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36
modal-analysis.bib
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@ -1,12 +1,38 @@
@article{wang11_extrac_real_modes_physic_matric,
author = {Tong Wang and Lingmi Zhang and Kong Fah Tee},
title = {Extraction of Real Modes and Physical Matrices From Modal
Testing},
journal = {Earthquake Engineering and Engineering Vibration},
volume = 10,
number = 2,
pages = {219-227},
year = 2011,
doi = {10.1007/s11803-011-0060-6},
url = {https://doi.org/10.1007/s11803-011-0060-6},
DATE_ADDED = {Tue Jul 9 15:51:21 2019},
}
@article{pastor12_modal_assur_criter,
author = {Miroslav Pastor and Michal Binda and Tom{\'a}{\v{s}} Har{\v{c}}arik},
title = {Modal Assurance Criterion},
journal = {Procedia Engineering},
volume = {48},
number = {nil},
pages = {543-548},
year = {2012},
doi = {10.1016/j.proeng.2012.09.551},
url = {https://doi.org/10.1016/j.proeng.2012.09.551},
DATE_ADDED = {Thu Jul 11 13:51:57 2019},
}
@book{ewins00_modal,
author = {Ewins, DJ},
title = {Modal testing: theory, practice and application},
year = 2000,
year = {2000},
publisher = {Wiley-Blackwell},
address = {Baldock, Hertfordshire, England Philadelphia, PA},
isbn = 0863802184,
journal = {Research studies Pre, 2nd ed., ISBN-13},
keywords = {favorite, identification},
pages = {978--0863802188},
isbn = {0863802184},
address = {Baldock, Hertfordshire, England Philadelphia, PA},
keywords = {favorite, identification},
}

174
modal-analysis.org
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@ -13,10 +13,11 @@
#+BIND: org-latex-image-default-option "scale=1"
#+BIND: org-latex-image-default-width ""
#+LaTeX_CLASS: scrreprt
#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]
#+LaTeX_HEADER_EXTRA: \input{preamble.tex}
#+LATEX_HEADER_EXTRA: \bibliography{modal-analysis.bib}
#+LATEX_CLASS: scrreprt
#+LATEX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]
#+LATEX_HEADER: \input{preamble.tex}
#+LATEX_HEADER_EXTRA: \input{preamble_extra.tex}
#+LATEX_HEADER_EXTRA: \bibliography{nass-uniaxial-model.bib}
#+BIND: org-latex-bib-compiler "biber"
@ -100,18 +101,37 @@ Prefix is =modal=
** TODO [#C] Section ref:ssec:modal_solid_body_first_check can probably be removed
As the solid body assumption is already verified in section ref:ssec:modal_solid_body_assumption.
* Glossary and Acronyms - Tables :ignore:
#+name: glossary
| label | name | description |
|-------+-------------------------+---------------------------------------------|
| psdx | \ensuremath{\Phi_{x}} | Power spectral density of signal $x$ |
| asdx | \ensuremath{\Gamma_{x}} | Amplitude spectral density of signal $x$ |
| cpsx | \ensuremath{\Phi_{x}} | Cumulative Power Spectrum of signal $x$ |
| casx | \ensuremath{\Gamma_{x}} | Cumulative Amplitude Spectrum of signal $x$ |
#+name: acronyms
| key | abbreviation | full form |
|------+--------------+----------------------------------|
| nass | NASS | Nano Active Stabilization System |
| asd | ASD | Amplitude Spectral Density |
| psd | PSD | Power Spectral Density |
| frf | FRF | Frequency Response Function |
| dof | DoF | Degree of freedom |
* Introduction :ignore:
In order to further improve the accuracy of the performance predictions, a model that better represents the micro-station dynamics is required.
A multi-body model, consisting of several rigid bodies connected by kinematic constraints (i.e. joints) and springs and damper elements, is a good candidate to model the micro-station.
A multi-body model, consisting of several rigid bodies connected by kinematic constraints (i.e. joints), springs and damper elements, is a good candidate to model the micro-station.
Even though the inertia of each solid body can easily be estimated from its geometry and its material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
The experimental modal analysis will be useful to verify that a multi-body model can represent accurately the dynamics of the micro-station and to help tuning the model.
The experimental modal analysis will be use to tune the model, and to verify that a multi-body model can represent accurately the dynamics of the micro-station.
The approach of tuning the multi-body model from measurements is illustrated in Figure ref:fig:modal_vibration_analysis_procedure.
First, a /response model/ is obtained, which corresponds to a set of frequency response functions computed from experimental measurements.
From this response model, and modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another one describing the mode shapes.
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is to say to tune the mass spring and damping properties of the considered solid bodies.
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is to say to tune the mass of the considering solid bodies, and the springs and dampers connecting the solid bodies.
#+begin_src latex :file modal_vibration_analysis_procedure.pdf
\begin{tikzpicture}
@ -167,7 +187,7 @@ It shows how complex the micro-station dynamics is, and the necessity of the dev
In order to perform an experimental modal analysis, a proper measurement setup is key.
This include using appropriate instrumentation (presented in Section ref:ssec:modal_instrumentation) and properly preparing the structure to be measured (Section ref:ssec:modal_test_preparation).
Then, the location of the measured motion (Section ref:ssec:modal_accelerometers) and the location of the hammer impacts (Section ref:ssec:modal_hammer_impacts) have to be chosen carefully.
Obtained force and acceleration signals are shown in Section ref:ssec:modal_measured_signals, and the quality of the measured data is verified.
Obtained force and acceleration signals are shown in Section ref:ssec:modal_measured_signals, and the quality of the measured data is checked.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -232,39 +252,37 @@ Finally, an /acquisition system/[fn:3] (figure ref:fig:modal_oros) is used to ac
<<ssec:modal_test_preparation>>
In order to obtain meaningful results, the modal analysis of the micro-station in performed /in-situ/.
To do so, all the micro-station stages controllers are turned "ON".
To do so, all the micro-station stage controllers are turned "ON".
This is especially important for stages for which the stiffness is provided by local feedback control, which is case for the air bearing spindle, and the translation stage.
If these local feedback control were turned OFF, this would have resulted in very low frequency modes difficult to measure in practice, and this would also have lead to decoupled dynamics which would not be the case in practice.
The top part representing the active stabilization stage has been disassembled in order to reduce the complexity of the dynamics and also because the active stabilization stage and the sample will be added in the multi-body model afterwards.
The top part representing the active stabilization stage has been disassembled as the active stabilization stage and the sample will be added in the multi-body model afterwards.
To perform the modal-analysis from the measured responses, the $n \times n$ frequency response function matrix $\mathbf{H}$ needs to be measured, where $n$ is the considered number of degrees of freedom.
The $H_{jk}$ element of this FRF matrix corresponds to the frequency response function from a force $F_k$ applied at DoF $k$ to the displacement of the structure $X_j$ at DoF $j$.
Measuring this FRF matrix is very time consuming as it requires to make $n^2$ measurements.
The $H_{jk}$ element of this acrfull:frf matrix corresponds to the frequency response function from a force $F_k$ applied at acrfull:dof $k$ to the displacement of the structure $X_j$ at acrshort:dof $j$.
