1 Experimental Setup
+
+
1 Experimental Setup
-The experimental Setup is schematically represented in Figure 1. +The experimental Setup is schematically represented in Figure 1.
@@ -70,21 +82,21 @@ The displacement of the mass (relative to the mechanical frame) is measured both
-
+
-
Figure 1: Schematic of the Experiment
+
-
Figure 2: Side View of the encoder
+
-
Figure 3: Front View of the encoder
@@ -92,8 +104,8 @@ The displacement of the mass (relative to the mechanical frame) is measured both
-
2 Huddle Test
+
+
-
2 Huddle Test
The goal in this section is the estimate the noise of both the encoder and the intereferometer. @@ -105,8 +117,8 @@ Ideally, a mechanical part would clamp the two together, we here suppose that th
-
2.1 Load Data
+
+
-
2.1 Load Data
load('mat/int_enc_huddle_test.mat', 'interferometer', 'encoder', 't'); @@ -121,11 +133,11 @@ encoder = detrend(encoder, 0);
-
2.2 Time Domain Results
+
+
2.2 Time Domain Results
-
+
-
Figure 4: Huddle test - Time domain signals
@@ -137,7 +149,7 @@ encoder = detrend(encoder, 0);
+
-
Figure 5: Huddle test - Time domain signals filtered with a LPF at 10Hz
@@ -145,8 +157,8 @@ encoder = detrend(encoder, 0);
-
2.3 Frequency Domain Noise
+
+
-
2.3 Frequency Domain Noise
Ts = 1e-4; @@ -158,7 +170,7 @@ win = hann(ceil(10/Ts));
+
-
Figure 6: Amplitude Spectral Density of the signals during the Huddle test
@@ -167,8 +179,8 @@ win = hann(ceil(10/Ts));
-
3 Comparison Interferometer / Encoder
+
+
-
3 Comparison Interferometer / Encoder
The goal here is to make sure that the interferometer and encoder measurements are coherent. @@ -176,8 +188,8 @@ We may see non-linearity in the interferometric measurement.
-
3.1 Load Data
+
+
-
3.1 Load Data
load('mat/int_enc_comp.mat', 'interferometer', 'encoder', 'u', 't'); @@ -193,18 +205,18 @@ u = detrend(u, 0);
-
3.2 Time Domain Results
+
+
3.2 Time Domain Results
-
+
-
Figure 7: One cycle measurement
+
-
Figure 8: Difference between the Encoder and the interferometer during one cycle
@@ -212,8 +224,8 @@ u = detrend(u, 0);
-
3.3 Difference between Encoder and Interferometer as a function of time
+
+
-
3.3 Difference between Encoder and Interferometer as a function of time
Ts = 1e-4; @@ -234,7 +246,7 @@ d_err_mean = d_err_mean - mean(d_err_mean);
+
-
Figure 9: Difference between the two measurement in the time domain, averaged for all the cycles
@@ -242,8 +254,8 @@ d_err_mean = d_err_mean - mean(d_err_mean);
-
3.4 Difference between Encoder and Interferometer as a function of position
+
+
-
3.4 Difference between Encoder and Interferometer as a function of position
Compute the mean of the interferometer measurement corresponding to each of the encoder measurement. @@ -262,7 +274,7 @@ i_mean_error = (i_mean - e_sorted);
+
Figure 10: Difference between the two measurement as a function of the measured position by the encoder, averaged for all the cycles
@@ -283,7 +295,7 @@ e_sorted_mean_over_period = mean(reshape(i_mean_error(i_init +
Figure 11: Non-Linearity of the Interferometer over the period of the wavelength
@@ -292,12 +304,12 @@ e_sorted_mean_over_period = mean(reshape(i_mean_error(i_init -4 Identification
+
+
-
4 Identification
-
4.1 Load Data
+
+
-
4.1 Load Data
load('mat/int_enc_id_noise_bis.mat', 'interferometer', 'encoder', 'u', 't'); @@ -313,8 +325,8 @@ u = detrend(u, 0);
->
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+ <>
+#+end_src
+
+** Load Data
+#+begin_src matlab
+ oc = load('./mat/identification_open_circuit.mat', 't', 'encoder', 'u');
+ sc = load('./mat/identification_short_circuit.mat', 't', 'encoder', 'u');
+#+end_src
+
