This
feature may be used to evaluate other models of the Almen Saturation Curve and validate
these against the True
Shape of Almen
Saturation Curves. One such model is an Avrami equation^{(2)}:

**f(x) =
a(1-exp(-bx ^{c}))**

Below are plots of this equation (see notes)

**Fig. 1** Run
1 (orange) is the true Almen Saturation Curve for the Wieland data. Run
2 (green) is the Avrami equation curve. For this
data set the Avrami model closely
approximates the true shape. It fails on other data sets as shown below.

**Fig. 2**
Run 1 (orange) is the true shape of the
Peenforming^{(2)}
data. Run 2 (green), the Avrami equation curve, is obviously wrong. The
calculated intensities are comparable. However, the Avrami equation exposure
time is 17 (seconds)
less than the true shape with a consequent four-fold error. It is unlikely that
the Avrami intensity can be qualified in an actual run.

**Notes:**

1.The Almen Saturation Curve Calculator is based on a single equation that is applicable to any data set of any number of points (the minimum of course is 4). Its development included evaluation of many standard models, including Avrami equations, all of which worked with some sets but failed in others.

With
careful analysis of data sets, it may be possible to apply certain equations to certain sets with acceptable results. However this approach is cumbersome,
inaccurate, prone to error and misinterpretation. The feature * User Defined
Equation* is provided as a tool for validating the accuracy of
other models of the Almen Saturation Curve.

2. Equations are entered in the upper box of the user window. It accepts a maximum of five coefficients (a to e), one independent variable (x) and the more common functions and operators (+, -, *, /, ^, (), exp, log, sin, cos, tan, atn, abs, sqr)

3. The table below the equation box accepts four entries for each coefficient:

**Guess**
- **(Required) **This is the starting value for the initial
nonlinear regression of coefficients. It
does not have to be accurate, however, the closer this value is to the actual
coefficient, the faster the regression. In Avrami equations, the coefficient
"a" is very close to the maximum arc height. Therefore, 15 and 11 were
selected for the Wieland and Peenforming data respectively. The coefficients
"b" and "c" range from <1 to <5. Therefore,
"1" was used for both in the two data sets.

**Ulimit**
- **(Required)** The coefficient upper limit. In
Almen Saturation curves, the coefficient "a" (arc height) cannot be
greater than 24 (.024 inch). +500 was used for a
larger degree of freedom. For the same reason, +10 was used for the exponents
"b" and "c". (Avoid
using a larger upper limit for exponents; it is unnecessary and may result in an
overflow error).

**Llimit**
- **(Required)**
The coefficient lower limit. Almen
arc heights never become negative, therefore Zero is the minimum. However, in
order to cover a larger regression space, -500 was used for coefficient
"a" and -10 for both "b" and "c".

**Fix**
- **(Optional)**
The number "1" (or any number other
than zero) in any box forces **Guess** to
a constant. Regression is prevented on flagged coefficients and the value is
used in drawing the curve. Useful in "what if" analysis, as well
as determining initial values in complex models that are difficult to resolve or
tend to diverge.

^{(1)}
Wieland, R. C., "A Statistical Analysis of the Shot Peening Intensity
Measurement", pp27-38,

Proceedings I.C.S.P.5, Oxford, 1993

^{(2)}
peenforming.squarespace.com