Christophe's review
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@ -21,47 +21,46 @@ colors = colororder;
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% As all key stages of the micro-station are expected to behave as solid bodies, only 6 acrshort:dof can be considered for each solid body.
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% However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrshort:dof) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section ref:ssec:modal_solid_body_assumption).
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% #+attr_latex: :options [t]{0.60\linewidth}
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% #+attr_latex: :options [b]{0.63\linewidth}
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% #+begin_minipage
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% #+name: fig:modal_location_accelerometers
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% #+caption: Position of the accelerometers
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% #+attr_latex: :width 0.99\linewidth :float nil
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% #+attr_latex: :width 0.95\linewidth :float nil
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% [[file:figs/modal_location_accelerometers.png]]
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% #+end_minipage
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% \hfill
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% #+attr_latex: :options [b]{0.38\linewidth}
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% #+attr_latex: :options [b]{0.36\linewidth}
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% #+begin_minipage
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% #+begin_scriptsize
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% #+name: tab:modal_position_accelerometers
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% #+caption: Positions in mm
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% #+latex: \centering
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% #+attr_latex: :environment tabularx :width \linewidth :placement [b] :align Xccc
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% #+attr_latex: :booktabs t :float nil :center nil
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% #+RESULTS:
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% | | $x$ | $y$ | $z$ |
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% |--------------+------+------+------|
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% | Low. Granite | -730 | -526 | -951 |
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% | Low. Granite | -735 | 814 | -951 |
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% | Low. Granite | 875 | 799 | -951 |
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% | Low. Granite | 865 | -506 | -951 |
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% | Up. Granite | -320 | -446 | -786 |
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% | Up. Granite | -480 | 534 | -786 |
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% | Up. Granite | 450 | 534 | -786 |
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% | Up. Granite | 295 | -481 | -786 |
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% | Translation | -475 | -414 | -427 |
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% | Translation | -465 | 407 | -427 |
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% | Translation | 475 | 424 | -427 |
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% | Translation | 475 | -419 | -427 |
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% | Tilt | -385 | -300 | -417 |
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% | Tilt | -420 | 280 | -417 |
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% | Tilt | 420 | 280 | -417 |
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% | Tilt | 380 | -300 | -417 |
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% | Spindle | -155 | -90 | -594 |
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% | Spindle | 0 | 180 | -594 |
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% | Spindle | 155 | -90 | -594 |
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% | Hexapod | -64 | -64 | -270 |
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% | Hexapod | -64 | 64 | -270 |
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% | Hexapod | 64 | 64 | -270 |
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% | Hexapod | 64 | -64 | -270 |
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% |-------------------+------+------+------|
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% | (17) Low. Granite | -730 | -526 | -951 |
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% | (18) Low. Granite | -735 | 814 | -951 |
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% | (19) Low. Granite | 875 | 799 | -951 |
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% | (20) Low. Granite | 865 | -506 | -951 |
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% | (13) Up. Granite | -320 | -446 | -786 |
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% | (14) Up. Granite | -480 | 534 | -786 |
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% | (15) Up. Granite | 450 | 534 | -786 |
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% | (16) Up. Granite | 295 | -481 | -786 |
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% | (9) Translation | -475 | -414 | -427 |
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% | (10) Translation | -465 | 407 | -427 |
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% | (11) Translation | 475 | 424 | -427 |
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% | (12) Translation | 475 | -419 | -427 |
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% | (5) Tilt | -385 | -300 | -417 |
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% | (6) Tilt | -420 | 280 | -417 |
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% | (7) Tilt | 420 | 280 | -417 |
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% | (8) Tilt | 380 | -300 | -417 |
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% | (21) Spindle | -155 | -90 | -594 |
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% | (22) Spindle | 0 | 180 | -594 |
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% | (23) Spindle | 155 | -90 | -594 |
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% | (1) Hexapod | -64 | -64 | -270 |
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% | (2) Hexapod | -64 | 64 | -270 |
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% | (3) Hexapod | 64 | 64 | -270 |
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% | (4) Hexapod | 64 | -64 | -270 |
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% #+latex: \captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm}
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% #+end_scriptsize
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% #+end_minipage
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@ -99,6 +98,7 @@ acc_pos = acc_pos(i, 2:4);
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% The "normalized" acrfull:asd of the two signals were computed and shown in Figure ref:fig:modal_asd_acc_force.
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% Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
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% These data are corresponding to a hammer impact in the vertical direction and to the measured acceleration in the $x$ direction by accelerometer $1$ (fixed to the micro-hexapod).
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% Similar results were obtained for all measured frequency response functions.
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@ -117,12 +117,12 @@ time = linspace(0, meas1_raw.Track1_X_Resolution*length(meas1_raw.Track1), lengt
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%% Raw measurement of the Accelerometer
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figure;
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hold on;
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plot(time-22.2, meas1_raw.Track2, 'DisplayName', '$X_{j}$ [$m/s^2$]');
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plot(time-22.2, 1e-3*meas1_raw.Track1, 'DisplayName', '$F_{k}$ [kN]');
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plot(time-22.2, meas1_raw.Track2, 'DisplayName', '$X_{1,x}$ [$m/s^2$]');
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plot(time-22.2, 1e-3*meas1_raw.Track1, 'DisplayName', '$F_{z}$ [kN]');
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hold off;
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xlabel('Time [s]');
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ylabel('Amplitude');
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xlim([0, 0.2]);
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xlim([0, 0.2])
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ylim([-2, 2]);
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legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
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@ -138,8 +138,8 @@ Noverlap = floor(Nfft/2); % Overlap for frequency analysis
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%% Normalized Amplitude Spectral Density of the measured force and acceleration
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figure;
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hold on;
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plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', '$X_{j}$');
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plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$F_{k}$');
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plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', '$\Gamma_{X_{1,x}}$');
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plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$\Gamma_{F_{z}}$');
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hold off;
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set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Normalized Spectral Density');
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@ -151,7 +151,7 @@ legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
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% #+name: fig:modal_raw_meas_asd
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% #+caption: Raw measurement of the accelerometer (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
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% #+caption: Raw measurement of the accelerometer 1 in the $x$ direction (blue) and of the force sensor at the Hammer tip (red) for an impact in the $z$ direction (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
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% #+attr_latex: :options [htbp]
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% #+begin_figure
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% #+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals}
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@ -168,7 +168,7 @@ legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
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% #+end_subfigure
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% #+end_figure
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% The frequency response function $H_{jk}$ from the applied force $F_{k}$ to the measured acceleration $X_j$ is then computed and shown Figure ref:fig:modal_frf_acc_force.