+** Transfer Functions
+#+begin_src matlab
+ Ts = 1e-4; % Sampling Time [s]
+ win = hann(ceil(10/Ts));
+#+end_src
+
+#+begin_src matlab
+ [tf_oc_est, f] = tfestimate(oc.u, oc.encoder, win, [], [], 1/Ts);
+ [co_oc_est, ~] = mscohere( oc.u, oc.encoder, win, [], [], 1/Ts);
+
+ [tf_sc_est, ~] = tfestimate(sc.u, sc.encoder, win, [], [], 1/Ts);
+ [co_sc_est, ~] = mscohere( sc.u, sc.encoder, win, [], [], 1/Ts);
+#+end_src
+
+#+begin_src matlab :exports none
+ figure;
+ hold on;
+ plot(f, co_oc_est, '-')
+ plot(f, co_sc_est, '-')
+ set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ ylabel('Coherence'); xlabel('Frequency [Hz]');
+ hold off;
+ xlim([0.5, 5e3]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/stiffness_force_sensor_coherence.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:stiffness_force_sensor_coherence
+#+caption:
+#+RESULTS:
+[[file:figs/stiffness_force_sensor_coherence.png]]
+
+
+#+begin_src matlab :exports none
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ ax1 = nexttile;
+ hold on;
+ plot(f, abs(tf_oc_est), '-', 'DisplayName', 'Open-Circuit')
+ plot(f, abs(tf_sc_est), '-', 'DisplayName', 'Short-Circuit')
+ set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+ ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
+ hold off;
+ ylim([1e-7, 3e-4]);
+ legend('location', 'southwest');
+
+ ax2 = nexttile;
+ hold on;
+ plot(f, 180/pi*angle(tf_oc_est), '-')
+ plot(f, 180/pi*angle(tf_sc_est), '-')
+ set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ ylabel('Phase'); xlabel('Frequency [Hz]');
+ hold off;
+ yticks(-360:90:360);
+ axis padded 'auto x'
+
+ linkaxes([ax1,ax2], 'x');
+ xlim([0.5, 5e3]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/stiffness_force_sensor_bode.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:stiffness_force_sensor_bode
+#+caption:
+#+RESULTS:
+[[file:figs/stiffness_force_sensor_bode.png]]
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ xlim([180, 280]);
+ exportFig('figs/stiffness_force_sensor_bode_zoom.pdf', 'width', 'small', 'height', 'tall');
+#+end_src
+
+#+name: fig:stiffness_force_sensor_bode_zoom
+#+caption: Zoom on the change of resonance
+#+RESULTS:
+[[file:figs/stiffness_force_sensor_bode_zoom.png]]
+
+#+begin_important
+ The change of resonance frequency / stiffness is very small and is not important here.
+#+end_important
+
+* Generated Number of Charge / Voltage
+** Introduction :ignore:
+Two stacks are used as actuator (in parallel) and one stack is used as sensor.
+
+The amplifier gain is 20V/V (Cedrat LA75B).
+
+** Matlab Init :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+ <>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+ <>
+#+end_src
+
+** Steps
+#+begin_src matlab
+ load('./mat/force_sensor_steps.mat', 't', 'encoder', 'u', 'v');
+#+end_src
+
+#+begin_src matlab
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+ nexttile;
+ plot(t, v);
+ xlabel('Time [s]'); ylabel('Measured voltage [V]');
+ nexttile;
+ plot(t, u);
+ xlabel('Time [s]'); ylabel('Actuator Voltage [V]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/force_sen_steps_time_domain.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:force_sen_steps_time_domain
+#+caption: Time domain signal during the 3 actuator voltage steps
+#+RESULTS:
+[[file:figs/force_sen_steps_time_domain.png]]
+
+Three steps are performed at the following time intervals:
+- 2.5, 23
+- 23.8, 35
+- 35.8, 50
+
+Fit function:
+#+begin_src matlab
+ f = @(b,x) b(1).*exp(b(2).*x) + b(3);
+#+end_src
+
+We are interested by the =b(2)= term, which is the time constant of the exponential.
