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| <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> | ||
| <html xmlns="http://www.w3.org/1999/xhtml"> | ||
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| <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> | ||
| <meta http-equiv="Content-Style-Type" content="text/css" /> | ||
| <meta name="generator" content="pandoc" /> | ||
| <title></title> | ||
| <style type="text/css">code{white-space: pre;}</style> | ||
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| <body> | ||
| <h1 id="homework-4">Homework #4</h1> | ||
| <h2 id="due-32817-by-1159pm">due 3/28/17 by 11:59pm</h2> | ||
| <p><em>Include all work as either an m-file script, m-file function, or example code included with ``` and document your code in the README.md file</em></p> | ||
| <ol style="list-style-type: decimal"> | ||
| <li><p>Create a new github repository called ‘04_linear_algebra’.</p> | ||
| <ol style="list-style-type: lower-alpha"> | ||
| <li><p>Add rcc02007 and pez16103 as collaborators.</p></li> | ||
| <li><p>Clone the repository to your computer.</p></li> | ||
| </ol></li> | ||
| <li><p>Create the 4x4 and 5x5 Hilbert matrix as H:</p></li> | ||
| </ol> | ||
| <ol style="list-style-type: lower-alpha"> | ||
| <li><p>What are the 2-norm, frobenius-norm, 0-norm and infinity-norm of the 4x4 and 5x5 Hilbert matrices?</p></li> | ||
| <li><p>What are the 2-norm, frobenius-norm, 0-norm and infinity-norm of the 4x4 and 5x5 inverse Hilbert matrices?</p></li> | ||
| <li><p>What are the condition numbers for the 2-norm, frobenius-norm, 0-norm and infinity-norm of the 4x4 and 5x5 Hilbert matrices?</p></li> | ||
| </ol> | ||
| <ol start="3" style="list-style-type: decimal"> | ||
| <li><p>Create an LU-decomposition function called <code>lu_tridiag.m</code> that takes 3 vectors as inputs and calculates the LU-decomposition of a tridiagonal matrix. The output should be 3 vectors, the diagonal of the Upper matrix, and the two off-diagonal vectors of the Lower and Upper matrices.</p> | ||
| <p><code>[ud,uo,lo]=lu_tridiag(e,f,g);</code></p></li> | ||
| <li><p>Use the output from <code>lu_tridiag.m</code> to create a forward substitution and back-substitution function called <code>solve_tridiag.m</code> that provides the solution of Ax=b given the vectors from the output of [ud,uo,lo]=lu_tridiag(e,f,g). <em>Note: do not use the backslash solver <code>\</code>, create an algebraic solution</em></p> | ||
| <p><code>x=solve_tridiag(ud,uo,lo,b);</code></p></li> | ||
| </ol> | ||
| <div class="figure"> | ||
| <img src="mass_springs.png" alt="Spring-mass system for problem 5" /> | ||
| <p class="caption">Spring-mass system for problem 5</p> | ||
| </div> | ||
| <ol start="5" style="list-style-type: decimal"> | ||
| <li>Create the stiffness matrix for the 4-mass system shown above for cases a-c. Calculate the condition of the stiffness matrices. What is the expected error when calculating the displacements of the 4 masses? Include the analysis and results in your README.</li> | ||
| </ol> | ||
| <ol style="list-style-type: lower-alpha"> | ||
| <li><p>K1=K2=K3=K4=1000 N/m</p></li> | ||
| <li><p>K1=K3=K4=1000 N/m, K2=1000e12 N/m</p></li> | ||
| <li><p>K1=K3=K4=1000 N/m, K2=1000e-12 N/m</p></li> | ||
| </ol> | ||
| <ol start="6" style="list-style-type: decimal"> | ||
| <li>Use <code>lu_tridiag.m</code> and <code>solve_tridiag.m</code> to solve for the displacements of hanging masses 1-4 shown above, if all masses are 1 kg.</li> | ||
| </ol> | ||
| <div class="figure"> | ||
| <img src="spring_mass.png" alt="Spring-mass system for analysis" /> | ||
| <p class="caption">Spring-mass system for analysis</p> | ||
| </div> | ||
| <ol start="7" style="list-style-type: decimal"> | ||
| <li><p>In the system shown above, determine the three differential equations for the position of masses 1, 2, and 3. Solve for the vibrational modes of the spring-mass system if k1=10 N/m, k2=k3=20 N/m, and k4=10 N/m. The masses are m1=1 kg, m2=2 kg and m3=4 kg. Create a function, <code>mass_spring_vibrate.m</code> that outputs the vibration modes and natural frequencies based upon the parameters, k1, k2, k3, and k4.</p></li> | ||
| <li><p>The curvature of a slender column subject to an axial load P can be modeled by</p></li> | ||
| </ol> | ||
| <p><span class="math inline">\(\frac{d^{2}y}{dx^{2}} + p^{2} y = 0\)</span></p> | ||
| <p>where <span class="math inline">\(p^{2} = \frac{P}{EI}\)</span></p> | ||
| <p>where E = the modulus of elasticity, and I = the moment of inertia of the cross section about its neutral axis.</p> | ||
| <p>This model can be converted into an eigenvalue problem by substituting a centered finite-difference approximation for the second derivative to give <span class="math inline">\(\frac{y_{i+1} -2y_{i} + y_{i-1} }{\Delta x^{2}}+ p^{2} y_{i}\)</span></p> | ||
| <p>where i = a node located at a position along the rod’s interior, and <span class="math inline">\(\Delta x\)</span> = the spacing between nodes. This equation can be expressed as <span class="math inline">\(y_{i-1} - (2 - \Delta x^{2} p^{2} )y_{i} y_{i+1} = 0\)</span> Writing this equation for a series of interior nodes along the axis of the column yields a homogeneous system of equations. (See 13.10 for 4 interior-node example)</p> | ||
| <p>Determine the eigenvalues for a 5-segment (4-interior nodes), 6-segment (5-interior nodes), and 10-segment (9-interior nodes). Using the modulus and moment of inertia of a pole for pole-vaulting ( <a href="http://people.bath.ac.uk/taf21/sports_whole.htm" class="uri">http://people.bath.ac.uk/taf21/sports_whole.htm</a>) E=76E9 Pa, I=4E-8 m^4, and L= 5m.</p> | ||
| <p>Include a table in the <code>README.md</code> that shows the following results: What are the largest and smallest loads in the beam based upon the different shapes? How many eigenvalues are there?</p> | ||
| <pre><code>| # of segments | largest load (N) | smallest load (N) | # of eigenvalues | | ||
| | --- | --- | --- | --- | | ||
| | 5 | ... | ... | ... | | ||
| | 6 | ... | ... | ... | | ||
| | 10 | ... | ... | ... |</code></pre> | ||
| <p>If the segment length (<span class="math inline">\(\Delta x\)</span>) approaches 0, how many eigenvalues would there be?</p> | ||
| </body> | ||
| </html> |
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