1 Experimental Setup
+
+
1 Experimental Setup
-A schematic of the test-bench is shown in Figure 1. +A schematic of the test-bench is shown in Figure 1.
@@ -105,31 +111,31 @@ The APA95ML has three stacks that can be used as actuator or as sensors.
-Pictures of the test bench are shown in Figure 2 and 3. +Pictures of the test bench are shown in Figure 2 and 3.
-
+
-
Figure 1: Schematic of the Setup
+
-
Figure 2: Picture of the Setup
+
-
Figure 3: Zoom on the APA
+
-Here are the equipment used in the test bench:
@@ -145,11 +151,504 @@ Here are the equipment used in the test bench:
-
2 Simscape model of the test-bench
+
+
+
+
+
+2 Estimation of electrical/mechanical relationships
- +In order to correctly model the piezoelectric actuator, we need to determine: +
+-
+
- \(g_a\): the ratio of the generated force \(F_a\) to the supply voltage \(V_a\) across the piezoelectric stack +
- \(g_s\): the ratio of the generated voltage \(V_s\) across the piezoelectric stack when subject to a strain \(\Delta h\) +
+We estimate \(g_a\) and \(g_s\) using different approaches: +
+ +
+
+
+2.1 Estimation from Data-sheet
+
+
+
+
+
+
+
++The stack parameters taken from the data-sheet are shown in Table 1. +
+ +| Parameter | +Unit | +Value | +
|---|---|---|
| Nominal Stroke | +\(\mu m\) | +20 | +
| Blocked force | +\(N\) | +4700 | +
| Stiffness | +\(N/\mu m\) | +235 | +
| Voltage Range | +\(V\) | +-20..150 | +
| Capacitance | +\(\mu F\) | +4.4 | +
| Length | +\(mm\) | +20 | +
| Stack Area | +\(mm^2\) | +10x10 | +
+Let’s compute the generated force +
+ ++The stroke is \(L_{\max} = 20\mu m\) for a voltage range of \(V_{\max} = 170 V\). +Furthermore, the stiffness is \(k_a = 235 \cdot 10^6 N/m\). +
+ ++The relation between the applied voltage and the generated force can be estimated as follows: +
+\begin{equation} + g_a = k_a \frac{L_{\max}}{V_{\max}} +\end{equation} + +
+
+
+ka = 235e6; % [N/m] +Lmax = 20e-6; % [m] +Vmax = 170; % [V] ++
+
+
+ka*Lmax/Vmax % [N/V] ++
+27.647 ++ + +
+From the parameters of the stack, it seems not possible to estimate the relation between the strain and the generated voltage. +
+
+
+
+2.2 Estimation from 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: +
+
+
+
+d33 = 300e-12; % Strain constant [m/V] +n = 80; % Number of layers per stack +ka = 235e6; % Stack stiffness [N/m] ++
+The ratio of the developed force to applied voltage is: +
+\begin{equation} +\label{org8300de8} + 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: +
+
+
+
+ga = d33*n*ka; % [N/V] ++
+ga = 5.6 [N/V] ++ + + +
+From (Fleming and Leang 2014) (page 123), the relation between relative displacement of the sensor stack and generated voltage is: +
+\begin{equation} +\label{org312083f} + 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: +
+
+
+
+d33 = 300e-12; % Strain constant [m/V] +n = 80; % Number of layers per stack +eT = 5.3e-9; % Permittivity under constant stress [F/m] +sD = 2e-11; % Compliance under constant electric displacement [m2/N] ++
+
+
+gs = d33/(eT*sD*n); % [V/m] ++
+gs = 35.4 [V/um] ++
+
+
+2.3 Estimation from Experiment
+
+
+
++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. +
+ ++Using the experimental identification, we can easily obtain the gain from the applied voltage to the generated displacement, but not to the generated force. +However, from the Simscape model, we can easily have the link from the generated force to the displacement, them we can computed \(g_a\). +
+ ++Similarly, it is fairly easy to experimentally obtain the gain from the stack displacement to the generated voltage across the stack. +To link that to the strain of the sensor stack, the simscape model is used. +
+
+
+
+2.3.1 From actuator voltage \(V_a\) to actuator force \(F_a\)
+
+
++The data from the identification test is loaded. +
+
+
+
+load('apa95ml_5kg_Amp_E505.mat', 't', 'um', 'y'); + +% Any offset value is removed: +um = detrend(um, 0); % Amplifier Input Voltage [V] +y = detrend(y , 0); % Mass displacement [m] ++
+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. +
