## 4 Results and Discussion  
### Cumulative Instantaneous Entropy, Counts, and Energy  
#### 4.1.1 Comparing Cyclic Evolutions and Values at Fracture  
Instantaneous entropy proposed in this work is calculated from AE signals by forming independent and new distributions of each individual AE signal's voltage values and employing Shannon's equation to estimate the distributions' disorders. Because instantaneous entropy is independent between waveforms, the total disorder due to fatigue damage is assumed to be measured by cumulative instantaneous entropy from the AE signals. To assess its utility, cumulative instantaneous entropy can be compared to cumulative counts and cumulative energy. First, it is best to observe the cumulative instantaneous entropy, counts, and energy behaviors against fatigue cycles up until specimen fracture as reported in Figure 7. This figure shows that all cumulative features slightly increase initially, stay constant for the majority of life, and then begin to increase near final fracture. This trend proves to be similar to fatigue stages in which crack growth is relatively constant for almost the entirety of fatigue and then sharply increases once a crack has initiated. In addition, MDD versus cycles for each experiment through fracture is depicted in Figure 8 showing that most of the damage measured via MDD is recorded in the last 2000-4000 cycles.  
Another point to consider is the final cumulative values of instantaneous entropy, counts, and energy at fracture. Theoretically, measures of damage should be of the same magnitude between identical structures that have been exhibited to the same level of fatigue loading even though uncertainty is inherent in material's property and measurements. Thus, any metric that represents the cumulative fatigue damage should be similar for all specimens with cracks of equal lengths and at fracture. Some variations, however, are expected in the final cumulative AE parameters at fracture due to inaccurate classification of AE noise signals, imprecise sensor placement which alters AE signal attenuation, and inherent microstructural differences between specimens.  
Figure 7: Cumulative trend for three parameters versus fatigue cycles for specimen 1 (S1) to specimen 8 (S8): (**a**) cumulative instantaneous entropy; (**b**) cumulative counts; and (**c**) cumulative energy.  
The spread of the final cumulative values can be quantified by the coefficient of variation (CV). The CV measures the relative standard deviation of a set of random numbers assumed to have come from a normal distribution. The following equation shows the CV in percent calculated based on the sample standard deviation, \(\sigma\), and the sample mean, \(\mu\):  
\[\text{CV}(\%)=\frac{\sigma}{\mu}\times 100\% \tag{4}\]  
A CV of 0 suggests all values are the same with no variability (i.e., no uncertainties), while wider data scatters will have larger CV values. Results for cumulative AE parameters are recorded in Table 3 where the CV for counts is the lowest at 18.2% followed by energy at 20.2%, while instantaneous entropy has the greatest CV of 22.4%.  
These values suggest cumulative instantaneous entropy has the most variability at fracture compared to cumulative counts and cumulative energy. Therefore, one could say instantaneous entropy may be most influenced by random variations in experiments compared to the other two traditional AE features. While it may seem advantageous for a parameter to have a lower CV, the higher variation for instantaneous entropy means that is it more sensitive to slight differences in AE signals. Thus, it could potentially be a better, more sensitive measure of damage if variations between experiments were reduced.