# Dynamic Measuring Method of Laser Beam Incident Angle for Laser Doppler Vibrometer
## 5 Error Analysis and Discussion
### Rotating Disk System Error
#### 5.3.3 Experimental Verification  
To verify the influence of the roll angle \(\alpha_{y}\) and the pitch angle \(\alpha_{x}\) on the measurement results, we changed the relative angle between the LDV laser beam and the rotating disk by adjusting the angle fine-tuning platforms 1 and 2. However, adjusting the angle fine-tuning platforms will cause the actual position of the laser spot on the rotating disk to change. To ensure that the relative position of the light spot \(\mathrm{P}^{\prime}\) and the desired point P does not change significantly, we need to adjust the displacement platforms 1 and 2. No matter which platform is adjusted, the constraint state of the dynamic measuring system will be changed. Changes in system constraints may bring unpredictable measurement effects.  
By adjusting the angle fine-tuning platform 1 to change the roll angle of the rotating disk, two roll angles are obtained: \(\alpha_{y}-0.1^{\circ}\) and \(\alpha_{y}+0.1^{\circ}\). Adjusting the angle fine-tuning platform 1 essentially changes \(\theta_{y}\) in Figure 12. According to Equation (25), \(\theta_{y}\) will cause the LDV signal to exhibit periodicity. Start the motor and measurement system, and record the signals provided by the rotating disk to the LDV in these abnormal installation conditions. Note that during each adjustment, it is necessary to release and lock the constraints of the measuring device. Subsequently, restore the roll angle of the dynamic measuring device to its initial state. When the motor speed \(\omega\) is 4000 rpm, the LDV velocity signals in three roll angle states are shown in Figure 16.  
The figure shows that under these three roll angles, the LDV measurement values all exhibit periodicity, with lower fluctuations in the initial state (red solid line in the figure), which is consistent with our expectations. For roll angles \(\alpha_{y}-0.1^{\circ}\) and \(\alpha_{y}+0.1^{\circ}\), their  
Figure 16: Velocity curves corresponding to different \(\alpha_{y}\).  
variation relative to the initial roll angle \(\alpha_{y}\) is 0.1\({}^{\circ}\). In theory, the amplitude fluctuation of two curves should be the same except for the phase. However, the actual roll angle increment may not have reached 0.1\({}^{\circ}\) due to system clearance. According to the average value of the last two curves, the incident angle \(\beta\) of the LDV laser beam is calculated to be 1.86963\({}^{\circ}\) and 1.86983\({}^{\circ}\), respectively. These two results are not much different from the 1.8695\({}^{\circ}\) corresponding to \(\alpha_{y}\) in Table 2. Therefore, it can be considered that the roll angle \(\alpha_{y}\) has little influence on the proposed dynamic measuring device.  
Adjusting the angle fine-tuning platform 2 will change the relative angle between the LDV beam and the rotating disk. This will result in amplitude changes in the LDV measurements without significantly changing the periodic fluctuation of the signal. Two pitch angles are obtained by adjusting the angle fine-tuning platform 2: \(\alpha_{x}-0.1^{\circ}\) and \(\alpha_{x}+0.1^{\circ}\). The velocity signals picked up by LDV are shown in Figure 17 when the motor speed is 4000 rpm.  
The figure shows that after changing the pitch angle \(\alpha_{x}\), the two newly obtained LDV curves significantly deviate from the initial one, but do not exhibit periodic fluctuations. According to the average value of the latter two curves, the incident angle \(\beta\) of the LDV laser beam is calculated to be 1.79751\({}^{\circ}\) and 1.95316\({}^{\circ}\), respectively. Compared with 1.86959\({}^{\circ}\) in the initial state, they are reduced by 0.07208\({}^{\circ}\) and increased by 0.08357\({}^{\circ}\), respectively. Both differences are less than the angle fine adjustment increment of 0.1\({}^{\circ}\). This is likely related to the system clearance and the position error of the laser spot, but it also indicates that \(\alpha_{x}\) has a significant impact on the dynamic measuring device.