+
+
+
+um = 10*um; % Stack Actuator Input Voltage [V] ++
+Then, the transfer function from the stack voltage to the vertical displacement is computed. +
+
+
+
+Ts = t(end)/(length(t)-1); +Fs = 1/Ts; + +win = hanning(ceil(1*Fs)); + +[tf_est, f] = tfestimate(um, y, win, [], [], 1/Ts); ++
+The gain from input voltage of the stack to the vertical displacement is determined: +
+
+
+
+
+g_d_Va = 4e-7; % [m/V] ++
+
+
+
+
Figure 4: Transfer function from actuator stack voltage \(V_a\) to vertical displacement of the mass \(d\)
++Then, the transfer function from forces applied by the stack actuator to the vertical displacement of the mass is identified from the Simscape model. +
+
+
+
+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, '/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); ++
+The DC gain the the identified dynamics +
+
+
+
+g_d_Fa = abs(dcgain(Gd)); % [m/N]
+
++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} + +
+
+
+ga = g_d_Va/g_d_Fa;
+
++ga = 33.7 [N/V] ++ + +
+The obtained comparison between the Simscape model and the identified dynamics is shown in Figure 5. +
+ + +
+
+
+
Figure 5: Comparison of the identified transfer function between actuator voltage \(V_a\) and vertical mass displacement \(d\)
+
+
+2.3.2 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. +
+
+
+
+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] ++
+Here, an amplifier with a gain of 20 is used. +
+
+
+
+u = 20*u; % Input Voltage of the Amplifier [V] ++
+Then, the transfer function from \(V_a\) to \(V_s\) is identified and its DC gain is estimated (Figure 6). +
+
+
+
+Ts = t(end)/(length(t)-1); +Fs = 1/Ts; + +win = hann(ceil(10/Ts)); + +[tf_est, f] = tfestimate(u, v, win, [], [], 1/Ts); ++
+
+
+
+g_Vs_Va = 0.022; % [V/V]
+
+
+
+
+
+
+
Figure 6: Transfer function from actuator stack voltage \(V_a\) to sensor stack voltage \(V_s\)
++Now the transfer function from the actuator stack voltage to the sensor stack strain is estimated using the Simscape model. +
+
+
+
+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); ++
+The gain from the actuator stack voltage to the sensor stack strain is estimated below. +
+
+
+
+G_dh_Va = abs(dcgain(Gf)); ++
+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} + +
+
+
+gs = g_Vs_Va/G_dh_Va; % [V/m] ++
+gs = 3.5 [V/um] ++ + + +
+
+
+
Figure 7: Comparison of the identified transfer function between actuator voltage \(V_a\) and sensor stack voltage
+
+
+2.4 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. +
+
+
+save('./matlab/mat/apa95ml_params.mat', 'ga', 'gs'); ++
+
3 Simscape model of the test-bench
+
+
The idea here is to model the test-bench using Simscape. @@ -160,23 +659,31 @@ Whereas the suspended mass and metrology frame can be considered as rigid bodies
-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 8) and imported into Simscape.
+ + +
+
-
+
Figure 8: Finite Element Model of the APA95ML
-
2.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
-
+
+
+
+
-
-
@@ -510,28 +1021,28 @@ Then, we extract the coordinates of the interface nodes.
-
-
+
+
+
3.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
+We first extract the stiffness and mass matrices.
-
K = extractMatrix('APA95ML_K.txt'); -M = extractMatrix('APA95ML_M.txt'); +K = readmatrix('APA95ML_K.CSV'); +M = readmatrix('APA95ML_M.CSV');
| 300000000.0 | +-1000.0 | -30000.0 | -8000.0 | --200.0 | --30.0 | --60000.0 | +-40.0 | +70000.0 | +300.0 | 20000000.0 | --4000.0 | -500.0 | -8 | +-30.0 | +-5000.0 | +5 | ||
| -30000.0 | -100000000.0 | -400.0 | -30.0 | -200.0 | --1 | -4000.0 | --8000000.0 | -800.0 | -7 | -|||||||||
| 8000.0 | -400.0 | +-1000.0 | 50000000.0 | --800000.0 | --300.0 | --40.0 | +-7000.0 | +800000.0 | +-20.0 | 300.0 | -100.0 | -5000000.0 | -40000.0 | -|||||
| -200.0 | -30.0 | --800000.0 | -20000.0 | -5 | -1 | --10.0 | --2 | --40000.0 | --300.0 | -|||||||||
| -30.0 | -200.0 | --300.0 | -5 | -40000.0 | -0.3 | --4 | --10.0 | -40.0 | -0.4 | -|||||||||
