Identification of Piezo parameters
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figs/APA95ML_nodes_2.png
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figs/compare_Gd_id_simscape.pdf
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figs/compare_Gf_id_simscape.pdf
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figs/gain_Va_to_Vs.pdf
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figs/gain_Va_to_d.pdf
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@ -67,18 +67,442 @@ Here are the equipment used in the test bench:
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- Low Noise Voltage Preamplifier from Ametek ([[file:doc/model_5113.pdf][doc]])
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- Low Noise Voltage Preamplifier from Ametek ([[file:doc/model_5113.pdf][doc]])
|
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#+end_note
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#+end_note
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||||||
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* Estimation of electrical/mechanical relationships
|
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** Introduction :ignore:
|
||||||
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In order to correctly model the piezoelectric actuator with Simscape, we need to determine:
|
||||||
|
1. $g_a$: the ratio of the generated force $F_a$ to the supply voltage $V_a$ across the piezoelectric stack
|
||||||
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2. $g_s$: the ratio of the generated voltage $V_s$ across the piezoelectric stack when subject to a strain $\Delta h$
|
||||||
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|
||||||
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We estimate $g_a$ and $g_s$ using different approaches:
|
||||||
|
1. Section [[sec:estimation_datasheet]]: $g_a$ is estimated from the datasheet of the piezoelectric stack
|
||||||
|
2. Section [[sec:estimation_piezo_params]]: $g_a$ and $g_s$ are estimated using the piezoelectric constants
|
||||||
|
3. Section [[sec:estimation_identification]]: $g_a$ and $g_s$ are estimated experimentally
|
||||||
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||||||
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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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||||||
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||||||
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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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||||||
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||||||
|
#+begin_src matlab :tangle no
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|
addpath('./matlab/mat/');
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|
addpath('./matlab/');
|
||||||
|
#+end_src
|
||||||
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|
||||||
|
#+begin_src matlab :eval no
|
||||||
|
addpath('./mat/');
|
||||||
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#+end_src
|
||||||
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|
||||||
|
** Estimation from Data-sheet
|
||||||
|
<<sec:estimation_datasheet>>
|
||||||
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|
||||||
|
The stack parameters taken from the data-sheet are shown in Table [[tab:stack_parameters]].
|
||||||
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|
||||||
|
#+name: tab:stack_parameters
|
||||||
|
#+caption: Stack Parameters
|
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|
| Parameter | Unit | Value |
|
||||||
|
|----------------+-----------+----------|
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||||||
|
| Nominal Stroke | $\mu m$ | 20 |
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|
| Blocked force | $N$ | 4700 |
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|
| Stiffness | $N/\mu m$ | 235 |
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|
| Voltage Range | $V$ | -20..150 |
|
||||||
|
| Capacitance | $\mu F$ | 4.4 |
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||||||
|
| Length | $mm$ | 20 |
|
||||||
|
| Stack Area | $mm^2$ | 10x10 |
|
||||||
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|
||||||
|
Let's compute the generated force
|
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|
|
||||||
|
The stroke is $L_{\max} = 20\mu m$ for a voltage range of $V_{\max} = 170 V$.
|
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|
Furthermore, the stiffness is $k_a = 235 \cdot 10^6 N/m$.
|
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|
||||||
|
The relation between the applied voltage and the generated force can be estimated as follows:
|
||||||
|
\begin{equation}
|
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|
g_a = k_a \frac{L_{\max}}{V_{\max}}
|
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|
\end{equation}
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
ka = 235e6; % [N/m]
|
||||||
|
Lmax = 20e-6; % [m]
|
||||||
|
Vmax = 170; % [V]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :results value replace
|
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|
ka*Lmax/Vmax % [N/V]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+RESULTS:
|
||||||
|
: 27.647
|
||||||
|
|
||||||
|
From the parameters of the stack, it seems not possible to estimate the relation between the strain and the generated voltage.
|
||||||
|
|
||||||
|
** Estimation from Piezoelectric parameters
|
||||||
|
<<sec:estimation_piezo_params>>
|
||||||
|
|
||||||
|
In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:
|
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|
#+begin_src matlab
|
||||||
|
d33 = 300e-12; % Strain constant [m/V]
|
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|
n = 80; % Number of layers per stack
|
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|
ka = 235e6; % Stack stiffness [N/m]
|
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|
#+end_src
|
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|
||||||
|
The ratio of the developed force to applied voltage is:
|
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|
#+name: eq:piezo_voltage_to_force
|
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|
\begin{equation}
|
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|
F_a = g_a V_a, \quad g_a = d_{33} n k_a
|
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|
\end{equation}
|
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|
where:
|
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|
- $F_a$: developed force in [N]
|
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|
- $n$: number of layers of the actuator stack
|
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|
- $d_{33}$: strain constant in [m/V]
|
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- $k_a$: actuator stack stiffness in [N/m]
|
||||||
|
- $V_a$: applied voltage in [V]
|
||||||
|
|
||||||
|
If we take the numerical values, we obtain:
|
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|
#+begin_src matlab
|
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|
ga = d33*n*ka; % [N/V]
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|
#+end_src
|
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|
|
||||||
|
#+begin_src matlab :results value replace :exports results :tangle no
|
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|
sprintf('ga = %.1f [N/V]', ga)
|
||||||
|
#+end_src
|
||||||
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|
||||||
|
#+RESULTS:
|
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|
: ga = 5.6 [N/V]
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|
||||||
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|
||||||
|
From cite:fleming14_desig_model_contr_nanop_system (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
|
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|
#+name: eq:piezo_strain_to_voltage
|
||||||
|
\begin{equation}
|
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|
V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
|
||||||
|
\end{equation}
|
||||||
|
where:
|
||||||
|
- $V_s$: measured voltage in [V]
|
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|
- $d_{33}$: strain constant in [m/V]
|
||||||
|
- $\epsilon^T$: permittivity under constant stress in [F/m]
|
||||||
|
- $s^D$: elastic compliance under constant electric displacement in [m^2/N]
|
||||||
|
- $n$: number of layers of the sensor stack
|
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- $\Delta h$: relative displacement in [m]
|
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|
||||||
|
If we take the numerical values, we obtain:
|
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|
#+begin_src matlab
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|
d33 = 300e-12; % Strain constant [m/V]
|
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|
n = 80; % Number of layers per stack
|
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|
eT = 5.3e-9; % Permittivity under constant stress [F/m]
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sD = 2e-11; % Compliance under constant electric displacement [m2/N]
|
||||||
|
#+end_src
|
||||||
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|
||||||
|
#+begin_src matlab
|
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|
gs = d33/(eT*sD*n); % [V/m]
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|
#+end_src
|
||||||
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|
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|
#+begin_src matlab :results value replace :exports results :tangle no
|
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|
sprintf('gs = %.1f [V/um]', 1e-6*gs)
|
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#+end_src
|
||||||
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|
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#+RESULTS:
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: gs = 35.4 [V/um]
|
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|
||||||
|
** Estimation from Experiment
|
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|
<<sec:estimation_identification>>
|
||||||
|
*** Introduction :ignore:
|
||||||
|
The idea here is to obtain the parameters $g_a$ and $g_s$ from the comparison of an experimental identification and the identification using Simscape.
