Add simplier model identification and comparison
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figs/apa300ml_comp_simpler_model.pdf
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figs/apa300ml_comp_simpler_model.pdf
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figs/apa300ml_comp_simpler_model.png
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figs/apa300ml_comp_simpler_model.png
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figs/souleille18_model_piezo.png
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index.org
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index.org
@ -778,7 +778,7 @@ Then, we extract the coordinates of the interface nodes.
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Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.
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** Piezoelectric parameters
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Parameters for the APA95ML:
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Parameters for the APA300ML:
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#+begin_src matlab
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d33 = 3e-10; % Strain constant [m/V]
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@ -790,8 +790,8 @@ Parameters for the APA95ML:
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#+end_src
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#+begin_src matlab
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na = 3; % Number of stacks used as actuator
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ns = 0; % Number of stacks used as force sensor
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na = 2; % Number of stacks used as actuator
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ns = 1; % Number of stacks used as force sensor
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#+end_src
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The ratio of the developed force to applied voltage is $d_{33} n k_a$ in [N/V].
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@ -802,7 +802,7 @@ We denote this constant by $g_a$ and:
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#+end_src
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#+RESULTS:
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: 1.88
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: 0.42941
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From cite:fleming14_desig_model_contr_nanop_system (page 123), the relation between relative displacement and generated voltage is:
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\[ V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h \]
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@ -1357,6 +1357,181 @@ Root locus
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xlabel('Real Part'); ylabel('Imaginary Part');
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#+end_src
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** Identification for a simpler model
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The goal in this section is to identify the parameters of a simple APA model from the FEM.
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This can be useful is a lower order model is to be used for simulations.
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The presented model is based on cite:souleille18_concep_activ_mount_space_applic.
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The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]).
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The parameters are shown in the table below.
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#+name: fig:souleille18_model_piezo
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#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
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[[file:./figs/souleille18_model_piezo.png]]
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#+caption: Parameters used for the model of the APA 100M
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| | Meaning |
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|-------+----------------------------------------------------------------|
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| $k_e$ | Stiffness used to adjust the pole of the isolator |
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| $k_1$ | Stiffness of the metallic suspension when the stack is removed |
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| $k_a$ | Stiffness of the actuator |
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| $c_1$ | Added viscous damping |
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The goal is to determine $k_e$, $k_a$ and $k_1$ so that the simplified model fits the FEM model.
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Three unknowns and three equations:
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\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
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\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
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\[ \gamma = \frac{dL}{f}(\omega=0) = \frac{1}{k_a + \frac{k_e k_1}{k_e + k_1}} \]
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#+begin_src matlab :exports none
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m = 10;
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#+end_src
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'APA300ML_test_bench';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
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io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m]
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io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N]
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io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
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io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
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io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1; % Stack Displacement [m]
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G = linearize(mdl, io);
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G.InputName = {'Fd', 'w', 'Fa'};
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G.OutputName = {'y', 'Fs', 'd'};
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#+end_src
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#+begin_src matlab
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alpha = abs(dcgain(G('y', 'Fa')));
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beta = abs(dcgain(G('y', 'Fd')));
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gamma = abs(dcgain(G('d', 'Fa')));
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#+end_src
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#+begin_src matlab
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na = 1;
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ns = 2;
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#+end_src
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Amplification Factor
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#+begin_src matlab
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A = abs(dcgain(G('y', 'Fa'))/dcgain(G('d', 'Fa')));
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#+end_src
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#+begin_src matlab
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ka = 0.8*(1/A)/gamma;
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ke = ka/(beta/alpha - 1);
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k1 = 1/beta - ke*ka/(ke + ka);
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable(1e-6*[ka; ke; k1], {'ka', 'ke', 'k1'}, {'Value [N/um]'}, ' %.1f ');
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#+end_src
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#+RESULTS:
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| | Value [N/um] |
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|----+--------------|
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| ka | 42.9 |
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| ke | 1.5 |
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| k1 | 0.4 |
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Adjust the damping in the system.
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#+begin_src matlab
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c1 = 1e2;
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#+end_src
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Analytical model of the simpler system:
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#+begin_src matlab
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Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ...
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[ 1 , k1 + c1*s + ke*ka/(ke + ka) , ke/(ke + ka) ;
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-ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke + ka)*(m*s^2 + c1*s + k1)];
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Ga.InputName = {'Fd', 'w', 'Fa'};
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Ga.OutputName = {'y', 'Fs'};
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#+end_src
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Adjust the DC gain for the force sensor;
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#+begin_src matlab
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lambda = dcgain(Ga('Fs', 'Fd'))/dcgain(G('Fs', 'Fd'));
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#+end_src
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#+begin_src matlab :exports none
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freqs = logspace(0, 5, 1000);
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figure;
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ay = subplot(2, 3, 1);
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title('$\displaystyle \frac{x_1}{w}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('y', 'w'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
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ax2 = subplot(2, 3, 2);
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title('$\displaystyle \frac{x_1}{f}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fa'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
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ax3 = subplot(2, 3, 3);
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title('$\displaystyle \frac{x_1}{F}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fd'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
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ax4 = subplot(2, 3, 4);
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title('$\displaystyle \frac{F_s}{w}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(lambda*G( 'Fs', 'w'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'w'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
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ax5 = subplot(2, 3, 5);
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title('$\displaystyle \frac{F_s}{f}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(lambda*G( 'Fs', 'Fa'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'Fa'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
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ax6 = subplot(2, 3, 6);
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title('$\displaystyle \frac{F_s}{F}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(lambda*G( 'Fs', 'Fd'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'Fd'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
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linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/apa300ml_comp_simpler_model.pdf', 'width', 'full', 'height', 'full');
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#+end_src
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#+name: fig:apa300ml_comp_simpler_model
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#+caption: Comparison of the Dynamics between the FEM model and the simplified one
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#+RESULTS:
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[[file:figs/apa300ml_comp_simpler_model.png]]
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* Flexible Joint
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** Introduction :ignore:
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