Reworked APA300ML section

This commit is contained in:
Thomas Dehaeze 2020-11-12 17:33:52 +01:00
parent d4579bf04d
commit 1a6304f135
114 changed files with 964 additions and 1339 deletions

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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
addpath('APA300ML/');
open('APA300ML.slx');
% Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
% We first extract the stiffness and mass matrices.
K = readmatrix('mat_K.CSV');
M = readmatrix('mat_M.CSV');
% #+caption: First 10x10 elements of the Mass matrix
% #+RESULTS:
% | 0.01 | -2e-06 | 1e-06 | 6e-09 | 5e-05 | -5e-09 | -0.0005 | -7e-07 | 6e-07 | -3e-09 |
% | -2e-06 | 0.01 | 8e-07 | -2e-05 | -8e-09 | 2e-09 | -9e-07 | -0.0002 | 1e-08 | -9e-07 |
% | 1e-06 | 8e-07 | 0.009 | 5e-10 | 1e-09 | -1e-09 | -5e-07 | 3e-08 | 6e-05 | 1e-10 |
% | 6e-09 | -2e-05 | 5e-10 | 3e-07 | 2e-11 | -3e-12 | 3e-09 | 9e-07 | -4e-10 | 3e-09 |
% | 5e-05 | -8e-09 | 1e-09 | 2e-11 | 6e-07 | -4e-11 | -1e-06 | -2e-09 | 1e-09 | -8e-12 |
% | -5e-09 | 2e-09 | -1e-09 | -3e-12 | -4e-11 | 1e-07 | -2e-09 | -1e-09 | -4e-10 | -5e-12 |
% | -0.0005 | -9e-07 | -5e-07 | 3e-09 | -1e-06 | -2e-09 | 0.01 | 1e-07 | -3e-07 | -2e-08 |
% | -7e-07 | -0.0002 | 3e-08 | 9e-07 | -2e-09 | -1e-09 | 1e-07 | 0.01 | -4e-07 | 2e-05 |
% | 6e-07 | 1e-08 | 6e-05 | -4e-10 | 1e-09 | -4e-10 | -3e-07 | -4e-07 | 0.009 | -2e-10 |
% | -3e-09 | -9e-07 | 1e-10 | 3e-09 | -8e-12 | -5e-12 | -2e-08 | 2e-05 | -2e-10 | 3e-07 |
% Then, we extract the coordinates of the interface nodes.
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
% 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:
% #+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:
d33*n*ka % [N/V]
% #+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:
1e-6*d33/(eT*sD*n) % [V/um]
% Stiffness
m = 0.001;
% The transfer function from vertical external force to the relative vertical displacement is identified.
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
% The inverse of its DC gain is the axial stiffness of the APA:
1e-6/dcgain(G) % [N/um]
% Resonance Frequency
% The resonance frequency is specified to be between 650Hz and 840Hz.
% This is also the case for the FEM model (Figure [[fig:apa300ml_resonance]]).
freqs = logspace(2, 4, 5000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude');
hold off;
% Amplification factor
% The amplification factor is the ratio of the vertical displacement to the stack displacement.
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
% The ratio of the two displacement is computed from the FEM model.
abs(dcgain(G(1,1))./dcgain(G(2,1)))
% #+RESULTS:
% : 5.0749
% This is actually correct and approximately corresponds to the ratio of the piezo height and length:
75/15
% Stroke
% Estimation of the actuator stroke:
% \[ \Delta H = A n \Delta L \]
% with:
% - $\Delta H$ Axial Stroke of the APA
% - $A$ Amplification factor (5 for the APA300ML)
% - $n$ Number of stack used
% - $\Delta L$ Stroke of the stack (0.1% of its length)
1e6 * 5 * 3 * 20e-3 * 0.1e-2
% Identification of the Dynamics from actuator to replace displacement
% We first set the mass to be approximately zero.
m = 0.01;
% The dynamics is identified from the applied force to the measured relative displacement.
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
Gh = -linearize(mdl, io);
% The same dynamics is identified for a payload mass of 10Kg.
m = 10;
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
Ghm = -linearize(mdl, io);
freqs = logspace(0, 4, 5000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gh, freqs, 'Hz'))), '-');
plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gh, freqs, 'Hz')))), '-', ...
'DisplayName', '$m = 0kg$');
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Ghm, freqs, 'Hz')))), '-', ...
'DisplayName', '$m = 10kg$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360);
ylim([-360 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
legend('location', 'southwest');
% #+name: fig:apa300ml_plant_dynamics
% #+caption: Transfer function from forces applied by the stack to the axial displacement of the APA
% #+RESULTS:
% [[file:figs/apa300ml_plant_dynamics.png]]
% The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure [[fig:apa300ml_dvf_root_locus]].
figure;
gains = logspace(0, 5, 500);
hold on;
plot(real(pole(Ghm)), imag(pole(G)), 'kx');
plot(real(tzero(Ghm)), imag(tzero(G)), 'ko');
for k = 1:length(gains)
cl_poles = pole(feedback(Ghm, gains(k)*s));
plot(real(cl_poles), imag(cl_poles), 'k.');
end
hold off;
axis square;
xlim([-500, 10]); ylim([0, 510]);
xlabel('Real Part'); ylabel('Imaginary Part');
% Identification of the Dynamics from actuator to force sensor
% Let's use 2 stacks as a force sensor and 1 stack as force actuator.
% The transfer function from actuator voltage to sensor voltage is identified and shown in Figure [[fig:apa300ml_iff_plant]].
m = 10;
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1;
Giff = -linearize(mdl, io);
freqs = logspace(0, 4, 5000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Giff, freqs, 'Hz'))), '-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Giff, freqs, 'Hz')))), '-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360);
ylim([-180 180]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:apa300ml_iff_plant