Measuring this acrshort:frf matrix is very time consuming as it requires to make $n^2$ measurements.
However thanks to the principle of reciprocity ($H_{jk} = H_{kj}$) and using the /point measurement/ ($H_{jj}$), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix $\mathbf{H}$ [[cite:&ewins00_modal chapt. 5.2]].
Therefore, a minimum set of $n$ frequency response functions needs to be measured.
This can be done either by measuring the response $X_{j}$ at a fixed DoF $j$ while applying forces $F_{i}$ for at all $n$ considered DoF, or by applying a force $F_{k}$ at a fixed DoF $k$ and measuring the response $X_{i}$ for all $n$ DoF.
This can be done either by measuring the response $X_{j}$ at a fixed acrshort:dof $j$ while applying forces $F_{i}$ for at all $n$ considered acrshort:dof, or by applying a force $F_{k}$ at a fixed acrshort:dof $k$ and measuring the response $X_{i}$ for all $n$ acrshort:dof.
It is however not advised to measure only one row or one column as one or more modes may be missed by an unfortunate choice of force or acceleration measured locations.
In this modal-analysis, it is chosen to measure the response of the structure at all considered DoF, and to excite the structure at one location in three directions in order to have some redundancy and to make sure that all modes are properly excited.
It is however not advised to measure only one row or one column as one or more modes may be missed by an unfortunate choice of force or acceleration measured locations (for instance if the force is applied at a vibration node of a particular mode).
In this modal-analysis, it is chosen to measure the response of the structure at all considered acrshort:dof, and to excite the structure at one location in three directions in order to have some redundancy and to make sure that all modes are properly energized.
** Location of the Accelerometers
<<ssec:modal_accelerometers>>
The location of the accelerometers fixed to the micro-station is essential as it defines where the dynamics is identified.
The location of the accelerometers fixed to the micro-station is essential as it defines where the dynamics is measured.
A total of 23 accelerometers are fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod.
The position of the accelerometers are visually shown on a CAD model in Figure ref:fig:modal_location_accelerometers and their precise locations with respect to a frame located at the point of interest are summarized in Table ref:tab:modal_position_accelerometers.
Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure ref:fig:modal_accelerometer_pictures.
As all key stages of the micro-station are foreseen to behave as solid bodies, only 6 DoF can be considered per solid body.
However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured DoF) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section ref:ssec:modal_solid_body_assumption).
As all key stages of the micro-station are foreseen to behave as solid bodies, only 6 acrshort:dof can be considered per solid body.
However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrshort:dof) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section ref:ssec:modal_solid_body_assumption).
#+attr_latex: :options [b]{0.68\linewidth}
#+begin_minipage
#+name: fig:modal_location_accelerometers
#+caption: Position of the accelerometers using SolidWorks
#+caption: Position of the accelerometers using SolidWorks in mm
#+attr_latex: :width 0.9\linewidth :float nil
[[file:figs/modal_location_accelerometers.png]]
#+end_minipage
@ -306,7 +324,7 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured DoF)
#+name: fig:modal_accelerometer_pictures
#+caption: Accelerometers fixed on the micro-station
#+caption: Accelerometers fixed on the micro-station stages
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_ty} $T_y$ stage}
@ -339,10 +357,10 @@ data2orgtable([[1:23]', 1000*acc_pos], {'Hexapod', 'Hexapod', 'Hexapod', 'Hexapo
<<ssec:modal_hammer_impacts>>
The chosen location of the hammer impact corresponds to the location of accelerometer number $11$ fixed to the translation stage.
It was chosen to match the location of one accelerometer, because a /point measurement/ (i.e. a measurement of $H_{kk}$) is necessary to be able to reconstruct the full FRF matrix [[cite:ewins00_modal]].
It was chosen to match the location of one accelerometer, because a /point measurement/ (i.e. a measurement of $H_{kk}$) is necessary to be able to reconstruct the full acrshort:frf matrix [[cite:ewins00_modal]].
The impacts are performed in three directions, which are shown in figures ref:fig:modal_impact_x, ref:fig:modal_impact_y and ref:fig:modal_impact_z.
This excitation point and the three considered directions allows to properly excite all the modes in the frequency band of interest and to provide good coherence for all the accelerometers as will be shown in the next section.
This excitation point with the three considered directions allows to properly energize all the modes in the frequency band of interest and to provide good coherence for all the accelerometers as will be shown in the next section.
#+name: fig:modal_hammer_impacts
#+caption: The three hammer impacts used for the modal analysis
@ -371,11 +389,11 @@ This excitation point and the three considered directions allows to properly exc
** Force and Response signals
<<ssec:modal_measured_signals>>
The force sensor and the accelerometers signals are shown in the time domain in Figure ref:fig:modal_raw_meas.
The force sensor of the instrumented hammer and the accelerometers signals are shown in the time domain in Figure ref:fig:modal_raw_meas.
Sharp "impacts" can be seen for the force sensor, indicating wide frequency band excitation.
For the accelerometer, many resonances can be seen on the right, indicating complex dynamics.
For the accelerometer, a much more complex signal can be observed, indicating complex dynamics.
The "normalized" amplitude spectral density of the two signals are computed and shown in Figure ref:fig:modal_asd_acc_force.
The "normalized" acrfull:asd of the two signals are computed and shown in Figure ref:fig:modal_asd_acc_force.
Conclusions based on the time domain signals can be clearly seen in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
#+begin_src matlab
@ -472,7 +490,7 @@ Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corres
%% Frequency Response Function between the force and the acceleration
figure;
plot(f, abs(G1));
xlabel('Frequency [Hz]'); ylabel('FRF [$m/s^2/N$]')
xlabel('Frequency [Hz]'); ylabel('acrshort:frf [$m/s^2/N$]')
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
xlim([0, 200]);
xticks([0:20:200]);
@ -521,7 +539,7 @@ exportFig('figs/modal_coh_acc_force.pdf', 'width', 'half', 'height', 'normal');
<<sec:modal_frf_processing>>
** Introduction :ignore:
All measurements where conducted and a $n \times p \times q$ Frequency Response Functions Matrix were computed with:
All measurements where conducted and a $n \times p \times q$ acrlongpl:frf Matrix were computed with:
- $n = 69$: the number of output measured accelerations (23 3-axis accelerometers)
- $p = 3$: the number of input force excitations
- $q = 801$: the number of frequency points $\omega_{i}$
@ -541,15 +559,14 @@ For each frequency point $\omega_{i}$, a 2D complex matrix is obtained that link
However, for the multi-body model being developed, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the hexapod.
Therefore, only $6 \times 6 = 36$ degrees of freedom are of interest.
The objective in this section is therefore to process the Frequency Response Matrix to reduce the number of measured acrshort:dof from 69 to 36.
The objective in this section is therefore to process the Frequency Response Matrix to reduce the number of measured DoFs from 69 to 36.
In order to be able to perform this reduction of measured acrshort:dof, the rigid body assumption first needs to be verified (Section ref:ssec:modal_solid_body_first_check).
In order to be able to perform this reduction of measured DoFs, the rigid body assumption first needs to be verified (Section ref:ssec:modal_solid_body_first_check).