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% The frequency response function from the applied force to the measured acceleration is then computed and shown Figure ref:fig:modal_frf_acc_force.
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% The quality of the obtained data can be estimated using the /coherence/ function (Figure ref:fig:modal_coh_acc_force).
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% Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest.
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@ -19,7 +19,7 @@ colors = colororder;
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% The motion of the rigid body of figure ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to the reference frame $\{O\}$.
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% The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation eqref:eq:modal_rotation_matrix.
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% The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation eqref:eq:modal_rotation_matrix [[cite:&ewins00_modal chapt. 4.3.2]].
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% \begin{equation}\label{eq:modal_compute_point_response}
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% \vec{\delta} p_{i} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{i} \\
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@ -52,30 +52,12 @@ colors = colororder;
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% \end{array}\right]
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% \end{equation}
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% Provided that the four sensors are properly located, the system of equation eqref:eq:modal_cart_to_acc can be solved by matrix inversion[fn:5].
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% The motion of the solid body expressed in a chosen frame $\{O\}$ can be determined using equation eqref:eq:modal_determine_global_disp.
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% Provided that the four sensors are properly located, the system of equation eqref:eq:modal_cart_to_acc can be solved by matrix inversion[fn:modal_5].
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% The motion of the solid body expressed in a chosen frame $\{O\}$ can be determined by inverting equation eqref:eq:modal_cart_to_acc.
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% Note that this matrix inversion is equivalent to resolving a mean square problem.
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% Therefore, having more accelerometers permits better approximation of the motion of a solid body.
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% \begin{equation}
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% \left[\begin{array}{c}
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% \delta p_x \\ \delta p_y \\ \delta p_z \\ \hline \delta\Omega_x \\ \delta\Omega_y \\ \delta\Omega_z
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% \end{array}\right] =
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% \left[\begin{array}{ccc|ccc}
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% 1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
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% 0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
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% 0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
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% & \vdots & & & \vdots & \\ \hline
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% 1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
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% 0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
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% 0 & 0 & 1 & p_{4y} & -p_{4x} & 0
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% \end{array}\right]^{-1} \left[\begin{array}{c}
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% \delta p_{1x} \\ \delta p_{1y} \\ \delta p_{1z} \\\hline \vdots \\\hline \delta p_{4x} \\ \delta p_{4y} \\ \delta p_{4z}
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% \end{array}\right] \label{eq:modal_determine_global_disp}
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% \end{equation}
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% From the CAD model, the position of the center of mass of each considered solid body is computed (see Table ref:tab:modal_com_solid_bodies).
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% From the CAD model, the position of the center of mass of each solid body is computed (see Table ref:tab:modal_com_solid_bodies).
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% The position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be determined.
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@ -110,19 +92,19 @@ model_com = reshape(table2array(readtable('mat/model_solidworks_com.txt', 'ReadV
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% #+name: tab:modal_com_solid_bodies
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% #+caption: Center of mass of considered solid bodies with respect to the "point of interest"
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% #+attr_latex: :environment tabularx :width 0.6\linewidth :align lXXX
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% #+attr_latex: :environment tabularx :width 0.55\linewidth :align Xccc
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% #+attr_latex: :center t :booktabs t
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% #+RESULTS:
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% | | $X$ [mm] | $Y$ [mm] | $Z$ [mm] |
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% |-------------------+----------+----------+----------|
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% | Bottom Granite | 45 | 144 | -1251 |
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% | Top granite | 52 | 258 | -778 |
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% | Translation stage | 0 | 14 | -600 |
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% | Tilt Stage | 0 | -5 | -628 |
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% | Spindle | 0 | 0 | -580 |
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% | Hexapod | -4 | 6 | -319 |
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% | | $X$ | $Y$ | $Z$ |
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% |-------------------+-----------------+------------------+--------------------|
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% | Bottom Granite | $45\,\text{mm}$ | $144\,\text{mm}$ | $-1251\,\text{mm}$ |
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% | Top granite | $52\,\text{mm}$ | $258\,\text{mm}$ | $-778\,\text{mm}$ |
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% | Translation stage | $0$ | $14\,\text{mm}$ | $-600\,\text{mm}$ |
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% | Tilt Stage | $0$ | $-5\,\text{mm}$ | $-628\,\text{mm}$ |
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% | Spindle | $0$ | $0$ | $-580\,\text{mm}$ |
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% | Hexapod | $-4\,\text{mm}$ | $6\,\text{mm}$ | $-319\,\text{mm}$ |
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% Using eqref:eq:modal_determine_global_disp, the frequency response matrix $\mathbf{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\mathbf{H}$.
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% Using eqref:eq:modal_cart_to_acc, the frequency response matrix $\mathbf{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\mathbf{H}$.
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% \begin{equation}\label{eq:modal_frf_matrix_com}
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% \mathbf{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
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@ -205,7 +187,7 @@ end
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% This also validates the reduction in the number of degrees of freedom from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof).
