## 4 The Dynamic Balance Experiment and Its Result Analysis  
The picture of the experimental system is shown as Figure 8. The motorized spindle is used to drive the rotating spindle with the dynamic balance mechanism to rotate synchronously. The sensors and signal acquisition system are used to collect the vibration signals, reference signal, and position signals. The pneumatic control system is used to drive the balance disks to reach the expected position. The software system is programmed by Labview language for signal acquisition, signal analysis, and output control. According to the principle and steps of influence coefficient method, the influence coefficient change condition is examined by a series of trial weight experiments. Step 1, Taken the balance speed of 4800 rpm as an example, the original vibration signals and reference signal are shown in Figure 9.  
Figure 8: Picture of the experimental system. Figure 9: The original vibration signals and reference signal of the spindle. Comparing with the reference signal, the unbalance vibration vectors of points 1 and 2 are separated from the testing vibration signals by the least square method [17]. The separating results of unbalance vibration vectors are listed as follows. \[\overset{\rightarrow}{V_{01}}=11.35\upmu\text{m}\angle-3.3^{\circ},\overset{ \rightarrow}{V_{02}}=26.74\upmu\text{m}\angle 47.4^{\circ}\]  
Step 2, The unbalance vector \(\overset{\rightarrow}{P_{1}}=400\text{g}\cdot\text{cm}\angle 0^{\circ}\) is exerted in correcting face 1, the spindle is run to the balance speed again, the testing signals of spindle are shown in Figure 10.  
The unbalance vibration signals separated from the signals in Figure 10 are listed as follows. \[\overset{\rightarrow}{V_{11}}=36.32\upmu\text{m}\angle-117.2^{\circ},\overset{ \rightarrow}{V_{12}}=69.66\upmu\text{m}\angle-135.1^{\circ}\]  
The influence coefficient of correcting face 1 relative to points 1 and 2 can be calculated as follows. \[\overset{\rightarrow}{R_{11}}=0.106\upmu\text{m}/(\text{g}\cdot\text{cm}) \angle-131.4^{\circ},\overset{\rightarrow}{R_{12}}=0.185\upmu\text{m}/(\text{ g}\cdot\text{cm})\angle-156.3^{\circ}\]  
Step 3, The unbalance vector \(\overset{\rightarrow}{P_{2}}=560\text{g}\cdot\text{cm}\angle 90^{\circ}\) is exerted in correcting face 2, the spindle is run to the balance speed again, and the testing signals of the spindle are shown in Figure 11.  
Figure 10: The original vibration signals and reference signal of the spindle after the first trial weight. The unbalance vibration signals separated from Figure 11 are listed as follows. Figure 9.0  
The influence coefficient of correcting face 2 relative to points 1 and 2 can be calculated as follows. Figure 9.1  
According to the above steps, the trial weight experiments are carried out once a month and the five times repeated experiment results are listed in Figure 9.2  
In Table 1, the correcting vectors are calculated according to the current influence coefficient, the residual unbalance vibration vectors are tested after correcting vectors are exerted. It can be seen from Table 1, affected by the factors of kinetic parameter variation of the spindle system, adjustment error, measurement, and signal analysis error etc, that the differences of these influence coefficients are obvious even at the same speed. The correcting vectors and residual unbalance vibration vectors are affected directly by the influence coefficient. Figure 9.3 The original vibration signals and reference signal of the spindle after the second trial weight. Based on the data in Table 1, the adaptive process of influence coefficient and the gain parameters are shown in Figure 9.4 For ease of calculation, the influence coefficient and gain parameters are expressed as the plural. In Table 2, the forgetting factors are set to be 0.5I\({}_{2\times 2}\), the weighting matrix H is set to be 8I\({}_{2\times 2}\), and weighting matrix \(Q\) set to be 8 OI\({}_{2\times 2}\).