Analysis and Synthesis of Engineering Surfaces in Bridge Manufacturing and Design - Manufacturing Systems Processes - page 32

FIGURE 1.12 Coefficient vectors for Example 1.

eigenfunctions which look alike, except for a phase shift. This is due to the fact that the phase angle of
each snapshot is different, hence introducing a second eigenfunction [70]. Recall that phase information
is typically lost with Fourier-based methods. Similarly, the fourth and fifth eigenfunctions correspond
to the high-frequency sinusoidal component, and the corresponding phase-shifted version. Note that each
one of these eigenfunctions can be studied individually to understand the nature of the fault patterns in
a complex signal. Sinusoidal components are isolated without any misinterpretation caused by the obscuring
of the data by other components. The linear trend is separated from the sinusoidal components. For example,
if a third frequency component is introduced, it would appear as an additional eigenfunction pair. This
might, for example, correspond to the appearance of harmonics due to a bearing misalignment in the
manufacturing machine. Its immediate identification is crucial for the diagnosis of the fault mechanism.
The coefficient vectors corresponding to each of the fundamental eigenfunctions are shown in Fig. 1.12 .
Note that the first coefficient vector shows the change in slope of the linear trend isolated in the form
of eigenfunction #1 ( Fig. 1.11 ). Using the eigenfunction and the corresponding coefficient vector, the
introduction of linear trends can be monitored separately from the rest of the significant components.
In addition, changes in the slope of existing linear trends can be monitored as well. The introduction of
slopes and/or changes in the slope of existing linear trends may be indicative of serious errors of form
in the system. Their accurate identification is crucial in identifying and eliminating the origin of the
errors. The remaining coefficient vectors correspond to the values of each sinusoidal eigenfunction along
the snapshots. This can be viewed as the value of each snapshot projected onto the eigenfunctions in the
new transform domain.
It is important to note that the Karhunen-Loève transform results in M 30 eigenfunctions. The

dimensionality in the new domain is determined by the rank of the covariance matrix, which is deter-
mined by the number of input snapshots. As shown in Table 1.11 , only a subset ( M 5) of these

eigenfunctions is significant, containing the majority of the total energy. As expected, the reconstruction
of the original profiles using these five eigenfunctions is perfect, as shown in Fig. 1.13 . Recall that the