Analysis and Synthesis of Engineering Surfaces in Bridge Manufacturing and Design - Manufacturing Systems Processes - page 34

FIGURE 1.14 Decomposition of signals from Example 2.

FIGURE 1.15 Coefficient vectors for Example 2.

change introduced in the multi-component signal. Note that the eigenvalue for the first eigenfunction is
not as significant as the first eigenvalue in the first example. The eigenvalue is related to the variance of the
coefficients, and hence is greater in the case of a linear trend with increasing slope. Note that the eigenvalues
for the second and third eigenfunctions (corresponding to the low-frequency sinusoidal component) are
higher than in the first example. This is due to the fact that the amplitude of the low-frequency sinusoidal
component increases during the monitoring process. This nonstationary change is reflected on the eigen-
values since they are computed based on the coefficient vectors. Recall that the coefficient vectors should
reflect the change in the amplitude of the low-frequency sinusoidal component. Notice that the shape of
the first pair of sinusoidal eigenfunctions remains the same as in the first example, since the fundamental
characteristics (e.g., frequency and phase) of the sinusoidal components do not change. Finally, the eigen-
values and shape for the second pair of eigenfunctions (corresponding to the high-frequency sinusoidal
component) remain the same as in the first example, since no change is introduced to this component.
The slight change in the eigenvalues (#4 and #5) is due to the presence of the noise component in the
second example.
The coefficient vectors corresponding to the five fundamental eigenfunctions are shown in Fig. 1.15 .
As expected, the coefficient vector corresponding to the linear eigenfunction (#l) indicates the change in
the offset level starting from the snapshot m 40. This is a crucial result, since it provides us with the

ability to determine the exact location and severity of the nonstationary fault pattern in the signal. Recall
that it is often very difficult to determine these two properties about nonstationary changes in a signal.
The coefficient vectors corresponding to low-frequency eigenfunctions (#2 and #3) indicate the change
in amplitude between snapshots m

40. This is also a crucial result, since such a nonsta-
tionary change would be typically averaged out using Fourier-based techniques. In our case, by moni-
toring the eigenfunction and the corresponding coefficient function simultaneously, we are able to detect

20 and m