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Using the experimental identification, we can easily obtain the gain from the applied voltage to the generated displacement, but not to the generated force.
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However, from the Simscape model, we can easily have the link from the generated force to the displacement, them we can computed $g_a$.
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||||||
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Similarly, it is fairly easy to experimentally obtain the gain from the stack displacement to the generated voltage across the stack.
|
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To link that to the strain of the sensor stack, the simscape model is used.
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|
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*** From actuator voltage $V_a$ to actuator force $F_a$
|
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The data from the identification test is loaded.
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#+begin_src matlab
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load('apa95ml_5kg_Amp_E505.mat', 't', 'um', 'y');
|
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|
% Any offset value is removed:
|
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|
um = detrend(um, 0); % Amplifier Input Voltage [V]
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y = detrend(y , 0); % Mass displacement [m]
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#+end_src
|
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|
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Now we add a factor 10 to take into account the gain of the voltage amplifier and thus obtain the voltage across the piezoelectric stack.
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#+begin_src matlab
|
||||||
|
um = 10*um; % Stack Actuator Input Voltage [V]
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|
#+end_src
|
||||||
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|
||||||
|
Then, the transfer function from the stack voltage to the vertical displacement is computed.
|
||||||
|
#+begin_src matlab
|
||||||
|
Ts = t(end)/(length(t)-1);
|
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|
Fs = 1/Ts;
|
||||||
|
|
||||||
|
win = hanning(ceil(1*Fs));
|
||||||
|
|
||||||
|
[tf_est, f] = tfestimate(um, y, win, [], [], 1/Ts);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
The gain from input voltage of the stack to the vertical displacement is determined:
|
||||||
|
#+begin_src matlab
|
||||||
|
g_d_Va = 4e-7; % [m/V]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(0, 4, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
hold on;
|
||||||
|
plot(f, abs(tf_est), 'DisplayName', 'Identification')
|
||||||
|
plot([f(2) f(end)], [g_d_Va g_d_Va], '--', 'DisplayName', sprintf('$d/V_a \\approx %.0g\\ m/V$', g_d_Va))
|
||||||
|
hold off;
|
||||||
|
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
||||||
|
xlabel('Frequency [Hz]'); ylabel('Amplitude $d/V_a$ [m/V]');
|
||||||
|
xlim([10, 5e3]); ylim([1e-8, 1e-5]);
|
||||||
|
legend('location', 'southwest');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/gain_Va_to_d.pdf', 'width', 'wide', 'height', 'normal');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:gain_Va_to_d
|
||||||
|
#+caption: Transfer function from actuator stack voltage $V_a$ to vertical displacement of the mass $d$
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/gain_Va_to_d.png]]
|
||||||
|
|
||||||
|
Then, the transfer function from forces applied by the stack actuator to the vertical displacement of the mass is identified from the Simscape model.
|
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|
#+begin_src matlab :exports none
|
||||||
|
% Load Parameters
|
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|
K = readmatrix('APA95ML_K.CSV');
|
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M = readmatrix('APA95ML_M.CSV');
|
||||||
|
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
|
||||||
|
|
||||||
|
% Define parameters just for the simulation
|
||||||
|
ga = 1; % [N/V]
|
||||||
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gs = 1; % [V/m]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
m = 5.5;
|
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|
|
||||||
|
%% Name of the Simulink File
|
||||||
|
mdl = 'piezo_amplified_3d';
|
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|
||||||
|
%% Input/Output definition
|
||||||
|
clear io; io_i = 1;
|
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|
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N]
|
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|
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
|
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|
||||||
|
Gd = linearize(mdl, io);
|
||||||
|
#+end_src
|
||||||
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|
||||||
|
The DC gain the the identified dynamics
|
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#+begin_src matlab
|
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g_d_Fa = abs(dcgain(Gd)); % [m/N]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :results value replace :exports results :tangle no
|
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|
sprintf('G_d_Fa = %.2g [m/N]', g_d_Fa)
|
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|
#+end_src
|
||||||
|
|
||||||
|
#+RESULTS:
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|
: G_d_Fa = 1.2e-08 [m/N]
|
||||||
|
|
||||||
|
|
||||||
|
And finally, the gain $g_a$ from the the actuator voltage $V_a$ to the generated force $F_a$ can be computed:
|
||||||
|
\begin{equation}
|
||||||
|
g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \frac{d}{V_a}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
ga = g_d_Va/g_d_Fa;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :results value replace :exports results :tangle no
|
||||||
|
sprintf('ga = %.1f [N/V]', ga)
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+RESULTS:
|
||||||
|
: ga = 33.7 [N/V]
|
||||||
|
|
||||||
|
The obtained comparison between the Simscape model and the identified dynamics is shown in Figure [[fig:compare_Gd_id_simscape]].