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%% Comparaison of the original accelerometer response and reconstructed response from the solid body response
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%% Comparison of the original accelerometer response and reconstructed response from the solid body response
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exc_names = {'$F_x$', '$F_y$', '$F_z$'};
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DOFs = {'x', 'y', 'z', '\theta_x', '\theta_y', '\theta_z'};
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@ -223,11 +205,10 @@ for i = 1:length(accs_i)
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hold on;
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for dir_i = 1:3
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plot(freqs, abs(squeeze(frf(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'DisplayName', sprintf('$a_{%i,%s}$', acc_i, DOFs{dir_i}));
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plot(freqs, abs(squeeze(frf(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'color', [colors(dir_i,:), 0.5], 'linewidth', 2.5, 'DisplayName', sprintf('$a_{%i,%s}$ - meas', acc_i, DOFs{dir_i}));
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end
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set(gca,'ColorOrderIndex',1)
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for dir_i = 1:3
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plot(freqs, abs(squeeze(frfs_A(3*(acc_i-1)+dir_i, exc_dir, :))), '--', 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(frfs_A(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'color', colors(dir_i, :), 'DisplayName', sprintf('$a_{%i,%s}$ - solid body', acc_i, DOFs{dir_i}));
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end
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hold off;
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@ -246,5 +227,6 @@ for i = 1:length(accs_i)
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set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
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xlim([0, 200]); ylim([1e-6, 3e-2]);
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xticks([0:20:200]);
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legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
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leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
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leg.ItemTokenSize(1) = 15;
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end
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@ -12,7 +12,7 @@ colors = colororder;
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% Number of modes determination
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% <<ssec:modal_number_of_modes>>
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% The acrshort:mif is here applied to the $n\times p$ acrshort:frf matrix where $n$ is a relatively large number of measurement DOFs (here $n=69$) and $p$ is the number of excitation DOFs (here $p=3$).
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% The acrshort:mif is applied to the $n\times p$ acrshort:frf matrix where $n$ is a relatively large number of measurement DOFs (here $n=69$) and $p$ is the number of excitation DOFs (here $p=3$).
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% The complex modal indication function is defined in equation eqref:eq:modal_cmif where the diagonal matrix $\Sigma$ is obtained from a acrlong:svd of the acrshort:frf matrix as shown in equation eqref:eq:modal_svd.
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% \begin{equation} \label{eq:modal_cmif}
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@ -46,7 +46,7 @@ end
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figure;
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hold on;
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for i = 1:size(MIF, 1)
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plot(freqs, squeeze(MIF(i, i, :)));
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plot(freqs, squeeze(MIF(i, i, :)), 'DisplayName', sprintf('MIF${}_%i$', i));
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end
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hold off;
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set(gca, 'Xscale', 'lin'); set(gca, 'Yscale', 'log');
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@ -54,6 +54,7 @@ xlabel('Frequency [Hz]'); ylabel('CMIF Amplitude');
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xticks([0:20:200]);
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xlim([0, 200]);
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ylim([1e-6, 2e-2]);
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ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
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% Verification of the modal model validity
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% <<ssec:modal_model_validity>>
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@ -70,7 +71,7 @@ ylim([1e-6, 2e-2]);
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% \end{bmatrix}_{n \times 2m}
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% \end{equation}
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% The full acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ can be synthesize using eqref:eq:modal_synthesized_frf.
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% The full acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ can be obtained using eqref:eq:modal_synthesized_frf.
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% \begin{equation}\label{eq:modal_synthesized_frf}
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% [\mathbf{H}_{\text{syn}}(\omega)]_{n\times n} = [\Phi]_{n\times2m} [\mathbf{H}_{\text{mod}}(\omega)]_{2m\times2m} [\Phi]_{2m\times n}^T
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@ -138,14 +139,14 @@ for i = 1:length(freqs)
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Hsyn(:, :, i) = eigen_vec_ext_M*diag(1./(diag(modal_a_ext_M).*(j*2*pi*freqs(i) - diag(eigen_val_ext_M))))*eigen_vec_ext_M.';
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end
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%% Derivate two times to to have the acceleration response
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%% Derivate two times to have the acceleration response
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for i = 1:size(Hsyn, 1)
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Hsyn(i, :, :) = squeeze(Hsyn(i, :, :)).*(j*2*pi*freqs).^2;
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end
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% The comparison between the original measured frequency response functions and those synthesized ones from the modal model is presented in Figure ref:fig:modal_comp_acc_frf_modal.
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% A comparison between original measured frequency response functions and synthesized ones from the modal model is presented in Figure ref:fig:modal_comp_acc_frf_modal.
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% Whether the obtained match is good or bad is quite arbitrary.
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% However, the modal model seems to be able to represent the coupling between different nodes and different directions, which is quite important from a control perspective.
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% This can be seen in Figure ref:fig:modal_comp_acc_frf_modal_3 that shows the frequency response function from the force applied on node 11 (i.e. on the translation stage) in the $y$ direction to the measured acceleration at node $2$ (i.e. at the top of the micro-hexapod) in the $x$ direction.
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@ -142,7 +142,7 @@ Experimental modal analysis will be used to tune the model, and to verify that a
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The tuning approach for the multi-body model based on measurements is illustrated in Figure ref:fig:modal_vibration_analysis_procedure.
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First, a /response model/ is obtained, which corresponds to a set of frequency response functions computed from experimental measurements.
|
||||
From this response model, the modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another describing the mode shapes.
|
||||
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considering solid bodies and the springs and dampers connecting the solid bodies.
|
||||
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considered solid bodies and the springs and dampers connecting the solid bodies.
|
||||
|
||||
#+begin_src latex :file modal_vibration_analysis_procedure.pdf
|
||||
\begin{tikzpicture}
|
||||
@ -221,10 +221,10 @@ The obtained force and acceleration signals are described in Section ref:ssec:mo
|
||||
<<m-init-other>>
|
||||
#+end_src
|
||||
|
||||
** Used Instrumentation
|
||||
** Instrumentation
|
||||
<<ssec:modal_instrumentation>>
|
||||
|
||||
Three type of equipment are essential for a good modal analysis.
|
||||
Three types of equipment are essential for a good modal analysis.
|
||||
First, /accelerometers/ are used to measure the response of the structure.
|
||||
Here, 3-axis accelerometers[fn:modal_1] shown in figure ref:fig:modal_accelero_M393B05 are used.
|
||||
These accelerometers were glued to the micro-station using a thin layer of wax for best results [[cite:&ewins00_modal chapt. 3.5.7]].
|
||||
@ -262,7 +262,7 @@ Finally, an /acquisition system/[fn:modal_3] (figure ref:fig:modal_oros) is used
|
||||
** Structure Preparation and Test Planing
|
||||
<<ssec:modal_test_preparation>>
|
||||
|
||||
To obtain meaningful results, the modal analysis of the micro-station in performed /in-situ/.
|
||||
To obtain meaningful results, the modal analysis of the micro-station is performed /in-situ/.