+
+#+begin_src matlab
+ tau = zeros(3,1);
+ V0 = zeros(3,1);
+#+end_src
+
+#+begin_src matlab
+ t_cur = t(2.5 < t & t < 23);
+ t_cur = t_cur - t_cur(1);
+ y_cur = v(2.5 < t & t < 23);
+
+ nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
+ B0 = [0.5, -0.15, 2.2]; % Choose Appropriate Initial Estimates
+ [B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’
+
+ tau(1) = 1/B(2);
+ V0(1) = B(3);
+#+end_src
+
+#+begin_src matlab
+ t_cur = t(23.8 < t & t < 35);
+ t_cur = t_cur - t_cur(1);
+ y_cur = v(23.8 < t & t < 35);
+
+ nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
+ B0 = [0.5, -0.15, 2.2]; % Choose Appropriate Initial Estimates
+ [B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’
+
+ tau(2) = 1/B(2);
+ V0(2) = B(3);
+#+end_src
+
+#+begin_src matlab
+ t_cur = t(35.8 < t);
+ t_cur = t_cur - t_cur(1);
+ y_cur = v(35.8 < t);
+
+ nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
+ B0 = [0.5, -0.15, 2.2]; % Choose Appropriate Initial Estimates
+ [B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’
+
+ tau(3) = 1/B(2);
+ V0(3) = B(3);
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+ data2orgtable([abs(tau), V0], {}, {'$tau$ [s]', '$V_0$ [V]'}, ' %.2f ');
+#+end_src
+
+#+RESULTS:
+| $tau$ [s] | $V_0$ [V] |
+|-----------+-----------|
+| 6.47 | 2.26 |
+| 6.76 | 2.26 |
+| 6.49 | 2.25 |
+
+With the capacitance being $C = 4.4 \mu F$, the internal impedance of the Speedgoat ADC can be computed as follows:
+#+begin_src matlab
+ Cp = 4.4e-6; % [F]
+ Rin = abs(mean(tau))/Cp;
+#+end_src
+
+#+begin_src matlab :results value replace :exports results
+ ans = Rin
+#+end_src
+
+#+RESULTS:
+: 1494100.0
+
+The input impedance of the Speedgoat's ADC should then be close to $1.5\,M\Omega$ (specified at $1\,M\Omega$).
+
+#+begin_important
+ How can we explain the voltage offset?
+#+end_important
+
+As shown in Figure [[fig:piezo_sensor_model_instrumentation]] (taken from cite:reza06_piezoel_trans_vibrat_contr_dampin), an input voltage offset is due to the input bias current $i_n$.
+
+#+name: fig:piezo_sensor_model_instrumentation
+#+caption: Model of a piezoelectric transducer (left) and instrumentation amplifier (right)
+[[file:figs/piezo_sensor_model_instrumentation.png]]
+
+The estimated input bias current is then:
+#+begin_src matlab
+ in = mean(V0)/Rin;
+#+end_src
+
+#+begin_src matlab :results value replace :exports results
+ ans = in
+#+end_src
+
+#+RESULTS:
+: 1.5119e-06
+
+An additional resistor in parallel with $R_{in}$ would have two effects:
+- reduce the input voltage offset
+ \[ V_{off} = \frac{R_a R_{in}}{R_a + R_{in}} i_n \]
+- increase the high pass corner frequency $f_c$
+ \[ C_p \frac{R_{in}R_a}{R_{in} + R_a} = \tau_c = \frac{1}{f_c} \]
+ \[ R_a = \frac{R_i}{f_c C_p R_i - 1} \]
+
+
+If we allow the high pass corner frequency to be equals to 3Hz:
+#+begin_src matlab
+ fc = 3;
+ Ra = Rin/(fc*C*Rin - 1);
+#+end_src
+
+#+begin_src matlab :results value replace :exports results
+ ans = Ra
+#+end_src
+
+#+RESULTS:
+: 79804
+
+With this parallel resistance value, the voltage offset would be:
+#+begin_src matlab
+ V_offset = Ra*Rin/(Ra + Rin) * in;
+#+end_src
+
+#+begin_src matlab :results value replace :exports results
+ ans = V_offset
+#+end_src
+
+#+RESULTS:
+: 0.11454
+
+Which is much more acceptable.
