nass-micro-station-measurements/modal-analysis/index.org

#+TITLE: Modal Analysis of the ID31 Micro-Station
:DRAWER:
#+STARTUP: overview

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#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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The goal is to experimentally extract a *Spatial Model* (mass, damping, stiffness) of the structure (shown on figure [[fig:nass_picture]]) in order to tune the Multi-Body model.

#+name: fig:nass_picture
#+caption: Picture of the ID31 Micro-Station. (1) Granite (2) Translation Stage (3) Tilt Stage (4) Hexapod (5) Dummy Mass
#+attr_html: :width 500px
[[file:img/nass_picture.png]]

The procedure is represented on figure [[fig:vibration_analysis_procedure]] where we go from left to right.

#+name: fig:vibration_analysis_procedure
#+caption: Vibration Analysis Procedure
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[[file:img/vibration_analysis_procedure.png]]

The steps are:
- we obtain a *Response Model* (Frequency Response Functions) from measurements (described [[file:measurement.org][here]])
- the response model is further converted into a *Modal Model* (Natural Frequencies and Mode Shapes) (described [[file:modal_extraction.org][here]])
- this is converted into a *Spatial Model* with the Mass/Damping/Stiffness matrices (described [[file:mathematical_model.org][here]])

Theses matrices will be used to tune the Simscape (multi-body) model.

The modes we want to identify are those in the frequency range between 0Hz and 150Hz.

#+name: tab:terminology_numbers
#+caption: Terminology for further analysis
|  <c>   | <l>                             |  <c>  |
| Symbol | Meaning                         | Value |
|--------+---------------------------------+-------|
|  $p$   | Number of solid body considered |   6   |
|  $m$   | Number of accelerometers        |  23   |
|  $n$   | Number of identified modes      |  21   |
|  $q$   | Number of frequency points      |  801  |
|  $s$   | Number of excitation            |   3   |

#+name: tab:terminology_elements
#+caption: Terminology for further analysis
|       <c>       | <l>                                  |
|     Symbol      | Meaning                              |
|-----------------+--------------------------------------|
|   $[\Lambda]$   | Complex eigen value matrix           |
|    $[\Psi]$     | Complex eigen vector matrix          |
|   $\omega_r$    | Eigen frequency of mode $r$ [rad/s]  |
|     $\xi_r$     | Modal damping for mode $r$           |
|  $\{\psi\}_r$   | Complex mode shape of mode $r$       |
| $[M], [C], [K]$ | Mass, damping and stiffness matrices |
|      $a_r$      | "Modal A" for mode $r$               |

The modal analysis of the ID31 Micro-station thus consists of several parts:
- [[file:measurement.org][Frequency Response Measurements]]
- [[file:frf_processing.org][Frequency Response Analysis and Processing]]
- [[file:modal_extraction.org][Modal Parameter Extraction]]
- [[file:mathematical_model.org][Derivation of Mathematical Model]]

#+name: fig:modal_test_procedure_vertical
#+caption: Modal Anslysis Procedure
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[[file:figs/modal_test_procedure_vertical.png]]