The coordinate transformation from accelerometers DoFs to the solid body 6 DoFs (three translations and three rotations) is performed in Section ref:ssec:modal_acc_to_solid_dof.
The coordinate transformation from accelerometers acrshort:dof to the solid body 6 acrshortpl:dof (three translations and three rotations) is performed in Section ref:ssec:modal_acc_to_solid_dof.
The $69 \times 3 \times 801$ frequency response matrix is then reduced to a $36 \times 3 \times 801$ frequency response matrix where the motion of each solid body is expressed with respect to its center of mass.
To further validate this reduction of DoF and the solid body assumption, the frequency response function at the accelerometer location are synthesized from the reduced frequency response matrix and are compared with the initial measurements in Section ref:ssec:modal_solid_body_assumption.
To further validate this reduction of acrshort:dof and the solid body assumption, the frequency response function at the accelerometer location are synthesized from the reduced frequency response matrix and are compared with the initial measurements in Section ref:ssec:modal_solid_body_assumption.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -575,7 +592,7 @@ To further validate this reduction of DoF and the solid body assumption, the fre
** First verification of the solid body assumption
<<ssec:modal_solid_body_first_check>>
In this section, it is shown that two accelerometers fixed to a /rigid body/ at positions $\vec{p}_1$ and $\vec{p}_2$ such that $\vec{p}_2 = \vec{p}_1 + \alpha \vec{x}$ will measure the same acceleration in the $\vec{x}$ direction.
In this section, it is shown that two accelerometers fixed to a /rigid body/ at positions $\vec{p}_1$ and $\vec{p}_2 = \vec{p}_1 + \alpha \vec{x}$ measure the same acceleration in the $\vec{x}$ direction.
Such situation is illustrated in Figure ref:fig:modal_aligned_accelerometers.
#+begin_src latex :file modal_aligned_accelerometers.pdf :results file raw silent
@ -617,7 +634,7 @@ Such situation is illustrated in Figure ref:fig:modal_aligned_accelerometers.
The motion of the rigid body of figure ref:fig:modal_aligned_accelerometers is here described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to a reference frame $\{O\}$.
The motion of points $p_1$ and $p_2$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ eqref:eq:modal_p1_p2_motion, with $\bm{\delta\Omega}$ defined in eqref:eq:modal_rotation_matrix.
The motion of points $p_1$ and $p_2$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ eqref:eq:modal_p1_p2_motion, with $\bm{\delta\Omega}$ defined in equation eqref:eq:modal_rotation_matrix.
\begin{subequations}\label{eq:modal_p1_p2_motion}
\begin{align}
@ -643,7 +660,7 @@ Considering only the $x$ direction, equation eqref:eq:modal_p1_p2_x_motion is ob
\end{align}
\end{subequations}
Because the two sensors are co-linearity in the $x$ direction, $p_{1y} = p_{2y}$ and $p_{1z} = p_{2z}$, and eqref:eq:modal_colinear_sensors_equal is obtained.
Because the two sensors are aligned in the $x$ direction, $p_{1y} = p_{2y}$ and $p_{1z} = p_{2z}$, and eqref:eq:modal_colinear_sensors_equal is obtained.
\begin{equation}\label{eq:modal_colinear_sensors_equal}
\boxed{\delta p_{x1} = \delta p_{x2}}
@ -652,12 +669,12 @@ Because the two sensors are co-linearity in the $x$ direction, $p_{1y} = p_{2y}$
It is therefore concluded that two position sensors fixed to a rigid body will measure the same quantity in the direction "in line" the two sensors.
Such property can be used to verify that the considered stages are indeed behaving as rigid body in the frequency band of interest.
From Table ref:tab:modal_position_accelerometers, the pairs of accelerometers that aligned in the X and Y directions can be identified.
Such property can be used to check if the considered stages are having non rigid body behavior in the frequency band of interest.
From Table ref:tab:modal_position_accelerometers, the accelerometer pairs that are aligned in the X and Y directions can be identified.
The response in the X direction of pairs of sensors aligned in the X direction are compared in Figure ref:fig:modal_solid_body_comp_x_dir.
A good match is observed up to 200Hz.
Similar result is obtained for the Y direction.
This therefore indicates that the considered bodies are behaving as solid bodes in the frequency range of interest.
This provides a first indication that the considered bodies are behaving as solid bodes in the frequency range of interest.
#+begin_src matlab
%% Load frequency response matrix
@ -756,7 +773,7 @@ exportFig('figs/modal_solid_body_comp_x_dir.pdf', 'width', 'full', 'height', 'ta
<<ssec:modal_acc_to_solid_dof>>
Let's consider the schematic shown in Figure ref:fig:modal_local_to_global_coordinates where the motion of a solid body is measured at 4 distinct locations (in $x$, $y$ and $z$ directions).
The goal here is to link these $4 \times 3 = 12$ measurements to the 6 DoFs of the solid body expressed in the frame $\{O\}$.
The goal here is to link these $4 \times 3 = 12$ measurements to the 6 acrshort:dof of the solid body expressed in the frame $\{O\}$.
#+begin_src latex :file modal_local_to_global_coordinates.pdf
\newcommand\irregularcircle[2]{% radius, irregularity
@ -815,8 +832,9 @@ Writing Eq. eqref:eq:modal_p1_p2_motion for the four displacement sensors in a m
\end{array}\right]
\end{equation}
Supposing that the four sensors are properly located such that the system of equation eqref:eq:modal_cart_to_acc can be solved, the motion of the solid body expressed in a chosen frame $\{O\}$ using the accelerometers attached to it can be determined using equation eqref:eq:modal_determine_global_disp.
Note that this inversion is equivalent to resolving a mean square problem.
Provided that the four sensors are properly located, the system of equation eqref:eq:modal_cart_to_acc can be solved by matrix inversion.
The motion of the solid body expressed in a chosen frame $\{O\}$ can be determined using equation eqref:eq:modal_determine_global_disp.
Note that this matrix inversion is equivalent to resolving a mean square problem.
Therefore, having more accelerometers permits to have a better approximation of the motion of the solid body.
@ -864,7 +882,7 @@ data2orgtable(1000*model_com', {'Bottom Granite', 'Top granite', 'Translation st
| Spindle | 0 | 0 | -580 |
| Hexapod | -4 | 6 | -319 |
Using eqref:eq:modal_determine_global_disp, the frequency response matrix $\bm{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) at the center of mass of each solid body can be computed from the initial FRF matrix $\bm{H}$.