+
+** Sinus
+#+begin_src matlab
+ load('./mat/force_sensor_sin.mat', 't', 'encoder', 'u', 'v');
+
+ u = u(t>25);
+ v = v(t>25);
+ encoder = encoder(t>25) - mean(encoder(t>25));
+ t = t(t>25);
+#+end_src
+
+The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure [[fig:force_sensor_sin_u]]).
+
+#+begin_src matlab :exports none
+ figure;
+ plot(t, u)
+ xlabel('Time [s]'); ylabel('Control Voltage [V]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/force_sensor_sin_u.pdf', 'width', 'normal', 'height', 'small');
+#+end_src
+
+#+name: fig:force_sensor_sin_u
+#+caption: Driving Voltage
+#+RESULTS:
+[[file:figs/force_sensor_sin_u.png]]
+
+The full stroke as measured by the encoder is:
+#+begin_src matlab :results value replace
+ max(encoder)-min(encoder)
+#+end_src
+
+#+RESULTS:
+: 5.005e-05
+
+Its signal is shown in Figure [[fig:force_sensor_sin_encoder]].
+
+#+begin_src matlab :exports none
+ figure;
+ plot(t, encoder)
+ xlabel('Time [s]'); ylabel('Encoder [m]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/force_sensor_sin_encoder.pdf', 'width', 'normal', 'height', 'small');
+#+end_src
+
+#+name: fig:force_sensor_sin_encoder
+#+caption: Encoder measurement
+#+RESULTS:
+[[file:figs/force_sensor_sin_encoder.png]]
+
+The generated voltage by the stack is shown in Figure
+
+#+begin_src matlab :exports none
+ figure;
+ plot(t, v)
+ xlabel('Time [s]'); ylabel('Force Sensor Output [V]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/force_sensor_sin_stack.pdf', 'width', 'normal', 'height', 'small');
+#+end_src
+
+#+name: fig:force_sensor_sin_stack
+#+caption: Voltage measured on the stack used as a sensor
+#+RESULTS:
+[[file:figs/force_sensor_sin_stack.png]]
+
+The capacitance of the stack is
+#+begin_src matlab
+ Cp = 4.4e-6; % [F]
+#+end_src
+
+The corresponding generated charge is then shown in Figure [[fig:force_sensor_sin_charge]].
+#+begin_src matlab :exports none
+ figure;
+ plot(t, 1e6*Cp*(v-mean(v)))
+ xlabel('Time [s]'); ylabel('Generated Charge [$\mu C$]');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/force_sensor_sin_charge.pdf', 'width', 'normal', 'height', 'small');
+#+end_src
+
+#+name: fig:force_sensor_sin_charge
+#+caption: Generated Charge
+#+RESULTS:
+[[file:figs/force_sensor_sin_charge.png]]
+
+
+The relation between the generated voltage and the measured displacement is almost linear as shown in Figure [[fig:force_sensor_linear_relation]].
