User:Eml4507.s13.team2.cc/R5

Honor Pledge
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions. MATLAB code was based on that from our own Problem 3.1 located here.

Given: A 2-Bar Truss System
The following 2-bar truss system is given in FEAD.F08 notes on p.5-4 and p.11-3.



The properties of each element are given in the table below.

Plot the eigenvectors
Plot the eigenvectors corresponding to the zero eigenvalues of the 2-bar truss system.

Given: A 4-Bar Truss System
The following 4-bar truss system including support is given in FEAD.F08 notes on p.21-3.



The properties of each element are given in the table below.

Plot the eigenvectors
Plot the eigenvectors corresponding to the zero eigenvalues for the truss system.

Finding the eigenvectors
The following MATLAB code is used to find the global stiffness matrix and solve for the eigenvalues and eigenvectors of the 2-bar truss.

The results of the code are given below, where K is the stiffness matrix, V is a matrix of column eigenvectors, and D is a diagonal matrix of the corresponding eigenvalues.

K = 0.5625   0.3248   -0.5625   -0.3248         0         0    0.3248    0.1875   -0.3248   -0.1875         0         0   -0.5625   -0.3248    3.0625   -2.1752   -2.5000    2.5000   -0.3248   -0.1875   -2.1752    2.6875    2.5000   -2.5000         0         0   -2.5000    2.5000    2.5000   -2.5000         0         0    2.5000   -2.5000   -2.5000    2.5000 V = -0.1322   0.3884   -0.6488    0.1712    0.6174   -0.0139   -0.0614   -0.9125   -0.1730    0.0809    0.3565   -0.0080   -0.4668   -0.0888   -0.4571    0.1009   -0.5409    0.5123    0.5182   -0.0860   -0.5050    0.2026   -0.4329   -0.4904   -0.4328   -0.0263   -0.1808   -0.7246   -0.0765   -0.4984    0.5522   -0.0235   -0.2287   -0.6228    0.0765    0.4984 D = -0.0000        0         0         0         0         0         0   -0.0000         0         0         0         0         0         0    0.0000         0         0         0         0         0         0    0.0000         0         0         0         0         0         0    1.4705         0         0         0         0         0         0   10.0295

Plotting the eigenvectors
Looking at the matrix D, the first 4 columns contain zero eigenvalues. Hence, the first 4 columns of the matrix V are the eigenvectors which correspond to zero eigenvalues. These eigenvectors are plotted with the following MATLAB code, repeated for each eigenvector.

The plotted eigenvectors are shown below.









Finding the eigenvectors
The following MATLAB code is used to find the global stiffness matrix and solve for the eigenvalues and eigenvectors of the 4-bar truss with support.

The results of the code are given below, where K is the stiffness matrix, V is a matrix of column eigenvectors, and D is a diagonal matrix of the corresponding eigenvalues. K = Columns 1 through 6 2.1213   2.1213         0         0   -2.1213   -2.1213    2.1213    8.1213         0   -6.0000   -2.1213   -2.1213         0         0    6.0000         0   -6.0000         0         0   -6.0000         0    6.0000         0         0   -2.1213   -2.1213   -6.0000         0    8.1213    2.1213   -2.1213   -2.1213         0         0    2.1213    8.1213         0         0         0         0         0         0         0         0         0         0         0   -6.0000  Columns 7 through 8 0        0         0         0         0         0         0         0         0         0         0   -6.0000         0         0         0    6.0000 V = Columns 1 through 6 0.2910  -0.6675    0.0645    0.2562   -0.5948    0.0000    0.1575    0.4183    0.4471    0.0177   -0.1586   -0.5000   -0.1462   -0.2866    0.5408    0.0639    0.4362    0.5000    0.1575    0.4183    0.4471    0.0177   -0.4362    0.5000   -0.1462   -0.2866    0.5408    0.0639    0.1586   -0.5000    0.5947    0.0374   -0.0292    0.2101    0.1586   -0.0000    0.3402   -0.1935    0.0975   -0.9150    0.0000   -0.0000    0.5947    0.0374   -0.0292    0.2101    0.4362   -0.0000  Columns 7 through 8 -0.0000   0.2150   -0.2887    0.4914   -0.2887    0.2764    0.2887   -0.2764    0.2887   -0.4914   -0.5774   -0.4914    0.0000   -0.0000    0.5774    0.2764 D = Columns 1 through 6 -0.0000        0         0         0         0         0         0   -0.0000         0         0         0         0         0         0   -0.0000         0         0         0         0         0         0    0.0000         0         0         0         0         0         0    3.8183         0         0         0         0         0         0   12.0000         0         0         0         0         0         0         0         0         0         0         0         0  Columns 7 through 8 0        0         0         0         0         0         0         0         0         0         0         0   12.0000         0         0   16.6670

Plotting the eigenvectors
Looking at the matrix D, the first 4 columns contain zero eigenvalues. Hence, the first 4 columns of the matrix V are the eigenvectors which correspond to zero eigenvalues. These eigenvectors are plotted with the following MATLAB code, repeated for each eigenvector.

The plotted eigenvectors are shown below.