# Practical Estimation of Machine Tool Spindle Dynamics for Maintenance Decision Making
## 3 Results
### Dynamic Spindle Test  
The dynamic spindle test presented additional challenges due to the vibrations produced by the rotating spindle. The \(x\) axis vibration time-series of a single trial of the 0 RPM and 10,000 RPM test immediately at impact from the IG are shown in Figure 16a,b for comparison. At high speed, a harmonic oscillation corresponding to the spindle rotational speed was observed in the acceleration signal. After the vibration response from the IG decayed, the harmonic component remained in the acceleration signal, as observed in Figure 16c,d, which displays a detailed view of the same signals 0.1 s later. Averaging the acceleration signal from 50 trials had a significant reduction on this noise, while retaining the characteristic response from the input of the IG. The averaging approach is illustrated in Figure 17 for the 10,000 RPM trial. Figure 18 shows a comparison of the average accelerometer signal collected from all 50 impact tests performed at each of the spindle speeds. The results are similar to the static spindle tests; however, the magnitude of the vibration response tended to decrease as the spindle speed increased. A discussion of the spindle speed effect on the vibration dynamics can be found in Section 4.5. The FRF was calculated for each trial for an in-depth investigation. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Mode 1**} & \multicolumn{4}{c}{**Mode 2**} & \multicolumn{4}{c}{**Mode 3**} \\ \cline{2-13}
**Test Condition** & \(\omega_{1}\) & \(m_{1}\) & \(k_{1}\) & \(\zeta_{1}\) & \(\omega_{2}\) & \(m_{2}\) & \(k_{2}\) & \(\zeta_{2}\) & \(\omega_{3}\) & \(m_{3}\) & \(k_{3}\) & \(\zeta_{3}\) \\  & **Hz** & **kg** & **Ns2/m** & **\%** & **Hz** & **kg** & **Ns2/m** & **\%** & **Hz** & **kg** & **Ns2/m** & **\%** \\ \hline Condition 1 & 179 & \(3.9\times 10^{-3}\) & 124 & 2.8 & 189 & \(1.3\times 10^{-2}\) & 448 & 1.3 & 231 & \(2.3\times 10^{-3}\  
The magnitude plot for the dynamic FRFs is shown in Figure 19. Similar to the static tests, close agreement was observed between the FRFs in the range up to 1000 Hz. One notable exception is the spike in magnitude at 167 Hz observed in the 10,000 RPM case corresponding to the spindle speed. The FRF magnitude tended to decrease in the range beyond 1000 Hz, with the exception of the 1000 RPM trial near 1100 Hz. At 1080 Hz, the 1000 RPM trial was the only speed at which the magnitude of this peak exceeded the static trial. Figure 16: Comparison of the acceleration response of a single trial for the static and dynamic spindles: (**a**) during the impulse from the IG at 0 RPM; (**b**) during the impulse from the IG at 10,000 RPM; (**c**) after the vibration decay at 0 RPM; (**d**) after the vibration decay at 10,000 RPM. Figure 17: Illustration of the averaging method for noise reduction at 10,000 RPM: (**a**) acceleration response signal for 50 trials during an impulse from the IG; (**b**) average of 50 trials, which retains the overall shape of the responses; (**c**) acceleration response signal for 50 trials; (**d**) average of 50 trials showing a reduced signal noise. Each colored line in the chart depicts an individual impact trial, while the pink color is reserved for the averaged value. The real and imaginary components of the FRF for each speed were considered in the range for characterization of the modes in this region. Figure 20 shows the real component for the FRF in the dynamic spindle tests, while Figure 21 shows the imaginary components. The modal characteristics are listed in Table 2 for each spindle speed.