## 3 Experimental Results, Analysis and Discussion
### Identification of the Failing Joint  
Obviously for a structure involving more than one joint, the main goal is to identify the one that is failing. For this purpose, data from tests with structure-B and specifically, damping factors and frequencies of the (two) modes shown in Tables 2 and 3 may be used as follows. From Figure 4 it is obvious that data resulting from FJ1-B tests feature broadly similar frequencies to those resulting from FJ2-B tests for the modes considered. But damping factors for these two modes are consistently higher for data resulting from FJ2-B tests than those from FJ1-B tests. This observation is significant but of rather limited practical interest. In fact, a conclusion on whether J1 or J2 is failing should be drawn based on setting thresholds for a single quantity, as opposed to using some combinatory reasoning. Ideally, one should include values of frequencies and damping factors into one feature, which should offer a clear indication as to which joint is failing. To introduce such a feature, one may attempt the following reasoning. The mode considered is thought of as corresponding to a trivial second order system like the simple harmonic oscillator in Figure 5. This is essentially a mass \(m\) moving due to the application of an external force \(f(t)\) while restricted by a spring of constant \(K\) and a damping element of constant \(C\) (also known as mass-spring-damper system, see Chapter 2 in [23]). Its resonant \(\omega n\) and damping factor \(\zeta\) may be easily computed (see Chapter 2, ref. [23]) as:  
\[\omega n=\sqrt{\frac{K}{m}},\zeta=\frac{C}{2\cdot\sqrt{K\cdot m}} \tag{2}\]  
Figure 5: The simple harmonic oscillator featuring a mass \(m\), moving due to the external force \(f\), while constrained by the damping element with constant \(C\) and the spring element with constant \(K\). The oscillator’s natural frequency \(\omega n\) and damping factor \(\zeta\) are in Formula (2). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Test Case—Run \#**} & \multicolumn{3}{c}{**Mode—I**} & \multicolumn{3}{c}{**Mode—II**} \\ \cline{2-7}  & \(\zeta\) & \(\omega n\) **(Hz)** & \(\omega n\)**/\(\zeta\)** & \(\zeta\) & \(\omega n\) **(Hz)** & \(\omega n\)**/\(\zeta\)** \\ \hline NJ-B \#1 & \(5.783\times 10^{-5}\) & 1255.5 & n/a & \(1.934\times 10^{-5}\) & 1641.7 & n/a \\ NJ-B \#2 & \(3.091\times 10^{-4}\) & 1228.6 & n/a & \(7.567\times 10^{-6}\) & 1606.3 & n/a \\ NJ-B \#3 & \(2.530\times 10^{-4}\) & 1213.3 & n/a & \(8.866\times 10^{-5}\) & 1587.3 & n/a \\ FJ1-B \#1 & \(4.735\times 10^{-4}\) & 1204.1 & 2,542,977.82 & \(1.114\times 10^{-5}\) & 1576 & 108,087,971.3 \\ FJ1-B \#2 & \(1.937\times 10^{-4}\) & 1197.9 & 6,184,305.62 & \(1.384\times 10^{-5}\) & 1566.7 & 86,553,468.21 \\ FJ1-B \#3 & \(3.316\times 10^{-5}\) & 1194.7 & 36,028,447.41 & \(1.080\times 10^{-5}\) & 1563 & 110,620,370.4 \\ FJ2-B \#1 & \(4.512\times 10^{-4}\) & 1192.4 & 2,642,730.49 & \(2.405\times 10^{-5}\) & 1558.6 & 49,580,041.58 \\ FJ2-B \#2 & \(2.692\times 10^{-4}\) & 1196.2 & 4,443,536.40 & \(1.589\times 10^{-5}\) & 1564.4 & 75,280,050.35 \\ FJ3-B \#3 & \(7.293\times 10^{-4}\) & 1195.6 & 1,639,380.22 & \(3.313\times 10^{-5}\) & 1564.6 & 36,088,137.64 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Testing scenarios (at 80 Hz–800 mVpp) for detecting and identifying failing joints in structure-B. The feature \(\omega n/\zeta\) depends only on the ratio between \(K\) and \(C\). In other words, the ratio \(\omega n/\zeta\) associates both damping and stiffness characteristics into a single quantity, which responds to variations in either \(K\) or \(C\). Specifically, if the resonant frequency of the considered mode varies little between data sets, the feature \(\omega n/\zeta\) is highly sensitive to damping variations.