| -60000.0 | --1 | --40.0 | -1 | -0.3 | 3000.0 | -7000.0 | -0.8 | --1 | -0.0003 | +5000000.0 | +400.0 | +-40000.0 | +||||||
| -30000.0 | +-7000.0 | +100000000.0 | +-200.0 | +-60.0 | +70.0 | +3000.0 | +3000.0 | +-8000000.0 | +-30.0 | +|||||||||
| -40.0 | +800000.0 | +-200.0 | +20000.0 | +-0.4 | +4 | +30.0 | +40000.0 | +7 | +-300.0 | +|||||||||
| 70000.0 | +-20.0 | +-60.0 | +-0.4 | +3000.0 | +1 | +-6000.0 | +10.0 | +8 | +-0.1 | +|||||||||
| 300.0 | +300.0 | +70.0 | +4 | +1 | +40000.0 | +-10.0 | +-10.0 | +30.0 | +0.1 | |||||||||
| 20000000.0 | -4000.0 | -300.0 | +3000.0 | +3000.0 | +30.0 | +-6000.0 | -10.0 | --4 | -7000.0 | 300000000.0 | -20000.0 | -3000.0 | -80.0 | -|||||
| -4000.0 | --8000000.0 | -100.0 | --2 | --10.0 | -0.8 | -20000.0 | -100000000.0 | --4000.0 | +2000.0 | +9000.0 | -100.0 | |||||||
| 500.0 | -800.0 | +-30.0 | 5000000.0 | --40000.0 | -40.0 | --1 | 3000.0 | --4000.0 | +40000.0 | +10.0 | +-10.0 | +2000.0 | 50000000.0 | -800000.0 | +-3000.0 | +-800000.0 | ||
| 8 | +-5000.0 | +400.0 | +-8000000.0 | 7 | -40000.0 | +8 | +30.0 | +9000.0 | +-3000.0 | +100000000.0 | +100.0 | +|||||||
| 5 | +-40000.0 | +-30.0 | -300.0 | -0.4 | -0.0003 | -80.0 | +-0.1 | +0.1 | -100.0 | -800000.0 | +-800000.0 | +100.0 | 20000.0 |
| 0.03 | +7e-08 | 2e-06 | --2e-07 | -1e-08 | -2e-08 | -0.0002 | +-3e-09 | +-0.0002 | +-6e-08 | -0.001 | -2e-07 | --8e-08 | --9e-10 | +8e-07 | +6e-07 | +-8e-09 | ||
| 2e-06 | +7e-08 | 0.02 | --5e-07 | -7e-09 | -3e-08 | -2e-08 | --3e-07 | -0.0003 | --1e-08 | -1e-10 | -||||||||
| -2e-07 | --5e-07 | -0.02 | --9e-05 | -4e-09 | --1e-08 | -2e-07 | --2e-08 | +-1e-06 | +9e-05 | +-3e-09 | +-4e-09 | +-1e-06 | -0.0006 | --5e-06 | -||||
| 1e-08 | -7e-09 | --9e-05 | -1e-06 | -6e-11 | -4e-10 | --1e-09 | -3e-11 | +-4e-08 | 5e-06 | -3e-08 | ||||||||
| 2e-08 | -3e-08 | -4e-09 | -6e-11 | -1e-06 | -2e-10 | --2e-09 | -2e-10 | --7e-09 | --4e-11 | -|||||||||
| 0.0002 | -2e-08 | --1e-08 | -4e-10 | -2e-10 | 2e-06 | --2e-06 | --1e-09 | --7e-10 | --9e-12 | +-1e-06 | +0.02 | +-3e-08 | +-4e-08 | +1e-08 | +1e-07 | +-2e-07 | +0.0003 | +1e-09 | +
| -3e-09 | +9e-05 | +-3e-08 | +1e-06 | +-3e-11 | +-3e-13 | +-7e-09 | +-5e-06 | +-3e-10 | +3e-08 | +|||||||||
| -0.0002 | +-3e-09 | +-4e-08 | +-3e-11 | +2e-06 | +6e-10 | +2e-06 | +-7e-09 | +-2e-09 | +7e-11 | +|||||||||
| -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 | +-1e-06 | +1e-07 | +-7e-09 | +2e-06 | +1e-08 | 0.03 | +4e-08 | -2e-06 | --1e-07 | --5e-09 | +8e-09 | ||
| 2e-07 | -0.0003 | --2e-08 | -3e-11 | -2e-10 | --1e-09 | --2e-06 | -0.02 | --8e-07 | --1e-08 | -|||||||||
| -8e-08 | --1e-08 | +8e-07 | -0.0006 | -5e-06 | +-2e-07 | +-5e-06 | -7e-09 | --7e-10 | --1e-07 | --8e-07 | +3e-09 | +4e-08 | 0.02 | -9e-05 | +-9e-07 | +-9e-05 | ||
| -9e-10 | -1e-10 | --5e-06 | +6e-07 | +-4e-08 | +0.0003 | +-3e-10 | +-2e-09 | +-2e-09 | +-2e-06 | +-9e-07 | +0.02 | +2e-08 | +||||||
| -8e-09 | +5e-06 | +1e-09 | 3e-08 | --4e-11 | --9e-12 | --5e-09 | --1e-08 | -9e-05 | +7e-11 | +2e-13 | +8e-09 | +-9e-05 | +2e-08 | 1e-06 |
| Total number of Nodes | -168959 | +7 |
| Number of interface Nodes | -13 | +7 |
| Number of Modes | -30 | +6 |
| Size of M and K matrices | -108 | +48 |
| 1.0 | -168947.0 | +40467.0 | 0.0 | -0.03 | 0.0 | +0.029997 |
| 2.0 | -168949.0 | +40469.0 | 0.0 | --0.03 | 0.0 | +-0.029997 |
| 3.0 | -168950.0 | +40470.0 | -0.035 | 0.0 | 0.0 | @@ -580,175 +1091,54 @@ Then, we extract the coordinates of the interface nodes.|
| 4.0 | -168951.0 | --0.028 | +40475.0 | +-0.015 | 0.0 | 0.0 |
| 5.0 | -168952.0 | --0.021 | +40476.0 | +-0.005 | 0.0 | 0.0 |
| 6.0 | -168953.0 | --0.014 | +40477.0 | +0.015 | 0.0 | 0.0 |
| 7.0 | -168954.0 | --0.007 | -0.0 | -0.0 | -||
| 8.0 | -168955.0 | -0.0 | -0.0 | -0.0 | -||
| 9.0 | -168956.0 | -0.007 | -0.0 | -0.0 | -||
| 10.0 | -168957.0 | -0.014 | -0.0 | -0.0 | -||
| 11.0 | -168958.0 | -0.021 | -0.0 | -0.0 | -||
| 12.0 | -168959.0 | +40478.0 | 0.035 | 0.0 | 0.0 | |
| 13.0 | -168960.0 | -0.028 | -0.0 | -0.0 | -
+
+
+
Figure 9: Interface Nodes chosen for the APA95ML
+
Using K, M and int_xyz, we can use the Reduced Order Flexible Solid simscape block.