Using eqref:eq:modal_determine_global_disp, the frequency response matrix $\bm{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\bm{H}$.
\begin{equation}\label{eq:modal_frf_matrix_com}
\bm{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
@ -882,7 +900,7 @@ Using eqref:eq:modal_determine_global_disp, the frequency response matrix $\bm{H
#+begin_src matlab
%% Frequency Response Matrix - Response expressed at the CoM of the solid bodies
FRFs_CoM = zeros(length(solid_names)*6, 3, 801);
acrshort:frfs_CoM = zeros(length(solid_names)*6, 3, 801);
for solid_i = 1:length(solid_names)
% Number of accelerometers fixed to this solid body
@ -902,37 +920,37 @@ for solid_i = 1:length(solid_names)
end
for exc_dir = 1:3
FRFs_CoM((solid_i-1)*6+1:solid_i*6, exc_dir, :) = A\squeeze(frf((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, :));
acrshort:frfs_CoM((solid_i-1)*6+1:solid_i*6, exc_dir, :) = A\squeeze(frf((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, :));
end
end
#+end_src
#+begin_src matlab :eval no
%% Save the computed FRF at the CoM
save('mat/frf_com.mat', 'FRFs_CoM');
%% Save the computed acrshort:frf at the CoM
save('mat/frf_com.mat', 'acrshort:frfs_CoM');
#+end_src
#+begin_src matlab :tangle no
%% Save the computed FRF at the CoM
save('matlab/mat/frf_com.mat', 'FRFs_CoM');
%% Save the computed acrshort:frf at the CoM
save('matlab/mat/frf_com.mat', 'acrshort:frfs_CoM');
#+end_src
** Verification of solid body assumption
<<ssec:modal_solid_body_assumption>>
From the response of one solid body along its 6 DoFs (from $\bm{H}_{\text{CoM}}$), and using eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any location, in particular at the location of the accelerometers fixed to this solid body.
From the response of one solid body along its 6 acrshortpl:dof (i.e. from $\bm{H}_{\text{CoM}}$), and using equation eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any location, in particular at the location of the four accelerometers fixed to this solid body (i.e. estimation of the initial $\bm{H}$ matrix).
Comparing the computed response of a particular accelerometer from $\bm{H}_{\text{CoM}}$ with the original measurements $\bm{H}$ is useful to check if the change of coordinate eqref:eq:modal_determine_global_disp works as expected, and if the solid body assumption is correct in the frequency band of interest.
#+begin_src matlab
FRFs_A = zeros(size(frf));
acrshort:frfs_A = zeros(size(frf));
% For each excitation direction
for exc_dir = 1:3
% For each solid
for solid_i = 1:length(solid_names)
v0 = squeeze(FRFs_CoM((solid_i-1)*6+1:(solid_i-1)*6+3, exc_dir, :));
W0 = squeeze(FRFs_CoM((solid_i-1)*6+4:(solid_i-1)*6+6, exc_dir, :));
v0 = squeeze(acrshort:frfs_CoM((solid_i-1)*6+1:(solid_i-1)*6+3, exc_dir, :));
W0 = squeeze(acrshort:frfs_CoM((solid_i-1)*6+4:(solid_i-1)*6+6, exc_dir, :));
% For each accelerometer attached to the current solid
for acc_i = solids.(solid_names{solid_i})
@ -941,7 +959,7 @@ for exc_dir = 1:3
% pos = acc_pos(acc_i, :).';
posX = [0 pos(3) -pos(2); -pos(3) 0 pos(1) ; pos(2) -pos(1) 0];
FRFs_A(3*(acc_i-1)+1:3*(acc_i-1)+3, exc_dir, :) = v0 + posX*W0;
acrshort:frfs_A(3*(acc_i-1)+1:3*(acc_i-1)+3, exc_dir, :) = v0 + posX*W0;
end
end
end
@ -950,7 +968,7 @@ end
The comparison is made for the 4 accelerometers fixed to the micro-hexapod in Figure ref:fig:modal_comp_acc_solid_body_frf.
The original frequency response functions and the ones computed from the CoM responses are well matching in the frequency range of interested.
Similar results are obtained for the other solid bodies, indicating that the solid body assumption is valid, and that a multi-body model can be used to represent the dynamics of the micro-station.
This also validates the reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3DoF) to 36 (6 solid bodies with 6 DoF).
This also validates the reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof).
#+begin_src matlab :exports none :results none
%% Comparaison of the original accelerometer response and reconstructed response from the solid body response
@ -975,7 +993,7 @@ for i = 1:length(accs_i)
end
set(gca,'ColorOrderIndex',1)
for dir_i = 1:3
plot(freqs, abs(squeeze(FRFs_A(3*(acc_i-1)+dir_i, exc_dir, :))), '--', 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(acrshort:frfs_A(3*(acc_i-1)+dir_i, exc_dir, :))), '--', 'HandleVisibility', 'off');
end
hold off;
@ -1016,7 +1034,7 @@ exportFig('figs/modal_comp_acc_solid_body_frf.pdf', 'width', 'full', 'height', '
The goal here is to extract the modal parameters describing the modes of station being studied, namely:
- the eigen frequencies and the modal damping (eigen values)
- the mode shapes (eigen vectors)
This is done from the FRF matrix previously extracted from the measurements.
This is done from the acrshort: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.
@ -1024,17 +1042,17 @@ This corresponds to how many modes are present in the frequency band of interest
In section ref:ssec:modal_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*.
The modal parameter extraction methods generally consists of *curve-fitting a theoretical expression for an individual acrshort: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 part of a single acrshort:frf curve
- works on a complete curve encompassing several resonances
- works on a set of many FRF plots all obtained from the same structure
- works on a set of many acrshort: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 ref:ssec:modal_parameter_extraction.
From the modal model, it is possible to obtain a graphic display of the mode shapes (section ref:ssec:modal_mode_shapes).
In order to validate the quality of the modal model, we will synthesize the FRF matrix from the modal model and compare it with the FRF measured (section ref:ssec:modal_model_validity).
In order to validate the quality of the modal model, we will synthesize the acrshort:frf matrix from the modal model and compare it with the acrshort:frf measured (section ref:ssec:modal_model_validity).
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.
@ -1063,11 +1081,11 @@ The modes of the structure are expected to be complex, however real modes are ea
** Determine the number of modes
<<ssec:modal_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.
The Mode Indicator Functions are usually used on $n\times p$ acrshort: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.
In these methods, the frequency dependent acrshort: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.
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 acrshort:frf data prior to modal analysis.
From the documentation of the modal software:
#+begin_quote
@ -1078,7 +1096,7 @@ 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:
The *Complex Mode Indicator Function* is defined simply by the SVD of the acrshort: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}
@ -1123,7 +1141,7 @@ exportFig('figs/modal_indication_function.pdf', 'width', 'wide', 'height', 'norm
[[file:figs/modal_indication_function.png]]
**** Composite Response Function
An alternative is the Composite Response Function $HH(\omega)$ defined as the sum of all the measured FRF:
An alternative is the Composite Response Function $HH(\omega)$ defined as the sum of all the measured acrshort:frf:
\begin{equation}
HH(\omega) = \sum_j\sum_kH_{jk}(\omega)
\end{equation}
@ -1161,8 +1179,8 @@ exportFig('figs/modal_composite_reponse_function.pdf', 'width', 'wide', 'height'
Modal identification are done within the Modal software of OROS.
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 time.
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.
We choose to use the "broad band" method as it permits to identify the modal parameters using all the acrshort:frf curves at the same time.
It takes into account the fact the the properties of all the individual curves are related by being from the same structure: all acrshort: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.