+
+#+begin_src matlab
+ b1 = encoder\(v-mean(v));
+#+end_src
+
+#+begin_src matlab :exports none
+ figure;
+ hold on;
+ plot(encoder, v-mean(v), 'DisplayName', 'Measured Voltage');
+ plot(encoder, encoder*b1, 'DisplayName', sprintf('Linear Fit: $U_s \\approx %.3f [V/\\mu m] \\cdot d$', 1e-6*abs(b1)));
+ hold off;
+ xlabel('Measured Displacement [m]'); ylabel('Generated Voltage [V]');
+ legend();
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/force_sensor_linear_relation.pdf', 'width', 'normal', 'height', 'small');
+#+end_src
+
+#+name: fig:force_sensor_linear_relation
+#+caption: Almost linear relation between the relative displacement and the generated voltage
+#+RESULTS:
+[[file:figs/force_sensor_linear_relation.png]]
+
+With a 16bits ADC, the resolution will then be equals to (in [nm]):
+#+begin_src matlab :results value replace
+ abs((20/2^16)/(b1/1e9))
+#+end_src
+
+#+RESULTS:
+: 3.9838
4.2 Identification
+
+
-
+
+
4.2 Identification
Ts = 1e-4; % Sampling Time [s] @@ -332,24 +344,436 @@ win = hann(ceil(10/Ts));
+
-
+
+
+
+5 Change of Stiffness due to Sensors stack being open/closed circuit
+
+
+
+
+
+5.1 Load Data
+
+
+
+
+oc = load('./mat/identification_open_circuit.mat', 't', 'encoder', 'u'); +sc = load('./mat/identification_short_circuit.mat', 't', 'encoder', 'u'); ++
+
+5.2 Transfer Functions
+
+
+
+
+
+Ts = 1e-4; % Sampling Time [s] +win = hann(ceil(10/Ts)); ++
+
+
+
+[tf_oc_est, f] = tfestimate(oc.u, oc.encoder, win, [], [], 1/Ts); +[co_oc_est, ~] = mscohere( oc.u, oc.encoder, win, [], [], 1/Ts); + +[tf_sc_est, ~] = tfestimate(sc.u, sc.encoder, win, [], [], 1/Ts); +[co_sc_est, ~] = mscohere( sc.u, sc.encoder, win, [], [], 1/Ts); ++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Figure 16: Zoom on the change of resonance
+
+
++The change of resonance frequency / stiffness is very small and is not important here. +
+ +
+
6 Generated Number of Charge / Voltage
+
+
+
++Two stacks are used as actuator (in parallel) and one stack is used as sensor. +
+ ++The amplifier gain is 20V/V (Cedrat LA75B). +
+
+
+
+6.1 Steps
+
+
+
+
+
+
+
+
+
+
+load('./mat/force_sensor_steps.mat', 't', 'encoder', 'u', 'v'); ++
+
+
+
+figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); +nexttile; +plot(t, v); +xlabel('Time [s]'); ylabel('Measured voltage [V]'); +nexttile; +plot(t, u); +xlabel('Time [s]'); ylabel('Actuator Voltage [V]'); ++
+
+
+
+
Figure 17: Time domain signal during the 3 actuator voltage steps
++Three steps are performed at the following time intervals: +
+-
+
- 2.5, 23 +
- 23.8, 35 +
- 35.8, 50 +
+Fit function: +
+
+
+
+f = @(b,x) b(1).*exp(b(2).*x) + b(3); ++
+We are interested by the b(2) term, which is the time constant of the exponential.
