# Efficient Estimation of Synthetic Indicators for the Assessment of Nonlinear Systems Quality
## 3 Results
### Laboratory Measures on Damaged Concrete Structures  
The validity of the technique was also verified in connection with an experiment to test for cracks in a concrete structure: the experiment was performed on a prism-shaped concrete specimen measuring 4 \(\times\) 4 \(\times\) 16 cm\({}^{3}\); a U-shaped notch with a cross-section of 4 \(\times\) 4 mm\({}^{2}\) was made halfway along the length of a 16 cm face of the specimen. The concrete matrix was produced using a CEMI 52.5 R cement, a water-to-cement ratio of 0.5 and a cement-to-sand ratio of 1:3 by weight. The specimen was damaged by subjecting it to a three-point bending loading procedure with controlled displacement rate at the edge of the notch. This procedure created a small crack at the end of the cycle whose size is of the order of 40% compared to the cross-section.  
Ultrasonic tests were performed on the damaged specimen: using Phenyl Salicy-late, two ultrasonic transducers (nominal center frequency at 55 kHz, diameter of 2 cm) were glued to the two ends of the specimen. An arbitrary waveform generator (TiePie handyscope HS5) was used to produce an exponential chirp signal in the frequency band from 40 to 60 kHz with a duration of about 0.4 s, sampled at 2.5 MHz. The amplitude of the source signals was increased from 10 mV to 1.5 V (before amplification). The signal was then amplified by a factor of 20 before feeding it to the Tx transducer (Falco-System power amplifier). The Rx transducer was connected to an oscilloscope for data acquisition (TiePie handyscope HS5 used as oscilloscope data logger). Figure 6 shows a schematic representation of the stressed bar and the measuring bench.  
For each of the input signal amplitudes, the Hammerstein model was estimated from the received signal by means of the PuC technique. For each amplitude, the ratio between energy of the signals associated with the third harmonic and with the fundamental was  
Figure 5: Damage index HI for different SNR values as a function of the value of \(\alpha\). (**a**) HI trends for two different values of input signal amplitude: amplitude R1 which guarantees SNR = 5 and amplitude R2 which guarantees SNR = 10 dB. (**b**) Comparison between the experimental HI curve for amplitude R1 (SNR = 5) and the curve obtained by applying the extrapolation procedure from R2 (SNR = 10) to R1 (SNR = 5).  
Figure 6: Schematic representation: (**a**) the stressed bar; (**b**) the measuring bench.  
then calculated: a change in the slope of the curve as the amplitude of the excitation signal increases highlights the onset of nonlinear behavior, and thus the presence of a microcrack.  
The results of the comparison are shown in the two panels in Figure 7: the left panel shows the \(E_{3}/E_{1}\) ratio for 20 different input signal amplitudes. The right panel shows the trend of the same ratio having considered the measurement results for only four amplitudes, the points of which are highlighted in the figure, and having extrapolated from this, via (12), the value of the ratio for the other amplitudes. The values reported with red points were estimated starting from the measured points 5, 10, 15 and 20. From each of these values the extrapolation for the four lower levels of input signal was evaluated. In both panels the dotted lines represent linear interpolations on the first group of points (from 1 to 5) and on the remaining 15 points. The abscissa of both panels is the amplitude of the signal received, measured as the maximum of the absolute value of the fundamental in the response to the excitation signal.  
Comparison of the two graphs shows substantial equivalence between the two trends, with the knee of the curve at the same amplitude range. Thus, the validity of the extrapolation technique proposed in this paper remains confirmed here as well.