Mechanical Spring Deflection

Some tests are run where a ball of a fixed mass dropped from a known height to land inelastically onto a spring whose maximum compression is recorded. The data collected from tests runs at several heights will be contained in a text file $($in the same folder as your Matlab code$)$ called CompressionVersusHeight.txt$,$ an example of which is available on CatCourses. Note that the data file can have any number of data points after the three header lines and any script that reads it must be able to read different versions without change to the script. A plot of the example data is shown.

Write a single$-$file Matlab script that reads the data from CompressionVersusHeight.txt$.$ Then, use Matlab's curve$-$fitting function to find a curve of the form $\backslash ($ x$=$A $\backslash$sqrt$\{$h$\}\backslash )$ that best fits the data. Make sure that your code assigns the value of the value of $\backslash ($ A $\backslash )$ to a variable called A$.$ On a single figure, plot the original data with markers and the fitted curve on the same plot, with a legend, appropriate axis labels and indicate clearly in the plot title the value obtained for $\backslash ($ A $\backslash ),$ with four digits after the decimal place.

Hint: This is most easily done using the custom curve fit technique discussed in the lecture notes.

A sample of the resulting plot for a slightly different data set is shown below. Write a single$-$file Matlab script that reads the data from CompressionVersusHeight.txt$.$ Then, use Matlab's curve$-$fitting function to find a curve of the form $\backslash ($ x$=$A $\backslash$sqrt$\{$h$\}\backslash )$ that best fits the data. Make sure that your code assigns the value of the value of $\backslash ($ A $\backslash )$ to a variable called A$.$ On a single figure, plot the original data with markers and the fitted curve on the same plot, with a legend, appropriate axis labels and indicate clearly in the plot title the value obtained for $\backslash ($ A $\backslash ),$ with four digits after the decimal place.

Hint: This is most easily done using the custom curve fit technique discussed in the lecture notes.

A sample of the resulting plot for a slightly different data set is shown below.

nata with $\backslash (\backslash$mathbf$\{$x$\}=\backslash$boldsymbol$\{\backslash$Delta$\}^\{*\}\backslash )$ cart$/$ h $)$ curve fit

$```$

$\%$Read the data from the file

data $=$ readmatrix$(\text{'}$CompressionVersusHeight$.$txt$\text{'},$ 'NumHeaderLines', $3)$;

$\%$Extract height $($h$)$ and compression $($x$)$ from the data

h $=$ data$($:$,1)$;

x $=$ data$($:$,2)$;

$\%$Define the curve fitting model

model $=$ fittype$(\text{'}$A $*$ sqrt$($h$)\text{'},$ 'independent', $\text{'}$h$\text{'},$ 'coefficients', $\text{'}$A$\text{'})$;

$\%$Perform the curve fitting

startPoint $=1$;

fitResult $=$ fit$($h$,$ x$,$ model, 'StartPoint', startPoint$)$;

$\%$Extract the coefficient A

A $=$ fitResult.A;

$\%$Generate the fitted curve for plotting

fitted$\_$x $=$ A $*$ sqrt$($h$)$;

$\%$Create the plot

figure;

plot$($h$,$ x$,\text{'}$gx$\text{'},$ 'DisplayName', 'Data'$)$; $\%$ Original data points

hold on;

plot$($h$,$ fitted$\_$x$,\text{'}$b$-\text{'},$ 'DisplayName', 'Fit'$)$; $\%$ Fitted curve

hold off;

$\%$Add labels, legend, and title

xlabel$(\text{'}$Height$,$ h $($m$)\text{'})$;

ylabel$(\text{'}$Compression$,$ x $($m$)\text{'})$;

legend$(\text{'}$show$\text{'})$;

title$($sprintf$(\text{'}$Data with x $=$ A $*$ sqrt$($h$)$ curve fit

A $=\%\mathrm{.}4$f$\text{'},$ A$))$;

$```$