@ -1181,7 +1199,7 @@ We can then run the modal analysis, and the software will identify the modal dam
The obtained modal parameters are:
- Resonance frequencies in Hertz
- Modal damping ratio in percentage
- (complex) Modes shapes for each measured DoF
- (complex) Modes shapes for each measured acrshort:dof
- Modal A and modal B which are parameters important for further normalization
#+begin_src matlab
@ -1255,7 +1273,7 @@ data2orgtable([(1:length(freqs_m))', freqs_m, damps_m], {}, {'Mode', 'Frequency
| 16 | 165.4 | 1.4 |
**** Theory
It seems that the modal analysis software makes the *assumption* of viscous damping for the model with which it tries to fit the FRF measurements.
It seems that the modal analysis software makes the *assumption* of viscous damping for the model with which it tries to fit the acrshort:frf measurements.
If we note $N$ the number of modes identified, then there are $2N$ eigenvalues and eigenvectors given by the software:
\begin{align}
@ -1277,7 +1295,7 @@ We would like to arrange the obtained modal parameters into two modal matrices:
\end{bmatrix}_{M \times N} \]
\[ \{\psi_i\} = \begin{Bmatrix} \psi_{i, 1_x} & \psi_{i, 1_y} & \psi_{i, 1_z} & \psi_{i, 2_x} & \dots & \psi_{i, 23_z} \end{Bmatrix}^T \]
$M$ is the number of DoF: here it is $23 \times 3 = 69$.
$M$ is the number of acrshort:dof: here it is $23 \times 3 = 69$.
$N$ is the number of mode
#+begin_src matlab
@ -1382,8 +1400,8 @@ They are probably *not well leveled*, so the granite is supported only by two Ai
<<ssec:modal_model_validity>>
There are two main ways to verify the validity of the modal model
- Synthesize FRF measurements that has been used to generate the modal model and compare
- Synthesize FRF that has not yet been measured. Then measure that FRF and compare
- Synthesize acrshort:frf measurements that has been used to generate the modal model and compare
- Synthesize acrshort:frf that has not yet been measured. Then measure that acrshort:frf and compare
From the modal model, we want to synthesize the Frequency Response Functions that has been used to build the modal model.
@ -1391,7 +1409,7 @@ Let's recall that:
- $M$ is the number of measured DOFs ($3 \times n_\text{acc}$)
- $N$ is the number of modes identified
We then have that the FRF matrix $[H_{\text{syn}}]$ can be synthesize using the following formula:
We then have that the acrshort:frf matrix $[H_{\text{syn}}]$ can be synthesize using the following formula:
#+begin_important
\begin{equation}
[H_{\text{syn}}(\omega)]_{M\times M} = [\Phi]_{M\times2N} \left[\frac{Q_r}{j\omega - s_r}\right]_{2N\times2N} [\Phi]_{2N\times M}^T
@ -1429,7 +1447,7 @@ Then
#+end_quote
#+begin_src matlab
%% Synthesize FRF from the modal model
%% Synthesize acrshort:frf from the modal model
Hsyn = zeros(acc_n*dir_n, acc_n*dir_n, length(freqs));
for i = 1:length(freqs)
@ -1484,7 +1502,7 @@ exportFig('figs/modal_comp_acc_frf_modal.pdf', 'width', 'wide', 'height', 'tall'
[[file:figs/modal_comp_acc_frf_modal.png]]
# #+name: fig:from11xto1x
# #+caption: Original FRF and synthesize FRF using the modal software. From force applied in the X direction to the acceleration of accelerometer 11 in the X direction
# #+caption: Original acrshort:frf and synthesize acrshort:frf using the modal software. From force applied in the X direction to the acceleration of accelerometer 11 in the X direction
# #+attr_latex: :width \linewidth
# [[file:img/modal_software/from11xto1x.jpg]]
@ -1579,4 +1597,4 @@ colors = colororder;
[fn:3]OROS OR36. 24bits signal-delta ADC.
[fn:2]Kistler 9722A2000. Sensitivity of $2.3\,mV/N$ and measurement range of $2\,kN$
[fn:1]PCB 393B05. Sensitivity is $10\,V/g$, measurement range is $0.5\,g$ and bandwidth is $0.7$ to $450\,\text{Hz}$.
[fn:1]PCB 356B18. Sensitivity is $1\,V/g$, measurement range is $\pm 5\,g$ and bandwidth is $0.5$ to $5\,\text{kHz}$.

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@ -1,9 +1,19 @@
% Created 2024-06-26 Wed 17:58
% Created 2024-10-24 Thu 09:36
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
\input{preamble.tex}
\bibliography{modal-analysis.bib}
\newacronym{nass}{NASS}{Nano Active Stabilization System}
\newacronym{asd}{ASD}{Amplitude Spectral Density}
\newacronym{psd}{PSD}{Power Spectral Density}
\newacronym{frf}{FRF}{Frequency Response Function}
\newacronym{dof}{DoF}{Degree of freedom}
\newglossaryentry{psdx}{name=\ensuremath{\Phi_{x}},description={{Power spectral density of signal $x$}}}
\newglossaryentry{asdx}{name=\ensuremath{\Gamma_{x}},description={{Amplitude spectral density of signal $x$}}}
\newglossaryentry{cpsx}{name=\ensuremath{\Phi_{x}},description={{Cumulative Power Spectrum of signal $x$}}}
\newglossaryentry{casx}{name=\ensuremath{\Gamma_{x}},description={{Cumulative Amplitude Spectrum of signal $x$}}}
\input{preamble_extra.tex}
\bibliography{nass-uniaxial-model.bib}
\author{Dehaeze Thomas}
\date{\today}
\title{Micro-Station - Modal Analysis}
@ -12,7 +22,7 @@
pdftitle={Micro-Station - Modal Analysis},
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs 29.3 (Org mode 9.6)},
pdfcreator={Emacs 29.4 (Org mode 9.6)},
pdflang={English}}
\usepackage{biblatex}
@ -24,15 +34,15 @@
\clearpage
In order to further improve the accuracy of the performance predictions, a model that better represents the micro-station dynamics is required.
A multi-body model, consisting of several rigid bodies connected by kinematic constraints (i.e. joints) and springs and damper elements, is a good candidate to model the micro-station.
A multi-body model, consisting of several rigid bodies connected by kinematic constraints (i.e. joints), springs and damper elements, is a good candidate to model the micro-station.
Even though the inertia of each solid body can easily be estimated from its geometry and its material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
The experimental modal analysis will be useful to verify that a multi-body model can represent accurately the dynamics of the micro-station and to help tuning the model.
The experimental modal analysis will be use to tune the model, and to verify that a multi-body model can represent accurately the dynamics of the micro-station.
The approach of tuning the multi-body model from measurements is illustrated in Figure \ref{fig:modal_vibration_analysis_procedure}.
First, a \emph{response model} is obtained, which corresponds to a set of frequency response functions computed from experimental measurements.
From this response model, and modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another one describing the mode shapes.
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is to say to tune the mass spring and damping properties of the considered solid bodies.
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is to say to tune the mass of the considering solid bodies, and the springs and dampers connecting the solid bodies.
\begin{figure}[htbp]
\centering
@ -56,13 +66,13 @@ It shows how complex the micro-station dynamics is, and the necessity of the dev
In order to perform an experimental modal analysis, a proper measurement setup is key.
