diff --git a/modal-analysis/data/oros_modal_manual.pdf b/modal-analysis/data/oros_modal_manual.pdf new file mode 100644 index 0000000..b7b9b4b Binary files /dev/null and b/modal-analysis/data/oros_modal_manual.pdf differ diff --git a/modal-analysis/figs/cmif.png b/modal-analysis/figs/cmif.png new file mode 100644 index 0000000..1b2e2b2 Binary files /dev/null and b/modal-analysis/figs/cmif.png differ diff --git a/modal-analysis/figs/cmif_compare.png b/modal-analysis/figs/cmif_compare.png new file mode 100644 index 0000000..082263c Binary files /dev/null and b/modal-analysis/figs/cmif_compare.png differ diff --git a/modal-analysis/figs/composite_response_function.png b/modal-analysis/figs/composite_response_function.png new file mode 100644 index 0000000..5cbbb62 Binary files /dev/null and b/modal-analysis/figs/composite_response_function.png differ diff --git a/modal-analysis/figs/modal_complexities.png b/modal-analysis/figs/modal_complexities.png index 67f5e21..20ae22a 100644 Binary files a/modal-analysis/figs/modal_complexities.png and b/modal-analysis/figs/modal_complexities.png differ diff --git a/modal-analysis/figs/modal_complexity_high.png b/modal-analysis/figs/modal_complexity_high.png index 02f4a84..95994f2 100644 Binary files a/modal-analysis/figs/modal_complexity_high.png and b/modal-analysis/figs/modal_complexity_high.png differ diff --git a/modal-analysis/figs/modal_complexity_small.png b/modal-analysis/figs/modal_complexity_small.png index 27daaa6..3e19fec 100644 Binary files a/modal-analysis/figs/modal_complexity_small.png and b/modal-analysis/figs/modal_complexity_small.png differ diff --git a/modal-analysis/frf_processing.org b/modal-analysis/frf_processing.org index 354b472..957bc40 100644 --- a/modal-analysis/frf_processing.org +++ b/modal-analysis/frf_processing.org @@ -41,6 +41,7 @@ #+PROPERTY: header-args:latex+ :output-dir figs :END: +* Introduction :ignore: The measurements have been conducted and we have computed the $n \times p \times q$ Frequency Response Functions Matrix with: - $n$: the number of measurements: $23 \times 3 = 69$ (23 accelerometers measuring 3 directions each) - $p$: the number of excitation inputs: $3$ diff --git a/modal-analysis/modal_extraction.html b/modal-analysis/modal_extraction.html index 48bc2fd..3afb636 100644 Binary files a/modal-analysis/modal_extraction.html and b/modal-analysis/modal_extraction.html differ diff --git a/modal-analysis/modal_extraction.org b/modal-analysis/modal_extraction.org index 32d2083..98b6840 100644 --- a/modal-analysis/modal_extraction.org +++ b/modal-analysis/modal_extraction.org @@ -41,14 +41,50 @@ #+PROPERTY: header-args:latex+ :output-dir figs :END: -The goal here is to extract the modal parameters describing the modes of station being studied. +* Introduction :ignore: +The goal here is to extract the modal parameters describing the modes of station being studied, namely: +- the eigen frequencies +- the modal damping +- the mode shapes (eigen vectors) +This is done from the FRF matrix previously extracted from the measurements. + + +In order to do the modal parameter extraction, we first have to estimate the order of the modal model we want to obtain. +This corresponds to how many modes are present in the frequency band of interest. +In section [[sec:number_of_modes]], we will use the Singular Value Decomposition and the Modal Indication Function to estimate the number of modes. + + +The modal parameter extraction methods generally consists of *curve-fitting a theoretical expression for an individual FRF to the actual measured data*. +However, there are multiple level of complexity: +- works on a part of a single FRF curve +- works on a complete curve encompassing several resonances +- works on a set of many FRF plots all obtained from the same structure +The third method is the most complex but gives better results. This is the one we will use in section [[sec:modal_extraction]]. + + +From the modal model, it is possible to obtain a graphic display of the mode shapes (section [[sec:mode_shape_display]]). + + +The modes of the structure are expected to be complex, however real modes are easier to work with when it comes to obtain a spatial model from the modal parameters. +We will thus study the complexity of those modes, in section [[sec:modal_complexity]], and see if we can estimate real modes from the complex modes. + + +The mode obtained from the modal software describe the motion of the structure at the position of each accelerometer for all the modes. +However, we would like to describe the motion of each stage (solid body) of the structure in its 6 DOFs. +This in done in section [[sec:global_mode_shapes]]. * ZIP file containing the data and matlab files :ignore: #+begin_src bash :exports none :results none if [ matlab/modal_extraction.m -nt data/modal_extraction.zip ]; then cp matlab/modal_extraction.m modal_extraction.m; zip data/modal_extraction \ - mat/data.mat \ + mat/frf_coh_matrices.mat \ + mat/frf_o.mat \ + mat/mode_shapes.txt \ + mat/mode_freqs.txt \ + mat/mode_damps.txt \ + mat/mode_modal_a.txt \ + mat/mode_modal_b.txt \ modal_extraction.m rm modal_extraction.m; fi @@ -67,31 +103,170 @@ The goal here is to extract the modal parameters describing the modes of station <> #+end_src -* TODO Part to explain how to choose the modes frequencies -- bro-band method used -- Stabilization Chart -- 21 modes +* Load Data +#+begin_src matlab + load('mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs'); + load('mat/frf_o.mat', 'FRFs_O'); +#+end_src -* Obtained Modal Parameters -From the modal analysis software, we can export the obtained modal parameters: +* Determine the number of modes + <> +** Singular Value Decomposition - Modal Indication Function +The Mode Indicator Functions are usually used on $n\times p$ FRF matrix where $n$ is a relatively large number of measurement DOFs and $p$ is the number of excitation DOFs, typically 3 or 4. + +In these methods, the frequency dependent FRF matrix is subjected to a singular value decomposition analysis which thus yields a small number (3 or 4) of singular values, these also being frequency dependent. + +These methods are used to *determine the number of modes* present in a given frequency range, to *identify repeated natural frequencies* and to pre process the FRF data prior to modal analysis. + +From the documentation of the modal software: +#+begin_quote +The MIF consist of the singular values of the Frequency response function matrix. The number of MIFs equals the number of excitations. +By the powerful singular value decomposition, the real signal space is separated from the noise space. Therefore, the MIFs exhibit the modes effectively. +A peak in the MIFs plot usually indicate the existence of a structural mode, and two peaks at the same frequency point means the existence of two repeated modes. +Moreover, the magnitude of the MIFs implies the strength of the a mode. +#+end_quote + +#+begin_important +The *Complex Mode Indicator Function* is defined simply by the SVD of the FRF (sub) matrix: +\begin{align*} + [H(\omega)]_{n\times p} &= [U(\omega)]_{n\times n} [\Sigma(\omega)]_{n\times p} [V(\omega)]_{p\times p}^H\\ + [CMIF(\omega)]_{p\times p} &= [\Sigma(\omega)]_{p\times n}^T [\Sigma(\omega)]_{n\times p} +\end{align*} +#+end_important + +We compute the Complex Mode Indicator Function. The result is shown on figure [[fig:cmif]]. +The exact same curve is obtained when computed using the OROS software. + +#+begin_src matlab + MIF = zeros(size(FRFs, 2), size(FRFs, 2), size(FRFs, 3)); + + for i = 1:length(freqs) + [~,S,~] = svd(FRFs(:, :, i)); + MIF(:, :, i) = S'*S; + end +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + for i = 1:size(MIF, 1) + plot(freqs, squeeze(MIF(i, i, :)), 'DisplayName', sprintf('MDIF - %i', i)); + end + hold off; + set(gca, 'Yscale', 'log'); + xlabel('Frequency [Hz]'); ylabel('CMIF Amplitude'); + xlim([1, 200]); + legend('location', 'southeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/cmif.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:cmif +#+CAPTION: Complex Mode Indicator Function +[[file:figs/cmif.png]] + +We can also compute the CMIF using the FRF matrix expressed in the same global frame. +We compare the two CMIF on figure [[fig:cmif_compare]]. + +They do not indicate the same resonance frequencies, especially around 110Hz. + +#+begin_src matlab + MIF_O = zeros(size(FRFs_O, 2), size(FRFs_O, 2), size(FRFs_O, 3)); + for i = 1:length(freqs) + [~,S,~] = svd(FRFs_O(:, :, i)); + MIF_O(:, :, i) = S'*S; + end +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + for i = 1:size(MIF, 1) + set(gca,'ColorOrderIndex',i) + plot(freqs, squeeze(MIF(i, i, :)), '-'); + set(gca,'ColorOrderIndex',i) + plot(freqs, squeeze(MIF_O(i, i, :)), '--'); + end + hold off; + set(gca, 'Yscale', 'log'); + xlabel('Frequency [Hz]'); ylabel('CMIF Amplitude'); + xlim([1, 200]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/cmif_compare.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:cmif_compare +#+CAPTION: Complex Mode Indicator Function - Original FRF: solid curves - Processed FRF: dashed curve +[[file:figs/cmif_compare.png]] + +** Composite Response Function +An alternative is the Composite Response Function $HH(\omega)$ defined as the sum of all the measured FRF: +\begin{equation} + HH(\omega) = \sum_j\sum_kH_{jk}(\omega) +\end{equation} + +Instead, we choose here to use the sum of the norms of the measured FRFs: +\begin{equation} + HH(\omega) = \sum_j\sum_k \left|H_{jk}(\omega) \right| +\end{equation} + +The result is shown on figure [[fig:composite_response_function]]. + +#+begin_src matlab + figure; + hold on; + plot(freqs, squeeze(sum(sum(abs(FRFs)))), '-k'); + plot(freqs, squeeze(sum(sum(abs(FRFs_O)))), '--k'); + hold off; + xlabel('Frequency [Hz]'); ylabel('Amplitude'); + xlim([1, 200]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/composite_response_function.