Change two damping values

matlab/test_apa_3_model_2dof.m

@@ -32,11 +32,11 @@ freqs = 5*logspace(0, 3, 1000);
 
 % Tuning of the APA model                                       :ignore:
 
-% 9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:test_apa_2dof_model_Simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics.
+% 9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:test_apa_2dof_model_simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics.
 
-% #+name: fig:test_apa_2dof_model_Simscape
+% #+name: fig:test_apa_2dof_model_simscape
 % #+caption: Schematic of the two degrees-of-freedom model of the APA300ML with input $V_a$ and outputs $d_e$ and $V_s$
-% [[file:figs/test_apa_2dof_model_Simscape.png]]
+% [[file:figs/test_apa_2dof_model_simscape.png]]
 
 
 %% Stiffness values for the 2DoF APA model
@@ -50,8 +50,8 @@ ka = 1.5*(ktot-k1); % Stiffness of the (two) actuator stacks [N/m]
 ke = 2*ka; % Stiffness of the Sensor stack [N/m]
 
 %% Damping values for the 2DoF APA model
-c1 = 20;   % Damping for the Shell [N/(m/s)]
-ca = 100;  % Damping of the actuators stacks [N/(m/s)]
+c1 = 5;   % Damping for the Shell [N/(m/s)]
+ca = 50;  % Damping of the actuators stacks [N/(m/s)]
 ce = 2*ca; % Damping of the sensor stack [N/(m/s)]
 
 %% Estimation ot the sensor and actuator sensitivities
@@ -69,7 +69,7 @@ n_hexapod.actuator = initializeAPA(...
     'Gs', 1    ... % Sensor constant [V/m]
 );
 
-c_granite = 0; % Do not take into account damping added by the air bearing
+c_granite = 50; % Do not take into account damping added by the air bearing
 
 % Run the linearization
 G_norm = linearize(mdl, io, 0.0, opts);

matlab/test_apa_4_model_flexible.m

@@ -44,7 +44,7 @@ n_hexapod.actuator = initializeAPA(...
     'ga', 1, ...
     'gs', 1);
 
-c_granite = 100; % Rought estimation of the damping added by the air bearing
+c_granite = 50; % Rought estimation of the damping added by the air bearing
 
 % Identify the dynamics
 G_norm = linearize(mdl, io, 0.0, opts);

test-bench-apa.org

@@ -1396,11 +1396,11 @@ Such a simple model has some limitations:
 
 **** Tuning of the APA model                                       :ignore:
 
-9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:test_apa_2dof_model_Simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics.
+9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:test_apa_2dof_model_simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics.
 
-#+name: fig:test_apa_2dof_model_Simscape
+#+name: fig:test_apa_2dof_model_simscape
 #+caption: Schematic of the two degrees-of-freedom model of the APA300ML with input $V_a$ and outputs $d_e$ and $V_s$
-[[file:figs/test_apa_2dof_model_Simscape.png]]
+[[file:figs/test_apa_2dof_model_simscape.png]]
 
 #+begin_src matlab
 %% Stiffness values for the 2DoF APA model
@@ -1414,8 +1414,8 @@ ka = 1.5*(ktot-k1); % Stiffness of the (two) actuator stacks [N/m]
 ke = 2*ka; % Stiffness of the Sensor stack [N/m]
 
 %% Damping values for the 2DoF APA model
-c1 = 20;   % Damping for the Shell [N/(m/s)]
-ca = 100;  % Damping of the actuators stacks [N/(m/s)]
+c1 = 5;   % Damping for the Shell [N/(m/s)]
+ca = 50;  % Damping of the actuators stacks [N/(m/s)]
 ce = 2*ca; % Damping of the sensor stack [N/(m/s)]
 #+end_src
 
@@ -1435,7 +1435,7 @@ n_hexapod.actuator = initializeAPA(...
     'Gs', 1    ... % Sensor constant [V/m]
 );
 
-c_granite = 0; % Do not take into account damping added by the air bearing
+c_granite = 50; % Do not take into account damping added by the air bearing
 
 % Run the linearization
 G_norm = linearize(mdl, io, 0.0, opts);
@@ -1460,7 +1460,7 @@ First, the mass $m$ supported by the APA300ML can be estimated from the geometry
 Both methods lead to an estimated mass of $m = 5.7\,\text{kg}$.
 
 Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,N/\mu m$ in Section ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure ref:fig:test_apa_frf_force.
-Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $20\,Ns/m$.
+Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $5\,Ns/m$.
 
 Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics[fn:5].
 Therefore, we have $k_e = 2 k_a$ and $c_e = 2 c_a$ as the actuator stack is composed of two stacks in series.
@@ -1477,11 +1477,11 @@ Knowing from eqref:eq:test_apa_tot_stiffness that the total stiffness is $k_{\te
 \end{equation}
 
 Then, $c_a$ (and therefore $c_e = 2 c_a$) can be tuned to match the damping ratio of the identified resonance.
-$c_a = 100\,Ns/m$ and $c_e = 200\,Ns/m$ are obtained.
+$c_a = 50\,Ns/m$ and $c_e = 100\,Ns/m$ are obtained.
 
 In the last step, $g_s$ and $g_a$ can be tuned to match the gain of the identified transfer functions.
 
-The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model_Simscape are summarized in Table ref:tab:test_apa_2dof_parameters.
+The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model_simscape are summarized in Table ref:tab:test_apa_2dof_parameters.
 
 #+name: tab:test_apa_2dof_parameters
 #+caption: Summary of the obtained parameters for the 2 DoF APA300ML model
@@ -1493,9 +1493,9 @@ The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model
 | $k_1$       | $0.38\,N/\mu m$  |
 | $k_e$       | $5.0\, N/\mu m$  |
 | $k_a$       | $2.5\,N/\mu m$   |
-| $c_1$       | $20\,Ns/m$       |
-| $c_e$       | $200\,Ns/m$      |
-| $c_a$       | $100\,Ns/m$      |
+| $c_1$       | $5\,Ns/m$        |
+| $c_e$       | $100\,Ns/m$      |
+| $c_a$       | $50\,Ns/m$       |
 | $g_a$       | $-2.58\,N/V$     |
 | $g_s$       | $0.46\,V/\mu m$  |
 
@@ -1632,7 +1632,7 @@ exportFig('figs/test_apa_2dof_comp_frf_force.pdf', 'width', 'half', 'height', 't
 **** Introduction                                                  :ignore:
 In this section, a /super element/ of the APA300ML is computed using a finite element software[fn:11].
 It is then imported into Simscape (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in ref:sec:test_apa_model_2dof.
-This procedure is illustrated in Figure ref:fig:test_apa_super_element_Simscape.
+This procedure is illustrated in Figure ref:fig:test_apa_super_element_simscape.
 Several /remote points/ are defined in the finite element model (here illustrated by colorful planes and numbers from =1= to =5=) and are then made accessible in the Simscape model as shown at the right by the "frames" =F1= to =F5=.
 
 For the APA300ML /super element/, 5 /remote points/ are defined.
@@ -1640,10 +1640,10 @@ Two /remote points/ (=1= and =2=) are fixed to the top and bottom mechanical int
 Two /remote points/ (=3= and =4=) are located across two piezoelectric stacks and are used to apply internal forces representing the actuator stacks.
 Finally, two /remote points/ (=4= and =4=) are located across the third piezoelectric stack, and will be used to measured the strain of the sensor stack.
 
-#+name: fig:test_apa_super_element_Simscape
+#+name: fig:test_apa_super_element_simscape
 #+attr_latex: :width 1.0\linewidth
 #+caption: Finite Element Model of the APA300ML with "remotes points" on the left. Simscape model with included "Reduced Order Flexible Solid" on the right.
-[[file:figs/test_apa_super_element_Simscape.png]]
+[[file:figs/test_apa_super_element_simscape.png]]
 
 **** Matlab Init                                          :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -1699,7 +1699,7 @@ n_hexapod.actuator = initializeAPA(...
     'ga', 1, ...
     'gs', 1);
 
-c_granite = 100; % Rought estimation of the damping added by the air bearing
+c_granite = 50; % Rought estimation of the damping added by the air bearing
 
 % Identify the dynamics
 G_norm = linearize(mdl, io, 0.0, opts);

test-bench-apa.tex

@@ -1,4 +1,4 @@
-% Created 2024-04-30 Tue 17:24
+% Created 2024-10-26 Sat 12:10
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -31,7 +31,7 @@
  pdftitle={Test Bench - Amplified Piezoelectric Actuator},
  pdfkeywords={},
  pdfsubject={},
- pdfcreator={Emacs 29.3 (Org mode 9.6)}, 
+ pdfcreator={Emacs 29.4 (Org mode 9.6)}, 
  pdflang={English}}
 \usepackage{biblatex}
 
@@ -553,19 +553,19 @@ Such a simple model has some limitations:
 \caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees-of-freedom model of the APA300ML, adapted from \cite{souleille18_concep_activ_mount_space_applic}}
 \end{figure}
 
-9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure \ref{fig:test_apa_2dof_model_Simscape}) well represents the identified dynamics in Section \ref{sec:test_apa_dynamics}.
+9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure \ref{fig:test_apa_2dof_model_simscape}) well represents the identified dynamics in Section \ref{sec:test_apa_dynamics}.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/test_apa_2dof_model_Simscape.png}
-\caption{\label{fig:test_apa_2dof_model_Simscape}Schematic of the two degrees-of-freedom model of the APA300ML with input \(V_a\) and outputs \(d_e\) and \(V_s\)}
+\includegraphics[scale=1]{figs/test_apa_2dof_model_simscape.png}
+\caption{\label{fig:test_apa_2dof_model_simscape}Schematic of the two degrees-of-freedom model of the APA300ML with input \(V_a\) and outputs \(d_e\) and \(V_s\)}
 \end{figure}
 
 First, the mass \(m\) supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale.
 Both methods lead to an estimated mass of \(m = 5.7\,\text{kg}\).
 
