Analysis and Synthesis of Engineering Surfaces in Bridge Manufacturing and Design - Manufacturing Systems Processes - page 39

FIGURE 1.21 A sample milling profile.

FIGURE 1.22 Power spectra of the dominant KL eigenvectors.

In the next section, we expound on this relationship, illustrating the unification with a design and
manufacturing example.

Conjunction of the Two Methods
To perform the proposed conjunction, we investigate the surface of a part generated using the milling
process described earlier in the chapter. In the following, the Karhunen-Loève technique is first used to
isolate and characterize the deterministic components in the milling surface profiles. Next, the milling
profiles are filtered by removing the fundamental KL eigenvectors. This step is followed by the application
of the fractal-wavelet technique to the filtered surfaces to represent the characteristics of the remaining
stochastic component.
We first analyze these surfaces using the Karhunen-Loève transform. M 20 measurements of such

a milling surface are collected over a 20 mm total traverse length, each containing N 256 data points.

The Karhunen-Loève transform results in five fundamental eigenvectors, adding up to 93 percent of the
total energy. A sample profile is shown in Fig. 1.21 .
The first eigenvector corresponds to the dominant frequency component and is identified clearly using
the transform. This component has the same frequency as the one identified earlier in the fractal-wavelet
analysis of the milling profiles. While the first dominant eigenvector captures the dominant characteristics
of the milling profiles, the KL analysis reveals additional modes which were not detected with the fractal-
wavelet technique. To confirm these results, we take advantage of the standard Fourier transform for
purely periodic signals: Fig. 1.22 shows the power spectrum of the eigenfunctions, indicating the main
frequency component at discrete frequency l

25, and two additional frequency components, at discrete

frequencies l 75. The first frequency component, identified with the first eigenvector, is

the component corresponding to the feed marks during the milling process. The second and third
frequency components are identified as harmonics of the first component, due to the tool vibrations

50 and l