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b/matlab/amplified_piezo_model.slx index d066673..ae9f985 100644 Binary files a/matlab/amplified_piezo_model.slx and b/matlab/amplified_piezo_model.slx differ diff --git a/org/amplified_piezoelectric_stack.org b/org/amplified_piezoelectric_stack.org index c34406c..08d4239 100644 --- a/org/amplified_piezoelectric_stack.org +++ 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)