This include using appropriate instrumentation (presented in Section \ref{ssec:modal_instrumentation}) and properly preparing the structure to be measured (Section \ref{ssec:modal_test_preparation}).
Then, the location of the measured motion (Section \ref{ssec:modal_accelerometers}) and the location of the hammer impacts (Section \ref{ssec:modal_hammer_impacts}) have to be chosen carefully.
Obtained force and acceleration signals are shown in Section \ref{ssec:modal_measured_signals}, and the quality of the measured data is verified.
Obtained force and acceleration signals are shown in Section \ref{ssec:modal_measured_signals}, and the quality of the measured data is checked.
\section{Used Instrumentation}
\label{ssec:modal_instrumentation}
Three equipment are key to perform a good modal analysis.
First, \emph{accelerometers} are used to measure the response of the structure.
Here, 3-axis accelerometers\footnote{PCB 393B05. Sensitivity is \(10\,V/g\), measurement range is \(0.5\,g\) and bandwidth is \(0.7\) to \(450\,\text{Hz}\).} shown in figure \ref{fig:modal_accelero_M393B05} are used.
Here, 3-axis accelerometers\footnote{PCB 356B18. Sensitivity is \(1\,V/g\), measurement range is \(\pm 5\,g\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} shown in figure \ref{fig:modal_accelero_M393B05} are used.
These accelerometers are glued to the micro-station using a thin layer of wax for best results \cite[chapt. 3.5.7]{ewins00_modal}.
\begin{figure}[htbp]
@ -97,44 +107,43 @@ Finally, an \emph{acquisition system}\footnote{OROS OR36. 24bits signal-delta AD
\label{ssec:modal_test_preparation}
In order to obtain meaningful results, the modal analysis of the micro-station in performed \emph{in-situ}.
To do so, all the micro-station stages controllers are turned ``ON''.
To do so, all the micro-station stage controllers are turned ``ON''.
This is especially important for stages for which the stiffness is provided by local feedback control, which is case for the air bearing spindle, and the translation stage.
If these local feedback control were turned OFF, this would have resulted in very low frequency modes difficult to measure in practice, and this would also have lead to decoupled dynamics which would not be the case in practice.
The top part representing the active stabilization stage has been disassembled in order to reduce the complexity of the dynamics and also because the active stabilization stage and the sample will be added in the multi-body model afterwards.
The top part representing the active stabilization stage has been disassembled as the active stabilization stage and the sample will be added in the multi-body model afterwards.
To perform the modal-analysis from the measured responses, the \(n \times n\) frequency response function matrix \(\mathbf{H}\) needs to be measured, where \(n\) is the considered number of degrees of freedom.
The \(H_{jk}\) element of this FRF matrix corresponds to the frequency response function from a force \(F_k\) applied at DoF \(k\) to the displacement of the structure \(X_j\) at DoF \(j\).
Measuring this FRF matrix is very time consuming as it requires to make \(n^2\) measurements.
The \(H_{jk}\) element of this \acrfull{frf} matrix corresponds to the frequency response function from a force \(F_k\) applied at \acrfull{dof} \(k\) to the displacement of the structure \(X_j\) at \acrshort{dof} \(j\).
Measuring this \acrshort{frf} matrix is very time consuming as it requires to make \(n^2\) measurements.
However thanks to the principle of reciprocity (\(H_{jk} = H_{kj}\)) and using the \emph{point measurement} (\(H_{jj}\)), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix \(\mathbf{H}\) \cite[chapt. 5.2]{ewins00_modal}.
Therefore, a minimum set of \(n\) frequency response functions needs to be measured.
This can be done either by measuring the response \(X_{j}\) at a fixed DoF \(j\) while applying forces \(F_{i}\) for at all \(n\) considered DoF, or by applying a force \(F_{k}\) at a fixed DoF \(k\) and measuring the response \(X_{i}\) for all \(n\) DoF.
This can be done either by measuring the response \(X_{j}\) at a fixed \acrshort{dof} \(j\) while applying forces \(F_{i}\) for at all \(n\) considered \acrshort{dof}, or by applying a force \(F_{k}\) at a fixed \acrshort{dof} \(k\) and measuring the response \(X_{i}\) for all \(n\) \acrshort{dof}.
It is however not advised to measure only one row or one column as one or more modes may be missed by an unfortunate choice of force or acceleration measured locations.
In this modal-analysis, it is chosen to measure the response of the structure at all considered DoF, and to excite the structure at one location in three directions in order to have some redundancy and to make sure that all modes are properly excited.
It is however not advised to measure only one row or one column as one or more modes may be missed by an unfortunate choice of force or acceleration measured locations (for instance if the force is applied at a vibration node of a particular mode).
In this modal-analysis, it is chosen to measure the response of the structure at all considered \acrshort{dof}, and to excite the structure at one location in three directions in order to have some redundancy and to make sure that all modes are properly energized.
\section{Location of the Accelerometers}
\label{ssec:modal_accelerometers}
The location of the accelerometers fixed to the micro-station is essential as it defines where the dynamics is identified.
The location of the accelerometers fixed to the micro-station is essential as it defines where the dynamics is measured.
A total of 23 accelerometers are fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod.
The position of the accelerometers are visually shown on a CAD model in Figure \ref{fig:modal_location_accelerometers} and their precise locations with respect to a frame located at the point of interest are summarized in Table \ref{tab:modal_position_accelerometers}.
Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure \ref{fig:modal_accelerometer_pictures}.
As all key stages of the micro-station are foreseen to behave as solid bodies, only 6 DoF can be considered per solid body.
However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured DoF) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section \ref{ssec:modal_solid_body_assumption}).
As all key stages of the micro-station are foreseen to behave as solid bodies, only 6 \acrshort{dof} can be considered per solid body.
However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrshort{dof}) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section \ref{ssec:modal_solid_body_assumption}).
\begin{minipage}[b]{0.68\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/modal_location_accelerometers.png}
\captionof{figure}{\label{fig:modal_location_accelerometers}Position of the accelerometers using SolidWorks}
\captionof{figure}{\label{fig:modal_location_accelerometers}Position of the accelerometers using SolidWorks in mm}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.31\linewidth}
\begin{center}
\captionof{table}{\label{tab:modal_position_accelerometers}Accelerometer positions}
\scriptsize
\begin{tabularx}{\linewidth}{Xcccc}
\toprule
@ -165,8 +174,6 @@ Hexapod & 64 & 64 & -270\\
Hexapod & 64 & -64 & -270\\
\bottomrule
\end{tabularx}
\captionof{table}{\label{tab:modal_position_accelerometers}Accelerometer positions}
\end{center}
\end{minipage}
@ -185,17 +192,17 @@ Hexapod & 64 & -64 & -270\\
\end{center}
\subcaption{\label{fig:modal_accelerometers_hexapod} Micro-Hexapod}
\end{subfigure}
\caption{\label{fig:modal_accelerometer_pictures}Accelerometers fixed on the micro-station}
\caption{\label{fig:modal_accelerometer_pictures}Accelerometers fixed on the micro-station stages}
\end{figure}
\section{Hammer Impacts}
\label{ssec:modal_hammer_impacts}
The chosen location of the hammer impact corresponds to the location of accelerometer number \(11\) fixed to the translation stage.