-
-
-2.2 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: -
-
-
-
-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] --
-The ratio of the developed force to applied voltage is: -
-\begin{equation} -\label{org1eea105} - 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: -
-
-
-
-d33*n*ka % [N/V] --
-5.64 -- - -
-From (Fleming and Leang 2014) (page 123), the relation between relative displacement of the sensor stack and generated voltage is: -
-\begin{equation} -\label{org7e93246} - 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: -
-
-
-
-1e-6*d33/(eT*sD*n) % [V/um] --
-23.438 --
-
2.3 Simscape Model
-
+
+
-3.2 Simscape Model
+
The flexible element is imported using the Reduced Order Flexible Solid Simscape block.
Internal Force block is added between the
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.
@@ -772,11 +1158,11 @@ One mass is fixed at one end of the piezo-electric stack actuator, the other end
-
2.4 Dynamics from Actuator Voltage to Vertical Mass Displacement
-
+
+
-
3.3 Dynamics from Actuator Voltage to Vertical Mass Displacement
+-The identified dynamics is shown in Figure 4. +The identified dynamics is shown in Figure 10.
@@ -793,19 +1179,19 @@ Ghm = -linearize(mdl, io);
-
+
Figure 4: Dynamics from \(F\) to \(d\) without a payload and with a 5kg payload
+Figure 10: Dynamics from \(F\) to \(d\) without a payload and with a 5kg payload
-
2.5 Dynamics from Actuator Voltage to Force Sensor Voltage
-
+
+
-
3.4 Dynamics from Actuator Voltage to Force Sensor Voltage
+-The obtained dynamics is shown in Figure 5. +The obtained dynamics is shown in Figure 11.
@@ -822,17 +1208,17 @@ Gfm = linearize(mdl, io);
-
+
Figure 5: Dynamics from \(F\) to \(F_m\) for \(m=0\) and \(m = 10kg\)
+Figure 11: Dynamics from \(F\) to \(F_m\) for \(m=0\) and \(m = 10kg\)
-
2.6 Save Data for further use
-
+
+
-
3.5 Save Data for further use
+save('matlab/mat/fem_simscape_models.mat', 'Ghm', 'Gfm')@@ -846,250 +1232,27 @@ Gfm = linearize(mdl, io);
-
-
-3 Estimation of piezoelectric parameters
-
-
-
-
-
-3.1 From actuator voltage to vertical displacement
-
-
--The data from the “noise test” and the identification test are loaded. -
-
-
-
-load('apa95ml_5kg_Amp_E505.mat', 't', 'um', 'y'); --
-Any offset value is removed: -
-
-
-
-um = detrend(um, 0); % Amplifier Input Voltage [V] -y = detrend(y , 0); % Mass displacement [m] --
-Now we add a factor 10 to take into account the gain of the voltage amplifier. -
-
-
-
-um = 10*um; % Stack Actuator Input Voltage [V] --
-
-
-Ts = t(end)/(length(t)-1); -Fs = 1/Ts; - -win = hanning(ceil(1*Fs)); --
-
-
-[tf_est, f] = tfestimate(um, y, win, [], [], 1/Ts);
-
--The gain from input voltage of the stack to the vertical displacement is determined: -
-
-
-
-gD = 4e-7; % [m/V] --
-
-
-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'); --
-Define parameters just for the simulation. Should not change any results -
-
-
-
-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] --
-
-
-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); --
-
-
-gF = abs(dcgain(Gd)); % [m/N] -ans = gF --
-6.1695e-09 -- - -
-\[ g_a = g_D/g_F \] -in [N/V] -
-
-
-
-ga = gD/gF -ans = ga --
-64.835 -- - -
-
-
-na = 2; ns = 1; d33 = 300e-12; n = 80; ka = 235e6; gL = 1.7; -na/(na+ns)*gL*d33*n*ka --
-6.392 -- - -
-
-
[ ]Why is there a factor 10 with the “theoretical estimation?”
