User:EML4500.f08.FEABBQ.Jayma/HW4

Notes for October 13, 2008
The four zero-eigenvalues of the $$ \underline{K}$$ matrix correspond to
 * 3 Rigid Body Modes
 * 1 Mechanism

HW 1
Plot the eigenvectors corresponding to the zero eigenvalues of the 2-bar truss system; interpret the results.

Before we do this assignment, it is important to understand the nature of these mode shapes. These mode shapes (that correspond to the zero eigenvalues) may come out as a linear combination of the pure mode shapes (pure rigid body motions and pure mechanism(s)). For example, take a body that has only two rigid body motions: horizontal and directly vertical. The linear combination of these two pure modes would be a diagonal motion. ''In Structural Mechanics, the eigenvalues represent the frequencies while the eigenvectors represent the mode shapes during free vibration. The lower frequencies (eigenvalues) tend to have more prominent vibration.''

Eigenvalue Problem: $$ \underline{K}\underline{v}=\lambda\underline{v} $$

Let {$$ \underline{u}_{1}, \underline{u}_{2}, \underline{u}_{3}, \underline{u}_{4} $$} be the pure eigenvectors corresponding to the four zero eigenvalues.

$$ \underline{K}_{_{6x6}}\underline{u}_{i_{6x1}}=0\cdot\underline{u}_{i_{6x1}}=\underline{0}_{_{6x1}} $$, where i=1,...,4.

A linear combination of {$$ \underline{u}_{i}, i=1,...,4$$}

$$ \sum_{i=1}^4 \alpha_{i_{1x1}} \underline{u}_{i_{6x1}} =: \underline{W}_{_{6x1}} $$, where $$ \alpha_{i} $$ are real numbers.

Note: the $$=:$$ means "equality definition."

$$ \underline{W} $$ is also an eigenvector corresponding to a zero eigenvalue: $$ \underline{K}\underline{W}=\underline{K}(\sum_{i=1}^4\alpha_{i} \underline{u}_{i})=\sum_{i=1}^4 \alpha_{i}(\underline{K}\underline{u}_{i})=\underline{0}=0\cdot\underline{W} $$

From the two-bar truss, we have the following output of $$ \underline{K} $$

And the following is the are the eigenvectors and eigenvalues, using the notation [V,D], where V are the eigenvectors and D are the eigenvalues.

The first four column vectors of V, which correspond to the four zero eigenvalues, are the eigenvectors for those zero eigenvalues. The matrix V is called the modal matrix.

The following are the four column vectors (eigenvectors) plotted as mode shapes. Note: they are shown in the order from the first column to the fourth column of the modal matrix.









HW2
Solve for $$ \underline{\overline{K}}\underline{v}=\lambda\underline{v} $$, the stiffness matrix for the constrained system below. Plot the eigenvectors corresponding to the zero eigenvalues for case (a).



The following is the Matlab code which solves for the global stiffness matrix, $$ \underline{K} $$, for the three bar truss in case (a) above.

The following is the output for the stiffness matrix of the three bar truss in case (a)

Using the method to solve for the eigenvectors of case(a), we get the following. Note: the first five column vectors correspond to zero eigenvalues.

The first five eigenvectors above correspond to zero eigenvalues, therefore their mode shapes are plotted below in order.











The Matlab code which solves for case (a) can be modified to solve for the stiffness matrix, $$ \underline{K} $$ for the four bar truss, case (b)

The following is the output for the stiffness matrix, $$ \underline{K} $$, for the four bar truss system.