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    Table of Contents

      -
    • Section 1:
    • -
    • Section 2:
    • +
    • Section 1:
    • +
    • Section 3:
    • Section [[]]:
    • Section [[]]:
    • Section [[]]:
    -
    -

    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.

    -
    +

    test_bench_apa_schematic.png

    Figure 1: Schematic of the Setup

    -
    +

    setup_picture.png

    Figure 2: Picture of the Setup

    -
    +

    setup_zoom.png

    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: +

    +
      +
    1. \(g_a\): the ratio of the generated force \(F_a\) to the supply voltage \(V_a\) across the piezoelectric stack
    2. +
    3. \(g_s\): the ratio of the generated voltage \(V_s\) across the piezoelectric stack when subject to a strain \(\Delta h\)
    4. +
    + +

    +We estimate \(g_a\) and \(g_s\) using different approaches: +

    +
      +
    1. Section 2.1: \(g_a\) is estimated from the datasheet of the piezoelectric stack
    2. +
    3. Section 2.2: \(g_a\) and \(g_s\) are estimated using the piezoelectric constants
    4. +
    5. Section 2.3: \(g_a\) and \(g_s\) are estimated experimentally
    6. +
    +
    + +
    +

    2.1 Estimation from Data-sheet

    +
    +

    + +

    + +

    +The stack parameters taken from the data-sheet are shown in Table 1. +

    + + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    Table 1: Stack Parameters
    ParameterUnitValue
    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]
    +
    +
    + + +
    +

    gain_Va_to_d.png +

    +

    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. +

    + + +
    +

    compare_Gd_id_simscape.png +

    +

    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]
    +
    +
    + + +
    +

    gain_Va_to_Vs.png +

    +

    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]
    +
    + + + +
    +

    compare_Gf_id_simscape.png +

    +

    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.

    + + +
    +

    APA95ML_FEM.png +

    +

    Figure 8: Finite Element Model of the APA95ML

    -
    -

    2.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates

    -
    +
    + +
    +

    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');
     
    - +@@ -202,131 +709,131 @@ M = extractMatrix('APA95ML_M.txt'); + - - - - + + + - - - + + + - - - - - - - - - - - - - - - + - - - + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - - + + + + - - - - - - - - - - - - - - - - - + + - - + - - - - + + + + - + + - + + + - + + + + + + + + + + + + - - - + + - + + @@ -334,7 +841,7 @@ M = extractMatrix('APA95ML_M.txt');
    Table 1: First 10x10 elements of the Stiffness matrixTable 2: First 10x10 elements of the Stiffness matrix
    300000000.0-1000.0 -30000.08000.0-200.0-30.0-60000.0-40.070000.0300.0 20000000.0-4000.0500.08-30.0-5000.05
    -30000.0100000000.0400.030.0200.0-14000.0-8000000.0800.07
    8000.0400.0-1000.0 50000000.0-800000.0-300.0-40.0-7000.0800000.0-20.0 300.0100.05000000.040000.0
    -200.030.0-800000.020000.051-10.0-2-40000.0-300.0
    -30.0200.0-300.0540000.00.3-4-10.040.00.4
    -60000.0-1-40.010.3 3000.07000.00.8-10.00035000000.0400.0-40000.0
    -30000.0-7000.0100000000.0-200.0-60.070.03000.03000.0-8000000.0-30.0
    -40.0800000.0-200.020000.0-0.4430.040000.07-300.0
    70000.0-20.0-60.0-0.43000.01-6000.010.08-0.1
    300.0300.070.04140000.0-10.0-10.030.00.1
    20000000.04000.0300.03000.03000.030.0-6000.0 -10.0-47000.0 300000000.020000.03000.080.0
    -4000.0-8000000.0100.0-2-10.00.820000.0100000000.0-4000.02000.09000.0 -100.0
    500.0800.0-30.0 5000000.0-40000.040.0-1 3000.0-4000.040000.010.0-10.02000.0 50000000.0800000.0-3000.0-800000.0
    8-5000.0400.0-8000000.0 740000.0830.09000.0-3000.0100000000.0100.0
    5-40000.0-30.0 -300.00.40.000380.0-0.10.1 -100.0800000.0-800000.0100.0 20000.0
    - +@@ -360,131 +867,131 @@ M = extractMatrix('APA95ML_M.txt'); + - - - - + + + - - - + + + - + - - - - - - - - - - - - - - - - - - - + + + + + - - - - - - - - - - - - + - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - - - - - + + + + + + - - + - - - - - - - - - - - - - - - + - + + - - - + + - + + - - - + + + + + + + + + + + + + + + + - - - - - + + + + + @@ -499,6 +1006,10 @@ Then, we extract the coordinates of the interface nodes. +

    +The interface nodes are shown in Figure 9 and their coordinates are listed in Table 4. +

