tdehaeze/nass-simscape
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Add analysis on soft granite suspension

Browse Source
This commit is contained in:
Thomas Dehaeze
2020-04-07 17:10:39 +02:00
parent 48b2d36418
commit 60c41f1cbc
12 changed files with 319 additions and 54 deletions
Download Patch File Download Diff File Expand all files Collapse all files

189
org/optimal_stiffness_disturbances.org
View File

@@ -389,12 +389,22 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
** Conclusion
#+begin_important
Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.
In order to conclude on the optimal stiffness that will yield the smallest sample vibration, one has to include the level of disturbances. This is done in Section [[sec:open_loop_budget_error]].
#+end_important
* Effect of granite stiffness
<<sec:granite_stiffness>>
** Introduction :ignore:
In this section, we wish to see if a soft granite suspension could help in reducing the effect of disturbances on the position error of the sample.
** Analytical Analysis
*** Simple mass-spring-damper model
Let's consider the system shown in Figure [[fig:2dof_system_granite_stiffness]] consisting of two stacked mass-spring-damper systems.
The bottom one represents the granite, and the top one all the positioning stages.
We want the smallest stage "deformation" $d = x^\prime - x$ due to ground motion $w$.
#+begin_src latex :file 2dof_system_granite_stiffness.pdf
\begin{tikzpicture}
% ====================
@@ -415,7 +425,7 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
% ====================
\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
\draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0) coordinate(dlow);
\draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$x_{w}$};
\draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$w$};
% ====================
% Micro Station
@@ -425,8 +435,8 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m$};
% Spring, Damper, and Actuator
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k$};
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c$};
\draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
\draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c$};
% Displacements
\draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0);
@@ -434,8 +444,8 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
% Legend
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
\end{scope}
% ====================
@@ -446,8 +456,8 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m^\prime$};
% Spring, Damper, and Actuator
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$};
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c^\prime$};
\draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$};
\draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c^\prime$};
% Displacements
\draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0) coordinate(dhigh);
@@ -455,44 +465,123 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
% Legend
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:2dof_system_granite_stiffness
#+caption: Figure caption
#+caption: Mass Spring Damper system consisting of a granite and a positioning stage
#+RESULTS:
[[file:figs/2dof_system_granite_stiffness.png]]
If we write the equation of motion of the system in Figure [[fig:2dof_system_granite_stiffness]], we obtain:
\begin{align}
m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\
ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (x_w - x)
ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (w - x)
\end{align}
If we note $d = x^\prime - x$, we obtain:
#+name: eq:plant_ground_transmissibility
\begin{equation}
\frac{d}{x_w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
\frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
\end{equation}
*** General Case
Let's now considering a general positioning stage defined by:
- $G^\prime(s) = \frac{F}{x}$: its mechanical "impedance"
- $D^\prime(s) = \frac{d}{x}$: its "deformation" transfer function
#+begin_src latex :file general_system_granite_stiffness.pdf
\begin{tikzpicture}
\def\massw{2.2} % Width of the masses
\def\massh{0.8} % Height of the masses
\def\spaceh{1.2} % Height of the springs/dampers
\def\dispw{0.3} % Width of the dashed line for the displacement
\def\disph{0.5} % Height of the arrow for the displacements
\def\bracs{0.05} % Brace spacing vertically
\def\brach{-10pt} % Brace shift horizontaly
% Mass
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle node[left=6pt]{$m$} (0.5*\massw, \spaceh+\massh);
% Spring, Damper, and Actuator
\draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
\draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c$};
% Ground
\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
% Groud Motion
\draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0);
\draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$w$};
% Displacements
\draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(dhigh);
\draw[->] (0.5*\massw+1.5*\dispw, \spaceh+\massh) -- ++(0, \disph) node[right]{$x$};
% Legend
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