-
-
-3.2 From actuator voltage to sensor Voltage
-
-
-
-
-
-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] --
-
-
-u = 20*u; % Input Voltage of the Amplifier [V] --
-
-
-Ts = t(end)/(length(t)-1); -Fs = 1/Ts; - -win = hann(ceil(10/Ts)); - -[tf_est, f] = tfestimate(u, v, win, [], [], 1/Ts); --
-
-
-gV = 0.022; % [V/V]
-
-
-
-
-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); --
-\(g_F\) in [m/N] -
-
-
-
-gF = abs(dcgain(Gf));
-ans = gF
-
--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] -
-
-
-
-gS = gV/gF/ga; -ans = gS --
-15749000.0 -- - -
-
-
-d33 = 300e-12; eT = 5.3e-9; sD = 2e-11; n = 80; -d33/(eT*sD*n) --
-35377000.0 --
-
4 Huddle Test
+
+
4 Huddle Test
-
-
4.1 Time Domain Data
+
+
4.1 Time Domain Data
-
-
+
Figure 6: Measurement of the Mass displacement during Huddle Test
+Figure 12: Measurement of the Mass displacement during Huddle Test
-
4.2 PSD of Measurement Noise
+
+
-
-4.2 PSD of Measurement Noise
Ts = t(end)/(length(t)-1); @@ -1105,20 +1268,20 @@ win = hanning(ceil(1*Fs));
+
Figure 7: Amplitude Spectral Density of the Displacement during Huddle Test
+Figure 13: Amplitude Spectral Density of the Displacement during Huddle Test
-
5 Identification of the dynamics from actuator to displacement
+
+
5 Identification of the dynamics from actuator to displacement
-
-
5.1 Load Data
+
+
-5.1 Load Data
The data from the “noise test” and the identification test are loaded. @@ -1165,8 +1328,8 @@ ht.u = 10*ht.u;
-
5.2 Comparison of the PSD with Huddle Test
+
+
-
-
5.2 Comparison of the PSD with Huddle Test
Ts = t(end)/(length(t)-1); @@ -1183,16 +1346,16 @@ win = hanning(ceil(1*Fs));
+
Figure 8: Comparison of the ASD for the identification test and the huddle test
+Figure 14: Comparison of the ASD for the identification test and the huddle test
-
5.3 Compute TF estimate and Coherence
+
+
-
-5.3 Compute TF estimate and Coherence
[tf_est, f] = tfestimate(um, -y, win, [], [], 1/Ts); @@ -1201,10 +1364,10 @@ win = hanning(ceil(1*Fs));
+
-
Figure 9: Coherence
+Figure 15: Coherence
@@ -1216,20 +1379,20 @@ Comparison with the FEM model
+
Figure 10: Comparison of the identified transfer function and the one estimated from the FE model
+Figure 16: Comparison of the identified transfer function and the one estimated from the FE model
-
6 Identification of the dynamics from actuator to force sensor
+
+
6 Identification of the dynamics from actuator to force sensor
-The transfer function from input voltage to output voltage are computed and shown in Figure 11. +The transfer function from input voltage to output voltage are computed and shown in Figure 17.
@@ -1311,14 +1474,14 @@ win = hann(ceil(10/Ts));
-
+
-
Figure 11: Comparison of the identified dynamics from voltage output to voltage input and the FEM
+Figure 17: Comparison of the identified dynamics from voltage output to voltage input and the FEM
-
6.1 System Identification
+
+
-
6.1 System Identification
w_z = 2*pi*111; % Zeros frequency [rad/s] @@ -1332,44 +1495,44 @@ Gi = G_inf*(s^2- +-
Figure 12: Identification of the IFF plant
+Figure 18: Identification of the IFF plant
-
6.2 Integral Force Feedback
+
+
-6.2 Integral Force Feedback
-
+
Figure 13: Root Locus for IFF
+Figure 19: Root Locus for IFF
-
7 Integral Force Feedback
+
+
7 Integral Force Feedback
-
+
-
Figure 14: Schematic of the test bench using IFF
+Figure 20: Schematic of the test bench using IFF
-
7.1 First tests with few gains
+
+
-
7.1 First tests with few gains
iff_g10 = load('apa95ml_iff_g10_res.mat', 'u', 't', 'y', 'v'); @@ -1394,24 +1557,24 @@ win = hann(ceil(10/Ts));
+
-
-
Figure 15: Coherence
+Figure 21: Coherence
+
Figure 16: Bode plot for different values of IFF gain
+Figure 22: Bode plot for different values of IFF gain
-
7.2 Second test with many Gains
+
+
-
7.2 Second test with many Gains
load('apa95ml_iff_test.mat', 'results'); @@ -1440,7 +1603,7 @@ g_iff = [0, 1, 5, 10, 50, 100];
+
@@ -1462,13 +1625,13 @@ f_end = 500; % [Hz]
-
+
->
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+ <>
+#+end_src
+
+#+begin_src matlab :tangle no
+ addpath('./matlab/mat/');
+ addpath('./matlab/');
+#+end_src
+
+#+begin_src matlab :eval no
+ addpath('./mat/');
+#+end_src
+
+** Estimation from Data-sheet
+<>
+
+The stack parameters taken from the data-sheet are shown in Table [[tab:stack_parameters]].