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:composite_response_function +#+CAPTION: Composite Response Function. Solid curve: computed from the original FRFs. Dashed curve: computed from the FRFs expressed in the global frame. +[[file:figs/composite_response_function.png]] + +* Modal parameter extraction + <> +** OROS - Modal software +Modal identification can be done within the [[https://www.oros.com/solutions/structural-dynamics/modal-analysis/][Modal software of OROS]]. The manual for the software is available [[file:data/oros_modal_manual.pdf][here]]. + +Several modal parameter extraction methods are available. +We choose to use the "broad band method" as it permits to identify the modal parameters using all the FRF curves at the same times. +It takes into account the fact the the properties of all the individual curves are related by being from the same structure: all FRF plots on a given structure should indicate the same values for the natural frequencies and damping factor of each mode. + +Such method also have the advantage of producing a unique and consistent model as direct output. + +In order to apply this method, we select the frequency range of interest and we give an estimate of how many modes are present. + +Then, it shows a stabilization charts, such as the one shown on figure [[fig:stabilization_chart]], where we have to manually select which modes to take into account in the modal model. + +#+name: fig:stabilization_chart +#+caption: Stabilization Chart +[[file:figs/stabilisation_chart.jpg]] + +We can then run the modal analysis, and the software will identify the modal damping and mode shapes at the selected frequency modes. + +** Exported modal parameters +The obtained modal parameters are: - the resonance frequencies - the modes shapes - the modal damping - the residues -These can be express as the *eigen matrices*: -\[ \Omega = \begin{bmatrix} - \omega_1^2 & & 0 \\ - & \ddots & \\ - 0 & & \omega_n^2 -\end{bmatrix}; \quad \Psi = \begin{bmatrix} - & & \\ - \{\psi_1\} & \dots & \{\psi_n\} \\ - & & -\end{bmatrix} \] -where $\bar{\omega}_r^2$ is the $r^\text{th}$ eigenvalue squared and $\{\phi\}_r$ is a description of the corresponding *mode shape*. - -The file containing the modal parameters is =mat/modes.asc=. Its first 20 lines as shown below. +They are all exported in a text file named =modes.asc=. +Its first 20 lines as shown below. #+begin_src bash :results output :exports results :eval no-export sed 20q mat/modes.asc | sed $'s/\r//' @@ -141,6 +316,7 @@ We split this big =modes.asc= file into sub text files using =bash=. The obtaine | =mat/mode_modal_a.txt= | modal residues at low frequency (to be checked) | | =mat/mode_modal_b.txt= | modal residues at high frequency (to be checked) | +** Importation of the modal parameters on Matlab Then we import the obtained =.txt= files on Matlab using =readtable= function. #+begin_src matlab shapes = readtable('mat/mode_shapes.txt', 'ReadVariableNames', false); % [Sign / Real / Imag] @@ -184,41 +360,61 @@ As the mode shapes are split into 3 parts (direction plus sign, real part and im #+end_src The obtained mode frequencies and damping are shown below. -#+begin_src matlab :exports both :results value table replace :post addhdr(*this*) - data2orgtable([freqs, damps], {}, {'Frequency [Hz]', 'Damping [%]'}, ' %.1f '); +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([(1:length(freqs))', freqs, damps], {}, {'Mode number', 'Frequency [Hz]', 'Damping [%]'}, ' %.1f '); #+end_src +#+name: tab:obtained_modes_freqs_damps +#+caption: Obtained eigen frequencies and modal damping #+RESULTS: -| Frequency [Hz] | Damping [%] | -|----------------+-------------| -| 11.4 | 8.7 | -| 18.5 | 11.8 | -| 37.6 | 6.4 | -| 39.4 | 3.6 | -| 54.0 | 0.2 | -| 56.1 | 2.8 | -| 69.7 | 4.6 | -| 71.6 | 0.6 | -| 72.4 | 1.6 | -| 84.9 | 3.6 | -| 90.6 | 0.3 | -| 91.0 | 2.9 | -| 95.8 | 3.3 | -| 105.4 | 3.3 | -| 106.8 | 1.9 | -| 112.6 | 3.0 | -| 116.8 | 2.7 | -| 124.1 | 0.6 | -| 145.4 | 1.6 | -| 150.1 | 2.2 | -| 164.7 | 1.4 | +| Mode number | Frequency [Hz] | Damping [%] | +|-------------+----------------+-------------| +| 1.0 | 11.4 | 8.7 | +| 2.0 | 18.5 | 11.8 | +| 3.0 | 37.6 | 6.4 | +| 4.0 | 39.4 | 3.6 | +| 5.0 | 54.0 | 0.2 | +| 6.0 | 56.1 | 2.8 | +| 7.0 | 69.7 | 4.6 | +| 8.0 | 71.6 | 0.6 | +| 9.0 | 72.4 | 1.6 | +| 10.0 | 84.9 | 3.6 | +| 11.0 | 90.6 | 0.3 | +| 12.0 | 91.0 | 2.9 | +| 13.0 | 95.8 | 3.3 | +| 14.0 | 105.4 | 3.3 | +| 15.0 | 106.8 | 1.9 | +| 16.0 | 112.6 | 3.0 | +| 17.0 | 116.8 | 2.7 | +| 18.0 | 124.1 | 0.6 | +| 19.0 | 145.4 | 1.6 | +| 20.0 | 150.1 | 2.2 | +| 21.0 | 164.7 | 1.4 | + +** Modal Matrices +We arrange the obtained modal parameters into matrices: +\[ \Omega = \begin{bmatrix} + \omega_1^2 & & 0 \\ + & \ddots & \\ + 0 & & \omega_n^2 +\end{bmatrix}; \quad \Psi = \begin{bmatrix} + & & \\ + \{\psi_1\} & \dots & \{\psi_n\} \\ + & & +\end{bmatrix} \] +with $n$ the number of identified modes and: +\[ \{\psi_1\} = \begin{Bmatrix} \psi_{1_x} & \psi_{2_x} & \dots & \psi_{23_x} & \psi_{1_y} & \dots & \psi_{1_z} & \dots & \psi_{23_z} \end{Bmatrix}^T \] + +#+begin_src matlab + eigen_value_M = diag(freqs*2*pi); + eigen_vector_M = reshape(modes, [mod_n, acc_n*dir_n])'; +#+end_src + +Each eigen vector is normalized: $\| \{\psi_i\}\} \|_2 = 1$ * Obtained Mode Shapes animations -One all the FRFs are obtained, we can estimate the modal parameters (resonance frequencies, modal shapes and modal damping) within the modal software. - -For that, multiple modal extraction techniques can be used (SIMO, MIMO, narrow band, wide band, ...). - -Then, it is possible to show the modal shapes with an animation. + <> +From the modal parameters, it is possible to show the modal shapes with an animation. Examples are shown on figures [[fig:mode1]] and [[fig:mode6]]. @@ -232,985 +428,9 @@ Animations of all the other modes are accessible using the following links: [[fi #+caption: Mode 6 [[file:img/modes/mode6.gif]] -* Compute the Modal Model -** Matlab Init :noexport:ignore: -#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) - <> -#+end_src +We can learn quite a lot from these mode shape animations. -#+begin_src matlab :exports none :results silent :noweb yes - <> -#+end_src - -** Position of the accelerometers -There are 23 accelerometers: -- 4 on the bottom granite -- 4 on the top granite -- 4 on top of the translation stage -- 4 on the tilt stage -- 3 on top of the spindle -- 4 on top of the hexapod - -The coordinates defined in the software are displayed below. - -#+begin_src bash :results output :exports results :eval no-export - sed -n 18,40p modal_analysis/acc_coordinates.txt | tac -- -#+end_src - -#+RESULTS: -#+begin_example - 1 1.0000e-001 1.0000e-001 1.1500e+000 0 Top - 2 1.0000e-001 -1.0000e-001 1.1500e+000 0 Top - 3 -1.0000e-001 -1.0000e-001 1.1500e+000 0 Top - 4 -1.0000e-001 1.0000e-001 1.1500e+000 0 Top - 5 4.0000e-001 4.0000e-001 9.5000e-001 0 inner - 6 4.0000e-001 -4.0000e-001 9.5000e-001 0 inner - 7 -4.0000e-001 -4.0000e-001 9.5000e-001 0 inner - 8 -4.0000e-001 4.0000e-001 9.5000e-001 0 inner - 9 5.0000e-001 5.0000e-001 9.0000e-001 0 outer - 10 5.0000e-001 -5.0000e-001 9.0000e-001 0 outer - 11 -5.0000e-001 -5.0000e-001 9.0000e-001 0 outer - 12 -5.0000e-001 5.0000e-001 9.0000e-001 0 outer - 13 5.5000e-001 5.5000e-001 5.5000e-001 0 top - 14 5.5000e-001 -5.5000e-001 5.5000e-001 0 top - 15 -5.5000e-001 -5.5000e-001 5.5000e-001 0 top - 16 -5.5000e-001 5.5000e-001 5.5000e-001 0 top - 17 9.5000e-001 9.5000e-001 4.0000e-001 0 low - 18 9.5000e-001 -9.5000e-001 4.0000e-001 0 low - 19 -9.5000e-001 -9.5000e-001 4.0000e-001 0 low - 20 -9.5000e-001 9.5000e-001 4.0000e-001 0 low - 21 2.0000e-001 2.0000e-001 8.5000e-001 0 bot - 22 0.0000e+000 -2.0000e-001 8.5000e-001 0 bot - 23 -2.0000e-001 2.0000e-001 8.5000e-001 0 bot -#+end_example - -#+name: tab:acc_location -#+caption: Location of each Accelerometer (using the normal coordinate frame with X aligned with the X ray) -| *Node number* | *Solid Body* | *Location* | *X* | *Y* | *Z* | -|---------------+-------------------+------------+-------+-------+------| -| 1 | Hexapod - Top | -X/-Y | -0.10 | -0.10 | 1.15 | -| 2 | | -X/+Y | -0.10 | 0.10 | 1.15 | -| 3 | | +X/+Y | 0.10 | 0.10 | 1.15 | -| 4 | | +X/-Y | 0.10 | -0.10 | 1.15 | -|---------------+-------------------+------------+-------+-------+------| -| 5 | Tilt - Top | -X/-Y | -0.40 | -0.40 | 0.95 | -| 6 | | -X/+Y | -0.40 | 0.40 | 0.95 | -| 7 | | +X/+Y | 0.40 | 0.40 | 0.95 | -| 8 | | +X/-Y | 0.40 | -0.40 | 0.95 | -|---------------+-------------------+------------+-------+-------+------| -| 9 | Translation - Top | -X/-Y | -0.50 | -0.50 | 0.90 | -| 10 | | -X/+Y | -0.50 | 0.50 | 0.90 | -| 11 | | +X/+Y | 0.50 | 0.50 | 0.90 | -| 12 | | +X/-Y | 0.50 | -0.50 | 0.90 | -|---------------+-------------------+------------+-------+-------+------| -| 13 | Top Granite | -X/-Y | -0.55 | -0.50 | 0.55 | -| 14 | | -X/+Y | -0.55 | 0.50 | 0.55 | -| 15 | | +X/+Y | 0.55 | 0.50 | 0.55 | -| 16 | | +X/-Y | 0.55 | -0.50 | 0.55 | -|---------------+-------------------+------------+-------+-------+------| -| 17 | Bottom Granite | -X/-Y | -0.95 | -0.90 | 0.40 | -| 18 | | -X/+Y | -0.95 | 0.90 | 0.40 | -| 19 | | +X/+Y | 0.95 | 0.90 | 0.40 | -| 20 | | +X/-Y | 0.95 | -0.90 | 0.40 | -|---------------+-------------------+------------+-------+-------+------| -| 21 | Spindle - Top | -X/-Y | -0.20 | -0.20 | 0.85 | -| 22 | | +0/+Y | 0.00 | 0.20 | 0.85 | -| 23 | | +X/-Y | 0.20 | -0.20 | 0.85 | - -** Define positions of the accelerometers on matlab -We define the X-Y-Z position of each sensor. -Each line corresponds to one accelerometer, X-Y-Z position in meter. -#+begin_src matlab - positions = [... - -0.10, -0.10, 1.15 ; ... - -0.10, 0.10, 1.15 ; ... - 0.10, 0.10, 1.15 ; ... - 0.10, -0.10, 1.15 ; ... - -0.40, -0.40, 0.95 ; ... - -0.40, 0.40, 0.95 ; ... - 0.40, 0.40, 0.95 ; ... - 0.40, -0.40, 0.95 ; ... - -0.50, -0.50, 0.90 ; ... - -0.50, 0.50, 0.90 ; ... - 0.50, 0.50, 0.90 ; ... - 0.50, -0.50, 0.90 ; ... - -0.55, -0.50, 0.55 ; ... - -0.55, 0.50, 0.55 ; ... - 0.55, 0.50, 0.55 ; ... - 0.55, -0.50, 0.55 ; ... - -0.95, -0.90, 0.40 ; ... - -0.95, 0.90, 0.40 ; ... - 0.95, 0.90, 0.40 ; ... - 0.95, -0.90, 0.40 ; ... - -0.20, -0.20, 0.85 ; ... - 0.00, 0.20, 0.85 ; ... - 0.20, -0.20, 0.85 ]; -#+end_src - - -#+begin_src matlab - figure; - hold on; - fill3(positions(1:4, 1), positions(1:4, 2), positions(1:4, 3), 'k', 'FaceAlpha', 0.5) - fill3(positions(5:8, 1), positions(5:8, 2), positions(5:8, 3), 'k', 'FaceAlpha', 0.5) - fill3(positions(9:12, 1), positions(9:12, 2), positions(9:12, 3), 'k', 'FaceAlpha', 0.5) - fill3(positions(13:16, 1), positions(13:16, 2), positions(13:16, 3), 'k', 'FaceAlpha', 0.5) - fill3(positions(17:20, 1), positions(17:20, 2), positions(17:20, 3), 'k', 'FaceAlpha', 0.5) - fill3(positions(21:23, 1), positions(21:23, 2), positions(21:23, 3), 'k', 'FaceAlpha', 0.5) - hold off; -#+end_src - -** Import the modal vectors on matlab -*** Mode1 -#+begin_src bash :results output :exports none :eval no-export - sed -n 12,80p modal_analysis/modes_propres_narband.asc -#+end_src - -#+RESULTS: -#+begin_example - 1X+: -9.34637e-002 4.52445e-002 - 1Y+: 2.33790e-001 1.41439e-003 - 1Z+: -1.73754e-002 6.02449e-003 - 2X+: -7.42108e-002 3.91543e-002 - 2Y+: 2.41566e-001 -1.44869e-003 - 2Z+: -5.99285e-003 2.10370e-003 - 4X+: -1.02163e-001 2.79561e-002 - 4Y+: 2.29048e-001 2.89782e-002 - 4Z+: -2.85130e-002 1.77132e-004 - 5X+: -8.77132e-002 3.34081e-002 - 5Y+: 2.14182e-001 2.14655e-002 - 5Z+: -1.54521e-002 1.26682e-002 - 6X+: -7.90143e-002 2.42583e-002 - 6Y+: 2.20669e-001 2.12738e-002 - 6Z+: 4.60755e-002 4.96406e-003 - 7X+: -7.79654e-002 2.58385e-002 - 7Y+: 2.06861e-001 3.48019e-002 - 7Z+: -1.78311e-002 -1.29704e-002 - 8X+: -8.49357e-002 3.55200e-002 - 8Y+: 2.07470e-001 3.59745e-002 - 8Z+: -7.66974e-002 -3.19813e-003 - 9X+: -7.38565e-002 1.95146e-002 - 9Y+: 2.17403e-001 2.01550e-002 - 9Z+: -1.77073e-002 -3.46414e-003 - 10X+: -7.77587e-002 2.36700e-002 - 10Y+: 2.35654e-001 -2.14540e-002 - 10Z+: 7.94165e-002 -2.45897e-002 - 11X+: -8.17972e-002 2.20583e-002 - 11Y+: 2.20906e-001 -4.30164e-003 - 11Z+: -5.60520e-003 3.10187e-003 - 12X+: -8.64261e-002 3.66022e-002 - 12Y+: 2.15000e-001 -5.74661e-003 - 12Z+: -1.22622e-001 4.11767e-002 - 13X+: -4.25169e-002 1.56602e-002 - 13Y+: 5.31036e-002 -1.73951e-002 - 13Z+: -4.07130e-002 1.26884e-002 - 14X+: -3.85032e-002 1.29431e-002 - 14Y+: 5.36716e-002 -1.80868e-002 - 14Z+: 1.00367e-001 -3.48798e-002 - 15X+: -4.25524e-002 1.46363e-002 - 15Y+: 5.19668e-002 -1.69744e-002 - 15Z+: 5.89747e-003 -2.32428e-003 - 16X+: -4.31268e-002 1.38332e-002 - 16Y+: 5.07545e-002 -1.53045e-002 - 16Z+: -1.04172e-001 3.17984e-002 - 17X+: -2.69757e-002 9.07955e-003 - 17Y+: 3.07837e-002 -9.44663e-003 - 17Z+: -7.63502e-003 1.68203e-003 - 18X+: -3.00097e-002 9.23966e-003 - 18Y+: 2.83585e-002 -8.97747e-003 - 18Z+: 1.52467e-001 -4.78675e-002 - 19X+: -2.70223e-002 6.16478e-003 - 19Y+: 3.06149e-002 -6.25382e-003 - 19Z+: -4.84888e-003 1.93970e-003 - 20X+: -2.90976e-002 7.13184e-003 - 20Y+: 3.36738e-002 -7.30875e-003 - 20Z+: -1.66902e-001 3.93419e-002 - 3X+: -9.40720e-002 3.93724e-002 - 3Y+: 2.52307e-001 0.00000e+000 - 3Z+: -1.53864e-002 -9.25720e-004 - 21X+: -7.91940e-002 4.39648e-002 - 21Y+: 2.04567e-001 9.49987e-003 - 21Z+: -1.56087e-002 7.08838e-003 - 22X+: -1.01070e-001 3.13534e-002 - 22Y+: 1.92270e-001 1.80423e-002 - 22Z+: 2.93053e-003 -1.97308e-003 - 23X+: -8.86455e-002 4.29906e-002 - 23Z+: -3.38351e-002 1.81362e-003 - 23Y-: -1.90862e-001 -2.53414e-002 -#+end_example - -#+begin_src matlab - mode1 = [... - -9.34637e-002+j*4.52445e-002, +2.33790e-001+j*1.41439e-003, -1.73754e-002+j*6.02449e-003; - -7.42108e-002+j*3.91543e-002, +2.41566e-001-j*1.44869e-003, -5.99285e-003+j*2.10370e-003; - -9.40720e-002+j*3.93724e-002, +2.52307e-001+j*0.00000e+000, -1.53864e-002-j*9.25720e-004; - -1.02163e-001+j*2.79561e-002, +2.29048e-001+j*2.89782e-002, -2.85130e-002+j*1.77132e-004; - -8.77132e-002+j*3.34081e-002, +2.14182e-001+j*2.14655e-002, -1.54521e-002+j*1.26682e-002; - -7.90143e-002+j*2.42583e-002, +2.20669e-001+j*2.12738e-002, +4.60755e-002+j*4.96406e-003; - -7.79654e-002+j*2.58385e-002, +2.06861e-001+j*3.48019e-002, -1.78311e-002-j*1.29704e-002; - -8.49357e-002+j*3.55200e-002, +2.07470e-001+j*3.59745e-002, -7.66974e-002-j*3.19813e-003; - -7.38565e-002+j*1.95146e-002, +2.17403e-001+j*2.01550e-002, -1.77073e-002-j*3.46414e-003; - -7.77587e-002+j*2.36700e-002, +2.35654e-001-j*2.14540e-002, +7.94165e-002-j*2.45897e-002; - -8.17972e-002+j*2.20583e-002, +2.20906e-001-j*4.30164e-003, -5.60520e-003+j*3.10187e-003; - -8.64261e-002+j*3.66022e-002, +2.15000e-001-j*5.74661e-003, -1.22622e-001+j*4.11767e-002; - -4.25169e-002+j*1.56602e-002, +5.31036e-002-j*1.73951e-002, -4.07130e-002+j*1.26884e-002; - -3.85032e-002+j*1.29431e-002, +5.36716e-002-j*1.80868e-002, +1.00367e-001-j*3.48798e-002; - -4.25524e-002+j*1.46363e-002, +5.19668e-002-j*1.69744e-002, +5.89747e-003-j*2.32428e-003; - -4.31268e-002+j*1.38332e-002, +5.07545e-002-j*1.53045e-002, -1.04172e-001+j*3.17984e-002; - -2.69757e-002+j*9.07955e-003, +3.07837e-002-j*9.44663e-003, -7.63502e-003+j*1.68203e-003; - -3.00097e-002+j*9.23966e-003, +2.83585e-002-j*8.97747e-003, +1.52467e-001-j*4.78675e-002; - -2.70223e-002+j*6.16478e-003, +3.06149e-002-j*6.25382e-003, -4.84888e-003+j*1.93970e-003; - -2.90976e-002+j*7.13184e-003, +3.36738e-002-j*7.30875e-003, -1.66902e-001+j*3.93419e-002; - -7.91940e-002+j*4.39648e-002, +2.04567e-001+j*9.49987e-003, -1.56087e-002+j*7.08838e-003; - -1.01070e-001+j*3.13534e-002, +1.92270e-001+j*1.80423e-002, +2.93053e-003-j*1.97308e-003; - -8.86455e-002+j*4.29906e-002, +1.90862e-001+j*2.53414e-002, -3.38351e-002+j*1.81362e-003]; -#+end_src - -*** Mode2 -#+begin_src bash :results output :exports none :eval no-export - sed -n 88,156p modal_analysis/modes_propres_narband.asc -#+end_src - -#+RESULTS: -#+begin_example - 1X+: 7.56931e-002 3.61548e-002 - 1Y+: 2.07574e-001 1.69205e-004 - 1Z+: 1.29733e-002 -6.78426e-004 - 2X+: 8.58732e-002 2.54470e-002 - 2Y+: 2.07117e-001 -1.31755e-003 - 2Z+: -2.13788e-003 -1.24974e-002 - 4X+: 7.09825e-002 3.66313e-002 - 4Y+: 2.09969e-001 1.11484e-002 - 4Z+: 9.19478e-003 3.47272e-002 - 5X+: 6.23935e-002 1.02488e-002 - 5Y+: 2.30687e-001 -3.58416e-003 - 5Z+: 3.27122e-002 -5.85468e-002 - 6X+: 7.61163e-002 -2.43630e-002 - 6Y+: 2.26743e-001 -1.15334e-002 - 6Z+: -6.20205e-003 -1.21742e-001 - 7X+: 8.01824e-002 -1.94769e-002 - 7Y+: 1.97485e-001 4.50105e-002 - 7Z+: -2.21170e-002 9.77052e-002 - 8X+: 6.19294e-002 8.15075e-003 - 8Y+: 2.03864e-001 4.45835e-002 - 8Z+: 2.55133e-002 1.36137e-001 - 9X+: 4.38135e-002 7.30537e-002 - 9Y+: 2.28426e-001 -6.58868e-003 - 9Z+: 1.16313e-002 5.09427e-004 - 10X+: 5.45770e-002 4.34251e-002 - 10Y+: 2.50823e-001 0.00000e+000 - 10Z+: -4.63460e-002 -4.76868e-002 - 11X+: 5.50987e-002 4.26178e-002 - 11Y+: 2.29394e-001 5.78236e-002 - 11Z+: 1.90158e-002 1.09139e-002 - 12X+: 4.98867e-002 7.30190e-002 - 12Y+: 2.07871e-001 4.57750e-002 - 12Z+: 6.69433e-002 9.00315e-002 - 13X+: 2.48819e-002 3.03222e-002 - 13Y+: -2.56046e-002 -3.34132e-002 - 13Z+: 2.13260e-002 2.58544e-002 - 14X+: 2.45706e-002 2.60221e-002 - 14Y+: -2.57723e-002 -3.35612e-002 - 14Z+: -5.71282e-002 -6.61562e-002 - 15X+: 2.68196e-002 2.83888e-002 - 15Y+: -2.57263e-002 -3.29627e-002 - 15Z+: -2.11722e-003 -3.37239e-003 - 16X+: 2.51442e-002 3.32558e-002 - 16Y+: -2.54372e-002 -3.25062e-002 - 16Z+: 5.65780e-002 7.64142e-002 - 17X+: 1.62437e-002 1.94534e-002 - 17Y+: -1.31293e-002 -2.05924e-002 - 17Z+: 1.05274e-003 3.59474e-003 - 18X+: 1.83431e-002 2.03836e-002 - 18Y+: -1.16818e-002 -1.86334e-002 - 18Z+: -8.66632e-002 -1.08216e-001 - 19X+: 1.62553e-002 1.79588e-002 - 19Y+: -1.28857e-002 -1.90512e-002 - 19Z+: 6.25653e-003 4.97733e-003 - 20X+: 1.63830e-002 2.03943e-002 - 20Y+: -1.48941e-002 -2.11717e-002 - 20Z+: 8.68045e-002 1.16491e-001 - 3X+: 8.17201e-002 2.36079e-002 - 3Y+: 2.15927e-001 1.61300e-002 - 3Z+: -5.48456e-004 2.55691e-002 - 21X+: 6.79204e-002 -5.55513e-002 - 21Y+: 2.32871e-001 2.33389e-002 - 21Z+: 1.34345e-002 -2.31815e-002 - 22X+: 4.02414e-002 -8.38957e-002 - 22Y+: 2.35273e-001 2.73256e-002 - 22Z+: -8.51632e-003 -7.49635e-003 - 23X+: 6.18293e-002 -5.99671e-002 - 23Z+: 1.63533e-002 6.09161e-002 - 23Y-: -2.37693e-001 -4.34204e-002 -#+end_example - -#+begin_src matlab - mode2 = [... - +7.56931e-002+j*3.61548e-002, +2.07574e-001+j*1.69205e-004, +1.29733e-002-j*6.78426e-004; - +8.58732e-002+j*2.54470e-002, +2.07117e-001-j*1.31755e-003, -2.13788e-003-j*1.24974e-002; - +8.17201e-002+j*2.36079e-002, +2.15927e-001+j*1.61300e-002, -5.48456e-004+j*2.55691e-002; - +7.09825e-002+j*3.66313e-002, +2.09969e-001+j*1.11484e-002, +9.19478e-003+j*3.47272e-002; - +6.23935e-002+j*1.02488e-002, +2.30687e-001-j*3.58416e-003, +3.27122e-002-j*5.85468e-002; - +7.61163e-002-j*2.43630e-002, +2.26743e-001-j*1.15334e-002, -6.20205e-003-j*1.21742e-001; - +8.01824e-002-j*1.94769e-002, +1.97485e-001+j*4.50105e-002, -2.21170e-002+j*9.77052e-002; - +6.19294e-002+j*8.15075e-003, +2.03864e-001+j*4.45835e-002, +2.55133e-002+j*1.36137e-001; - +4.38135e-002+j*7.30537e-002, +2.28426e-001-j*6.58868e-003, +1.16313e-002+j*5.09427e-004; - +5.45770e-002+j*4.34251e-002, +2.50823e-001+j*0.00000e+000, -4.63460e-002-j*4.76868e-002; - +5.50987e-002+j*4.26178e-002, +2.29394e-001+j*5.78236e-002, +1.90158e-002+j*1.09139e-002; - +4.98867e-002+j*7.30190e-002, +2.07871e-001+j*4.57750e-002, +6.69433e-002+j*9.00315e-002; - +2.48819e-002+j*3.03222e-002, -2.56046e-002-j*3.34132e-002, +2.13260e-002+j*2.58544e-002; - +2.45706e-002+j*2.60221e-002, -2.57723e-002-j*3.35612e-002, -5.71282e-002-j*6.61562e-002; - +2.68196e-002+j*2.83888e-002, -2.57263e-002-j*3.29627e-002, -2.11722e-003-j*3.37239e-003; - +2.51442e-002+j*3.32558e-002, -2.54372e-002-j*3.25062e-002, +5.65780e-002+j*7.64142e-002; - +1.62437e-002+j*1.94534e-002, -1.31293e-002-j*2.05924e-002, +1.05274e-003+j*3.59474e-003; - +1.83431e-002+j*2.03836e-002, -1.16818e-002-j*1.86334e-002, -8.66632e-002-j*1.08216e-001; - +1.62553e-002+j*1.79588e-002, -1.28857e-002-j*1.90512e-002, +6.25653e-003+j*4.97733e-003; - +1.63830e-002+j*2.03943e-002, -1.48941e-002-j*2.11717e-002, +8.68045e-002+j*1.16491e-001; - +6.79204e-002-j*5.55513e-002, +2.32871e-001+j*2.33389e-002, +1.34345e-002-j*2.31815e-002; - +4.02414e-002-j*8.38957e-002, +2.35273e-001+j*2.73256e-002, -8.51632e-003-j*7.49635e-003; - +6.18293e-002-j*5.99671e-002, +2.37693e-001+j*4.34204e-002, +1.63533e-002+j*6.09161e-002] -#+end_src - -*** Mode3 -#+begin_src bash :results output :exports none :eval no-export - sed -n 164,232p modal_analysis/modes_propres_narband.asc -#+end_src - -#+RESULTS: -#+begin_example - 1X+: 1.34688e-001 -6.65071e-002 - 1Y+: 1.55316e-002 1.01277e-002 - 1Z+: -5.88466e-002 1.14294e-002 - 2X+: 1.53934e-001 -9.76990e-003 - 2Y+: 7.17487e-003 1.11925e-002 - 2Z+: -4.57205e-002 7.26573e-003 - 4X+: 1.37298e-001 -5.24661e-002 - 4Y+: 1.19427e-003 -5.39240e-002 - 4Z+: -1.25915e-002 5.38133e-003 - 5X+: 2.43192e-001 -3.17374e-002 - 5Y+: -2.15730e-001 -7.69941e-004 - 5Z+: -1.56268e-001 1.44118e-002 - 6X+: -7.27705e-002 -3.54943e-003 - 6Y+: -2.47706e-001 2.66480e-003 - 6Z+: -1.21590e-001 1.06054e-002 - 7X+: -7.25870e-002 -4.62024e-003 - 7Y+: 2.27073e-001 -3.69315e-002 - 7Z+: 1.22611e-001 -6.67337e-003 - 8X+: 2.32731e-001 -2.85516e-002 - 8Y+: 2.35389e-001 -3.81905e-002 - 8Z+: 5.35574e-002 4.30394e-004 - 9X+: 2.64170e-001 -2.67367e-002 - 9Y+: -2.56227e-001 3.97957e-005 - 9Z+: -1.95398e-001 2.23549e-002 - 10X+: -1.66953e-002 -7.95698e-003 - 10Y+: -2.66547e-001 -2.17687e-002 - 10Z+: 1.56278e-002 2.23786e-003 - 11X+: -3.42364e-002 -9.30205e-003 - 11Y+: 2.52340e-001 -7.47237e-003 - 11Z+: -9.51643e-004 3.64798e-003 - 12X+: 2.97574e-001 0.00000e+000 - 12Y+: 2.23170e-001 -1.37831e-002 - 12Z+: 1.06266e-001 2.30324e-003 - 13X+: 2.67178e-002 -4.15723e-004 - 13Y+: 6.75423e-003 -2.18428e-003 - 13Z+: -1.69423e-002 3.12395e-003 - 14X+: -1.12283e-002 2.86316e-004 - 14Y+: 5.08225e-003 -2.14053e-003 - 14Z+: 2.18339e-002 -3.25204e-003 - 15X+: -1.17948e-002 6.82873e-004 - 15Y+: 1.94914e-002 -2.42151e-003 - 15Z+: 2.68660e-003 -2.92104e-004 - 16X+: 1.19490e-002 1.72236e-005 - 16Y+: 1.83552e-002 -2.71289e-003 - 16Z+: -2.70914e-002 4.84164e-003 - 17X+: 1.00173e-002 -5.80552e-005 - 17Y+: -3.87262e-003 -1.19607e-003 - 17Z+: -8.53809e-003 1.48424e-003 - 18X+: -1.22262e-002 5.13096e-004 - 18Y+: -5.73905e-003 -1.07659e-003 - 18Z+: 3.51730e-002 -6.13814e-003 - 19X+: -1.43735e-002 -4.78552e-004 - 19Y+: 2.31135e-002 -6.30554e-004 - 19Z+: 1.80171e-003 -1.98835e-004 - 20X+: 9.17792e-003 5.36661e-004 - 20Y+: 2.18969e-002 -5.81759e-004 - 20Z+: -3.72117e-002 5.35813e-003 - 3X+: 1.61551e-001 1.65478e-002 - 3Y+: -4.12527e-004 -5.60909e-002 - 3Z+: -9.00640e-003 3.50754e-003 - 21X+: 3.38754e-002 -3.38703e-002 - 21Y+: -2.20843e-002 2.78581e-002 - 21Z+: -8.79541e-002 -3.67473e-003 - 22X+: 3.93064e-002 4.69476e-002 - 22Y+: -1.69132e-002 -1.04606e-002 - 22Z+: -1.85351e-002 1.33750e-003 - 23X+: 3.60396e-002 -2.46238e-002 - 23Z+: 3.57722e-003 3.64827e-003 - 23Y-: 1.92038e-002 6.65895e-002 -#+end_example - -#+begin_src matlab - mode3 = [... - +1.34688e-001-j*6.65071e-002, +1.55316e-002+j*1.01277e-002, -5.88466e-002+j*1.14294e-002; - +1.53934e-001-j*9.76990e-003, +7.17487e-003+j*1.11925e-002, -4.57205e-002+j*7.26573e-003; - +1.61551e-001+j*1.65478e-002, -4.12527e-004-j*5.60909e-002, -9.00640e-003+j*3.50754e-003; - +1.37298e-001-j*5.24661e-002, +1.19427e-003-j*5.39240e-002, -1.25915e-002+j*5.38133e-003; - +2.43192e-001-j*3.17374e-002, -2.15730e-001-j*7.69941e-004, -1.56268e-001+j*1.44118e-002; - -7.27705e-002-j*3.54943e-003, -2.47706e-001+j*2.66480e-003, -1.21590e-001+j*1.06054e-002; - -7.25870e-002-j*4.62024e-003, +2.27073e-001-j*3.69315e-002, +1.22611e-001-j*6.67337e-003; - +2.32731e-001-j*2.85516e-002, +2.35389e-001-j*3.81905e-002, +5.35574e-002+j*4.30394e-004; - +2.64170e-001-j*2.67367e-002, -2.56227e-001+j*3.97957e-005, -1.95398e-001+j*2.23549e-002; - -1.66953e-002-j*7.95698e-003, -2.66547e-001-j*2.17687e-002, +1.56278e-002+j*2.23786e-003; - -3.42364e-002-j*9.30205e-003, +2.52340e-001-j*7.47237e-003, -9.51643e-004+j*3.64798e-003; - +2.97574e-001+j*0.00000e+000, +2.23170e-001-j*1.37831e-002, +1.06266e-001+j*2.30324e-003; - +2.67178e-002-j*4.15723e-004, +6.75423e-003-j*2.18428e-003, -1.69423e-002+j*3.12395e-003; - -1.12283e-002+j*2.86316e-004, +5.08225e-003-j*2.14053e-003, +2.18339e-002-j*3.25204e-003; - -1.17948e-002+j*6.82873e-004, +1.94914e-002-j*2.42151e-003, +2.68660e-003-j*2.92104e-004; - +1.19490e-002+j*1.72236e-005, +1.83552e-002-j*2.71289e-003, -2.70914e-002+j*4.84164e-003; - +1.00173e-002-j*5.80552e-005, -3.87262e-003-j*1.19607e-003, -8.53809e-003+j*1.48424e-003; - -1.22262e-002+j*5.13096e-004, -5.73905e-003-j*1.07659e-003, +3.51730e-002-j*6.13814e-003; - -1.43735e-002-j*4.78552e-004, +2.31135e-002-j*6.30554e-004, +1.80171e-003-j*1.98835e-004; - +9.17792e-003+j*5.36661e-004, +2.18969e-002-j*5.81759e-004, -3.72117e-002+j*5.35813e-003; - +3.38754e-002-j*3.38703e-002, -2.20843e-002+j*2.78581e-002, -8.79541e-002-j*3.67473e-003; - +3.93064e-002+j*4.69476e-002, -1.69132e-002-j*1.04606e-002, -1.85351e-002+j*1.33750e-003; - +3.60396e-002-j*2.46238e-002, -1.92038e-002-j*6.65895e-002, +3.57722e-003+j*3.64827e-003]; -#+end_src - -*** Mode4 -#+begin_src bash :results output :exports none :eval no-export - sed -n 240,308p modal_analysis/modes_propres_narband.asc -#+end_src - -#+RESULTS: -#+begin_example - 1X+: -1.02501e-001 -1.43802e-001 - 1Y+: -1.07971e-001 5.61418e-004 - 1Z+: 1.87145e-001 -1.03605e-001 - 2X+: -9.44764e-002 -1.36856e-001 - 2Y+: -1.04428e-001 5.27790e-003 - 2Z+: 1.60710e-001 -7.74212e-002 - 4X+: -9.17242e-002 -1.36656e-001 - 4Y+: -1.34249e-001 -1.03884e-002 - 4Z+: 1.92123e-001 -1.25627e-001 - 5X+: 1.05875e-002 -1.03886e-001 - 5Y+: -8.26338e-002 3.58498e-002 - 5Z+: 2.55819e-001 -6.94290e-003 - 6X+: -4.58970e-002 -1.33904e-002 - 6Y+: -9.41660e-002 4.99682e-002 - 6Z+: 1.28276e-001 4.59685e-002 - 7X+: -6.01521e-002 -1.30165e-002 - 7Y+: 2.56439e-003 -6.78141e-002 - 7Z+: 5.03428e-002 -1.59420e-001 - 8X+: -1.00895e-002 -8.80550e-002 - 8Y+: 1.26327e-002 -8.14444e-002 - 8Z+: 1.59506e-001 -2.05360e-001 - 9X+: -3.04658e-003 -1.57921e-001 - 9Y+: -8.23501e-002 4.82748e-002 - 9Z+: 1.69315e-001 1.22804e-002 - 10X+: -8.25875e-002 -7.31038e-002 - 10Y+: -1.08668e-001 3.56364e-002 - 10Z+: 8.28567e-002 -4.49596e-003 - 11X+: -1.06792e-001 -6.95394e-002 - 11Y+: 3.77195e-002 -7.65410e-002 - 11Z+: 8.00590e-003 -2.32461e-002 - 12X+: -4.84292e-002 -1.45790e-001 - 12Y+: 1.03862e-002 -7.31212e-002 - 12Z+: 1.78122e-001 -1.00939e-001 - 13X+: -3.49891e-002 -6.20969e-003 - 13Y+: -1.18504e-002 -1.94225e-002 - 13Z+: 4.13007e-002 7.67087e-003 - 14X+: -3.55795e-002 1.16708e-003 - 14Y+: -1.68128e-002 -1.82344e-002 - 14Z+: 3.92416e-002 -3.64434e-002 - 15X+: -3.45304e-002 3.78185e-003 - 15Y+: -7.62559e-003 -2.24241e-002 - 15Z+: 6.28286e-003 -1.32711e-002 - 16X+: -9.95646e-003 -6.04395e-003 - 16Y+: -8.73465e-003 -2.20807e-002 - 16Z+: 3.56946e-002 1.69231e-002 - 17X+: -9.32661e-003 -5.51944e-003 - 17Y+: -1.91087e-002 -9.09191e-003 - 17Z+: 4.04981e-002 8.38685e-004 - 18X+: -2.84456e-002 4.02762e-003 - 18Y+: -2.20044e-002 -8.86197e-003 - 18Z+: 4.43051e-002 -5.21033e-002 - 19X+: -3.27019e-002 3.59765e-003 - 19Y+: 2.93163e-003 -2.05064e-002 - 19Z+: -1.77289e-002 -1.29477e-002 - 20X+: -1.08474e-002 -5.78419e-003 - 20Y+: 3.86759e-003 -1.91642e-002 - 20Z+: 2.10135e-002 3.18051e-002 - 3X+: -9.11657e-002 -1.36611e-001 - 3Y+: -1.78165e-001 -3.47193e-002 - 3Z+: 2.37121e-001 -4.96494e-002 - 21X+: -1.34808e-002 -9.69121e-003 - 21Y+: 1.25218e-002 -2.71411e-002 - 21Z+: 2.76673e-001 0.00000e+000 - 22X+: 1.96744e-003 4.90797e-003 - 22Y+: -9.82609e-004 -3.31065e-002 - 22Z+: 1.79246e-001 -3.33238e-002 - 23X+: -1.08728e-002 -8.80278e-003 - 23Z+: 2.30814e-001 -8.33151e-002 - 23Y-: 1.15217e-002 4.01143e-002 -#+end_example - -#+begin_src matlab - mode4 = [... - -1.02501e-001-j*1.43802e-001, -1.07971e-001+j*5.61418e-004, +1.87145e-001-j*1.03605e-001; - -9.44764e-002-j*1.36856e-001, -1.04428e-001+j*5.27790e-003, +1.60710e-001-j*7.74212e-002; - -9.11657e-002-j*1.36611e-001, -1.78165e-001-j*3.47193e-002, +2.37121e-001-j*4.96494e-002; - -9.17242e-002-j*1.36656e-001, -1.34249e-001-j*1.03884e-002, +1.92123e-001-j*1.25627e-001; - +1.05875e-002-j*1.03886e-001, -8.26338e-002+j*3.58498e-002, +2.55819e-001-j*6.94290e-003; - -4.58970e-002-j*1.33904e-002, -9.41660e-002+j*4.99682e-002, +1.28276e-001+j*4.59685e-002; - -6.01521e-002-j*1.30165e-002, +2.56439e-003-j*6.78141e-002, +5.03428e-002-j*1.59420e-001; - -1.00895e-002-j*8.80550e-002, +1.26327e-002-j*8.14444e-002, +1.59506e-001-j*2.05360e-001; - -3.04658e-003-j*1.57921e-001, -8.23501e-002+j*4.82748e-002, +1.69315e-001+j*1.22804e-002; - -8.25875e-002-j*7.31038e-002, -1.08668e-001+j*3.56364e-002, +8.28567e-002-j*4.49596e-003; - -1.06792e-001-j*6.95394e-002, +3.77195e-002-j*7.65410e-002, +8.00590e-003-j*2.32461e-002; - -4.84292e-002-j*1.45790e-001, +1.03862e-002-j*7.31212e-002, +1.78122e-001-j*1.00939e-001; - -3.49891e-002-j*6.20969e-003, -1.18504e-002-j*1.94225e-002, +4.13007e-002+j*7.67087e-003; - -3.55795e-002+j*1.16708e-003, -1.68128e-002-j*1.82344e-002, +3.92416e-002-j*3.64434e-002; - -3.45304e-002+j*3.78185e-003, -7.62559e-003-j*2.24241e-002, +6.28286e-003-j*1.32711e-002; - -9.95646e-003-j*6.04395e-003, -8.73465e-003-j*2.20807e-002, +3.56946e-002+j*1.69231e-002; - -9.32661e-003-j*5.51944e-003, -1.91087e-002-j*9.09191e-003, +4.04981e-002+j*8.38685e-004; - -2.84456e-002+j*4.02762e-003, -2.20044e-002-j*8.86197e-003, +4.43051e-002-j*5.21033e-002; - -3.27019e-002+j*3.59765e-003, +2.93163e-003-j*2.05064e-002, -1.77289e-002-j*1.29477e-002; - -1.08474e-002-j*5.78419e-003, +3.86759e-003-j*1.91642e-002, +2.10135e-002+j*3.18051e-002; - -1.34808e-002-j*9.69121e-003, +1.25218e-002-j*2.71411e-002, +2.76673e-001+j*0.00000e+000; - +1.96744e-003+j*4.90797e-003, -9.82609e-004-j*3.31065e-002, +1.79246e-001-j*3.33238e-002; - -1.08728e-002-j*8.80278e-003, -1.15217e-002-j*4.01143e-002, +2.30814e-001-j*8.33151e-002]; -#+end_src - -*** All modes -#+begin_src matlab -mode_shapes = zeros(23, 3, 10); - -mode_shapes(:, :, 1) = [... --9.34637e-002+j*4.52445e-002, +2.33790e-001+j*1.41439e-003, -1.73754e-002+j*6.02449e-003; --7.42108e-002+j*3.91543e-002, +2.41566e-001-j*1.44869e-003, -5.99285e-003+j*2.10370e-003; --9.40720e-002+j*3.93724e-002, +2.52307e-001+j*0.00000e+000, -1.53864e-002-j*9.25720e-004; --1.02163e-001+j*2.79561e-002, +2.29048e-001+j*2.89782e-002, -2.85130e-002+j*1.77132e-004; --8.77132e-002+j*3.34081e-002, +2.14182e-001+j*2.14655e-002, -1.54521e-002+j*1.26682e-002; --7.90143e-002+j*2.42583e-002, +2.20669e-001+j*2.12738e-002, +4.60755e-002+j*4.96406e-003; --7.79654e-002+j*2.58385e-002, +2.06861e-001+j*3.48019e-002, -1.78311e-002-j*1.29704e-002; --8.49357e-002+j*3.55200e-002, +2.07470e-001+j*3.59745e-002, -7.66974e-002-j*3.19813e-003; --7.38565e-002+j*1.95146e-002, +2.17403e-001+j*2.01550e-002, -1.77073e-002-j*3.46414e-003; --7.77587e-002+j*2.36700e-002, +2.35654e-001-j*2.14540e-002, +7.94165e-002-j*2.45897e-002; --8.17972e-002+j*2.20583e-002, +2.20906e-001-j*4.30164e-003, -5.60520e-003+j*3.10187e-003; --8.64261e-002+j*3.66022e-002, +2.15000e-001-j*5.74661e-003, -1.22622e-001+j*4.11767e-002; --4.25169e-002+j*1.56602e-002, +5.31036e-002-j*1.73951e-002, -4.07130e-002+j*1.26884e-002; --3.85032e-002+j*1.29431e-002, +5.36716e-002-j*1.80868e-002, +1.00367e-001-j*3.48798e-002; --4.25524e-002+j*1.46363e-002, +5.19668e-002-j*1.69744e-002, +5.89747e-003-j*2.32428e-003; --4.31268e-002+j*1.38332e-002, +5.07545e-002-j*1.53045e-002, -1.04172e-001+j*3.17984e-002; --2.69757e-002+j*9.07955e-003, +3.07837e-002-j*9.44663e-003, -7.63502e-003+j*1.68203e-003; --3.00097e-002+j*9.23966e-003, +2.83585e-002-j*8.97747e-003, +1.52467e-001-j*4.78675e-002; --2.70223e-002+j*6.16478e-003, +3.06149e-002-j*6.25382e-003, -4.84888e-003+j*1.93970e-003; --2.90976e-002+j*7.13184e-003, +3.36738e-002-j*7.30875e-003, -1.66902e-001+j*3.93419e-002; --7.91940e-002+j*4.39648e-002, +2.04567e-001+j*9.49987e-003, -1.56087e-002+j*7.08838e-003; --1.01070e-001+j*3.13534e-002, +1.92270e-001+j*1.80423e-002, +2.93053e-003-j*1.97308e-003; --8.86455e-002+j*4.29906e-002, -3.38351e-002+j*1.81362e-003, +1.90862e-001+j*2.53414e-002]; - -mode_shapes(:, :, 2) = [... -+7.56931e-002+j*3.61548e-002, +2.07574e-001+j*1.69205e-004, +1.29733e-002-j*6.78426e-004; -+8.58732e-002+j*2.54470e-002, +2.07117e-001-j*1.31755e-003, -2.13788e-003-j*1.24974e-002; -+8.17201e-002+j*2.36079e-002, +2.15927e-001+j*1.61300e-002, -5.48456e-004+j*2.55691e-002; -+7.09825e-002+j*3.66313e-002, +2.09969e-001+j*1.11484e-002, +9.19478e-003+j*3.47272e-002; -+6.23935e-002+j*1.02488e-002, +2.30687e-001-j*3.58416e-003, +3.27122e-002-j*5.85468e-002; -+7.61163e-002-j*2.43630e-002, +2.26743e-001-j*1.15334e-002, -6.20205e-003-j*1.21742e-001; -+8.01824e-002-j*1.94769e-002, +1.97485e-001+j*4.50105e-002, -2.21170e-002+j*9.77052e-002; -+6.19294e-002+j*8.15075e-003, +2.03864e-001+j*4.45835e-002, +2.55133e-002+j*1.36137e-001; -+4.38135e-002+j*7.30537e-002, +2.28426e-001-j*6.58868e-003, +1.16313e-002+j*5.09427e-004; -+5.45770e-002+j*4.34251e-002, +2.50823e-001+j*0.00000e+000, -4.63460e-002-j*4.76868e-002; -+5.50987e-002+j*4.26178e-002, +2.29394e-001+j*5.78236e-002, +1.90158e-002+j*1.09139e-002; -+4.98867e-002+j*7.30190e-002, +2.07871e-001+j*4.57750e-002, +6.69433e-002+j*9.00315e-002; -+2.48819e-002+j*3.03222e-002, -2.56046e-002-j*3.34132e-002, +2.13260e-002+j*2.58544e-002; -+2.45706e-002+j*2.60221e-002, -2.57723e-002-j*3.35612e-002, -5.71282e-002-j*6.61562e-002; -+2.68196e-002+j*2.83888e-002, -2.57263e-002-j*3.29627e-002, -2.11722e-003-j*3.37239e-003; -+2.51442e-002+j*3.32558e-002, -2.54372e-002-j*3.25062e-002, +5.65780e-002+j*7.64142e-002; -+1.62437e-002+j*1.94534e-002, -1.31293e-002-j*2.05924e-002, +1.05274e-003+j*3.59474e-003; -+1.83431e-002+j*2.03836e-002, -1.16818e-002-j*1.86334e-002, -8.66632e-002-j*1.08216e-001; -+1.62553e-002+j*1.79588e-002, -1.28857e-002-j*1.90512e-002, +6.25653e-003+j*4.97733e-003; -+1.63830e-002+j*2.03943e-002, -1.48941e-002-j*2.11717e-002, +8.68045e-002+j*1.16491e-001; -+6.79204e-002-j*5.55513e-002, +2.32871e-001+j*2.33389e-002, +1.34345e-002-j*2.31815e-002; -+4.02414e-002-j*8.38957e-002, +2.35273e-001+j*2.73256e-002, -8.51632e-003-j*7.49635e-003; -+6.18293e-002-j*5.99671e-002, +1.63533e-002+j*6.09161e-002, +2.37693e-001+j*4.34204e-002]; - -mode_shapes(:, :, 3) = [... -+1.34688e-001-j*6.65071e-002, +1.55316e-002+j*1.01277e-002, -5.88466e-002+j*1.14294e-002; -+1.53934e-001-j*9.76990e-003, +7.17487e-003+j*1.11925e-002, -4.57205e-002+j*7.26573e-003; -+1.61551e-001+j*1.65478e-002, -4.12527e-004-j*5.60909e-002, -9.00640e-003+j*3.50754e-003; -+1.37298e-001-j*5.24661e-002, +1.19427e-003-j*5.39240e-002, -1.25915e-002+j*5.38133e-003; -+2.43192e-001-j*3.17374e-002, -2.15730e-001-j*7.69941e-004, -1.56268e-001+j*1.44118e-002; --7.27705e-002-j*3.54943e-003, -2.47706e-001+j*2.66480e-003, -1.21590e-001+j*1.06054e-002; --7.25870e-002-j*4.62024e-003, +2.27073e-001-j*3.69315e-002, +1.22611e-001-j*6.67337e-003; -+2.32731e-001-j*2.85516e-002, +2.35389e-001-j*3.81905e-002, +5.35574e-002+j*4.30394e-004; -+2.64170e-001-j*2.67367e-002, -2.56227e-001+j*3.97957e-005, -1.95398e-001+j*2.23549e-002; --1.66953e-002-j*7.95698e-003, -2.66547e-001-j*2.17687e-002, +1.56278e-002+j*2.23786e-003; --3.42364e-002-j*9.30205e-003, +2.52340e-001-j*7.47237e-003, -9.51643e-004+j*3.64798e-003; -+2.97574e-001+j*0.00000e+000, +2.23170e-001-j*1.37831e-002, +1.06266e-001+j*2.30324e-003; -+2.67178e-002-j*4.15723e-004, +6.75423e-003-j*2.18428e-003, -1.69423e-002+j*3.12395e-003; --1.12283e-002+j*2.86316e-004, +5.08225e-003-j*2.14053e-003, +2.18339e-002-j*3.25204e-003; --1.17948e-002+j*6.82873e-004, +1.94914e-002-j*2.42151e-003, +2.68660e-003-j*2.92104e-004; -+1.19490e-002+j*1.72236e-005, +1.83552e-002-j*2.71289e-003, -2.70914e-002+j*4.84164e-003; -+1.00173e-002-j*5.80552e-005, -3.87262e-003-j*1.19607e-003, -8.53809e-003+j*1.48424e-003; --1.22262e-002+j*5.13096e-004, -5.73905e-003-j*1.07659e-003, +3.51730e-002-j*6.13814e-003; --1.43735e-002-j*4.78552e-004, +2.31135e-002-j*6.30554e-004, +1.80171e-003-j*1.98835e-004; -+9.17792e-003+j*5.36661e-004, +2.18969e-002-j*5.81759e-004, -3.72117e-002+j*5.35813e-003; -+3.38754e-002-j*3.38703e-002, -2.20843e-002+j*2.78581e-002, -8.79541e-002-j*3.67473e-003; -+3.93064e-002+j*4.69476e-002, -1.69132e-002-j*1.04606e-002, -1.85351e-002+j*1.33750e-003; -+3.60396e-002-j*2.46238e-002, +3.57722e-003+j*3.64827e-003, -1.92038e-002-j*6.65895e-002]; - -mode_shapes(:, :, 