353 lines
13 KiB
Org Mode
353 lines
13 KiB
Org Mode
#+TITLE: Flexible Joint - Measurement of the Bending Stiffness
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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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* Introduction :ignore:
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The goal is to design a test bench to measure the bending stiffness of a flexible joint with 1% accuracy.
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* Finite Element Model
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<<sec:fem>>
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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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From the Finite Element Model, the stiffnesses and strokes of the flexible joint have been computed.
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#+begin_src matlab
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ka = 94e6; % Axial Stiffness [N/m]
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ks = 13e6; % Shear Stiffness [N/m]
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kb = 5; % Bending Stiffness [Nm/rad]
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kt = 260; % Torsional Stiffness [Nm/rad]
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#+end_src
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#+begin_src matlab
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Fa = 469; % Axial Force before yield [N]
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Fs = 242; % Shear Force before yield [N]
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Fb = 0.118; % Bending Force before yield [Nm]
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Ft = 1.508; % Torsional Force before yield [Nm]
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#+end_src
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#+begin_src matlab
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Xa = Fa/ka; % Axial Stroke before yield [m]
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Xs = Fs/ks; % Shear Stroke before yield [m]
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Xb = Fb/kb; % Bending Stroke before yield [rad]
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Xt = Ft/kt; % Torsional Stroke before yield [rad]
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([1e-6*ka, Fa, 1e6*Xa; 1e-6*ks, Fs, 1e6*Xs], {'Axial', 'Shear'}, {'Stiffness [N/um]', 'Max Force [N]', 'Stroke [um]'}, ' %.0f ');
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#+end_src
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#+RESULTS:
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| | Stiffness [N/um] | Max Force [N] | Stroke [um] |
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|-------+------------------+---------------+-------------|
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| Axial | 94 | 469 | 5 |
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| Shear | 13 | 242 | 19 |
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([kb, 1e3*Fb, 1e3*Xb; kt, 1e3*Ft, 1e3*Xt], {'Bending', 'Torsional'}, {'Stiffness [Nm/rad]', 'Max Torque [Nmm]', 'Stroke [mrad]'}, ' %.0f ');
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#+end_src
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#+RESULTS:
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| | Stiffness [Nm/rad] | Max Torque [Nmm] | Stroke [mrad] |
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|-----------+--------------------+------------------+---------------|
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| Bending | 5 | 118 | 24 |
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| Torsional | 260 | 1508 | 6 |
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* Setup
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<<sec:setup>>
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Let's say a force is applied on top of the flexible joint with a distance $H_F$ with the joint's center.
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The displacement of the flexible joint is also measure at an height $H_D$ from the joint's center.
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#+name: fig:flexible_joint_test_bench_bending_setup
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#+caption: Figure caption
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[[file:figs/flexible_joint_test_bench_bending_setup.png]]
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* Effect of Bending
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<<sec:bending>>
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The torque applied is:
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\begin{equation}
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M_b = F \cdot H_F
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\end{equation}
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The flexible joint is experiencing a rotation $R_b$ due to the torque $M_b$:
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\begin{equation}
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R_b = \frac{M_b}{k_b} = \frac{F \cdot H_F}{k_b}
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\end{equation}
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This rotation is then measured by the displacement sensor.
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The measured displacement is:
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\begin{equation}
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D_b = H_D \tan(R_b) = H_D \tan\left( \frac{F \cdot H_F}{k_b} \right) \label{eq:bending_meaured_disp}
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\end{equation}
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* Computation of the bending stiffness
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From equation eqref:eq:bending_meaured_disp, we can compute the bending stiffness:
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\begin{equation}
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k_b = \frac{\tan^{-1}\left( \frac{D_b}{H_D} \right)}{F \cdot H_F}
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\end{equation}
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And therefore, to precisely measure $k_b$, we need to:
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- precisely measure the motion $D_b$
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- precisely measure the applied force $F$
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- precisely know the height from the flexible joint's center to the measurement point $H_D$
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- precisely know the height from the flexible joint's center to the force application point $H_F$
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If there are estimation errors for $H_D$ or $H_F$ as shown in Figure [[fig:bending_effect_error_vertical]], this will induce an error for the estimation of the stiffness.
