## 3 Results and Discussion
### The Dynamic Response of the Masses  
To check the calibrated measurement by mass cancellation and the determined function, \(HI_{pp}(f)\) is reapplied to the already measured masses under vibration. \(AM_{testobj. }(f)\) is calculated by Equation (13), the results are shown in Figure 8. As in Figure 5 the ideal values of the measured masses (Section 2.5) are shown with dashed lines and the data points of each test bench are marked blue and orange. The deviations from the ideal expected behavior of a real mass in Figure 8 are much smaller than the uncorrected ones shown in Figure 5. The better representation of the magnitude of AM is mainly due to the elimination of the mass \(m_{sensor}\) (Equation (13)). The smaller deviation from the ideal behavior of the phase of AM is due to the determination of the _measurement systems FRF_\(HI_{pp}\) (Equation (13)). The uniform reduction in the deviations shows that the correction using a frequency-dependent function HI is suitable for achieving the desired results. In order to quantitatively evaluate the improvements of the measurement results by the presented calibration, the deviation of the test results are to be determined. Furthermore, the deviation shall be compared to the procedures of [PERSON] [26] and [PERSON] et al. [25]. The deviation determined in each case is given by the difference of AM and the weighed masses  
Figure 8: Calibrated apparent mass \(AM_{testobj. }\) of freely vibrating masses over frequency. Figure 7: Average residual e (Equation (19)) of \(HI_{pp}(f)\) over degree of fitting polynoma for the low frequency (**a**) and high frequency (**b**) test bench. The red circle marks the polynomial degree whose relative change of the residual is less than 1%. \(m_{i}\) (Section 2.5) by Equation (22). Since the masses vary, the relative deviation is determined by dividing the absolute deviation by the corresponding mass resulting in Equation (23). \[e_{AM,abs}=\frac{1}{N}\sum_{n=1}^{N}|AM_{n}(\omega)-m_{i}| \tag{22}\]  
\[e_{AM,rel}=\frac{1}{N}\sum_{n=1}^{N}\frac{|AM_{n}(\omega)-m_{i}|}{m_{i}} \tag{23}\]  
The deviation of the results is examined for the following approaches: static calibration only, mass cancellation by Ewins only, mass cancellation and _measurement systems FRF_\(HI_{pp}\) and mass cancellation by [PERSON] et al. and _measurement systems FRF_\(HI_{pp,fit}\) and mass cancellation determined over multiple masses as presented here in this publication. For the determination of the average deviation \(e_{AM}\) by the static calibration \(AM_{n}=AM_{measured}\) is used and or the mass cancellation only, \(AM_{n}=AM_{measured}-m_{sensor}\). For the mass cancellation and _measurement systems FRF_ as used by [PERSON] \(AM_{testobj}\), from Equation (18) are used, the function \(HI_{pp,fit}\) is calculated only by the mass of the sensor and adapter. For comparison, the procedure addressed in this publication is listed with the measurement of different added masses (Table 1). The resulting absolute and relative average deviation \(e_{AM}\) by by Equation (22) and by Equation (23) for each of the four approaches is shown in Table 2.  
By only statically calibrated sensors without consideration the influence of fixtures and measuring devices of the test bench a clear deviation of test results can be seen. The deviation in this case is particularly due to the mass of the sensors and adapters and the size thus also corresponds roughly to the mass \(m_{sensor}\) (Table 1). The deviation on the low frequency test bench deviates from the mass \(m_{sensor,lowfreq}\), this indicates a uniform deviation of the determined AM, which then leads to a deviation resulting over all measured masses. Since the tested masses on the high frequency test bench are lower than the mass of the adapter and the sensors, it results in a very high relative deviation of the measurement results of over 250 %. For this reason, the deviation due to mass cancellation on the high frequency test rig decreases a lot. The approach of [PERSON] et al.