# Dynamic Measuring Method of Laser Beam Incident Angle for Laser Doppler Vibrometer
## 5 Error Analysis and Discussion
### Laser Spot Position Error  
Due to the untouchability of the laser beam, there may be errors in accurately locating the laser spot position, leading to the failure of dynamic measurement. According to the characteristics of the probability distribution, laser spot P\({}^{\prime}\), which deviates from the ideal position P, will be concentrated within a circle centered on point P with a radius of \(R_{0}\), as shown in Figure 9a,b. Assuming the distance from the actual laser spot P\({}^{\prime}\) to the rotation center O of the rotating disk is \(R\) and \(\angle\)POP\({}^{\prime}=\zeta\), the following constraint relationships hold. \[\left\{ \begin{array}{l}\mathrm{abs}(\xi)\leq\arctan(R_{0}/r)\\ r-R_{0}\leq R\leq r+R_{0}\end{array} \right. \tag{10}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \(\omega\) **(rpm)** & **(mm/s)** & \(\beta\) (\({}^{\circ}\)) \\ \hline
1000 & 162.632 & 1.8736 \\
1500 & 243.768 & 1.8722 \\
2000 & 325.035 & 1.8723 \\
2500 & 406.169 & 1.8717 \\
3000 & 487.117 & 1.8705 \\
3500 & 568.201 & 1.8703 \\
4000 & 649.145 & 1.8696 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dynamic measurement results of \(\overline{V_{\rm m}}\) and \(\beta\). The tangential velocity at the expected measurement point P is denoted as \(V_{\rm T}\). The tangential velocity provided at the actual measurement point P\({}^{\prime}\) and the actual measurement velocity of LDV are recorded as \(V_{\rm TR}\) and \(V_{\rm mR}\), respectively. In Figure 9c, the lines P\({}^{\prime}\)Q and P\({}^{\prime}\)M are in the same direction as the velocities \(V_{\rm T}\) and V\({}_{\rm TR}\), respectively, P\({}^{\prime}\)N is in the same direction as the actual laser beam, and NQ and MQ are both perpendicular to P\({}^{\prime}\)Q. The following relationships exist in triangles P\({}^{\prime}\)NQ, P\({}^{\prime}\)MQ, and NQM. \[\left\{ \begin{array}{l}\overline{\rm{NQ}}=\overline{\rm{P}^{\prime}Q}/\tan( \beta)\\ \overline{\rm{P}^{\prime}N}=\overline{\rm{P}^{\prime}Q}/\sin(\beta)\end{array} \right. \tag{11}\]  
\[\left\{ \begin{array}{l}\overline{\rm{MQ}}=\overline{\rm{P}^{\prime}Q}\cdot \tan(\xi)\\ \overline{\rm{P}^{\prime}M}=\overline{\rm{P}^{\prime}Q}/\cos(\xi)\end{array} \right. \tag{12}\]  
\[\overline{\rm{NM}}=\sqrt{\overline{\rm{NQ}}^{2}+\overline{\rm{MQ}}^{2}} \tag{13}\]  
Then, in the triangle NP\({}^{\prime}\)M,  
\[\cos(\angle N{\rm{P}^{\prime}M})=\frac{\overline{\rm{P}^{\prime}M}^{2}+ \overline{\rm{P}^{\prime}N}^{2}-\overline{\rm{NM}}^{2}}{2\overline{\rm{P}^{ \prime}M}\cdot\overline{\rm{P}^{\prime}N}}=\sin\beta\cdot\cos\xi \tag{14}\]  
The actual measured value \(V_{\rm mR}\) of LDV in this state is expressed by Equation (15). \[V_{\rm mR}=V_{\rm TR}\cos(\angle N{\rm{P}^{\prime}M})=\omega R\cdot\sin\beta \cdot\cos\xi \tag{15}\]  
Combining the physical meaning of Equation (7), the actual angle \(\beta_{\rm mR}\) is shown in Equation (16). \[\beta_{\rm mR}{=\arcsin(V_{\rm mR}/V_{\rm T})}=\arcsin(\sin(\beta)\cdot\cos( \xi)\cdot R/r) \tag{16}\]  
According to the monotonicity of trigonometric function and inverse trigonometric function in Equation (16): (a) when \(R=r-R_{0}\) and \(\xi=\pm R_{0}/r\), \(\beta_{\rm mR}\) takes the minimum value of \(\beta_{\rm mR1}\); (b) when \(R=r+R_{0}\) and \(\xi=0\), \(\beta_{\rm mR}\) takes the maximum value of \(\beta_{\rm mR2}\). The calculation process is shown in Equation (17). \[\left\{ \begin{array}{l}\beta_{\rm mR1}=\arcsin(\sin(\beta)\cdot\cos(R_{0}/ r)\cdot(1-R_{0}/r))\\ \beta_{\rm mR2}=\arcsin(\sin(\beta)\cdot(1+R_{0}/r))\end{array} \right. \tag{17}\]  
Define \(\lambda\) represents the position deviation ratio of P\({}^{\prime}\), and \(\lambda=R_{0}/r\), and the value range of \(\lambda\) is [0, 1]. Due to \(\cos(\lambda)\leq 1\), \(1-\lambda\leq 1\), and \(1+\lambda\geq 1\), when analyzing dynamic measuring errors, only considering \(\beta_{\rm mR2}\) is sufficient. Based on the above analysis, the maximum measurement error \(\Delta\beta_{\rm mR}\), caused by laser spot position error, is represented by Equation (18).