+
+#+name: tab:stack_parameters
+#+caption: Stack Parameters
+| Parameter | Unit | Value |
+|----------------+-----------+----------|
+| Nominal Stroke | $\mu m$ | 20 |
+| Blocked force | $N$ | 4700 |
+| Stiffness | $N/\mu m$ | 235 |
+| Voltage Range | $V$ | -20..150 |
+| Capacitance | $\mu F$ | 4.4 |
+| Length | $mm$ | 20 |
+| Stack Area | $mm^2$ | 10x10 |
+
+Let's compute the generated force
+
+The stroke is $L_{\max} = 20\mu m$ for a voltage range of $V_{\max} = 170 V$.
+Furthermore, the stiffness is $k_a = 235 \cdot 10^6 N/m$.
+
+The relation between the applied voltage and the generated force can be estimated as follows:
+\begin{equation}
+ g_a = k_a \frac{L_{\max}}{V_{\max}}
+\end{equation}
+
+#+begin_src matlab
+ ka = 235e6; % [N/m]
+ Lmax = 20e-6; % [m]
+ Vmax = 170; % [V]
+#+end_src
+
+#+begin_src matlab :results value replace
+ 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
+<>
+
+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
+ ka = 235e6; % Stack stiffness [N/m]
+#+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
+ ga = d33*n*ka; % [N/V]
+#+end_src
+
+#+begin_src matlab :results value replace :exports results :tangle no
+ sprintf('ga = %.1f [N/V]', ga)
+#+end_src
+
+#+RESULTS:
+: ga = 5.6 [N/V]
+
+
+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
+ d33 = 300e-12; % Strain constant [m/V]
+ n = 80; % Number of layers per stack
+ eT = 5.3e-9; % Permittivity under constant stress [F/m]
+ sD = 2e-11; % Compliance under constant electric displacement [m2/N]
+#+end_src
+
+#+begin_src matlab
+ gs = d33/(eT*sD*n); % [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 = 35.4 [V/um]
+
+** Estimation from Experiment
+<>
+*** 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.
+
+Using the experimental identification, we can easily obtain the gain from the applied voltage to the generated displacement, but not to the generated force.
+However, from the Simscape model, we can easily have the link from the generated force to the displacement, them we can computed $g_a$.
+
+Similarly, it is fairly easy to experimentally obtain the gain from the stack displacement to the generated voltage across the stack.
+To link that to the strain of the sensor stack, the simscape model is used.
+
+*** From actuator voltage $V_a$ to actuator force $F_a$
+The data from the identification test is loaded.
+#+begin_src matlab
+ load('apa95ml_5kg_Amp_E505.mat', 't', 'um', 'y');
+
+ % Any offset value is removed:
+ 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 and thus obtain the voltage across the piezoelectric stack.
+#+begin_src matlab
+ um = 10*um; % Stack Actuator Input Voltage [V]
+#+end_src
+
+Then, the transfer function from the stack voltage to the vertical displacement is computed.
+#+begin_src matlab
+ Ts = t(end)/(length(t)-1);
+ 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.
+#+begin_src matlab :exports none
+ % Load Parameters
+ K = readmatrix('APA95ML_K.CSV');
+ 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]
+ gs = 1; % [V/m]
+#+end_src
+
+#+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, '/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
+
+The DC gain the the identified dynamics
+#+begin_src matlab
+ g_d_Fa = abs(dcgain(Gd)); % [m/N]
+#+end_src
+
+#+begin_src matlab :results value replace :exports results :tangle no
+ sprintf('G_d_Fa = %.2g [m/N]', g_d_Fa)
+#+end_src
+
+#+RESULTS:
+: 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
:PROPERTIES:
:header-args:matlab+: :tangle matlab/simscape_model.m
:header-args:matlab+: :comments org :mkdirp yes
:END:
<>
+
** Introduction :ignore:
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.
-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:
#+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
We first extract the stiffness and mass matrices.