|
||||
To do so, all the micro-station stage controllers are turned "ON".
|
||||
This is especially important for stages for which the stiffness is provided by local feedback control, such as the air bearing spindle, and the translation stage.
|
||||
If these local feedback controls were turned OFF, this would have resulted in very low-frequency modes that were difficult to measure in practice, and it would also have led to decoupled dynamics, which would not be the case in practice.
|
||||
@ -274,7 +274,7 @@ The $H_{jk}$ element of this acrfull:frf matrix corresponds to the frequency res
|
||||
Measuring this acrshort:frf matrix is time consuming as it requires to make $n \times n$ measurements.
|
||||
However, due to the principle of reciprocity ($H_{jk} = H_{kj}$) and using the /point measurement/ ($H_{jj}$), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix $\mathbf{H}$ [[cite:&ewins00_modal chapt. 5.2]].
|
||||
Therefore, a minimum set of $n$ frequency response functions is required.
|
||||
This can be done either by measuring the response $X_{j}$ at a fixed acrshort:dof $j$ while applying forces $F_{i}$ for at all $n$ considered acrshort:dof, or by applying a force $F_{k}$ at a fixed acrshort:dof $k$ and measuring the response $X_{i}$ for all $n$ acrshort:dof.
|
||||
This can be done either by measuring the response $X_{j}$ at a fixed acrshort:dof $j$ while applying forces $F_{i}$ at all $n$ considered acrshort:dof, or by applying a force $F_{k}$ at a fixed acrshort:dof $k$ and measuring the response $X_{i}$ for all $n$ acrshort:dof.
|
||||
|
||||
It is however not advised to measure only one row or one column, as one or more modes may be missed by an unfortunate choice of force or acceleration measurement location (for instance if the force is applied at a vibration node of a particular mode).
|
||||
In this modal analysis, it is chosen to measure the response of the structure at all considered acrshort:dof, and to excite the structure at one location in three directions in order to have some redundancy, and to ensure that all modes are properly identified.
|
||||
@ -404,7 +404,7 @@ For the accelerometer, a much more complex signal can be observed, indicating co
|
||||
|
||||
The "normalized" acrfull:asd of the two signals were computed and shown in Figure ref:fig:modal_asd_acc_force.
|
||||
Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
|
||||
These data are corresponding to an hammer impact in the vertical direction and to the measured acceleration in the $x$ direction by accelerometer $1$ (fixed to the micro-hexapod).
|
||||
These data are corresponding to a hammer impact in the vertical direction and to the measured acceleration in the $x$ direction by accelerometer $1$ (fixed to the micro-hexapod).
|
||||
Similar results were obtained for all measured frequency response functions.
|
||||
|
||||
#+begin_src matlab
|
||||
@ -568,9 +568,9 @@ For each frequency point $\omega_{i}$, a 2D complex matrix is obtained that link
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
However, for the multi-body model being developed, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the micro-hexapod.
|
||||
However, for the multi-body model, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the micro-hexapod.
|
||||
Therefore, only $6 \times 6 = 36$ degrees of freedom are of interest.
|
||||
Therefore, the objective of this section is to to process the Frequency Response Matrix to reduce the number of measured acrshort:dof from 69 to 36.
|
||||
Therefore, the objective of this section is to process the Frequency Response Matrix to reduce the number of measured acrshort:dof from 69 to 36.
|
||||
|
||||
The coordinate transformation from accelerometers acrshort:dof to the solid body 6 acrshortpl:dof (three translations and three rotations) is performed in Section ref:ssec:modal_acc_to_solid_dof.
|
||||
The $69 \times 3 \times 801$ frequency response matrix is then reduced to a $36 \times 3 \times 801$ frequency response matrix where the motion of each solid body is expressed with respect to its center of mass.
|
||||
@ -679,29 +679,11 @@ Writing this in matrix form for the four points gives eqref:eq:modal_cart_to_acc
|
||||
\end{equation}
|
||||
|
||||
Provided that the four sensors are properly located, the system of equation eqref:eq:modal_cart_to_acc can be solved by matrix inversion[fn:modal_5].
|
||||
The motion of the solid body expressed in a chosen frame $\{O\}$ can be determined using equation eqref:eq:modal_determine_global_disp.
|
||||
The motion of the solid body expressed in a chosen frame $\{O\}$ can be determined by inverting equation eqref:eq:modal_cart_to_acc.
|
||||
Note that this matrix inversion is equivalent to resolving a mean square problem.
|
||||
Therefore, having more accelerometers permits better approximation of the motion of a solid body.
|
||||
|
||||
\begin{equation}
|
||||
\left[\begin{array}{c}
|
||||
\delta p_x \\ \delta p_y \\ \delta p_z \\ \hline \delta\Omega_x \\ \delta\Omega_y \\ \delta\Omega_z
|
||||
\end{array}\right] =
|
||||
\left[\begin{array}{ccc|ccc}
|
||||
1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
|
||||
0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
|
||||
0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
|
||||
& \vdots & & & \vdots & \\ \hline
|
||||
1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
|
||||
0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
|
||||
0 & 0 & 1 & p_{4y} & -p_{4x} & 0
|
||||
\end{array}\right]^{-1} \left[\begin{array}{c}
|
||||
\delta p_{1x} \\ \delta p_{1y} \\ \delta p_{1z} \\\hline \vdots \\\hline \delta p_{4x} \\ \delta p_{4y} \\ \delta p_{4z}
|
||||
\end{array}\right] \label{eq:modal_determine_global_disp}
|
||||
\end{equation}
|
||||
|
||||
|
||||
From the CAD model, the position of the center of mass of each considered solid body is computed (see Table ref:tab:modal_com_solid_bodies).
|
||||
From the CAD model, the position of the center of mass of each solid body is computed (see Table ref:tab:modal_com_solid_bodies).
|
||||
The position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be determined.
|
||||
|
||||
#+begin_src matlab
|
||||
@ -762,7 +744,7 @@ data2orgtable(1000*model_com', {'Bottom Granite', 'Top granite', 'Translation st
|
||||
| Spindle | $0$ | $0$ | $-580\,\text{mm}$ |
|
||||
| Hexapod | $-4\,\text{mm}$ | $6\,\text{mm}$ | $-319\,\text{mm}$ |
|
||||
|
||||
Using eqref:eq:modal_determine_global_disp, the frequency response matrix $\mathbf{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\mathbf{H}$.