+
+
+
+tau = zeros(3,1); +V0 = zeros(3,1); ++
+
+
+t_cur = t(2.5 < t & t < 23); +t_cur = t_cur - t_cur(1); +y_cur = v(2.5 < t & t < 23); + +nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function +B0 = [0.5, -0.15, 2.2]; % Choose Appropriate Initial Estimates +[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’ + +tau(1) = 1/B(2); +V0(1) = B(3); ++
+
+
+t_cur = t(23.8 < t & t < 35); +t_cur = t_cur - t_cur(1); +y_cur = v(23.8 < t & t < 35); + +nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function +B0 = [0.5, -0.15, 2.2]; % Choose Appropriate Initial Estimates +[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’ + +tau(2) = 1/B(2); +V0(2) = B(3); ++
+
+
+t_cur = t(35.8 < t); +t_cur = t_cur - t_cur(1); +y_cur = v(35.8 < t); + +nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function +B0 = [0.5, -0.15, 2.2]; % Choose Appropriate Initial Estimates +[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’ + +tau(3) = 1/B(2); +V0(3) = B(3); ++
| \(tau\) [s] | +\(V_0\) [V] | +
|---|---|
| 6.47 | +2.26 | +
| 6.76 | +2.26 | +
| 6.49 | +2.25 | +
+With the capacitance being \(C = 4.4 \mu F\), the internal impedance of the Speedgoat ADC can be computed as follows: +
+
+
+
+Cp = 4.4e-6; % [F] +Rin = abs(mean(tau))/Cp; ++
+1494100.0 ++ + +
+The input impedance of the Speedgoat’s ADC should then be close to \(1.5\,M\Omega\) (specified at \(1\,M\Omega\)). +
+ +
+
+
++How can we explain the voltage offset? +
+ ++As shown in Figure 18 (taken from (Reza and Andrew 2006)), an input voltage offset is due to the input bias current \(i_n\). +
+ + +
+
+
+
+
Figure 18: Model of a piezoelectric transducer (left) and instrumentation amplifier (right)
++The estimated input bias current is then: +
+
+
+
+in = mean(V0)/Rin;
+
++1.5119e-06 ++ + +
+An additional resistor in parallel with \(R_{in}\) would have two effects: +
+-
+
- reduce the input voltage offset +\[ V_{off} = \frac{R_a R_{in}}{R_a + R_{in}} i_n \] +
- increase the high pass corner frequency \(f_c\) +\[ C_p \frac{R_{in}R_a}{R_{in} + R_a} = \tau_c = \frac{1}{f_c} \] +\[ R_a = \frac{R_i}{f_c C_p R_i - 1} \] +
+If we allow the high pass corner frequency to be equals to 3Hz: +
+
+
+
+fc = 3; +Ra = Rin/(fc*C*Rin - 1); ++
+79804 ++ + +
+With this parallel resistance value, the voltage offset would be: +
+
+
+
+V_offset = Ra*Rin/(Ra + Rin) * in; ++
+0.11454 ++ + +
+Which is much more acceptable. +
+
+
+6.2 Sinus
+
+
+
+
+
+load('./mat/force_sensor_sin.mat', 't', 'encoder', 'u', 'v'); + +u = u(t>25); +v = v(t>25); +encoder = encoder(t>25) - mean(encoder(t>25)); +t = t(t>25); ++
+The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure 19). +
+ + +
+
+
+
+
Figure 19: Driving Voltage
++The full stroke as measured by the encoder is: +
+
+
+
+max(encoder)-min(encoder)
+
++5.005e-05 ++ + +
+Its signal is shown in Figure 20. +
+ + +
+
+
+
+
Figure 20: Encoder measurement
++The generated voltage by the stack is shown in Figure +
+ + +
+
+
+
+
Figure 21: Voltage measured on the stack used as a sensor
++The capacitance of the stack is +
+
+
+
+Cp = 4.4e-6; % [F] ++
+The corresponding generated charge is then shown in Figure 22. +
+ +
+
+
+
+
+
Figure 22: Generated Charge
++The relation between the generated voltage and the measured displacement is almost linear as shown in Figure 23. +
+ +
+
+
+
+b1 = encoder\(v-mean(v)); ++
+
+
+
+
Figure 23: Almost linear relation between the relative displacement and the generated voltage
++With a 16bits ADC, the resolution will then be equals to (in [nm]): +
+
+
+
+abs((20/2^16)/(b1/1e9)) ++
+3.9838 ++ + +
Bibliography
+ +
-
diff --git a/index.org b/index.org
index 3efded5..647de5c 100644
--- a/index.org
+++ b/index.org
@@ -404,3 +404,408 @@ The period of the non-linearity seems to be $1.53 \mu m$ which corresponds to th
#+caption:
#+RESULTS:
[[file:figs/identification_dynamics_bode.png]]
+
+* Change of Stiffness due to Sensors stack being open/closed circuit
+** Matlab Init :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+ <Created: 2020-10-25 dim. 09:06
+Created: 2020-10-29 jeu. 09:59