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For 1% accuracy estimation of $k_b$, we can write the following approximate requirements:
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| | Accuracy |
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|--------------------------+----------|
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| Force Measurement | 1% |
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| Displacement Measurement | 1% |
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| $H_D$ | 1% |
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| $H_F$ | 1% |
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#+name: fig:bending_effect_error_vertical
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#+caption: Error in the estimation of the height of the force sensor and displacement sensor
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[[file:figs/bending_effect_error_vertical.png]]
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* Effect of Shear
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<<sec:shear>>
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The effect of Shear on the measured displacement is simply:
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\begin{equation}
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D_s = \frac{F}{k_s}
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\end{equation}
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We would like to have this displacement much smaller than the displacement induced by the bending effects:
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\begin{equation}
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D_b \gg D_s
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\end{equation}
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Which is equivalent as to have:
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\begin{equation}
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H_D \tan\left( \frac{F \cdot H_F}{k_b} \right) \gg \frac{F}{k_s}
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\end{equation}
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Here to simplify, we suppose $FH_F/k_b \ll 1$ (which is the case in practice), and we suppose $H_D = H_F = H$.
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The obtained condition is then:
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\begin{equation}
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H \gg \frac{k_b}{k_s}
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\end{equation}
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable(1e3*[1e1*sqrt(kb/ks), 1e2*sqrt(kb/ks), 1e3*sqrt(kb/ks)], {'$H\,[mm]$'}, {'10% error', '1% error', '0.1% error'}, ' %.0f ');
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#+end_src
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#+name: tab:effect_shear
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#+caption: Height $H$ to have less than certain amount of error due to shear effects
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#+attr_latex: :environment tabularx :width \linewidth :align lXXX
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#+attr_latex: :center t :booktabs t :float t
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#+RESULTS:
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| | 10% error | 1% error | 0.1% error |
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|-----------+-----------+----------+------------|
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| $H\,[mm]$ | 6 | 62 | 620 |
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In order to limit the effect of shear of less than 1%, the height from the joint's center to the force application point and to the measurement point should be larger than 62mm.
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#+begin_src matlab
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H = 62e-3; % [m]
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#+end_src
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* Effect of Torsion
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<<sec:torsion>>
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If the application force is not aligned with the vertical axis of the flexible joint, this will induce a torsion motion that will induce a measurement error (Figure [[fig:bending_effect_torsion]]).
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#+name: fig:bending_effect_torsion
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#+caption: Horizontal position error of the force sensor and displacement sensor
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[[file:figs/bending_effect_torsion.png]]
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Let's note the offset of the force sensor $\epsilon_{F,y}$ and the offset of the measurement point $\epsilon_{D,y}$.
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The vertical torque (torsion) will be equal to:
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\begin{equation}
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M_t = F \cdot \epsilon_{F,y}
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\end{equation}
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And the induced torsion:
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\begin{equation}
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R_t = \frac{M_t}{k_t} = \frac{F \cdot \epsilon_{F,y}}{k_t}
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\end{equation}
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The effect on the measured displacement is:
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\begin{equation}
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D_t = \epsilon_{D,y} \tan \left( R_t \right) = \epsilon_{D,y} \tan\left( \frac{F \cdot \epsilon_{F,y}}{k_t} \right)
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\end{equation}
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And we would like to have:
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\begin{equation}
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D_b \gg D_t
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\end{equation}
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Which is equivalent as to have:
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\begin{equation}
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H_D \tan\left( \frac{F \cdot H_F}{k_b} \right) \gg \epsilon_{D,y} \tan\left( \frac{F \cdot \epsilon_{F,y}}{k_t} \right)
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\end{equation}
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Supposing $H_F = H_D = H$ and $\epsilon_{F,y} = \epsilon_{D,y} = \epsilon_{y}$, the condition becomes:
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\begin{equation}
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\epsilon_{y} \ll H \sqrt{\frac{k_t}{k_b}}
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\end{equation}
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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H = 62e-3; % [m]
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data2orgtable(1e3*[1e-1*H*sqrt(kt/kb), 1e-2*H*sqrt(kt/kb), 1e-3*H*sqrt(kt/kb)], {'$\epsilon_y\,[mm]$'}, {'10% error', '1% error', '0.1% error'}, ' %.1f ');
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#+end_src
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#+name: tab:effect_torsion
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#+caption: Maximum lateral position error $\epsilon_y$ to have less than certain amount of error due to torsion effects
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#+attr_latex: :environment tabularx :width \linewidth :align lXXX
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#+attr_latex: :center t :booktabs t :float t
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#+RESULTS:
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| | 10% error | 1% error | 0.1% error |
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|--------------------+-----------+----------+------------|
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| $\epsilon_y\,[mm]$ | 44.7 | 4.5 | 0.4 |
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For 1% error, the lateral positioning errors $\epsilon_y$ for both the force sensor and the displacement sensor should be less than 4.5mm.
