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1 Force on plane truss The figure below depicts a plane truss having 21 embers (the numbered lines) connecting 12 joints (the circles labeled with letters). The indicated loads in tons, are applied at joints B. E,F,1 and and we want to determine the resulting force on each member of the truss. For the truss to be instaticequilibrium there must be no net force horizontally or vertically at any joint. Thus. we can determine the member forces equating the horizontal forces tot the left and right each joint, and similarly equating the vertical forces upward and downward at each joint. For the joints this would give equations, which more than the unknown factors to be determined. For the truss to be statically determinate that is, for there to be a unique solution, assume that joint Air rigidly fixed both horizontally and vertically and that joint Lis fixed vertically. 4 8 G 12 16 3 7 11 \13 15 17 19 20 (A 2 6 10 14 18 21 X W T V 30 15 10 20 30 (a) Resolving the member forces into horizontal and vertical components, determine set of equations for each joint B-K so that the truss static equilibrium Construct the resulting sparse matrix for this problem using sparse commano in MATLAB where you give the list of row clumn (i.j) and the values non-zero coefficients The resulting matrix should be You can assume that the diagional components of the truss make: 45 degree an with respect to the horizontal and vertical components (b) spy plot of the sparse matrix from part (a). (c) Solve the linear system from part (a) and report the force values for the components f, 14, f9, fis /16, f17 and f21- 2. ageRank Download the bsusurfer. function from the course webpage This modified version of NCM function surfer that restricts the webpagesi visits to those that contain the word boisestate in their address (note thati is by no mesans perfect). (a) Using the bsusurfer function collect 1000 webpages. This task will take awhile to complete, so it is best to do it once and save the results. You can do this with the MATI commands [U,G] beusurfer(n): save BSUSurferResults G: When you want to use and later, you can simply use the command load BSUSurferResults which loads U and G into the workspace (b) Display the connectivity matrix for this collection of webpages using the spy command and compute and display its sparsity ratio in the title of the plot. (c) Using the function pagerank posted on the course webpage, compute the PageRank of the webpages from (a) with the default damping factor o 0.85 (the book and function call this value p). (d) Report the 10 pages with the highest PageRank and the 10 pages with the lowest PageR- ank. By default the pagerank function displays the 10 pages with the highest PageRank You need figure out how to make also display the lowest ranked pages. (e) Repeat part (c) and (d), but with clamping factor of o 0.98. (f) Compare the results of part (e) with part (d). (Note that compare mesans to actually write something messningful about the differences or similarities you see). For reference see the dides con PageRank posted on the course webpage and refer to Section 2.11 the NCM book. 3. Curve fittin The temperature dependence of the reaction rabe coefficient of a hemical reaction often modeled by the Arrhenius equation k Aexp(-En/(RT)) (1) where is the reaction rate Ais the pre exponential factor En the activation energy, Ris the universal gas constant, and Tis the absolute temperature (K) Experimental data for a particular reaction yield the following results 786 797 810 (a) Use polyfit toget least squares fit of this data to obtain values for and Ea for the reaction. Take R: 8314J/kmol/K° Report the two values and En that you find (b) Plot the data points On the same plot. plot the best- fit curve of your data. Add a legend. axes label: and title to your plot. (c) What is the reaction rate this process at T 3K°? Use fprintf and format string print out your answer. Use floating point notation and show significant figures after the decimal place. 4. Polynomial least squares fitting, Download the data file H20density dat cested on the course webesite. This file contains the measured values of the density of a sas turated liquid water as function of temperature. (a) Determinethe coefficients the lenst squares polynomial Pr that fits the data for degrees n=0,1,2,3, & 4 (use temperature as the independent variable). Report these values. (b) For each polynomial Pn(T) you obtain, also compute the maximum error as defined by where T1 the temperature data. is the density data and Nis the number of data points in the data file. To compute the error. you can use the norm function. For example, err(n) norn(pn rho,inf) where pn the polynomial you found evaluated at the data points Ti and e are the mensured values of density. Report each of the four errors in your write-up. 2 (c) Make niceplot ofthe data and the 1th degree polynomial PAIT that you found evaluated at several values over the range of temperature given. Include the original data in your plot and add legend, axes labels and title to your plot. 5. Oscillatory data fitting, NCM, Chapter 5. Problem 5.8 6. itting planetary orbits. NCM. Chapter Problem 5.12

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4.

a.
function weights = barywghts(x)
    X = repmat(x(:), [1 numel(x)]);
    xi = reshape(X(~eye(size(X))),size(X,1),size(X,2)-1);
    xj = X(:,1:end-1);
    weights = 1./prod(xj - xi, 2);
end

b.
function p = baryinterp(x,f,t)
x = x(:);
t = t(:);
f = f(:);

m = numel(t);
w = barywghts(x);
p = zeros(size(t));

for i=1:m
    interpPt = t(i) == x;
    if any(interpPt)
       temp = f(interpPt);
       p(i) = temp(1);
    else
       deno = sum(w./(t(i)-x),1);
       nume = sum((w.*f)./(t(i)-x),1);
       p(i) = nume/deno;
    end
end
end...
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