#+begin_src matlab
- K = extractMatrix('APA95ML_K.txt');
- M = extractMatrix('APA95ML_M.txt');
+ K = readmatrix('APA95ML_K.CSV');
+ M = readmatrix('APA95ML_M.CSV');
#+end_src
#+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
#+RESULTS:
-| 300000000.0 | -30000.0 | 8000.0 | -200.0 | -30.0 | -60000.0 | 20000000.0 | -4000.0 | 500.0 | 8 |
-| -30000.0 | 100000000.0 | 400.0 | 30.0 | 200.0 | -1 | 4000.0 | -8000000.0 | 800.0 | 7 |
-| 8000.0 | 400.0 | 50000000.0 | -800000.0 | -300.0 | -40.0 | 300.0 | 100.0 | 5000000.0 | 40000.0 |
-| -200.0 | 30.0 | -800000.0 | 20000.0 | 5 | 1 | -10.0 | -2 | -40000.0 | -300.0 |
-| -30.0 | 200.0 | -300.0 | 5 | 40000.0 | 0.3 | -4 | -10.0 | 40.0 | 0.4 |
-| -60000.0 | -1 | -40.0 | 1 | 0.3 | 3000.0 | 7000.0 | 0.8 | -1 | 0.0003 |
-| 20000000.0 | 4000.0 | 300.0 | -10.0 | -4 | 7000.0 | 300000000.0 | 20000.0 | 3000.0 | 80.0 |
-| -4000.0 | -8000000.0 | 100.0 | -2 | -10.0 | 0.8 | 20000.0 | 100000000.0 | -4000.0 | -100.0 |
-| 500.0 | 800.0 | 5000000.0 | -40000.0 | 40.0 | -1 | 3000.0 | -4000.0 | 50000000.0 | 800000.0 |
-| 8 | 7 | 40000.0 | -300.0 | 0.4 | 0.0003 | 80.0 | -100.0 | 800000.0 | 20000.0 |
+| 300000000.0 | -1000.0 | -30000.0 | -40.0 | 70000.0 | 300.0 | 20000000.0 | -30.0 | -5000.0 | 5 |
+| -1000.0 | 50000000.0 | -7000.0 | 800000.0 | -20.0 | 300.0 | 3000.0 | 5000000.0 | 400.0 | -40000.0 |
+| -30000.0 | -7000.0 | 100000000.0 | -200.0 | -60.0 | 70.0 | 3000.0 | 3000.0 | -8000000.0 | -30.0 |
+| -40.0 | 800000.0 | -200.0 | 20000.0 | -0.4 | 4 | 30.0 | 40000.0 | 7 | -300.0 |
+| 70000.0 | -20.0 | -60.0 | -0.4 | 3000.0 | 1 | -6000.0 | 10.0 | 8 | -0.1 |
+| 300.0 | 300.0 | 70.0 | 4 | 1 | 40000.0 | -10.0 | -10.0 | 30.0 | 0.1 |
+| 20000000.0 | 3000.0 | 3000.0 | 30.0 | -6000.0 | -10.0 | 300000000.0 | 2000.0 | 9000.0 | -100.0 |
+| -30.0 | 5000000.0 | 3000.0 | 40000.0 | 10.0 | -10.0 | 2000.0 | 50000000.0 | -3000.0 | -800000.0 |
+| -5000.0 | 400.0 | -8000000.0 | 7 | 8 | 30.0 | 9000.0 | -3000.0 | 100000000.0 | 100.0 |
+| 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
@@ -129,16 +553,16 @@ We first extract the stiffness and mass matrices.
#+caption: First 10x10 elements of the Mass matrix
#+RESULTS:
-| 0.03 | 2e-06 | -2e-07 | 1e-08 | 2e-08 | 0.0002 | -0.001 | 2e-07 | -8e-08 | -9e-10 |
-| 2e-06 | 0.02 | -5e-07 | 7e-09 | 3e-08 | 2e-08 | -3e-07 | 0.0003 | -1e-08 | 1e-10 |
-| -2e-07 | -5e-07 | 0.02 | -9e-05 | 4e-09 | -1e-08 | 2e-07 | -2e-08 | -0.0006 | -5e-06 |
-| 1e-08 | 7e-09 | -9e-05 | 1e-06 | 6e-11 | 4e-10 | -1e-09 | 3e-11 | 5e-06 | 3e-08 |
-| 2e-08 | 3e-08 | 4e-09 | 6e-11 | 1e-06 | 2e-10 | -2e-09 | 2e-10 | -7e-09 | -4e-11 |
-| 0.0002 | 2e-08 | -1e-08 | 4e-10 | 2e-10 | 2e-06 | -2e-06 | -1e-09 | -7e-10 | -9e-12 |
-| -0.001 | -3e-07 | 2e-07 | -1e-09 | -2e-09 | -2e-06 | 0.03 | -2e-06 | -1e-07 | -5e-09 |
-| 2e-07 | 0.0003 | -2e-08 | 3e-11 | 2e-10 | -1e-09 | -2e-06 | 0.02 | -8e-07 | -1e-08 |
-| -8e-08 | -1e-08 | -0.0006 | 5e-06 | -7e-09 | -7e-10 | -1e-07 | -8e-07 | 0.02 | 9e-05 |
-| -9e-10 | 1e-10 | -5e-06 | 3e-08 | -4e-11 | -9e-12 | -5e-09 | -1e-08 | 9e-05 | 1e-06 |
+| 0.03 | 7e-08 | 2e-06 | -3e-09 | -0.0002 | -6e-08 | -0.001 | 8e-07 | 6e-07 | -8e-09 |
+| 7e-08 | 0.02 | -1e-06 | 9e-05 | -3e-09 | -4e-09 | -1e-06 | -0.0006 | -4e-08 | 5e-06 |
+| 2e-06 | -1e-06 | 0.02 | -3e-08 | -4e-08 | 1e-08 | 1e-07 | -2e-07 | 0.0003 | 1e-09 |
+| -3e-09 | 9e-05 | -3e-08 | 1e-06 | -3e-11 | -3e-13 | -7e-09 | -5e-06 | -3e-10 | 3e-08 |
+| -0.0002 | -3e-09 | -4e-08 | -3e-11 | 2e-06 | 6e-10 | 2e-06 | -7e-09 | -2e-09 | 7e-11 |
+| -6e-08 | -4e-09 | 1e-08 | -3e-13 | 6e-10 | 1e-06 | 1e-08 | 3e-09 | -2e-09 | 2e-13 |
+| -0.001 | -1e-06 | 1e-07 | -7e-09 | 2e-06 | 1e-08 | 0.03 | 4e-08 | -2e-06 | 8e-09 |
+| 8e-07 | -0.0006 | -2e-07 | -5e-06 | -7e-09 | 3e-09 | 4e-08 | 0.02 | -9e-07 | -9e-05 |
+| 6e-07 | -4e-08 | 0.0003 | -3e-10 | -2e-09 | -2e-09 | -2e-06 | -9e-07 | 0.02 | 2e-08 |
+| -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.