|
||||
Using eqref:eq:modal_cart_to_acc, the frequency response matrix $\mathbf{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\mathbf{H}$.
|
||||
|
||||
\begin{equation}\label{eq:modal_frf_matrix_com}
|
||||
\mathbf{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
|
||||
@ -852,7 +834,7 @@ Similar results were obtained for the other solid bodies, indicating that the so
|
||||
This also validates the reduction in the number of degrees of freedom from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof).
|
||||
|
||||
#+begin_src matlab :exports none :results none
|
||||
%% Comparaison of the original accelerometer response and reconstructed response from the solid body response
|
||||
%% Comparison of the original accelerometer response and reconstructed response from the solid body response
|
||||
exc_names = {'$F_x$', '$F_y$', '$F_z$'};
|
||||
DOFs = {'x', 'y', 'z', '\theta_x', '\theta_y', '\theta_z'};
|
||||
|
||||
@ -870,11 +852,10 @@ for i = 1:length(accs_i)
|
||||
|
||||
hold on;
|
||||
for dir_i = 1:3
|
||||
plot(freqs, abs(squeeze(frf(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'DisplayName', sprintf('$a_{%i,%s}$', acc_i, DOFs{dir_i}));
|
||||
plot(freqs, abs(squeeze(frf(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'color', [colors(dir_i,:), 0.5], 'linewidth', 2.5, 'DisplayName', sprintf('$a_{%i,%s}$ - meas', acc_i, DOFs{dir_i}));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
for dir_i = 1:3
|
||||
plot(freqs, abs(squeeze(frfs_A(3*(acc_i-1)+dir_i, exc_dir, :))), '--', 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(frfs_A(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'color', colors(dir_i, :), 'DisplayName', sprintf('$a_{%i,%s}$ - solid body', acc_i, DOFs{dir_i}));
|
||||
end
|
||||
hold off;
|
||||
|
||||
@ -893,7 +874,8 @@ for i = 1:length(accs_i)
|
||||
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
|
||||
xlim([0, 200]); ylim([1e-6, 3e-2]);
|
||||
xticks([0:20:200]);
|
||||
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
end
|
||||
#+end_src
|
||||
|
||||
@ -902,7 +884,7 @@ exportFig('figs/modal_comp_acc_solid_body_frf.pdf', 'width', 'full', 'height', '
|
||||
#+end_src
|
||||
|
||||
#+name: fig:modal_comp_acc_solid_body_frf
|
||||
#+caption: Comparaison of the original accelerometer response (solid curves) and the reconstructed response from the solid body response (dashed curves). Accelerometers 1 to 4 corresponding to the micro-hexapod are shown.
|
||||
#+caption: Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4 corresponding to the micro-hexapod are shown. Input is a hammer force applied on the micro-hexapod in the $x$ direction
|
||||
#+RESULTS:
|
||||
[[file:figs/modal_comp_acc_solid_body_frf.png]]
|
||||
|
||||
@ -920,7 +902,7 @@ In order to perform the modal parameter extraction, the order of the modal model
|
||||
This is achived using the acrfull:mif in section ref:ssec:modal_number_of_modes.
|
||||
|
||||
In section ref:ssec:modal_parameter_extraction, the modal parameter extraction is performed.
|
||||
The graphical display of the mode shapes can be computed from the modal model, which is quite quite useful for physical interpretation of the modes.
|
||||
The graphical display of the mode shapes can be computed from the modal model, which is quite useful for physical interpretation of the modes.
|
||||
|
||||
To validate the quality of the modal model, the full acrshort:frf matrix is computed from the modal model and compared to the initial measured acrshort:frf (section ref:ssec:modal_model_validity).
|
||||
|
||||
@ -947,7 +929,7 @@ To validate the quality of the modal model, the full acrshort:frf matrix is comp
|
||||
|
||||
** Number of modes determination
|
||||
<<ssec:modal_number_of_modes>>
|
||||
The acrshort:mif is here applied to the $n\times p$ acrshort:frf matrix where $n$ is a relatively large number of measurement DOFs (here $n=69$) and $p$ is the number of excitation DOFs (here $p=3$).
|
||||
The acrshort:mif is applied to the $n\times p$ acrshort:frf matrix where $n$ is a relatively large number of measurement DOFs (here $n=69$) and $p$ is the number of excitation DOFs (here $p=3$).
|
||||
|
||||
The complex modal indication function is defined in equation eqref:eq:modal_cmif where the diagonal matrix $\Sigma$ is obtained from a acrlong:svd of the acrshort:frf matrix as shown in equation eqref:eq:modal_svd.
|
||||
\begin{equation} \label{eq:modal_cmif}
|
||||
@ -998,6 +980,7 @@ ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
|
||||
exportFig('figs/modal_indication_function.pdf', 'width', 'normal', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+begin_figure
|
||||
#+attr_latex: :options [b]{0.70\linewidth}
|
||||
#+begin_minipage
|
||||
#+name: fig:modal_indication_function
|
||||
@ -1033,11 +1016,12 @@ exportFig('figs/modal_indication_function.pdf', 'width', 'normal', 'height', 'no
|
||||
#+latex: \captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Identified modes}
|
||||
#+end_scriptsize
|
||||
#+end_minipage
|
||||
#+end_figure
|
||||
|
||||
** Modal parameter extraction
|
||||
<<ssec:modal_parameter_extraction>>
|
||||
|
||||
Generally, modal identification consists of curve-fitting a theoretical expression to the actual measured acrshort:frf data.
|
||||
Generally, modal identification is using curve-fitting a theoretical expression to the actual measured acrshort:frf data.
|
||||
However, there are multiple levels of complexity, from fitting of a single resonance, to fitting a complete curve encompassing several resonances and working on a set of many acrshort:frf plots all obtained from the same structure.
|
||||
|
||||
Here, the last method is used because it provides a unique and consistent model.
