## Linear Dynamics

FEA is an extremely important tool for the analysis of the dynamics of structures. We have conducted FEA-based dynamic analyses of multi-storied buildings and verified the results with physical measurements. We have also conducted dynamic analyses on components for the International Space Station (ISS). We routinely evaluate the Eigenvalues (Modal Frequencies) and Eigenvectors (Mode Shapes) of systems and components in the design stage. We then use these values to assess the likely response to internal and external loads. Such analyses can identify and lead to the correction of problems without the expense of building prototypes. Some terms that are used in linear dynamics include:

• Modal frequencies – the so-called “natural frequencies” of a system or component.
• Mode Shapes – are the shapes that a structure of component takes when vibrating at a modal frequency.

## Response Prediction

The eigenvalues and eigenvectors tell us the preferred frequencies and shapes at which a system or component vibrates, but they do not tell us anything about the magnitude of the response to an applied load. Depending on the nature of the loading, several procedures can be used to compute the response. The response computation procedures using the eigenvalues and eigenvectors include:

• ## Harmonic

• This procedure computes the response of the system to a harmonic load such as a piece of rotating equipment. The response to loading at a series of frequencies, such as would be observed during a swept sine test, is also computed using this method.
• ## Power Spectral Density

• Loads that can best be described statistically, such as transportation-induced loading, are typically applied using the Power Spectral Density (PSD) function as a system input. The output for this analysis is the mean deflection and stress. Statistical limits on the expected deflection and stresses can then be computed from the mean.
• ## Response Spectrum

• The Response (or Shock) Spectrum technique is used to compute the maximum deflection and stress due to some event. This procedure is commonly used in seismic analysis and in the duplication of shock tests.
• ## Time History

• This procedure computes the response of a system to a specified loading based upon its time history. In general, the harmonic and response spectrum techniques (above) represent shorthand methods of arriving at the same conclusions reached by the Time History method.

## Transient Dynamic Analysis

Objects move, and although in most cases they can be analyzed either through static techniques or through frequency response techniques, sometimes it is necessary to analyze their complete motion. PMI has a suite of tools that include Algor, ANSYS and LS-DYNA for highly non-linear analyses. We have performed a multitude of transient analyses, some of which included non-linearities such as contact and non-linear materials. We have also performed these analyses on very large models. One example was a complete laptop computer model that contained 650,000 elements that was used to perform drop testing. This model required multiple load steps to assemble the components in order to provide the correct pre-stresses prior to performing the drop test. The model also included fracture models for the LCD screen and for the board mounted components.