95 lines
4.2 KiB
Org Mode
95 lines
4.2 KiB
Org Mode
#+TITLE: Modal Analysis - Derivation of Mathematical Models
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:DRAWER:
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#+STARTUP: overview
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ../index.html
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#+HTML_LINK_UP: ./index.html
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#+HTML_MATHJAX: align: center tagside: right font: TeX
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:latex+ :output-dir figs
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:END:
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* Type of Model
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The model that we want to obtain is a *multi-body model*.
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It is composed of several *solid bodies connected with springs and dampers*.
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The solid bodies are represented with different colors on figure [[fig:nass_solidworks]].
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In the simscape model, the solid bodies are:
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- the granite (1 or 2 solids)
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- the translation stage
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- the tilt stage
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- the spindle and slip-ring
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- the hexapod
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#+name: fig:nass_solidworks
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#+caption: CAD view of the ID31 Micro-Station
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#+attr_html: :width 800px
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[[file:img/nass_solidworks.png]]
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However, each of the DOF of the system may not be relevant for the modes present in the frequency band of interest.
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For instance, the translation stage may not vibrate in the Z direction for all the modes identified. Then, we can block this DOF and this simplifies the model.
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The modal identification done here will thus permit us to determine *which DOF can be neglected*.
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* Some notes about constraining the number of degrees of freedom
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We want to have the two eigen matrices.
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They should have the same size $n \times n$ where $n$ is the number of modes as well as the number of degrees of freedom.
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Thus, if we consider 21 modes, we should restrict our system to have only 21 DOFs.
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Actually, we are measured 6 DOFs of 6 solids, thus we have 36 DOFs.
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From the mode shapes animations, it seems that in the frequency range of interest, the two marbles can be considered as one solid.
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We thus have 5 solids and 30 DOFs.
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In order to determine which DOF can be neglected, two solutions seems possible:
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- compare the mode shapes
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- compare the FRFs
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The question is: in which base (frame) should be express the modes shapes and FRFs?
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Is it meaningful to compare mode shapes as they give no information about the amplitudes of vibration?
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| Stage | Motion DOFs | Parasitic DOF | Total DOF | Description of DOF |
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|---------+-------------+---------------+-----------+--------------------|
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| Granite | 0 | 3 | 3 | |
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| Ty | 1 | 2 | 3 | Ty, Rz |
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| Ry | 1 | 2 | 3 | Ry, |
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| Rz | 1 | 2 | 3 | Rz, Rx, Ry |
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| Hexapod | 6 | 0 | 6 | Txyz, Rxyz |
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|---------+-------------+---------------+-----------+--------------------|
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| | 9 | 9 | 18 | |
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#+TBLFM: $4=vsum($2..$3)
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#+TBLFM: @>$2..$>=vsum(@I..@II)
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