|
||||
@ -1107,7 +1091,7 @@ In order to synthesize the full acrshort:frf matrix, the eigenvectors $\phi_r$ a
|
||||
\end{bmatrix}_{n \times 2m}
|
||||
\end{equation}
|
||||
|
||||
The full acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ can be synthesize using eqref:eq:modal_synthesized_frf.
|
||||
The full acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ can be obtained using eqref:eq:modal_synthesized_frf.
|
||||
|
||||
\begin{equation}\label{eq:modal_synthesized_frf}
|
||||
[\mathbf{H}_{\text{syn}}(\omega)]_{n\times n} = [\Phi]_{n\times2m} [\mathbf{H}_{\text{mod}}(\omega)]_{2m\times2m} [\Phi]_{2m\times n}^T
|
||||
@ -1175,7 +1159,7 @@ for i = 1:length(freqs)
|
||||
Hsyn(:, :, i) = eigen_vec_ext_M*diag(1./(diag(modal_a_ext_M).*(j*2*pi*freqs(i) - diag(eigen_val_ext_M))))*eigen_vec_ext_M.';
|
||||
end
|
||||
|
||||
%% Derivate two times to to have the acceleration response
|
||||
%% Derivate two times to have the acceleration response
|
||||
for i = 1:size(Hsyn, 1)
|
||||
Hsyn(i, :, :) = squeeze(Hsyn(i, :, :)).*(j*2*pi*freqs).^2;
|
||||
end
|
||||
@ -1259,7 +1243,7 @@ exportFig('figs/modal_comp_acc_frf_modal_3.pdf', 'width', 'third', 'height', 'no
|
||||
#+end_src
|
||||
|
||||
#+name: fig:modal_comp_acc_frf_modal
|
||||
#+caption: Comparison of the measured FRF with the synthesized FRF from the modal model.
|
||||
#+caption: Comparison of the measured FRF with the FRF synthesized from the modal model.
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:modal_comp_acc_frf_modal_1}From $F_{11,z}$ to $a_{11,z}$}
|
||||
@ -1287,7 +1271,7 @@ exportFig('figs/modal_comp_acc_frf_modal_3.pdf', 'width', 'third', 'height', 'no
|
||||
|
||||
In this study, a modal analysis of the micro-station was performed.
|
||||
Thanks to an adequate choice of instrumentation and proper set of measurements, high quality frequency response functions can be obtained.
|
||||
The obtained frequency response functions indicate that the dynamics of the micro-station are complex, which is expected from a heavy stack stage architecture.
|
||||
The obtained frequency response functions indicate that the dynamics of the micro-station is complex, which is expected from a heavy stack stage architecture.
|
||||
It shows a lot of coupling between stages and different directions, and many modes.
|
||||
|
||||
By measuring 12 degrees of freedom on each "stage", it could be verified that in the frequency range of interest, each stage behaved as a rigid body.
|
||||
|
||||
Binary file not shown.
@ -1,4 +1,4 @@
|
||||
% Created 2024-11-14 Thu 10:42
|
||||
% Created 2025-03-25 Tue 21:57
|
||||
% Intended LaTeX compiler: pdflatex
|
||||
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
|
||||
|
||||
@ -44,7 +44,7 @@ Experimental modal analysis will be used to tune the model, and to verify that a
|
||||
The tuning approach for the multi-body model based on measurements is illustrated in Figure \ref{fig:modal_vibration_analysis_procedure}.
|
||||
First, a \emph{response model} is obtained, which corresponds to a set of frequency response functions computed from experimental measurements.
|
||||
From this response model, the modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another describing the mode shapes.
|
||||
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considering solid bodies and the springs and dampers connecting the solid bodies.
|
||||
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considered solid bodies and the springs and dampers connecting the solid bodies.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -69,10 +69,10 @@ In order to perform an experimental modal analysis, a suitable measurement setup
|
||||
This includes using appropriate instrumentation (presented in Section \ref{ssec:modal_instrumentation}) and properly preparing the structure to be measured (Section \ref{ssec:modal_test_preparation}).
|
||||
Then, the locations of the measured motions (Section \ref{ssec:modal_accelerometers}) and the locations of the hammer impacts (Section \ref{ssec:modal_hammer_impacts}) have to be chosen carefully.
|
||||
The obtained force and acceleration signals are described in Section \ref{ssec:modal_measured_signals}, and the quality of the measured data is assessed.
|
||||
\section{Used Instrumentation}
|
||||
\section{Instrumentation}
|
||||
\label{ssec:modal_instrumentation}
|
||||
|
||||
Three type of equipment are essential for a good modal analysis.
|
||||
Three types of equipment are essential for a good modal analysis.
|
||||
First, \emph{accelerometers} are used to measure the response of the structure.
|
||||
Here, 3-axis accelerometers\footnote{PCB 356B18. Sensitivity is \(1\,V/g\), measurement range is \(\pm 5\,g\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} shown in figure \ref{fig:modal_accelero_M393B05} are used.
|
||||
These accelerometers were glued to the micro-station using a thin layer of wax for best results \cite[chapt. 3.5.7]{ewins00_modal}.
|
||||
@ -108,7 +108,7 @@ Finally, an \emph{acquisition system}\footnote{OROS OR36. 24bits signal-delta AD
|
||||
\section{Structure Preparation and Test Planing}
|
||||
\label{ssec:modal_test_preparation}
|
||||
|
||||
To obtain meaningful results, the modal analysis of the micro-station in performed \emph{in-situ}.
|
||||
To obtain meaningful results, the modal analysis of the micro-station is performed \emph{in-situ}.
|
||||
To do so, all the micro-station stage controllers are turned ``ON''.
|
||||
This is especially important for stages for which the stiffness is provided by local feedback control, such as the air bearing spindle, and the translation stage.
|
||||
If these local feedback controls were turned OFF, this would have resulted in very low-frequency modes that were difficult to measure in practice, and it would also have led to decoupled dynamics, which would not be the case in practice.
|
||||
@ -120,7 +120,7 @@ The \(H_{jk}\) element of this \acrfull{frf} matrix corresponds to the frequency
|
||||
Measuring this \acrshort{frf} matrix is time consuming as it requires to make \(n \times n\) measurements.
