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@ -10,98 +10,12 @@ addpath('./mat/'); % Path for data
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%% Colors for the figures
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colors = colororder;
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% Location of the Accelerometers
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% <<ssec:modal_accelerometers>>
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% The location of the accelerometers fixed to the micro-station is essential because it defines where the dynamics is measured.
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% A total of 23 accelerometers were fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod.
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% The positions of the accelerometers are visually shown on a CAD model in Figure ref:fig:modal_location_accelerometers and their precise locations with respect to a frame located at the point of interest are summarized in Table ref:tab:modal_position_accelerometers.
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% Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure ref:fig:modal_accelerometer_pictures.
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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 [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 0.95\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]{0.36\linewidth}
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% #+begin_minipage
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% #+begin_scriptsize
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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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% | | $x$ | $y$ | $z$ |
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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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% #+name: fig:modal_accelerometer_pictures
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% #+caption: Accelerometers fixed on the micro-station stages
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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_accelerometers_ty} $T_y$ stage}
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% #+attr_latex: :options {0.49\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :height 6cm
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% [[file:figs/modal_accelerometers_ty.jpg]]
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% #+end_subfigure
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% #+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_hexapod} Micro-Hexapod}
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% #+attr_latex: :options {0.49\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :height 6cm
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% [[file:figs/modal_accelerometers_hexapod.jpg]]
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% #+end_subfigure
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% #+end_figure
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%% Load Accelerometer positions
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acc_pos = readtable('mat/acc_pos.txt', 'ReadVariableNames', false);
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acc_pos = table2array(acc_pos(:, 1:4));
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[~, i] = sort(acc_pos(:, 1));
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acc_pos = acc_pos(i, 2:4);
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% Force and Response signals
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% <<ssec:modal_measured_signals>>
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% The force sensor of the instrumented hammer and the accelerometer signals are shown in the time domain in Figure ref:fig:modal_raw_meas.
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% Sharp "impacts" can be observed for the force sensor, indicating wide frequency band excitation.
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% For the accelerometer, a much more complex signal can be observed, indicating complex dynamics.
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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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%% Load raw data
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meas1_raw = load('mat/meas_raw_1.mat');
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@ -148,31 +62,6 @@ xticks([0:20:200]);
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ylim([0, 1])
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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 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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% #+attr_latex: :options {0.49\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :width 0.95\linewidth
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% [[file:figs/modal_raw_meas.png]]
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% #+end_subfigure
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% #+attr_latex: :caption \subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)}
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% #+attr_latex: :options {0.49\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :width 0.95\linewidth
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% [[file:figs/modal_asd_acc_force.png]]
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% #+end_subfigure
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% #+end_figure
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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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%% Compute the transfer function and Coherence
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[G1, f] = tfestimate(meas1_raw.Track1, meas1_raw.Track2, win, Noverlap, Nfft, Fs);
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[coh1, ~] = mscohere( meas1_raw.Track1, meas1_raw.Track2, win, Noverlap, Nfft, Fs);
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@ -10,57 +10,6 @@ addpath('./mat/'); % Path for data
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%% Colors for the figures
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colors = colororder;
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% #+name: fig:modal_local_to_global_coordinates
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% #+caption: Schematic of the measured motions of a solid body
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% #+RESULTS:
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% [[file:figs/modal_local_to_global_coordinates.png]]
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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 [[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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% \end{equation}
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% \begin{equation}\label{eq:modal_rotation_matrix}
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% \bm{\delta\Omega} = \begin{bmatrix}
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% 0 & -\delta\Omega_z & \delta\Omega_y \\
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% \delta\Omega_z & 0 & -\delta\Omega_x \\
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% -\delta\Omega_y & \delta\Omega_x & 0
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% \end{bmatrix}
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% \end{equation}
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% Writing this in matrix form for the four points gives eqref:eq:modal_cart_to_acc.
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% \begin{equation}\label{eq:modal_cart_to_acc}
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% \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] =
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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] \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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% \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: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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% 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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%% Load frequency response matrix
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load('frf_matrix.mat', 'freqs', 'frf');
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@ -88,39 +37,6 @@ save('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
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%% Extract the CoM of considered solid bodies
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model_com = reshape(table2array(readtable('mat/model_solidworks_com.txt', 'ReadVariableNames', false)), [3, 6]);
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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 0.55\linewidth :align Xccc
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% #+attr_latex: :center t :booktabs t
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% #+RESULTS:
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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_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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% \frac{D_{1,T_x}}{F_x}(\omega_i) & \frac{D_{1,T_x}}{F_y}(\omega_i) & \frac{D_{1,T_x}}{F_z}(\omega_i) \\
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% \frac{D_{1,T_y}}{F_x}(\omega_i) & \frac{D_{1,T_y}}{F_y}(\omega_i) & \frac{D_{1,T_y}}{F_z}(\omega_i) \\
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% \frac{D_{1,T_z}}{F_x}(\omega_i) & \frac{D_{1,T_z}}{F_y}(\omega_i) & \frac{D_{1,T_z}}{F_z}(\omega_i) \\
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% \frac{D_{1,R_x}}{F_x}(\omega_i) & \frac{D_{1,R_x}}{F_y}(\omega_i) & \frac{D_{1,R_x}}{F_z}(\omega_i) \\
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% \frac{D_{1,R_y}}{F_x}(\omega_i) & \frac{D_{1,R_y}}{F_y}(\omega_i) & \frac{D_{1,R_y}}{F_z}(\omega_i) \\
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% \frac{D_{1,R_z}}{F_x}(\omega_i) & \frac{D_{1,R_z}}{F_y}(\omega_i) & \frac{D_{1,R_z}}{F_z}(\omega_i) \\
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% \frac{D_{2,T_x}}{F_x}(\omega_i) & \frac{D_{2,T_x}}{F_y}(\omega_i) & \frac{D_{2,T_x}}{F_z}(\omega_i) \\
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% \vdots & \vdots & \vdots \\
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% \frac{D_{6,R_z}}{F_x}(\omega_i) & \frac{D_{6,R_z}}{F_y}(\omega_i) & \frac{D_{6,R_z}}{F_z}(\omega_i)
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% \end{bmatrix}
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% \end{equation}
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%% Frequency Response Matrix - Response expressed at the CoM of the solid bodies
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frfs_CoM = zeros(length(solid_names)*6, 3, 801);
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@ -149,14 +65,6 @@ end
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%% Save the computed FRF at the CoM
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save('mat/frf_com.mat', 'frfs_CoM');
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% Verification of solid body assumption
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% <<ssec:modal_solid_body_assumption>>
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% From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from $\mathbf{H}_{\text{CoM}}$), and using equation eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered location.
