# Practical Estimation of Machine Tool Spindle Dynamics for Maintenance Decision Making
## 3 Results  
### Validation of the Mechanical Impulse Generator  
The results of 50 tap trials for the conventional impact hammer are shown in Figure 4a, while the corresponding results collected from the force plate dynamometer are shown in Figure 4b. The force hammer produced a range of impulse magnitudes. The lowest recorded peak force was 175 N, while the largest was 414 N. This spread was due to the manual operation of the hammer. The force plate signal displayed a clear oscillation after the initial impulse due to the system dynamics, which are unrelated to the IG. To reduce the impact of these unrelated dynamics, the transfer function between the hammer and force plate was calculated using the FRF methods described in Section 2.4. Figure 5a shows the magnitude plot of the transfer function, while Figure 5b shows the coherence plot calculated for the 50 trials, which displayed values near unity up to 5000 Hz. The transfer function displayed a characteristic peak at 3538 Hz, which corresponded to the period of oscillations of 0.28 ms observed in Figure 4b. Figure 3: Pedagogical diagram of the FRF for a single degree-of-freedom system with important features for the modal characterization indicated: (**a**) real component with indicated features \(\omega_{n1}\) and \(\omega_{n2}\); (**b**) imaginary component with indicated features \(\omega_{n}\) and \(A\). The force plate response from 50 IG trials was collected, and the transfer function in Figure 5 was used to reduce the effect of the force plate dynamics. Figure 8 shows the measured impulse for a single trial of the IG. Some undesirable oscillation was observed in the measured force signal beyond the initial impulse; however, these effects were lessened through averaging over 50 trials. Further discussion of these oscillations is found in Section 4.1 of this article. A smooth and symmetric curve was needed to model the physical characteristics of the impulse. Some authors have suggested a truncated sine to represent the impulse [19], which can be generated by convolving an appropriate cosine function with a square window. However, the square window truncation is known to introduce artifacts in the frequency domain after applying the Fourier transform. In this article, the impulse was modeled with the Gaussian function shown in Equation (10), since it preserves the smoothness and symmetrical properties of the truncated sine while also avoiding the frequency domain problems resulting from the truncation. \[g(t)=ae^{-(\frac{t-b}{c})^{2}}+d \tag{10}\]  
In Equation (10), the independent variable \(t\) is the time in seconds, and the constants \(a,b,c\), and \(d\) characterize the magnitude and duration of the impulse. Fitting was accomplished using the MATLAB Curve Fitting Toolbox 23.2. Figure 6a shows the IG results across 50 trials for the fitted Gaussian impulse and in Figure 6b for the direct force plate measurements. The IG was found to produce a reliable impulse at 360 N, which is comparable to the average hammer force of 300 N. The impulse duration for both devices was also found to be comparable. The average force plate response for 50 trials from the hammer and the IG are shown in Figure 7. Both time-series showed close agreement with one another. Figure 4: Measured force signal from 50 manual conventional hammer strikes: (**a**) hammer signal time-series; (**b**) force plate time-series. Each colored line in the chart depicts an individual impact trial. Figure 5: Transfer function for the force plate response: (**a**) magnitude plot; (**b**) coherence plot. Figure 9 shows a comparison of the impulses from the conventional hammer and the fitted impulses of the IG across 50 trials each. The shaded regions in Figure 9 indicate the  
Figure 8: Impulse magnitude measured by the force plate after applying the transfer function to reduce the effect of force plate dynamics. A single trial is plotted of the force generated using the IG. A Gaussian impulse was fitted to the force plate data to characterize each impulse.