|
||||
However, due to the principle of reciprocity (\(H_{jk} = H_{kj}\)) and using the \emph{point measurement} (\(H_{jj}\)), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix \(\mathbf{H}\) \cite[chapt. 5.2]{ewins00_modal}.
|
||||
Therefore, a minimum set of \(n\) frequency response functions is required.
|
||||
This can be done either by measuring the response \(X_{j}\) at a fixed \acrshort{dof} \(j\) while applying forces \(F_{i}\) for at all \(n\) considered \acrshort{dof}, or by applying a force \(F_{k}\) at a fixed \acrshort{dof} \(k\) and measuring the response \(X_{i}\) for all \(n\) \acrshort{dof}.
|
||||
This can be done either by measuring the response \(X_{j}\) at a fixed \acrshort{dof} \(j\) while applying forces \(F_{i}\) at all \(n\) considered \acrshort{dof}, or by applying a force \(F_{k}\) at a fixed \acrshort{dof} \(k\) and measuring the response \(X_{i}\) for all \(n\) \acrshort{dof}.
|
||||
|
||||
It is however not advised to measure only one row or one column, as one or more modes may be missed by an unfortunate choice of force or acceleration measurement location (for instance if the force is applied at a vibration node of a particular mode).
|
||||
In this modal analysis, it is chosen to measure the response of the structure at all considered \acrshort{dof}, and to excite the structure at one location in three directions in order to have some redundancy, and to ensure that all modes are properly identified.
|
||||
@ -234,7 +234,7 @@ For the accelerometer, a much more complex signal can be observed, indicating co
|
||||
|
||||
The ``normalized'' \acrfull{asd} of the two signals were computed and shown in Figure \ref{fig:modal_asd_acc_force}.
|
||||
Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
|
||||
These data are corresponding to an hammer impact in the vertical direction and to the measured acceleration in the \(x\) direction by accelerometer \(1\) (fixed to the micro-hexapod).
|
||||
These data are corresponding to a hammer impact in the vertical direction and to the measured acceleration in the \(x\) direction by accelerometer \(1\) (fixed to the micro-hexapod).
|
||||
Similar results were obtained for all measured frequency response functions.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
@ -295,9 +295,9 @@ For each frequency point \(\omega_{i}\), a 2D complex matrix is obtained that li
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
However, for the multi-body model being developed, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the micro-hexapod.
|
||||
However, for the multi-body model, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the micro-hexapod.
|
||||
Therefore, only \(6 \times 6 = 36\) degrees of freedom are of interest.
|
||||
Therefore, the objective of this section is to to process the Frequency Response Matrix to reduce the number of measured \acrshort{dof} from 69 to 36.
|
||||
Therefore, the objective of this section is to process the Frequency Response Matrix to reduce the number of measured \acrshort{dof} from 69 to 36.
|
||||
|
||||
The coordinate transformation from accelerometers \acrshort{dof} to the solid body 6 \acrshortpl{dof} (three translations and three rotations) is performed in Section \ref{ssec:modal_acc_to_solid_dof}.
|
||||
The \(69 \times 3 \times 801\) frequency response matrix is then reduced to a \(36 \times 3 \times 801\) frequency response matrix where the motion of each solid body is expressed with respect to its center of mass.
|
||||
@ -351,29 +351,11 @@ Writing this in matrix form for the four points gives \eqref{eq:modal_cart_to_ac
|
||||
\end{equation}
|
||||
|
||||
Provided that the four sensors are properly located, the system of equation \eqref{eq:modal_cart_to_acc} can be solved by matrix inversion\footnote{As this matrix is in general non-square, the Moore–Penrose inverse can be used instead.}.
|
||||
The motion of the solid body expressed in a chosen frame \(\{O\}\) can be determined using equation \eqref{eq:modal_determine_global_disp}.
|
||||
The motion of the solid body expressed in a chosen frame \(\{O\}\) can be determined by inverting equation \eqref{eq:modal_cart_to_acc}.
|
||||
Note that this matrix inversion is equivalent to resolving a mean square problem.
|
||||
Therefore, having more accelerometers permits better approximation of the motion of a solid body.
|
||||
|
||||
\begin{equation}
|
||||
\left[\begin{array}{c}
|
||||
\delta p_x \\ \delta p_y \\ \delta p_z \\ \hline \delta\Omega_x \\ \delta\Omega_y \\ \delta\Omega_z
|
||||
\end{array}\right] =
|
||||
\left[\begin{array}{ccc|ccc}
|
||||
1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
|
||||
0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
|
||||
0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
|
||||
& \vdots & & & \vdots & \\ \hline
|
||||
1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
|
||||
0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
|
||||
0 & 0 & 1 & p_{4y} & -p_{4x} & 0
|
||||
\end{array}\right]^{-1} \left[\begin{array}{c}
|
||||
\delta p_{1x} \\ \delta p_{1y} \\ \delta p_{1z} \\\hline \vdots \\\hline \delta p_{4x} \\ \delta p_{4y} \\ \delta p_{4z}
|
||||
\end{array}\right] \label{eq:modal_determine_global_disp}
|
||||
\end{equation}
|
||||
|
||||
|
||||
From the CAD model, the position of the center of mass of each considered solid body is computed (see Table \ref{tab:modal_com_solid_bodies}).
|
||||
From the CAD model, the position of the center of mass of each solid body is computed (see Table \ref{tab:modal_com_solid_bodies}).
|
||||
The position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be determined.
|
||||
|
||||
\begin{table}[htbp]
|
||||
@ -393,7 +375,7 @@ Hexapod & \(-4\,\text{mm}\) & \(6\,\text{mm}\) & \(-319\,\text{mm}\)\\
|
||||
\end{tabularx}
|
||||
\end{table}
|
||||
|
||||
Using \eqref{eq:modal_determine_global_disp}, the frequency response matrix \(\mathbf{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response at the center of mass of each solid body \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) can be computed from the initial \acrshort{frf} matrix \(\mathbf{H}\).
|
||||
Using \eqref{eq:modal_cart_to_acc}, the frequency response matrix \(\mathbf{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response at the center of mass of each solid body \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) can be computed from the initial \acrshort{frf} matrix \(\mathbf{H}\).
