QFD: House of Quality Template

Running around the web, I could not find a quality HOQ template that was not locked, protected, or complete for my needs. So I decided to create one and share it. The attached template is created in Excel 2007 and does not contain any macros or VB code. Nothing is locked or protected, and it can be resized at will (though this may require a little cleanup afterwards). Drop down lists are generated using the Data Validation tool, in-cell charts through the use of the REPT function, cell highlighting using conditional formatting and the line charts are simple excel graphs, updated automatically, that are aligned with the cells.

Click on the picture for a larger image. You can download the MS Excel 2007 template here and the OOo 3.1.1 template here.

I place this template into the public domain. Feel free to use, modify, redistribute, etc. Though if you do find it useful, let me know.


Caveat lector — All work and ideas presented here may not be accurate and should be verified before application.

Calculating Mooney-Rivlin Constants

In this post we will look at the procedure for determining the Mooney-Rivlin constants from simple tensile test data of an elastomeric solid. The definition and derivation of the material model is left to others. For our purposes, all we need to know is that the material model yields a predicted engineering stress under simple tension of

where C1 and C2 are the Constants that need to determine and is the stretch ratio, defined as the ratio of the stretched length to the initial length of the sample. This can be defined in terms of the measured strain in a simple tensile test

We’ll start with a set pf published stress strain data for a 40 Shore A material from GLS corporation. The stress strain curve from the literature is shown below.

Using g3data we can extract the points and tabulate them. For simplicity, we also convert the strain to stretch ratio and the stress to SI units. The data file can be downloaded here.

Strain            Stress (psi)                                                    Stretch Ratio () Stress (Pa)      
0 0   1 0
0.5 141.2   1.5 9.74 x 105
1.0 204.9   2 1.41 X 106
2.0 299.8   3 2.07 x 106
3.0 367.6   4 2.53 x 106

Usign R to perform the regression is easy:

1. Read in the data from the file G7940.dat.

> SS_Data <- read.table("c:/G7940.dat", header = TRUE)

2. Examine the imported data.

> SS_Data
  Strain  Stress
1    0.0       0
2    0.5  974000
3    1.0 1410000
4    2.0 2070000
5    3.0 2530000

3. The "attach" command allows us to access the variables directly without having to reference the original structure.

> attach(SS_Data)

4. A quick plot of the imported data can then be generated.

> plot(Strain, Stress)

5. Since the MR model uses the stretch ratio, not the strain, we convert the strains and then plot the stress vs. stretch ratio.

> Stretch = Strain + 1
> plot (Stretch, Stress)


6. Now for the curve fitting itself. We use the "nls" function which stands for "Nonlinear Least Squares". We provide the model as given in the equation at the beginning of the post where C1 & C2 are the two constants we wish to determine, guesses for the initial values of those constants, and request that the trace is provided. Then we ask for a summary of the results. This prints the fitted values for our two constants along with some other helpful data.

> MR.fit <- nls(Stress ~ (2*C1+2*C2/Stretch)*(Stretch-1/(Stretch^2)),
+ start=list(C1=100,C2=100), trace=T)
1.361224e+13 :  100 100 
2117860971 :  239897.5 335348.9 
> summary(MR.fit)
Formula: Stress ~ (2 * C1 + 2 * C2/Stretch) * (Stretch - 1/(Stretch^2))
Parameters:
   Estimate Std. Error t value Pr(>|t|)    
C1   239898       8001   29.98 8.15e-05 ***
C2   335349      23840   14.07 0.000778 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 26570 on 3 degrees of freedom

Number of iterations to convergence: 1 
Achieved convergence tolerance: 5.802e-07 

7. Lastly, we would like to see how our curve fit matches the data. First we extract the coefficients into a vector "C" With the plot shown above still open, we add the curve and clean it up with a title. Notice the method for calling the coefficients.

> C <- coef(MR.fit)
> curve((2*C[1]+2*C[2]/x)*(x-1/(x^2)), from=1.0, to = 4.0, add=TRUE)
> title(main="Mooney-Rivlin Fit to Simple Tensile Data")


Caveat lector — All work and ideas presented here may not be accurate and should be verified before application.

Basic R – Univariate Dataset Graphics in R, Exploratory Data Analysis

As a continuation of this post, we continue on the analysis of univariate data by generating graphical views of the Michelson speed of light data.

The simplest method to view the distribution is the stem and leaf plot.


> stem(C$V1)

  The decimal point is 1 digit(s) to the left of the |

  2996 | 2
  2996 | 5
  2997 | 222444
  2997 | 566666788999
  2998 | 000001111111111223344444444
  2998 | 5555555566677778888888888999
  2999 | 0011233444
  2999 | 55566667888
  3000 | 000
  3000 | 7

We can also create a boxplot of the data. Notice the method to insert a mathematical expression in teh axis label.


> boxplot(C)
> title(ylab=expression(paste(m/sec," ",plain( x )," ",10^3)))
> title("Boxplot of Speed of Light")

Which gives
01

And a histogram gives us a graphical view similar to the stem an leaf.


> hist(C$V1, main="Histogram of Data", xlab=expression(paste(m/sec," ",plain( x )," ",10^3)))

02

It’s always a good idea to generate a Q-Q Plot to check for normality.


> qqnorm(C$V1)

03


Caveat lector — All work and ideas presented here may not be accurate and should be verified before application.