*3.2. Data Processing*

Data processing from the experiment was done using a script in Matlab software. Several main parts were included. The first was the use of an exponential window. Windowing is a common necessary signal processing technique used in modern data acquisition systems [28,29]. The exponential window is a time domain weighting function that has been elaborated for use with transient-type signals, such as those of impact testing. Used correctly, the exponential window can minimise leakage errors on lightly damped signals and can also improve the signal-to-noise ratio. The effect of the exponential window (Figure 5) is to increase the apparent damping of the measured system. The exponential window [30] is an exponential function as defined in equation (2) where the parameter *f* is the last point value of the window, *n* is the number of samples and *i* is index of the exponential function:

$$y\_i = e^{\frac{\ln f}{n-1}}, i = 1 \dots n \tag{2}$$

The time variable for the exponential function starts at zero, regardless if a pretrigger delay is used in the data acquisition. When impact testing lightly damped blade structures, the purpose of the exponential window is to reduce the effects of leakage by forcing the data to meet the requirements of a completely observed transient response signal more closely. By definition, a fully observed transient signal must begin and end within the measured time record. For lightly damped blade systems, the response of the structure will typically continue beyond the time period of data collection as shown in Figure 6a. Since the response does not decay to near zero at the end of the time record, the exponential window is applied to reduce the signal at the end of the time record to approximately 1%. The signal according to Figure 6a after the window has been applied shows Figure 6b. The windowed signal more closely represents an observable transient.

**Figure 6.** Measured response signal of a lightly damped blade system for impact excitation and its modification by the exponential window.

This data was processed by a fast Fourier transform (FFT) algorithm in a complex form to evaluate the transfer function of the system. That is, the FFT analysis was performed for the impact force course and its corresponding measured response. Subsequently, the ratio of the output to the input of the system was used to obtain the transfer function:

$$Y\_i =ABS \left(\frac{FFT\_{out}}{FFT\_{force}}\right); i = 1 \ldots n\_\prime \tag{3}$$

where *FFTout* is the resulting FFT analysis complex vector from the response data and *FFTforce* is the resulting FFT analysis complex vector from the force pulse data. The calculation works using complex arithmetic. The measurement of each turbocharger blade was performed five times to reduce possible measurement errors, and these five measurements were included in the data processing, at the end of which five calculated FFT spectra were always averaged. Since the time window was 0.2 s, the frequency resolution of the FFT is 5 Hz, and thus, the uncertainty of determination of each spectral component is 2.5 Hz, which is completely satisfactory for the determination of the blade disc mistuning.