It was chosen to match the location of one accelerometer, because a \emph{point measurement} (i.e. a measurement of \(H_{kk}\)) is necessary to be able to reconstruct the full FRF matrix \cite{ewins00_modal}.
It was chosen to match the location of one accelerometer, because a \emph{point measurement} (i.e. a measurement of \(H_{kk}\)) is necessary to be able to reconstruct the full \acrshort{frf} matrix \cite{ewins00_modal}.
The impacts are performed in three directions, which are shown in figures \ref{fig:modal_impact_x}, \ref{fig:modal_impact_y} and \ref{fig:modal_impact_z}.
This excitation point and the three considered directions allows to properly excite all the modes in the frequency band of interest and to provide good coherence for all the accelerometers as will be shown in the next section.
This excitation point with the three considered directions allows to properly energize all the modes in the frequency band of interest and to provide good coherence for all the accelerometers as will be shown in the next section.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
@ -222,11 +229,11 @@ This excitation point and the three considered directions allows to properly exc
\section{Force and Response signals}
\label{ssec:modal_measured_signals}
The force sensor and the accelerometers signals are shown in the time domain in Figure \ref{fig:modal_raw_meas}.
The force sensor of the instrumented hammer and the accelerometers signals are shown in the time domain in Figure \ref{fig:modal_raw_meas}.
Sharp ``impacts'' can be seen for the force sensor, indicating wide frequency band excitation.
For the accelerometer, many resonances can be seen on the right, indicating complex dynamics.
For the accelerometer, a much more complex signal can be observed, indicating complex dynamics.
The ``normalized'' amplitude spectral density of the two signals are computed and shown in Figure \ref{fig:modal_asd_acc_force}.
The ``normalized'' \acrfull{asd} of the two signals are computed and shown in Figure \ref{fig:modal_asd_acc_force}.
Conclusions based on the time domain signals can be clearly seen in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
\begin{figure}[htbp]
@ -267,7 +274,7 @@ Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which co
\chapter{Frequency Analysis}
\label{sec:modal_frf_processing}
All measurements where conducted and a \(n \times p \times q\) Frequency Response Functions Matrix were computed with:
All measurements where conducted and a \(n \times p \times q\) \acrlongpl{frf} Matrix were computed with:
\begin{itemize}
\item \(n = 69\): the number of output measured accelerations (23 3-axis accelerometers)
\item \(p = 3\): the number of input force excitations
@ -289,15 +296,14 @@ For each frequency point \(\omega_{i}\), a 2D complex matrix is obtained that li
However, for the multi-body model being developed, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the hexapod.
Therefore, only \(6 \times 6 = 36\) degrees of freedom are of interest.
The objective in this section is therefore to process the Frequency Response Matrix to reduce the number of measured \acrshort{dof} from 69 to 36.
The objective in this section is therefore to process the Frequency Response Matrix to reduce the number of measured DoFs from 69 to 36.
In order to be able to perform this reduction of measured \acrshort{dof}, the rigid body assumption first needs to be verified (Section \ref{ssec:modal_solid_body_first_check}).
In order to be able to perform this reduction of measured DoFs, the rigid body assumption first needs to be verified (Section \ref{ssec:modal_solid_body_first_check}).
The coordinate transformation from accelerometers DoFs to the solid body 6 DoFs (three translations and three rotations) is performed in Section \ref{ssec:modal_acc_to_solid_dof}.
The coordinate transformation from accelerometers \acrshort{dof} to the solid body 6 \acrshortpl{dof} (three translations and three rotations) is performed in Section \ref{ssec:modal_acc_to_solid_dof}.
The \(69 \times 3 \times 801\) frequency response matrix is then reduced to a \(36 \times 3 \times 801\) frequency response matrix where the motion of each solid body is expressed with respect to its center of mass.
To further validate this reduction of DoF and the solid body assumption, the frequency response function at the accelerometer location are synthesized from the reduced frequency response matrix and are compared with the initial measurements in Section \ref{ssec:modal_solid_body_assumption}.
To further validate this reduction of \acrshort{dof} and the solid body assumption, the frequency response function at the accelerometer location are synthesized from the reduced frequency response matrix and are compared with the initial measurements in Section \ref{ssec:modal_solid_body_assumption}.
\section{First verification of the solid body assumption}
\label{ssec:modal_solid_body_first_check}
@ -364,7 +370,7 @@ This therefore indicates that the considered bodies are behaving as solid bodes
\label{ssec:modal_acc_to_solid_dof}
Let's consider the schematic shown in Figure \ref{fig:modal_local_to_global_coordinates} where the motion of a solid body is measured at 4 distinct locations (in \(x\), \(y\) and \(z\) directions).
The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 DoFs of the solid body expressed in the frame \(\{O\}\).
The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 \acrshort{dofs} of the solid body expressed in the frame \(\{O\}\).
\begin{figure}[htbp]
\centering
@ -417,6 +423,7 @@ From the CAD model, the position of the center of mass of each considered solid
Then, the position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be derived.
\begin{table}[htbp]
\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the ``point of interest''}
\centering
\begin{tabularx}{0.6\linewidth}{lXXX}
\toprule
@ -430,11 +437,9 @@ Spindle & 0 & 0 & -580\\
Hexapod & -4 & 6 & -319\\
\bottomrule
\end{tabularx}
\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the ``point of interest''}
\end{table}
Using \eqref{eq:modal_determine_global_disp}, the frequency response matrix \(\bm{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) at the center of mass of each solid body can be computed from the initial FRF matrix \(\bm{H}\).
Using \eqref{eq:modal_determine_global_disp}, the frequency response matrix \(\bm{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) at the center of mass of each solid body can be computed from the initial \acrshort{frf} matrix \(\bm{H}\).
\begin{equation}\label{eq:modal_frf_matrix_com}
\bm{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
@ -453,14 +458,14 @@ Using \eqref{eq:modal_determine_global_disp}, the frequency response matrix \(\b
\section{Verification of solid body assumption}
\label{ssec:modal_solid_body_assumption}
From the response of one solid body along its 6 DoFs (from \(\bm{H}_{\text{CoM}}\)), and using \eqref{eq:modal_cart_to_acc}, it is possible to compute the response of the same solid body at any location, in particular at the location of the accelerometers fixed to this solid body.
From the response of one solid body along its 6 \acrshort{dofs} (from \(\bm{H}_{\text{CoM}}\)), and using \eqref{eq:modal_cart_to_acc}, it is possible to compute the response of the same solid body at any location, in particular at the location of the accelerometers fixed to this solid body.
Comparing the computed response of a particular accelerometer from \(\bm{H}_{\text{CoM}}\) with the original measurements \(\bm{H}\) is useful to check if the change of coordinate \eqref{eq:modal_determine_global_disp} works as expected, and if the solid body assumption is correct in the frequency band of interest.
The comparison is made for the 4 accelerometers fixed to the micro-hexapod in Figure \ref{fig:modal_comp_acc_solid_body_frf}.
The original frequency response functions and the ones computed from the CoM responses are well matching in the frequency range of interested.