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* Full stroke measured displacement and applied force as a function of $H$
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Applying a force with a large height $H$ means the induced rotation (for constant force) will be larger.
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This also means that the measured displacement $D_b$ will also be larger.
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Note that we here suppose the force axis is co-linear with the measurement axis ($H_F = H_D = H$).
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Let's compute:
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- $D_b$ as a function of $H$
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\[ D_b \approx H \tan (R_b) \]
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- the applied force $F_{\text{max}}$ to induce the maximum rotation as a function of $H$
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\[ F_{\text{max}} \approx \frac{X_b \cdot k_b}{H} \]
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#+begin_src matlab :exports none
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H = linspace(0, 100e-3, 1000);
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Db = H*tan(Xb);
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Fmax = Xb*kb./H;
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#+end_src
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#+begin_src matlab :exports none
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figure;
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yyaxis left
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plot(1e3*H, Fmax);
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ylabel('Maximum Force [$N$]');
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ylim([0, 100]);
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yyaxis right
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plot(1e3*H, 1e6*Db);
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ylabel('Measurement Range [$\mu m$]');
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xlabel('Offset $H$ [$mm$]');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/force_motion_function_H.pdf', 'width', 'wide', 'height', 'normal');
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#+end_src
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#+name: fig:force_motion_function_H
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#+caption: Applied force $F_{\text{max}}$ and measured displacement $D_b$ as a function of $H$
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#+RESULTS:
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[[file:figs/force_motion_function_H.png]]
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With an offset of 62mm, we obtained values shown in Table [[tab:disp_force_range]].
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#+begin_src matlab :exports none
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H = 62e-3; % [m]
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Db = H*tan(Xb); % [m]
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Fmax = Xb*kb./H; % [N]
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([1e3*Db, Fmax], {}, {'$D_b\,[mm]$', '$F_m\,[N]$'}, ' %.1f ');
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#+end_src
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#+name: tab:disp_force_range
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#+caption: Maximum displacement and maximum applied force for $H = 62\,[mm]$
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#+attr_latex: :environment tabularx :width 0.3\linewidth :align lX
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#+attr_latex: :center t :booktabs t :float t
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#+RESULTS:
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| $D_b\,[mm]$ | $F_m\,[N]$ |
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|-------------+------------|
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| 1.5 | 1.9 |
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* Negligible bending of the supporting bar
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This should be confirmed with FEM.
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* Conclusion
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#+name: tab:conclusion_force_disp
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#+caption: Conclusions in terms of forces and displacement measurements
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#+attr_latex: :environment tabularx :width \linewidth :align lXX
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#+attr_latex: :center t :booktabs t :float t
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| | Range | Accuracy |
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|--------------------------+--------+-------------|
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| Force Measurement | 2 N | 1% = 0.02 N |
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| Displacement Measurement | 1.5 mm | 1% = 15 um |
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#+name: tab:conclusion_positioning
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#+caption: Conclusions in terms of required positioning accuracy
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#+attr_latex: :environment tabularx :width \linewidth :align lXXX
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#+attr_latex: :center t :booktabs t :float t
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| | Value | Precision | Comment |
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|--------------+-------+------------+---------------------------------|
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| $H_D$ | 62mm | | For negligible Shear |
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| $H_F$ | 62mm | | Same |
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| $\epsilon_y$ | 0 | 4.5mm | For negligible Torsion |
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| $\epsilon_z$ | 0 | 1% = 0.6mm | For torque estimation precision |
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Load cells:
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- https://www.te.com/usa-en/product-CAT-FLS0020.html?q=&n=510872&d=681051%20564247&type=products&samples=N&inStoreWithoutPL=false&instock=N#mdp-tabs-content
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- https://www.digikey.com/en/products/detail/honeywell-sensing-and-productivity-solutions/FSS005WNSB/6056404
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Displacement sensors:
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- http://www.vaco-france.com/outillage-de-metrologie/comparateur-a-cadran-de-precision.htm
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- https://fr.rs-online.com/web/p/comparateurs/1940101/
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