|
||||
|
||||
\begin{equation}\label{eq:modal_frf_matrix_com}
|
||||
\mathbf{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
|
||||
@ -424,7 +406,7 @@ This also validates the reduction in the number of degrees of freedom from 69 (2
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/modal_comp_acc_solid_body_frf.png}
|
||||
\caption{\label{fig:modal_comp_acc_solid_body_frf}Comparaison of the original accelerometer response (solid curves) and the reconstructed response from the solid body response (dashed curves). Accelerometers 1 to 4 corresponding to the micro-hexapod are shown.}
|
||||
\caption{\label{fig:modal_comp_acc_solid_body_frf}Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4 corresponding to the micro-hexapod are shown. Input is a hammer force applied on the micro-hexapod in the \(x\) direction}
|
||||
\end{figure}
|
||||
|
||||
\chapter{Modal Analysis}
|
||||
@ -436,12 +418,12 @@ In order to perform the modal parameter extraction, the order of the modal model
|
||||
This is achived using the \acrfull{mif} in section \ref{ssec:modal_number_of_modes}.
|
||||
|
||||
In section \ref{ssec:modal_parameter_extraction}, the modal parameter extraction is performed.
|
||||
The graphical display of the mode shapes can be computed from the modal model, which is quite quite useful for physical interpretation of the modes.
|
||||
The graphical display of the mode shapes can be computed from the modal model, which is quite useful for physical interpretation of the modes.
|
||||
|
||||
To validate the quality of the modal model, the full \acrshort{frf} matrix is computed from the modal model and compared to the initial measured \acrshort{frf} (section \ref{ssec:modal_model_validity}).
|
||||
\section{Number of modes determination}
|
||||
\label{ssec:modal_number_of_modes}
|
||||
The \acrshort{mif} is here applied to the \(n\times p\) \acrshort{frf} matrix where \(n\) is a relatively large number of measurement DOFs (here \(n=69\)) and \(p\) is the number of excitation DOFs (here \(p=3\)).
|
||||
The \acrshort{mif} is applied to the \(n\times p\) \acrshort{frf} matrix where \(n\) is a relatively large number of measurement DOFs (here \(n=69\)) and \(p\) is the number of excitation DOFs (here \(p=3\)).
|
||||
|
||||
The complex modal indication function is defined in equation \eqref{eq:modal_cmif} where the diagonal matrix \(\Sigma\) is obtained from a \acrlong{svd} of the \acrshort{frf} matrix as shown in equation \eqref{eq:modal_svd}.
|
||||
\begin{equation} \label{eq:modal_cmif}
|
||||
@ -459,6 +441,7 @@ The obtained \acrshort{mif} is shown on Figure \ref{fig:modal_indication_functio
|
||||
A total of 16 modes were found between 0 and \(200\,\text{Hz}\).
|
||||
The obtained natural frequencies and associated modal damping are summarized in Table \ref{tab:modal_obtained_modes_freqs_damps}.
|
||||
|
||||
\begin{figure}
|
||||
\begin{minipage}[b]{0.70\linewidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,scale=1]{figs/modal_indication_function.png}
|
||||
@ -494,11 +477,12 @@ Mode & Frequency & Damping\\
|
||||
\captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Identified modes}
|
||||
\end{scriptsize}
|
||||
\end{minipage}
|
||||
\end{figure}
|
||||
|
||||
\section{Modal parameter extraction}
|
||||
\label{ssec:modal_parameter_extraction}
|
||||
|
||||
Generally, modal identification consists of curve-fitting a theoretical expression to the actual measured \acrshort{frf} data.
|
||||
Generally, modal identification is using curve-fitting a theoretical expression to the actual measured \acrshort{frf} data.
|
||||
However, there are multiple levels of complexity, from fitting of a single resonance, to fitting a complete curve encompassing several resonances and working on a set of many \acrshort{frf} plots all obtained from the same structure.
|
||||
|
||||
Here, the last method is used because it provides a unique and consistent model.
|
||||
@ -568,7 +552,7 @@ In order to synthesize the full \acrshort{frf} matrix, the eigenvectors \(\phi_r
|
||||
\end{bmatrix}_{n \times 2m}
|
||||
\end{equation}
|
||||
|
||||
The full \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) can be synthesize using \eqref{eq:modal_synthesized_frf}.
|
||||
The full \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) can be obtained using \eqref{eq:modal_synthesized_frf}.
|
||||
|
||||
\begin{equation}\label{eq:modal_synthesized_frf}
|
||||
[\mathbf{H}_{\text{syn}}(\omega)]_{n\times n} = [\Phi]_{n\times2m} [\mathbf{H}_{\text{mod}}(\omega)]_{2m\times2m} [\Phi]_{2m\times n}^T
|
||||
@ -603,7 +587,7 @@ This can be seen in Figure \ref{fig:modal_comp_acc_frf_modal_3} that shows the f
|
||||
\end{center}
|
||||
\subcaption{\label{fig:modal_comp_acc_frf_modal_3}From $F_{11,y}$ to $a_{2,x}$}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:modal_comp_acc_frf_modal}Comparison of the measured FRF with the synthesized FRF from the modal model.}
|
||||
\caption{\label{fig:modal_comp_acc_frf_modal}Comparison of the measured FRF with the FRF synthesized from the modal model.}
|
||||
\end{figure}
|
||||
|
||||
\chapter{Conclusion}
|
||||
@ -611,7 +595,7 @@ This can be seen in Figure \ref{fig:modal_comp_acc_frf_modal_3} that shows the f
|
||||
|
||||
In this study, a modal analysis of the micro-station was performed.
|
||||
Thanks to an adequate choice of instrumentation and proper set of measurements, high quality frequency response functions can be obtained.
|
||||
The obtained frequency response functions indicate that the dynamics of the micro-station are complex, which is expected from a heavy stack stage architecture.
|
||||
The obtained frequency response functions indicate that the dynamics of the micro-station is complex, which is expected from a heavy stack stage architecture.
|
||||
It shows a lot of coupling between stages and different directions, and many modes.
|
||||
|
||||
By measuring 12 degrees of freedom on each ``stage'', it could be verified that in the frequency range of interest, each stage behaved as a rigid body.
|
||||
|
||||
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