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(1, 4, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
hold on;
|
||||||
|
plot(f, abs(tf_est), 'DisplayName', 'Identification')
|
||||||
|
plot(f, ga*abs(squeeze(freqresp(Gd, f, 'Hz'))), 'DisplayName', 'Simscape Model')
|
||||||
|
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
||||||
|
ylabel('Amplitude $d/V_a$ [m/V]'); xlabel('Frequency [Hz]');
|
||||||
|
hold off;
|
||||||
|
xlim([1, 5e3]); ylim([1e-9, 1e-5]);
|
||||||
|
legend('location', 'southwest');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/compare_Gd_id_simscape.pdf', 'width', 'wide', 'height', 'normal');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:compare_Gd_id_simscape
|
||||||
|
#+caption: Comparison of the identified transfer function between actuator voltage $V_a$ and vertical mass displacement $d$
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/compare_Gd_id_simscape.png]]
|
||||||
|
|
||||||
|
*** From stack strain $\Delta h$ to generated voltage $V_s$
|
||||||
|
Now, the gain from the stack strain $\Delta h$ to the generated voltage $V_s$ is estimated.
|
||||||
|
|
||||||
|
We can determine the gain from actuator voltage $V_a$ to sensor voltage $V_s$ thanks to the identification.
|
||||||
|
Using the simscape model, we can have the transfer function from the actuator voltage $V_a$ (using the previously estimated gain $g_a$) to the sensor stack strain $\Delta h$.
|
||||||
|
|
||||||
|
Finally, using these two values, we can compute the gain $g_s$ from the stack strain $\Delta h$ to the generated Voltage $V_s$.
|
||||||
|
|
||||||
|
Identification data is loaded.
|
||||||
|
#+begin_src matlab
|
||||||
|
load('apa95ml_5kg_2a_1s.mat', 't', 'u', 'v');
|
||||||
|
|
||||||
|
u = detrend(u, 0); % Input Voltage of the Amplifier [V]
|
||||||
|
v = detrend(v, 0); % Voltage accross the stack sensor [V]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
Here, an amplifier with a gain of 20 is used.
|
||||||
|
#+begin_src matlab
|
||||||
|
u = 20*u; % Input Voltage of the Amplifier [V]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
Then, the transfer function from $V_a$ to $V_s$ is identified and its DC gain is estimated (Figure [[fig:gain_Va_to_Vs]]).
|
||||||
|
#+begin_src matlab
|
||||||
|
Ts = t(end)/(length(t)-1);
|
||||||
|
Fs = 1/Ts;
|
||||||
|
|
||||||
|
win = hann(ceil(10/Ts));
|
||||||
|
|
||||||
|
[tf_est, f] = tfestimate(u, v, win, [], [], 1/Ts);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
g_Vs_Va = 0.022; % [V/V]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(1, 4, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
hold on;
|
||||||
|
plot(f, abs(tf_est), 'DisplayName', 'Identification')
|
||||||
|
plot([f(2); f(end)], [g_Vs_Va; g_Vs_Va], '--', 'DisplayName', sprintf('$V_s/V_a \\approx %.0g\\ m/V$', g_Vs_Va))
|
||||||
|
hold off;
|
||||||
|
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
||||||
|
ylabel('Amplitude $V_s/V_a$ [V/V]'); xlabel('Frequency [Hz]');
|
||||||
|
xlim([5e-1 5e3]); ylim([1e-3, 1e1]);
|
||||||
|
legend('location', 'northwest');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/gain_Va_to_Vs.pdf', 'width', 'wide', 'height', 'normal');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:gain_Va_to_Vs
|
||||||
|
#+caption: Transfer function from actuator stack voltage $V_a$ to sensor stack voltage $V_s$
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/gain_Va_to_Vs.png]]
|
||||||
|
|
||||||
|
|
||||||
|
Now the transfer function from the actuator stack voltage to the sensor stack strain is estimated using the Simscape model.
|
||||||
|
#+begin_src matlab
|
||||||
|
m = 5.5;
|
||||||
|
|
||||||
|
%% Name of the Simulink File
|
||||||
|
mdl = 'piezo_amplified_3d';
|
||||||
|
|
||||||
|
%% Input/Output definition
|
||||||
|
clear io; io_i = 1;
|
||||||
|
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage [V]
|
||||||
|
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Stack displacement [m]
|
||||||
|
|
||||||
|
Gf = linearize(mdl, io);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
The gain from the actuator stack voltage to the sensor stack strain is estimated below.
