From 4963f620e8545e4e268932bd6778e478ba9e956a Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Mar 2017 15:31:30 -0400 Subject: [PATCH] added HW 5 --- HW5/README.html | 53 ++++ HW5/README.md | 83 +++++ HW5/README.pdf | Bin 0 -> 172036 bytes HW5/lu_tridiag.m | 12 + HW5/solve_tridiag.m | 11 + HW5/spring_mass.png | Bin 0 -> 5202 bytes HW5/spring_mass.svg | 299 ++++++++++++++++++ HW5/test_arrays.m | 8 + lecture_10/octave-workspace | Bin 134485 -> 153 bytes lecture_12/octave-workspace | Bin 11 -> 153 bytes lecture_15/eigenvalues.gif | Bin 0 -> 2616889 bytes lecture_15/lecture_15.ipynb | 159 ++++++---- lecture_15/octave-workspace | Bin 936 -> 153 bytes lecture_15/output/00001.png | Bin 57294 -> 28343 bytes lecture_15/output/00002.png | Bin 58296 -> 29519 bytes lecture_15/output/00003.png | Bin 58114 -> 29621 bytes lecture_15/output/00004.png | Bin 56973 -> 29834 bytes lecture_15/output/00005.png | Bin 58013 -> 29817 bytes lecture_15/output/00006.png | Bin 57890 -> 29742 bytes lecture_15/output/00007.png | Bin 58398 -> 29850 bytes lecture_15/output/00008.png | Bin 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lecture_15/output/00099.png | Bin 57725 -> 29676 bytes lecture_15/output/00100.png | Bin 57667 -> 29581 bytes "lecture_15/\340\244\267\001" | 0 "lecture_15/\360&\341\001" | 0 .../lecture_16-checkpoint.ipynb | 6 + lecture_16/compiled_data.csv | 233 ++++++++++++++ lecture_16/lecture_16.ipynb | 54 ++++ lecture_16/octave-workspace | Bin 0 -> 153 bytes 119 files changed, 849 insertions(+), 69 deletions(-) create mode 100644 HW5/README.html create mode 100644 HW5/README.md create mode 100644 HW5/README.pdf create mode 100644 HW5/lu_tridiag.m create mode 100644 HW5/solve_tridiag.m create mode 100644 HW5/spring_mass.png create mode 100644 HW5/spring_mass.svg create mode 100644 HW5/test_arrays.m create mode 100644 lecture_15/eigenvalues.gif create mode 100644 "lecture_15/\340\244\267\001" create mode 100644 "lecture_15/\360&\341\001" create mode 100644 lecture_16/.ipynb_checkpoints/lecture_16-checkpoint.ipynb create mode 100644 lecture_16/compiled_data.csv create mode 100644 lecture_16/lecture_16.ipynb create mode 100644 lecture_16/octave-workspace diff --git a/HW5/README.html b/HW5/README.html new file mode 100644 index 0000000..96595a9 --- /dev/null +++ b/HW5/README.html @@ -0,0 +1,53 @@ + + + + + + + + + + + +

Homework #5

+

due 3/28/17 by 11:59pm

+

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

+
    +
  1. Create a new github repository called ‘linear_algebra’.

    +
      +
    1. Add rcc02007 and pez16103 as collaborators.

    2. +
    3. Clone the repository to your computer.

    4. +
  2. +
  3. Create an LU-decomposition function called lu_tridiag.m 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.

    +

    [ud,uo,lo]=lu_tridiag(e,f,g);

  4. +
  5. Use the output from lu_tridiag.m to create a forward substitution and back-substitution function called solve_tridiag.m that provides the solution of Ax=b given the vectors from the output of [ud,uo,lo]=lu_tridiag(e,f,g). Note: do not use the backslash solver \, create an algebraic solution

    +

    x=solve_tridiag(ud,uo,lo,b);

  6. +
  7. Test your function on the matrices A3, A4, …, A10 generated with test_arrays.m Solving for b=ones(N,1); where N is the size of A. In your README.md file, compare the norm of the error between your result and the result of AN.

  8. +
+
| size of A | norm(error) |
+|-----------|-------------|
+|  3 | ... |
+| 4 | ... |
+
+Spring-mass system for analysis +

Spring-mass system for analysis

+
+
    +
  1. 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. Determine the eigenvalues and natural frequencies.

  2. +
  3. The curvature of a slender column subject to an axial load P (Fig. P13.10) can be modeled by

  4. +
+

\(\frac{d^{2}y}{dx^{2}} + p^{2} y = 0\)

+

where \(p^{2} = \frac{P}{EI}\)

+

where E = the modulus of elasticity, and I = the moment of inertia of the cross section about its neutral axis.