\begin{scope}[shift={(0, \spaceh+\massh)}]
\node[piezo={2.2}{1.5}{6}, anchor=south] (piezo) at (0, 0){};
\draw[->] (0,0)node[branch]{} -- ++(0, -0.6)node[above right]{$F$}
\draw[<->] (1.1+0.5*\dispw,0) -- node[midway, right]{$d$} ++(0,1.5);
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
($(piezo.south west) + (-10pt, 0)$) -- ($(piezo.north west) + (-10pt, 0)$) %
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:general_system_granite_stiffness
#+caption: Mass Spring Damper representing the granite and a general representation of positioning stages
#+RESULTS:
[[file:figs/general_system_granite_stiffness.png]]
The equation of motion are:
\begin{align}
ms^2 x &= (cs + k) (x - w) - F \\
F &= G^\prime(s) x \\
d &= D^\prime(s) x
\end{align}
where:
- $F$ is the force applied by the position stages on the granite mass
#+begin_important
We can express $d$ as a function of $w$:
\begin{equation}
\frac{d}{w} = \frac{D^\prime(s) (cs + k)}{ms^2 + cs + k + G^\prime(s)}
\end{equation}
This is the transfer function that we would like to minimize.
#+end_important
Let's verify this formula for a simple mass/spring/damper positioning stage.
In that case, we have:
\begin{align*}
D^\prime(s) &= \frac{d}{x} = \frac{- m^\prime s^2}{m^\prime s^2 + c^\prime s + k^\prime} \\
G^\prime(s) &= \frac{F}{x} = \frac{m^\prime s^2(c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}
\end{align*}
And finally:
\begin{equation}
\frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
\end{equation}
which is the same as the previously derived equation.
** Soft Granite
Let's initialize a soft granite that will act as an isolation stage from ground motion.
Let's initialize a soft granite and see how the sensitivity to disturbances will change.
#+begin_src matlab
initializeGranite('K', 5e5*ones(3,1), 'C', 5e3*ones(3,1));
#+end_src
#+begin_src matlab
Ks = logspace(3,9,7); % [N/m]
#+end_src
#+begin_src matlab :exports none
Gdr = {zeros(length(Ks), 1)};
#+end_src
#+begin_src matlab
for i = 1:length(Ks)
initializeNanoHexapod('k', Ks(i));
@@ -504,6 +593,11 @@ Let's initialize a soft granite that will act as an isolation stage from ground
#+end_src
** Effect of the Granite transfer function
From Figure [[fig:opt_stiff_soft_granite_Dw]], we can see that having a "soft" granite suspension greatly lowers the sensitivity to ground motion.
The sensitivity is indeed lowered starting from the resonance of the granite on its soft suspension (few Hz here).
From Figures [[fig:opt_stiff_soft_granite_Frz]] and [[fig:opt_stiff_soft_granite_Fd]], we see that the change of granite suspension does not change a lot the sensitivity to both direct forces and stage vibrations.
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
@@ -520,9 +614,18 @@ Let's initialize a soft granite that will act as an isolation stage from ground
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
legend('location', 'southeast');
legend('location', 'southwest');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/opt_stiff_soft_granite_Dw.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:opt_stiff_soft_granite_Dw
#+caption: Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves) ([[./figs/opt_stiff_soft_granite_Dw.png][png]], [[./figs/opt_stiff_soft_granite_Dw.pdf][pdf]])
[[file:figs/opt_stiff_soft_granite_Dw.png]]
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
@@ -539,9 +642,51 @@ Let's initialize a soft granite that will act as an isolation stage from ground
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
legend('location', 'southeast');
legend('location', 'southwest');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/opt_stiff_soft_granite_Frz.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:opt_stiff_soft_granite_Frz
#+caption: Change of sensibility to Spindle vibrations when using a stiff Granite (solid curves) and a soft Granite (dashed curves) ([[./figs/opt_stiff_soft_granite_Frz.png][png]], [[./figs/opt_stiff_soft_granite_Frz.pdf][pdf]])
[[file:figs/opt_stiff_soft_granite_Frz.png]]
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
for i = 1:length(Ks)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gd{i}( 'Ez', 'Fdz'), freqs, 'Hz'))), '-', ...
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gdr{i}('Ez', 'Fdz'), freqs, 'Hz'))), '--', ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('$E_{z}/F_{d,z}$ [m/N]'); xlabel('Frequency [Hz]');
legend('location', 'northeast');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/opt_stiff_soft_granite_Fd.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:opt_stiff_soft_granite_Fd
#+caption: Change of sensibility to direct forces when using a stiff Granite (solid curves) and a soft Granite (dashed curves) ([[./figs/opt_stiff_soft_granite_Fd.png][png]], [[./figs/opt_stiff_soft_granite_Fd.pdf][pdf]])
[[file:figs/opt_stiff_soft_granite_Fd.png]]
** Conclusion
#+begin_important
Having a soft granite suspension could greatly improve the sensitivity the ground motion and thus the level of sample vibration if it is found that ground motion is the limiting factor.
#+end_important
* Open Loop Budget Error
<<sec:open_loop_budget_error>>