Civil Engineering Reference
In-Depth Information
The first technique uses a subset of the traditional surface analysis techniques to remove the deter-
ministic components from the surface profiles. It then analyzes the remaining stochastic profile by means
of a multi-resolution wavelet decomposition and quantification with fractal measures. This method of
analysis departs from common usage of wavelet transforms by introducing a fractal interpretation. It
also provides a complementary technique for surface synthesis, enabling the integration of manufacturing
information earlier in the design process. The second technique uses an orthogonal decomposition via
the Karhunen-Loève transform to decompose surface profiles into their significant components, and
provides a means to monitor individual patterns on surface profiles. The fault status of individual patterns
is indicated with this method, with an equal capability to handle various different fault characteristics.
The information obtained from the second technique provides a clear understanding of the fault mech-
anisms; the understanding gained will help designers identify the source of the faults and remedy the
problems by redesign and/or process control. The technique allows a means of reconstructing an estimate
of surface profiles prior to manufacturing to enable prediction during the design stage.
There are clear distinctions between the two techniques. The advantage of the fractal technique is in
its ability to distinguish between stationary or nonstationary stochastic profiles by means of a minimum
set of measures. The removal of deterministic trends precedes the fractal analysis of the surfaces. Measures
are used to indicate the nature and severity of the deterministic components. However, changes in
deterministic components cannot be detected and monitored with this method. The Karhunen-Loève
technique has the advantage of decomposing the surface profiles into all of its significant patterns, and
monitoring all of the patterns simultaneously, including deterministic and nonstationary patterns, even
in the presence of highly stochastic noise. However, the method currently does not allow the monitoring
of the stochastic component of a signal. As we discuss below, the conjunction of these two methods is
one of the avenues being pursued by the authors.
In the following sections, the details of each method are presented in the context of analysis for surface
fault monitoring and synthesis for surface fault prediction. Specifically, the philosophy, models, and the
mathematical foundations of each method are presented, followed by applications to numerical signals,
as well as to design examples using experimental surface profiles.
1.3
Fractal-Wavelet Method
The first novel technique [57, 59, 58, 67] incorporates traditional spectral-based surface analysis tech-
niques [7] with a fractal-wavelet analysis of stochastic surfaces [76] in the form of a superposition model.
This formulation sets the stage for the first component of our vision: characterizing structure in the
stochastic nature of manufactured surfaces.
Manufactured surfaces are composed of a kaleidoscopic amalgam of error structures. While the
autocorrelation and power spectrum approaches are effective in diagnosing the presence or absence of
certain trends, they must be supplemented by additional methods like regression analysis to extract
structural parameters, and to synthesize surfaces. One effective tool that is well tailored to deal with the
complexity of manufactured surfaces is the fractal dimension. To accomplish our objectives of analysis
and synthesis of manufactured surfaces, we use fractal dimensions in conjunction with wavelet trans-
forms. We now survey the background literature on these two tools. Then we present our geometric
interpretation of fractal dimensions, followed by a discussion of the pertinence and origin of fractal
structures in manufacturing with specific reference to metal cutting processes.
Literature Background: Fractals
The subject of fractals and fractal dimensions was engendered by Mandelbrot in the classical reference
to the subject [36]. Several natural objects exhibit fractal characteristics, e.g., coastlines, clouds, and
geological formations [51]. There has been an unprecedented surge of applications for fractals in widely
diverse fields, ranging from economic data analysis [36], through analysis of lightning patterns [66], to
structural studies of human lungs [36].