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% In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements $\mathbf{H}$.
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% This is what is done here to check whether the solid body assumption is correct in the frequency band of interest.
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%% Compute the FRF at the accelerometer location from the CoM reponses
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frfs_A = zeros(size(frf));
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@ -179,14 +87,6 @@ for exc_dir = 1:3
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end
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end
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% The comparison is made for the 4 accelerometers fixed on the micro-hexapod (Figure ref:fig:modal_comp_acc_solid_body_frf).
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% The original frequency response functions and those computed from the CoM responses match well in the frequency range of interest.
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% Similar results were obtained for the other solid bodies, indicating that the solid body assumption is valid and that a multi-body model can be used to represent the dynamics of the micro-station.
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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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%% 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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@ -10,27 +10,6 @@ addpath('./mat/'); % Path for data
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%% Colors for the figures
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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 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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% [CMIF(\omega)]_{p\times p} = [\Sigma(\omega)]_{p\times n}^T [\Sigma(\omega)]_{n\times p}
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% \end{equation}
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% \begin{equation} \label{eq:modal_svd}
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% [H(\omega)]_{n\times p} = [U(\omega)]_{n\times n} [\Sigma(\omega)]_{n\times p} [V(\omega)]_{p\times p}^H
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% \end{equation}
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% The acrshort:mif therefore yields to $p$ values that are also frequency dependent.
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% A peak in the acrshort:mif plot indicates the presence of a mode.
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% Repeated modes can also be detected when multiple singular values have peaks at the same frequency.
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% The obtained acrshort:mif is shown on Figure ref:fig:modal_indication_function.
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% A total of 16 modes were found between 0 and $200\,\text{Hz}$.
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% The obtained natural frequencies and associated modal damping are summarized in Table ref:tab:modal_obtained_modes_freqs_damps.
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%% Load frequency response matrix
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load('frf_matrix.mat', 'freqs', 'frf');
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@ -56,33 +35,6 @@ 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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% To check the validity of the modal model, the complete $n \times n$ acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ is first synthesized from the modal parameters.
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% Then, the elements of this acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ that were already measured can be compared to the measured acrshort:frf matrix $\mathbf{H}$.
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% In order to synthesize the full acrshort:frf matrix, the eigenvectors $\phi_r$ are first organized in matrix from as shown in equation eqref:eq:modal_eigvector_matrix.
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% \begin{equation}\label{eq:modal_eigvector_matrix}
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% \Phi = \begin{bmatrix}
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% & & & & &\\
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% \phi_1 & \dots & \phi_N & \phi_1^* & \dots & \phi_N^* \\
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% & & & & &
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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 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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% \end{equation}
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% With $\mathbf{H}_{\text{mod}}(\omega)$ a diagonal matrix representing the response of the different modes eqref:eq:modal_modal_resp.
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% \begin{equation}\label{eq:modal_modal_resp}
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% \mathbf{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m}
|
||||
% \end{equation}
|
||||
|
||||
|
||||
%% Load modal parameters
|
||||
shapes_m = readtable('mat/mode_shapes.txt', 'ReadVariableNames', false); % [Sign / Real / Imag]
|
||||
freqs_m = table2array(readtable('mat/mode_freqs.txt', 'ReadVariableNames', false)); % in [Hz]
|
||||
@ -144,14 +96,6 @@ for i = 1:size(Hsyn, 1)
|
||||
Hsyn(i, :, :) = squeeze(Hsyn(i, :, :)).*(j*2*pi*freqs).^2;
|
||||
end
|
||||
|
||||
|
||||
|
||||
% 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.
|
||||
% Whether the obtained match is good or bad is quite arbitrary.
|
||||
% 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.
|
||||
% 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.
|
||||
|
||||
|
||||
acc_o = 11; dir_o = 3;
|
||||
acc_i = 11; dir_i = 3;
|
||||
|
||||
|
||||
@ -22,7 +22,7 @@
|
||||
#+BIND: org-latex-bib-compiler "biber"
|
||||
|
||||
#+PROPERTY: header-args:matlab :session *MATLAB*
|
||||
#+PROPERTY: header-args:matlab+ :comments org
|
||||
#+PROPERTY: header-args:matlab+ :comments no
|
||||
#+PROPERTY: header-args:matlab+ :exports none
|
||||
#+PROPERTY: header-args:matlab+ :results none
|
||||
#+PROPERTY: header-args:matlab+ :eval no-export
|
||||
|
||||
Reference in New Issue
Block a user