188 lines
6.5 KiB
Org Mode
188 lines
6.5 KiB
Org Mode
#+TITLE: Flexible Joint - Test Bench
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:DRAWER:
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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_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
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#+BIND: org-latex-image-default-option "scale=1"
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#+BIND: org-latex-image-default-width ""
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#+LaTeX_CLASS: scrreprt
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#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full]
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#+LaTeX_HEADER_EXTRA: \input{preamble.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:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :noweb yes
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results file raw replace
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#+PROPERTY: header-args:latex+ :buffer no
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#+PROPERTY: header-args:latex+ :tangle no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports results
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
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:END:
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#+begin_export html
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<hr>
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<p>This report is also available as a <a href="./test-bench-flexible-joints.pdf">pdf</a>.</p>
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<hr>
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#+end_export
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* Flexible Joints - Requirements
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| | *Specification* |
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|-------------------+-----------------|
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| Axial Stiffness | > 200 [N/um] |
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| Shear Stiffness | > 1 [N/um] |
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| Bending Stiffness | < 100 [Nm/rad] |
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| Torsion Stiffness | < 500 [Nm/rad] |
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| Bending Stroke | > 1 [mrad] |
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| Torsion Stroke | > 5 [urad] |
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* Test Bench Description
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** Introduction :ignore:
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The main characteristic of the flexible joint that we want to measure is its bending stiffness $k_{R_x} \approx k_{R_y}$.
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To do so, a test bench is used.
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Specifications of the test bench to precisely measure the bending stiffness are described in this section.
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The basic idea is to measured the angular deflection of the flexible joint as a function of the applied torque.
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#+name: fig:test-bench-schematic
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#+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints
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[[file:figs/test-bench-schematic.png]]
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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** Flexible joint Geometry
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The flexible joint used for the Nano-Hexapod is shown in Figure [[fig:flexible_joint_geometry]].
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Its bending stiffness is foreseen to be $k_{R_y}\approx 20\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx 20\,mrad$.
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#+name: fig:flexible_joint_geometry
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#+caption: Geometry of the flexible joint
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[[file:figs/flexible_joint_geometry.png]]
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The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 20\,mm$.
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Let's define the parameters on Matlab.
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#+begin_src matlab
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kRx = 20; % Bending Stiffness [Nm/rad]
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Rxmax = 20e-3; % Bending Stroke [rad]
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h = 20e-3; % Height [m]
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#+end_src
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** Required external applied force
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The bending $\theta_y$ of the flexible joint due to the force $F_x$ is:
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\begin{equation}
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\theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
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\end{equation}
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Therefore, the applied force to test the full range of the flexible joint is:
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\begin{equation}
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F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
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\end{equation}
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#+begin_src matlab
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Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N]
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#+end_src
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And we obtain:
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#+begin_src matlab :results value raw replace :exports results
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sprintf('\\begin{equation} F_{max} = %.1f\\, [N] \\end{equation}', Fxmax)
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#+end_src
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#+RESULTS:
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\begin{equation} F_{max} = 20.0\, [N] \end{equation}
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The measurement range of the force sensor should then be higher than $20\,N$.
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** Required actuator stroke and sensors range
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The flexible joint is designed to allow a bending motion of $\pm 20\,mrad$.
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The corresponding actuator stroke to impose such motion is:
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\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
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#+begin_src matlab
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dxmax = h*tan(Rxmax);
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#+end_src
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#+begin_src matlab :results value raw replace :exports results
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sprintf('\\begin{equation} d_{max} = %.1f\\, [mm] \\end{equation}', 1e3*dxmax)
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#+end_src
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#+RESULTS:
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\begin{equation} d_{max} = 0.4\, [mm] \end{equation}
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In order to test the full range of the flexible joint, the stroke of the actuator should be higher than $0.4\,mm$.
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The measurement range of the displacement sensor should also be higher than $0.4\,mm$.
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** First try with the APA95ML
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The APA95ML as a stroke of $100\,\mu m$ and the encoder in parallel can easily measure the required stroke.
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Suppose the full stroke of the APA can be used to bend the flexible joint (ideal case), the measured force will be:
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#+begin_src matlab
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Fxmax = kRx*100e-6/h^2; % Force at maximum stroke [N]
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#+end_src
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#+begin_src matlab :results value raw replace :exports results
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sprintf('\\begin{equation} F_{max} = %.1f\\, [N] \\end{equation}', Fxmax)
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#+end_src
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#+RESULTS:
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\begin{equation} F_{max} = 5.0\, [N] \end{equation}
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And the tested angular range is:
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#+begin_src matlab
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Rmax = tan(100e-6/h);
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#+end_src
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#+begin_src matlab :results value raw replace :exports results
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sprintf('\\begin{equation} \\theta_{max} = %.1f\\, [mrad] \\end{equation}', 1e3*Rmax)
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#+end_src
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#+RESULTS:
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\begin{equation} \theta_{max} = 5.0\, [mrad] \end{equation}
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** Test Bench
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#+name: fig:test-bench-schematic
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#+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints
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[[file:figs/test-bench-schematic.png]]
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#+begin_note
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- Manual Translation Stage
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- Load Cell TE Connectivity [[file:doc/A700000007147087.pdf][FC2231-0000-0010-L]]
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- Encoder: Renishaw [[file:doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf][Resolute 1nm]]
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#+end_note
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* Experimental measurement
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