Similar results are obtained for the other solid bodies, indicating that the solid body assumption is valid, and that a multi-body model can be used to represent the dynamics of the micro-station.
This also validates the reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3DoF) to 36 (6 solid bodies with 6 DoF).
This also validates the reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3acrshort:dof) to 36 (6 solid bodies with 6 \acrshort{dof}).
\begin{figure}[htbp]
\centering
@ -475,7 +480,7 @@ The goal here is to extract the modal parameters describing the modes of station
\item the eigen frequencies and the modal damping (eigen values)
\item the mode shapes (eigen vectors)
\end{itemize}
This is done from the FRF matrix previously extracted from the measurements.
This is done from the \acrshort{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.
@ -483,30 +488,30 @@ This corresponds to how many modes are present in the frequency band of interest
In section \ref{ssec:modal_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 \textbf{curve-fitting a theoretical expression for an individual FRF to the actual measured data}.
The modal parameter extraction methods generally consists of \textbf{curve-fitting a theoretical expression for an individual \acrshort{frf} to the actual measured data}.
However, there are multiple level of complexity:
\begin{itemize}
\item works on a part of a single FRF curve
\item works on a part of a single \acrshort{frf} curve
\item works on a complete curve encompassing several resonances
\item works on a set of many FRF plots all obtained from the same structure
\item works on a set of many \acrshort{frf} plots all obtained from the same structure
\end{itemize}
The third method is the most complex but gives better results. This is the one we will use in section \ref{ssec:modal_parameter_extraction}.
From the modal model, it is possible to obtain a graphic display of the mode shapes (section \ref{ssec:modal_mode_shapes}).
In order to validate the quality of the modal model, we will synthesize the FRF matrix from the modal model and compare it with the FRF measured (section \ref{ssec:modal_model_validity}).
In order to validate the quality of the modal model, we will synthesize the \acrshort{frf} matrix from the modal model and compare it with the \acrshort{frf} measured (section \ref{ssec:modal_model_validity}).
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.
\section{Determine the number of modes}
\label{ssec:modal_number_of_modes}
\paragraph{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.
The Mode Indicator Functions are usually used on \(n\times p\) \acrshort{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.
In these methods, the frequency dependent \acrshort{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 \textbf{determine the number of modes} present in a given frequency range, to \textbf{identify repeated natural frequencies} and to pre-process the FRF data prior to modal analysis.
These methods are used to \textbf{determine the number of modes} present in a given frequency range, to \textbf{identify repeated natural frequencies} and to pre-process the \acrshort{frf} data prior to modal analysis.
From the documentation of the modal software:
\begin{quote}
@ -517,7 +522,7 @@ Moreover, the magnitude of the MIFs implies the strength of the a mode.
\end{quote}
\begin{important}
The \textbf{Complex Mode Indicator Function} is defined simply by the SVD of the FRF (sub) matrix:
The \textbf{Complex Mode Indicator Function} is defined simply by the SVD of the \acrshort{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}
@ -534,7 +539,7 @@ The result is shown on Figure \ref{fig:modal_indication_function}.
\end{figure}
\paragraph{Composite Response Function}
An alternative is the Composite Response Function \(HH(\omega)\) defined as the sum of all the measured FRF:
An alternative is the Composite Response Function \(HH(\omega)\) defined as the sum of all the measured \acrshort{frf}:
\begin{equation}
HH(\omega) = \sum_j\sum_kH_{jk}(\omega)
\end{equation}
@ -558,8 +563,8 @@ The result is shown on figure \ref{fig:modal_composite_reponse_function}.
Modal identification are done within the Modal software of OROS.
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 time.
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.
We choose to use the ``broad band'' method as it permits to identify the modal parameters using all the \acrshort{frf} curves at the same time.
It takes into account the fact the the properties of all the individual curves are related by being from the same structure: all \acrshort{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 \textbf{unique and consistent model} as direct output.
@ -580,13 +585,14 @@ The obtained modal parameters are:
\begin{itemize}
\item Resonance frequencies in Hertz
\item Modal damping ratio in percentage
\item (complex) Modes shapes for each measured DoF
\item (complex) Modes shapes for each measured \acrshort{dof}
\item Modal A and modal B which are parameters important for further normalization
\end{itemize}
The obtained mode frequencies and damping are shown in Table \ref{tab:modal_obtained_modes_freqs_damps}.
\begin{table}[htbp]
\caption{\label{tab:modal_obtained_modes_freqs_damps}Obtained eigen frequencies and modal damping}
\centering
\scriptsize
\begin{tabularx}{0.35\linewidth}{ccc}
@ -611,12 +617,10 @@ Mode & Frequency [Hz] & Damping [\%]\\
16 & 165.4 & 1.4\\
\bottomrule
\end{tabularx}
\caption{\label{tab:modal_obtained_modes_freqs_damps}Obtained eigen frequencies and modal damping}
\end{table}
\paragraph{Theory}
It seems that the modal analysis software makes the \textbf{assumption} of viscous damping for the model with which it tries to fit the FRF measurements.
It seems that the modal analysis software makes the \textbf{assumption} of viscous damping for the model with which it tries to fit the \acrshort{frf} measurements.
If we note \(N\) the number of modes identified, then there are \(2N\) eigenvalues and eigenvectors given by the software:
\begin{align}
@ -638,7 +642,7 @@ We would like to arrange the obtained modal parameters into two modal matrices:
\end{bmatrix}_{M \times N} \]
\[ \{\psi_i\} = \begin{Bmatrix} \psi_{i, 1_x} & \psi_{i, 1_y} & \psi_{i, 1_z} & \psi_{i, 2_x} & \dots & \psi_{i, 23_z} \end{Bmatrix}^T \]
\(M\) is the number of DoF: here it is \(23 \times 3 = 69\).
\(M\) is the number of \acrshort{dof}: here it is \(23 \times 3 = 69\).
\(N\) is the number of mode
Each eigen vector is normalized: \(\| \{\psi_i\} \|_2 = 1\)
@ -726,8 +730,8 @@ They are probably \textbf{not well leveled}, so the granite is supported only by
There are two main ways to verify the validity of the modal model
\begin{itemize}
\item Synthesize FRF measurements that has been used to generate the modal model and compare
\item Synthesize FRF that has not yet been measured. Then measure that FRF and compare
\item Synthesize \acrshort{frf} measurements that has been used to generate the modal model and compare
\item Synthesize \acrshort{frf} that has not yet been measured. Then measure that \acrshort{frf} and compare
\end{itemize}
From the modal model, we want to synthesize the Frequency Response Functions that has been used to build the modal model.
@ -738,7 +742,7 @@ Let's recall that:
\item \(N\) is the number of modes identified
\end{itemize}
We then have that the FRF matrix \([H_{\text{syn}}]\) can be synthesize using the following formula:
We then have that the \acrshort{frf} matrix \([H_{\text{syn}}]\) can be synthesize using the following formula:
\begin{important}
\begin{equation}
[H_{\text{syn}}(\omega)]_{M\times M} = [\Phi]_{M\times2N} \left[\frac{Q_r}{j\omega - s_r}\right]_{2N\times2N} [\Phi]_{2N\times M}^T

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