|
||||||
|
#+begin_src matlab
|
||||||
|
G_dh_Va = abs(dcgain(Gf));
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :results value replace :exports results :tangle no
|
||||||
|
sprintf('G_dh_Va = %.2g [m/V]', G_dh_Va);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+RESULTS:
|
||||||
|
: G_dh_Va = 6.2e-09 [m/V]
|
||||||
|
|
||||||
|
And finally, the gain $g_s$ from the sensor stack strain to the generated voltage can be estimated:
|
||||||
|
\begin{equation}
|
||||||
|
g_s = \frac{V_s}{\Delta h} = \frac{V_s}{V_a} \frac{V_a}{\Delta h}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
gs = g_Vs_Va/G_dh_Va; % [V/m]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :results value replace :exports results :tangle no
|
||||||
|
sprintf('gs = %.1f [V/um]', 1e-6*gs)
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+RESULTS:
|
||||||
|
: gs = 3.5 [V/um]
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(1, 4, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
hold on;
|
||||||
|
plot(f, abs(tf_est), 'DisplayName', 'Identification')
|
||||||
|
plot(f, gs*abs(squeeze(freqresp(Gf, f, 'Hz'))), 'DisplayName', 'Simscape Model')
|
||||||
|
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
||||||
|
ylabel('Amplitude $V_s/V_a$ [V/V]'); xlabel('Frequency [Hz]');
|
||||||
|
hold off;
|
||||||
|
xlim([1, 5e3]); ylim([1e-3, 1e1]);
|
||||||
|
legend('location', 'southwest');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/compare_Gf_id_simscape.pdf', 'width', 'wide', 'height', 'normal');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:compare_Gf_id_simscape
|
||||||
|
#+caption: Comparison of the identified transfer function between actuator voltage $V_a$ and sensor stack voltage
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/compare_Gf_id_simscape.png]]
|
||||||
|
|
||||||
|
** Conclusion
|
||||||
|
The obtained parameters $g_a$ and $g_s$ are not consistent between the different methods.
|
||||||
|
|
||||||
|
The one using the experimental data are saved and further used.
|
||||||
|
#+begin_src matlab :tangle no
|
||||||
|
save('./matlab/mat/apa95ml_params.mat', 'ga', 'gs');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none :eval no
|
||||||
|
save('./mat/apa95ml_params.mat', 'ga', 'gs');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
* Simscape model of the test-bench
|
* Simscape model of the test-bench
|
||||||
:PROPERTIES:
|
:PROPERTIES:
|
||||||
:header-args:matlab+: :tangle matlab/simscape_model.m
|
:header-args:matlab+: :tangle matlab/simscape_model.m
|
||||||
:header-args:matlab+: :comments org :mkdirp yes
|
:header-args:matlab+: :comments org :mkdirp yes
|
||||||
:END:
|
:END:
|
||||||
<<sec:simscape_model>>
|
<<sec:simscape_model>>
|
||||||
|
|
||||||
** Introduction :ignore:
|
** Introduction :ignore:
|
||||||
The idea here is to model the test-bench using Simscape.
|
The idea here is to model the test-bench using Simscape.
|
||||||
|
|
||||||
Whereas the suspended mass and metrology frame can be considered as rigid bodies in the frequency range of interest, the Amplified Piezoelectric Actuator (APA) is flexible.
|
Whereas the suspended mass and metrology frame can be considered as rigid bodies in the frequency range of interest, the Amplified Piezoelectric Actuator (APA) is flexible.
|
||||||
|
|
||||||
To model the APA, a Finite Element Model (FEM) is used and imported into Simscape.
|
To model the APA, a Finite Element Model (FEM) is used (Figure [[fig:APA95ML_FEM]]) and imported into Simscape.
|
||||||
|
|
||||||
|
#+name: fig:APA95ML_FEM
|
||||||
|
#+caption: Finite Element Model of the APA95ML
|
||||||
|
[[file:figs/APA95ML_FEM.png]]
|
||||||
|
|
||||||
** Matlab Init :noexport:ignore:
|
** Matlab Init :noexport:ignore:
|
||||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||||
@ -101,8 +525,8 @@ To model the APA, a Finite Element Model (FEM) is used and imported into Simscap
|
|||||||
** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
|
** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
|
||||||
We first extract the stiffness and mass matrices.
|
We first extract the stiffness and mass matrices.
|
||||||
#+begin_src matlab
|
#+begin_src matlab
|
||||||
K = extractMatrix('APA95ML_K.txt');
|
K = readmatrix('APA95ML_K.CSV');
|
||||||
M = extractMatrix('APA95ML_M.txt');
|
M = readmatrix('APA95ML_M.CSV');
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
#+begin_src matlab :exports results :results value table replace :tangle no
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
||||||
@ -111,16 +535,16 @@ We first extract the stiffness and mass matrices.