    +
    Table 2: First 10x10 elements of the Mass matrixTable 3: First 10x10 elements of the Mass matrix
    0.037e-08 2e-06-2e-071e-082e-080.0002-3e-09-0.0002-6e-08 -0.0012e-07-8e-08-9e-108e-076e-07-8e-09
    2e-067e-08 0.02-5e-077e-093e-082e-08-3e-070.0003-1e-081e-10
    -2e-07-5e-070.02-9e-054e-09-1e-082e-07-2e-08-1e-069e-05-3e-09-4e-09-1e-06 -0.0006-5e-06
    1e-087e-09-9e-051e-066e-114e-10-1e-093e-11-4e-08 5e-063e-08
    2e-083e-084e-096e-111e-062e-10-2e-092e-10-7e-09-4e-11
    0.00022e-08-1e-084e-102e-10 2e-06-2e-06-1e-09-7e-10-9e-12-1e-060.02-3e-08-4e-081e-081e-07-2e-070.00031e-09
    -3e-099e-05-3e-081e-06-3e-11-3e-13-7e-09-5e-06-3e-103e-08
    -0.0002-3e-09-4e-08-3e-112e-066e-102e-06-7e-09-2e-097e-11
    -6e-08-4e-091e-08-3e-136e-101e-061e-083e-09-2e-092e-13
    -0.001-3e-072e-07-1e-09-2e-09-2e-06-1e-061e-07-7e-092e-061e-08 0.034e-08 -2e-06-1e-07-5e-098e-09
    2e-070.0003-2e-083e-112e-10-1e-09-2e-060.02-8e-07-1e-08
    -8e-08-1e-088e-07 -0.00065e-06-2e-07-5e-06 -7e-09-7e-10-1e-07-8e-073e-094e-08 0.029e-05-9e-07-9e-05
    -9e-101e-10-5e-066e-07-4e-080.0003-3e-10-2e-09-2e-09-2e-06-9e-070.022e-08
    -8e-095e-061e-09 3e-08-4e-11-9e-12-5e-09-1e-089e-057e-112e-138e-09-9e-052e-08 1e-06
    @@ -510,28 +1021,28 @@ Then, we extract the coordinates of the interface nodes. - + - + - + - +
    Total number of Nodes1689597
    Number of interface Nodes137
    Number of Modes306
    Size of M and K matrices10848
    - - +
    Table 3: Coordinates of the interface nodes
    +@@ -556,23 +1067,23 @@ Then, we extract the coordinates of the interface nodes. - + - + - + - + - + @@ -580,175 +1091,54 @@ Then, we extract the coordinates of the interface nodes. - - + + - - + + - - + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + - - - - - - - -
    Table 4: Coordinates of the interface nodes
    1.0168947.040467.0 0.00.03 0.00.029997
    2.0168949.040469.0 0.0-0.03 0.0-0.029997
    3.0168950.040470.0 -0.035 0.0 0.0
    4.0168951.0-0.02840475.0-0.015 0.0 0.0
    5.0168952.0-0.02140476.0-0.005 0.0 0.0
    6.0168953.0-0.01440477.00.015 0.0 0.0
    7.0168954.0-0.0070.00.0
    8.0168955.00.00.00.0
    9.0168956.00.0070.00.0
    10.0168957.00.0140.00.0
    11.0168958.00.0210.00.0
    12.0168959.040478.0 0.035 0.0 0.0
    13.0168960.00.0280.00.0
    + +
    +

    APA95ML_nodes_1.png +

    +

    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.

    @@ -758,10 +1148,6 @@ To model the actuator, an 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);
    -
    +

    dynamics_act_disp_comp_mass.png

    -

    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);
    -
    +

    dynamics_force_force_sensor_comp_mass.png

    -

    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

    -
    +

    huddle_test_time_domain.png

    -

    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));
     
    -
    +

    huddle_test_pdf.png

    -

    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

    - +

    • [ ] List of equipment
    • @@ -1130,8 +1293,8 @@ win = hanning(ceil(1*Fs)); E505 with gain of 10.

    -
    -

    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));
     
    -
    +

    apa95ml_5kg_PI_pdf_comp_huddle.png

    -

    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));
     
    -
    +

    apa95ml_5kg_PI_coh.png

    -

    Figure 9: Coherence

    +

    Figure 15: Coherence

    @@ -1216,20 +1379,20 @@ Comparison with the FEM model

    -
    +

    apa95ml_5kg_pi_comp_fem.png

    -

    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

    - +

    Two measurements are performed: @@ -1291,7 +1454,7 @@ Gfem_a_ss = exp(-s/1

    -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));
    -
    +

    bode_plot_force_sensor_voltage_comp_fem.png

    -

    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 
     
     
    -
    +

    iff_plant_identification_apa95ml.png

    -

    Figure 12: Identification of the IFF plant

    +

    Figure 18: Identification of the IFF plant

    -
    -

    6.2 Integral Force Feedback

    +
    +

    6.2 Integral Force Feedback

    -
    +

    root_locus_iff_apa95ml_identification.png

    -

    Figure 13: Root Locus for IFF

    +

    Figure 19: Root Locus for IFF

    -
    -

    7 Integral Force Feedback

    +
    +

    7 Integral Force Feedback

    - +

    -
    +

    test_bench_apa_schematic_iff.png

    -

    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));
     
    -
    +

    iff_first_test_coherence.png

    -

    Figure 15: Coherence

    +

    Figure 21: Coherence

    -
    +

    iff_first_test_bode_plot.png

    -

    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];
     
    -
    +

    iff_results_bode_plots.png

    @@ -1462,13 +1625,13 @@ f_end = 500; % [Hz]
    -
    +

    iff_results_bode_plots_identification.png

    -
    +

    iff_results_root_locus.png

    @@ -1483,7 +1646,7 @@ f_end = 500; % [Hz]

    Author: Dehaeze Thomas

    -

    Created: 2020-11-24 mar. 09:17

    +

    Created: 2020-11-24 mar. 13:54

    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) + <> +#+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

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