@@ -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');
#+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
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
#+RESULTS:
-| Total number of Nodes | 168959 |
-| Number of interface Nodes | 13 |
-| Number of Modes | 30 |
-| Size of M and K matrices | 108 |
+| Total number of Nodes | 7 |
+| Number of interface Nodes | 7 |
+| Number of Modes | 6 |
+| Size of M and K matrices | 48 |
#+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 ');
#+end_src
+#+name: tab:apa95ml_nodes_coordinates
#+caption: Coordinates of the interface nodes
#+RESULTS:
-| Node i | Node Number | x [m] | y [m] | z [m] |
-|--------+-------------+--------+-------+-------|
-| 1.0 | 168947.0 | 0.0 | 0.03 | 0.0 |
-| 2.0 | 168949.0 | 0.0 | -0.03 | 0.0 |
-| 3.0 | 168950.0 | -0.035 | 0.0 | 0.0 |
-| 4.0 | 168951.0 | -0.028 | 0.0 | 0.0 |
-| 5.0 | 168952.0 | -0.021 | 0.0 | 0.0 |
-| 6.0 | 168953.0 | -0.014 | 0.0 | 0.0 |
-| 7.0 | 168954.0 | -0.007 | 0.0 | 0.0 |
-| 8.0 | 168955.0 | 0.0 | 0.0 | 0.0 |
-| 9.0 | 168956.0 | 0.007 | 0.0 | 0.0 |
-| 10.0 | 168957.0 | 0.014 | 0.0 | 0.0 |
-| 11.0 | 168958.0 | 0.021 | 0.0 | 0.0 |
-| 12.0 | 168959.0 | 0.035 | 0.0 | 0.0 |
-| 13.0 | 168960.0 | 0.028 | 0.0 | 0.0 |
+| Node i | Node Number | x [m] | y [m] | z [m] |
+|--------+-------------+--------+-------+-----------|
+| 1.0 | 40467.0 | 0.0 | 0.0 | 0.029997 |
+| 2.0 | 40469.0 | 0.0 | 0.0 | -0.029997 |
+| 3.0 | 40470.0 | -0.035 | 0.0 | 0.0 |
+| 4.0 | 40475.0 | -0.015 | 0.0 | 0.0 |
+| 5.0 | 40476.0 | -0.005 | 0.0 | 0.0 |
+| 6.0 | 40477.0 | 0.015 | 0.0 | 0.0 |
+| 7.0 | 40478.0 | 0.035 | 0.0 | 0.0 |
+
+#+name: fig:APA95ML_nodes_1
+#+caption: Interface Nodes chosen for the APA95ML
+[[file:figs/APA95ML_nodes_1.png]]
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
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.
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.
#+begin_src matlab
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')
#+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)
- <>
-#+end_src
-
-#+begin_src matlab :exports none :results silent :noweb yes
- <>
-#+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
:PROPERTIES:
:header-args:matlab+: :tangle matlab/huddle_test.m
+
@@ -1483,7 +1646,7 @@ f_end = 500; % [Hz]
-
diff --git a/index.org b/index.org
index 1c3670e..27dca3c 100644
--- a/index.org
+++ b/index.org
@@ -67,18 +67,442 @@ Here are the equipment used in the test bench:
- Low Noise Voltage Preamplifier from Ametek ([[file:doc/model_5113.pdf][doc]])
#+end_note
+* Estimation of electrical/mechanical relationships
+** Introduction :ignore:
+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
+2. $g_s$: the ratio of the generated voltage $V_s$ across the piezoelectric stack when subject to a strain $\Delta h$
+
+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
+
+** 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: 2020-11-24 mar. 09:17
+Created: 2020-11-24 mar. 13:54