Civil Engineering Reference
In-Depth Information
where
p
o
(
x
) is called the outlier component. The next step is to identify each component, define it, and
establish methods to compute the relevant parameters for analysis and synthesis.
The linear trend is identified using autocorrelation and power spectrum methods [57, 67]. It is
manifested as a slow decay in the autocorrelation, and as a peak at zero frequency in the power spectrum.
A linear regression technique is used to compress the trend information in two parameters, I.e., an
intercept
p
to
, and a slope
s
t
.
p
t
x
()
p
t
0
s
t
x
0
x
.
(1.13)
A periodic trend exhibits a periodic autocorrelation, and appears as a peak at the underlying frequency
in the power spectrum. The periodic component is estimated from the surface profile by using a nonlinear
regression procedure. The following model is proposed for evaluating the periodic component
p
p
(
x
):
P
p
x
()
p
p
0
d
a
sin
(
2
f
r
x
L
)
0
xL
(1.14)
where
p
p
0 is an offset (from an arbitrary datum of zero),
d
a
, is the amplitude, and
f
r
is the frequency.
Nonlinear regression methods are used to evaluate the parameters [27, 52]. Initial estimates for
p
p
0 and
d
a
are set equal to 0, and the frequency corresponding to the spectral peak is used as a preliminary
estimate for
f
r
.
The fractal component is the remnant from the experimental profile, after subtracting the linear and
periodic trends. The outlier data points are corrected using the following rule (57]:
If the absolute value of a data point deviates from the mean by more than twice the standard deviation
of the entire data set, then the value is replaced by that of its predecessor. As explained above, the fractal-
wavelet method is used to characterize the fractal component in terms of its fractal dimension and
magnitude factor. The analysis (extraction of relevant profile parameters from experimental data) and
synthesis (regeneration of profiles from the computed profile parameters) is demonstrated for two
different processes: milling and grinding.
Milling Profiles
The first process selected for study is a face milling operation, with a flycutter [57]. The flycutter is a
single-point tool used to machine light cuts on flat workpieces. Several sources of error are possible in
this process: errors from roughing, clamping the workpiece, hardness variations, tool wear, vibrations,
and so on.
Experimental Layout
A Bridgeport vertical milling machine is selected. The cutting tool is high speed steel. Work material is
Aluminum 2024, with no cutting fluid used. Three process parameters are chosen for this experiment:
the cutting speed
s
, the feed
f
, and the depth of cut
d
. Milling is performed at two levels for each factor,
leading to factorial experiment. The values for the selected factors and levels are shown in
Table 1.1
.
The experiment is replicated twice. The layout of the design of experiments is shown in
Table 1.2
.
Note
that the replicate number is indicated by Roman numerals I and II, while the test number is indicated
by the letters A through H. In most of the following tables a particular replicate and test number
combination is denoted “REP.No”, e.g., I.A, II.D.
2
3
TABLE 1.1
Factors and Levels for Milling Experiments
Level/Factor
s(fpm)
f(in./rev.)
d(in.)
300
0.025
0.02
100
0.010
0.01