+

This model can be converted into an eigenvalue problem by substituting a centered finite-difference approximation for the second derivative to give \(\frac{y_{i+1} − 2y_{i} + y_{i−1} }{\Delta x^{2}}+ p^{2} y_{i}\)

+

where i = a node located at a position along the rod’s interior, and \(\Delta x\) = the spacing between nodes. This equation can be expressed as \(y_{i−1} − (2 − \Delta x^{2} p^{2} )y_{i} + y_{i+1} = 0\) 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)

+

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 ( http://people.bath.ac.uk/taf21/sports_whole.htm) E=76E9 Pa, I=4E-8 m^4, and L= 5m.

+

Include a table in the README.md that shows the following results: What are the largest and smallest eigenvalues for the beam? How many eigenvalues are there?

+
| # of segments | largest | smallest | # of eigenvalues |
+| --- | --- | --- | --- |
+| 5 | ... | ... | ... |
+| 6 | ... | ... | ... |
+| 10 | ... | ... | ... |
+

If the segment length approaches 0, how many eigenvalues would there be?

+ + diff --git a/HW5/README.md b/HW5/README.md new file mode 100644 index 0000000..a8495ea --- /dev/null +++ b/HW5/README.md @@ -0,0 +1,83 @@ +# Homework #5 +## due 3/28/17 by 11:59pm + +*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* + +1. Create a new github repository called 'linear_algebra'. + + a. Add rcc02007 and pez16103 as collaborators. + + b. Clone the repository to your computer. + +2. Create an LU-decomposition function called `lu_tridiag.m` 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. + + ```[ud,uo,lo]=lu_tridiag(e,f,g);``` + +3. Use the output from `lu_tridiag.m` to create a forward substitution and +back-substitution function called `solve_tridiag.m` that provides the solution of +Ax=b given the vectors from the output of [ud,uo,lo]=lu_tridiag(e,f,g). *Note: do not use +the backslash solver `\`, create an algebraic solution* + + ```x=solve_tridiag(ud,uo,lo,b);``` + +4. Test your function on the matrices A3, A4, ..., A10 generated with `test_arrays.m` +Solving for `b=ones(N,1);` where N is the size of A. In your `README.md` file, compare +the norm of the error between your result and the result of AN\b. + +``` +| size of A | norm(error) | +|-----------|-------------| +| 3 | ... | +| 4 | ... | +``` + +![Spring-mass system for analysis](spring_mass.png) + +5. 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. Determine +the eigenvalues and natural frequencies. + +6. The curvature of a slender column subject to an axial load P (Fig. P13.10) can be +modeled by + +$\frac{d^{2}y}{dx^{2}} + p^{2} y = 0$ + +where $p^{2} = \frac{P}{EI}$ + +where E = the modulus of elasticity, and I = the moment of inertia of the cross section +about its neutral axis. + +This model can be converted into an eigenvalue problem by +substituting a centered finite-difference approximation for the second derivative to give +$\frac{y_{i+1} -2y_{i} + y_{i-1} }{\Delta x^{2}}+ p^{2} y_{i}$ + +where i = a node located at a position along the rod’s interior, and $\Delta x$ = the +spacing between nodes. This equation can be expressed as $y_{i-1} - (2 - \Delta x^{2} +p^{2} )y_{i} y_{i+1} = 0$ 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) + +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 ( +[http://people.bath.ac.uk/taf21/sports_whole.htm](http://people.bath.ac.uk/taf21/sports_whole.htm)) +E=76E9 Pa, I=4E-8 m^4, and L= 5m. + +Include a table in the `README.md` that shows the following results: +What are the largest and smallest eigenvalues for the beam? How many eigenvalues are +there? + +``` +| # of segments | largest | smallest | # of eigenvalues | +| --- | --- | --- | --- | +| 5 | ... | ... | ... | +| 6 | ... | ... | ... | +| 10 | ... | ... | ... | +``` + +If the segment length ($\Delta x$) approaches 0, how many eigenvalues would there be? diff --git a/HW5/README.pdf b/HW5/README.