4) = [... --1.02501e-001-j*1.43802e-001, -1.07971e-001+j*5.61418e-004, +1.87145e-001-j*1.03605e-001; --9.44764e-002-j*1.36856e-001, -1.04428e-001+j*5.27790e-003, +1.60710e-001-j*7.74212e-002; --9.11657e-002-j*1.36611e-001, -1.78165e-001-j*3.47193e-002, +2.37121e-001-j*4.96494e-002; --9.17242e-002-j*1.36656e-001, -1.34249e-001-j*1.03884e-002, +1.92123e-001-j*1.25627e-001; -+1.05875e-002-j*1.03886e-001, -8.26338e-002+j*3.58498e-002, +2.55819e-001-j*6.94290e-003; --4.58970e-002-j*1.33904e-002, -9.41660e-002+j*4.99682e-002, +1.28276e-001+j*4.59685e-002; --6.01521e-002-j*1.30165e-002, +2.56439e-003-j*6.78141e-002, +5.03428e-002-j*1.59420e-001; --1.00895e-002-j*8.80550e-002, +1.26327e-002-j*8.14444e-002, +1.59506e-001-j*2.05360e-001; --3.04658e-003-j*1.57921e-001, -8.23501e-002+j*4.82748e-002, +1.69315e-001+j*1.22804e-002; --8.25875e-002-j*7.31038e-002, -1.08668e-001+j*3.56364e-002, +8.28567e-002-j*4.49596e-003; --1.06792e-001-j*6.95394e-002, +3.77195e-002-j*7.65410e-002, +8.00590e-003-j*2.32461e-002; --4.84292e-002-j*1.45790e-001, +1.03862e-002-j*7.31212e-002, +1.78122e-001-j*1.00939e-001; --3.49891e-002-j*6.20969e-003, -1.18504e-002-j*1.94225e-002, +4.13007e-002+j*7.67087e-003; --3.55795e-002+j*1.16708e-003, -1.68128e-002-j*1.82344e-002, +3.92416e-002-j*3.64434e-002; --3.45304e-002+j*3.78185e-003, -7.62559e-003-j*2.24241e-002, +6.28286e-003-j*1.32711e-002; --9.95646e-003-j*6.04395e-003, -8.73465e-003-j*2.20807e-002, +3.56946e-002+j*1.69231e-002; --9.32661e-003-j*5.51944e-003, -1.91087e-002-j*9.09191e-003, +4.04981e-002+j*8.38685e-004; --2.84456e-002+j*4.02762e-003, -2.20044e-002-j*8.86197e-003, +4.43051e-002-j*5.21033e-002; --3.27019e-002+j*3.59765e-003, +2.93163e-003-j*2.05064e-002, -1.77289e-002-j*1.29477e-002; --1.08474e-002-j*5.78419e-003, +3.86759e-003-j*1.91642e-002, +2.10135e-002+j*3.18051e-002; --1.34808e-002-j*9.69121e-003, +1.25218e-002-j*2.71411e-002, +2.76673e-001+j*0.00000e+000; -+1.96744e-003+j*4.90797e-003, -9.82609e-004-j*3.31065e-002, +1.79246e-001-j*3.33238e-002; --1.08728e-002-j*8.80278e-003, +2.30814e-001-j*8.33151e-002, -1.15217e-002-j*4.01143e-002]; - -mode_shapes(:, :, 5) = [... -+3.55328e-001+j*0.00000e+000, +6.67612e-002+j*5.48020e-002, +3.03237e-002+j*5.29473e-004; -+3.16372e-001-j*3.84091e-002, +5.27472e-002+j*5.88474e-002, +2.86305e-002-j*1.74805e-002; -+3.00803e-001-j*1.36309e-002, +7.04883e-002+j*1.24492e-001, +7.23329e-002+j*2.33738e-002; -+3.32527e-001-j*2.26876e-004, +9.82263e-002+j*1.20397e-001, +9.86580e-002+j*3.55048e-002; -+4.96498e-002+j*2.31008e-002, +9.79716e-002+j*1.42500e-002, -1.15121e-001-j*3.59085e-002; -+1.41924e-001+j*2.16209e-002, +8.76030e-002+j*6.39650e-003, -8.75727e-002-j*3.71261e-002; -+1.41522e-001+j*1.96964e-002, -1.01959e-001+j*4.10992e-004, +2.14744e-001+j*4.91249e-002; -+4.33170e-002+j*1.84481e-002, -8.24640e-002+j*3.42475e-003, +2.32281e-001+j*5.40699e-002; -+1.47782e-001+j*4.93091e-002, +8.75397e-002+j*7.75318e-004, -6.80833e-002-j*9.72902e-003; -+2.00055e-001+j*3.81689e-002, +8.06886e-002+j*1.19008e-002, -1.40810e-002-j*1.12625e-002; -+1.96526e-001+j*3.87737e-002, -8.42766e-002+j*9.20233e-003, +1.02951e-001+j*3.37680e-002; -+1.25035e-001+j*4.67796e-002, -8.81307e-002+j*5.81039e-004, +7.94320e-002+j*2.19736e-002; -+2.03946e-002+j*2.50162e-002, +7.93788e-002-j*1.40794e-002, -4.15470e-002+j*4.95855e-004; -+6.56876e-002-j*2.14826e-002, +8.21523e-002-j*1.94792e-002, +3.44089e-002+j*2.32727e-003; -+5.98960e-002-j*2.17160e-002, +4.74914e-002+j*2.31386e-002, +3.58704e-002+j*1.13591e-003; -+1.91580e-002+j*2.15329e-002, +5.14631e-002+j*1.70019e-002, -3.90820e-002-j*9.20853e-003; -+8.98876e-003+j*2.56390e-002, +7.93497e-002-j*2.34846e-002, -5.61039e-002-j*3.03271e-003; -+5.72051e-002-j*2.29477e-002, +8.76985e-002-j*2.73606e-002, +5.13896e-002+j*8.58341e-003; -+6.71099e-002-j*2.67997e-002, +1.80119e-002+j*4.02601e-002, +6.40092e-002+j*7.11273e-003; -+1.48349e-002+j*2.57533e-002, +1.35189e-002+j*3.69351e-002, -3.31024e-002-j*1.65471e-002; -+2.52156e-002-j*4.65138e-003, +7.94380e-004+j*1.56790e-002, +1.01067e-002-j*3.35468e-003; -+3.30841e-002+j*6.73531e-003, +1.69827e-004+j*9.48996e-003, +8.16746e-002+j*1.39885e-002; -+2.91404e-002-j*2.43138e-003, +1.38761e-001+j*3.67410e-002, -4.17833e-003+j*8.35629e-004]; - -mode_shapes(:, :, 6) = [... -+3.76055e-001-j*2.00477e-002, -2.18528e-002-j*2.46738e-001, -2.01369e-002-j*2.48481e-002; -+3.73870e-001-j*6.46353e-002, -3.66940e-002-j*2.49113e-001, -9.20410e-002-j*2.72520e-002; -+3.71070e-001-j*4.42411e-003, +5.79596e-002-j*2.58311e-001, +2.62275e-003+j*9.26708e-003; -+3.82440e-001+j*0.00000e+000, +3.30266e-002-j*2.08256e-001, +7.22558e-002+j*1.47204e-002; --3.73785e-003+j*3.81592e-002, +4.74482e-002+j*5.41354e-002, -5.78227e-002-j*8.70215e-002; -+6.08721e-002+j*7.20835e-002, +5.73258e-002+j*5.38176e-002, -3.34121e-002-j*8.54509e-002; -+5.96968e-002+j*8.31733e-002, -7.64618e-002-j*4.19504e-002, +3.48953e-002+j*4.79635e-002; --1.41532e-003+j*3.79318e-002, -6.79853e-002-j*3.88324e-002, +6.64542e-002+j*9.20955e-002; -+2.96044e-002+j*7.83613e-002, +4.91378e-002+j*4.74080e-002, -2.15758e-002-j*5.51292e-002; -+3.52566e-002+j*1.00976e-001, +4.16302e-002+j*5.95793e-002, -1.83944e-002-j*5.09313e-002; -+3.49287e-002+j*1.10404e-001, -5.46487e-002-j*3.95054e-002, +1.88545e-002+j*3.82156e-002; -+1.70173e-002+j*8.68762e-002, -5.52511e-002-j*4.46179e-002, +7.63555e-003+j*6.17646e-002; --8.22469e-004-j*1.81402e-002, -1.96193e-003-j*2.76646e-002, -1.97033e-003+j*1.06615e-002; --1.16438e-003-j*2.68725e-002, -1.11909e-003-j*2.38894e-002, +1.50332e-003+j*1.31644e-002; --1.77060e-003-j*2.77807e-002, -1.01121e-003-j*2.37147e-002, -2.45798e-003-j*1.41886e-002; --1.85500e-003-j*2.20304e-002, -4.77462e-004-j*2.19932e-002, -6.85097e-003-j*7.04903e-003; --6.77197e-004-j*1.56812e-002, -2.19412e-003-j*2.65284e-002, +3.96653e-004+j*2.95178e-002; --1.14513e-003-j*1.99551e-002, -2.35828e-003-j*2.88909e-002, +3.70211e-003+j*2.02423e-002; --1.16798e-003-j*2.31638e-002, -4.75969e-004-j*2.23413e-002, -2.07031e-003-j*3.41651e-002; --1.99807e-004-j*2.09301e-002, -2.40560e-004-j*1.78264e-002, -9.91090e-003-j*3.12664e-002; -+9.69966e-003+j*2.95222e-002, -1.19231e-002+j*5.23077e-003, -9.79113e-003-j*4.50715e-002; -+1.36018e-002+j*2.24850e-002, -9.79427e-003+j*9.84679e-003, +2.18456e-002-j*1.57858e-002; -+1.09214e-002+j*2.93115e-002, +2.43859e-002+j*3.35745e-002, -1.07735e-002+j*1.21178e-002]; - -mode_shapes(:, :, 7) = [... -+4.18739e-001-j*2.21696e-002, +6.52636e-002-j*1.29498e-001, -1.90337e-002+j*9.61505e-003; -+4.27024e-001-j*3.03566e-002, +4.22791e-002-j*1.32892e-001, -1.65941e-002+j*2.17638e-002; -+4.66694e-001+j*1.38778e-017, +7.51273e-002-j*1.57468e-001, -2.91618e-002+j*4.11516e-002; -+4.18867e-001+j*2.39590e-002, +7.51007e-002-j*1.13378e-001, -7.11238e-003+j*2.72692e-002; --3.26299e-002+j*2.12910e-002, -4.80710e-002+j*8.82799e-002, +5.82558e-002-j*9.16159e-002; --4.61031e-002+j*8.08300e-002, -2.80650e-002+j*7.37925e-002, +3.54903e-002-j*3.02341e-002; --5.14543e-002+j*7.23863e-002, +1.91387e-002-j*7.65710e-002, -3.92182e-002+j*5.06089e-002; --3.35647e-002+j*2.16695e-002, +1.88798e-002-j*6.19822e-002, -5.08508e-002+j*8.78653e-002; --4.93941e-002+j*6.46940e-002, -2.90637e-002+j*7.66360e-002, +5.24626e-002-j*6.16359e-002; --5.85364e-002+j*5.59044e-002, -2.63581e-002+j*5.43466e-002, +2.90174e-002-j*3.98049e-002; --6.07978e-002+j*5.54585e-002, +3.16828e-002-j*8.54810e-002, -3.44914e-002-j*4.03684e-003; --6.18120e-002+j*7.90821e-002, +2.38807e-002-j*5.62399e-002, -5.04609e-002+j*3.27008e-002; -+1.57670e-002-j*3.20728e-003, -5.98007e-003-j*3.10416e-003, -1.67104e-003-j*1.59089e-003; -+1.79638e-002-j*7.40235e-003, -8.20109e-003-j*2.36675e-004, -1.69812e-003+j*1.67596e-003; -+1.87809e-002-j*8.55724e-003, -2.81069e-003-j*3.03393e-003, -5.71319e-003-j*2.26161e-003; -+1.88829e-002-j*8.50228e-003, -5.28373e-003-j*2.52386e-003, -1.48737e-002-j*5.08140e-003; -+2.28380e-002-j*7.51769e-003, -7.57170e-003-j*2.75553e-003, -6.10380e-003+j*5.15562e-003; -+2.08600e-002-j*6.11732e-003, -4.73105e-003-j*3.30979e-003, -2.51369e-003+j*5.42921e-003; -+2.00008e-002-j*5.92617e-003, -3.50988e-003-j*4.55853e-003, -5.43701e-003-j*6.57229e-003; -+2.28149e-002-j*8.22905e-003, -4.83167e-003-j*3.10509e-003, -2.10958e-002-j*1.33421e-002; --1.83145e-002+j*2.76844e-002, +5.61668e-003-j*1.41226e-002, +2.20876e-002-j*2.27446e-002; --5.42112e-003+j*2.22444e-002, -4.20426e-005-j*8.78901e-003, +5.54714e-003+j*2.71564e-002; --1.70108e-002+j*2.83751e-002, -1.44473e-002+j*4.50880e-002, -5.26736e-003-j*5.75716e-003]; - -mode_shapes(:, :, 8) = [... --1.40928e-001+j*1.28570e-001, +2.95471e-001-j*1.35692e-001, -6.61656e-002+j*2.95705e-002; --1.56673e-001+j*5.19030e-002, +3.08231e-001-j*1.41453e-001, -9.70918e-002+j*4.95018e-002; --1.72505e-001+j*2.26273e-002, +3.97224e-001-j*2.77556e-017, -1.29223e-001+j*4.47412e-002; --1.69978e-001+j*1.16284e-001, +3.37516e-001+j*7.69873e-003, -8.49480e-002+j*2.17071e-002; -+5.38303e-004-j*3.59916e-003, -6.72455e-002-j*2.06230e-002, +2.66448e-002+j*4.10505e-002; --7.96526e-003-j*7.76851e-002, -2.63530e-002-j*3.75474e-002, -1.05984e-001+j*9.32474e-002; -+4.72518e-002-j*1.00199e-001, -5.50664e-002+j*1.50246e-001, -8.50976e-002+j*2.78531e-002; -+7.58419e-003-j*8.61594e-003, -9.02101e-002+j*1.58224e-001, +6.03081e-002-j*1.24162e-001; -+2.99027e-002-j*5.27128e-002, -8.80464e-002-j*2.99113e-004, +9.02851e-002-j*3.99771e-002; -+3.23132e-002-j*5.87278e-002, +3.81174e-002-j*3.69992e-002, +1.16643e-002+j*6.21068e-002; -+5.79795e-002-j*8.33565e-002, -1.22448e-003+j*8.81473e-002, +8.40150e-002-j*1.16264e-002; -+1.35399e-002-j*3.80303e-002, -9.58200e-002+j*1.46531e-001, +1.06769e-001-j*8.97034e-002; -+5.34299e-004+j*1.35179e-002, +8.71327e-004-j*6.41448e-003, +3.33208e-002-j*2.12545e-002; --2.79263e-004+j*5.08578e-003, -1.45476e-003-j*7.65161e-003, +6.98235e-002-j*2.45395e-002; --9.22822e-005+j*7.03205e-003, -5.62836e-003-j*2.79991e-003, +3.99717e-002-j*8.30891e-003; -+1.87833e-002+j*3.26772e-003, -4.86774e-003-j*4.32297e-003, +5.97375e-002-j*1.77542e-002; -+1.14169e-002+j*5.70930e-003, -8.23489e-003-j*4.53684e-003, +3.14016e-002-j*2.50637e-002; -+1.15995e-003+j*5.79180e-003, -6.69740e-003-j*4.66433e-003, +8.17695e-002-j*2.78384e-002; -+5.23838e-004+j*6.46432e-003, +3.36104e-003-j*4.42572e-003, +3.64589e-002+j*5.74796e-004; -+1.57042e-002+j*5.94177e-003, -1.49670e-003-j*4.22955e-003, +8.68520e-002-j*1.43981e-002; -+8.00706e-004-j*2.91734e-002, +1.20708e-002+j*4.24081e-002, -5.91796e-002+j*4.00346e-002; --1.91799e-003-j*1.37294e-002, +9.85285e-003+j*3.17934e-002, -1.78010e-001+j*7.91267e-002; -+3.57271e-003-j*3.09959e-002, -5.77781e-002-j*1.27957e-002, +1.31025e-002+j*1.92303e-002]; - -mode_shapes(:, :, 