|
|||||||
|
|
||||||
#+caption: First 10x10 elements of the Stiffness matrix
|
#+caption: First 10x10 elements of the Stiffness matrix
|
||||||
#+RESULTS:
|
#+RESULTS:
|
||||||
| 300000000.0 | -30000.0 | 8000.0 | -200.0 | -30.0 | -60000.0 | 20000000.0 | -4000.0 | 500.0 | 8 |
|
| 300000000.0 | -1000.0 | -30000.0 | -40.0 | 70000.0 | 300.0 | 20000000.0 | -30.0 | -5000.0 | 5 |
|
||||||
| -30000.0 | 100000000.0 | 400.0 | 30.0 | 200.0 | -1 | 4000.0 | -8000000.0 | 800.0 | 7 |
|
| -1000.0 | 50000000.0 | -7000.0 | 800000.0 | -20.0 | 300.0 | 3000.0 | 5000000.0 | 400.0 | -40000.0 |
|
||||||
| 8000.0 | 400.0 | 50000000.0 | -800000.0 | -300.0 | -40.0 | 300.0 | 100.0 | 5000000.0 | 40000.0 |
|
| -30000.0 | -7000.0 | 100000000.0 | -200.0 | -60.0 | 70.0 | 3000.0 | 3000.0 | -8000000.0 | -30.0 |
|
||||||
| -200.0 | 30.0 | -800000.0 | 20000.0 | 5 | 1 | -10.0 | -2 | -40000.0 | -300.0 |
|
| -40.0 | 800000.0 | -200.0 | 20000.0 | -0.4 | 4 | 30.0 | 40000.0 | 7 | -300.0 |
|
||||||
| -30.0 | 200.0 | -300.0 | 5 | 40000.0 | 0.3 | -4 | -10.0 | 40.0 | 0.4 |
|
| 70000.0 | -20.0 | -60.0 | -0.4 | 3000.0 | 1 | -6000.0 | 10.0 | 8 | -0.1 |
|
||||||
| -60000.0 | -1 | -40.0 | 1 | 0.3 | 3000.0 | 7000.0 | 0.8 | -1 | 0.0003 |
|
| 300.0 | 300.0 | 70.0 | 4 | 1 | 40000.0 | -10.0 | -10.0 | 30.0 | 0.1 |
|
||||||
| 20000000.0 | 4000.0 | 300.0 | -10.0 | -4 | 7000.0 | 300000000.0 | 20000.0 | 3000.0 | 80.0 |
|
| 20000000.0 | 3000.0 | 3000.0 | 30.0 | -6000.0 | -10.0 | 300000000.0 | 2000.0 | 9000.0 | -100.0 |
|
||||||
| -4000.0 | -8000000.0 | 100.0 | -2 | -10.0 | 0.8 | 20000.0 | 100000000.0 | -4000.0 | -100.0 |
|
| -30.0 | 5000000.0 | 3000.0 | 40000.0 | 10.0 | -10.0 | 2000.0 | 50000000.0 | -3000.0 | -800000.0 |
|
||||||
| 500.0 | 800.0 | 5000000.0 | -40000.0 | 40.0 | -1 | 3000.0 | -4000.0 | 50000000.0 | 800000.0 |
|
| -5000.0 | 400.0 | -8000000.0 | 7 | 8 | 30.0 | 9000.0 | -3000.0 | 100000000.0 | 100.0 |
|
||||||
| 8 | 7 | 40000.0 | -300.0 | 0.4 | 0.0003 | 80.0 | -100.0 | 800000.0 | 20000.0 |
|
| 5 | -40000.0 | -30.0 | -300.0 | -0.1 | 0.1 | -100.0 | -800000.0 | 100.0 | 20000.0 |
|
||||||
|
|
||||||
|
|
||||||
#+begin_src matlab :exports results :results value table replace :tangle no
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
||||||
@ -129,16 +553,16 @@ We first extract the stiffness and mass matrices.
|
|||||||
|
|
||||||
#+caption: First 10x10 elements of the Mass matrix
|
#+caption: First 10x10 elements of the Mass matrix
|
||||||
#+RESULTS:
|
#+RESULTS:
|
||||||
| 0.03 | 2e-06 | -2e-07 | 1e-08 | 2e-08 | 0.0002 | -0.001 | 2e-07 | -8e-08 | -9e-10 |
|
| 0.03 | 7e-08 | 2e-06 | -3e-09 | -0.0002 | -6e-08 | -0.001 | 8e-07 | 6e-07 | -8e-09 |
|
||||||
| 2e-06 | 0.02 | -5e-07 | 7e-09 | 3e-08 | 2e-08 | -3e-07 | 0.0003 | -1e-08 | 1e-10 |
|
| 7e-08 | 0.02 | -1e-06 | 9e-05 | -3e-09 | -4e-09 | -1e-06 | -0.0006 | -4e-08 | 5e-06 |
|
||||||
| -2e-07 | -5e-07 | 0.02 | -9e-05 | 4e-09 | -1e-08 | 2e-07 | -2e-08 | -0.0006 | -5e-06 |
|
| 2e-06 | -1e-06 | 0.02 | -3e-08 | -4e-08 | 1e-08 | 1e-07 | -2e-07 | 0.0003 | 1e-09 |
|
||||||
| 1e-08 | 7e-09 | -9e-05 | 1e-06 | 6e-11 | 4e-10 | -1e-09 | 3e-11 | 5e-06 | 3e-08 |
|
| -3e-09 | 9e-05 | -3e-08 | 1e-06 | -3e-11 | -3e-13 | -7e-09 | -5e-06 | -3e-10 | 3e-08 |
|
||||||
| 2e-08 | 3e-08 | 4e-09 | 6e-11 | 1e-06 | 2e-10 | -2e-09 | 2e-10 | -7e-09 | -4e-11 |
|
| -0.0002 | -3e-09 | -4e-08 | -3e-11 | 2e-06 | 6e-10 | 2e-06 | -7e-09 | -2e-09 | 7e-11 |
|
||||||
| 0.0002 | 2e-08 | -1e-08 | 4e-10 | 2e-10 | 2e-06 | -2e-06 | -1e-09 | -7e-10 | -9e-12 |
|
| -6e-08 | -4e-09 | 1e-08 | -3e-13 | 6e-10 | 1e-06 | 1e-08 | 3e-09 | -2e-09 | 2e-13 |
|
||||||
| -0.001 | -3e-07 | 2e-07 | -1e-09 | -2e-09 | -2e-06 | 0.03 | -2e-06 | -1e-07 | -5e-09 |
|
| -0.001 | -1e-06 | 1e-07 | -7e-09 | 2e-06 | 1e-08 | 0.03 | 4e-08 | -2e-06 | 8e-09 |
|
||||||
| 2e-07 | 0.0003 | -2e-08 | 3e-11 | 2e-10 | -1e-09 | -2e-06 | 0.02 | -8e-07 | -1e-08 |
|
| 8e-07 | -0.0006 | -2e-07 | -5e-06 | -7e-09 | 3e-09 | 4e-08 | 0.02 | -9e-07 | -9e-05 |
|
||||||
| -8e-08 | -1e-08 | -0.0006 | 5e-06 | -7e-09 | -7e-10 | -1e-07 | -8e-07 | 0.02 | 9e-05 |
|
| 6e-07 | -4e-08 | 0.0003 | -3e-10 | -2e-09 | -2e-09 | -2e-06 | -9e-07 | 0.02 | 2e-08 |
|
||||||
| -9e-10 | 1e-10 | -5e-06 | 3e-08 | -4e-11 | -9e-12 | -5e-09 | -1e-08 | 9e-05 | 1e-06 |
|
| -8e-09 | 5e-06 | 1e-09 | 3e-08 | 7e-11 | 2e-13 | 8e-09 | -9e-05 | 2e-08 | 1e-06 |
|
||||||
|
|
||||||
|
|
||||||
Then, we extract the coordinates of the interface nodes.
