User:Eml4500.f08.ateam.carr/HW4

Eigenvalue Analysis - Continuation
The links below illustrate the application of Eigenvalue Modal Analysis to real world examples:


 * Frequency analysis of shell structures with contact
 * Frequency Solutions with Contact

The first link demonstrates various modes of the oscillations in a shell structure. The second link is an example of rigid body modes with zero eigenvalues.

In order to generate the plots of modal analysis, the Eigenvalue is simply multiplied by a sinusoidal function. For example:

$$d(t)=\lambda cos(\omega t)$$

where $$ \omega= \frac{2\pi}{f}$$

f is the frequency of vibration.undefined

Assembly of Element Stiffness Matrix into Global Stiffness Matrix
Consider the previous example of the two bar truss and recall the element force-displacement relationship:

$$k^{e}d^{e}=f^{e}$$

We have previously proved the equilibrium of Node 2 from the free body diagrams of Element 1 and Element 2.

$$d_{3}=d_3^{(1)}=d_1^{(2)}$$

$$d_{4}=d_4^{(1)}=d_2^{(2)}$$

For Node 2, identify the global degree of freedom to element degree of freedom for both Element 1 and 2.

Equilibrium of Node 2


$$ \sum F_x=0=-f_{3}^{(1)}-f_{1}^{(2)} $$

$$ \sum F_y=0=P-f_{4}^{(1)}-f_{2}^{(2)} $$

Next replace each f above with the individual element kd relationship.