## 3 Results and Discussion
### Calibration of the Measurement System  
The determination of the _measurement systems_\(FRF\)\(HI_{pp}\) is given by Equation (18). The masses of the sensor at each test bench are derived at Section 2.5. Each weighted real constant mass configuration (Table 1) is divided by the corresponding measured complex value of \(AM_{meas.}(f)\). The resulting values for the low and high frequency test bench are marked in frequency domain as data points (Figure 6). At each evaluated frequency, 12 data points resulting from four different mass configurations with each three reputations are shown.  
The magnitude of \(abs(HI)\) for the low frequency test bench is slightly above the ideal magnitude value of one, while \(abs(HI)\) for the high frequency test bench is decreasing from a value of 1.05 to 0.85. To determine HI, the mass \(m_{sensor}\) has already been subtracted. The phase difference behaves as the inverse of \(AM_{meas.}(f)\) shown in Figure 5. The deviation from the ideal magnitude one and phase difference zero show the necessity to use the calibration function \(HI_{pp}\), as introduced by [PERSON] [27]. The pure mass cancellation of Ewins [26] is not enough to calculate the deviation from the ideal result for the given test benches, even though both test stands are statically calibrated.  
Figure 5: Apparent mass \(AM_{meas.}\) of freely vibration masses over frequency.  
The data points of \(HI_{pp}\) scatter around a center value depending on the frequency. A continuous FRF has to be formulated. A polynomial function allows a flexible determination when the behavior is unknown [35]. Using a polynomial function, however, cannot be recommended to extrapolate results at the far ends of the determined data [35]. The polynomial function is determined individually for the magnitude and phase angle, and then combined to the complex function \(HI_{pp}(f)\) in Euler form. In this way, the HI function can be represented in a shorter notation than if always the higher polynomial degree is used for both magnitude and phase angle. The high amount of data points \(k\) theoretically allows the determination of a polynomial of a high degree of \(k-1\)[36]. The data to be described can be expressed by a function of much lower polynomial degree. For this, the residual between the data points \(HI_{pp,n}\) and the function \(HI_{pp,fit}\) can be minimized [36].  
\[e=\frac{1}{N}\sum_{m=1}^{N}|HI_{pp,n}(\omega)-HI_{pp,fit}(\omega)| \tag{19}\]  
The average residual e can be calculated by Equation (19) for each function \(HI_{pp,fit}\). Figure 7 shows the average residual over the degree of the polynomial of the argument and the modulus. The average residual is calculated from the summed up difference between each data point and the function, with a given polynomial degree divided by the amount of data points \(k\).  
As a compromise between a simple description versus the accuracy of the data, the lowest polynomial degree is chosen, whose relative change of the residual to the next polynomial degree is less than 1% (marked as red circle at Figure 7). The two following functions describe the resulting function \(HI_{pp}(f)\) for each test bench. The resulting functions are marked as dashed lines in Figure 6 and qualitatively fit the data.  
\[HI_{pp,fit,lowfreq}(f)=(1.0196\ -5.7312\cdot 10^{-5}\cdot f)\cdot exp (i\cdot(-0.52767\ -0.1353\cdot f\ +0.01676\cdot f^{2}\ -0.001087\cdot f^{3}\\ +3.5122\cdot 10^{-5}\cdot f^{4}-4.4507\cdot 10^{-7}\cdot f^{5})) \tag{20}\]  
\[HI_{pp,fit,highfreq}(f)=(1.056\ -3.1385\cdot 10^{-4}\cdot f\ -8.9521\cdot 10^{- 7}\cdot f^{2}\ +4.0439\cdot 10^{-9}\cdot f^{3}\ -5.3453\cdot 10^{-12}\cdot f^{4})\\ \cdot exp(i\cdot(-0.02695\ -0.0021295\cdot f\ +9.3418\cdot 10^{- 6}\cdot f^{2}-2.2897\cdot 10^{-8}\cdot f^{3}\ +2.4072\cdot 10^{-11}\cdot f^{4})) \tag{21}\]  
Figure 6: _Measurement systems FRF \(HI_{pp}\)_ over frequency of both test benches.