# Practical Estimation of Machine Tool Spindle Dynamics for Maintenance Decision Making
## 4 Discussion
### Oscillations Observed in the IG Force Profile  
The simple mechanical design of the IG has many benefits, which were explored in Section 2.2. However, as was observed from the force plate measurements, as shown in Figure 8, the IG displayed some undesirable characteristics related to the selection of the springs and impact mass in the device. The force plate measurements collected from all 50 validation trials with the IG are shown in Figure 25a, while the average of these measurements is shown in Figure 25b. In the force plot, there were two features that deviated from the Gaussian impulse, which was expected from the IG: (1) a high-frequency oscillation observed near the initial impulse and (2) a low-frequency series of undulations that rippled after the initial impulse. The spectrum of the input force for the hammer signal, the IG collected from the force plate, and the Gaussian-fitted impulse from the IG were plotted, as shown in Figure 26, for the frequency analysis. In this plot, the high-frequency content from the IG signal was located near 5000 Hz, beyond the range of the results discussed in this study. The low-frequency undulations appeared as a broad band of spectral content in the range from 0 to 1000 Hz. The fitted impulse does capture the overall shape of the impulse resulting from the IG, which is supported by the agreement of the FRF plots observed in Figure 13. Judicious selection of the spring and mass elements inside the IG is expected to reduce the presence of these low-frequency effects, while the selection of materials is expected to be a valid method to dampen the high-frequency oscillations. Figure 24: Detailed view of the imaginary component of the FRF for each of the directly measured and Gaussian-fitted static spindle test conditions. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \multirow{2}{*}{**Test Condition**} & \multicolumn{4}{c}{**Mode 1**} & \multicolumn{4}{c}{**Mode 2**} & \multicolumn{4}{c}{**Mode 3**} \\ \cline{2-11}  & \(\mathbf{\omega_{1}}\) & \(\mathbf{m_{1}}\) & \(\mathbf{k_{1}}\) & \(\mathbf{\zeta_{1}}\) & \(\mathbf{\omega_{2}}\) & \(\mathbf{m_{2}}\) & \(\mathbf{k_{2}}\)  
The design of the IG was refined to enhance the force profile of the device and make it match more closely with the expected Gaussian impulse. Three adjustments were made to the design. First, the tolerance between the housing and the plunger of the impact tip was changed from a free-running clearance fit to a tighter locational clearance fit. This tolerancing change allowed for the free movement of the plunger without any perceivable play in the device, which could cause undesired undulations from excessive looseness. Second, a longer spring was selected to ensure that the impacting mass never came out of contact with the spring. The loss of contact caused an abrupt change in dynamics of the spring-mass system, which should be avoided. With the longer spring, there was always a small amount of preload stored in the system to avoid this problem. Third, viscous oil was added inside the housing of the IG to dampen vibrations of the mass after the initial impact. The validation results following the procedures were repeated following the same procedures as defined in Section 2.1.1 for the refined IG. The new input force magnitude for 50 trials is shown in Figure 27a, and the average results of the 50 trials are shown in Figure 27b. The average magnitude of the impulse increased from 360 to 410 N due to the increased preload from the longer spring. The undulations following the initial impulse were observed to be less significant in the redesigned IG. Figure 26: Spectrum comparison of the input force for the conventional hammer, the IG force plate measurements, and the Gaussian impulse fitted to the force plate measurements. Figure 25: Input force magnitudes recorded on the force plate from the mechanical IG: (**a**) all 50 impact trials; (**b**) average result of 50 trials. Each colored line in the chart depicts an individual impact trial.