% #+caption: Transfer function from actuator to force sensor
% #+RESULTS:
% [[file:figs/apa300ml_iff_plant.png]]
% For root locus corresponding to IFF is shown in Figure [[fig:apa300ml_iff_root_locus]].
figure;
gains = logspace(0, 5, 500);
hold on;
plot(real(pole(Giff)), imag(pole(Giff)), 'kx');
plot(real(tzero(Giff)), imag(tzero(Giff)), 'ko');
for k = 1:length(gains)
cl_poles = pole(feedback(Giff, gains(k)/s));
plot(real(cl_poles), imag(cl_poles), 'k.');
end
hold off;
axis square;
xlim([-500, 10]); ylim([0, 510]);
xlabel('Real Part'); ylabel('Imaginary Part');
% Identification for a simpler model
% The goal in this section is to identify the parameters of a simple APA model from the FEM.
% This can be useful is a lower order model is to be used for simulations.
% The presented model is based on cite:souleille18_concep_activ_mount_space_applic.
% The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]).
% The parameters are shown in the table below.
% #+name: fig:souleille18_model_piezo
% #+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
% [[file:./figs/souleille18_model_piezo.png]]
% #+caption:Parameters used for the model of the APA 100M
% | | Meaning |
% |-------+----------------------------------------------------------------|
% | $k_e$ | Stiffness used to adjust the pole of the isolator |
% | $k_1$ | Stiffness of the metallic suspension when the stack is removed |
% | $k_a$ | Stiffness of the actuator |
% | $c_1$ | Added viscous damping |
% The goal is to determine $k_e$, $k_a$ and $k_1$ so that the simplified model fits the FEM model.
% \[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
% \[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
% If we can fix $k_a$, we can determine $k_e$ and $k_1$ with:
% \[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \]
% \[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]
m = 10;
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m]
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N]
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1; % Stack Displacement [m]
G = linearize(mdl, io);
G.InputName = {'Fd', 'w', 'Fa'};
G.OutputName = {'y', 'Fs', 'd'};
% From the identified dynamics, compute $\alpha$ and $\beta$
alpha = abs(dcgain(G('y', 'Fa')));
beta = abs(dcgain(G('y', 'Fd')));
% $k_a$ is estimated using the following formula:
ka = 0.8/abs(dcgain(G('y', 'Fa')));
% The factor can be adjusted to better match the curves.
% Then $k_e$ and $k_1$ are computed.
ke = ka/(beta/alpha - 1);
k1 = 1/beta - ke*ka/(ke + ka);
% #+RESULTS:
% | | Value [N/um] |
% |----+--------------|
% | ka | 40.5 |
% | ke | 1.5 |
% | k1 | 0.4 |
% The damping in the system is adjusted to match the FEM model if necessary.
c1 = 1e2;
% The analytical model of the simpler system is defined below:
Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ...
[ 1 , k1 + c1*s + ke*ka/(ke + ka) , ke/(ke + ka) ;
-ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke + ka)*(m*s^2 + c1*s + k1)];
Ga.InputName = {'Fd', 'w', 'Fa'};
Ga.OutputName = {'y', 'Fs'};
% And the DC gain is adjusted for the force sensor:
F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd'));
% The dynamics of the FEM model and the simpler model are compared in Figure [[fig:apa300ml_comp_simpler_model]].
freqs = logspace(0, 5, 1000);
figure;
tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('y', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/w$ [m/m]');
ylim([1e-6, 1e2]);
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/f$ [m/N]');
ylim([1e-14, 1e-6]);
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/F$ [m/N]');
ylim([1e-14, 1e-4]);
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/w$ [m/m]');
ylim([1e2, 1e8]);
ax5 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/f$ [m/N]');
ylim([1e-4, 1e1]);
ax6 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/F$ [m/N]');
ylim([1e-7, 1e2]);
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
% #+name: fig:apa300ml_comp_simpler_model
% #+caption: Comparison of the Dynamics between the FEM model and the simplified one
% #+RESULTS:
% [[file:figs/apa300ml_comp_simpler_model.png]]
% The simplified model has also been implemented in Simscape.
% The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure [[fig:apa300ml_comp_simpler_simscape]].
%% Name of the Simulink File
mdl = 'APA300ML_simplified';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m]
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]
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
Gs = linearize(mdl, io);
Gs.InputName = {'Fd', 'w', 'Fa'};
Gs.OutputName = {'y', 'Fs'};
freqs = logspace(0, 5, 1000);
figure;
tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gs('y', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/w$ [m/m]');
ylim([1e-6, 1e2]);
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/f$ [m/N]');
ylim([1e-14, 1e-6]);
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/F$ [m/N]');
ylim([1e-14, 1e-4]);
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/w$ [m/m]');
ylim([1e2, 1e8]);
ax5 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/f$ [m/N]');
ylim([1e-4, 1e1]);
ax6 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/F$ [m/N]');
ylim([1e-7, 1e2]);
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');

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