|
Then, we extract the coordinates of the interface nodes.
|
||||||
@ -146,100 +570,47 @@ Then, we extract the coordinates of the interface nodes.
|
|||||||
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
|
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
|
The interface nodes are shown in Figure [[fig:APA95ML_nodes_1]] and their coordinates are listed in Table [[tab:apa95ml_nodes_coordinates]].
|
||||||
|
|
||||||
#+begin_src matlab :exports results :results value table replace :tangle no
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
||||||
data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
|
data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
#+RESULTS:
|
#+RESULTS:
|
||||||
| Total number of Nodes | 168959 |
|
| Total number of Nodes | 7 |
|
||||||
| Number of interface Nodes | 13 |
|
| Number of interface Nodes | 7 |
|
||||||
| Number of Modes | 30 |
|
| Number of Modes | 6 |
|
||||||
| Size of M and K matrices | 108 |
|
| Size of M and K matrices | 48 |
|
||||||
|
|
||||||
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
||||||
data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
|
data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
|
#+name: tab:apa95ml_nodes_coordinates
|
||||||
#+caption: Coordinates of the interface nodes
|
#+caption: Coordinates of the interface nodes
|
||||||
#+RESULTS:
|
#+RESULTS:
|
||||||
| Node i | Node Number | x [m] | y [m] | z [m] |
|
| Node i | Node Number | x [m] | y [m] | z [m] |
|
||||||
|--------+-------------+--------+-------+-------|
|
|--------+-------------+--------+-------+-----------|
|
||||||
| 1.0 | 168947.0 | 0.0 | 0.03 | 0.0 |
|
| 1.0 | 40467.0 | 0.0 | 0.0 | 0.029997 |
|
||||||
| 2.0 | 168949.0 | 0.0 | -0.03 | 0.0 |
|
| 2.0 | 40469.0 | 0.0 | 0.0 | -0.029997 |
|
||||||
| 3.0 | 168950.0 | -0.035 | 0.0 | 0.0 |
|
| 3.0 | 40470.0 | -0.035 | 0.0 | 0.0 |
|
||||||
| 4.0 | 168951.0 | -0.028 | 0.0 | 0.0 |
|
| 4.0 | 40475.0 | -0.015 | 0.0 | 0.0 |
|
||||||
| 5.0 | 168952.0 | -0.021 | 0.0 | 0.0 |
|
| 5.0 | 40476.0 | -0.005 | 0.0 | 0.0 |
|
||||||
| 6.0 | 168953.0 | -0.014 | 0.0 | 0.0 |
|
| 6.0 | 40477.0 | 0.015 | 0.0 | 0.0 |
|
||||||
| 7.0 | 168954.0 | -0.007 | 0.0 | 0.0 |
|
| 7.0 | 40478.0 | 0.035 | 0.0 | 0.0 |
|
||||||
| 8.0 | 168955.0 | 0.0 | 0.0 | 0.0 |
|
|
||||||
| 9.0 | 168956.0 | 0.007 | 0.0 | 0.0 |
|
#+name: fig:APA95ML_nodes_1
|
||||||
| 10.0 | 168957.0 | 0.014 | 0.0 | 0.0 |
|
#+caption: Interface Nodes chosen for the APA95ML
|
||||||
| 11.0 | 168958.0 | 0.021 | 0.0 | 0.0 |
|
[[file:figs/APA95ML_nodes_1.png]]
|
||||||
| 12.0 | 168959.0 | 0.035 | 0.0 | 0.0 |
|
|
||||||
| 13.0 | 168960.0 | 0.028 | 0.0 | 0.0 |
|
|
||||||
|
|
||||||
Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.
|
Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.
