From 5a157db8a10a47236a99ddbe06cf7cddfec3ceea Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Mon, 25 May 2020 10:25:13 +0200 Subject: [PATCH] Compare analytical model and simscape model --- docs/figs/comp_simscape_analytical.pdf | 2822 ++++++++++++++++++++++++ docs/figs/comp_simscape_analytical.png | Bin 0 -> 196928 bytes matlab/amplified_piezo_model.slx | Bin 28792 -> 28664 bytes org/amplified_piezoelectric_stack.org | 92 + 4 files changed, 2914 insertions(+) create mode 100644 docs/figs/comp_simscape_analytical.pdf create mode 100644 docs/figs/comp_simscape_analytical.png diff --git a/docs/figs/comp_simscape_analytical.pdf b/docs/figs/comp_simscape_analytical.pdf new file mode 100644 index 0000000..f5b61f1 --- /dev/null +++ b/docs/figs/comp_simscape_analytical.pdf @@ -0,0 +1,2822 @@ +%PDF-1.4 +% +1 0 obj +<< +/Producer (Apache FOP Version 2.4.0-SNAPSHOT: PDFDocumentGraphics2D) +/CreationDate (D:20200525102326+02'00') +>> +endobj +2 0 obj +<< +/Type /ExtGState +/CA 0.2509804 +>> 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b/org/amplified_piezoelectric_stack.org @@ -200,6 +200,98 @@ exportFig('figs/amplified_piezo_root_locus.pdf', 'width', 'wide', 'height', 'tal #+RESULTS: [[file:figs/amplified_piezo_root_locus.png]] +** Analytical Results +If we apply the Newton's second law of motion on the top mass, we obtain: +\[ ms^2 x_1 = F + k_1 (w - x_1) + k_e (x_e - x_1) \] + +Then, we can write that the measured force $F_s$ is equal to: +\[ F_s = k_a(w - x_e) + f = -k_e (x_1 - x_e) \] +which gives: +\[ x_e = \frac{k_a}{k_e + k_a} w + \frac{1}{k_e + k_a} f + \frac{k_e}{k_e + k_a} x_1 \] + +Re-injecting that into the previous equations gives: +\[ x_1 = F \frac{1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + f \frac{\frac{k_e}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \] +\[ F_s = - F \frac{\frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_e k_a}{k_e + k_a} \Big( \frac{ms^2}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) - f \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \] + +#+begin_src matlab + Ga = 1/(m*s^2 + k1 + ke*ka/(ke + ka)) * ... + [ 1 , k1 + ke*ka/(ke + ka) , ke/(ke + ka) ; + -ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke+ka)*(m*s^2 + k1)]; + Ga.InputName = {'F', 'w', 'f'}; + Ga.OutputName = {'x1', 'Fs'}; +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(1, 4, 1000); + + figure; + + ax1 = subplot(2, 3, 1); + title('$\displaystyle \frac{x_1}{w}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('x1', 'w'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Ga('x1', 'w'), freqs, 'Hz'))), 'k--'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]'); + + ax2 = subplot(2, 3, 2); + title('$\displaystyle \frac{x_1}{f}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('x1', 'f'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Ga('x1', 'f'), freqs, 'Hz'))), 'k--'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]'); + + ax3 = subplot(2, 3, 3); + title('$\displaystyle \frac{x_1}{F}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('x1', 'F'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Ga('x1', 'F'), freqs, 'Hz'))), 'k--'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]'); + + ax4 = subplot(2, 3, 4); + title('$\displaystyle \frac{F_s}{w}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('Fs', 'w'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'w'), freqs, 'Hz'))), 'k--'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]'); + + ax5 = subplot(2, 3, 5); + title('$\displaystyle \frac{F_s}{f}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('Fs', 'f'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'f'), freqs, 'Hz'))), 'k--'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]'); + + ax6 = subplot(2, 3, 6); + title('$\displaystyle \frac{F_s}{F}$') + hold on; + plot(freqs, abs(squeeze(freqresp(G('Fs', 'F'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'F'), freqs, 'Hz'))), 'k--'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + + linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_simscape_analytical.pdf', 'width', 'full', 'height', 'full'); +#+end_src + +#+name: fig:comp_simscape_analytical +#+caption: Comparison of the Identified Simscape Dynamics (solid) and the Analytical Model (dashed) +#+RESULTS: +[[file:figs/comp_simscape_analytical.png]] + * Rotating X-Y platform ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)