9) = [... -+1.58897e-002+j*3.23763e-002, -1.23332e-001-j*3.20376e-002, +6.78860e-002-j*1.28743e-002; -+5.67179e-003+j*4.26539e-002, -1.20726e-001-j*4.15603e-002, +7.66846e-002-j*1.24290e-002; -+2.24198e-002+j*3.45953e-002, -1.03213e-001-j*4.97049e-002, +5.49175e-002-j*5.50883e-003; -+2.68792e-002+j*2.97222e-002, -1.17598e-001-j*3.13791e-002, +5.59736e-002-j*1.71122e-002; -+1.39733e-002-j*1.56260e-002, +1.43952e-001+j*2.28119e-002, -4.56377e-003+j*4.88790e-002; --2.18507e-002-j*1.25664e-002, +6.15387e-002-j*6.31793e-003, +3.05342e-002+j*3.24595e-002; --7.85412e-003-j*1.85600e-002, +1.25733e-001+j*1.77063e-002, +5.34635e-002+j*4.72260e-003; --1.10454e-002-j*2.13217e-002, +1.26440e-001+j*1.96001e-002, -5.62624e-002-j*1.07192e-002; -+5.84467e-003-j*4.07134e-002, +5.16711e-003+j*4.70857e-002, -1.93010e-001+j*6.79213e-003; -+5.31962e-002-j*1.11322e-002, +3.23294e-001-j*2.12981e-002, +2.14310e-001+j*4.85898e-003; --2.44580e-002-j*1.33220e-002, +4.08800e-001+j*2.09082e-002, +5.11454e-001+j*0.00000e+000; --2.51332e-002-j*1.42719e-003, +8.97105e-002+j*4.85852e-002, -1.27426e-001-j*9.59723e-003; --2.08176e-003+j*1.37185e-002, -3.99530e-002+j*2.11895e-002, -1.17813e-001+j*7.60972e-002; --1.34824e-002+j*7.11258e-003, -4.19473e-002+j*1.83590e-002, -2.07198e-002-j*2.51991e-002; --6.84747e-003+j*8.45921e-003, -3.33872e-002+j*1.71496e-002, +6.99867e-002-j*6.93158e-002; --2.64313e-002+j*5.08903e-003, -3.03569e-002+j*1.29946e-002, -1.39115e-001+j*5.72459e-002; --2.90186e-002+j*1.48257e-002, -5.55429e-002+j*2.74156e-002, -1.62035e-001+j*8.04187e-002; --2.05855e-002+j*1.11922e-002, -6.58789e-002+j*3.20524e-002, -1.02263e-002-j*4.24087e-002; --1.40204e-002+j*8.01102e-003, -5.72647e-002+j*2.37484e-002, +1.75053e-001-j*9.63667e-002; --3.50818e-002+j*1.41152e-002, -5.19701e-002+j*2.31951e-002, -1.15951e-001+j*2.91582e-002; --5.78005e-003-j*7.05841e-003, +8.29016e-002+j*1.36984e-002, +4.03470e-003+j*4.03325e-002; --1.39928e-002-j*1.14088e-002, +8.05288e-002+j*1.51031e-002, +1.12255e-002+j*3.21224e-002; --1.02276e-002-j*8.35724e-003, -4.89246e-003+j*1.67800e-002, +7.80514e-002+j*1.53467e-002]; - -mode_shapes(:, :, 10) = [... -+3.33349e-002-j*4.89606e-003, -8.67138e-002-j*1.69402e-002, +2.87366e-002-j*1.66842e-002; -+2.95730e-002-j*6.10477e-004, -9.24590e-002-j*1.92562e-002, +5.21162e-002-j*1.31811e-002; -+3.26966e-002+j*1.03975e-002, -8.55682e-002-j*4.71847e-002, +3.99404e-002+j*8.59358e-003; -+3.45452e-002-j*3.05951e-003, -7.73823e-002-j*2.32199e-002, +2.30960e-002-j*7.49928e-003; --3.72461e-003-j*5.40336e-003, +1.80151e-001-j*1.42898e-002, -8.63921e-003+j*1.95638e-002; --1.11285e-002-j*2.22175e-003, +6.92355e-002-j*1.24144e-002, +8.02097e-002-j*6.81531e-003; --2.99885e-002-j*4.21951e-004, +8.19709e-002+j*1.22484e-002, +3.97531e-002+j*1.33874e-002; --1.59231e-002-j*3.96929e-003, +9.31295e-002+j*9.75532e-003, -4.01947e-002-j*5.17841e-003; --7.32828e-003-j*2.93496e-002, +4.39909e-002+j*1.25298e-002, -8.87525e-002+j*2.05359e-002; -+8.20167e-002+j*6.86693e-004, +4.25475e-001-j*2.36494e-002, +3.37034e-001-j*2.44199e-002; --2.68694e-002+j*3.82921e-003, +4.79292e-001+j*1.35903e-002, +5.06762e-001+j*0.00000e+000; -+1.40565e-002-j*5.41957e-003, +1.17563e-001+j*2.50398e-002, -5.97219e-002+j*5.92813e-004; -+2.30939e-002+j*1.16755e-002, +4.11136e-003+j*1.65726e-002, +2.41863e-002+j*5.06658e-002; --4.92960e-003+j*9.24082e-003, -5.44667e-003+j*1.41983e-002, -3.73537e-002-j*1.87263e-002; -+9.68682e-003+j*1.11791e-002, -5.13436e-003+j*1.29205e-002, -4.88765e-002-j*4.97821e-002; --1.66711e-003+j*1.03280e-002, +9.57955e-004+j*1.28350e-002, +3.30268e-002+j*5.44211e-002; -+5.90540e-003+j*1.67113e-002, +1.34280e-002+j*2.62111e-002, +4.38613e-002+j*7.17028e-002; -+4.97752e-003+j*1.47634e-002, +1.56773e-002+j*3.16026e-002, -5.27493e-002-j*3.38315e-002; -+2.83485e-003+j*1.14816e-002, +9.80676e-003+j*2.51504e-002, -6.78645e-002-j*9.90875e-002; -+5.62294e-003+j*1.84035e-002, +1.03717e-002+j*2.37801e-002, +2.58497e-002+j*3.75352e-002; --8.75236e-003-j*4.71723e-003, +6.99107e-002+j*1.51894e-002, +3.46273e-002+j*1.48547e-002; --1.72822e-002-j*8.75192e-003, +6.96759e-002+j*1.59783e-002, +8.07917e-002+j*1.97809e-002; --1.29601e-002-j*5.61834e-003, +1.76126e-002+j*4.62761e-003, +6.64667e-002+j*1.61199e-002]; -#+end_src - -** Define a point for each solid body -We define accelerometer indices used to define the motion of each solid body (2 3-axis accelerometer are enough). -#+begin_src matlab - stages = [17, 19; % Bottom Granite - 13, 15; % Top Granite - 9, 11; % Ty - 5, 7; % Ry - 21, 22; % Spindle - 1, 3]; % Hexapod -#+end_src - -We define the origin point ${}^AO_B$ of the solid body $\{B\}$. -Here we choose the middle point between the two accelerometers. -This could be define differently (for instance by choosing the center of mass). -#+begin_src matlab - AOB = zeros(3, size(stages, 1)); - - for i = 1:size(stages, 1) - AOB(:, i) = mean(positions(stages(i, :), 1:3))'; - end -#+end_src - -Then we compute the positions of the sensors with respect to the previously defined origin for the frame $\{B\}$: ${}^BP_1$ and ${}^BP_2$. -#+begin_src matlab - BP1 = zeros(3, size(stages, 1)); - BP2 = zeros(3, size(stages, 1)); - - for i = 1:size(stages, 1) - BP1(:, i) = positions(stages(i, 1), 1:3)' - AOB(:, i); - BP2(:, i) = positions(stages(i, 2), 1:3)' - AOB(:, i); - end -#+end_src - -Let's define one absolute frame $\{A\}$ and one frame $\{B\}$ fixed w.r.t. the solid body. -We note ${}^AO_B$ the position of origin of $\{B\}$ expressed in $\{A\}$. - -We are measuring with the accelerometers the absolute motion of points $P_1$ and $P_2$: ${}^Av_{P_1}$ and ${}^Av_{P_2}$. - -Let's note ${}^BP_1$ and ${}^BP_2$ the (known) coordinates of $P_1$ and $P_2$ expressed in the frame $\{B\}$. - -Then we have: -\begin{align} - {}^Av_{P_1} &= {}^Av_{O_B} + {}^A\Omega^\times {}^AR_B {}^BP_1 \\ - {}^Av_{P_2} &= {}^Av_{O_B} + {}^A\Omega^\times {}^AR_B {}^BP_2 -\end{align} - -And we obtain -\begin{align} - {}^A\Omega^\times {}^AR_B &= \left( {}^Av_{P_2} - {}^Av_{P_1} \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1}\\ - {}^Av_{O_B} &= {}^Av_{P_1} - \left( {}^Av_{P_2} - {}^Av_{P_1} \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1} {}^BP_1 -\end{align} - -#+begin_src matlab - AVOB = zeros(3, size(stages, 1)); - ARB = zeros(3, 3, size(stages, 1)); - for i = 1:size(stages, 1) - AVOB(:, i) = mode1(stages(i, 1), :)' - (mode1(stages(i, 2), :)' - mode1(stages(i, 1), :)')*pinv(BP2(:, i) - BP1(:, i))*BP1(:, i); - ARB(:, :, i) = (mode1(stages(i, 2), :)' - mode1(stages(i, 1), :)')*pinv(BP2(:, i) - BP1(:, i)); - end -#+end_src - -** Argand Diagram -For mode 1 - -#+begin_src matlab - figure; - hold on; - for i=1:size(mode1, 1) - plot([0, real(mode1(i, 1))], [0, imag(mode1(i, 1))], '-k') - plot([0, real(mode1(i, 2))], [0, imag(mode1(i, 2))], '-k') - plot([0, real(mode1(i, 3))], [0, imag(mode1(i, 3))], '-k') - - % plot([0, real(mode2(i, 1))], [0, imag(mode2(i, 1))], '-r') - % plot([0, real(mode2(i, 2))], [0, imag(mode2(i, 2))], '-r') - % plot([0, real(mode2(i, 3))], [0, imag(mode2(i, 3))], '-r') - - % plot([0, real(mode3(i, 1))], [0, imag(mode3(i, 1))], '-b') - % plot([0, real(mode3(i, 2))], [0, imag(mode3(i, 2))], '-b') - % plot([0, real(mode3(i, 3))], [0, imag(mode3(i, 3))], '-b') - end - - for i=1:size(AVOB, 2) - plot([0, real(AVOB(1, i))], [0, imag(AVOB(1, i))], '-r') - plot([0, real(AVOB(2, i))], [0, imag(AVOB(2, i))], '-r') - plot([0, real(AVOB(3, i))], [0, imag(AVOB(3, i))], '-r') - end - - % ang=0:0.01:2*pi; - % radius1 = max(max(sqrt(real(mode1).^2+imag(mode1).^2))); - % plot(radius1*cos(ang), radius1*sin(ang), '-k'); - - % radius2 = max(max(sqrt(real(mode2).^2+imag(mode2).^2))); - % plot(radius2*cos(ang), radius2*sin(ang), '-r'); - - % radius3 = max(max(sqrt(real(mode3).^2+imag(mode3).^2))); - % plot(radius3*cos(ang), radius3*sin(ang), '-b'); - - hold off; - axis manual equal -#+end_src - -** TEST: animate first mode -#+begin_src matlab - figure; - hold on; - fill3(positions(1:4, 1), positions(1:4, 2), positions(1:4, 3), 'k', 'FaceAlpha', 0.5) - fill3(positions(5:8, 1), positions(5:8, 2), positions(5:8, 3), 'k', 'FaceAlpha', 0.5) - fill3(positions(9:12, 1), positions(9:12, 2), positions(9:12, 3), 'k', 'FaceAlpha', 0.5) - fill3(positions(13:16, 1), positions(13:16, 2), positions(13:16, 3), 'k', 'FaceAlpha', 0.5) - fill3(positions(17:20, 1), positions(17:20, 2), positions(17:20, 3), 'k', 'FaceAlpha', 0.5) - fill3(positions(21:23, 1), positions(21:23, 2), positions(21:23, 3), 'k', 'FaceAlpha', 0.5) - hold off; -#+end_src - -#+NAME: fig:mode_shapes -#+HEADER: :tangle no :exports results :results value file raw replace :noweb yes -#+begin_src matlab - rec = polyshape([-2 -2 2 2],[-3 3 3 -3]); - h = figure; - filename = 'figs/mode_shapes.gif'; - n = 20; - - for i = 1:n - axis manual equal - - Dm = real(V(1:3, 5)*cos(2*pi*i/n)); - rec_i = rotate(rec, 180/pi*Dm(3)); - rec_i = translate(rec_i, 10*Dm(1), 10*Dm(2)); - plot(rec_i); - - xlim([-3, 3]); ylim([-4, 4]); - - set(h, 'visible', 'off'); - set(h, 'pos', [0, 0, 500, 500]); - drawnow; - - % Capture the plot as an image - frame = getframe(h); - im = frame2im(frame); - [imind,cm] = rgb2ind(im,256); - - % Write to the GIF File - if i == 1 - imwrite(imind,cm,filename,'gif','DelayTime',0.1,'Loopcount',inf); - else - imwrite(imind,cm,filename,'gif','DelayTime',0.1,'WriteMode','append'); - end - end - - set(h, 'visible', 'on'); - ans = filename; -#+end_src - -** From 6 translations to translation + rotation -Let's define one absolute frame $\{A\}$ and one frame $\{B\}$ fixed w.r.t. the solid body. -We note ${}^AO_B$ the position of origin of $\{B\}$ expressed in $\{A\}$. - -We are measuring with the accelerometers the absolute motion of points $P_1$ and $P_2$: ${}^AP_1$ and ${}^AP_2$. - -Let's note ${}^BP_1$ and ${}^BP_2$ the (known) coordinates of $P_1$ and $P_2$ expressed in the frame $\{B\}$. - -Then we have: -\begin{align} - {}^AP_1 &= {}^AO_B + {}^AR_B {}^BP_1 \\ - {}^AP_2 &= {}^AO_B + {}^AR_B {}^BP_2 -\end{align} - -And we obtain -\begin{align} - {}^AR_B &= \left( {}^AP_2 - {}^AP_1 \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1}\\ - {}^AO_B &= {}^Av_{P_1} - \left( {}^AP_2 - {}^AP_1 \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1} {}^BP_1 -\end{align} - -* Problem with AirLoc System -The mode shape of the first mode at 11Hz (figure [[fig:mode1]]) seems to indicate that this corresponds to a suspension mode. +For instance, the mode shape of the first mode at 11Hz (figure [[fig:mode1]]) seems to indicate that this corresponds to a suspension mode. This could be due to the 4 Airloc Levelers that are used for the granite (figure [[fig:airloc]]). @@ -1221,161 +441,95 @@ This could be due to the 4 Airloc Levelers that are used for the granite (figure They are probably *not well leveled*, so the granite is supported only by two Airloc. -* Setup -#+name: fig:nass-modal-test -#+caption: Position and orientation of the accelerometer used -[[file:figs/nass-modal-test.png]] +* Modal Complexity + <> +A method of displaying *modal complexity* is by plotting the elements of the eigenvector on an *Argand diagram* (complex plane), such as the ones shown in figure [[fig:modal_complexity_small]]. -* Mode extraction and importation -First, we split the big =modes.asc= files into sub text files using =bash=. -#+begin_src bash :results none - sed '/^\s*[0-9]*[XYZ][+-]:/!d' modal_analysis_updated/modes.asc > mat/mode_shapes.txt - sed '/freq/!d' modal_analysis_updated/modes.asc | sed 's/.* = \(.*\)Hz/\1/' > mat/mode_freqs.txt - sed '/damp/!d' modal_analysis_updated/modes.asc | sed 's/.* = \(.