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fdb31a30d55578b3387b633dcd150b76d0860848 GIT binary patch literal 172036 zcmd42W2|UFwC8zj+qP}nwr$(Cb&qY^wr%4c+g9J#)34K+p5*n1nPgV#!>ZazRcclC z-oN^<&kwLMpgm_0(&DXC>|atdKptYa~BH&CMGs^g8yz%^kSAa zE~ZWd^kO!KE~X-;#`Y$rP<(t)&Mr=-hPF^1n}0Mk?YGAf{a))Z2;dJ0hTQUa*ye-T zE=9Z7F|suL8e!?p!;&!W6FfB*iTL?cbyt{=JWR;Z_IsmT{Zw^%`Sp4{-kVaXO+ylUXlzOE8peziW0E$zBxlcRY_(>7Q|vI-P#GJE30c6n7NP zy(mjApOnn%Ri>b6n>>j)m-son`wn$)Q=xknBAb+iq;N8T{NzX1Pf$6^nQUBC0?yWL z>GEDDb~!65)*Z4$TIrB30G~&E0>_%N?xynQpwV8l-h*4(%;1FS>hHv36IuRkiTXO{ zQbSbMk-xU9?sOBLsx@DY7j@<@+p59!fcz!CPAWH2b&R&a6Ts*CBxrf+{GG*WlW7D; 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zwI7vuH~Z^1Lc)>mH@cJG1GF8)|HR@)jsTlA)VYScW{KF|^M|$?rvGXZQ#@1{9UHm-b*T(B|cv7_8p{M7i{ z&o`NaKUK**TEc#`3{giOKF1)M)&R+PU3+YPKh-8arVJP2vo@h3Q;`)!!&%Q?kH0;# zJH-Wke#Z-M|CW$upPb&|I}?5WCH6MAF3-h6*fiq?5cBVQ>x}Xt*4Xynw3b?z&(+yx zRxFNWNJa2QrC2!!&q{!A@)+C4mmq=9|Lw|9_$O' to search the manual index.\n", "\n", - " 5 0\n", - " 0 15\n", - "\n" + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" ] } ], "source": [ - "m=40; % mass in kg\n", - "k=200; % spring constant in N/m\n", - "\n", - "K=[2*k/m,-k/m;-k/m,2*k/m]\n", - "\n", - "[v,e]=eig(K)" + "help eig" ] }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -240,59 +251,38 @@ "name": "stdout", "output_type": "stream", "text": [ - "ans =\n", - "\n", - " -10.607\n", - " 10.607\n", + "K =\n", "\n", - "ans =\n", + " 10 -5\n", + " -5 10\n", "\n", - " -10.607\n", - " 10.607\n", - "\n" - ] - } - ], - "source": [ - "K*v(:,2)\n", - "e(2,2)*v(:,2)" - ] - }, - { - "cell_type": "code", - "execution_count": 26, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ "v =\n", "\n", - " 0.78868 -0.57735 0.21132\n", - " -0.57735 -0.57735 0.57735\n", - " 0.21132 0.57735 0.78868\n", + " -0.70711 -0.70711\n", + " -0.70711 0.70711\n", "\n", "e =\n", "\n", "Diagonal Matrix\n", "\n", - " 1.2679 0 0\n", - " 0 3.0000 0\n", - " 0 0 4.7321\n", + " 5 0\n", + " 0 15\n", "\n" ] } ], "source": [ - "[v,e]=eig(A)" + "m=40; % mass in kg\n", + "k=200; % spring constant in N/m\n", + "\n", + "K=[2*k/m,-k/m;-k/m,2*k/m]\n", + "\n", + "[v,e]=eig(K)" ] }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -303,22 +293,20 @@ "text": [ "ans =\n", "\n", - " 1.00000\n", - " -0.73205\n", - " 0.26795\n", + " -10.607\n", + " 10.607\n", "\n", "ans =\n", "\n", - " 1.00000\n", - " -0.73205\n", - " 0.26795\n", + " -10.607\n", + " 10.607\n", "\n" ] } ], "source": [ - "A*v(:,1)\n", - "e(1,1)*v(:,1)" + "K*v(:,2)\n", + "e(2,2)*v(:,2)" ] }, { @@ -364,7 +352,7 @@ }, { "cell_type": "code", - "execution_count": 55, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -411,7 +399,7 @@ }, { "cell_type": "code", - "execution_count": 57, + "execution_count": 9, "metadata": { "collapsed": false }, @@ -563,6 +551,39 @@ "![solution to principle stresses](stress_soln.svg)" ] }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "e =\n", + "\n", + " -10\n", + " 0\n", + " 10\n", + "\n" + ] + } + ], + "source": [ + "e=eig(s)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, { "cell_type": "code", "execution_count": null, diff --git a/lecture_15/octave-workspace 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