 Then, the axial stiffness of the shell was estimated at \(k_1 = 0.38\,N/\mu m\) in Section \ref{ssec:test_apa_meas_dynamics} from the frequency of the anti-resonance seen on Figure \ref{fig:test_apa_frf_force}.
-Similarly, \(c_1\) can be estimated from the damping ratio of the same anti-resonance and is found to be close to \(20\,Ns/m\).
+Similarly, \(c_1\) can be estimated from the damping ratio of the same anti-resonance and is found to be close to \(5\,Ns/m\).
 
 Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics\footnote{Note that this is not completely correct as it was shown in Section \ref{ssec:test_apa_stiffness} that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.}.
 Therefore, we have \(k_e = 2 k_a\) and \(c_e = 2 c_a\) as the actuator stack is composed of two stacks in series.
@@ -582,11 +582,11 @@ Knowing from \eqref{eq:test_apa_tot_stiffness} that the total stiffness is \(k_{
 \end{equation}
 
 Then, \(c_a\) (and therefore \(c_e = 2 c_a\)) can be tuned to match the damping ratio of the identified resonance.
-\(c_a = 100\,Ns/m\) and \(c_e = 200\,Ns/m\) are obtained.
+\(c_a = 50\,Ns/m\) and \(c_e = 100\,Ns/m\) are obtained.
 
 In the last step, \(g_s\) and \(g_a\) can be tuned to match the gain of the identified transfer functions.
 
-The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_model_Simscape} are summarized in Table \ref{tab:test_apa_2dof_parameters}.
+The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_model_simscape} are summarized in Table \ref{tab:test_apa_2dof_parameters}.
 
 \begin{table}[htbp]
 \centering
@@ -598,9 +598,9 @@ The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_mode
 \(k_1\) & \(0.38\,N/\mu m\)\\
 \(k_e\) & \(5.0\, N/\mu m\)\\
 \(k_a\) & \(2.5\,N/\mu m\)\\
-\(c_1\) & \(20\,Ns/m\)\\
-\(c_e\) & \(200\,Ns/m\)\\
-\(c_a\) & \(100\,Ns/m\)\\
+\(c_1\) & \(5\,Ns/m\)\\
+\(c_e\) & \(100\,Ns/m\)\\
+\(c_a\) & \(50\,Ns/m\)\\
 \(g_a\) & \(-2.58\,N/V\)\\
 \(g_s\) & \(0.46\,V/\mu m\)\\
 \bottomrule
@@ -634,7 +634,7 @@ This indicates that this model represents well the axial dynamics of the APA300M
 \label{sec:test_apa_model_flexible}
 In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used}.
 It is then imported into Simscape (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in \ref{sec:test_apa_model_2dof}.
-This procedure is illustrated in Figure \ref{fig:test_apa_super_element_Simscape}.
+This procedure is illustrated in Figure \ref{fig:test_apa_super_element_simscape}.
 Several \emph{remote points} are defined in the finite element model (here illustrated by colorful planes and numbers from \texttt{1} to \texttt{5}) and are then made accessible in the Simscape model as shown at the right by the ``frames'' \texttt{F1} to \texttt{F5}.
 
 For the APA300ML \emph{super element}, 5 \emph{remote points} are defined.
@@ -644,8 +644,8 @@ Finally, two \emph{remote points} (\texttt{4} and \texttt{4}) are located across
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=1.0\linewidth]{figs/test_apa_super_element_Simscape.png}
-\caption{\label{fig:test_apa_super_element_Simscape}Finite Element Model of the APA300ML with ``remotes points'' on the left. Simscape model with included ``Reduced Order Flexible Solid'' on the right.}
+\includegraphics[scale=1,width=1.0\linewidth]{figs/test_apa_super_element_simscape.png}
+\caption{\label{fig:test_apa_super_element_simscape}Finite Element Model of the APA300ML with ``remotes points'' on the left. Simscape model with included ``Reduced Order Flexible Solid'' on the right.}
 \end{figure}
 
 \paragraph{Identification of the Actuator and Sensor constants}