*\)\%/\1/' > mat/mode_damps.txt - sed '/modal A/!d' modal_analysis_updated/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_a.txt - sed '/modal B/!d' modal_analysis_updated/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_b.txt +To evaluate the complexity of the modes, we plot a polygon around the extremities of the individual vectors. +The obtained area of this polygon is then compared with the area of the circle which is based on the length of the largest vector element. The resulting ratio is used as an *indication of the complexity of the mode*. + +A mode with small complexity is shown on figure [[fig:modal_complexity_small]] whereas an highly complex mode is shown on figure [[fig:modal_complexity_high]]. +The complexity of all the modes are compared on figure [[fig:modal_complexities]]. + +#+begin_src matlab :exports none + figure; + mod_i = 1; + i_max = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); + radius = max(abs(eigen_vector_M(:, mod_i))); + theta = linspace(0, 2*pi, 100); + + hold on; + plot(radius*cos(theta), radius*sin(theta), '-'); + plot(real(eigen_vector_M(i_max, mod_i)), imag(eigen_vector_M(i_max, mod_i)), '-'); + plot(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i)), 'ko'); + hold off; + xlabel('Real Part'); ylabel('Imaginary Part'); + title(sprintf('Mode %i', mod_i)); + axis manual equal #+end_src -Then we import them on Matlab. -#+begin_src matlab - shapes = readtable('mat/mode_shapes.txt', 'ReadVariableNames', false); % [Sign / Real / Imag] - freqs = table2array(readtable('mat/mode_freqs.txt', 'ReadVariableNames', false)); % in [Hz] - damps = table2array(readtable('mat/mode_damps.txt', 'ReadVariableNames', false)); % in [%] - modal_a = table2array(readtable('mat/mode_modal_a.txt', 'ReadVariableNames', false)); % [Real / Imag] - modal_a = complex(modal_a(:, 1), modal_a(:, 2)); - modal_b = table2array(readtable('mat/mode_modal_b.txt', 'ReadVariableNames', false)); % [Real / Imag] - modal_b = complex(modal_b(:, 1), modal_b(:, 2)); +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/modal_complexity_small.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png") + <> #+end_src -We guess the number of modes identified from the length of the imported data. -#+begin_src matlab - acc_n = 23; % Number of accelerometers - dir_n = 3; % Number of directions - dirs = 'XYZ'; +#+NAME: fig:modal_complexity_small +#+CAPTION: Modal Complexity of one mode with small complexity +[[file:figs/modal_complexity_small.png]] - mod_n = size(shapes,1)/acc_n/dir_n; % Number of modes +#+begin_src matlab :exports none + mod_i = 8; + i_max = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); + radius = max(abs(eigen_vector_M(:, mod_i))); + theta = linspace(0, 2*pi, 100); + + figure; + hold on; + plot(radius*cos(theta), radius*sin(theta), '-'); + plot(real(eigen_vector_M(i_max, mod_i)), imag(eigen_vector_M(i_max, mod_i)), '-'); + plot(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i)), 'ko'); + hold off; + xlabel('Real Part'); ylabel('Imaginary Part'); + title(sprintf('Mode %i', mod_i)); + axis manual equal #+end_src -As the mode shapes are split into 3 parts (direction plus sign, real part and imaginary part), we aggregate them into one array of complex numbers. -#+begin_src matlab - T_sign = table2array(shapes(:, 1)); - T_real = table2array(shapes(:, 2)); - T_imag = table2array(shapes(:, 3)); +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/modal_complexity_high.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src - modes = zeros(mod_n, acc_n, dir_n); +#+NAME: fig:modal_complexity_high +#+CAPTION: Modal Complexity of one higly complex mode +[[file:figs/modal_complexity_high.png]] +#+begin_src matlab :exports none + modes_complexity = zeros(mod_n, 1); for mod_i = 1:mod_n - for acc_i = 1:acc_n - % Get the correct section of the signs - T = T_sign(acc_n*dir_n*(mod_i-1)+1:acc_n*dir_n*mod_i); - for dir_i = 1:dir_n - % Get the line corresponding to the sensor - i = find(contains(T, sprintf('%i%s',acc_i, dirs(dir_i))), 1, 'first')+acc_n*dir_n*(mod_i-1); - modes(mod_i, acc_i, dir_i) = str2num([T_sign{i}(end-1), '1'])*complex(T_real(i),T_imag(i)); - end - end + i = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); + area_complex = polyarea(real(eigen_vector_M(i, mod_i)), imag(eigen_vector_M(i, mod_i))); + area_circle = pi*max(abs(eigen_vector_M(:, mod_i)))^2; + modes_complexity(mod_i) = area_complex/area_circle; end + + figure; + plot(1:mod_n, modes_complexity, 'ok'); + ylim([0, 1]); + xlabel('Mode Number'); ylabel('Modal Complexity'); #+end_src -The obtained mode frequencies and damping are shown below. -#+begin_src matlab :exports both :results value table replace :post addhdr(*this*) - data2orgtable([freqs, damps], {}, {'Frequency [Hz]', 'Damping [%]'}, ' %.1f '); +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/modal_complexities.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") + <> #+end_src -#+RESULTS: -| Frequency [Hz] | Damping [%] | -|----------------+-------------| -| 11.4 | 8.7 | -| 18.5 | 11.8 | -| 37.6 | 6.4 | -| 39.4 | 3.6 | -| 54.0 | 0.2 | -| 56.1 | 2.8 | -| 69.7 | 4.6 | -| 71.6 | 0.6 | -| 72.4 | 1.6 | -| 84.9 | 3.6 | -| 90.6 | 0.3 | -| 91.0 | 2.9 | -| 95.8 | 3.3 | -| 105.4 | 3.3 | -| 106.8 | 1.9 | -| 112.6 | 3.0 | -| 116.8 | 2.7 | -| 124.1 | 0.6 | -| 145.4 | 1.6 | -| 150.1 | 2.2 | -| 164.7 | 1.4 | +#+NAME: fig:modal_complexities +#+CAPTION: Modal complexity for each mode +[[file:figs/modal_complexities.png]] -* Positions of the sensors -We process the file exported from the =modal= software containing the positions of the sensors using =bash=. -#+begin_src bash :results none - cat modal_analysis_updated/id31_nanostation_modified.cfg | grep NODES -A 23 | sed '/\s\+[0-9]\+/!d' | sed 's/\(.*\)\s\+0\s\+.\+/\1/' > mat/acc_pos.txt -#+end_src - -We then import that on =matlab=, and sort them. -#+begin_src matlab - acc_pos = readtable('mat/acc_pos.txt', 'ReadVariableNames', false); - acc_pos = table2array(acc_pos(:, 1:4)); - [~, i] = sort(acc_pos(:, 1)); - acc_pos = acc_pos(i, 2:4); -#+end_src - -The positions of the sensors relative to the point of interest are shown below. -#+begin_src matlab :exports both :results value table replace :post addhdr(*this*) - data2orgtable(1000*acc_pos, {}, {'x [mm]', 'y [mm]', 'z [mm]'}, ' %.0f '); -#+end_src - -#+RESULTS: -| x [mm] | y [mm] | z [mm] | -|--------+--------+--------| -| -64 | -64 | -296 | -| -64 | 64 | -296 | -| 64 | 64 | -296 | -| 64 | -64 | -296 | -| -385 | -300 | -417 | -| -420 | 280 | -417 | -| 420 | 280 | -417 | -| 380 | -300 | -417 | -| -475 | -414 | -427 | -| -465 | 407 | -427 | -| 475 | 424 | -427 | -| 475 | -419 | -427 | -| -320 | -446 | -786 | -| -480 | 534 | -786 | -| 450 | 534 | -786 | -| 295 | -481 | -786 | -| -730 | -526 | -951 | -| -735 | 814 | -951 | -| 875 | 799 | -951 | -| 865 | -506 | -951 | -| -155 | -90 | -594 | -| 0 | 180 | -594 | -| 155 | -90 | -594 | - -* Solids -We consider the following solid bodies: -- Bottom Granite -- Top Granite -- Translation Stage -- Tilt Stage -- Spindle -- Hexapod - -We create a structure =solids= that contains the accelerometer number of each solid bodies (as shown on figure [[fig:nass-modal-test]]). -#+begin_src matlab - solids = {}; - solids.granite_bot = [17, 18, 19, 20]; - solids.granite_top = [13, 14, 15, 16]; - solids.ty = [9, 10, 11, 12]; - solids.ry = [5, 6, 7, 8]; - solids.rz = [21, 22, 23]; - solids.hexa = [1, 2, 3, 4]; - - solid_names = fields(solids); -#+end_src - -* From local coordinates to global coordinates for the mode shapes +* BKMK From local coordinates to global coordinates for the mode shapes + <> +** Mathematical description #+begin_src latex :file local_to_global_coordinates.pdf :post pdf2svg(file=*this*, ext="png") :exports results \newcommand\irregularcircle[2]{% radius, irregularity \pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}} @@ -1469,6 +623,7 @@ and then we obtain the velocity and rotation of the solid in the wanted frame $\ This inversion is equivalent to a mean square problem. +** Matlab Implementation #+begin_src matlab mode_shapes_O = zeros(mod_n, length(solid_names), 6); @@ -1490,8 +645,8 @@ This inversion is equivalent to a mean square problem. end #+end_src -* Modal Matrices -We want to obtain the two following matrices: +** Modal Matrices +We arrange the obtained modal parameters into matrices: \[ \Omega = \begin{bmatrix} \omega_1^2 & & 0 \\ & \ddots & \\ @@ -1501,100 +656,30 @@ We want to obtain the two following matrices: \{\psi_1\} & \dots & \{\psi_n\} \\ & & \end{bmatrix} \] +with +\[ \{\psi_1\} = \begin{Bmatrix} \psi_{1_x} & \psi_{2_x} & \dots & \psi_{6_x} & \psi_{1_y} & \dots & \psi_{1\Omega_x} & \dots & \psi_{6\Omega_z} \end{Bmatrix}^T \] -- [ ] How to add damping to the eigen value matrix? +#+begin_warning + How to add damping to the eigen matrices? +#+end_warning #+begin_src matlab eigen_value_M = diag(freqs*2*pi); eigen_vector_M = reshape(mode_shapes_O, [mod_n, 6*length(solid_names)])'; #+end_src -\[ \{\psi_1\} = \begin{Bmatrix} \psi_{1_x} & \psi_{2_x} & \dots & \psi_{6_x} & \psi_{1_x} & \dots & \psi_{1\Omega_x} & \dots & \psi_{6\Omega_z} \end{Bmatrix}^T \] +** TODO Normalization of mode shapes? +We normalize each column of the eigen vector matrix. -* Modal Complexity -A method of displaying *modal complexity* is by plotting the elements of the eigenvector on an *Argand diagram*, such as the ones shown in figure [[fig:modal_complexity_small]]. +Then, each eigenvector as a norm of 1. -To evaluate the complexity of the modes, we plot a polygon around the extremities of the individual vectors. -The obtained area of this polygon is then compared with the area of the circle which is based on the length of the largest vector element. The resulting ratio is used as an indication of the complexity of the mode. - -A little complex mode is shown on figure [[fig:modal_complexity_small]] whereas an highly complex mode is shown on figure [[fig:modal_complexity_high]]. -The complexity of all the modes are compared on figure [[fig:modal_complexities]]. - -#+begin_src matlab :exports none - mod_i = 1; - i_max = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); - radius = max(abs(eigen_vector_M(:, mod_i))); - theta = linspace(0, 2*pi, 100); - - figure; - hold on; - plot(radius*cos(theta), radius*sin(theta), '-'); - plot(real(eigen_vector_M(i_max, mod_i)), imag(eigen_vector_M(i_max, mod_i)), '-'); - plot(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i)), 'ko'); - hold off; - xlabel('Real Part'); ylabel('Imaginary Part'); - title(sprintf('Mode %i', mod_i)); - axis manual equal +#+begin_src matlab + eigen_vector_M = eigen_vector_M./vecnorm(eigen_vector_M); #+end_src -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/modal_complexity_small.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:modal_complexity_small -#+CAPTION: Modal Complexity of one mode with small complexity -[[file:figs/modal_complexity_small.png]] - -#+begin_src matlab :exports none - mod_i = 8; - i_max = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); - radius = max(abs(eigen_vector_M(:, mod_i))); - theta = linspace(0, 2*pi, 100); - - figure; - hold on; - plot(radius*cos(theta), radius*sin(theta), '-'); - plot(real(eigen_vector_M(i_max, mod_i)), imag(eigen_vector_M(i_max, mod_i)), '-'); - plot(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i)), 'ko'); - hold off; - xlabel('Real Part'); ylabel('Imaginary Part'); - title(sprintf('Mode %i', mod_i)); - axis manual equal -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/modal_complexity_high.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:modal_complexity_high -#+CAPTION: Modal Complexity of one higly complex mode -[[file:figs/modal_complexity_high.png]] - -#+begin_src matlab :exports none - modes_complexity = zeros(mod_n, 1); - for mod_i = 1:mod_n - i = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); - area_complex = polyarea(real(eigen_vector_M(i, mod_i)), imag(eigen_vector_M(i, mod_i))); - area_circle = pi*max(abs(eigen_vector_M(:, mod_i)))^2; - modes_complexity(mod_i) = area_complex/area_circle; - end - - figure; - plot(1:mod_n, modes_complexity, 'ok'); - ylim([0, 1]); - xlabel('Mode Number'); ylabel('Modal Complexity'); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/modal_complexities.