\), the normal line Z\({}_{1}\) of the rotating disk rotates to Z\({}_{1}\)\({}^{\prime}\). For the roll angle \(\alpha_{y}\), it is a variable in this process. At two moments, \(t_{1}\) and \(t_{2}\), \(\alpha_{y}\) reaches the minimum and maximum values \(\alpha_{y1}\) and \(\alpha_{y2}\), respectively. In the following, \(\alpha_{yqt}\) will be used to represent the roll angle \(\alpha_{y}\) of the rotating disk at any time \(t\). Figure 12: When the roll angle is the largest, the positional relationship between the rotating disk and the LDV laser beam are in the X\({}_{2}\) and \(-\)Y\({}_{2}\) directions, respectively. Figure 13: Schematic diagram of the reciprocating motion of the rotating disk. Ideally, the laser spot on the rotating disk should be at point P, with the corresponding rotation radius of \(r\). However, due to the reciprocating motion of the rotating disk, the actual laser spot P\({}^{\prime}\) will fluctuate periodically within the range of points B and A. This causes the distance \(r_{t}\) from the laser spot P\({}^{\prime}\) to the center O to change at any time \(t\), and its range is [\(r_{t1}\), \(r_{t2}\)]. The radius \(r_{ti}\) represents the distance from the actual laser spot P\({}^{\prime}\) to the center \(O\) at time \(t_{i}\), \(i=1\), \(2\), and the calculation method is shown in Equation (19). \[\left\{ \begin{array}{l}r_{t}=r/\cos(\alpha_{yt})\\ r_{t1}=r/\cos(\alpha_{yt1})\\ r_{t2}=r/\cos(\alpha_{yt2})\end{array} \right. \tag{19}\]  
where \(\alpha_{yt1}=\theta\), \(\alpha_{yt2}=\theta+2\varphi\), and \(\alpha_{yt}=(\alpha_{yt2}+\alpha_{yt1})\cdot\sin(\omega t)/2\). The circumferential velocity \(V_{t}\) provided by point P\({}^{\prime}\) at time \(t\) can be expressed as \(V_{t}=\omega\cdot r_{t}\), then the first velocity component \(V_{\rm m1}\) picked up by LDV can be calculated by Equation (20). \[V_{\rm m1}=V_{t}\cdot\sin(\beta)=\frac{\omega r\cdot\sin(\beta)}{\cos((\theta+ \varphi)\sin(\omega t))} \tag{20}\]  
In addition, the roll angle \(\alpha_{yt}\) causes the rotating disk to move periodically in space with \(T\), affecting the LDV measurement. Viewed from the \(-\)Y\({}_{2}\) direction, the laser spot P\({}^{\prime}\) reciprocates between points A and B, and its amplitude \(A_{\rm m}\) can be calculated according to Equation (21). \[A_{\rm m}=(s_{t2}-s_{t1})/2=(h_{t2}-h_{t1})/(2\cos(\beta)) \tag{21}\]  
where  
\[\left\{ \begin{array}{l}h_{t2}=r\cdot\tan(\alpha_{yt2})\\ h_{t1}=r\cdot\tan(\alpha_{yt1})\end{array} \right.\]  
The displacement introduced by the reciprocating motion of the rotating disk can be expressed by Equation (22). \[S(t)=A_{\rm m}\sin(\omega t+\psi) \tag{22}\]  
Without a loss of generality, let the initial phase \(\psi=0\). Based on the differential relationship between velocity and displacement, the velocity component \(V_{\rm m2}\) generated by reciprocating motion is shown in Equation (23).