|
||||||
|
|
||||||
** Piezoelectric parameters
|
|
||||||
In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:
|
|
||||||
#+begin_src matlab
|
|
||||||
d33 = 300e-12; % Strain constant [m/V]
|
|
||||||
n = 80; % Number of layers per stack
|
|
||||||
eT = 1.6e-8; % Permittivity under constant stress [F/m]
|
|
||||||
sD = 1e-11; % Compliance under constant electric displacement [m2/N]
|
|
||||||
ka = 235e6; % Stack stiffness [N/m]
|
|
||||||
C = 5e-6; % Stack capactiance [F]
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
The ratio of the developed force to applied voltage is:
|
|
||||||
#+name: eq:piezo_voltage_to_force
|
|
||||||
\begin{equation}
|
|
||||||
F_a = g_a V_a, \quad g_a = d_{33} n k_a
|
|
||||||
\end{equation}
|
|
||||||
where:
|
|
||||||
- $F_a$: developed force in [N]
|
|
||||||
- $n$: number of layers of the actuator stack
|
|
||||||
- $d_{33}$: strain constant in [m/V]
|
|
||||||
- $k_a$: actuator stack stiffness in [N/m]
|
|
||||||
- $V_a$: applied voltage in [V]
|
|
||||||
|
|
||||||
If we take the numerical values, we obtain:
|
|
||||||
#+begin_src matlab :results replace value
|
|
||||||
d33*n*ka % [N/V]
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+RESULTS:
|
|
||||||
: 5.64
|
|
||||||
|
|
||||||
From cite:fleming14_desig_model_contr_nanop_system (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
|
|
||||||
#+name: eq:piezo_strain_to_voltage
|
|
||||||
\begin{equation}
|
|
||||||
V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
|
|
||||||
\end{equation}
|
|
||||||
where:
|
|
||||||
- $V_s$: measured voltage in [V]
|
|
||||||
- $d_{33}$: strain constant in [m/V]
|
|
||||||
- $\epsilon^T$: permittivity under constant stress in [F/m]
|
|
||||||
- $s^D$: elastic compliance under constant electric displacement in [m^2/N]
|
|
||||||
- $n$: number of layers of the sensor stack
|
|
||||||
- $\Delta h$: relative displacement in [m]
|
|
||||||
|
|
||||||
If we take the numerical values, we obtain:
|
|
||||||
#+begin_src matlab :results replace value
|
|
||||||
1e-6*d33/(eT*sD*n) % [V/um]
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+RESULTS:
|
|
||||||
: 23.438
|
|
||||||
|
|
||||||
** Simscape Model
|
** Simscape Model
|
||||||
The flexible element is imported using the =Reduced Order Flexible Solid= Simscape block.
|
The flexible element is imported using the =Reduced Order Flexible Solid= Simscape block.
|
||||||
|
|
||||||
To model the actuator, an =Internal Force= block is added between the nodes 3 and 12.
|
To model the actuator, an =Internal Force= block is added between the nodes 3 and 12.
|
||||||
A =Relative Motion Sensor= block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.
|
A =Relative Motion Sensor= block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.
|
||||||
|
|
||||||
- [ ] Add schematic of the model with interface nodes
|
|
||||||
|
|
||||||
One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
|
One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
|
||||||
#+begin_src matlab
|
#+begin_src matlab
|
||||||
m = 5;
|
m = 5;
|
||||||
@ -354,236 +725,6 @@ The obtained dynamics is shown in Figure [[fig:dynamics_force_force_sensor_comp_
|
|||||||
save('mat/fem_simscape_models.mat', 'Ghm', 'Gfm')
|
save('mat/fem_simscape_models.mat', 'Ghm', 'Gfm')
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
* TODO Estimation of piezoelectric parameters
|
|
||||||
** Matlab Init :noexport:ignore:
|
|
||||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
||||||
<<matlab-dir>>
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab :exports none :results silent :noweb yes
|
|
||||||
<<matlab-init>>
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab :tangle no
|
|
||||||
addpath('./matlab/mat/');
|
|
||||||
addpath('./matlab/');
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab :eval no
|
|
||||||
addpath('./mat/');
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
** From actuator voltage to vertical displacement
|
|
||||||
The data from the "noise test" and the identification test are loaded.
|
|
||||||
#+begin_src matlab
|
|
||||||
load('apa95ml_5kg_Amp_E505.mat', 't', 'um', 'y');
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
Any offset value is removed:
|
|
||||||
#+begin_src matlab
|
|
||||||
um = detrend(um, 0); % Amplifier Input Voltage [V]
|
|
||||||
y = detrend(y , 0); % Mass displacement [m]
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
Now we add a factor 10 to take into account the gain of the voltage amplifier.
|
|
||||||
#+begin_src matlab
|
|
||||||
um = 10*um; % Stack Actuator Input Voltage [V]
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab
|
|
||||||
Ts = t(end)/(length(t)-1);
|
|
||||||
Fs = 1/Ts;
|
|
||||||
|
|
||||||
win = hanning(ceil(1*Fs));
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab
|
|
||||||
[tf_est, f] = tfestimate(um, y, win, [], [], 1/Ts);
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
The gain from input voltage of the stack to the vertical displacement is determined:
|
|
||||||
#+begin_src matlab
|
|
||||||
gD = 4e-7; % [m/V]
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab :exports none
|
|
||||||
freqs = logspace(0, 4, 1000);
|
|
||||||
|
|
||||||
figure;
|
|
||||||
hold on;
|
|
||||||
plot(f, abs(tf_est))
|
|
||||||
plot([f(2) f(end)], [gD gD])
|
|
||||||
hold off;
|
|
||||||
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
|
||||||
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel', []);
|
|
||||||
xlim([10, 5e3]);
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab
|
|
||||||
K = extractMatrix('APA95ML_K.txt');
|
|
||||||
M = extractMatrix('APA95ML_M.txt');
|
|
||||||
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
Define parameters just for the simulation. Should not change any results
|
|
||||||
#+begin_src matlab
|
|
||||||
d33 = 1; % Strain constant [m/V]
|
|
||||||
n = 1; % Number of layers per stack
|
|
||||||
eT = 1; % Permittivity under constant stress [F/m]
|
|
||||||
sD = 1; % Compliance under constant electric displacement [m2/N]
|
|
||||||
ka = 1; % Stack stiffness [N/m]
|
|
||||||
C = 1; % Stack capactiance [F]
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab
|
|
||||||
m = 5;
|
|
||||||
|
|
||||||
%% Name of the Simulink File
|
|
||||||
mdl = 'piezo_amplified_3d';
|
|
||||||
|
|
||||||
%% Input/Output definition
|
|
||||||
clear io; io_i = 1;
|
|
||||||
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N]
|
|
||||||
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
|
|
||||||
|
|
||||||
Gd = linearize(mdl, io);
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab :results value replace
|
|
||||||
gF = abs(dcgain(Gd)); % [m/N]
|
|
||||||
ans = gF
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+RESULTS:
|
|
||||||
: 6.1695e-09
|
|
||||||
|
|
||||||
\[ g_a = g_D/g_F \]
|
|
||||||
in [N/V]
|
|
||||||
#+begin_src matlab :results value replace
|
|
||||||
ga = gD/gF
|
|
||||||
ans = ga
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+RESULTS:
|
|
||||||
: 64.835
|
|
||||||
|
|
||||||
#+begin_src matlab :results value replace
|
|
||||||
na = 2; ns = 1; d33 = 300e-12; n = 80; ka = 235e6; gL = 1.7;
|
|
||||||
na/(na+ns)*gL*d33*n*ka
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+RESULTS:
|
|
||||||
: 6.392
|
|
||||||
|
|
||||||
- [ ] Why is there a factor 10 with the "theoretical estimation?"