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:modal_complexities -#+CAPTION: Modal complexity for each mode -[[file:figs/modal_complexities.png]] +#+begin_important + Should we do such normalization? +#+end_important * Some notes about constraining the number of degrees of freedom We want to have the two eigen matrices. @@ -1628,13 +713,6 @@ Is it meaningful to compare mode shapes as they give no information about the am #+TBLFM: $4=vsum($2..$3) #+TBLFM: @>$2..$>=vsum(@I..@II) -* TODO Normalization of mode shapes? -We normalize each column of the eigen vector matrix. -Then, each eigenvector as a norm of 1. -#+begin_src matlab - eigen_vector_M = eigen_vector_M./vecnorm(eigen_vector_M); -#+end_src - * Compare Mode Shapes Let's say we want to see for the first mode which DOFs can be neglected. In order to do so, we should estimate the motion of each stage in particular directions. @@ -1681,332 +759,4 @@ If we look at the z motion for instance, we will find that we cannot neglect tha test = mode_shapes_O(10, 2, :)/norm(squeeze(mode_shapes_O(10, 2, :))); #+end_src -* Importation of measured FRF curves -There are 24 measurements files corresponding to 24 series of impacts: -- 3 directions, 8 sets of 3 accelerometers -For each measurement file, the FRF and coherence between the impact and the 9 accelerations measured. - -In reality: 4 sets of 10 things - -#+begin_src matlab - a = load('mat/meas_frf_coh_1.mat'); -#+end_src - -#+begin_src matlab - figure; - - ax1 = subplot(2, 1, 1); - hold on; - plot(a.FFT1_AvXSpc_2_1_RMS_X_Val, a.FFT1_AvXSpc_2_1_RMS_Y_Mod) - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - ylabel('Amplitude'); - title(sprintf('From %s, to %s', FFT1_AvXSpc_2_1_RfName, FFT1_AvXSpc_2_1_RpName)) - - ax2 = subplot(2, 1, 2); - hold on; - plot(a.FFT1_AvXSpc_2_1_RMS_X_Val, a.FFT1_AvXSpc_2_1_RMS_Y_Phas) - hold off; - ylim([-180, 180]); yticks(-180:90:180); - xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); - set(gca, 'xscale', 'log'); - - linkaxes([ax1,ax2],'x'); - xlim([1, 200]); -#+end_src - -* Importation of measured FRF curves to global FRF matrix -FRF matrix $n \times p$: -- $n$ is the number of measurements: $3 \times 24$ -- $p$ is the number of excitation inputs: 3 - -23 measurements: 3 accelerometers - -\begin{equation} - \text{FRF}(\omega_i) = \begin{bmatrix} - \frac{D_{1_x}}{F_x}(\omega_i) & \frac{D_{1_x}}{F_y}(\omega_i) & \frac{D_{1_x}}{F_z}(\omega_i) \\ - \frac{D_{1_y}}{F_x}(\omega_i) & \frac{D_{1_y}}{F_y}(\omega_i) & \frac{D_{1_y}}{F_z}(\omega_i) \\ - \frac{D_{1_z}}{F_x}(\omega_i) & \frac{D_{1_z}}{F_y}(\omega_i) & \frac{D_{1_z}}{F_z}(\omega_i) \\ - \frac{D_{2_x}}{F_x}(\omega_i) & \frac{D_{2_x}}{F_y}(\omega_i) & \frac{D_{2_x}}{F_z}(\omega_i) \\ - \vdots & \vdots & \vdots \\ - \frac{D_{23_z}}{F_x}(\omega_i) & \frac{D_{23_z}}{F_y}(\omega_i) & \frac{D_{23_z}}{F_z}(\omega_i) \\ -\end{bmatrix} -\end{equation} - -#+begin_src matlab - n_meas = 24; - n_acc = 23; - - dirs = 'XYZ'; - - % Number of Accelerometer * DOF for each acccelerometer / Number of excitation / frequency points - FRFs = zeros(3*n_acc, 3, 801); - COHs = zeros(3*n_acc, 3, 801); - - % Loop through measurements - for i = 1:n_meas - % Load the measurement file - meas = load(sprintf('mat/meas_frf_coh_%i.mat', i)); - - % First: determine what is the exitation (direction and sign) - exc_dir = meas.FFT1_AvXSpc_2_1_RMS_RfName(end); - exc_sign = meas.FFT1_AvXSpc_2_1_RMS_RfName(end-1); - % Determine what is the correct excitation sign - exc_factor = str2num([exc_sign, '1']); - if exc_dir ~= 'Z' - exc_factor = exc_factor*(-1); - end - - % Then: loop through the nine measurements and store them at the correct location - for j = 2:10 - % Determine what is the accelerometer and direction - [indices_acc_i] = strfind(meas.(sprintf('FFT1_H1_%i_1_RpName', j)), '.'); - acc_i = str2num(meas.(sprintf('FFT1_H1_%i_1_RpName', j))(indices_acc_i(1)+1:indices_acc_i(2)-1)); - - meas_dir = meas.(sprintf('FFT1_H1_%i_1_RpName', j))(end); - meas_sign = meas.(sprintf('FFT1_H1_%i_1_RpName', j))(end-1); - % Determine what is the correct measurement sign - meas_factor = str2num([meas_sign, '1']); - if meas_dir ~= 'Z' - meas_factor = meas_factor*(-1); - end - - % FRFs(acc_i+n_acc*(find(dirs==meas_dir)-1), find(dirs==exc_dir), :) = exc_factor*meas_factor*meas.(sprintf('FFT1_H1_%i_1_Y_ReIm', j)); - % COHs(acc_i+n_acc*(find(dirs==meas_dir)-1), find(dirs==exc_dir), :) = meas.(sprintf('FFT1_Coh_%i_1_RMS_Y_Val', j)); - - FRFs(find(dirs==meas_dir)+3*(acc_i-1), find(dirs==exc_dir), :) = exc_factor*meas_factor*meas.(sprintf('FFT1_H1_%i_1_Y_ReIm', j)); - COHs(find(dirs==meas_dir)+3*(acc_i-1), find(dirs==exc_dir), :) = meas.(sprintf('FFT1_Coh_%i_1_RMS_Y_Val', j)); - end - end - freqs = meas.FFT1_Coh_10_1_RMS_X_Val; -#+end_src - -* Analysis of some FRFs -#+begin_src matlab - acc_i = 3; - acc_dir = 1; - exc_dir = 1; - - figure; - - ax1 = subplot(2, 1, 1); - hold on; - plot(freqs, abs(squeeze(FRFs(acc_dir+3*(acc_i-1), exc_dir, :)))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - ylabel('Amplitude'); - - ax2 = subplot(2, 1, 2); - hold on; - plot(freqs, mod(180+180/pi*phase(squeeze(FRFs(acc_dir+3*(acc_i-1), exc_dir, :))), 360)-180); - hold off; - ylim([-180, 180]); yticks(-180:90:180); - xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); - set(gca, 'xscale', 'log'); - - linkaxes([ax1,ax2],'x'); - xlim([1, 200]); -#+end_src - -* Composite Response Function -We here sum the norm instead of the complex numbers. - -#+begin_src matlab - HHx = squeeze(sum(abs(FRFs(:, 1, :)))); - HHy = squeeze(sum(abs(FRFs(:, 2, :)))); - HHz = squeeze(sum(abs(FRFs(:, 3, :)))); - HH = squeeze(sum([HHx, HHy, HHz], 2)); -#+end_src - -#+begin_src matlab - exc_dir = 3; - - figure; - hold on; - for i = 1:3*n_acc - plot(freqs, abs(squeeze(FRFs(i, exc_dir, :))), 'color', [0, 0, 0, 0.2]); - end - plot(freqs, abs(HHx)); - plot(freqs, abs(HHy)); - plot(freqs, abs(HHz)); - plot(freqs, abs(HH), 'k'); - hold off; - set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); - xlabel('Frequency [Hz]'); ylabel('Amplitude'); - xlim([1, 200]); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/composite_response_function.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:composite_response_function -#+CAPTION: Composite Response Function -[[file:figs/composite_response_function.png]] - -* TODO Singular Value Decomposition - Modal Indication Function -Show the same plot as in the modal software. -This helps to identify double modes. - -From the documentation of the modal software: -#+begin_quote -The MIF consist of the singular values of the Frequency response function matrix. -The number of MIFs equals the number of excitations. -By the powerful singular value decomposition, the real signal space is separated from the noise space. -Therefore, the MIFs exhibit the modes effectively. -A peak in the MIFs plot usually indicate the existence of a structural mode, and two peaks at the same frequency point means the existence of two repeated modes. -Moreover, the magnitude of the MIFs implies the strength of the a mode. -#+end_quote - -* From local coordinates to global coordinates with the FRFs -#+begin_src matlab - % Number of Solids * DOF for each solid / Number of excitation / frequency points - FRFs_O = zeros(length(solid_names)*6, 3, 801); - - for exc_dir = 1:3 - for solid_i = 1:length(solid_names) - solids_i = solids.(solid_names{solid_i}); - - A = zeros(3*length(solids_i), 6); - for i = 1:length(solids_i) - A(3*(i-1)+1:3*i, 1:3) = eye(3); - - A(3*(i-1)+1:3*i, 4:6) = [0 acc_pos(i, 3) -acc_pos(i, 2) ; -acc_pos(i, 3) 0 acc_pos(i, 1) ; acc_pos(i, 2) -acc_pos(i, 1) 0]; - end - - for i = 1:801 - FRFs_O((solid_i-1)*6+1:solid_i*6, exc_dir, i) = A\FRFs((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, i); - end - end - end -#+end_src - -* Analysis of some FRF in the global coordinates -#+begin_src matlab - solid_i = 6; - dir_i = 1; - exc_dir = 1; - - figure; - - ax1 = subplot(2, 1, 1); - hold on; - plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :)))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - ylabel('Amplitude'); - - ax2 = subplot(2, 1, 2); - hold on; - plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 360)-180); - hold off; - ylim([-180, 180]); yticks(-180:90:180); - xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); - set(gca, 'xscale', 'log'); - - linkaxes([ax1,ax2],'x'); - xlim([1, 200]); -#+end_src - -* Compare global coordinates to local coordinates -#+begin_src matlab - solid_i = 1; - acc_dir_O = 6; - acc_dir = 3; - exc_dir = 3; - - figure; - - ax1 = subplot(2, 1, 1); - hold on; - for i = solids.(solid_names{solid_i}) - plot(freqs, abs(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :)))); - end - plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), '-k'); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - ylabel('Amplitude'); - - ax2 = subplot(2, 1, 2); - hold on; - for i = solids.(solid_names{solid_i}) - plot(freqs, mod(180+180/pi*phase(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :))), 360)-180); - end - plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), 360)-180, '-k'); - hold off; - ylim([-180, 180]); yticks(-180:90:180); - xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); - set(gca, 'xscale', 'log'); - - linkaxes([ax1,ax2],'x'); - xlim([1, 200]); -#+end_src - -* Verify that we find the original FRF from the FRF in the global coordinates -From the computed FRF of the Hexapod in its 6 DOFs, compute the FRF of the accelerometer 1 fixed to the Hexapod during the measurement. - -#+begin_src matlab - FRF_test = zeros(801, 3); - for i = 1:801 - FRF_test(i, :) = FRFs_O(31:33, 1, i) + cross(FRFs_O(34:36, 1, i), acc_pos(1, :)'); - end -#+end_src - -#+begin_src matlab :exports none - figure; - - ax1 = subplot(3, 1, 1); - hold on; - plot(freqs, abs(squeeze(FRFs(1, 1, :)))); - plot(freqs, abs(squeeze(FRF_test(:, 1))), '--k'); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - xlim([1, 200]); - title('FRF $\frac{D_{1x}}{F_x}$'); - - ax2 = subplot(3, 1, 2); - hold on; - plot(freqs, abs(squeeze(FRFs(2, 1, :)))); - plot(freqs, abs(squeeze(FRF_test(:, 2))), '--k'); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - ylabel('Amplitude'); - xlim([1, 200]); - title('FRF $\frac{D_{1y}}{F_x}$'); - - ax3 = subplot(3, 1, 3); - hold on; - plot(freqs, abs(squeeze(FRFs(3, 1, :)))); - plot(freqs, abs(squeeze(FRF_test(:, 3))), '--k'); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); - xlim([1, 200]); - legend({'Original Measurement', 'Recovered Measurement'}, 'Location', 'southeast'); - title('FRF $\frac{D_{1z}}{F_x}$'); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/compare_original_meas_with_recovered.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+NAME: fig:compare_original_meas_with_recovered -#+CAPTION: Comparison of the original measured FRFs with the recovered FRF from the FRF in the common cartesian frame -[[file:figs/compare_original_meas_with_recovered.png]] - -#+begin_important - The reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF) seems to work well. - - This confirms the fact that this stage, for that mode is indeed behaving as a solid body. - This should be verified for all the stages for modes with high resonance frequencies. -#+end_important - -* TODO Synthesis of FRF curves +* TODO Synthesis of FRF curves from the modal parameters