|
|
||||||
|
|
||||||
#+begin_src matlab :exports none
|
|
||||||
freqs = logspace(1, 4, 1000);
|
|
||||||
|
|
||||||
figure;
|
|
||||||
hold on;
|
|
||||||
plot(f, abs(tf_est))
|
|
||||||
plot(f, ga*abs(squeeze(freqresp(Gd, f, 'Hz'))))
|
|
||||||
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
|
||||||
ylabel('Amplitude'); xlabel('Frequency [Hz]');
|
|
||||||
hold off;
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
** From actuator voltage to sensor Voltage
|
|
||||||
#+begin_src matlab
|
|
||||||
load('apa95ml_5kg_2a_1s.mat', 't', 'u', 'v');
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab
|
|
||||||
u = detrend(u, 0); % Input Voltage of the Amplifier [V]
|
|
||||||
v = detrend(v, 0); % Voltage accross the stack sensor [V]
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab
|
|
||||||
u = 20*u; % Input Voltage of the Amplifier [V]
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab
|
|
||||||
Ts = t(end)/(length(t)-1);
|
|
||||||
Fs = 1/Ts;
|
|
||||||
|
|
||||||
win = hann(ceil(10/Ts));
|
|
||||||
|
|
||||||
[tf_est, f] = tfestimate(u, v, win, [], [], 1/Ts);
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab
|
|
||||||
gV = 0.022; % [V/V]
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab :exports none
|
|
||||||
freqs = logspace(1, 4, 1000);
|
|
||||||
|
|
||||||
figure;
|
|
||||||
hold on;
|
|
||||||
plot(f, abs(tf_est))
|
|
||||||
plot([f(2); f(end)], [gV; gV])
|
|
||||||
hold off;
|
|
||||||
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
|
||||||
ylabel('Amplitude'); xlabel('Frequency [Hz]');
|
|
||||||
ylim([1e-3, 1e1]);
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+begin_src matlab
|
|
||||||
m = 5;
|
|
||||||
|
|
||||||
%% Name of the Simulink File
|
|
||||||
mdl = 'piezo_amplified_3d';
|
|
||||||
|
|
||||||
%% Input/Output definition
|
|
||||||
clear io; io_i = 1;
|
|
||||||
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N]
|
|
||||||
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Stack displacement [m]
|
|
||||||
|
|
||||||
Gf = linearize(mdl, io);
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
$g_F$ in [m/N]
|
|
||||||
#+begin_src matlab :results value replace
|
|
||||||
gF = abs(dcgain(Gf));
|
|
||||||
ans = gF
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+RESULTS:
|
|
||||||
: 2.1546e-10
|
|
||||||
|
|
||||||
Finally, we compute the gain from strain of the sensor stack to its generated voltage:
|
|
||||||
\[ g_S = \frac{g_V}{g_F g_a} \]
|
|
||||||
Gs in [V/m]
|
|
||||||
#+begin_src matlab :results value replace
|
|
||||||
gS = gV/gF/ga;
|
|
||||||
ans = gS
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+RESULTS:
|
|
||||||
: 15749000.0
|
|
||||||
|
|
||||||
#+begin_src matlab :results value replace
|
|
||||||
d33 = 300e-12; eT = 5.3e-9; sD = 2e-11; n = 80;
|
|
||||||
d33/(eT*sD*n)
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+RESULTS:
|
|
||||||
: 35377000.0
|
|
||||||
|
|
||||||
#+begin_src matlab :exports none
|
|
||||||
freqs = logspace(1, 4, 1000);
|
|
||||||
|
|
||||||
figure;
|
|
||||||
hold on;
|
|
||||||
plot(f, abs(tf_est))
|
|
||||||
plot(f, ga*gS*abs(squeeze(freqresp(Gf, f, 'Hz'))))
|
|
||||||
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
|
||||||
ylabel('Amplitude'); xlabel('Frequency [Hz]');
|
|
||||||
hold off;
|
|
||||||
ylim([1e-3, 1e1]);
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
* Huddle Test
|
* Huddle Test
|
||||||
:PROPERTIES:
|
:PROPERTIES:
|
||||||
:header-args:matlab+: :tangle matlab/huddle_test.m
|
:header-args:matlab+: :tangle matlab/huddle_test.m
|
||||||
|